C6770 NMR Spectroscopy of Biomolecules
Luk ´aˇs ˇZ´ıdek, Radovan Fiala, Pavel Kadeˇr ´avek
December 17, 2024
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Contents
NMR as a tool for structural biology 1
1.1 Aim of the course . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Structure and structure determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 Macromolecular structure from interatomic distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 List of applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.5 When can a structural biologist use NMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
NMR as a physical phenomenon 9
1.6 Spin and magnetic moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.7 Nuclear magnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
NMR as a method reflecting chemistry 13
2.1 Nuclei in the test-tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 No magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Static magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 Creating a signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.5 Nuclei are affected by their environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.6 Interactions with close nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.7 Interactions with electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.8 Interactions between nuclei mediated by electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Spin alchemy 23
3.1 Pulse sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2 Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3 Evolution of coherences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.4 NMR experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.5 Echoes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.6 Refocusing echo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.7 Decoupling echo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.8 Simultaneous echo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Correlated spectroscopy 37
4.1 Polarization transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2 HSQC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.3 Pulsed field gradients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Technical issues 45
5.1 Content of the lecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.2 Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.3 Constant time evolution∗
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.4 Pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
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5.5 Offset effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.6 From oscillations to spectrum: Fourier transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.7 Quadrature detection and demodulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.8 Complex signal in the indirect dimension(s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.9 Time-proportional phase incrementation∗
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.10 Phase cycling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.11 Quadrature artifacts and CYCLOPS∗
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.12 Pulse imperfections and EXORCYCLE∗
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.13 Axial peaks∗
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.14 Cleaning and frequency discrimination by pulsed field gradients∗
. . . . . . . . . . . . . . . . . . . . . 65
5.15 Preservation of equivalent pathways∗
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Protein heteronuclear techniques 69
6.1 Proteins and NMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.2 One-dimensional proton spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
6.3 1
H-15
N two-dimensional correlation (HSQC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
6.4 Triple resonance experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6.5 Sequential assignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.6 Degenerated sequencing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Protein homonuclear techniques 79
7.1 Weak and strong J-coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
7.2 COSY∗
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
7.3 Magnetic equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
7.4 Isotropic mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
7.5 TOCSY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
7.6 NOESY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
From spectra to protein structure 87
8.1 Heteronuclear-edited homonuclear correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
8.2 Side-chain assignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
8.3 NMR data for structure determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
8.4 Distances and nuclear Overhauser effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
8.5 Torsion angles and three-bond J-couplings∗
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
8.6 Orientation in space and residual dipolar couplings∗
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
8.7 Building a structural model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Molecular motions and NMR relaxation 103
9.1 Real power of biomolecular NMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
9.2 Connection between molecular motions and NMR relaxation . . . . . . . . . . . . . . . . . . . . . . . . 103
9.3 Rotational diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
9.4 Quantitative description of relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
9.5 Spectral density function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Measurement of relaxation rates 113
10.1 General principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
10.2 Inversion recovery experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
10.3 Measurement of the transverse relaxation rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
10.4 Measurement of the steady state Nuclear Overhauser Effect . . . . . . . . . . . . . . . . . . . . . . . . 115
10.5 Measurement of the relaxation rates in biomacromolecules . . . . . . . . . . . . . . . . . . . . . . . . . 116
CONTENTS v
Analysis of internal motions 121
11.1 Internal motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
11.2 Motional parameters from NMR measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
11.3 Chemical exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
11.4 Effect of chemical exchange on the transverse magnetization . . . . . . . . . . . . . . . . . . . . . . . . 123
11.5 Effect of chemical exchange on the longitudinal magnetization . . . . . . . . . . . . . . . . . . . . . . . 128
Intermolecular interactions 129
12.1 Studies of intermolecular interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
12.2 Titration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
12.3 Saturation transfer difference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
12.4 Isotope-edited and isotope-filtered NOE experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Nucleic acids 139
13.1 Chemical composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
13.2 Torsion angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
13.3 Helical parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
13.4 A and B double helix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
13.5 Atoms in nucleic acids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
13.6 Exchangeable an non-exchangeable protons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
vi CONTENTS
Lecture 1a
NMR as a tool for structural biology
1.1 Aim of the course
The aim of our course (C6770 NMR Spectroscopy of Biomolecules) is to give students interested in structural biology
an insight into methods of NMR spectroscopy applicable to biologically important molecules (mostly proteins and
nucleic acids). Principles of methods are explained in details, but using the classical physics instead of the quantum
mechanics. It should be emphasized that classical physics describes NMR correctly as long as we discuss macroscopic
samples, consisting of large numbers of molecules. Graphical description and explanation in text is preferred to
derivations based on series of plain equations. The theoretical background is limited, those interested in the theory of
the nuclear magnetic resonance (based on the quantum mechanics) may choose the course C5320 Theoretical concepts
of NMR.
Most topics of our course, except for the NMR spectroscopy of nucleic acids, are described very well in Cavanagh
et al., Protein NMR spectroscopy, 2nd. ed., Academic Press 2006. However, the book is based on the quantum
mechanical approach and is not easy to read for a beginner without a solid physical background. General ideas and
technical issues are explained in Keeler, Understanding NMR spectroscopy, 2nd. ed., Wiley 2010, and in Levitt, Spin
dynamics, 2nd. ed., Wiley 2008. Both books are written for non-experts and present NMR in a very understandable
manner, but they do not cover applications to proteins and nucleic acids. The mentioned limitations of the available
literature were the major motivation to write this study text. The intention was to complement the lectures by writing
a short review of NMR techniques applicable to biomacromolecules. The main purpose of the lectures is to explain
the basic ideas in an interactive manner. The text should summarize the topics in a more fluent form and to provide
a broader context.
1.2 Structure and structure determination
It should not surprise anybody that the structure is what a structural biologist is mostly interested in. Therefore, we
have a good reason to start our discussion of the nuclear magnetic resonance with the topic of structure determination.
There are of course good reasons to start somewhere else (e.g., to show applications where no other method can
compete with NMR), but I feel an obligation to acknowledge the importance of the structural information right at the
beginning.
Before we start talking about structure determination, we should ask what the word structure means. The question
may sound silly but it is really useful to define this term clearly if we wish to avoid possible confusions. When structural
biologists talk about the structure they usually refer to a model of little balls representing atoms and sticks representing
covalent chemical bonds connecting the atoms. The way how the atoms are connected is described by the chemical
term configuration. Limitations of the described model are obvious. The model is rigid and cannot describe molecular
motions that are always present or chemical reactivity of the molecule.
Macromolecules of our interest, i.e., proteins and nucleic acids, are composed of a small number of basic building
blocks, residues. The configuration is almost completely described by the sequence of the residues in the studied
molecule. The configuration of the individual building blocks is well defined and depends very little on the actual
1
2
position of the residue in the macromolecule. In other words, the sticks have a certain length and fit to holes drilled into
the balls, and there is very little that we can do about it. Even the way how to assemble the individual residues is given,
so our task is only to find out how to join the residues together. Various sequencing techniques have been designed
in to determine the order of residues in a particular macromolecular chain. Once we build the chain of residues, and
do some little adjustments like connecting correct disulfide bridges in proteins, the (chemical) configuration of our
molecule is defined.
Knowing the configuration, we have enough information to draw a pretty scheme of our macromolecule, in the
same way as the authors of the organic chemistry textbooks describe somewhat smaller molecules. Note that organic
chemists call such pictures ”structures”, they use the term structure at a level equivalent to the configuration. In the
case of small molecules, the drawing is giving a fairly good idea about the shape of the molecule (described by the
ball-and-stick model). The situation is different if the molecule has a size of a protein. Why? It is possible to rotate
more or less freely about most chemical bonds without changing the configuration. An astronomic number of possible
ball-and-stick models can be generated by the rotation about various bonds. If we need to refer to one of them, we
use the term conformation. Now it should be clear that the term structure is typically a synonym of conformation in
the structural biology, unlike in the organic chemistry (where structure usually refers to the configuration).
Now, when we know that the structure determination is nothing else than the process of finding the correct
conformation of the studied molecule, we can briefly discuss the mathematical description of the structures. Position
of every ball in our model can be described by three coordinates. Very often, the familiar Cartesian x, y, and z
coordinates are used. Cartesian coordinates are very transparent but they also present some problems. If the molecule
moves or rotates a little bit, the numbers describing its Cartesian coordinates completely change. It is difficult to say
if two molecules are identical just by looking at their Cartesian coordinate lists. Another way of numerical description
of the molecule is to list all bond length, angles between bonds, and torsions about the bonds (in terms of so-called
dihedral angles or torsion angles).1
The latter way represents internal coordinates of the molecule, independent of the
position or orientation of the whole molecule. It should not surprise you that if six external coordinates (three defining
position of the molecule in the chosen coordinate system and three describing overall orientation of the molecule with
respect to the axes) are added to the internal coordinates, we obtain a list of numbers exactly as long as the list of
Cartesian coordinates. The internal coordinates can be divided into those describing configuration of the molecule (i.e.,
bond length and bond angles) and those describing conformation (torsion angles). This is useful because configuration
can be taken as granted and the whole structure can be described by torsion angles only.
1.3 Macromolecular structure from interatomic distances
Positions of individual atoms in biologically important macromolecules are most typically determined by techniques
that somehow depend on the diffraction of waves: X-ray crystallography, electron microscopy, neutron diffraction. The
general requirement is that the wavelength must be similar to the distances between atoms (or shorter). Obviously,
NMR spectroscopy does not meet this criterion. There are waves used in NMR spectroscopy, as we discuss later ,
but their wavelength is several decimeters. How can such waves provide information about atomic distances? The
answer is: NMR is an indirect method. This is a big disadvantage because the NMR structure determination is less
straightforward and based on less complete data than when structures are solved using the diffraction techniques. But
the indirect nature of the NMR structure determination is also a great advantage because limitations of the diffraction
techniques do not apply to the NMR spectroscopy. Let us now compare the direct and indirect approaches to the
structure determination. Instead of equations and technical argument, we use a story.
Imagine you are a sailor and you visit a beautiful Pacific island with friendly people. You wish to live there for
ever. You burn your yacht, crash your smartphone, and build a shanty. After couple of days, you find out that the
inhabitants do not have any map of their land. You decide to draw one for them.
There are several ways how to draw a map. For example, you can make a hot-air balloon, inflate it, sail over your
island, and draw a map when watching the landscape from the balloon’s basket. Or you can just walk from one site
to another one, count your steps and measure distances between caps, bays, mountains, and other sites that should
appear in the map.
1Note that mathematical definitions of the torsion and dihedral angles differ in the sign.
1.4. LIST OF APPLICATIONS 3
The first approach requires a long preparation but makes drawing the map rather easy. It resembles the diffraction
techniques. The second approach does not need any preparation. You can start striding immediately. However, it
takes a long time to walk between all sites and the distances measured in this way will not be too precise. It is also not
so straightforward to construct a map from distances as to draw what you see from the sky. You have to draw circles
of radii corresponding to the measured distances. Positions of the important points are given by intersections of the
circles and every error in the measured distances makes the intersections less accurate. Accuracy can be improved if
you use more circles for every point, but it would be time consuming to compete with the first approach.
The latter approach resembles the NMR structure determination. The NMR methods discussed in our course are
applied to molecules in solution.2
You just need to dissolve your molecule in a buffer and add some heavy water as
a sort of internal standard. However, the measurement itself requires a whole set of different experiments, and often
takes several weeks. Like in the case of the sailor, the more experiment you run, the more accurate structural model
you obtain. Analysis of the measured data is also more tedious than it was in the case of crystallography. Finally, it is
very hard to determine the structure of molecules larger than approximately 25 kDa. Not only the measured spectra
are more complex, but it is difficult to get any signal at all. Currently, NMR is used as a structure determination
method for small proteins, individual domains of large proteins, and structurally interesting fragments of nucleic acids.
Approximately 15 % structures in the databases were obtained from NMR data.
In order to explain how the NMR structure determination really works, we have to modify the sailor’s story a little
bit. We let our sailor use a more sophisticated means of measuring distances by giving him a set of walkie-talkies,
tunable to different frequency channels (Figure 1.1). The sailor can give them to his friends living at various sites of
the island. Each of these ”radio-operators” can estimate how far her or his neighbors are by evaluating signal intensity.
The procedure is simple. A guy sitting at a certain place tries to tune in all his fellows that he can hear, and writes
down how strong the signal is (your operator locates the position of your mobile phone in a similar manner). Then
the sailor collects the notes from every walkie-talkie and begins drawing the map.
The process of NMR structure determination is very similar to the walkie-talkie tale. We make the atomic nuclei
transmit radio waves and then we measure how much of the signal has been passed to the nuclei in their vicinity.
But such an experiment is not where we usually start. It certainly helped the sailor if he first made something like
a telephone directory of his walkie-talkie operators: a list of their names and transmitting frequencies of the devices
given to them. It is almost a rule that the NMR structure determination (actually, any NMR study of a biologically
interesting macromolecule) starts with ”writing a telephone directory”. It is not so easy because nuclei do not have
their frequencies written on their backs. We have a list of nuclei defined by the amino-acid (or nucleotide) sequence
in one hand and a list of measured frequencies in the other hand. It is our task to find which frequency belongs
to which nucleus. There is a whole suite of NMR experiments designed just for this purpose and analysis of their
results is a routine well known to anybody who uses NMR as a structure determination tool. The process of writing
”nuclear phone directory” has been given a self-explanatory name assignment. Only when the assignment is done, the
structural data starts to make sense.3
It can take longer to assign frequencies to (almost) all nuclei than to collect
information about distances among nuclei. If you later feel that I spend too much time talking about the assignment
experiments which tell us nothing about the structure,4
please, be patient a remember what I just told you.
1.4 List of applications
We have described NMR as a method of structure determination, working reasonably well for smaller proteins and
nucleic acid fragments. However, NMR is a much more flexible technique. Other applications are perhaps even more
useful than building a structural model. In many cases, the assignment provides a basis for a variety of structural and
functional studies. What looked just as an additional effort in the context of structure determination, opens doors to
many interesting experiments. Let us now list at least some of them.
2NMR spectroscopy of biomacromolecules in solid state is a rapidly developing field and is already able to determine structures with
atomic resolution.
3In theory, we do not need to assign frequencies to nuclei in advance. A structural model can be obtained just from a set of internuclear
frequencies. However, protocols following this idea are currently less successful than the ”orthodox approaches” described in the rest of
this book.
4This is not quite fair, useful structural information can be obtained already from the assigned frequencies.
4
A B
C D
Figure 1.1: Walkie-talkie method of distance measurement. Technically, you decided to use PMR 446 (personal mobile radio operating
at 446 MHz). Your walkie-talkies are tuned to four frequencies (four channels) that distinguish individual devices: 446.11875 MHz,
446.13125 MHz, 446.14375 MHz, and 446.15625 MHz. The transmitted and received frequencies are shown on the walkie-talkie displays in
red and black, respectively (for better visibility, only three digits following 446 MHz are displayed). When switched to the receiver mode,
the displays show also the strength of the received signal (the indicator).
1.4. LIST OF APPLICATIONS 5
• Is my protein folded? We start our overview with a very simple but very useful application. It is an experiment
telling us if the prepared protein has a well-defined structure, is only partially folded, or is present as a random
coil. This information can be obtained in several minutes. The spectra provide atomic resolution (unlike the
circular dichroism spectra, giving only the percentage of the secondary structure elements). The atomic resolution
may seem useless when the assignment is not yet available, but it allows us to count signals corresponding to
the observed amino acids. If we count as many signals as we expect, and if their distribution is typical for a
well folded protein, we know that the protein is likely to be prepared correctly. What I have said should already
convince you that this experiment is a very useful test of the sample. It is always run prior to more complex
NMR experiments. It is also a good idea to record a simple NMR spectrum before we try to crystallize a protein
(if the protein is not too large).
• 3D structure determination has been already described in Section 1.2.
• Specific structural details. We already know that NMR data are different from data provided by diffraction
techniques. Each NMR frequency can be attributed to an individual nucleus (and to the effects of its neighbors).
On the other hand, diffraction of X-rays at each atom contributes to all reflections in a predictable way. This is
a great advantage of X-ray crystallography. The diffraction pattern can be back-transformed into the electron
density map of the molecule. However, the local nature of the NMR data can be an advantage under certain
circumstances. There are cases when we are not interested so much in the whole structure but we would like to
learn more about some detail (e.g., a mutated active center of an enzyme, or a binding site occupied by various
ligands). We can save time and look just at the interesting site instead of solving the structure completely.
Of course, the assignment must be known in advance, it is only the structural part of the study that can be
simplified.
• Intermolecular interactions. Once the assignment is completed, interactions of a protein with another molecule
(I call it ”ligand” in the following text but it can be almost anything: metal ions, small molecules, other proteins,
nucleic acids) can be monitored very quickly. We take the protein and titrate it with the ligand. After each
addition of the ligand, a simple NMR experiment is run. Such experiments do not need to take more than an
hour. A qualitative information at the atomic-level resolution is obtained immediately from the spectra. Nuclei
that change frequency upon addition are likely to be close to the binding site. A real structural information
requires distance measurements, like when determining the whole structure. But NMR makes our life easier
again. It is possible to distinguish intramolecular and intermolecular distances using a clever combination of
isotope labeling and design of the NMR experiment. This approach eliminates many ambiguities and produces
highly reliable information about intermolecular contacts.
• Molecular motions. Crystal structures and EM images of proteins reflect protein flexibility to some degree. Less
defined electron density may indicate internal motions. Missing loops are likely to be flexible. NMR methods
tell us much more about the molecular dynamics. Hydrodynamic behavior of macromolecules may seem not too
interesting from a biological point of view. Still, it is useful to know how a molecule moves in solution because
the hydrodynamic properties depend on the size and shape of the molecule. This type of information is useful
when we need to check if a molecule is a monomer or a dimer, if it is present in complex with another large
molecule or not, etc. Internal motions are often essential for the biological function of biomolecules. Motions
on various time scales can be distinguished and described quantitatively with atomic resolution. Motions on the
time scale of seconds (and slower) can be studied in real time as you can expect. More interestingly, very fast
motions (faster than 10−9
s) can be described relatively easily in terms of their amplitude and often frequency (at
the atomic resolution, of course). It is also an advantage of NMR that the signals affected by the fast motions are
averaged but not missing. Various experiments have been designed to study slower, microsecond to millisecond,
motions.
• Kinetics and thermodynamics. The fact that NMR can be applied to kinetic studies has been already mentioned
in the previous paragraph. It is very easy to monitor molecular processes at various temperatures. Activation
energies can be obtained at least in some cases. In principle, NMR spectroscopy can also provide thermodynamic
data such as binding constants. However, the typical concentration range used in NMR experiments (higher
than 100 µM) is rather high to study interactions with high affinities.
6
• In vivo measurements. Mild conditions of the NMR measurements offer the possibility to study molecules in
the living cells. The major limitation is relatively a low sensitivity of the method and a high complexity of
the molecular mixture inside the cell. Small molecules have been studied in applications like metabolic profile
monitoring or intracellular pH measurements. It is currently also possible to measure in vivo NMR spectra of
proteins and nucleic acids, although at higher than physiological concentrations. This allows us to estimate how
are molecules influenced by the cellular environment and to check if spectra obtained in vitro are the same as
those measured in the cell. Recently published in-cell studies already addressed important biological issues.
• Spatial resolution. So far, we looked at applications where NMR spectra were measured in the entire volume of
the sample. It is also possible to monitor the distribution of certain type of nucleus in space. Magnetic resonance
imaging, used in medical diagnostics, is a typical example. Requirements for spectral and spatial resolutions can
be combined, and in situ spectra (limited to a small volume) can be measured.
1.5 When can a structural biologist use NMR
I have tried to describe NMR as a wonderful method offering so much useful information and you may ask why it is
not used more frequently by biochemists and biologists. I must admit that NMR has also its requirements and that
some of them can be tough for a particular biological macromolecule. The common source of the limitations is an
inherently low sensitivity. So, how should a biomolecular sample look like to be ready for an NMR study in general
(and for the structure determination, in particular)?
• Solubility. Sensitivity of the method requires relatively high concentrations. Samples of 1 mM macromolecules
promise good results, higher concentrations are better. Working with concentrations below 0.1 mM is a challenge.
• Stability. Not only the molecule of interest should be well soluble, but it should also stay in solution for a long
time. Ideally, the sample survives for month at room temperature. Molecules that are decomposed in less than
a week are not suited for a long-term study such as structure determination. Measurements at temperatures
close to 0 ◦
C are possible but their sensitivity is much worse (especially for large molecules).
• Stable isotope labeling. We do not see all nuclei in NMR. Almost all hydrogen and all phosphorus nuclei are active
but only every hundredth nucleus of carbon is visible. The most abundant isotope 12
C is stable and silent in NMR
experiments. Natural samples contain about 1 % of the isotope 13
C. It is also stable but can be observed in NMR
experiments. Finally, trace amounts of a third isotope are present in Nature. The third isotope, 14
C, is created
by cosmic rays in atmosphere, undergoes a radioactive decay, and is silent in NMR experiments. Unfortunately,
the natural amounts of 13
C can be observed only in the most sensitive NMR experiments. The situation is even
worse for nitrogen, and oxygen is completely useless for biomolecular application. Therefore, artificial molecules
of proteins and nucleic acids are often prepared for NMR experiments by replacing the most abundant, NMR
silent, isotope 12
C with the NMR active isotope 13
C. Also, 14
N is replaced by a more suitable 15
N. This effects
NMR spectroscopy to proteins that are expressed in well behaving host organisms (often bacteria) and to the
synthetically available molecules.
• Scaling up production. It follows from the previous paragraph, that proteins for NMR studies must be prepared
in milligram amounts and with completely changed ratio of isotopes. Fortunately, recombinant methods of
molecular biology nicely fit our needs. As most proteins are prepared in a recombinant form anyway, the protein
sample preparation does not present any extraordinary requirements. Samples of nucleic acids are not obtained
so easily and labeling with NMR active isotopes is relatively expensive in this case.
• Size. The sensitivity of the NMR methods for large molecules is very poor. It is not the molecular mass but the
rate of tumbling in solution that is critical. Large rigid molecules (or rigid molecules in more viscous solvents,
e.g. in cold water) tumbles slowly and give weaker NMR signal. The sensitivity can be regained by using special
tricks. Molecules of a relative mass close to one million have been successfully studied. On the other hand, the
process of full structure determination is very difficult and has been applied to only few monomeric proteins
larger that ∼30 kDa so far.
1.5. WHEN CAN A STRUCTURAL BIOLOGIST USE NMR 7
• Separation from impurities. Purity of the NMR sample is usually not a big problem. Obviously, signals of impurities
are observed in the spectra, may obscure signals of our interest, and should be eliminated. Paramagnetic
compounds (typically various metal ions) should be avoided or removed from the sample. In theory, molecular
oxygen dissolved in water may interfere as a paramagnetic molecule, but its presence causes no problems when
working with large molecules. Bacterial contamination represents more serious danger than chemical impurities.
Addition of an antimicrobial agent, such as sodium azide, is necessary to keep samples stable for a long time.
• Salt content. A high ionic strength should be avoided for purely technical reasons. A decrease of sensitivity can
be noticed with concentration of salt above 100 mM. Higher than 0.5 M salt concentrations should not be used
at all. NMR spectroscopists just do not like salty stuff.
8
Lecture 1b
NMR as a physical phenomenon
1.6 Spin and magnetic moment
Nuclear magnetism is a phenomenon exact description of which requires an advanced physical theory of quantum
electrodynamics. Yet, we can use very simple pictures to understand the basic ideas. Let us start with a really na¨ıve
model of the hydrogen atom shown in Figure 1.2A. Let us imagine an electron orbiting a proton like our Earth orbits
the Sun. The electron is charged and its circular motion is equivalent to the electric current (electric current in a wire
is nothing else than traveling electrons). This circular current induces a magnetic field like in an electric motor. But
this is not all. The Earth is also spinning about its own axes. Let us assume for a moment that the electron is spinning
in the same manner. The spin is also a circular motion and induces its own magnetic field. If we pull the electron
out of the atom, the orbital motion (and the orbital magnetism) disappears, but the spin (and the spin magnetism)
remains. If we put the electron inside a magnet, we can observe its spin magnetism (actually, this is how the spin of
electrons was discovered).
Magnetism of the electrons is quantitatively described by the magnetic moment. It is a vector property. Magnitude
of the magnetic moment describes strength of the magnetic field generated by the electron, and the direction of the
magnetic moment defines orientation of this magnetic field. The magnetic moment is directly proportional to the
angular momentum.5
Behavior of the magnetic moment and of the angular momentum is therefore similar (except for
the numeric value) and we will inspect both vectors together.
As any vector, the magnetic moment or the angular momentum can be decomposed into three perpendicular
components oriented along the x, y, and z axes. Perhaps a more convenient way of description is the spherical
representation, where the radius of the sphere defines the magnitude and two angles define direction in the same way
as the geographical latitude and longitude define the position on the Earth’s surface.
How does a spinning electron move in a magnetic field? Well, in a similar way as a spinning top moves in the
gravitational field of the Earth. If you steal such a toy from children and set it spinning, there is a little chance
that the axis of rotation is precisely perpendicular to the Earth’s surface. More often, the axis is slightly askew.
The gravitation pulls the spinning top down, but it does not fall to the ground immediately. Instead, the angular
momentum rotates the spinning axis so that it undergoes a slow precession motion. You do a similar thing when you
ride your bicycle. Your body does not stay straight up but it is swinging from left to right. Thanks to the angular
momentum of your wheels you do not fall down, you are only forced to continually change direction. It should not
surprise you that magnetic moments of free electrons are precessing in static homogeneous magnetic fields.
In order to proceed in our description of electrons in a magnetic field, we must borrow some results from the
quantum theory.6
So far we have not noticed any difference between the behavior of the electrons and our everyday
experience. One might get a feeling that we do not need the quantum theory at all. This feelings disappear as soon as
we realize one big difference between electrons and macroscopic rotating bodies such as the Earth or the spinning top.
Spinning of the electron cannot be stopped. It cannot be accelerated or slowed down. The angular momentum of the
5Angular momentum is a vector oriented along the axis of rotation and its absolute value is rmv, were r is radius of the circular orbit
and m and v are mass and velocity of the electron, respectively.
6It is interesting that the quantum mechanics itself does not explain why the spin exists. Only in combination with the special theory
of relativity, the quantum theory predicts spin as a necessary property of particles like electrons.
9
10
A B
⃗B
θ ⃗µ = γ⃗I
C ⃗B
⃗ω
⃗µ = γ⃗I
Figure 1.2: A, A na¨ıve model of hydrogen atom. B, A magnetic particle in a homogeneous magnetic field ⃗B0. C, Precession of a magnetic
dipolar moment in a homogeneous magnetic field ⃗B0.
spinning top is introduced by external forces of our fingers but the angular momentum of the electron is an inherent
property of the particle. It is even misleading to take the spin as a kind of rotation motion. Experiments and theory
show that electrons are not spinning, they simply have spin. I agree that the idea of an angular momentum7
without
physical rotation is very strange but we have to live with it.
Another consequence of the quantum theory that deserves our attention is the relation between the electron’s spin
and magnetic moment. In the classical physics, the proportionality constant is equal to one half of the charge-to-mass
ratio. The quantum theory gives a value twice as big in the first approximation. Further corrections are needed to
account for interactions of the electron with its own electromagnetism, but these effects are small and can be easily
calculated. In the case of the electron, the quantum theory can predict the magnetic moment with an astonishing
precision of eleven digits.
The quantum theory of course describes the electron and its spin in more details. For example, it tells us that
only some physical quantities can be measured simultaneously. It also shows that two particular orientations of the
magnetic moment in a magnetic fields are special. They are called stationary states and traditionally labeled α and
β. We do not need such details for the description of NMR because we are not interested in isolated particles but in
macroscopic samples. It is not only unnecessary, but confusing and physically incorrect (but unfortunately frequent
in the literature8
) to apply the quantum theory of single particles to NMR samples.
1.7 Nuclear magnetism
In spite of our deep interest in atomic nuclei, we talked about electrons so far. The only reason was that electrons
behave as simple particles, nicely described by quantum electrodynamics. Because the word ”simple” may suggest
various things, physicists call the electron and other well-behaving particles the Dirac particles.9
On the other hand,
nuclei are composite particles and their theoretical description is more complex. Even the nucleus of the hydrogen
atom, a single proton, is a composite particle. Protons and neutrons consist of Dirac particles, quarks. Each quark
possesses its own spin and magnetic moment. Quarks are held together by very strong interactions, mediated by
gluons. Calculation of the magnetic moments of protons and neutrons, and consequently of the whole nuclei, is very
difficult, as the resulting magnetic moment is a combination of spin and orbital moments of quarks and gluons. One
of the following three general results can be obtained.
7The word ”spin” is just a short name of the inherent angular momentum of a particle – do not confuse it with the angular momentum
describing the orbital motion of an electron around the nucleus.
8The quantum theory can be used to describe macroscopic samples but it must incorporate the statistical approach. The textbooks
recommended in Section 1.1 use the quantum theory correctly.
9Paul Dirac laid the foundations for the development of quantum electrodynamics.
1.7. NUCLEAR MAGNETISM 11
1. The magnetic moments are canceled completely and the nucleus does not show any magnetism. The most
abundant isotope of carbon, 12
C, composed of six protons and six neutrons, is a typical example.
2. The magnetic moments are combined such that the magnetic behavior of the nucleus resembles magnetic behavior
of a Dirac particle. The nucleus has the same spin as electrons. As quantum theory describes this type of particles
with the spin quantum numbers ±1
2 , a term spin-1/2 nuclei is used. It follows from the theory that these nuclei
have spherically symmetric electric field, which is a great advantage for NMR measurements. Typical examples
are isotopes 1
H, 13
C, 15
N, 31
P.
3. The third category includes all other combinations. Such nuclei have larger angular momenta (spins) and
asymmetric electric fields. They are not suitable for biomolecular NMR investigations. The most abundant
isotope of nitrogen (14
N) is an example.
It follows from the previous discussion that we can ignore the first and third type of nuclei10
and treat nuclei of
the second type as Dirac-like particles. Such an approach leads to correct qualitative conclusions but the numerical
values must be taken as empirical constants.
Now it is a good time to turn our attention to the quantitative issues. Fortunately, we do not need heavy math
to understand biomolecular applications NMR spectroscopy.11
It is good to remember three simple equations relating
four physical quantities (i) the spin or the angular momentum ⃗I, (ii) the magnetic moment ⃗µ and its (iii) energy E
and (iv) the angular precession frequency ⃗ω in a magnetic field of the magnetic induction ⃗B (Figure 1.2B,C):
⃗µ = γ⃗I, (1.1)
E = − ⃗B · ⃗µ = −γ ⃗B · ⃗I, (1.2)
⃗ω = −γ ⃗B, (1.3)
where the symbol γ describes the proportionality constant between spin and magnetic moment, known as the
magnetogyric ratio.
We have already discussed the relation between ⃗µ and ⃗I (Eq. 1.1) for the electron and the nuclei. The value of
γ is provided by the quantum mechanics with a great precision for the Dirac particles and should be determined
experimentally for the Dirac-like particles, including nuclei.
Eq. 1.2 defines the energy of the magnetic moment in a magnetic field. The classical physics shows that a magnet
in a magnetic field has the lowest energy if its magnetic moment is oriented along the magnetic field (parallel with
the vector of the magnetic induction ⃗B. The expression ⃗B · ⃗µ is a short version of the sum Bxµx + Byµy + Bzµz (Bx
is the x component of the vector ⃗B etc.). Traditionally, the z axis is defined by the direction of the magnetic field ⃗B0
inside the strong magnet of the NMR spectrometer. Therefore, the Bz component of ⃗B0 must be equal to the total
length of the vector,12
B0, while Bx and By must be equal to zero. Eq. 1.2 can be simply written as
E = −Bµz = −γB0Iz = −γB0I cos θ, (1.4)
where θ is the angle between the magnetic moment ⃗µ and the vector of the magnetic induction ⃗B.
Finally, Eq. 1.3 shows that the angular precession frequency ⃗ω is proportional to the external magnetic field and
that the proportionality constant is again γ. Note that the angular precession frequency is presented as a vector
quantity in Eq. 1.3. The magnitude |ω| is the speed of the precession (in the units of radian per second, precession
frequency in the units of Hertz is |ω|/(2π)), whereas the direction of ⃗ω defines the axis of the precession motion. The
minus sign reminds us that the axis of precession is oriented opposite to the magnetic induction.
In summary, Eqs. 1.1–1.3 tell us that the spin and the magnetic moment are intrinsic properties of the nucleus
but the energy E and precession frequency ⃗ω obviously depend on the external magnetic field.
10Only the nucleus of heavy hydrogen (deuterium) 2H is somewhat exceptional. Its electric field is not spherically symmetric, but is
relatively small and the deuterium signal is observed in some biomolecular applications.
11Much more mathematics is needed if we ask why the NMR experiments work as they do. Another course at Masaryk University, C5320
Theoretical concepts of NMR, tries to ask such questions.
12The total length or magnitude, usually called absolute value or amplitude in physical literature, is labeled simply by the letter
symbolizing the vector written without the arrow (B0 in our case), or by the letter surrounded with the absolute value symbols (|B0|).
12
Lecture 2
NMR as a method reflecting chemistry
2.1 Nuclei in the test-tube
After discussing the nuclear magnetism as a phenomenon, we can look at it from a more practical point of view.
We are now interested not in microscopic processes inside isolated atoms but in a measurable macroscopic quantity
describing magnetism of the whole sample. Such quantity is called total magnetization and labeled with the symbol
⃗M. As we moved from microscopic particles to macroscopic samples, the classical (non-quantum) physics should be
sufficient to describe what we observe.
In physics, the magnetization is defined as the sum of magnetic moments per unit volume. Magnetic moments
of the same nucleus in different molecules have the same magnitude but different orientations. The orientation in
a coordinate frame formed by the axes x, y, z is described by two angles (Figure 2.1), (i) the angle θ between the
magnetic moment and the z axis induction vector of the external magnetic field ⃗B (the inclination, an analogy of
the geographic latitude, but measured from the ”pole”), and (ii) the angle ϕ describing the azimuth, i.e., the angle
telling us how much is the component of the vector perpendicular to z rotated from the direction y in the xy plane (an
analogy of the geographic longitude). Therefore, the result of the addition of many little vectors of magnetic moments
of individual nuclei depends on their angles θ and ϕ.
Real chemical and biological samples contain huge numbers of nuclei. The only practical approach to such a large
collection of nuclei is a statistical description. Let us ask a typical statistical question: What are the most probable
angles θ and ϕ? The answer depends on the magnetic field, describing forces acting on the magnetic moments. In the
following paragraphs, we explore three types of magnetic environments.
z
x
y
⃗r
θ
ϕ
Figure 2.1: Orientation of a vector ⃗r (red) described by spherical coordinates ϕ, θ (black) and by the Cartesian coordinates x, y, z. The
longitudinal direction (z) is shown in green, the transverse plane (xy) is shown in blue.
13
14
A B C
⃗M
⃗B0
Figure 2.2: A, a schematic representation of an NMR sample. Dots represent molecules, arrows represent magnetic moments (only one
magnetic moment per molecule is shown for the sake of simplicity, like e.g. in compressed 13C16O2). B, Distribution of magnetic moments
in the absence of a magnetic field. The molecules are superimposed to make the distribution of magnetic moments visible. C, distribution
of magnetic moments (black arrows) in a homogeneous magnetic field ⃗B0. The cyan arrow represents the macroscopic magnetization. Even
if all black arrows precess with the same frequency about ⃗B0, the cyan arrow remains static.
2.2 No magnetic field
First, we can consider a sample of a biologically interesting molecule sitting on our laboratory bench (Figure 2.2A).
If we ignore magnetism of the Earth and all other magnetic fields that surround us every day, the nuclear magnetic
moments can be described as completely uncorrelated vectors, oriented randomly, pointing to all directions in space
with the same probability (Figure 2.2B). No wonder that such sample has a zero total magnetization.
2.3 Static magnetic field
Second, we put the sample inside a static, homogeneous magnetic field defined by the magnetic induction vector ⃗B0
and ask again: What are the most probable angles θ and ϕ if the axis z is given by the direction of ⃗B0? The answer
is simple for ϕ: all values of ϕ (all azimuths) are equally probable because the energy of a magnetic moment in ⃗B0
does not depend on ϕ (Eq. 1.4). As a consequence, components of magnetic vectors perpendicular to ⃗B0 (along x and
y axes in our coordinate frame) are uniformly distributed and therefore they sum to zero. Only the components of
magnetic moments parallel with ⃗B0 depend on ⃗B0. These components are described only by the angle θ. Therefore,
we have to statistically analyze θ in order to evaluate the only non-zero component of the sum of magnetic moments,
the component along the z axis of our coordinate frame.
On one hand, we have seen (Eq. 1.4) that the energy of the magnetic moment in a magnetic field depends on the
angle θ. Eq. 1.2 shows that the orientation of the magnetic moment parallel with ⃗B0, i.e. orientation with θ = 0, has
the lowest energy. One might expect that magnetic moments of nuclei should have this most favored orientation, at
least in the classical physics. On the other hand, there is only one orientation with θ = 0 (the magnetic moments
pointing to the ”northern pole”), but there are many possible orientations of magnetic moments for θ = 90 ◦
(magnetic
moment pointing to the ”equator”). We can see that we obtained two contradictory answers: if we ignore statistics
and consider only the energy, the most probable θ is zero, if we ignore the energy and consider only statistics, the most
probable θ is 90 ◦
. The correct balance between energy and statistics is described by the Boltzmann law. Nuclei are
not isolated from the rest of the world. They interact with the environment full of randomly moving molecules. This
stochastic motion is manifested as the temperature, and represents a rich source of energy for the nuclear magnets.
According to the Boltzmann law, fraction of magnetic moments in each orientation is indirectly proportional to the
exponent of the ratio of the magnetic energy to the thermal energy. If we study biological molecules in liquid solutions,
2.4. CREATING A SIGNAL 15
we work at temperatures high compared to the magnetic energies of nuclei in the strongest magnets. Therefore, the
exponent in the Boltzmann law is a very small number and the exponential dependence is very close to a linear relation.
Going through the somewhat tedious calculation1
yields the average value of cos θ at the thermal equilibrium:
cos θeq =
|µ||B0|
3kBT
, (2.1)
where kB = 1.38 × 10−23
JK−1
is the Boltzmann constant and T is the thermodynamic temperature. For the sake
of simplicity, we assumed that the molecule contains only one type of nuclei with a magnetic moment, for example
protons. As shown in Figure 2.2C, the distribution of magnetic moments is polarized in the direction of ⃗B0 (the most
favored orientation of a magnetic moment).
If we use the value of |µ| provided by the quantum mechanics (
√
3γℏ/2, where ℏ is the Planck constant divided by
2π, ℏ = 1.054 × 10−34
J s), we can express the z component of the magnetization at the equilibrium as
Meq
z = N|µ|cos θeq = N
γ2
ℏ2
4kBT
, (2.2)
where N is the number of magnetic nuclei per unit volume.
The x and y components of the magnetization
Meq
x = N|µ|sin θeq · cos ϕeq, (2.3)
Meq
y = N|µ|sin θeq · sin ϕeq (2.4)
are equal to zero because cos ϕ and sin ϕ average to zero when all values of ϕ are equally probable. The magnetization
vector pointing in the direction z tells us that the magnetic moments are polarized along the z axis (so-called
longitudinal polarization).
Note that we summed moments undergoing precession and obtained a macroscopic magnetization that does not
move at all (Figure 2.2C). Although all summed magnetic moments precess at the same frequency and the total
magnetization is not zero in this case, the rate of precession cannot be measured.
2.4 Creating a signal
We see that having a magnet is not enough to measure the precession frequency. We also need something that would
disturb the macroscopic magnetization from its equilibrium orientation along ⃗B0. Not surprisingly, this ”something”
is again a magnetic field. Let us try to orient the additional field, described by the induction ⃗B1, perpendicular
to the original external field (now we need to distinguish two independent external fields, the original field ⃗B0 and
the new one ⃗B1). The magnetic force of ⃗B1 would pull the magnetic moments from their equilibrium orientations.
Unfortunately, static ⃗B1 would not work as we wish. It is shown in Figure 2.3. The magnetic moments precess around
⃗B0, so they would feel the force of ⃗B1 pulling them from their equilibrium orientation at the beginning but back to
the equilibrium orientation a fraction of microsecond later. The static ⃗B1 would perhaps introduce fast fluctuations
but not a net change in the total magnetization.
Obviously, ⃗B1 must be synchronized with the precession caused by ⃗B0. For example, the magnetic field ⃗B1 could
be a field that rotates at the frequency which resonates with the precession frequency of nuclear magnetic moments.
This is why we talk about the nuclear magnetic resonance. In such case, the precessing magnetic moment would feel
a constant pull in one direction. This direction would precess together with the magnetic moment but would always
point away from the equilibrium orientation. It is difficult to imagine the effect of the synchronized ⃗B1. Physicists like
to make things simpler and so they introduced a coordinate system rotating together with the precessing magnetic
moment. It is like jumping on a carousel that rotates with the precession frequency. Once we jump on it, the magnetic
moments seem not to move any more and also ⃗B1 looks like a static field (Figure 2.4A). If ⃗B1 is static, it must have the
same effect as ⃗B0, it must introduce a circular (precession) motion of the magnetic moments. The total magnetization
1The following equation is derived in Course C5320 Theoretical concepts of NMR and the derivation is presented in the study material
for the course.
16
A
⃗M
⃗B0
B
⃗M
⃗Bstatic
⃗B0
C ⃗M
⃗Bstatic
⃗B0
Figure 2.3: Distribution of magnetic moments in the presence of an external homogeneous magnetic field ⃗B0 (vertical violet arrow) is
such that the macroscopic magnetization of nuclei (shown in cyan) is oriented along ⃗B0 (A). Application of an another static magnetic field
⃗B1 rotates the magnetization away from the original vertical orientation down in a clockwise direction (B). However, the magnetization also
precesses about ⃗B0. After a half-turn precession (C), the clockwise rotation by the additional magnetic field ⃗B1 returns the magnetization
towards its original vertical direction. Therefore, a static field cannot be used to turn the magnetization from the vertical direction to a
perpendicular orientation.
is not parallel to ⃗B1 but perpendicular to it. So, it does not stay in its original position, defined by ⃗B0, but it starts
to rotate about a new axis defined by ⃗B1. The rotation of the magnetization is a result of the additional precession
introduced by ⃗B1, and has the same frequency as the new precession. Outside the rotating frame, we observe two
superimposed rotations about two perpendicular axes. In our spherical representation of vectors, the tip of the arrow
symbolizing the total magnetization draws spirals on a spherical surface. Such motion has been given a name nutation
by astronomers.
I must admit that the current technology does not allow us to rotate magnets at the precession frequency of
nuclei. Fortunately, a magnetic field oscillating in one direction has a very similar effect if it is much weaker than ⃗B0
(Figure 2.4B). Radio waves represent a readily available source of a magnetic field oscillating in one direction with
a frequency resonating with the precession frequency of the magnetic moments. Application of a radio wave for a
well chosen period of time, known as the radio-wave pulse, thus rotates the magnetization vector from the z direction
to the xy plane. After the pulse, the magnetic moments are not polarized vertically, but horizontally (Figure 2.4C).
The direction of the horizontal, or transverse polarization then rotates with the precession frequency of the polarized
magnetic moments. In other words, the Mx and My components of the magnetization vector oscillate as cosine and
sine functions with the frequency equal to the precession frequency of individual nuclei. The cosine and sine functions
are actually shifted in time by a phase factor depending on the phase of the radio wave, as shown in Figure 2.5.
We should stress that the radio-wave pulse did not create the transverse (horizontal) polarization from an unpolarized
distribution. It only rotated the already existing longitudinal (vertical) polarization induced by the much
stronger static field ⃗B0. Therefore, the magnitude of the magnetization rotating in the xy plane is proportional to
|B0| according to the Boltzmann law (Eq. 2.2).
Bringing the magnetization to the xy plane is called creation of coherence in the NMR literature. The term
coherence has its origin in the statistical quantum mechanics, discussed in detail in Course C5320 Theoretical concepts
of NMR. At this moment, we use the word ”coherence” as a synonym for transverse polarization (we extend its meaning
when we take into account mutual interactions between nuclei). The coherence is preserved as long as the magnetic
moments move together, coherently. It does not take too long (usually seconds or less) to lose the coherence and, finally,
to restore the equilibrium distribution of magnetic moments. The loss of coherence and return to the equilibrium are
known as relaxation. We discuss relaxation in more detail later in our course. Before doing so, we explain other issues
important for NMR spectroscopy and often ignore the relaxation effects for the sake of simplicity. However, we should
never forget that relaxation exists and influences all NMR experiments.
2.4. CREATING A SIGNAL 17
A
⃗M
⃗B1
B
⃗M
⃗B0
⃗Bradio
C
⃗M
⃗B0
Figure 2.4: Effect of the radio waves on the bulk magnetization (A, in the rotating coordinate frame; B, in the laboratory coordinate
frame) and distribution of magnetic moments after application of the radio-wave pulse (C). The thin purple line shows oscillation of the
magnetic induction vector of the radio waves, the cyan trace shows evolution of the magnetization during irradiation. If the perpendicular
magnetic field oscillates with a frequency equal to the precession frequency of the magnetization, it rotates the magnetization clockwise
when it is tilted to the right, but counter-clockwise when the magnetization is tilted to the left. Therefore, the magnetization is more
and more tilted down from the original vertical direction. The total duration of the irradiation by the radio wave was chosen so that
the magnetization is rotated to the plane perpendicular to ⃗B0 (cyan arrow). Note that the ratio | ⃗B0|/| ⃗Bradio| is much higher in a real
experiment.
Mx
t
My
t
Figure 2.5: Values of Mx (blue) and My (red) components of a magnetization vector rotating with the initial phase of 60 ◦. The values
decay due to a loss of coherence.
18
A
θ
B
θ
C
θ
Figure 2.6: A, Classical description of interaction of a spin magnetic moment of the observed nucleus (shown in cyan) with a spin
magnetic moment of another nucleus (shown in green). The thick purple arrow represents ⃗B0, the thin green induction lines represent the
magnetic field ⃗B2 of the green nucleus (the small green arrows indicate its direction). The black line represents the internuclear vector
⃗r. As the molecule rotates, the cyan nucleus moves from a position where the field of the spin magnetic moment of the green nucleus ⃗B2
has the opposite direction than ⃗B0 (A), through a position where ⃗B2 is perpendicular to ⃗B0 (B), to the position where ⃗B2 has the same
direction as ⃗B0 (C).
2.5 Nuclei are affected by their environment
So far, we have discussed a very simple situation. Nuclei were placed into a homogeneous magnetic field of the magnet
sitting in our laboratory. The result was a precession motion at a frequency given (1) by the internal structure of the
nucleus (it defines γ) and (2) by the strength of our magnet (it determines ⃗B0). The observable magnetization, tilted
to the transverse plane by a pulse of a radio wave, rotated et the same frequency. Clearly, such frequency carried
no information about the molecule built of atoms containing our nuclei. The chemical and biological applications are
only possible because the molecular environment modifies the magnetic field felt by the nuclei. We will look at the
most interesting effects of the molecular environment in this section.
2.6 Interactions with close nuclei
Magnetic dipolar moments of the nuclei experience not only the external magnetic fields. The nuclei themselves are
also sources of little magnetic fields. The effective magnetic field perceived by a certain nucleus is modified by its
nuclear neighbors. The effect of a close nucleus depends on the mutual orientation of both nuclei, i.e., the observed
nucleus (shown in cyan in Figure 2.6) and its neighbor (shown in green in Figure 2.6) modifying the field felt by the
observed nucleus. Let us suppose that the neighbor’s magnetic moment is pointing up (the vertical green arrow in
Figure 2.6). If the neighbor is placed in a direction perpendicular to the induction of the external field (Figure 2.6A),
the magnetic field generated by the neighbor at the site of the observed nucleus is oriented opposite to the external
field, and the effective field is decreased. If the neighbor is positioned in the direction of the external field, its magnetic
moment creates a field that is added to the external field (Figure 2.6C). Quantitatively, the field created by the neighbor
nucleus is proportional to 3 cos2
θ − 1, where θ is the angle between a line connecting the nuclei and the direction of
the external field. Graphically, the direction of the neighbor’s field is described by the shape of the magnetic induction
lines of the neighbor’s magnetic dipole. The effects are exactly opposite if the neighbor’s magnetic moment is pointing
down. This type of interaction is called dipole-dipole coupling (or dipolar coupling) in the NMR literature.
The effects of the close nuclei can be observed directly as shifts of the precession frequency if the molecules are
ordered in space (in crystals, or at least partially in liquid crystals). If the molecule freely rotates and if the rate of the
rotation is the same in all directions (isotropic rotation), the position of the neighbor varies in time and the effects of
the neighbor’s magnetic field average to zero (if we calculate the average value of cos2
θ for all possible values of θ, we
find that is equal to 1/3, and therefore the average of 3 cos2
θ −1 is zero). Does it mean that no information about the
molecular structure can be obtained? Not at all. The fluctuating electromagnetic fields resulting from the molecular
2.7. INTERACTIONS WITH ELECTRONS 19
tumbling can be viewed as electromagnetic waves (or photons) of certain frequency. The frequency of the molecular
rotation changes randomly as the molecule collides with other molecules, and once a while it matches the precession
frequency of the neighbor nucleus. When this happens, the x and y components of the magnetic induction of the
stochastic waves generated by the molecular rotation resemble the radio waves applied by us: the frequency of the
oscillation in the xy plane resonates with the precession frequency of the close nucleus and rotates its magnetization
as shown in Figure 2.4 until a next collision changes the frequency of the molecular rotation. Such exchange of
energy between nuclei is more probable if the nuclei are closer in space. Distances between nuclei can be estimated
by measuring the efficiency of the described energy exchange, known as the nuclear Overhauser effect in the NMR
literature. The interactions with magnetic moments of nuclear neighbors thus represent the physical nature of the
allegory about the sailor and his walkie-talkie method (Section 1.3). Later we also show that dipole-dipole interactions
varying due to the molecular motions cause relaxation (loss of coherence and return to the equilibrium distribution of
magnetic moments).
2.7 Interactions with electrons
Every nucleus in a molecule is surrounded by electrons. The magnetic fields of electrons significantly modify the field
felt by the nucleus. Interactions with the spin magnetic moments of electrons are not so important because most
electrons are paired and their spin magnetic moments canceled.2
What is more interesting is the effect of the orbital
magnetic moments. The orbital magnetism is induced by the external magnetic field and is always oriented against it.
The external field is shielded by the paired electrons. Therefore, the precession frequency of a nucleus is decreased by
orbital magnetic moments of electrons of the same atom (Figure 2.7A). Nuclei are also influenced by orbital magnetic
moments of electrons of other atoms. The effect then depends on the orientation of the whole molecule in the magnetic
field (Figure 2.7B,C). The precession frequency can be slowed down (shielding) but also speeded up (deshielding). As
a result, different protons (or other nuclides) in the same molecule have slightly different frequencies. Therefore, the
frequency of the applied radio wave can match only one precession frequency. Deviations of precession frequencies of
the other nuclei from the radio-wave frequency are called frequency offsets and usually labeled Ω. The orientation of the
molecule in the magnetic field influences the interaction with the orbital magnetic moments of electrons in a similar
manner as described for the dipole-dipole interactions in Section 2.6, including relaxation. However, the isotropic
molecular tumbling does not eliminate the shielding. As a result, the precession frequency is shifted even in perfectly
spherical molecules. This phenomenon is known as the chemical shift (chemical because it does not reflect nuclear
properties but molecular structure). The chemical shift is a sensitive probe of the structure of the molecule because it
depends on the distribution of electrons in the molecule. Empirical values are routinely used as the key identifiers in
the process of determining chemical configuration of small organic molecules. Unfortunately, interpretation of subtle
chemical shift variations is very difficult and the use of chemical shifts in structure determination of biomacromolecules
is limited.
2.8 Interactions between nuclei mediated by electrons
We described how magnetic moments modify the external magnetic field felt by their nuclear neighbors. The little
magnetic moments of nuclei also interact with the magnetic moments of electrons forming bonds between atoms.
Therefore, nuclei interact not only directly, as described in Section 2.6, but also through a cascade of interactions
with magnetic moments of electrons forming chemical bonds between them. This type of interaction is known as the
J-coupling.
The origin of the J-coupling is interesting. The dominant component of the J-coupling is the Fermi contact
interaction, which can be explained as follows. It is not surprising that magnetic moments of nuclei interact with spin
magnetic moments of electrons, but this is just a regular dipole-dipole interaction. We already learned in Section 2.6
that the shape of the magnetic field of a nuclear dipole is described by magnetic induction lines of the dipole, and
that the numerical value is given by 3 cos2
θ − 1, which averages to zero if the nucleus freely rotates in the external
magnetic field (Figure 2.8A–C). This is true, but with one exception, depicted in Figure 2.8D.
2Unpaired electrons are mostly observed in proteins containing metal ions. They affect nuclei through the same mechanism as described
in Section 2.6, but the interaction is much stronger.
20
A B C
Figure 2.7: A, Classical description of interaction of an observed magnetic moment with the orbital magnetic moment of an electron of
the same atom. The observed nucleus and the electron are shown in cyan and red, respectively. The thick purple arrow represents ⃗B0, the
thin purple induction lines represent the magnetic field of the electron (the small purple arrows indicate its direction). The electron in ⃗B0
moves in a circle shown in red, direction of the motion is shown as the red arrow. The field of the orbital magnetic moment of the electron
in the same atom decreases the total field in the place of the observed nucleus (the small purple arrow in the place of the cyan nucleus is
pointing down). B, Interaction of an observed magnetic moment with the orbital magnetic moment of an electron of the another atom (its
nucleus is shown in gray). In the shown orientation of the molecule, the field of the orbital magnetic moment of the electron in the other
atom increases the total field in the place of the observed nucleus (the small purple arrow close the cyan nucleus is pointing up). C, As the
molecule rotates, the cyan nucleus moves to a position where the field of the orbital magnetic moment of the electron in the other atom
starts to decrease the total field (the induction lines reverse their direction in the place of the cyan nucleus).
If the electron is present exactly at the nucleus, the vector of the electron spin magnetic moment ⃗µe has the
same direction as ⃗Be and the magnetic energy is proportional to the scalar product of the vectors of the magnetic
moments of the electron and the nucleus (−⃗µn · ⃗µe). This energy is lowest when the magnetic moments are parallel,
and obviously does not depend on the orientation of the molecules (the scalar product is a scalar, not a vector).
Therefore, this interaction is often called scalar coupling. The exact co-localization of the electron and the nucleus
may look strange in our classical (non-quantum) discussion of NMR, but the interaction between the nucleus and the
electron inside the nucleus can be simulated by a hypothetical current loop giving the correct magnetic moment when
treated classically. The direction of the magnetic induction lines inside such loop (inside the nucleus) is parallel with
the magnetic moment of the electron.
When we discussed interactions between nuclear magnetic moments in Section 2.6, we did not pay any attention
to the field inside the nucleus because no other nucleus can be found there, nuclei are always separated in molecules.
But this restriction does not apply to electrons! The shape of the atomic s-orbital suggests that the electron can be
found with a non-zero probability exactly at the position of the nucleus (Figure 2.8E).
Let us now look at a pair of two nuclei connected by a single bond, interacting with two electrons with a non-zero
probability to be present exactly at the positions of the nuclei (Figure 2.8F,G). The electrons in pairs that form
chemical bonds have the opposite spin (red arrows in Figure 2.8F,G). It has the following consequence.
If the magnetic moment of the neighbor (shown in green in Figure 2.8F,G) is pointing up (Figure 2.8F), it makes
the −z orientation (pointing down) of the spin magnetic moment of the electron at the place of the cyan nucleus more
favorable. Such a magnetic moment is a source of a magnetic field ⃗Be oriented in the −z direction at the site of the
cyan nucleus. Any magnetic field causes precession of magnetic moments. The magnetic moment of the cyan nucleus
would precess about an axis given by the vector of the angular frequency ⃗ωe = −γ ⃗Be. For a magnetic field pointing
down, this corresponds to a counterclockwise precession if γ of the cyan nucleus is positive. The described precession
caused by the electron’s field is added to the much faster precession caused by ⃗B0. As the electron generates a field
in a direction opposite to ⃗B0, the electron’s field slows down the precession. To make a long story short, the +z
orientation of the neighbor’s magnetic field decreases the precession frequency of the observed nucleus.
If the magnetic moment of the neighbor is pointing down (Figure 2.8G), the effect is opposite. The electron colocalized
with the observed nucleus increases precession frequency of the magnetic moment of the observed nucleus
(the field of the electron ⃗Be has the same direction as ⃗B0).
We have analyzed only the vertical direction of nuclear magnetic moments when discussing Figure 2.8F,G. Other
directions have similar effects, they introduce additional counterclockwise precession for nuclei with γ > 0 (and
2.8. INTERACTIONS BETWEEN NUCLEI MEDIATED BY ELECTRONS 21
A
θ
B
θ
C
θ
D
E F G
Figure 2.8: J-coupling. A–C, magnetic fields of an electron (red induction lines) outside the nucleus (cyan), influencing the nucleus as
expected for regular dipole-dipole interactions. D, magnetic field of an electron localized at the position of the nucleus. E, probability
of finding an electron in the hydrogen atom at particular coordinates is described by the probability density. The probability density
described by the orbital 1s (depicted as a sphere) has a non-zero value at the position of the nucleus (shown in cyan). Therefore, there is
a non-zero probability of finding electron (red circle) exactly at the site of the nucleus. The field produced at the site of the nucleus by
the electron’s magnetic moment (red arrow) does not depend on the orientation of the atom if the positions of the nucleus and electron
coincide. Therefore, the interaction of the nucleus with the electron is not averaged to zero if the atom rotates isotropically. F and G, the
probability density described by the sigma orbitals (depicted as an ellipsoid) in molecules has also non-zero values at the sites of nuclei.
The spin states of the electrons in the bonding sigma orbital are opposite (indicated by the opposite direction of the red arrows) and
perturbed by the magnetic moment of the nuclei. The parallel orientations of magnetic moments is energetically favorable for a nucleus
and an electron sharing its position. As a consequence, precession frequency of the cyan nucleus is lower in Panel F and higher in Panel G
(see text for details).
clockwise precession for nuclei with γ < 0). However, if the precession frequencies of the cyan and green nuclei differ,
the horizontal polarization of the green nucleus rotates quickly in a coordinate frame of the cyan nucleus. Therefore,
the effect of the horizontal polarization of the green nucleus on the precession frequency of the cyan nucleus fluctuates
rapidly and is quickly averaged to zero.
The described electron-mediated interactions of course depend on the electron distribution in the molecule. The
existence of an interaction so closely related to the chemical structure is obviously very useful. When we discuss
the NMR methodology in details, we find out that the J-couplings are the corner stones of almost all modern NMR
techniques.
22
Lecture 3
Spin alchemy
3.1 Pulse sequences
Section 2.4 introduced a very valuable servant, radio waves. The oscillating field of a short pulse of the radio wave
helped us to disturb equilibrium distribution of nuclear magnetic moments and created an observable signal. The
magic power of the radio-wave pulses is much greater than simply exciting magnetic moments from their equilibrium.
Combinations of exactly spaced pulses allow us to create, destroy, and convert polarizations of nuclei with an ease
that overcomes all dreams of alchemists. This is why the modern NMR methodology is sometimes described as spin
alchemy. The sequences of radio-wave pulses and delays, called simply pulse sequences in the NMR literature, may
be very complex. However, they are built from relatively simple building blocks or modules. Knowledge of the basic
modules allows us to understand, analyze, and design very sophisticated pulse sequences. In this lecture, we learn the
most important modules.
3.2 Excitation
We start with the very fundamental excitation module. It is very simple, consisting of a single 90◦
pulse. We
know that a single 90◦
pulse applied to a sample with the equilibrium distribution of magnetic moments turns the
magnetization vector from the vertical orientation to the xy plane. The obtained horizontally polarized sample is not
at the equilibrium, it is excited. A coherence is created and a rotation of the horizontal (transverse) magnetization
vector at the precession frequency of the observed nucleus can be detected.
The excitation pulse also plays an important role in the formal description of polarization and magnetization. The
magnetic induction ⃗B1 of the radio wave defines the x axis of our rotating coordinate system, as ⃗B0 defines the z axis
(of rotating and laboratory coordinate frames). However, the orientation of the ⃗B1 axis is somewhat arbitrary because
the phase of the radio wave is not well defined (it depends on length of the cables, distance of the transmitter antenna
from the sample, various instrumental time delays etc.). Unfortunately, the typical convention (see Figure 3.5) is a bit
confusing. We assume that the phase is 180 ◦
for nuclei with γ > 0 (e.g. 1
H or 13
C) and 0 ◦
for nuclei with γ < 0 (e.g.
15
N). Then, the nutation frequency is ⃗ω1 = +γ ⃗B1 = +|γ| ⃗B1 (opposite convention to the precession frequency!) for
nuclei with γ > 0 and ⃗ω1 = −γ ⃗B1 = +|γ| ⃗B1 (the same convention as the precession frequency) for nuclei with γ < 0.
A practical consequence is that an ideal excitation pulse generates a pure −y component of magnetization for nuclei
with any sign of γ.
3.3 Evolution of coherences
The coherence, introduced by the ⃗B1 field, evolves under the influence of magnetic fields acting on the observed
nucleus, and the macroscopic magnetization of the sample varies. In order to track the changes, it is useful to
decompose periodic changes of the overall magnetization into individual contributions with different frequencies.
• The first contributions are rotations of magnetizations of different nuclides (1
H, 13
C, 15
N, 31
P). Each nuclide
23
24
is characterized by a unique value of γ and the magnetization vector rotates about ⃗B0 with the frequency
⃗ω0 = −γ ⃗B0 (Eq. 1.3).
• The second contributions are differences in magnetizations of the same nuclides in different chemical groups.
Such nuclides are surrounded by different distributions of electrons resulting in different chemical shifts. As
a consequence, precession frequencies ω0 and frequency offsets Ω of magnetic moments of the same nuclide in
different groups slightly vary.
• The third contributions are effects of the J-coupling with other nuclei. This contributions are not easy to grasp.
In order to get some insight, we analyze evolution of magnetization more mathematically below.
We start by recalling the fact that the equilibrium macroscopic magnetization of e.g. a particular proton in a protein
is proportional to the equilibrium polarization of its magnetic moments in the ensemble of protein molecules. The
application of a 90◦
pulse at a frequency very close to the 1
H precession frequency rotates the equilibrium polarization
of 1
H to the horizontal plane. According to a convention described in Section 5.4, the axis of the rotation is the same
as the direction of ⃗B1 for protons and other nuclei with positive γ. Therefore, the 1
H magnetic moments are polarized
in the direction −y after the pulse. Note that polarization only changed its direction. The size of the polarization
stays the same until the relaxation changes it (we neglect the relaxation effects in our explanation for the sake of
simplicity).
The polarization of 1
H magnetic moments then starts to rotate in the horizontal plane with its frequency offset ΩH.
A mathematical description of the rotation of the magnetization vector is the same as the mathematical description
of a vector (x, y, 0) describing motion of a point1
on a circle around the axis z with the angular frequency ω. Velocity
of such a point in the direction x and y is given by
vx = −ω y vy = ω x. (3.1)
In the absence of the J-coupling, the changes of magnetizations are described by the same equations. However, if
1
H is chemically bonded to 13
C, the evolution of the proton magnetization depends on the longitudinal polarization
of 13
C. A polarization along +z slows down the rotation of the proton polarization (makes it less negative, which in
the case of proton increases the frequency offset according to the standard convention). For a single proton magnetic
moment ⃗µH coupled to a vertically oriented 13
C magnetic moment (via a single chemical bond), we could write the
”velocities” (mathematically time derivatives dµH,x/dt and dµH,y/dt) as
dµH,x
dt
= − ΩH +
πJ
µC
µC,z µH,y
dµH,y
dt
= ΩH +
πJ
µC
µC,z µH,x. (3.2)
But in reality, we observe magnetizations of large ensembles of nuclei. The correct equations are therefore
dµH,x
dt
= −ΩH µH,y −
πJ
µC
µH,yµC,z
dµH,y
dt
= +ΩH µH,x +
πJ
µC
µH,xµC,z. (3.3)
Note that ensemble averaging µH,x · µC,z cannot be written as µH,x · µC,z because orientations of the vectors ⃗µH
and ⃗µC are not independent (the nuclei are coupled). We see that the distribution of proton magnetic moments in
the xy plane is described by four quantities, µH,x, µH,y, µH,x µC,z, and µH,y µC,z. The familiar quantities µH,x and
µH,y are called in-phase coherences. They correspond to the x and y components of the macroscopic magnetization
of the proton (MH,x, MH,y). The new quantities µH,x µC,z, and µH,y µC,z are called anti-phase coherences. They
do not contribute to the measurable macroscopic magnetization (are ”invisible”). Attempts to symbolize coherences
graphically are presented in Table 3.1. To avoid drawing of the several polarization vectors, we introduce simple arrows
(the last two columns in Table 3.1) specifying polarizations of the magnetic moments of protons attached to 13
C with
the magnetic moment polarized in the +z direction (dashed arrow) and in the −z direction (solid arrow).
Although the J-coupling is a quantum phenomenon, Eq. 3.3 (together with the complementary equations describing
evolution of µH,x µC,z and µH,y µC,z) can be solved classically.2
The solution shows the following:
1If we draw the vector magnetic moment as an arrow, the rotating point is the arrowhead.
2The solution is presented in the study material for Course C5320 Theoretical concepts of NMR.
3.4. NMR EXPERIMENT 25
• Chemical shift causes the transverse polarization of nuclei in different chemical groups to rotate about ⃗B0 with
different frequencies (Figure 3.1). The size of the transverse magnetization is preserved (except for relaxation
effects, neglected here), only the direction changes. Any direction in the xy plane can be expressed as a linear
combination of x and y components of the magnetization. Both components oscillate as the magnetization
rotates, and these oscillations are detected as the NMR signal. Accordingly, the in-phase coherence can be
decomposed into x and y components. The chemical shift does not convert in-phase coherences to anti-phase
coherences or vice versa.
• J-coupling causes oscillations between ”visible” in-phase and ”invisible” anti-phase coherences. It makes the
transverse polarization and magnetization of nuclei in the same chemical group to disappear and reappear
periodically. The oscillations are easy to followed if the proton frequency offset is set to zero (Figure 3.2). As a
consequence of the J-coupling, the total 1
H magnetization is zero at the time equal to 1/(2J). The polarization
of 1
H is lost at this moment, but the distribution of magnetic moments is still coherent! If we follow the evolution
further, we find out that the original coherence is restored. The frequency of the oscillation is given by the value
of the interaction constant J.
• Simultaneous effects of the chemical shift and the J-coupling is presented in Figure 3.3.
In the following sections, we use the solid and dashed arrows introduced in Table 3.1 to analyze evolution of
coherences in various modules of pulse sequences.
3.4 NMR experiment
After discussing the physical phenomena important for understanding nuclear magnetic resonance, we proceed to more
technical issues. The use of NMR spectroscopy in chemistry and structural biology relies on our ability to measure
precession frequencies of nuclear magnetic moments in the studied molecules. We already know that this requires
a strong static homogeneous magnetic field and a much weaker magnetic field oscillating with frequencies close to
the measured precession frequencies. By accident, the required frequencies are in the same range as those used in
frequency modulated radio broadcasting. We see that the instrument able to measure precession frequencies, the NMR
spectrometer, must contain three components: magnet (as a source of the static magnetic field ⃗B0), radio transmitter
(as a generator of the oscillating field ⃗B1), and radio receiver (as a detector).
Scenario of the basic NMR experiment is rather simple. We put our sample inside a strong homogeneous magnet
and let it reach thermal equilibrium. Then we switch on our radio transmitter for a precisely defined time period.
This event is called the radio-frequency pulse. Duration of the pulse should be such that the pulse rotates the total
magnetization just by 90 ◦
in order to achieve the highest sensitivity.3
Finally, we switch off the transmitter and turn
on the radio receiver. Our sample is now a source of magnetic field oscillations. These oscillations have frequencies
very close to the frequency of the applied radio pulse (only nuclei resonating with this frequency were excited from
their equilibrium states), but not exactly identical. Slight variations in frequency have their origin mostly in the
electron shielding (the chemical shift). Nuclei in different chemical moieties are shielded to different extent and feel
slightly different magnetic field (see Section 2.5). Now we see why the external field ⃗B0 must be very homogeneous.
Variations of ⃗B0 would cause the precession frequencies to vary as well. Even relatively small variability in the inherent
frequencies of nuclei would completely mask the tiny shielding effects of electrons.
The megahertz radio frequency is modulated by the variable shielding, causing deviations ranging from zero up to
several kilohertz. In a similar manner, megahertz radio waves are modulated with the audio frequencies in the FM
radio broadcasting. The megahertz frequency serves as a carrier for the kilohertz frequencies generated by the electron
environment of nuclei in the case of NMR or by vibrations of musical instruments or human voice in the case of the
radio. The NMR receiver does the same job as an ordinary radio receiver. It subtracts the carrier frequency from the
signal (technical details are discussed in Section 5.7). The remaining lower frequencies called frequency offsets Ω, are
converted to a digital signal and recorded. The saved file has the same format as a typical audio record. If plotted,
it shows a sum of sine (or cosine) curves. The amplitude of oscillations gradually decreases, as the nuclear magnetic
moments lose their coherence, and as they return back to their equilibrium distribution. NMR spectroscopists use
3We learn soon that the first pulse is followed by a whole sequence of carefully timed pulses in all biomolecular experiments.
26
Table 3.1: Examples of two graphical representations of in-phase and anti-phase coherences: as hypothetical extreme polarizations and
as arrows (used in Figure 3.6). Cyan arrows represent magnetic moments resonating with the applied radio wave (magnetic moments of 1H
in our example). Green arrows represent magnetic moments that do not resonate with the radio frequency (magnetic moments of 13C or
15N in our example). The solid and dashed orange arrows presented in the last two columns correspond to partial distributions of proton
magnetic moments, shown in cyan in the second column. The direction of the arrow is given by the direction of the cyan arrows, the type
(dashed or solid) of the arrow is given by the direction of the green arrows in the second column (up or down, respectively).
Coherence depicted as polarizations depicted as arrows decomposed arrows
In-phase
−y
x
z
−y
x
z
y
x
Anti-phase
x
−y
z
−y
x
z
x
y
3.4. NMR EXPERIMENT 27
−1
−1
−1
−1
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
2
2
2
2
3
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24
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24
x:
y:
x:
y:
x
y
In−phase:
Anti−phase:
Figure 3.1: Coherence evolution due to the chemical shift. Several snapshots of a coherence evolving in the xy plane are shown in the
circles in the middle. The evolution of the x and y components of the in-phase (red) and anti-phase (blue) coherences are plotted above
and below the snapshots, respectively. The in-phase coherence (blue bar) coincides with the transverse polarization of magnetic moments
(cyan arrow), the size of which is preserved in the presence of the chemical shifts. The anti-phase coherence does not evolve.
28
−1
−1
−1
−1
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
2
2
2
2
3
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23
23
23
23
24
24
24
24
x:
y:
x:
y:
In−phase:
Anti−phase:
x
y
Figure 3.2: Coherence evolution due to the J-coupling. Several snapshots of a coherence evolving in the xy plane are shown in the circles
in the middle. The evolution of the x and y components of the in-phase (red) and anti-phase (blue) coherences are plotted above and
below the snapshots, respectively. The in-phase coherence (blue bar) evolves into the anti-phase coherence (red bars). The orientation of
the anti-phase coherence is given by the direction of the dashed bar. The direction of the transverse polarization in the absence of the
J-coupling is shown as the cyan arrow. The solid and dashed arrows, used in this text to describe evolution of coherences, are shown in
orange. The blue and red bars, representing the in-phase and anti-phase coherences, are projections of the orange arrows to the directions
parallel and perpendicular to the cyan arrow.
3.4. NMR EXPERIMENT 29
−1
−1
−1
−1
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
2
2
2
2
3
3
3
3
4
4
4
4
5
5
5
5
6
6
6
6
7
7
7
7
8
8
8
8
9
9
9
9
10
10
10
10
11
11
11
11
12
12
12
12
13
13
13
13
14
14
14
14
15
15
15
15
16
16
16
16
17
17
17
17
18
18
18
18
19
19
19
19
20
20
20
20
21
21
21
21
22
22
22
22
23
23
23
23
24
24
24
24
x:
y:
In−phase:
Anti−phase:
x:
y:
x
y
Figure 3.3: Coherence evolution due to the chemical shift and J-coupling. Several snapshots of a coherence evolving in the xy plane are
shown in the circles in the middle. The evolution of the x and y components of the in-phase (red) and anti-phase (blue) coherences are
plotted above and below the snapshots, respectively. The in-phase coherence (blue bar) evolves into the anti-phase coherence (red bars).
The orientation of the anti-phase coherence is given by the direction of the dashed bar. The direction of the transverse polarization in the
absence of the J-coupling is shown as the cyan arrow. The solid and dashed arrows, used in this text to describe evolution of coherences,
are shown in orange. The blue and red bars, representing the in-phase and anti-phase coherences, are projections of the orange arrows to
the directions parallel and perpendicular to the cyan arrow.
30
Signalintensity
t
Spectrumintensity
f
ν1
ν2
ν3
Figure 3.4: Signal (left) and frequency spectrum (right) with three precession frequencies.
the term free induction decay, or simply FID, for the damped oscillating signal. When the system of nuclei arrives
sufficiently close to the equilibrium, a new experiment can start. Waiting for the return to equilibrium is typically
slightly longer than one second in the case of biological macromolecules.
The output of the NMR experiments is an audio record. We can therefore easily listen to the NMR data. Well,
listening is not a typical way of analyzing chemical data. We would like to present our records in some graphical
representation. Simple plotting does not make us happy. The damped sine curves interfere and several sinusoids already
produce a mess. Fortunately, the signal, recorded as a function of time, can be easily converted to a spectrum, which
is a function of frequency. The mathematical procedure converting FID to to a spectrum is Fourier transformation.
The principle of Fourier transformation is described in Section 5.6. Her we only briefly discuss some features of the
obtained spectra.
The really attractive feature of Fourier transformation is its additivity. Two or more cosine curves oscillating at
different frequencies just show two or more peaks in the spectrum. Figure 3.4 documents that the additivity allows
us to convert a signal that is very difficult to interpret (three interfering cosine curves in the left plot) to a spectrum
clearly showing presence of three frequencies and allowing us to read their values with a good accuracy (maxima of
three peaks in the right plot). The width of each peak is given by the damping rate – the faster a cosine curve decays,
the broader the resonance peak is.
Interactions described in Sections 2.6–2.8 affect the shape of the spectrum in a clear manner, as listed below and
depicted in Table 3.2.
• Dipole-dipole interactions make the peak broader (contribute to relaxation) but do not shift the average frequency
(peak maximum).
• Chemical shift shifts the average frequency (peak maximum) and its dependence on molecular orientation makes
the peak broader (contributes to relaxation). The size of the shift is given by the electron distribution in the
molecule and is linearly proportional to B0 (because the effect is shielding B0). Therefore it is presented in
relative units (ppm). In principle, the chemical shift should be equal to (ω0 + γB0)/(−γB0). In practice,
it is reported relative to a reference signal from a standard compound (ω0 − ω0,ref )/ω0,ref . The standards
used in biochemistry are sodium 3-(trimethylsilyl)propane-1-sulfonate (13
C1
H3)3SiCH2CH2CH2SO−
3 Na+
, liquid
ammonia 15
NH3, and 85% phosphoric acid H3
31
PO4. Chemical shift is the value reported in the literature as a
property of the given compound. It is given by the molecular structure, is influenced by chemical and physical
conditions (temperature, ionic strength, pH) but does not depend on the experimental setup (on B0 or frequency
of radio waves used in the experiment). As −γB0 is negative for nuclei with positive γ, peaks of the most shielded
nuclei (with low values of the chemical shift and positive Ω) appear in the right-hand side of the spectrum, i.e.
the Ω values increase from left to right, and the chemical shift values increase from right to left (Figure 3.5).
As the spectrometers do not distinguish sign of the frequency, the chemical shift values also increase from right
3.5. ECHOES 31
to left in spectra of nuclei with negative γ, and Ω values increase from right to left, which is, strictly speaking,
inconsistent with the general convention used in physics.
• J-coupling splits the peak to a doublet4
but does not contribute to broadening (does not change as the molecule
tumbles5
) The appearance of additional frequency in the spectrum (observing two peaks instead of one) is a
direct consequence of the oscillation6
between ”visible” in-phase coherence and ”invisible” anti-phase coherence,
presented in Figures 3.1 and 3.3. Frequencies of individual peaks of the doublets are the same as rotation
frequencies of the solid and dashed arrows introduced in Section 3.3 (Figure 3.5). If one nucleus is coupled to
more nuclear magnetic moments, more complex multiplets appear in the spectra.7
The frequency differences
(splittings) are given by the electron distribution in the molecule and do not change with B0 (J-coupling does
not depend on the external field). Therefore they are presented in units of frequency (Hz).
In summary, frequencies observed in a signal converted to a spectrum by Fourier transformation are given by a
combination of three effects: the external magnetic field, chemical shift, and J-coupling. The signs of the frequencies
(senses of rotation of coherences described by the solid and dashed arrows introduced in Section 3.3) depend on signs
of γ and Ω. Figure 3.5 shows the evolution for all combinations of positive and negative γ and Ω and also summarizes
(sometimes confusing) conventions used in NMR spectroscopy.
3.5 Echoes
So far, we have discussed only the very basic NMR experiment: the application of the 90◦
excitation pulse followed
by detecting the evolution of the magnetization influenced by the effects discussed in Section 3.3. Now we proceed
to more complex experiments, needed to study biomacromolecules. In such experiments, more or less complex series
of pulses is applied before we start the detection. In this section, we explore symmetric modules of pulse sequences,
called echoes.
The echoes are built of two delays of the same length τ. The delays may be separated by 180◦
pulses acting
on individual nuclei. Schematic drawings of the modules are presented in the left section of Figure 3.6. In a model
compound containing coupled 1
H and 13
C nuclei, three combinations of 180◦
pulses are possible: a pulse resonating
with the 1
H frequency (Figure 3.6B), a pulse resonating with the 13
C frequency (Figure 3.6C), and both such pulses
applied at the same time (Figure 3.6D). Let us analyze what effects these three modules have when applied just after
the excitation pulse acting on 1
H. A graphical analysis is shown in the right section of Figure 3.6. The first row
(Figure 3.6A) shows a combined effect of chemical shift and J coupling in the absence of any 180◦
pulse. It was
discussed already in Section 3.3. Figure 3.6A extends the previous discussion to a molecule containing three 1
H–13
C
groups. Analysis during modules including one or two 180◦
pulses is presented in the following sections.
3.6 Refocusing echo
.
Let us start with the echo involving a 180◦
pulse is applied to the excited nucleus (Figure 3.6B). We know that the
magnetic moment distribution polarized in the −y direction rotates with the frequency offset Ω (the difference between
the frequency of the radio waves and the precession frequency modulated by the chemical shift), and is further split
into two components. One (represented by the dashed arrow) is accelerated and the other one (represented by the
solid arrow) is slowed down by the effect of 13
C. The third column of circles in Figure 3.6 shows the directions of the
arrows right before applying the 180◦
pulse. The slower arrow rotated by angle of (Ω − πJ)τ, while the faster arrow
4This is true for Dirac-like nuclei with sufficiently different precession frequencies.
5J-coupling does not contribute to peak broadening via molecular tumbling, but it may cause relaxation (broaden the peaks) if the
structure of the molecule and/or distribution of magnetic moments changes during the experiment.
6The mathematical relation between sine/cosine oscillations and the number of peaks in the spectrum obtained by Fourier transformation
is discussed in more detail in Section 5.7.
7E.g., spectrum of three mutually coupled nuclei consists of three quartets (doublets of doublets), i.e., of twelve peaks. However, the
central peaks may overlap if the interactions with both neighbors are of similar size, and the quartet may look like a triplet with doubled
intensity of the middle peak.
32
Table 3.2: Effects of interactions of magnetic moments with magnetic fields in the molecule on the shape of a spectrum. The external
magnetic field is assumed to be ideally homogeneous, the molecular tumbling isotropic, the data acquisition is supposed to be continuous
and infinitely long.
Interaction mechanism effect spectrum
coherent relaxation
none (bare nucleus)
dipolar coupling direct dipole-dipole interaction through space no yes
chemical shift shielding by electrons (orbital magnetic moment) yes yes
J-coupling mediated by electrons in covalent bonds yes no
all combination of all yes yes
3.6. REFOCUSING ECHO 33
x
y
x
y
x
Laboratory frame
t > 0
x
y
x
y
x
y
x
y
x
y
x
y
y
x
y
x
y
γ < 0, Ω < 0 γ < 0, Ω > 0 γ > 0, Ω > 0
ωradiot
Ωt
−ωradiot + π
Rotating frame
ω0t
Ωt
0−ωradio+ωradio
ω−ωradio
0
Ω− ++
0
−
0
+ −+
0
−
+ δref
− + δref
− + δref
− + δref
− δ
γ > 0, Ω < 0
0
t = 0
t > 0
−+ωradio
−0+0+
x
y
ω0t ±πJt
J > 0J < 0
±πJt
Figure 3.5: Conventions used in NMR spectroscopy when describing the effect of radio-wave pulses and the evolution of coherences,
and when presenting the spectra. Evolution of coherences symbolized by the solid and dashed orange arrows (introduced in Table 3.1)
for nuclei with different signs of γ and Ω is shown in the circles. The first row of circles represents the polarization immediately after
applying a very short 90◦ radio-wave pulse (at t = 0). The second and third row show directions of coherences and of ⃗B1 at time t the
laboratory and rotating coordinate frame, respectively. The direction of the transverse polarization in the absence of the J-coupling is
shown as the cyan arrow. The z axis is defined by the direction of ⃗B0. The oscillating radio-wave magnetic field is decomposed into two
counter-rotating components. The purple arrows indicate the direction of the resonant component ⃗B1 (+ ⃗Bradio for γ < 0 and − ⃗Bradio for
γ > 0). The absolute value of ⃗B0 is supposed to be much greater that the amplitude of ⃗B1. Note the convention to add a phase of 180 ◦ to
the direction of ⃗B1 (i.e., to revert the direction) for γ > 0. The resulting spectra (after Fourier transformation and applying the necessary
phase correction, which is 90 ◦ in the presented cases), are plotted below the circles. Arrows above the spectra assign individual peaks to
the solid and dashed arrows. Note the convention to plot the frequency axis from the right to the left for nuclei with γ < 0.
34
N
or13C 15N
or13
C 15
N
or13
C 15
N
1
H
1H
1H
1H
or13
C 15
Figure 3.6: Left, schematic drawings of the pulse sequences of echoes. 90◦ and 180◦ radio-wave pulses are depicted as narrow and wide
bars, respectively. Letters below the pulse sequences refer to the columns in the right section of the figure. Right, analysis of spin echoes
for 1H and 13C (15N) in an isolated –CH– (or –NH–) group. In individual rows, evolution of magnetization vectors in the xy plane is shown
for three protons (distinguished by colors) with slightly different precession frequency due to the different chemical shifts δ. The protons
are bonded to 13C. The magnetic moment distributions are represented by solid and dashed arrows using the convention introduced in
Table 3.1. The first column shows magnetization vectors at the beginning of the echo (after the initial 90◦ pulse at the proton frequency,
letter b below the pulse sequences), the second column shows magnetization vectors in the middle of the first delay τ, the third and fourth
columns, respectively, show magnetization immediately before and after the 180◦ pulse(s) in the middle of the echo (letters c and d below
the pulse sequences, respectively), the fifths column shows magnetization vectors in the middle of the second delay τ, the sixth column
shows magnetization vectors at the end of the echo (letter e below the pulse sequences). Row A corresponds to an experiment when no
180◦ pulse is applied, row B corresponds to the echo with the 180◦ pulse applied at the proton frequency, row C corresponds to the echo
with the 180◦ pulse applied at the 13C frequency, and row D corresponds to the echo with the 180◦ pulses applied at both frequencies.
The x-axis points down, the y-axis points to the right.
3.7. DECOUPLING ECHO 35
rotated by angle of (Ω + πJ)τ. The 180◦
pulse flips the arrows by rotation about the x axis. The flipped arrows form
a mirror image to the original ones with respect to the xz plane. The dashed arrow is less advanced after flipping.
But it still rotates in the same direction with a higher frequency. In a similar way, the solid arrow is more advanced
but rotates with a lower frequency. The angles between arrows in the third column and the original direction −y are
the same as the angles between arrows in the fourth column and the direction +y. Therefore, solid and dashed arrows
meet in the original y direction at the end of the echo (the sixth column). The dashed arrow had a little longer way
to go, but being faster it caught on the slower solid arrow exactly at the end of the module. Note that both arrows
pass the y axis in that moment regardless of the actual Ω value. This is why the module is called an ”echo”. Can you
see the magic? Effects of both chemical shift and scalar coupling are canceled at the end of the echo. The 180◦
pulse
now acts as a refocusing pulse: effects of the chemical shift and scalar coupling are refocused at the end of the echo.
3.7 Decoupling echo
What if the 180◦
pulse is applied to the other nucleus, which was not excited (13
C in our example)? Again, we know
the evolution up to the pulse. What happens if we invert the distribution of 13
C magnetic moments by the 180◦
pulse right in the middle (Figure 3.6C)? The faster fraction of proton magnetic moments that was accelerated by
13
C magnetic moments pointing down is now slowed down because the protons are coupled to 13
C with magnetic
moments pointing up, and vice versa. In our graphical representation, solid arrows become dashed and the dashed
arrows become solid. The originally slower fraction of proton magnetic moments would rotate faster after the 180◦
pulse and would catch the originally accelerated fraction just at the end of the module (Figure 3.6C). Now we see
another magic. The nuclear magnetic moments interact and there is no way how we can turn off their interaction.
But we can play a trick with the magnetic moments and prepare the same distribution as we would have obtained if
no coupling were present. This is why the described inversion 180◦
pulse has been given the name decoupling pulse.
3.8 Simultaneous echo
We know how to refocus both chemical shift and scalar coupling and we know how to refocus evolution of the coupling
only. How can we refocus just the chemical shift evolution? The answer is the echo with two 180◦
pulses, each hitting
one of the nuclei (Figure 3.6D). The proton pulse flips arrows representing proton coherences and the 13
C pulse inverts
longitudinal polarizations of 13
C nuclei (solid arrows change to dashed ones and vice versa). As a result, the average
direction of dashed and solid arrows is refocused at the end of the echo but the difference due to the coupling is
preserved (the handicapped arrows were made slower by the inversion of longitudinal polarization of 13
C). The magic
is here again. The same scalar coupling evolution is observed as it would be at the and of a delay long 2τ with no
pulses applied and no chemical shift evolution present.
36
Lecture 4
Correlated spectroscopy
4.1 Polarization transfer
The three echoes described in Sections 3.6–3.8 give us what looks like a supernatural power. We turn on and off physical
interaction as we want, at least seemingly. Some applications are obvious, like measurement of the chemical shift
without complications introduced by the coupling. I would like to show you another application of great importance.
It is a real alchemy, we are going to change protons to carbon nuclei, of course at the level of magnetic properties.
The pulse sequence module presented in Figure 4.1A is an extension of the third echo (Figure 3.6D). The trick
requires that the dashed and solid arrows evolve to exactly opposite directions. In other words, that the distribution
of the magnetic moments is the anti-phase coherence (see the second row of Table 3.1). It can be achieved by choosing
the total length of echo equal to 1/(2J) which means τ = 1/(4J). Then, the solid and dashed arrows in the sixth circle
of Figure 3.6D evolve to the +x and −x directions, respectively. In the moment when the chemical shift is refocused
but the J-coupling is showing its full glory, two 90◦
pulses are applied, one at the 1
H frequency and the other one at
the 13
C frequency. As the solid and dashed arrows, describing distribution of proton magnetic moments, are aligned
along ±x, the magnetic field of the radio waves cannot be applied in the x direction of the rotating coordinate frame.
The desired radio-wave magnetic field oriented in the y direction can be obtained by shifting the phase of the radio
wave by 90 ◦
from that of the excitation pulse. In the NMR literature, pulse of a radio wave with 90 ◦
phase shift is
called simply a y-pulse. Phase of the 13
C pulse is arbitrary. The proton 90◦
pulse rotates the distribution of proton
magnetic moments so that the cyan arrows in the second row of Table 3.1 are aligned along the ±z direction. The
13
C 90 ◦
pulse rotates the distribution of 13
C magnetic moments. This of course creates transverse polarization of the
13
C magnetic moments. But we are now interested in a different effect.
The fractions of the 13
C magnetic moments represented by the green arrows in the second row of Table 3.1 rotated
by the 13
C 90 ◦
pulse become distributed around the +y and −y direction. Such distribution is an ”invisible” 13
C
anti-phase coherence. It immediately starts to rotate about ⃗B0 with the 13
C precession frequency (modulated by the
13
C chemical shift). As 13
C nuclei are coupled to 1
H, the two green distributions also evolve in a similar manner as
described in Section 3.3 for the cyan distribution of proton magnetic moments. The green and cyan distributions are
correlated. Therefore, the green cluster coupled to the cyan cluster that happened to be pointing down evolves faster
than the other green cluster. As a consequence, the green clusters with originally opposite populations (anti-phase
coherence) starts to redistribute and after t = 1/(2J) they merge into a distribution polarized in a single direction.
The 13
C in-phase coherence is created. In such a way, the distribution of 13
C magnetic moments oscillates between
”invisible” anti-phase and ”visible” in-phase coherences, in addition to the overall rotation with the 13
C frequency
offset. This process resembles the evolution started from the proton in-phase coherence described in Section 3.3, with
the following differences.
• The overall rotation is modulated by the chemical shift of 13
C (not of 1
H).
• The 13
C in-phase coherence oscillates as a sine curve because the evolution started from the anti-phase coherence
(i.e., from a zero contribution of the in-phase coherence, as the sine curve starts from the zero value). In contrast,
the 1
H in-phase coherence oscillated as a cosine curve after the excitation of 1
H because the evolution started
37
38
A
H1
N15
C13
or
1
4J
1
4J
N15
C13
or
H1
y
B
1
H
N15
C13
or
Figure 4.1: INEPT pulse sequence applied to 1H and 13C or 15N (A) and direct excitation of 13C or 15N (B). The narrow and wide
rectangles represent 90◦ and 180◦ radio wave pulses, respectively. The label y above the pulse indicates irradiation by a radio wave with
the phase shifted by 90 ◦ relative to the first pulse in the sequence. Distributions of magnetic moments are shown schematically above the
pulse sequences for time instants labeled by the red arrows. The distributions are shown in a coordinate frame with the z axis pointing up
and the x axis pointing to the right.
4.2. HSQC 39
A
2
1
4J
1
4J
N15
C13
or
t1
H1
1
4J
1
4J
ty
f hg i
B
1
4J
1
4J
t2
1
4J
1
4J
N15
C13
or
t1
H1
y
Figure 4.2: HSQC experiment. A, basic HSQC pulse sequence. B, general idea of the decoupling in the direct dimension. The decoupling
pulses applied to proton and to 13C (or 15N) are shown in cyan and green, respectively. Other symbols are used as explained in Figure 4.1.
from the in-phase coherence (i.e., from a maximum contribution of the in-phase coherence, as the cosine curve
starts from the maximum).
• The transverse polarization of 13
C magnetic moments correlated with proton magnetic moments during the echo
depends on the equilibrium distribution of proton magnetic moments because proton was excited. The original
equilibrium distribution is approximately proportional to the value of γ which is four times higher for 1
H than
for 13
C. Therefore, the size of the measured signal is four times higher that the signal measured in an experiment
with a direct excitation of 13
C (a 13
C 90 ◦
pulse immediately followed by detection, Figure 4.1B). It gave the
module a name INEPT (insensitive nuclei enhanced by polarization transfer). The sensitivity improvement is
even bigger (tenfold) in the case of 15
N.
The original purpose of the described trick was to improve sensitivity of carbon measurements: as discussed above,
the dependence on the magnetogyric ratio results in a four-fold increase of the signal.1
An application much more
important in biomolecular NMR is the topic of the next section.
4.2 HSQC
Let us build the following pulse sequence from the modules that we learned: Excitation of 1
H, INEPT to 13
C, decoupling
echo, INEPT to 1
H, and acquisition (Figure 4.2). During the first INEPT, the proton polarization is converted to
the anti-phase 13
C coherence, the second INEPT converts the anti-phase 13
C coherence back to the in-phase proton
coherence.
1In reality, the 13C 90 ◦ pulse also generates coherence corresponding to the direct excitation of 13C. However, this coherence can be
filtered out of the signal by phase cycling, described in Section 5.10.
40
Figure 4.3: Schematic representation of multiple quantum coherences. Correlations of magnetic moment distributions are depicted by
color coding sets of magnetic moments of coupled 1H (cyan) and 13C or 1N N (green) nuclei in the same molecules. In the absence of radio
waves, the magnetic moment distributions oscillates between individual multiple quantum coherences with the frequency offsets ΩH + ΩC
and ΩH − ΩC (or ΩH + ΩN and ΩH − ΩN)
What looks like a useless exercise at the first glance, is actually a basic form of a very important pulse sequence.
One advantage of the pulse sequence is that protons are excited at the beginning and protons are directly detected
in the end. We discussed in Section 4.1 that the signal depends on the original polarization of the excited nucleus,
which is proportional to the magnetogyric ratio γ, and that, as a consequence, exciting protons instead of 13
C yields
four-fold improvement in the sensitivity. The signal amplitude is also proportional to the absolute value of the detected
magnetic moment and to the frequency of the rotation.2
As both parameters are approximately proportional to γ,
the signal is proportional to γ3
. The noise is proportional
√
γ,3
therefore the signal-to-noise ratio, which is the true
measure of sensitivity, is proportional to γ3/2
. The resulting theoretical improvement in sensitivity, compared to the
direct detection, is 32-fold for 13
C and 315-fold for 15
N.
The sensitivity enhancement is already impressive, but the HSQC pulse sequence offers more. The described pulse
sequence allows us to measure both 1
H frequency and 13
C (or 15
N) frequency in a single experiment and in a correlated
manner. Let us now try to understand how is it possible.
We emphasized in Section 4.1 that it is important to apply the radio waves with a phase corresponding to ⃗B1
perpendicular to the direction of the solid and dashed arrows describing anti-phase polarization. This is why we had
to apply the second proton 90◦
pulse with a 90◦
phase shift (as a ”y” pulse). Therefore, the effect of the second
13
C 90 ◦
pulse also depends on the length of the delay between the 13
C 90 ◦
pulses. If we apply the second pulse
immediately after the first one, all 13
C anti-phase coherence is converted back to the proton in-phase coherence. If we
wait, the arrows symbolizing the distribution of magnetic moments rotates start to rotate about ⃗B0. The rate of the
rotation is the 13
C frequency offset, given by the 13
C chemical shift. It is useful to decompose the solid and dashed
arrows into two directions in the xy plane, one parallel to the 13
C ⃗B1 field and the other perpendicular. Our NMR
experiment relies on two facts: (i) only the perpendicular component is affected by the second 13
C pulse, applied in
the x direction, and (ii) how much of the arrow is in the perpendicular direction depends on the length of the delay
between 13
C 90 ◦
pulses (t1 in Figure 4.2). Let us know explain both facts in more detail.
• Only perpendicular component, affected by the second 13
C pulse, yields the signal. The second 13
C pulse rotates
the perpendicular components of the arrows to the ±z direction and the parallel components stay in the xy
plane. The simultaneous proton pulse creates proton anti-phase coherence from the perpendicular components,
which evolves to the ”visible” in-phase coherence in a reversed process to what was described in Section 4.1. The
magnetic moment distribution corresponding to the parallel components of the arrows are converted to multiple
quantum coherences, correlated distributions of 1
H and 13
C magnetic moments (Figure 4.3) that rotate in the xy
plane and do not contribute to the signal (are not converted to the observable proton polarization in the given
experiment). The fact that only single-quantum (not multiple quantum) coherences contribute to the signal is
encoded in the name of the experiment: Heteronuclear Single-Quantum Coherence spectroscopy (HSQC).
• The signal depends on the delay t1 in Figure 4.2. The INEPT resulted in arrows perpendicular to ⃗B1 of the first
13
C pulse (direction ±x). The perpendicular component therefore evolves following a cosine function with the
2The signal is an electromotive force (voltage) induced by an oscillating flux of the magnetic field (generated by the rotating transverse
magnetization) through the detector loop. The electromotive force is proportional to the change of the flux in time. If Φ oscillates as
Φ(t) = Φ0 sin(ωt), its change is the time derivative dΦ(t)/dt = ωΦ0 cos(ωt), where both Φ0 and ω are proportional to γ.
3Square of the noise is proportional to the window of detected frequencies, which depends linearly on γ.
4.2. HSQC 41
t2 f2 f2
t1
f1
Figure 4.4: Principle of 2D spectroscopy. The recorded signal is shown in red, the result of the first Fourier transformation in magenta,
and the final spectrum, obtained by two Fourier transformations, in blue.
argument ΩCt1, where ΩC is the carbon frequency offset and t1 is the length of the delay between the 13
C pulses.
As a result, the amount of proton anti-phase coherence converted to proton polarization is modulated by cosine
of the 13
C frequency offset. This amount represents the amplitude of the signal, the length of the transverse
magnetization vector that rotates during the measurement and generates electromotive force in the detector.
The critical trick is to repeat the measurement many times while incrementing the delay t1 in the middle. We
then obtain a whole array of 1D spectra, of cosine curves with frequencies modulated by the proton chemical shift. The
spectra stored in a digital form, not as a continuous function, but as a list of values of voltage measured at discrete
time points labeled t2. The spectra have the same frequency (proton frequency offset ΩH), but different amplitude.
As t1 grows, the argument in cos(ΩCt1) also increases and therefore the amplitude is modulated by cos(ΩCt1). The
array of the spectra can be stored as two two-dimensional digital data, a table of signal intensities measured for all
combination of t1 and t2 values.
To sum up, we obtain a two-dimensional signal, which is frequency modulated by the proton chemical shift in t2 and
amplitude modulated by the 13
C chemical shift in t1. An example is shown in Figure 4.4. The acquired two-dimensional
array is then converted to a two-dimensional spectrum by applying the Fourier transformation twice, once with t2 as
the time variable, and once with t1 as the time variable.
The obtained information is quite different from what we learn from separate one-dimensional 1
H and 13
C spectra.
The 13
C frequency can be measured only because 1
H and 13
C nuclei are coupled. Therefore, the described experiment
works selectively for carbons directly bonded to protons and for protons directly bonded to carbons. But the coupling
introduces also some problems because it splits frequencies of both nuclei due to the oscillations between in-phase and
anti-phase coherences. The splitting of peaks caused by the J-coupling is an undesired complication in most spectra. It
is easy to remove the effect of coupling from the indirect dimension – simply by applying a decoupling echo during the
incremented delay (the cyan pulse in Figure 4.2). This trick cannot be played when we directly record the spectrum.
We need to remove the coupling effect at each moment during the acquisition, not just at the and of some delay. We
need to use a brute force approach. We do not apply just one decoupling echo, we apply many little decoupling echoes,
each as short as possible. The result is that the coupling is always refocused if we ignore tiny evolution during each
little echo. Note that we do not apply a separate pulse any more. The sample is now bombarded by a long train of
42
A
0
B
B
Figure 4.5: Magnetic field gradients in the vertical (z) direction. A, magnetic induction lines and the corresponding schematic drawings
of the gradients (used to present the gradients in pulse sequence diagrams) are shown in purple and black, respectively. B, local transverse
polarizations (magnetization) at different positions in the sample tube for increasing gradients (indicated by the black schematic drawings
below the sample tubes). The arrows representing the transverse polarization (magnetization) vectors are color-coded so that blue corresponds
to Mx, red corresponds to −Mx, and white corresponds to ±My. The round shape of the gradient symbols indicates that the
gradients were applied with smoothly changing magnetic field inhomogeneity.
decoupling pulses during the whole acquisition (Figure 4.2B). Power of these pulses must be moderate because energy
is absorbed by the sample for a long time, and the sample could get overheated.
Unfortunately, during the described series of many 180 ◦
pulses various imperfections accumulate. Therefore such
a train of pulses is never applied in practice. Great effort has been dedicated to designing sequences of relatively weak
pulses having the best possible performance. Pulses rotating magnetization by angles different from 180 ◦
or 90 ◦
are
used and their phases are varied in complicated manners. Decoupling can be optimized either for the width of the range
of the affected frequencies or for the highest efficiency. One of the most popular decoupling trains, known as GARP,
has been designed with the help of numerical optimization to meet the former requirement, other decoupling trains
(e.g. MLEV) provide better efficiency (lower residual scalar coupling) in a somewhat narrower range of frequencies.
Obviously, all decoupling methods discussed here require that the decoupled and observed nuclei must differ in the
resonance frequency so that they can be manipulated separately.
The area of pulse train applications is not limited to the decoupling. Polarization transfer based on pulse trains
will be an important topic of our later discussion (in Section 7.4).
4.3 Pulsed field gradients
Resonance frequencies of nuclei depend on properties of the molecule (inherent properties of nuclei and interactions
of nuclei with their microscopic environment) and on the external magnetic field. The external magnetic field is what
we control and the molecular properties is what we study. We try to keep the external magnetic field as homogeneous
as possible so that all nuclei feel the same external field ⃗B0 and their frequencies are modulated by their molecular
environment only. Now we learn a trick of the spin alchemy which is based on violating this paradigm.
It is possible to create a magnetic field that is inhomogeneous in a controlled way. We will discuss an example when
4.3. PULSED FIELD GRADIENTS 43
τττ τ
a b
z
Unwanted
Wanted
G
Figure 4.6: Gradient echoes. Black rectangles and round shapes indicate pluses of radio waves and magnetic field gradients, respectively.
Evolution of the phase of the desired and undesired transverse coherence (describing direction of the corresponding transverse polarization)
is shown as green and red lines, respectively. Values of the phase at different positions in the sample tubes correspond to the distances of
the green and red lines from the central black line.
the field is linearly increasing along the z axis (Figure 4.5). A linear gradient of magnetic field (or simply ”a gradient”
in the NMR jargon) is applied in the z direction. The nuclei close to the bottom of the sample tube feel a weaker
magnetic field and have a lower resonance frequency while the nuclei close to the top feel a stronger field and have a
higher resonance frequency in such case. We can say that frequency carries information about position along the z axis.
In the NMR experiments, the gradients are applied for a limited time, as short pulses of magnetic field perturbation.
Therefore, they are described as pulsed field gradients in the NMR literature. Moreover, the gradients are usually not
applied as a sudden perturbation of a given strength (magnetic field change per unit distance), but the inhomogeneity
is increased and then decreased smoothly (cf. the shape of symbols representing gradients in Figure 4.5).
If the gradient in the z direction is applied when the total magnetization vector rotates in the xy plane, nuclei
at different height of the sample acquire different frequencies of rotation. In the individual slices, the coherence is
preserved. But after a while, vectors of local transverse polarization (magnetization) rotating at different frequencies
in different slices of the sample would point to all possible directions and they would no longer add up to a measurable
total magnetization. The gradient destroys the bulk (macroscopic) transverse magnetization and the NMR signal.
The longitudinal polarizations are not influenced.
At the first glance, it seems that dephasing of coherences and the consequent loss of the signal are completely useless
and should be avoided in NMR experiments. It is not true, gradients are very useful if they are applied correctly. The
first trick is to apply gradients that destroy coherences we are not interested in. Such gradients have cleaning effects
and remove unwanted contributions from the spectra.
Another trick is to recover the magnetization back. If we apply the same gradient for the same time, but in the
opposite direction (−z), later in the pulse sequence, the magnetic vectors are refocused and the signal appears again.
We see how an echo can be created from two opposite gradients. There are also other ways of creating gradient echoes,
presented in Figure 4.6. Instead of using two opposite gradients, two identical gradients can be applied during the
refocusing echo (described in Section 3.6), one in the first half of the echo and the other one in the other half (echo
”a” in Figure 4.6). The gradients do nothing else but adding another source of frequency variation, on the top of the
chemical shift and scalar coupling effects. Magnetic moments of the nuclei affected by the 180◦
pulse in the middle
44
of the echo get always refocused, no matter what was the origin of the frequency variability. On the other hand,
magnetic moments of nuclei not affected by the 180◦
pulse (e.g., 13
C or 15
N nuclei if radio ways are applied at the
proton frequency) feel two identical gradients and get dephased. We see the selective cleaning effect of the gradient
echo, e.g. preserving the 1
H coherences but destroying unwanted 13
C coherences. Gradients incorporated into the
decoupling echo (described in Section 3.7) have exactly the opposite effect (echo ”b” in Figure 4.6). In this spirit,
gradients are frequently added to the echoes in the pulse sequence to clean imperfections of the used pulses. Application
of a cleaning gradient and of gradient echoes in a real NMR experiment is presented in Section ?? (Figure 5.12A).
Lecture 5
Technical issues
5.1 Content of the lecture
Up to now, we tried to understand the basic principles of NMR spectroscopy without going into technical details.
Anybody who intends to actually run an NMR experiment must learn the technical issues at some point. In our
course, we are getting to this point right now. This lecture is dedicated to technical details that are critical for optimal
performance of NMR experiments. Parameters of basic elements of pulse sequences are discussed in the following
sections. The number of parameters that must be controlled and of the technical tricks that are routinely applied
to obtain high quality spectra is large, and the time of our lectures is limited. Therefore, only really fundamental
technical issues are discussed in the class. Additional technical details are also included in the following text so that
an interested reader can gain a deeper insight. Sections discussing such more advanced topics are labeled with the
asterisks.
5.2 Delays
Delay is a very easy trick of spin alchemy, it is simply waiting and doing nothing. The nuclei only feel ⃗B0 modulated by
their environment during the waiting periods. Evolution of transverse magnetization during a delay was discussed in
Section 3.3 and depicted graphically in Figure 3.6A. The only parameter describing the delay is thus its length. In multidimensional
experiments, we distinguish constant delays and incremented delays, introducing amplitude modulation
by chemical shift1
in the indirect dimensions. Length of the incremented delays must be defined very precisely.
The fact that we do not measure time continuously in the indirect dimensions, but only in discrete time increments,
and that the number of the increments is limited, has important consequences:2
• Consequence of recording spectra for discrete time increments. As shown in Figure 5.1A, the same set of discrete
data points (blue dots) spaced by the time increment ∆t fits equally well the cosine curve with the angular
frequency offset Ω1 and the cosine curve with the angular frequency offset Ω1 + 2π/∆t. This documents that
angular frequency can be distinguished only within a window of the width of 2π/∆t, i.e. in the range from
−π/∆t to +π/∆t. If the frequency is measured in Hz, the window width is 1/∆t, ranging from −1/(2∆t) to
+1/(2∆t).
• Consequence of limited number of time increments. As shown in Figure 5.1B, termination of data acquisition
before the signal decays to the noise results in artificial wiggles around peaks in the spectrum. The wiggles are
suppressed if the acquired signal is multiplied by a function that decays to zero prior to the Fourier transformation.
This trick, called apodization, is applied almost always in multidimensional spectra.
1J-coupling also modulates the signal if decoupling is not applied.
2The same actually applies to the directly detected signal. It is converted to a digital form (i.e., to a discrete series of data points) and
the acquisition time is also limited.
45
46
A
Signal
t
Spectrumintensity
ω
Ω1 = Ω2 − 2π/∆t
B
Signal
t
Spectrumintensity
ω
Ω
Figure 5.1: Consequences of recording spectra for discrete time increments (A) and of limited number of time increments (B).
5.3. CONSTANT TIME EVOLUTION∗
47
H1
C13
t1
2
n
2J
n
2J
t1
n
J
t1
2
t1
1
4J
1
4J
t2
1
4J
1
4J
GARP
y
CC CC
CC
CHCH CH CH
Figure 5.2: 1H,13C HSQC experiment with constant-time JCC evolution in the indirect dimension. The symbols are used as explained
in Figure 4.2.
5.3 Constant time evolution∗
The constant and incremented delays can be combined. A simple example, known as constant time evolution is
presented in Figure 5.2. The depicted pulse sequence addresses one limitation of the HSQC experiment, namely the
J-coupling between 13
C nuclei. If a molecule contains several J-coupled 13
C nuclei with similar chemical shifts, the
effect of the 13
C-13
C J-coupling cannot be eliminated by the decoupling echo because we cannot selectively invert
distribution of the magnetic moments of the coupled partner. A solution is to combine the evolution period t1 (a
decoupling echo of incremented duration shown in red in Figure 5.2) with a simultaneous echo affecting all 13
C nuclei
(blue). The simultaneous echo is actually formed by a single 180 ◦
pulse of a radio wave with frequency sufficiently close
to precession frequencies of all 13
C nuclei. If the total length of the echo is an integer multiple of JCC, where JCC is the
13
C-13
C coupling constant (assumed to be identical for all carbon pairs), the original 13
C coherence (anti-phase in our
case) is restored at the end of the echo. It is because the in-phase and anti-phase 13
C coherences evolve as cos(πJt)
during both echoes3
of the complete length of t = n/JCC, and therefore cos(πJt) = cos(nπ) = ±1 (no modulation
by t1). The chemical shift of 13
C evolves during the decoupling echo, resulting in the modulation by cos(ΩCt1, but
it is refocused during the simultaneous echo. Therefore, the modulation by cos(ΩCt1 is preserved. When analyzing
the effect of the J-couplings of 13
C with 1
H, we follow two lines: ”1
H” and ”13
C”. The first (red) echo is decoupling,
the second one (blue) is refocusing (now we do not consider the 13
C-13
C J-coupling), therefore the analyzed 13
C-1
H
J-coupling does not have any effect at the end of the second echo. In summary, we were able to keep the modulation by
the chemical shift of 13
C and remove the effects of both 13
C-1
H and 13
C-13
C J-couplings, without actually decoupling
the second one.
Another, very frequent, application of the constant time evolution is the incorporation of the incremented delay
t1 (introducing modulation of the signal by the frequency of 15
N) into a rather long INEPT transferring polarization
between 15
N and 13
C in triple resonance experiments, discussed in Section 6.4. An example is presented in Figure 6.5.
3No pulse is applied to 13C during t1 (see the red arrow below the pulse sequence in Figure 5.2), therefore both 13C chemical shift and
the J-coupling between 13C evolve freely. During the following simultaneous echo (blue arrow below the pulse sequence in Figure 5.2), the
evolution due to the 13C chemical shift is refocused, but the evolution due to the J-coupling is preserved.
48
5.4 Pulses
The pulse of radio waves is the fundamental element of NMR experiments as we learned in Section 2.4. Each pulse of
an NMR experiment must be precisely defined by setting the following parameters.
• Frequency of the radio wave is the first important parameter of each pulse, the unit is Hertz. It is chosen to match
the precession frequency of magnetic moments of the observed nucleus, therefore it depends on the magnetic
induction B0 (cf. Eq. 1.3). Spectrometers used for biomolecular applications are equipped with magnets of the
induction 9 T–24 T (the highest value available in 2020 was 28 T), it corresponds to 400 MHz–1 GHz for protons
(values for 13
C and 15
N are four-times and ten-times lower, respectively).
• Phase is another fundamental, although somewhat arbitrary parameter of the radio wave. The absolute phase
is not clearly defined, it depends on exact timing, length of the cables etc. As mentioned in Section 3.2 and
shown in Figure 3.5, the phase of the radio wave applied during the first pulse is assumed to be 180 ◦
and 0 ◦
for
nuclei with γ > 0 and γ < 0, respectively, so that the rotation axis (direction of ⃗ω1) is oriented along the x axis
regardless of the sign of γ. The relative phase (difference in phase compared to the first pulse) is well defined
and should be set precisely. A change of the phase of the actual radio wave oscillating in certain direction has
a similar effect as applying a field in a different direction in the rotating coordinate system: the phase shifts of
0 ◦
, 90 ◦
, 180 ◦
, and 270 ◦
correspond to a ⃗ω1 in the x, y, −x, and −y directions, respectively. Therefore, the
pulses with phase shifts of 0 ◦
, 90 ◦
, 180 ◦
, and 270 ◦
are are traditionally labeled as x, y, −x, and −y pulses in
the literature, respectively.
• Phase modulation The pulse can be divided into a certain number of short rectangular pulses (Figure 5.3). Phase
of the short pulses can be varied during irradiation (Figure 5.3B–D). A gradual change of the phase is nothing
else than another oscillation. This trick allows us to modify frequency of ⃗B1 without re-tuning the actual carrier
frequency generated by the synthesizer.
Power of the radio wave P (energy of the radiation per unit time) is proportional of the square of its amplitude
|⃗B1|. The absolute unit of power is Watt, a relative logarithmic unit decibell (dB) is often used in practice. The
relative power is often calculated as attenuation of some reference (maximum) power. The relative and absolute
units are related by the following equations.
20 log10
|⃗B1|2
|⃗B1|1
= 10 log10
|⃗B1|2
2
|⃗B1|2
1
= 10 log10
P2
P1
= attenuation/dB. (5.1)
Typical power for excitation pulses ranges between 10 W and 1000 W, this is the number we need to know to set
up the experiment. However, we are not interested in the power in Watts, but in the effect on the magnetization
vector. Therefore, we do not report power in Watts when we describe the NMR experiment in the literature.
Instead, we present the frequency ω1 (more accurately ω1/(2π)) in kHz. The actual value of the power determines
the length of the pulse, as discussed bellow.
• Length (duration) of the radio-wave pulse must be chosen so that the pulse has a desired effect. If the radio-wave
frequency exactly matches the precession frequency, the total angle of rotation (flip angle) is a product of γ, B1
(proportional to the square root of the power as discussed above), and duration tp. Once we choose the power,
B1 is defined. We must calibrate the time tp (length of the pulse) to achieve the desired effect. Pulses with
γB1tp = π/2 (i.e., 90 ◦
) cause maximum excitation (ideally complete conversion of Mz to −My), pulses with
γB1tp = π (i.e., 180 ◦
) turn Mz to the opposite direction −Mz. The flipped Mz corresponds to the inverted
distribution of magnetic moments.4
A pulse of a flip angle equal to 360 ◦
returns magnetization to its original
position. The length of the 360◦
pulse is the inverse value of ω1/(2π), i.e. of the aforementioned equivalent of
power in units of Hz. Apparently, the same effect (flip angle) can be achieved by increasing B1 or tp. However,
this is true only on resonance, when the radio-wave frequency exactly matches the precession frequency. If the
4The Boltzmann law seems to require that the temperature is lower than absolute zero (or higher than infinity) under such circumstances.
But do not panic, the Second Law of Thermodynamics is not violated because the inverted populations are not in thermal equilibrium.
5.5. OFFSET EFFECTS 49
A B C D
Figure 5.3: Phase-modulated pulses. A radio wave irradiation is divided into 16 pulses of different phases, distinguished by different
colors. The blue and red colors correspond to 0◦ and 180◦ phases (x and −x pulses), respectively, whereas white corresponds to 90◦ and
270◦ phases (±y pulses). Magnetic moments irradiated by the pulses B, C, and D experience additional frequency of one, two, and four
cycles per the pulse length, respectively. For example, if the pulse length is 100 µs, the pulses B, C, and D, respectively, resonate with the
frequency higher by 10 kHz, 20 kHz, and 40 kHz than the carrier frequency.
radio-wave frequency differs from the precession frequency, effects of long pulses deviate more from the desired
performance than those of short ones. Therefore, we often choose the highest possible power to be able to shorten
the pulse as much as possible. The limits are rather technical. The energy of the radio waves is absorbed by
the sample and temperature may increase if the instrument is not able to cool the sample efficiently (note that
1000 W is a typical power of a microwave oven). Sample may overheat or even get evaporated! Eventually, the
transmitter coil could be burned which would not make the lab manager happy. Therefore, the pulses cannot
be so short that the deviations from the resonance would be negligible. We discuss the consequences of being
off-resonance in Section 5.5.
• Amplitude modulation. So-far, we discussed only rectangular pulses, generated simply by switching the radio
transmitter on and off. We can also increase the power gradually,5
so that the radio-pulse is amplitude modulated
like in the AM broadcasting. As the amplitude, forming an envelope of the pulse, has a certain shape, the term
shaped pulses is used for the amplitude-modulated pulses. Careful choice of the shape can lead to a pulse
which affects almost ideally magnetic moments in a certain region of frequency offsets and has a negligible
effect on magnetic moments with frequency offsets outside this region. Such selective performance is discussed in
Section 5.5. In combination with the phase modulation, we have the liberty to affect various ranges of frequencies
selectively.
5.5 Offset effects
If the radio-wave frequency exactly matches the precession frequency of a certain nucleus, the effect of the pulse is
pretty clear. The frequency offset Ω is zero and ⃗B0 has no effect on the evolution. The effect of ⃗B1 on a system in
equilibrium can be described simply as a rotation about the x axis in the rotating coordinate system, as shown in
Figure 5.4A.
If a precession frequency of a certain nucleus slightly differs from the radio-wave frequency of ⃗B1, the evolution
of the magnetization vector during the pulse is not easy to describe. In a coordinate frame rotating with the same
frequency as ⃗B1 (i.e., axis z is defined by the direction of ⃗B0 and axis x is defined by the direction of ⃗B1), the
magnetization rotates about two axes, as shown in Figure 5.4B. One axis is ⃗B1 (or more precisely ⃗ω1), the other one
is ⃗Ω, with the magnitude given by the frequency offset and direction given by ⃗B0. Formally, such motion can be
described as a rotation about an effective field ⃗Beff , defined in Figure 5.4B. As a consequence, the orientation of the
magnetization is not perpendicular to both ⃗B1 and ⃗B0. The difference between the on-resonance and off-resonance
pulses (the difference between Figure 5.4A and Figure 5.4B) is called the offset effect.
Due to the offset effects, signals (oscillating induced electromotive forces) of different nuclei differ in amplitude and
phases, which decreases sensitivity and introduces distortions of spectra. But the offset effects have also some positive
consequences. Variation of power (combined with the corresponding change of length in order to keep the same flip
angle) allows us to introduce certain selectivity. We start with a non-selective pulse that should excite two nuclei,
5In practice, the pulse is composed of a certain number of short rectangular pulses differing in the amplitude of the radio wave.
50
Table 5.1: Dependence of excitation efficiency on frequency offsets for various amplitude modulations of radio wave pulses. The lengths
and amplitudes (shapes) are plotted in the real ratios, blue and red correspond to the phase of 0◦ and 180◦, respectively. The efficiency
of excitation is plotted in blue, ranging from zero (Mz = Meq, Mx = My = M2
x + M2
y = 0, magnetization vector in the z direction,
no excitation) to one (Mz = 0, M2
x + M2
y = Meq, magnetization vector in the xy plane). The deviations of the x and y components
of the magnetization vector from the desired −y direction are plotted in red. The range of the frequency offsets is −30 kHz to +30 kHz.
The arrows above the plot indicate average frequency offsets of 13Cα (assumed to be on-resonance, light green), carbonyl 13C (red), and
methyl 13C (dark green) at 111.75 T (corresponding to the proton resonance frequency of 500 MHz). The lengths and amplitude of the
hard rectangular pulse correspond to 10 µs and |ω1| = 25 kHz. The lengths (64.5 µs) and amplitude (|ω1| = 9.675 kHz) of the selective
rectangular pulse are chosen so that the frequency offset of carbonyl 13C (15 kHz) is equal to
√
15|ω1|. The lengths of the Q5 and EBURP2
pulses (300 µs both) were chosen so that the aliphatic 13C nuclei are excited and carbonyl 13C magnetic moments remain in the equilibrium.
The amplitudes of the Q5 and EBURP2 pulses were set so that the pulses rotate the magnetization by 90◦ when applied on resonance.
NMR-Sim (Pavel Kessler) was used to calculate the effects of shaped pulses.
Name shape offset-dependent effect
rectangular (hard) 0
1
0
2π
rectangular (selective) 0
1
0
2π
Q5 0
1
0
2π
EBURP2 0
1
0
2π
5.5. OFFSET EFFECTS 51
Table 5.2: Dependence of inversion efficiency on frequency offsets for various amplitude modulations of radio wave pulses. The lengths
and amplitudes (shapes) are plotted in the real ratios, blue and red correspond to the phase of 0◦ and 180◦, respectively. The efficiency
of inversion is plotted in blue, ranging from zero (Mz = Meq) to one (Mz = −Meq). The range of the frequency offsets is −30 kHz to
+30 kHz. The arrows above the plot indicate average frequency offsets of 13Cα (assumed to be on-resonance, light green), carbonyl 13C
(red), and methyl 13C (dark green) at 111.75 T (corresponding to the proton resonance frequency of 500 MHz). The lengths and amplitude
of the hard rectangular pulse correspond to 20 µs and |ω1| = 25 kHz. The lengths (57.7 µs) and amplitude (|ω1| = 4.33 kHz) of the selective
rectangular pulse are chosen so that the frequency offset of carbonyl 13C (15 kHz) is equal to
√
3|ω1|. The lengths of the Q3 and IBURP2
pulses (250 µs and 300 µs, respectively) were chosen so that the aliphatic 13C nuclei are inverted and carbonyl 13C magnetic moments
remain in the equilibrium. The amplitudes of the Q3 and IBURP2 pulses were set so that the pulses rotate the magnetization by 180◦
when applied on resonance. NMR-Sim (Pavel Kessler) was used to calculate the effects of shaped pulses.
Name shape offset-dependent effect
rectangular (hard) 0
1
rectangular (selective) 0
1
Q3 0
1
IBURP2 0
1
52
Table 5.3: Dependence of refocusing efficiency on frequency offsets for various amplitude modulations of radio wave pulses. The lengths
and amplitudes (shapes) are plotted in the real ratios, blue and red correspond to the phase of 0◦ and 180◦, respectively. The efficiency
of inversion is plotted in blue, ranging from zero (My = −Meq) to one (My = Meq). The deviations of the x and y components of the
magnetization vector from the desired y direction are plotted in red. The range of the frequency offsets is −30 kHz to +30 kHz. The arrows
above the plot indicate average frequency offsets of 13Cα (assumed to be on-resonance, light green), carbonyl 13C (red), and methyl 13C
(dark green) at 111.75 T (corresponding to the proton resonance frequency of 500 MHz). The lengths and amplitude of the hard rectangular
pulse correspond to 20 µs and |ω1| = 25 kHz. The lengths (57.7 µs) and amplitude (|ω1| = 4.33 kHz) of the selective rectangular pulse
are chosen so that the frequency offset of carbonyl 13C (15 kHz) is equal to
√
3|ω1|. The lengths of the Q3 and REBURP pulses (250 µs
and 300 µs, respectively) were chosen so that the aliphatic 13C nuclei are refocused and rotation of the carbonyl 13C magnetization is
unaffected. The amplitudes of the Q3 and REBURP pulses were set so that the pulses rotate the magnetization by 180◦ when applied on
resonance. NMR-Sim (Pavel Kessler) was used to calculate the effects of shaped pulses.
Name shape offset-dependent effect
rectangular (hard) 0
1
0
2π
rectangular (selective) 0
1
0
2π
Q3 0
1
0
2π
REBURP 0
1
0
2π
5.5. OFFSET EFFECTS 53
A
⃗M
⃗B1
B
⃗M
−⃗Ω/γ
⃗B1
⃗Beff
Figure 5.4: Evolution of the magnetization vectors with precession frequency exactly matching the used radio frequency (left) and slightly
off-resonance (right). The evolution is shown in a coordinate frame rotating with ⃗ωrot = −⃗ωradio.
e.g. 13
Cα
and carbonyl 13
C, simultaneously. If the pulse is applied with a high power, the condition |B1| ≫ |Ω/γ| is
fulfilled sufficiently for a broad range of frequency offsets Ω. Within this range of frequencies, ⃗Beff is close enough to
⃗B1 and magnetization vectors of all nuclei are rotated close to the desired orientation. Pulses of lower powers have
this ”on-resonance window” narrower. In the case of 13
Cα
and carbonyl 13
C, the window must cover 140 ppm to
affect both types of nuclei efficiently. Let us now suppose that we wish to rotate magnetization of certain nuclei (e.g.
13
Cα
) by 90 ◦
and leave other nuclei (e.g. carbonyl 13
C) untouched. In other words, we are looking for a pulse that
rotates 13
Cα
magnetization by 90 ◦
and carbonyl 13
C magnetization by 360 ◦
. The solution is presented in Figure 5.5.
Finding of the correct value of |B1| = |Ω|/(
√
15γ), or |ω1| = |Ω|/
√
15, for the given Ω (120 ppm in the case of 13
Cα
and carbonyl 13
C, corresponding e.g. to 60 kHz at 11.75 T) is just a matter of applying the Pythagorean theorem.
Figure 5.6 shows a similar solution for a 180 ◦
pulse selectively inverting longitudinal polarization of carbonyl 13
C but
not of 13
Cα
. In this case, the condition is |B1| = |Ω|/(
√
3γ), or |ω1| = |Ω|/
√
3.
Figure 5.6 suggests that a pulse with |ω1| = |Ω|/
√
3 and resonating exactly with the carbonyl precession frequency
can be used to selectively decouple carbonyl 13
C during an incremented delay when the signal is modulated by the
13
Cα
frequency (in a similar fashion as the cyan pulse in t1 in Figure 4.2 decouples 15
N). The longitudinal polarization
of carbonyl 13
C should be inverted and transverse polarization (or anti-phase coherence) of 13
Cα
should be returned to
its original direction. This is true, and we discuss pulse sequences containing such decoupling later (see e.g. Figure 6.5).
However, there is a caveat here. If a selective carbonyl pulse of a duration tp is applied, the 13
Cα
coherence is returned
to its original direction (Figure 5.6. If the selective carbonyl pulse is not applied, the 13
Cα
coherence rotates with its
frequency offset Ω). Therefore, application of the pulse reduces the total rotation during the incremented delay by
Ωtp. This reduction corresponds to a phase shift in the modulation (sometimes called the Bloch-Siegert shift of the
second kind).6
We should admit that the selection described in Figures 5.5 and 5.6 is rather rough as no sharp border exists
between ”on-resonance” and ”off-resonance”. The offset effects increase gradually and the efficiency of the pulse
( M2
x + M2
y /M for excitation, −Mz/M for inversion, see Tables 5.1 and 5.2, respectively) gradually decreases (with
additional oscillations). Therefore, only magnetic moments with relatively narrow and sufficiently distant frequency
6It is not difficult to calculate the phase shift in our case. The selective pulse rotates the carbonyl magnetization by 180 ◦ = π rad.
Therefore, ω1tp = π and tp = π/ω1. As mentioned above, the radio wave power is chosen so that |ω1| = |Ω|/
√
3. Consequently,
tp = π/ω1 = ±
√
3π/Ω, showing that the ”missing” angle of the rotation of the 13Cα coherence is Ωtp = ±
√
3π.
54
A
⃗M
⃗B1
B
⃗M
−⃗Ω/γ
⃗B1
⃗Beff
Figure 5.5: Evolution of the magnetization vectors with precession frequency exactly matching the used radio frequency (left) and with a frequency
offset Ω (right), for ω1 = Ω/
√
15. If ω1 rotates magnetization of the former nucleus by 90 ◦
, then ωeff =
√
1 + 15Ω = 4Ω rotates magnetization of
the latter nucleus by 4 × 90 ◦
= 360 ◦
, i.e., by the full circle. The evolution is shown in a coordinate frame rotating with ⃗ωrot = −⃗ωradio. In both
cases, magnetization rotates about the thick purple arrow with the angular frequency proportional to the length of the arrow.
offsets can be distinguished. Fortunately, this is true only for rectangular pulses, generated simply by switching the
transmitter on and off. The amplitude modulated, or shaped pulses provide much better selectivity. Careful choice
of the shape (or, more efficiently, computer optimization) can lead to a pulse which efficiency is almost constant in
a certain region of frequencies and than it rapidly fades out. Performance of amplitude-modulated pulses providing
selective excitation, inversion, and refocusing is compared in Tables 5.1–5.3. The effects of the pulses presented in
Tables 5.1–5.3 are calculated for a typical example of selectively manipulated nuclei in biomacromolecules, for 13
C
in aliphatic and carbonyl groups in proteins. The tables show that different amplitude modulations can be used for
similar purposes with slightly different outcomes.
5.6 From oscillations to spectrum: Fourier transformation
Fourier transformation is crucial for processing NMR signal. Therefore, we discuss its principle starting from the very
basic idea.
Certain people, including the author of this text, have hard time to remember their shoe size. Some shops take this
handicap into account. They keep a special device, a board with the footprints of various sizes. Imagine somebody
really silly, who starts with the smallest size and keeps comparing his foot with the footprints on the board to the
largest size. He ranks each trial on a scale of comfort ranging from zero to 100 %. Clearly, the shoe size that fits
his foot gets high comfort mark, perhaps two closest sizes get some low mark, and the other sizes are ranked as zero
comfort. This is roughly the same procedure as used by computers converting FID to a spectrum.
The procedure described in the previous paragraph closely resembles Fourier transformation. Fourier transformation
does not compare footprints but sine (or cosine) curves (Figure 5.7). Instead of the board with various footprints,
Fourier transformation works with a set of testing sine (or cosine) functions of gradually increasing frequency. Each
trial represents calculating product of the tested curve (experimental data points shown in red in Figure 5.7) with
one of the testing curve (blue curves in Figure 5.7). In case of different frequencies, the oscillations are uncorrelated
and the products at individual points (shown in magenta in Figure 5.7) oscillate around zero. The total product (sum
over all points) is equal or very close to zero. On the other hand, if the frequency of the tested curve matches the
frequency of the testing curve, the oscillations coincide and we obtain some positive number as the product at each
5.7. QUADRATURE DETECTION AND DEMODULATION 55
A
⃗M
⃗B1
B
⃗M
−⃗Ω/γ
⃗B1
⃗Beff
Figure 5.6: Evolution of the magnetization vectors with precession frequency exactly matching the used radio frequency (left) and with a frequency
offset Ω (right), for ω1 = Ω/
√
3. If ω1 rotates magnetization of the former nucleus by 180 ◦
, then ωeff =
√
1 + 3Ω = 2Ω rotates magnetization of
the latter nucleus by 2 × 180 ◦
= 360 ◦
, i.e., by the full circle. The evolution is shown in a coordinate frame rotating with ⃗ωrot = −⃗ωradio. In both
cases, magnetization rotates about the thick purple arrow with the angular frequency proportional to the length of the arrow.
point (with the exception of points where both curves pass zero). The total product is a large number. Then we plot
the total product as a function of frequency of the testing curve (green plot in Figure 5.7). The obtained plot has the
familiar shape of a resonance curve with a peak at the frequency that matched the frequency of the testing curve. We
obtained a spectrum.
5.7 Quadrature detection and demodulation
Precession of the magnetization vector in the sample induces a signal oscillating with the same frequency ω0 in the
coil of the NMR probe. The signal generated in the coil and amplified in the preamplifier is split into two channels,
labeled a and b here. The signal in each channel is mixed with a reference wave supplied by the radio-frequency
synthesizer. The reference waves have the same frequency −ωradio in both channels, but their phases are shifted by
90 ◦
. Let us assume that the signal oscillates as a cosine function cos(ω0t) and that the reference wave in the first
channel is a cosine wave cos(−ωradiot) and that the reference wave in the second channel is a sine wave − sin(−ωradiot).
Mathematically, splitting the signal and mixing it with the reference wave can be described as
cos(ω0t) →
1
2 cos(ω0t) → 1
2 cos(ω0t) cos(−ωradiot) channel a
1
2 cos(ω0t) → −1
2 cos(ω0t) sin(−ωradiot) channel b
(5.2)
Basic trigonometric identities show that the result of mixing in the first channel is a sum of a high-frequency cosine
wave cos((ω0 − ωradio)t) and a low-frequency cosine wave cos((ω0 + ωradio)t) = cos(Ωt), while the result of mixing in
the second channel is a difference of the corresponding sine waves:
1
2
cos(ω0t) cos(−ωradiot) =
1
4
cos((ω0 − ωradio)t) +
1
4
cos((ω0 + ωradio)t), (5.3)
−
1
2
cos(ω0t) sin(−ωradiot) = −
1
4
sin((ω0 − ωradio)t) +
1
4
sin((ω0 + ωradio)t). (5.4)
56
f1 f2 f3 f4 f5
ν = f3
t t
ν = ?
f5
Σ = 0
f4
Σ = 0
f3
Σ = 50
f2
Σ = 0
f1
Σ = 0
Figure 5.7: The basic principle of Fourier transformation presented for experimental data (red) and five testing functions (blue).
Experimental data multiplied by the testing functions are displayed as magenta dots, sums (integrals) of the products as magenta lines.
The sums are plotted in the right diagram and connected with the green lines.
The high-frequency waves are filtered out by a low-pass filter, resulting in signals oscillating with a low frequency
ω0 +ωradio = Ω. The procedure, similar to the demodulation in an ordinary radio receiver, thus produces audio signals
in both channels
cos(ω0t) →
1
2 cos(ω0t) → 1
2 cos(ω0t) cos(−ωradiot) → 1
4 cos(Ωt) channel a
1
2 cos(ω0t) → −1
2 cos(ω0t) sin(−ωradiot) → 1
4 sin(Ωt) channel b
(5.5)
The signal also has some amplitude, therefore, we replace the factor of 1/4 by an amplitude A and write
a = A cos(Ωt) b = A sin(Ωt). (5.6)
The described manipulation is called quadrature detection and the unit performing it is called the receiver. The
outputs of the receiver are converted to digital data (series of numbers describing values of the signal at discrete,
equally spaced time points). It is convenient to treat the outputs of the individual channels as a real and imaginary
component of a single complex number, but physically they are stored just as series of two numbers in the computer.
A very useful trick is to play with the order of the stored numbers. The four basic options are
data storing option: a, b conventionally labeled: x
b,−a y
−a,−b −x
−b, a −y
The given storage option is described as the receiver phase in the literature. It is not an accident that the
same symbols are used for the phase of the radio wave transmitted during the pulse and for the receiver phase.
Choosing the right storage option (setting the receiver phase) allows us to remove the effect of changing the pulse
phase. For example, a signal recorded immediately after an ideal 90◦
pulse of phase x (by definition) oscillates
as a = A sin(Ωt), b = −A cos(Ωt) (the magnetization starts to rotate from the −y direction). If we run the same
experiment but with the y phase of the first pulse, the signal oscillates as a = A cos(Ωt), b = A sin(Ωt). However, if we
5.8. COMPLEX SIGNAL IN THE INDIRECT DIMENSION(S) 57
use the option y to store the data, the record is the same as in the previous experiment: b = A sin(Ωt), −a = −A cos(Ωt).
We see that the same signal is obtained if the receiver phase matches the transmitter phase.
The motivation to store both sine- and cosine-oscillating data is that the cosine function itself does not tell us if Ω
is positive or negative because cos(Ωt) = cos(−Ωt). In other words, we do not know if the true ω0 is slightly higher or
lower than ωref . This ambiguity is reflected by the Fourier transformation which produces a spectrum with two peaks,
one is true and one is false (Figure 5.8A). To resolve the ambiguity and to identify the correct peak, we need both
cos(Ωt) and sin(Ωt). A complex variant of the Fourier transformations converts the pair of cos(Ωt) and sin(Ωt) into
a spectrum with a single peak at the right Ω, and produces also an imaginary component which is usually discarded
(Figure 5.8B).
5.8 Complex signal in the indirect dimension(s)
The incrementation of t1 in the HSQC experiment, as presented in Figure 4.2, introduces amplitude modulation of
the signal by the 13
C (or 15
N) frequency offset and consequently allows us to obtain a 2D correlated spectrum. The
cosine dependence of amplitude on ΩC cannot distinguish the sign of ΩC, as explained in Section 5.7. It is therefore
desirable to obtain both cosine and sine modulations also in the indirect dimension.
It is possible to introduce the complex numbers in the indirect dimension by repeating the measurement twice for
each value of t1, each time with a different phase of the radio waves applied during a 90◦ 13
C pulse. We describe
the procedure for the pulse sequence shown in Figure 5.9A. First we acquire the signal with the same phase (x by
definition) of both 90◦ 13
C pulses. As described in Section 4.2, only the ±y components (perpendicular to the direction
of B1) of the arrows symbolizing anti-phase 13
C coherence contribute to the signal. These components are modulated
by cos ΩCt1. We store the outputs of the quadrature receiver channels a and b as records r1 and r2, respectively.
Then, we repeat the acquisition with the phase of the first pulse shifted by 90◦
(phase y) and store the outputs of the
channels a and b as records r3 and r4, respectively. The y pulse creates anti-phase coherence described by arrows in
the ±x direction, but the second pulse selects the ±y components. Such selected coherence, shifted by 90 ◦
from the
former one, oscillates as sin ΩCt1 and can be used as an imaginary component of the signal. We than increase t1 and
repeat the cycle. This approach, known as hypercomplex or States-Haberkorn-Ruben method, provides both cosine
and sine amplitude modulation in the indirect dimension. If we label these modulated amplitudes Ac and As, we can
describe the acquired signal as
r1 = Aca = cos(ΩCt1) cos(ΩHt2) channel a, pulse x
r2 = Acb = cos(ΩCt1) sin(ΩHt2) channel b, pulse x
r3 = Asa = sin(ΩCt1) cos(ΩHt2) channel a, pulse y
r4 = Asb = sin(ΩCt1) sin(ΩHt2) channel b, pulse y



(Ac + iAs)(a + ib), (5.7)
or in a table
x/y receiver acquired as stored as records rj
+x +x a : Ac sin(ΩHt2) r1, r2 = Ac sin(ΩHt2), Ac cos(ΩHt2)
b : Ac cos(ΩHt2)
+y +x a : As sin(ΩHt2) r3, r4 = As sin(ΩHt2), As cos(ΩHt2)
b : As cos(ΩHt2)
We must be careful when processing the hypercomplex data. In the case of 1D spectra, processing of the signal
treated as series of complex numbers is simple. Complex Fourier transformation converts the data to the spectrum,
stored again as a complex number, but only the real part is interesting. Symbolically,
a + ib −→ X + iY, (5.8)
where X is the desired real spectrum and Y is its imaginary counterpart.
In the case of a hypercomplex 2D signal, applying the complex Fourier transformation twice is not what we want:
(Ac + iAs)(a + ib) −→ (Ac + iAs)(X2 + iY2) −→ (X1 + iY1)(X2 + iY2) = X1X2 − Y1Y2 + i(X1Y2 + Y1X2). (5.9)
58
A
Signalinsinglechannel
t
Spectrumintensity
ω
Ω−Ω
B
Signalinchannel1
t
Realintensity
ω
Ω
Signalinchannel2
t
Imaginaryintensity
ω
Ω
Figure 5.8: Fourier transformation of signal detected using a single channel (A) and two channels (B). Left panels show outputs of the
receivers, right panels show spectra obtained by Fourier transformations. Data treated as real and imaginary numbers are shown in blue
and red, respectively. The correct frequency offset is Ω.
5.9. TIME-PROPORTIONAL PHASE INCREMENTATION∗
59
What is stored as the real component of the signal is actually a mixture of the desired real peaks X1X2 and
”double-imaginary” peaks Y1Y2 of ugly shapes. The correct procedure is to do the Fourier transformation in t2,
discard the imaginary part of the spectrum, and do the Fourier transformation in t1:
(Ac + iAs)(a + ib) −→ (Ac + iAs)(X2 + iY2) −→ (Ac + iAs)X2 −→ (X1 + iY1)X2 = X1X2 + iY1X2. (5.10)
5.9 Time-proportional phase incrementation∗
Positive and negative frequencies can be also discriminated in the following manner. Let us assume that the time
increment ∆t in a reference experiment not distinguishing positive and negative frequencies has been chosen so that all
angular frequencies are within the range −π/∆t to +π/∆t (cf. Section 5.2). Then the following changes are applied
• The number of increments is doubled and the increment is shortened to ∆t/2. This increases the spectral width
to the range from −2π/∆t to +2π/∆t, but all really occurring frequencies are in the central region from −π/∆t
to +π/∆t.
• The phase of the 90◦
pulse before or after the incremented period is increased by 90◦
in each time increment.
The result can be analyzed as follows. Incrementation of the 90◦
pulse phase decreases the argument of the
modulating function by 90◦
, from Ωk∆t/2 to
Ωk
∆t
2
− k
π
2
= Ω −
π
∆t
k
∆t
2
, (5.11)
i.e., all frequencies are decreased by one quarter of the spectral width. Therefore, all really occurring frequencies
are shifted from the central region to the range from −2π/∆t to 0, where all angular frequencies have the same sign.
The other half of the spectrum (from 0 to +2π/∆t is empty and can be discarded. Then, we obtain a spectrum with
the same number of points as originally, but with the same sign of the frequencies.
This procedure is known as time-proportional phase incrementation (TPPI). The States-Haberkorn-Ruben method
(Section 5.8) and TPPI are actually very similar. In both cases, the frequency discrimination requires doubling
of the experimental time (repeated measurement in the case of the States-Haberkorn-Ruben method and doubling
increments in the case of TPPI). The even and odd increments in the TPPI method are equivalent to the cosine and
sine modulation:
cos Ωk
∆t
2
− k
π
2
= cos Ωk
∆t
2
cos k
π
2
+ sin Ωk
∆t
2
sin k
π
2
=
± cos Ωk ∆t
2 for even k
± sin Ωk ∆t
2 for odd k
. (5.12)
The advantage of TPPI is that artifacts known as axial peaks (discussed in Section 5.13) are moved to the edge
of the spectrum. TPPI is often combined with the States-Haberkorn-Ruben method by setting the phase of the 90◦
pulse to x, y, −x, −y and the receiver phase to x, x, −x, −x for the subsequent time increments.
5.10 Phase cycling
We discussed in Sections 5.8 and 5.9 how changing the pulse phase between acquisitions of separately stored records
provides complex modulation in the indirect dimension. In real experiments, phases are often changed when acquiring
data that are then summed and stored as a single record. This trick is known as phase cycling and it is applied in
order to suppress unwanted signals. The unwanted signals may be due to magnetic moments really present in the
sample (of water protons, of 13
C or 15
N nuclei without protons attached) or due to various artifacts.
When phase cycling is applied, the experiments are repeated for each increment of t1 with different phases of some
radio-wave pulses, and the acquired signals are combined. Repeating the measurements of course extends the overall
time of the experiments. However, this drawback is not as serious as it may appear. In many cases, the sensitivity of
60
A
1
4J
1
4J
t2
1
4J
1
4J
N15
C13
or
t1
H1
x y/
GARP
y
B
1
4J
1
4J
t2
1
4J
1
4J
N15
C13
or
t1
H1
2φ
2φ
y
GARP
3φ
=y,−y,y,−y,y,−y,y,−yφ1
=x,−x,x,−x,x,−x,x,−xφ1 =x,x,−x,−x,x,x,−x,−x
φ3=x,x,x,x,−x,−x,−x,−x
/ /1φ φ11φ1φ
receiver phase: x,−x,−x,x,x,−x,−x,x
Figure 5.9: A, Recording hypercomplex HSQC spectra. The label x/y indicates repeated acquisition with the phase of the given pulse
set first to 0 ◦ (x) and than to 90◦ (y), in order to obtain a cosine-modulated and sine-modulated 1D records for each t1 increment. B
HSQC experiment with phase cycling. The pulses with cycled phases are labeled ϕ1, ϕ2, ϕ3, and the actual phases during the cycles are
listed below the sequence. In order to add signals with the same signs, the individual signals are acquired with receiver phases indicated
below the sequence. Other symbols are used as explained in Figure 4.2.
5.11. QUADRATURE ARTIFACTS AND CYCLOPS∗
61
the measurement requires to sum results of several measurements anyway, in order to achieve a sufficient signal-to-noise
ratio. Usually, the signals recorded with individual settings of the phases are not stored separately but directly added
to the data acquired with the preceding phase setting. Phase of the desired signal acquired with different phases of
the pulses may vary. Therefore, the phase of the acquired signal has to be adjusted before it is added to the sum of
the signals recorded in the previous runs. As described in Section 5.7, signal with the same phase can be obtained by
changing the receiver phase (the way how outputs of individual channels of the receiver are stored).
In the example shown in Figure 5.9B, the phases of 13
C (or 15
N) 90◦
pulses are cycled in order to suppress the
signal of protons not attached to 13
C (or 15
N), e.g. of protons in water or in buffer components, that are not enriched
in 13
C (or 15
N). Let us first analyze the evolution of the wanted polarizations of protons attached to 13
C (or 15
N),
converted to the anti-phase 13
C (or 15
N) coherence and back by two INEPT modules. The phase of the first 90◦
pulse
is shifted by 180 ◦
after every run, which inverts direction of the arrows describing the anti-phase coherence and finally
changes the sign of the signal (i.e., shifts the signal by 180 ◦
). The phase of the second 90◦
pulse is cycled in a same
manner and with the same effect, but after every second run. The result of such two nested phase cycles is that the
phases of signal acquired in the second, third, sixth, and seventh runs are shifted by 180 ◦
and has to be adjusted,
which is achieved by storing the data in a manner described by the receiver phase −x. The undesired polarization of
protons in water or buffer of course does not sense the changes of 13
C (or 15
N) pulses and produces signal with the
same phase. Alternation of the receiver phase between x and −x thus cancels every two consecutive signals of the
uncoupled protons.
When the phase cycling is applied in a two-dimensional experiment, as discussed in this section, we should carefully
distinguish different purposes of repeating the data acquisition:
• In order to apply phase cycling and to improve signal-to-noise ratio, the signal is acquired repeatedly with various
phases of certain radio-wave pulses. The individual signals are called transients or scans in the NMR literature.
Usually, transients (scans) are not stored separately, but combined (summed after necessary phase adjustment).
Quadrature receivers of standard NMR spectrometers supply two output signals with phases shifted by 90 ◦
,
therefore each transient represents a complex signal with real and imaginary component.
• In order to introduce the second dimension, the signal is acquired repeatedly with increasing (or decreasing)
length of the delay t1. The individual signals are called increments in the NMR literature. The increments are
stored separately as an array of one-dimensional data (data matrix). Each increment is stored as two records
(sums of transients) with phases shifted by 90 ◦
, providing (after appropriate phase correction) real and imaginary
component of the data in the direct dimension.
• In order to achieve frequency discrimination in the indirect dimension (using the States-Haberkorn-Ruben
method described in Section 5.8), each increment is recorded twice with a different phase of a certain pulse.
The individual increments are called cosine modulated and sine modulated in the NMR literature. The StatesHaberkorn-Ruben
method of frequency discrimination is based on changing a pulse phase (like phase cycling),
but (unlike phase cycling) data collected with different phases are stored separately. The real components of
the cosine- and sine-modulated increments provide the real and imaginary component of the data in the indirect
dimension as described in Section 5.8.
The most common applications of phase cycling are described in the following sections.
5.11 Quadrature artifacts and CYCLOPS∗
Phase cycling is important in suppressing various instrumental artifacts. For example, two channels of the receiver
are never identical. They differ in the amplitudes (imbalance) and in the background direct current. Instead of the
perfect signal
A sin(Ωt), −A cos(Ωt) (5.13)
we obtain an imperfect signal
62
A
⃗M
⃗B1
B
⃗M
−⃗Ω/γ
⃗B1
⃗Beff
Figure 5.10: Evolution of refocusing 180◦
pulse on magnetization vectors with precession frequency exactly matching the used radio frequency
(A) and with a frequency offset Ω (B), for ω1 = Ω/
√
3. If ω1 rotates magnetization of the former nucleus by 180 ◦
, then ωeff =
√
1 + 3Ω = 2Ω
rotates magnetization of the latter nucleus by 2 × 180 ◦
= 360 ◦
, i.e., by the full circle. The evolution is shown in a coordinate frame rotating with
⃗ωrot = −⃗ωradio. In both cases, magnetization rotates about the thick purple arrow with the angular frequency proportional to the length of the
arrow.
D + (A + ∆) sin(Ωt), −D + (A − ∆)(− cos(Ωt)). (5.14)
The consequence of the direct current D is a presence of so-called quadrature glitch (an artificial peak at Ω = 0)
and the consequences of the imbalance ∆ are so-called quadrature images (artificial peaks at the false frequency −Ω).
The imperfections can be removed by cycling the phases of the transmitter (during the excitation pulse) and of
the receiver simultaneously, this phase cycle is called CYCLOPS:
transmitter acquired as receiver stored as
+x a : +D + (+A + ∆) sin(Ωt) +x +D + (+A + ∆) sin(Ωt), −D + (−A + ∆) cos(Ωt)
b : −D − (+A − ∆) cos(Ωt)
+y a : +D + (+A + ∆) cos(Ωt) +y −D + (+A − ∆) sin(Ωt), −D + (−A − ∆) cos(Ωt)
b : −D + (+A − ∆) sin(Ωt)
−x a : +D − (+A + ∆) sin(Ωt) −x −D + (+A + ∆) sin(Ωt), +D + (−A + ∆) cos(Ωt)
b : −D + (+A − ∆) cos(Ωt)
−y a : +D − (+A + ∆) cos(Ωt) −y +D + (+A − ∆) sin(Ωt), +D + (−A − ∆) cos(Ωt)
b : −D − (+A − ∆) sin(Ωt)
sum: 4A sin(Ωt), −4A cos(Ωt)
5.12 Pulse imperfections and EXORCYCLE∗
Phase cycling also removes imperfection of pulses. For example, each 180◦
pulse has a limited power and it affects
ideally only magnetic moments exactly on resonance (with the precession frequency exactly matching frequency of
the applied radio waves). All other magnetic moments experience some offset effects, as discussed in Section 5.5. The
extreme cases of ideal performance (on resonance) and no effect (off resonance) are presented in Figure 5.10A and B,
respectively. Let us now analyze how phase cycling affects the offset effects.
Let us analyze as an example a signal of uncoupled proton detected after a refocusing echo (Figure 3.6B). If the
180◦
pulse is applied on resonance, the echo refocuses the magnetization to My (see Section 3.6), which then yields
5.12. PULSE IMPERFECTIONS AND EXORCYCLE∗
63
H1
H1
H1
H1
N15
C13
or
N15
C13
or
N15
C13
or
N15
C13
or
A
B
C
D
a b c d ea
y
y
Figure 5.11: Spin echoes with the 180◦ 1H pulse applied with a 90◦the phase shift, as a y pulse. The symbols are explained in Figure 3.6.
Note that 180◦ 1H −x and −y pulses have the same effect as the −x and −y pulses, respectively, and that all phases of the 180◦ 13C (or
15N) have the same effect.
a signal oscillating as −A sin(Ωt) in channel a (detecting the Mx component of the magnetization vector) and as
A cos(Ωt) in channel b (detecting the −My component). If the 180◦
pulse is applied off resonance, it does not have
any effect and the magnetization freely evolves during the echo. Therefore, the detected signal is not a function of the
time t (measured from switching the receiver on), it is a function of time t + 2τ, where τ is the length of the one of
the delays constituting the echo. The off-resonance signal thus yields a signal oscillating as A sin(Ωt + ϕ) in channel
a and as −A cos(Ωt + ϕ) in channel b, where ϕ = 2Ωτ is a constant phase shift, distorting the signal.
What if we repeat the experiment with the phase of the 180◦
pulse shifted by 90 ◦
(a ”y” pulse)? If a 180◦
y pulse is
applied on resonance, it flips the Mx components to −Mx in the middle of the echo and leaves My as My (Figure 5.11).
At the end of the echo, the magnetization is refocused in the −y direction. Therefore the signal oscillates as A sin(Ωt)
in channel a and as −A cos(Ωt) in channel b during the detection. Again, the magnetic moments that are off resonance
are not influenced by the pulse and yield a signal oscillating as A sin(Ωt + ϕ) in channel a and as −A cos(Ωt + ϕ) in
channel b, as described above.
The following analysis shows that the desired on-resonance signal is preserved and the undesired phase-shifted
off-resonance signal is canceled if the transmitter phase cycles as x, y and the receiver phase cycles as x, −x
On resonance:
transmitter receiver acquired as stored as
+x +x a : −A sin(Ωt) −A sin(Ωt), +A cos(Ωt)
b : +A cos(Ωt)
+y −x a : +A sin(Ωt) −A sin(Ωt), +A cos(Ωt)
b : −A cos(Ωt)
sum: −2A sin(Ωt), 2A cos(Ωt)
64
Off resonance:
transmitter receiver acquired as stored as
+x +x a : +A sin(Ωt + ϕ) +A sin(Ωt + ϕ), −A cos(Ωt + ϕ)
b : −A cos(Ωt + ϕ)
+y −x a : +A sin(Ωt + ϕ) −A sin(Ωt + ϕ), +A cos(Ωt + ϕ)
b : −A cos(Ωt + ϕ)
sum: 0
Another common imperfection is inaccurate calibration of the pulse. A miscalibrated 180◦
pulse rotates the
magnetization vector too much or too little. For example, if the 180◦
x pulse in Figure 3.6B is longer by ∆tp, it does
not flip −My exactly to My, but it yields My cos(ω1∆tp) + Mz sin(ω1∆tp). If the same miscalibrated 180◦
pulse is
applied with a phase shifted by 180 ◦
(as a −x pulse), it rotates My in the opposite direction and the unwanted Mz
component thus has the opposite orientation: My cos(ω1∆tp) − Mz sin(ω1∆tp). If this happens in a more complex
pulse sequence, where a 90◦
y pulse is applied after the echo, the artificial Mz components is converted to Mx and
yields an undesired signal, shifted by 90 ◦
from the desired one. Fortunately, the unwanted Mx component can be
removed by cycling the transmitter as x, −x without changing the receiver phase, as shown below.
Desired My:
transmitter receiver acquired as stored as
+x +x a : −A cos(ω1∆tp) sin(Ωt) −A cos(ω1∆tp) sin(Ωt), +A cos(ω1∆tp) cos(Ωt)
b : +A cos(ω1∆tp) cos(Ωt)
−x +x a : −A cos(ω1∆tp) sin(Ωt) −A cos(ω1∆tp) sin(Ωt), +A cos(ω1∆tp) cos(Ωt)
b : +A cos(ω1∆tp) cos(Ωt)
sum: −2A cos(ω1∆tp) sin(Ωt), 2A cos(ω1∆tp) cos(Ωt)
Undesired Mz (−→ Mx after application of a 90◦
y pulse):
transmitter receiver acquired as stored a
+x +x a : −A sin(ω1∆tp) sin(Ωt) −A sin(ω1∆tp) sin(Ωt), −A
b : −A sin(ω1∆tp) cos(Ωt)
−x +x a : +A sin(ω1∆tp) sin(Ωt) +A sin(ω1∆tp) sin(Ωt), +A
b : +A sin(ω1∆tp) cos(Ωt)
sum: 0
The phase cycles suppressing offset and miscalibration effects are combined into a four-step cycle removing various
”ghost peaks”, called EXORCYCLE:
transmitter: x receiver: x
y −x
−x x
−y −x
5.13 Axial peaks∗
Another source of undesired signals is relaxation (return to the equilibrium distribution of magnetic moments). For
example, magnetic moments of uncoupled protons in the HSQC experiments (Figure 5.9) after the first INEPT module
are polarized in the direction y (they are not affected by the 90◦
y pulse). During t1, they start to redistribute due to
the relaxation and the equilibrium longitudinal (vertical) polarization starts to build up. As a consequence, a certain
Mz component is created. It is inverted to −Mz by the cyan pulse in Figure 4.2, but it is not modulated by any
frequency during t1 (it does not rotate about ⃗B0). The next proton 90◦
pulse flips the artificial −Mz component it
to +My, its evolution is refocused during the second INEPT, and it yields a signal oscillating with proton frequency
during t2. However, amplitude of this signal is not modulated by the 13
C or 15
N frequency. Therefore, the Fourier
transformation converts this signal to an axial peak, i.e. a peak at zero 13
C or 15
N frequency in the two-dimensional
spectrum. The origin of the axial peak closely resembles the miscalibration artifacts (unwanted Mz component
5.14. CLEANING AND FREQUENCY DISCRIMINATION BY PULSED FIELD GRADIENTS∗
65
converted to measurable signal by a 90◦
pulse). Therefore, the axial peaks are removed by the same cycle as the
miscalibration artifacts (by cycling the transmitter as x, −x without changing the receiver phase). In Figure 5.9B,
this cycle is combined (nested) with cycling the 90◦
pulses discussed in Section 5.10.
5.14 Cleaning and frequency discrimination by pulsed field gradients∗
Pulsed field gradients as tools allowing us to suppress unwanted coherences have been introduced in Section 4.3.
Application of a cleaning gradient and of gradient echoes in a real NMR experiment is presented in Figure 5.12A
(magenta and cyan symbols, respectively). Note that the cleaning (magenta) gradient is applied when no coherence
(transverse polarization) should be present. The cyan gradients are applied during the simultaneous echoes and refocus
coherences that evolve due to the J-coupling.
Figure 5.12A also shows another, more tricky use of gradients implemented in an improved version of the HSQC
experiment (blue/red and green symbols). The idea is to apply one gradient during the time when the desired coherence
rotates with the frequency of 13
C (or 15
N) and another gradient during the time when the total magnetization rotates
with the frequency of protons. In order to do it, we must generate a space in the pulse sequence by including a
refocusing echo (a typical example of using refocusing echoes in situation when we need more space but do not want to
change evolution). The two applied gradients are not identical, they change the magnetic fields to different extent. The
deviations of the field must be exactly in the ratio of resonance frequencies of 13
C and 1
H. Then, the gradients form a
heteronuclear gradient echo. Note what happens to various coherences of protons. The coherence which contributed
to the polarization transferred to 13
C and back experiences the gradients as an echo and gets refocused. On the
other hand, population of protons whose polarization was not transferred to 13
C (e.g. protons of water that are not
13
C-bonded) feels just two gradients of different strengths and its coherence is destroyed. The gradient echo makes
the experiment selective for protons correlated with carbons and suppresses the signal of uncorrelated protons.
The green and blue/red gradients in Figure 5.12A have also and important impact on acquiring the hypercomplex
data. The gradients are applied in echoes after t1 and before t2 when coherences rotate with the proton and 13
C (or
15
N) frequency offsets, respectively. The green gradient is applied only in one half of the echo, so the precession of
the magnetic moments is not refocused, but made dependent on their position along the z axis. It is no longer the
y component of the coherence at point ”f” what is selected because individual magnetic moments rotated further by
some angle ϕ1 (see Figure 4.5B). This angle is proportional to γC and to the gradient strength, depends on the actual
position of the magnetic moment, and adds as a phase to the coherence rotation angle ΩCt1. As a consequence, the
signal is modulated by cos(ΩCt1 + ϕ1), were the horizontal bar indicates that we have to sum values of the cosine
all magnetic moments in the sample (differing because ϕ1 depends on the vertical position). If no blue/red gradient
were applied in Figure 5.12A, the signal would be negligible because ϕ1 adopts all values between 0 ◦
and 180 ◦
(see
Figure 4.5B), and the sum of the cosines of all angles (actually the average value of the cosine function) is zero. The
blue/red gradient adds a similar phase ϕ2 to ΩHt2. We assume that ϕ2 depends on the position exactly like ϕ1 (i.e.,
that the molecules do dot diffuse to a different position in the short time between the gradients), so ϕ1 = ϕ2 if the
gradient strengths are set to be in the ratio γH : γC. Therefore, the outputs of the channels of the quadrature receivers
are
a : Acos(ΩCt1 + ϕ1) cos(ΩHt2 + ϕ2), (5.15)
b : Acos(ΩCt1 + ϕ1) sin(ΩHt2 + ϕ2). (5.16)
In order to see how the gradients refocus, we apply basic trigonometric relations:
a :
A
2
cos(ΩCt1 + ΩHt2 + ϕ1 + ϕ2) −
A
2
cos(ΩCt1 − ΩHt2 + ϕ1 − ϕ2), (5.17)
b :
A
2
sin(ΩCt1 + ΩHt2 + ϕ1 + ϕ2) −
A
2
sin(ΩCt1 − ΩHt2 + ϕ1 − ϕ2). (5.18)
Strengths of the green and blue gradients are set to be in the ratio γH : γC, so ϕ1 = ϕ2. The cosine function with
ϕ1 + ϕ2 = 2ϕ1 averages to zero, and (returning to the products of goniometric functions)
66
A
1
4J
1
4J
H1
N15
C13
or
1
4J
1
4J
t2
t1
y
Gz
GARP
x y/
e hfb c da g i
B
1=−x,x,−x,x,−x,x,−x,x and receiver phase −x,x,x,−x,−x,x,x,−x for even increments of
1
4J
1
4J
H1
N15
C13
or
=x,−x,x,−x,x,−x,x,−x and receiver phase x,−x,−x,x,x,−x,−x,x for odd increments of
2t
4J
1
4J
1
4J
1
4J
1
φ1
1t
y
Gz
/x−x
y
y
=x,x,y,y,−x,−x,−y,−y
φ1
φ
GARP
2φ
2φ
t1
t1
j k lda hg ieb c f
Figure 5.12: Gradient enhanced HSQC experiment (A) and its PEP version (B). Cleaning gradients and gradient echoes are shown in
magenta and cyan, respectively. The heteronuclear gradient echo consists of a gradient shown in green, applied during the refocusing echo
between time instants ”f” and ”g” (when the coherence evolves with the 13C or 15N frequency), and of another gradient applied during
the last echo (when transverse polarization evolves with the proton frequency). The latter gradient is shown in blue and red, depending
on the phase of the second 13C or 15N 90◦ pulse. Other symbols are used as explained in Figure 5.9
5.14. CLEANING AND FREQUENCY DISCRIMINATION BY PULSED FIELD GRADIENTS∗
67
a :
A
2
cos(ΩCt1 − ΩHt2) =
A
2
cos(ΩCt1) cos(ΩHt2) +
A
2
sin(ΩCt1) sin(ΩHt2) =
Ac
2
cos(ΩHt2) +
As
2
sin(ΩHt2), (5.19)
b :
A
2
sin(ΩCt1 − ΩHt2) =
A
2
sin(ΩCt1) cos(ΩHt2) −
A
2
cos(ΩCt1) sin(ΩHt2) =
As
2
cos(ΩHt2) −
Ac
2
sin(ΩHt2). (5.20)
Using terminology introduced in Section 5.8, complex Fourier transform of the acquired signal is
−→
1
2
(AcX2 + AsY2) +
i
2
(AsX2 − AcY2) −→
1
2
(X1X2 + Y1Y2) +
i
2
(Y1X2 − X1Y2). (5.21)
This closely resembles Eq. 5.9, showing that the heteronuclear gradient echo in Figure 5.12A provides discrimination
between positive and negative ΩC without recording each increment twice, but with the undesired shape of the peaks
due to the Y1Y2 contribution, as discussed in Section 5.8. The remedy is to repeat the acquisition with the usual 90◦
phase shift of the 90◦ 13
C (or 15
N) pulse (red ”y” in Figure 5.12A) and with reversed direction of the second gradient
(shown in red in Figure 5.12A). The signal modulation in the repeated experiment is
a : Asin(ΩCt1 + ϕ1) cos(ΩHt2 + ϕ2) =
A
2
sin(ΩCt1 + ΩHt2 + ϕ1 + ϕ2) +
A
2
sin(ΩCt1 − ΩHt2 + ϕ1 − ϕ2), (5.22)
b : Asin(ΩCt1 + ϕ1) sin(ΩHt2 + ϕ2) = −
A
2
cos(ΩCt1 + ΩHt2 + ϕ1 + ϕ2) +
A
2
cos(ΩCt1 − ΩHt2 + ϕ1 − ϕ2). (5.23)
Strengths of the green and red gradients are now set to be in the ratio γH : −γC, so ϕ1 = −ϕ2. The cosine function
with ϕ1 − ϕ2 = 2ϕ1 averages to zero, and (returning to the products of goniometric functions)
a :
A
2
sin(ΩCt1 + ΩHt2) =
A
2
sin(ΩCt1) cos(ΩHt2) +
A
2
cos(ΩCt1) sin(ΩHt2) =
As
2
cos(ΩHt2) +
Ac
2
sin(ΩHt2), (5.24)
b : −
A
2
cos(ΩCt1+ΩHt2) = −
A
2
cos(ΩCt1) cos(ΩHt2)+
A
2
sin(ΩCt1) sin(ΩHt2) = −
Ac
2
cos(ΩHt2)+
As
2
sin(ΩHt2). (5.25)
If we store the data with the y receiver phase (b, −a), the result of both runs can be summarized as
x/y receiver acquired as stored as records rj
+x +x a : +Ac
2 cos(ΩHt2) + As
2 sin(ΩHt2) r1, r2 =
b : +As
2 cos(ΩHt2) − Ac
2 sin(ΩHt2) +Ac
2 cos(ΩHt2) + As
2 sin(ΩHt2), +As
2 cos(ΩHt2) − Ac
2 sin(ΩHt2)
+y +y a : +As
2 cos(ΩHt2) + Ac
2 sin(ΩHt2) r3, r4 =
b : −Ac
2 cos(ΩHt2) + As
2 sin(ΩHt2) −Ac
2 cos(ΩHt2) + As
2 sin(ΩHt2), −As
2 cos(ΩHt2) − Ac
2 sin(ΩHt2)
The records r1, r2, r3, and r4 are then combined to obtain the desired hypercomplex set:
r1 − r3 = Ac cos(ΩHt2)
−r2 − r4 = Ac sin(ΩHt2)
r2 − r4 = As cos(ΩHt2)
r1 + r3 = As sin(ΩHt2)



(Ac + iAs)(cos(ΩHt2) + i sin(ΩHt2)). (5.26)
The hypercomplex Fourier transformation is performed exactly as described by Eq. 5.10. The experiment presented
in Figure 5.12A thus may seem equivalent to the States-Haberkorn-Ruben method introduced in Section 5.8. However,
the States-Haberkorn-Ruben method is more sensitive. Although the hypercomplex data obtained by both methods are
identical, including signal intensity, there is an important difference between them. When the States-Haberkorn-Ruben
approach is applied, each record (r1, r2, r3, and r4), acquired with some noise σ, directly provides one component of the
hypercomplex data. However, the experiment presented in Figure 5.12A provides such components as combinations
(sums or differences) of two records with independent noise. Adding or subtracting data data with comparable but
independent random noise increases the noise level by a factor of
√
2. Therefore, the signal-to-noise ratio of the data
obtained as described in Figure 5.12A is lower by a factor of
√
2 than the data obtained as described in Figure 5.9.
68
5.15 Preservation of equivalent pathways∗
In order to use gradients with the same theoretical sensitivity (neglecting relaxation) as provided by the StatesHaberkorn-Ruben
method, a trick known as preservation of equivalent pathways (PEP) can be employed. It is achieved
by inserting one more INEPT module followed by 90◦ 1
H pulse before detection (Figure 5.12B). This trick works only
for isolated spin pairs such as 1
H-13
C or 1
H-15
N (e.g. not for CH2 or NH2 groups). As discussed in Section 4.2, the
second pair of 90◦ 1
H and 13
C (or 15
N) pulses acts differently on y and x components of the 13
C (or 15
N) anti-phase
coherences. The PEP method tracks both components and eventually converts them to measurable signal.
• The y component is converted to 1
H anti-phase coherence (time point ”h” in Figure 5.12), which evolves to
the ”visible” 1
H in-phase coherence (x polarization) at point ”i”. In Figure 5.12A, signal detection starts at
this moment. In Figure 5.12B, the proton x polarization is rotated to the −z direction, where it survives the
additional INEPT step, and is finally returned to the xy plane by the last 90◦ 1
H x pulse. The experiment
presented in Figure 5.12B is repeated with the direction of the last (red) gradient and phase of the second 13
C
(or 15
N) 90◦
pulse inverted. This results in alternation of the proton polarization between −y and +y direction
at the beginning of detection (point ”l”).
• The x component is converted to ”invisible” multiple-quantum coherence (the longitudinal polarization of proton
magnetic moments correlated with the 13
C (or 15
N) transverse polarization is flipped to the xy). The multiplequantum
coherence is a special correlation between 1
H and 13
C (or 15
N) transverse polarizations, rotating with
the sum and difference of their frequency offsets. The simultaneous echo refocuses both rotations and returns
the multiple-quantum coherence to its original state. In Figure 5.12A, it is never recovered. In Figure 5.12B,
the pair of 90◦ 1
H and 13
C (or 15
N) y pulses converts the multiple-quantum coherence to proton anti-phase
coherence. The 13
C (or 15
N) pulse must have the y phase so that it affects the component which was untouched
by the previous 90◦
pulse. The proton 90◦
pulse must also have the y phase to keep the proton magnetization
in the xy plane. The created proton anti-phase coherence evolves to the in-phase coherence (x polarization) at
the point ”i”. This pathway of polarization transfer is insensitive to the phase alteration of the second 13
C (or
15
N) 90◦
pulse, which does not affect the x component of the 13
C (or 15
N) in-phase coherence.
In summary, the PEP method produces signal composed of two components. The first component results in a
signal produced by proton magnetization starting its rotation from −y and y direction depending on the phase of the
second 13
C (or 15
N) 90◦
pulse. The second component results in a signal produced by proton magnetization starting
always from direction x. Schematically, the result (with the second component, recovered by PEP, shown in green)
can be summarized as
−x/x receiver acquired as stored as records rj
−x +x a : +As cos(ΩHt2) + Ac sin(ΩHt2) r1, r2 =
b : −Ac cos(ΩHt2)+As sin(ΩHt2) +As cos(ΩHt2) + Ac sin(ΩHt2), −Ac cos(ΩHt2)+As sin(ΩHt2)
+x +x a : +As cos(ΩHt2) − Ac sin(ΩHt2) r3, r4 =
b : +Ac cos(ΩHt2)+As sin(ΩHt2) +As cos(ΩHt2) − Ac sin(ΩHt2), +Ac cos(ΩHt2)+As sin(ΩHt2)
If we subtract the signals detected with the ”blue” and ”red” setting in Figure 5.12B, we obtain the first component,
and when we add them, we obtain the second component. After shifting the phase of the first component (i.e. replacing
the order of the stored sums of records r1 + r3 and r2 + r4), we obtain a series of data with the same signal-to-noise
ratio (doubled signal and noise scaled by
√
2) as in the experiment presented in Figure 5.9:
r1 − r3 = 2Ac cos(ΩHt2)
−r2 + r4 = 2Ac sin(ΩHt2)
r2 + r4 = 2As cos(ΩHt2)
r1 + r3 = 2As sin(ΩHt2)



2(Ac + iAs)(cos(ΩHt2) + i sin(ΩHt2)). (5.27)
Lecture 6
Protein heteronuclear techniques
6.1 Proteins and NMR
We spend a lot of time talking about proteins when we discuss applications of NMR to biologically interesting macromolecules.
Proteins are well-suited for NMR studies and most features of biomolecular NMR can be explained using
proteins as a model case. What makes proteins so good for NMR? The answer is hidden in the chemical configuration
of proteins.
Proteins are composed of amino-acid residues forming a linear chain. The residues are joint by amide bonds, the
special term peptide bond is used in the context of proteins and peptides. Let us now compare the sketched chemical
configuration with the pattern of magnetic interactions among atomic nuclei in a protein.
We learned two types of interactions between nuclei in Section 2.5. The direct, through-space or dipolar, interactions
average to zero in common, i.e., isotropic, solutions and we do not need to care about them now. The J-coupling,
mediated by electrons in chemical bonds, deserves our attention. The armory of spin alchemy allows us to control
the effect of heteronuclear J-couplings easily. In the context of NMR, heteronuclear systems consist of nuclei which
greatly differ in the precession frequency. Pairs like 1
H–13
C, 1
H–15
N, 13
C–15
N are clearly heteronuclear. Moreover,
the frequencies of aliphatic, aromatic, and carbonyl 13
C are so different that pairs of these carbon nuclei behave like
heteronuclear systems in most experiments. The experiments based on the heteronuclear correlation include the HSQC
spectroscopy, introduced in Section 4.2, and its extensions, discussed bellow. Homonuclear J-couplings are harder to
control, and require other approaches than those based on the HSQC pulse sequence. Therefore, it is important to
identify networks of the homonuclear J-couplings in proteins. When we define such interaction network, it is sufficient
to consider directly bonded nuclei for most isotopes. Protons are exceptionally sensitive and J-couplings between
protons separated by up to three bonds need to be taken into account. Nuclei forming a J-coupling network are
described as a spin system (Figure 6.1).
What are the spin systems of proteins? The 15
N nuclei are isolated and well spread in the protein molecule
(Figure 6.1A). Every amino acid contains at least one nitrogen atom and no N–N bonds are present. The aliphatic 13
C
nuclei form an individual spin system for each residue, these spin systems are separated by peptide bonds (Figure 6.1B).
Distribution of carbonyl 13
C (Figure 6.1C) resembles that of 15
N. The 13
C nuclei in aromatic side-chains form isolated
spin systems (Figure 6.1D). Amide and aliphatic protons constitute an individual spin system for each residue like
aliphatic 13
C nuclei. These spin systems are separated by carbonyl (Figure 6.1E). Spin systems of aromatic protons
(cyan in Figure 6.1D) are separated from the aliphatic spin systems by γ-carbon.
The following features of the protein spin systems are the keys to the success of protein NMR spectroscopy:
• Spin systems of individual amino-acid residues in the polypeptide chain are separated. This allows us to analyze
the individual residues separately, which greatly simplifies interpretation and facilitates the side-chain
assignment, described in Section 8.2.
• Spin systems of individual amino-acid residues in the polypeptide chain are connected by one-bond heteronuclear
J-couplings. Using spin echoes, we can explore the heteronuclear J-couplings selectively. For example, we can
distinguish J-coupling of 15
N to carbonyl 13
C from the J-coupling between 15
N and aliphatic 13
C. This greatly
facilitates the sequential assignment, described in Section 6.5.
69
70
C
H
H
C
C
C
C
C
C
H
H
H
H C
H
H
C
H
H
C
H
H
H
C C
C
C C
C
HH
H H
H
O
O
O
C
C
C
H
H
HH
H
H
H
N
NH
H
C
NH
H
H
H C
H
H
C
H
H
C
H
H
H
C C
C
C C
C
HH
H H
H
O
O
O
C
C
C
H
H
HH
H
H
H
N
NH
H
C
C
N
C
H
H
H
H
H
H
H
H
H
H
H
C C
C
C C
C
HH
H H
H
O
O
O
H
H
HH
H
H
H
N
NH
H
C
C
C
C
N
C
C
H
H
H
H C
H
H
C
H
H
C
H
H
H
HH
H H
H
O
O
O
C
C
C
H
H
HH
H
H
H
N
N
C
C
C
C
N
C
C
C
C
C C C
C
C C
C
O
O
O
C
C
C
N
N
N
C
C C C
C
C
CC
C C C
C C
C
CC
H
H
H
H
H H
H
HH
H
H
H
H
H H
H
H
H
HH
HH
H H
H
A B C D E
C
C
C
Figure 6.1: Spin systems in proteins. Isolated spin systems of the following nuclei in three consecutive residues are distinguished by
different colors: 15N (A), aliphatic 13C (B), carbonyl 13C (C), aromatic 13C (D), and 1H.
In the following sections, we explore the protein NMR experiments, starting from the simple ones and continuing
with pulse sequences of increasing complexity.
6.2 One-dimensional proton spectra
The simple one-dimensional experiment that does not include more than excitation of protons and acquisition of their
signal is the basic tool of NMR of small molecules. One-dimensional proton spectra are also acquired when studying
proteins. However, the 1D protein spectroscopy relies on somewhat more sophisticated pulse sequences than the
mentioned single-pulse experiment. The major reason is trivial: the biomolecular proton-detection experiments must
address one important issue, presence of water. Concentration of water protons in pure water is 110 M, five order of
magnitude higher than concentration of protons in the studied proteins. The strong signal of water protons would
obscure most of other signals in the spectrum without effective suppression. There are several ways to removing water
signal from spectra:
• Removing oscillations of water protons from already recorded spectra. If the carrier frequency is chosen to match
the water resonance frequency, the signal of water protons does not oscillate or oscillates with very low frequency
offset of several Hertz – compared to kilohertz frequency offsets of other protons. The slow oscillation can be
removed by filtering the recorded and digitized data. In practice, a low-pass filter is applied to obtain a pure
signal of water, and it is subtracted from the overall signal.
• Presaturation is a method of disqualifying water protons prior to excitation (Figure 6.2A). We irradiate the
sample for a long time (approximately one second) with a weak radio-frequency field before we apply the
90◦
excitation pulse. The weak radiation selectively affects protons resonating at the carrier frequency, i.e.,
protons of water. The irradiated protons experience two opposite effects. On one hand, relaxation destroys
coherence and drives the magnetic moments to the equilibrium distribution, polarized along z. On the other
hand, the irradiating field continuously rotates the polarization from the z direction and thus restores coherence.
Finally, the distribution of magnetic moments reaches a steady state with polarization depending on the ratios
of ω1 to the relaxation rates. If ω1 is much greater than the relaxation rates, the steady state distribution is
not polarized in any direction. Rotation of such distribution by the 90◦
excitation pulse does not create any
transverse polarization. Therefore, signals of water protons do not appear in spectra. The major advantage of
presaturation is its simplicity and the fact that is applied before excitation. As the acquisition starts immediately
after the excitation 90◦
pulse, the signal intensity is proportional to the number of magnetic moments, i.e. to
the protein concentration.
6.3. 1
H-15
N TWO-DIMENSIONAL CORRELATION (HSQC) 71
A
1
t
H
B
Gz
t
H1
−x−x
Figure 6.2: Presaturation (A) and WATERGATE (B) water suppression techniques. The radio waves selectively affecting protons of
water are shown in cyan (a long very weak irradiation in panel A and a pair of selective 90◦ pulses with phase −x in panel B). Cleaning
gradients are shown in magenta.
• WATERGATE is an example of a more elaborate approach. The excitation pulse is followed by a refocusing
echo with identical gradients in both delays (Figure 6.2B). Such sequence works as a gradient echo for all nuclei.
What makes it a water suppression tool is a couple of selective 90◦
pulses surrounding the refocusing 180 ◦
pulse
in the middle of the echo. If the 90◦
pulses are applied with a phase opposite to the phase of the 180 ◦
pulse, the
effect of the refocusing echo is canceled. If the carrier frequency matches the water resonance frequency, protons
of water feel two gradients, both of which destroy the water signal. All nuclei not affected by the selective 90◦
pulses experience a gradient echo and their signal get safely refocused.
We know how to get proton one-dimensional spectra, but what can we learn from them? On one hand, such simple
experiments are very sensitive and very quick to run. We obtain a spectrum within several seconds or minutes. No
isotope labeling is needed, so we can study proteins of any origin. On the other hand, we do not learn too much from
the spectra. Thousands of relatively broad spectral lines heavily overlap, only a few protons are resolved. An example
of such spectrum is presented in Figure 6.3. The peaks between −1 ppm and 5 ppm are signals of aliphatic protons,
those with the lowest chemical shift values correspond to the methyl protons that are most shielded, some of them are
well resolved. Peaks between 6 and 11 ppm are mostly signals of amide and aromatic protons.
In spite of the limited resolution, the one-dimensional spectra provide useful information: they serve as a quick
check of quality and quantity of the protein sample (the first application in Section 1.4). An important signature
of the 1D spectrum in Figure 6.3A is a broad distribution of the amide peaks covering the whole range of chemical
shifts between 10 ppm and the aromatic region (approximately 6–7.5 ppm). The peaks of methyl groups also exhibit
a broad distribution starting with negative chemical shifts. The large dispersion of signals tells us that the protein
has a unique three-dimensional structure (probably is correctly folded). A comparison of chemical shift distribution
of methyl groups in well-ordered and disordered (unfolded by thermal denaturation) protein is shown in Figure 6.3B.
As the spectrum shows peaks of all protons that are not exchanged rapidly with water, all impurities are visible.
For example, the spectrum in Figure 6.3A revealed traces of glycerol. The 1D spectrum is also ideal for checking
protein quantity because the area under the peaks is proportional to the concentration.1
This does not apply to
more advanced pulse sequences, where the magnetization has time to relax between excitation and detection. Onedimensional
spectra are also useful when quick changes of protein structure are monitored (denaturation or folding
studies), as documented in Figure 6.3B.
6.3 1
H-15
N two-dimensional correlation (HSQC)
The two-dimensional HSQC experiment correlating 1
H and 15
N nuclei (Figure 6.4) is the basic element of the most
popular strategy of protein investigation by NMR. The experiment is relatively sensitive (typical time requirements
range between 15 minutes and two hours). Suppression of water signal is important. It is achieved by combination of
the tricks introduced in Section 6.2 with phase cycling (Section 5.10) and pulsed-field gradients (Section 4.3).
As the amide protons slowly dissociate and exchange with the protons of water, it is useful to keep their magnetic
moments in the equilibrium polarization during measurement. It is because sensitivity of the experiments is
1Strictly speaking, this is true only if water suppression is not based on additional radio wave pulses applied after excitation. As
mentioned above, presaturation should be used if the spectra are used to determine protein concentration.
72
A
10
10
5
5
0
0
δ(1H) / ppm
δ(15N)/ppm
B
-5.0e+05
0.0e+00
5.0e+05
1.0e+06
1.5e+06
2.0e+06
2.5e+06
3.0e+06
3.5e+06
0.20.40.60.81.01.2
Intensity
δ(1
H) / ppm
36.9 °C
42.4 °C
47.9 °C
53.4 °C
58.9 °C
64.5 °C
70.0 °C
75.6 °C
81.1 °C
86.7 °C
92.3 °C
97.9 °C
103.5 °C
109.1 °C
114.7 °C
-5.0e+05
0.0e+00
5.0e+05
1.0e+06
1.5e+06
2.0e+06
2.5e+06
3.0e+06
3.5e+06
0.20.40.60.81.01.2
Intensity
δ(1
H) / ppm
36.9 °C
42.4 °C
47.9 °C
53.4 °C
58.9 °C
64.5 °C
70.0 °C
75.6 °C
81.1 °C
86.7 °C
92.3 °C
97.9 °C
103.5 °C
109.1 °C
114.7 °C
-5.0e+05
0.0e+00
5.0e+05
1.0e+06
1.5e+06
2.0e+06
2.5e+06
3.0e+06
3.5e+06
0.20.40.60.81.01.2
Intensity
36.9 °C
42.4 °C
47.9 °C
53.4 °C
58.9 °C
64.5 °C
70.0 °C
75.6 °C
81.1 °C
86.7 °C
92.3 °C
97.9 °C
103.5 °C
109.1 °C
114.7 °C
Figure 6.3: A, comparison of proton 1D (top) and 1H-15N 2D HSQC (bottom) spectra of the same protein (mouse Major Urinary Protein
I consisting of 162 amino acids). The large peak in the 1D spectrum close to 5 ppm is incompletely suppressed signal of water, the sharp
peaks between 3 and 4 ppm are signals of an impurity (glycerol not completely removed during sample preparation). B, methyl region of
proton 1D spectra recorded during heating (left) and subsequent cooling (right) of barley Nonspecific Lipid Transfer Protein 1b. Native
protein (low temperatures in the left panel) has a wider distribution of chemical shifts and broader peaks than protein unfolded by thermal
denaturation (high temperatures). The right panel shows that the denaturation is irreversible as the peaks remain sharp and the chemical
shift distribution narrow after cooling the denatured sample.
6.3. 1
H-15
N TWO-DIMENSIONAL CORRELATION (HSQC) 73
A
t
1
H
4J
1
4J
1
z
2y
G
GARP
−x −x
t1
x y/
1
4J
1
4J
N15
ihb c fd ga e
B
1
H
4J
1
4J
1 −x
2t
4J
1
4J
1
4J
1
4J
1y
Gz
/x−x
y
y
=x,x,y,y,−x,−x,−y,−y
φ1
φ1
=x,−x,x,−x,x,−x,x,−x and receiver phase x,−x,−x,x,x,−x,−x,x for odd increments of
=−x,x,−x,x,−x,x,−x,x and receiver phase −x,x,x,−x,−x,x,x,−x for even increments of
GARP
2φ
2φ
t1
t1
1φ
t1
N15
j k lb c eda f g ih
Figure 6.4: 1H-15N versions of HSQC experiments presented in Figure 5.12 with water suppression and water ”flip-back” pulses. The
cyan symbols represents weak 90◦ pulses of radio waves selectively affecting magnetic moments of water. The cyan pulses applied after
time point d and before time point h are ”flip-back” pulses controlling magnetic moments of water so that they are in their equilibrium
polarization at the beginning of t2. The cyan pulses applied after time points h and i are parts of the WATERGATE module, together
with the gradients shown in magenta (cf. Figure 6.2B). Other symbols are used as explained in Figure 5.12
74
proportional to Mz prior to the first pulse. Therefore, amide protons, including those coming from water, should
have their magnetic moments polarized along z at the beginning of the pulse sequence. The pulse sequences presented
in Figure 6.4 include so-called ”flip-back” pulses that control the polarization of water at the beginning of the
measurement.
Another effect related to the water signal is the radiation damping. The concentration of water protons in protein
samples is so high that its magnetization, if flipped to the xy plane, induces a large current in the receiver coil. This
current, in turn, induces an oscillating magnetic field that cannot be neglected. As the frequency of the induced
oscillating current is given by the precession frequency of magnetic moments of protons in water, the resulting field
resonates with the precession, and rotates the water magnetization toward the z direction. In general, radiation
damping may be useful as it re-established the equilibrium polarization of water. However, if the ”flib-back” pulses
are applied, radiation damping interferes with their effect. Therefore, we suppress radiation damping by applying
pulsed-filed gradients immediately before and after 90◦
pulses and also during t1 in the pulse sequences presented in
Figure 6.4 (note that the weak gradients applied during t1 form a gradient echo for the 15
N anti-phase coherence, but
not for the proton transverse polarization).
Resolution in the two-dimensional 1
H-15
N HSQC spectra is dramatically improved in comparison to the onedimensional
spectra. Not only the signals are spread into two dimensions, but the number of peaks is greatly reduced
(see Figure 6.3). Instead of thousands proton peaks of the one-dimensional spectra, only protons directly attached to
nitrogen are visible. Since most residues have just one N–H group, namely the backbone amide, the 1
H-15
N correlation
is a wonderful fingerprint of the protein. Dispersion of proton chemical shifts in the 1
H-15
N two-dimensional spectrum
is a much better indicator of a regular fold of the protein than the 1D 1
H spectrum. Well-folded protein usually give a
resolution that allows us to count the peaks. If the number of peaks corresponds to what we expect from the sequence,
it is likely that the protein is complete and present in a single form. In order to predict the number of expected peaks,
the following simple rules can be applied.
• Expect one peak for each residue with the exception of prolines and the N-terminal amino acid.2
• Add one peak for each tryptophan and arginine.
• Add two peaks for each asparagine and glutamine.
It is impossible to assign peaks to individual amino acids in the sequence just from the 1
H-15
N HSQC spectrum.
Still, rough estimates of number of amino acids of a certain type can be made. Side-chain amides of asparagines
and glutamines appear as pairs of peaks of the same 15
N frequency in the ”upper right corner” of the spectrum.3
Tryptophan indole protons (i.e., side-chain, not backbone) can be often identified as their signal are found in the
”lower left corner” of the spectrum, separated from most backbone peaks. The only observable signal of the arginine
side chain has a 15
N frequency outside the typical range of backbone 15
N. The spectral width usually covers only
the backbone frequencies. It does not mean that the signals outside the spectral width are invisible. It is a special
feature of working with digital data that signals outside the covered frequency range are folded into the spectrum.
The apparent position of the folded peak depends on the chosen spectral width. The arginine side-chain peaks can be
distinguished by slightly changing the spectral width – they ”travel” along the 15
N axis while the unfolded peaks sit
at their place.4
6.4 Triple resonance experiments
The tricks of transferring polarization and incrementing the length of a chosen delay can be applied twice in a pulse
sequence. If the polarization is transferred via INEPT first from 1
H to 15
N and then (via another INEPT) to aliphatic
13
C (in position α) and modulation with 15
N and 13
C frequencies is introduced by appropriate delay incrementation,
2N–H group in the middle of backbone is a secondary amide but the N-terminal NH+
3 is an amino group. Amino-protons readily
dissociate and are exchanged with water protons before the spectrum is acquired. On the other hand, exchange half-times of amide protons
vary from seconds to weeks. Therefore, amide protons appear in the spectrum but amino-protons do not.
3It is a tradition to plot NMR spectra with the chemical shift increasing from right to left and from top to bottom.
4Backbone peaks close to the edge can be also folded if the spectral width is too narrow. It is easy to distinguish folded backbone peaks
of frequencies just outside the spectral with from the arginine side chain resonating approximately one spectral width away.
6.5. SEQUENTIAL ASSIGNMENT 75
N15
C13
CO13
1
4J
t1
2
t2
1
4JNC
1
4JNC
1
4JNC
t3
1
4JHN
1
4JHN
1
4JHN
t1
2
1
4JNC
H1
GARP
y
y
α
HN
y
Figure 6.5: Basic idea of an HNCA triple resonance experiment. The pulse sequence can be viewed as and extension of the 1H-15N
HSQC experiment consisting of two INEPT modules (black and green) and the t1 delay introducing amplitude modulation by the 15N
frequency. Another pair of INEPT modules (blue and cyan) separated by another incremented delay t2 (magenta) is inserted in the middle
of the HSQC pulse sequence. The blue INEPT step is actually combined with t1 to shorten the pulse sequence and reduce signal loss due
to relaxation. This is a typical example of the constant-time evolution, introduced in Section 5.3. The blue and cyan INEPT modules
transfer polarization from 15N further to 13Cα, creating a more complex 13C anti-phase coherence that evolves during t2 with the 13Cα
frequency offset. Therefore, the signal gets amplitude-modulated by the 13Cα chemical shift during t2. Interactions with three neighbors
of 13Cα (1H, 15N, and carbonyl 13C) are decoupled by the magenta 180◦ pulses, decoupling of the interaction with 13Cβ is not applied in
the presented pulse sequence (the J-couplings therefore evolves during t2 which somewhat limits the resolution).
the 1
H-15
N-13
C correlation is measured. The described scenario is just an example of a triple resonance experiment,
correlating nuclei of three different elements. The triple resonance experiments are three-dimensional, which allows
further improvement of resolution. This improvement is not achieved for free. Triple resonance experiments require
15
N and 13
C isotope labeling of the sample. Time needed to measure a three-dimensional triple resonance spectrum
typically varies between 10 hours and two days.5
The sensitivity is lower because the 15
N-13
C scalar coupling is
relatively weak and therefore requires long INEPT modules, when a significant fraction of the coherences relax. In
spite of these drawbacks, triple-resonance experiments represent the standard approach to the assignment of frequencies
in proteins consisting of 100–200 amino acids.
Although the resolution enhancement is useful, it is not the major benefit of the triple resonance experiments.
The major virtue of the triple resonance experiments is the choice of the polarization transfer direction. The N–H
groups can be correlated with their own Cα
or with Cα
of the preceding residue (using polarization transfer from 15
N
to carbonyl 13
C and then to 13
Cα
). This opens a door to the assignment of peaks to the backbone nuclei.
6.5 Sequential assignment
This section is dedicated to the first step of any detailed study of proteins using NMR, to the assignment of frequencies
observed in spectra to nuclei in the molecule. Our goal is to make a long list of nuclei with their frequencies. We
will need the list in future, when we run NMR experiments focused on structure, motions, or other properties of the
studied protein. We will use the list to identify atoms observed in the spectra as peaks at certain frequencies in the
same way as we use a phone directory when we need to specify the person we want to call by a telephone number.
The first step in sorting observed frequencies and relating them to the known chemical configuration is the sequential
assignment. We need to learn which of the observed resonance frequencies correspond to particular atoms in all residues
in the amino-acid sequence. The most popular sequential assignment approach is based on the triple resonance
experiments, introduced in Section 6.4. First we describe the general idea of the sequential assignment using an
example of the simplest triple resonance experiments applied to a short peptide. Then we look at the standard
experiments and at the problem of identifying sequential fragments in real-life samples.
5If sufficient concentrations of the protein are available, advanced techniques of data processing allow us to work with incomplete sets
of increments and the measurement time can be reduced to values typical for 2D spectra.
76
1 331 3 1 33 2 1 33 2 2 1 33 2 21 1 33 2 214 1 33 2 2144
H
2
1 3
4
1
gfedca b
C=O
CH−R’
N
C=O
CH−R
NH
Figure 6.6: Principle of sequential assignment based on the combination of HNCA and HN(CO)CA. Details are discussed in the text.
The basic idea of the sequential assignment utilizing triple resonance experiments is summarized in Figure 6.6.
In addition to the already discussed 1
H-15
N HSQC spectrum, we perform two 3D experiments. The arrows in the
protein backbone fragment indicate three correlations. The black arrow corresponds to the backbone amide 1
H–15
N
correlation, observed in the 1
H-15
N HSQC spectrum. The blue arrows mark the correlation observed in a triple
resonance experiment employing polarization transfers from backbone amide 1
H to 15
N, than to 13
C of α-carbon,
and back. It is a convention to name the triple resonance experiments based on the polarization transfer pathway.
According to this convention, the blue experiment is described as HNCA (H–N–Cα
). The red arrow shows the
correlation obtained in another triple resonance experiment, in HN(CO)CA.6
The acronym indicates that polarization
is transferred from backbone amide 1
H to 15
N, then to carbonyl 13
C, using a 13
C-13
C coupling to the α-carbon of
the preceding amino acid, and back. The parentheses tell us that the polarization was passed to carbonyl 13
C but its
frequency was not measured (i.e., frequency modulation with carbonyl 13
C frequency was not introduced).
The obtained spectra are shown in the upper half of the figure. The studied molecule was a penta-peptide, showing
four peaks in the 1
H-15
N HSQC spectrum. Each peak corresponds to one backbone amide. We do not know which
peak corresponds to which residue, so we give them some arbitrary numbers. The cubes in Figure 6.6 represent
three-dimensional spectra obtained by the triple resonance experiment. The peaks in the HNCA spectrum are blue
while the peaks in the HN(CO)CA spectrum are red. If we look at each cube from top and project the peaks to
the base of the cube, we see a pattern identical to the black peaks in the two-dimensional 1
H-15
N HSQC spectrum.
In other words, all peaks can be found in four columns above the 1
H-15
N correlation. The position of each peak in
its column is given by the 13
C frequency of the α-carbon correlated to the particular backbone amide. The vertical
positions differ between corresponding columns of the blue and red spectra because the blue spectrum (HNCA) shows
correlation of 13
Cα
with the amide of the same amino-acid but the red HN(CO)CA spectrum shows correlation of Cα
with the amide of the following residue. Of course, for each peak in the blue spectrum, a peak with the same position
must exist somewhere in the red spectrum.
6Planarity of the peptide bond makes the two-bond J coupling between 15N and 13C of α-carbon of the preceding residue relatively
strong (∼ 7 Hz). As a consequence, peaks shown in the red color in Figure 6.6 are often observed in the HNCA experiment as well. If the
sensitivity is high enough to observe the peaks for all residues, the HNCA spectrum contains all information needed for the assignment.
However, both HNCA and HN(CO)CA experiments are usually run.
6.6. DEGENERATED SEQUENCING 77
The assignment process, shown in the bottom half of Figure 6.6, is nothing else than a search for peaks of the same
vertical position. Let us start with the blue peak sitting above the amide labeled as ”1”. We find the same vertical
position for the red peak sitting above the amide number 3. Let us cut the columns above the amide peaks labeled 1
and 3 from the cubes and place them side-by-side (step a). At this point we know that amide number 3 follows amide
number 1 in the sequence. So let us look for amide 3 in the blue spectrum (step b). It has a red peak of the same
vertical position above amide 2 (step c). We can add the blue peak above amide 2 (step d) but we cannot find any red
peak of the same vertical position. We arrived at the end of the sequence, there is no amide following the α-carbon
observed as the last blue peak. But we are not done yet. We did not begin with the first residue, we just started with
a residue arbitrarily labeled as ”number 1”. So we must search for the residue preceding amide 1. We need to add
the red peak number 1 (step e) and look for a blue peak with the same vertical position. It is the peak sitting above
amide 4 (step f). Finally we add the red peak number 4 (step g) and our sequential assignment is complete. We see
all five α-carbons (five different vertical positions corresponding to five frequencies of 13
Cα
) in the final plot and we
can read the sequence (in arbitrary numbers of the amide peaks) as 4-1-3-2.
The sequential assignment shown in Figure 6.6 worked pretty well. Problem starts if two different α-carbons have
identical frequency. The assignment is ambiguous. This is why improved experiments were designed. One of them,
known as HNCACB, extends the correlation to β-carbons. An extra INEPT is added to pass polarization from 13
Cα
to
13
Cβ
. It is not a four-dimensional experiment, the polarization transfer is set to be only 50 % efficient. The polarization
is split between 13
Cα
and 13
Cβ
before the carbon frequency is measured, and joint again before it is transferred back
to amide protons. As a result, two carbon peaks appear in each column (with the exception of glycines), one at the
13
Cα
frequency, the other one at the 13
Cβ
. A complementary correlation with the Cα
and Cβ
of the preceding residue
is achieved in the CBCA(CO)NH7
experiment. What are the advantages of employing the β-carbons?
• The probability that both 13
Cα
and 13
Cβ
have identical frequency in two residues is much lower than the
probability that two α-carbons have the same frequency.
• The range of 13
Cβ
frequencies is much broader (approximately from 15 to 75 ppm) than the range of 13
Cα
frequencies (40 to 75 ppm).
• Typical frequencies of β-carbons are different for different types of amino acids.
There are also some disadvantages of the HNCACB and CBCA(CO)NH experiments, compared to HNCA and
HN(CO)CA:
• HNCACB and CBCA(CO)NH are less sensitive. First, polarization is split between two carbons, so each peak has
only half intensity. In addition, HNCACB and CBCA(CO)NH are more complex than HNCA and HN(CO)CA
and their sensitivity is therefore even worse due to the relaxation.
• Frequency of 13
Cα
is measured more precisely in HNCA and HN(CO)CA because a narrower spectral width is
needed.
As a consequence of the discussed advantages and disadvantages, the full set of triple resonance experiments is
usually measured. The sequential assignment relies mostly on the HNCACB and CBCA(CO)NH spectra while the
HNCA and HN(CO)CA help when sensitivity or resolution of 13
Cα
is critical.
6.6 Degenerated sequencing
As mentioned in Section 6.5, the typical frequencies of β-carbons vary for different types of amino acids. Alanines can
be distinguished by 13
Cβ
frequency lower than 25 ppm. There is a group of amino acids with 13
Cβ
frequency between
7Did you expect HN(CO)CACB? The different acronym shows that a different polarization transfer pathway was used. Going from
amide proton down to β-carbon via carbonyl would be too long, the experiment would be less sensitive. CBCA(CO)NH is a ”one-way”
experiment. Hα and Hβ are excited, and polarization is transferred to the aliphatic carbons, carbonyl, nitrogen, and finally to the amide
proton. The pulse sequence is rather tricky. The HN(CO)CACB experiment is also used in some cases. For example, protons attached
to carbons are replaced by deuterons when larger proteins are studied in order to reduce the dipole-dipole interactions and thus suppress
relaxation. Therefore, the sequence cannot start by the excitation of Hα and Hβ.
78
25 and 36 ppm (glutamic acid, glutamine, histidine, tryptophan, lysine, arginine, methionine, proline, valine, and
reduced cysteine). Another group of residues have 13
Cβ
resonating between 36 and 45 ppm (aspartic acid, asparagine,
phenylalanine, tyrosine, leucine, isoleucine, and cysteine forming disulfide bridges). Threonine and serine have 13
Cβ
frequency in a range typical for 13
Cα
(55 to 70 ppm) and glycine has no β-carbon. We see that the triple resonance
experiment cannot sequence a protein unambiguously, but that a degenerated sequence, with the 20-letter amino-acid
alphabet reduced to five letters, can be obtained. A hypothetical sequence
P A C D E F G H I K L M N Q R S T V W Y
can be read as
O A O X O X G O X O X O X O O U U O O X
where U=(S or T), O=(C,E,H,K,M,P,Q,R,V, or W) and X=(D,F,I,L,N, or Y). Some residues can be further
distinguished by higher 13
Cα
frequencies (isoleucine, valine, proline) but this indication is less reliable. On the other
hand, 13
Cα
close to 45 ppm is really unique.
The partial sequencing is used routinely. It is never possible to obtain sequential connectivity for the whole protein
unambiguously. For example, proline always breaks the sequential assignment based on the described experiments,
as it has no amide protons. The real-life sequential assignment is a combination of establishing connectivity in short
fragments, identifying them based on the degenerated sequencing, and looking for bridges between them.
Lecture 7
Protein homonuclear techniques
7.1 Weak and strong J-coupling
After discussing the experiments based on selective manipulations of magnetic moments of different nuclides, we turn
our attention to NMR techniques designed to explore homonuclear systems. First we discuss experiments based on
the homonuclear J-coupling. We start by asking how the J-coupling between nuclei of the same nuclide differs from
the J-coupling between different nuclides.
If the nuclei of the same type (nuclide) differ in chemical shift (they are present in different chemical groups and
therefore influenced by different distributions of electrons in their proximity), then their mutual J-coupling has the
same effect as described in Section 3.3. The only difference is that cyan and green do not distinguish distributions
of magnetic moments of protons and 13
C (or 13
N) nuclei, but of two protons in different chemical groups. If the
differences in frequencies of J-coupled nuclei are not significantly lower than their interaction constant J, the coupling
has a more complex effect on the spectrum. To understand the difference, we should return to the explanation of the
origin of the J-coupling, presented in Section 2.8.
We have shown in Section 2.8 that the magnetic moment of the J-coupled nucleus oriented in a certain direction
makes the opposite direction of a magnetic moment of an electron at the site of the observed nucleus more favorable.
As a result, the magnetic moment of nucleus 2 directly covalently bonded to nucleus 2 induces a counterclockwise1
”secondary precession” (nutation) of the magnetic moment of nucleus 1 about the direction given by the magnetic
moment of nucleus 1. This is shown for the magnetic moments of nucleus 2 oriented in the z and −z direction in
Figure 2.8, but it should be true for any orientation. What is then special about the z direction?
If the precession frequencies of nuclei 1 and 2 differ, the horizontal polarization of nucleus 2 rotates in a coordinate
frame of the observed nucleus 1. Let us consider nucleus 1 resonating with the radio wave (Ω1 = 0) coupled to
nucleus 2 with a different precession frequency and therefore Ω2 ̸= 0. If |Ω2| ≫ 2πJ, the vector describing polarization
(magnetization) of nucleus 2 makes many cycles of rotation about the z axis before conversion of the in-phase coherence
of nucleus 1 to the anti-phase coherence starts to be significant. Therefore, the effect of the horizontal polarization
of nucleus 1 on nucleus 2 fluctuates rapidly and is quickly averaged to zero. The oscillations between ”visible” and
”invisible” coherences of the observed nucleus are influenced only by the vertical (longitudinal) polarization of the
neighbor. This is called weak coupling in the NMR literature.
If the precession frequencies of nuclei 1 and 2 are similar, the magnetization vector of nucleus 2 rotates slowly and
the anti-phase coherence of nucleus 1 evolves significantly already during the first cycle of rotation of the magnetization
vector of nucleus 2. As a consequence, the rotational averaging is not complete and the horizontal polarization of
the neighbor also influences the spectrum. The interaction between such nuclei is known as a strong coupling. A
correct interpretation of the strong coupling is very important when spectra of smaller compounds are analyzed by an
organic chemist. The strong coupling also influences spectra of proteins, but usually does not have an impact on the
interpretation.
1The sense of rotation is counterclockwise for nuclei with γ > 0 connected by a single covalent bond. The change of sign of γ reverts
the sense of rotation. In addition, the sense of rotation is opposite for a coupling between nuclei separated by two chemical bonds, but the
same for nuclei three bonds apart.
79
80
7.2 COSY∗
A spectrum showing homonuclear correlation of two protons can be obtained very easily (Figure 7.1A), by running an
experiment consisting of only two 90◦
pulses and called simply COSY (COrrelated SpectroscopY). After an excitation
90◦
pulse, the signal amplitude is modulated by the chemical shift and J-coupling evolution by incrementing the delay
t1. The second 90◦
pulse (with the ”x” phase) acts on both interacting protons, like the pair of 90◦
pulses at the end
of INEPT (Section 4.1). This pulse
1. returns the y components of polarizations (in-phase coherences) of both nuclei to the invisible z magnetizations,
2. converts the x component of anti-phase coherences to invisible multiple-quantum coherences (note that the ”solid
and dashed arrows” in the ±x directions are not affected, but the pulse rotates the polarization of the neighbour
to the xy plane, creating the multiple-quantum coherences),
3. converts the y component of anti-phase coherence of the first proton into the −y component of the anti-phase
coherence of the second proton and vice versa (exactly like the last two 90◦
pulses of INEPT),
4. leaves the x components of polarizations (in-phase coherences) untouched.
Only the last two components contribute to the signal. The signal is created by coherences of both protons
rotating in the xy with the frequency offsets (given by chemical shifts) and further oscillating between ”visible” inphase
coherences (magnetizations) and ”invisible” anti-phase coherences. Moreover, the amplitude of the signal is
modulated by the same processes during t1. The result is a spectrum with rather complex shapes of peaks. It should
not surprise us that a simple experiment yields a complex spectrum. The tricks of spin alchemy (echoes) suppress
effects of individual interactions to simplify the spectra, and we did not applied any echo in Figure 7.1A.
The complex pattern of the spectrum carries useful information, but we are not going to discuss it. We just stress
the most important feature: because the second 90◦
exchanged anti-phase coherences of the protons, the spectrum
shows coherences of protons that interact via the J coupling.
7.3 Magnetic equivalence
The extreme case of a strong coupling is a pair of nuclei with the same chemical shift (and not differing in interactions
with other nuclei). Such nuclei are called magnetically equivalent.
Evolution of the coherences of magnetically equivalent nuclei is interesting. As frequencies of both nuclei are
the same, we can work in a rotating coordinate frame that is on-resonance for both nuclei. If a 90◦
pulse creates
polarizations of in the −y direction, they do not rotate (i.e., they rotate with a zero frequency offset). However, the
individual magnetic moments precess at different frequencies depending on the orientation of the magnetic moment
of the neighbor nucleus because the nuclei are coupled.
In order to simplify discussion of the J-coupling between magnetically equivalent nuclei, we decompose the magnetic
moments vectors into their x, y, and z components. As mentioned in Section 7.1, the z component of the magnetic
moment of directly bonded nucleus 2 induces a counterclockwise rotation of the x and y components of the magnetic
moment of nucleus 1 about the z axis. This conclusions can be generalized to all components of both nuclei (nucleus
1 also influences nucleus 2). There are six combinations of mutually interacting perpendicular components in a pair of
nuclei. In order to analyze such a complex system, we explore rotations for a very short time dt, i.e. by a very short
angle dα = ωdt, where ω is proportional to the size of the given perpendicular component of the neighbor’s magnetic
moment. As an example, we analyze the mutual effect of the x component of the magnetic moment of nucleus 1 and z
component of the magnetic moment of nucleus 2. Rotation of a vector ⃗a oriented in the x direction by a small angle α
about z does not change the x coordinate of the vector significantly because cos(dα) ≈ 0 for very small dα. However,
the y coordinate of the vector increases from zero to |a|dα because sin(dα) ≈ dα for very small dα (if the angles are
expressed in radians). We can therefore write the evolution of the coordinates as (a0, 0, 0) → (a0, a0dα, 0), or, in terms
of the changes in time, as (dax/dt, day/dt, daz/dt) = (0, a0dα/dt, 0) = (0, a0ω, 0). In our case, a0 is equal to the x
component of the magnetic moment of nucleus 1 and ω is proportional to the z component of the magnetic moment
of nucleus 2 at the given time. We can therefore write the change as (dµ1x/dt, dµ1y/dt, dµ1z/dt) = (0, gµ1xµ2z, 0),
7.3. MAGNETIC EQUIVALENCE 81
A
t1
t2
x y/
cba d
B
y y
t
y y y yy
1
x y/ t2y yy y y
c da b
C
t mτ1
t2
y/x
fba ec d
Figure 7.1: Homonuclear correlation experiments: COSY (A), TOCSY (B), and NOESY (C). The TOCSY mixing train is depicted as a
series of 180◦ y pulses (wide cyan bars), more elaborate pulse trains are used in practice. Other symbols are used as explained in Figure 4.2.
82
where g is the proportionality constant (coupling). Nucleus 2 is also influenced by nucleus 1. A similar analysis for
a small rotation of µ2z about µ1x provides (dµ2x/dt, dµ2y/dt, dµ2z/dt) = (0, −gµ1xµ2z, 0). The expression for all
perpendicular combinations are written in the following table
Combination µ1x, µ2y µ1x, µ2z µ1y, µ2x µ1y, µ2z µ1z, µ2x µ1z, µ2y
change of ⃗µ1 0, 0, −gµ1xµ2y 0, +gµ1xµ2z, 0 0, 0, +gµ1xµ2y −gµ1yµ2x, 0, 0 0, −gµ1yµ2z, 0 +gµ1yµ2x, 0, 0
change of ⃗µ2 0, 0, +gµ1xµ2y 0, −gµ1xµ2z, 0 0, 0, −gµ1xµ2y +gµ1yµ2x, 0, 0 0, +gµ1yµ2z, 0 −gµ1yµ2x, 0, 0
change of ⃗µ1 + ⃗µ2 0, 0, 0 0, 0, 0 0, 0, 0 0, 0, 0 0, 0, 0 0, 0, 0
The last line of the table reveals a remarkable fact. Although the two magnetic moments influence each other,
the sums µ1x + µ2x, µ1y + µ2y, and µ1z + µ2z do not change (their time derivatives are equal to zero). As the bulk
magnetization ⃗M = ⃗M1 + ⃗M2 consists of magnetic moments of both nuclei, its components Mx, My, and Mz keep
their original values in the coordinate frame rotating with the resonance frequency. Our treatment is equivalent to a
more elegant analysis, describing the mutual rotation in terms of vector products of ⃗µ1 and ⃗µ2, and showing that ⃗µ1
and ⃗µ2 rotate about the axis given by ⃗µ1 + ⃗µ2.
The practical consequence is that the J-coupling does not have any impact on the spectrum. It explains why we
see just one peak (not two peaks) in a spectrum of water, despite the fact that water contains two protons strongly
interacting via the J coupling.
7.4 Isotropic mixing
The fact that the overall magnetization of magnetically equivalent nuclei does not change during evolution has a
very interesting implication. Let us imagine that we can selectively excite just one of the equivalent nuclei. This
may seem impossible, but we later describe how we can achieve something at least very similar to selectively created
polarizations in a magnetically equivalent system. If nucleus 1 is polarized in the −y direction and nucleus 2 is left
with its equilibrium distribution of magnetic moments, the original in-phase −y coherence of nucleus 1 evolves to the
anti-phase x coherence. In the case of a heteronuclear pair, analyzed in Section 3.3, we observed oscillations between
”visible” in-phase and ”invisible” anti-phase coherences. What happens in the case of a magnetically equivalent
pair? The components of total magnetization, including My component must remain constant. In order to keep the
magnetization constant, a y component of the magnetization of nucleus 2 must be generated to compensate for the
loss of the contribution of nucleus 1. To make a long story short, if one of magnetically equivalent nuclei is excited,
its polarization is also transferred to its partner.
The described polarization transfer among magnetically equivalent nuclei is not limited to the y components of
magnetization. It applies to the x and z components as well because magnetic moments are influenced by all directions
of the neighbor’s magnetic moments. Therefore this type of polarization transfer is known as isotropic mixing.
The isotropic mixing is even more interesting in larger homonuclear systems when not all nuclei are mutually
coupled. For example, 1
Hδ
and 1
Hζ
in phenylalanine interact with 1
Hϵ
via an efficient three-bond J-coupling, but
1
Hδ
interacts only negligibly with 1
Hζ
(they are separated by four bonds). If 1
Hδ
is selectively excited and coherences
evolve as in a magnetically equivalent system, the original in-phase −y coherence of 1
Hδ
evolves and the magnetization
of 1
Hϵ
is converted to the y polarization of 1
Hϵ
, which contributes to the total My. But the sum of the y components
of 1
Hδ
and 1
Hϵ
is not constant because 1
Hϵ
is also coupled to 1
Hζ
. The y component of the 1
Hϵ
polarization is further
transferred to the y polarization of 1
Hζ
. It is the total My of all three protons that does not change. The described
transfer of the y polarization to all three nuclei of the spin system has a very useful consequence: all signals of the
spin system are correlated in the spectra, not only those of directly coupled nuclei.
7.5 TOCSY
The discussion in Section 7.4 may lead to the conclusion that isotropic mixing is not very useful in correlated spectroscopy
of proteins. Either the system is magnetically equivalent but then all nuclei have identical frequency and
there is no correlation to be observed (like the three protons in methyl groups), or the system is not magnetically
equivalent and the isotropic mixing does not work. Clearly, we meet the later case in proteins. Aliphatic carbon (or
proton) frequencies are not identical, the differences are large enough to destroy the coherences in the xy plane very
quickly.
7.6. NOESY 83
The spin alchemist never gives up. The seemingly incompatible requirements of resolved frequencies and isotropic
mixing are fulfilled with the help of pulse trains similar to those used for decoupling. The only difference is that
the pulse train now affects all nuclei. The sequence of very short echoes hitting both nuclei of any pair in the spin
system acts as a sequence of the simultaneous echoes. Therefore, the chemical shift is kept refocused during the whole
pulse train but the coupling is active. This is exactly what we wanted, all nuclei have apparently identical (i.e., zero)
frequency offset, and they are coupled as in a magnetically equivalent spin system.
The described trick is combined with the incremented evolution of t1 in a pulse sequence called TOCSY (TOtally
Correlated SpectroscopY), presented in Figure 7.1B. Spin systems of nuclei with different chemical shifts evolve as
described in Section 7.1. Then, a series of short echoes is applied. Let us assume that the TOCSY pulse train is
applied with the 90◦
phases of the radio waves (”y” pulses).2
As a consequence, the pulses keep the y components of
the magnetization intact and rotate other coherences about the y axis. During the pulse train, the y polarizations of all
nuclei are mixed as described in Section 7.4. Because the trains contain many (hundreds) of pulses, the imperfections
of the pulses and stochastic molecular motions randomize the direction of the polarizations in the xz plane (an effect
similar to the loss of coherence in the xy plane during evolution in the ⃗B0 field). Therefore, we assume that only the y
coherences, ”locked” in the y direction of the rotating frame, survive the TOCSY mixing pulse train and contribute to
the signal.3
The amplitude of the signal is modulated by the frequency offsets in t1. For example, if the polarization
of phenylalanine 1
Hδ
stays in the −y direction at the end of t1 and polarizations of 1
Hϵ
and 1
Hζ
just happen to
evolve to the x direction during t1, the only polarization entering the mixing train is that of 1
Hδ
. During the train,
the polarization is transferred to 1
Hϵ
and 1
Hζ
, so three polarizations evolve during t2 with frequencies of 1
Hδ
, 1
Hϵ
,
and 1
Hζ
. However, intensities of all three signals correspond to the magnitude of the y polarization of 1
Hδ
. As
t1 is incremented, the amplitudes of the signals are modulated by frequency offsets of all three protons. Therefore,
correlations in individual spin systems show in the spectra as cross-peaks at the frequency offsets of all nuclei in both
dimensions (e.g. six peaks for three nonequivalent aromatic protons of phenylalanine).
7.6 NOESY
So far, we have discussed the correlated spectroscopy based on the J-coupling. Now we explore a correlated spectroscopy
utilizing another interaction, the direct dipole-dipole coupling introduced in Section 2.6. As discussed in
Section 2.6, the effects of the dipole-dipole coupling on precession frequencies are canceled to zero in isotropic liquids.
Nevertheless, the dipole-dipole interactions contribute to the relaxation. In this section, we explain how the relaxation
can be employed to introduce correlation in the spectra.
The basic NOESY experiment is a sequence of three pulses separated by two delays (Figure 7.1C). Each pulse
rotates the magnetization by 90 ◦
about the x axis in the coordinate frame rotating about the z axis with the precession
frequency of the observed nucleus. The first pulse brings the magnetization to a direction perpendicular to the external
static field ⃗B0, under which influence it rotates with the precession frequency of the nucleus. The second pulse continues
the rotation by another 90 ◦
. The effect of the second pulse also depends on the length of the incremented delay t1.
We assume that we observe two protons, one exactly on resonance (Ω1 = 0) and the other one with such a frequency
offset Ω2 that it rotates by 90 ◦
during each increment of t1 (in other words, the increment is π/(2Ω2)).
We first analyze evolution of two protons that are far apart and their magnetic moments do not interact. If the
first 90◦
pulse is sufficiently strong, we can assume that both nuclei are polarized in the −y direction at the end of
the pulse. In the coordinate frame rotating with the precession frequency of proton 1, polarization of proton 2 rotates
during t1 in the xy plane with the frequency offset Ω2. It is useful to decompose the magnetization vector of proton 2
( ⃗M2) into the M2x and M2y component. The M2y component evolves from its original negative value following a cosine
function (cos(Ω2t1)) as the magnetization rotates. The evolution of the M2x component is described by a sine function
(sin(Ω2t1)). Only the M2y component is affected by the second pulse, applied in the x direction (the same phase as
the first pulse). As a result, the −y magnetization is rotated further to the −z direction, the +y magnetization is
returned to the +z direction, and the ±x magnetization stays in the xy plane.
The second pulse in our sequence is followed by another delay. It is usually relatively long, let us set its duration
to 0.2 s in our example. What happens to the magnetization rotated to the −z direction? The external field ⃗B0 has
2180◦ pulses are depicted in Figure 7.1B, but more elaborate pulse trains are used in reality.
3If the other coherences are not destroyed completely, their contribution can be removed by phase cycling.
84
no net effect, so it should stay intact during the delay. But we should be careful because the magnetization in the −z
direction does not correspond to the equilibrium distribution of magnetic moments. Therefore, relaxation should be
also considered. Due to the relaxation, the magnetic moments polarized in the −z direction slowly redistribute to the
equilibrium +z polarization. A typical rate constant of this process is 1 s−1
. The return to the equilibrium distribution
during 0.2 s with the rate of 1 s−1
can be calculated as 1 − 2e−1×0.2
≈ −0.64. Details of such calculations will be
discussed later, in lectures dedicated to relaxation. Now we just accept the result, telling us that a large portion of
magnetization (64 %) stays in the −z direction.
And what about the M2x component, that stayed in the xy plane? It continues to rotate about the z axis but
relaxation reduces its size. This component corresponds to the transverse polarization. In large rigid molecules, the
transverse polarization relaxes much faster than the longitudinal one. If we assume an exponential decay with a rate
constant of 20 s−1
, we can calculate the loss of the M2x component as e−20×0.2
≈ 0.02. For the sake of simplicity, we
neglect the 2 % leftover and assume that the M2x component decayed during the delay completely.
The third pulse would then rotate the ∓z component to the ±y direction and the receiver would detect rotation of
the magnetization about the z axis with the frequency offset of the observed nucleus. Directions of the magnetization
vectors of both protons at certain time points marked in Figure 7.1C are listed below for the first four increments of
t1.
Increment proton 1 proton 2
at c at d at e at f at c at d at e at f
0 −y −z −z +y −y −z −z +y
1 −y −z −z +y +x +x 0 0
2 −y −z −z +y +y +z +z −y
3 −y −z −z +y −x −x 0 0
The magnetization of proton 1 stays in the y direction at the beginning of t2, it corresponds to the amplitude
modulation with a zero frequency offset. The magnetization of proton 2 at the beginning of t2 (at point f) oscillates
between +y and −y as expected for an amplitude modulation with the nonzero frequency offset Ω2. The amplitude
of the detected signal is modulated by the frequency offset Ω2 because the fraction of the magnetization which did
not experience the second pulse relaxed almost completely during τm. We obtain a two dimensional spectrum with
the amplitude modulation in the indirect dimension. However, there is no correlation in the spectrum. The signal of
proton 1 does not oscillate in t1 or t2, and the signal of proton 2 is modulated in t1 with a frequency equal to the
frequency of signal oscillation in t2 (the frequency offset Ω2).
In order to achieve correlation, protons 1 and 2 must be close enough in space so that their magnetic moments
can interact directly (dipolar coupling). Let us now explore what happens in such a case to the z component of the
magnetization during τm. Proton 1 is always polarized in the −z direction, which is not the equilibrium polarization.
Actually, the z component of the magnetization of proton 1 is exactly opposite to its equilibrium value: M1z(d) = −Meq
1z
Therefore, its magnetic moments are redistributed during τm to return to the equilibrium. This is the process of
relaxation, driven by local magnetic fields. If proton 2 is close in space, interactions with its magnetic moments also
contribute to the relaxation. The result of the relaxation is that the z component of the magnetization of proton 1
at the time point e in Figure 7.1C is somewhere between −Meq
1z and Meq
1z . Where exactly, it depends (also) on the
magnetic moment distribution of proton 2 because what matters is the deviation of the total magnetization ⃗M1 + ⃗M2
from the equilibrium.4
But the distribution (polarization) of proton 2 oscillates as t1 increases. Therefore, the exact
amplitude of the signal of proton 1, proportional to the actual value of M1z(e), oscillates with the frequency of proton
4The behavior of magnetic moments during the pulse sequence presented in Figure 7.1C resembles the following experiment. Imagine
the temperature of our laboratory is 20 ◦C. We put an iron block cooled to −20 ◦C on a bench. The temperature of the block starts to
increase until it reaches 20 ◦C. We repeat the experiment several times and record the temperature after ten minutes. The temperature
should be the same in all experiments. This is an analogy of the evolution of the longitudinal polarization of proton 1 during τm. Then
we repeat the experiment with another block of iron, cooled to various temperatures: −20 ◦C, 0 ◦C, 20 ◦C, 0 ◦C. Now the temperature
recorded in subsequent experiments should oscillate. This resembles the effect of the t1 incrementation on the longitudinal polarization
of proton 2. In the third series of experiments, we use two iron blocks, one cooled always to 0 ◦C, and the other one cooled to different
temperatures: −20 ◦C, 0 ◦C, 20 ◦C, 0 ◦C. We place the blocks on the bench so that they touch each other, and record the temperatures
of both blocks after ten minutes. Now the blocks exchange heat and thus influence each other’s temperature. For example in the third
experiment, when the temperatures of the first and second blocks are −20 ◦C and 20 ◦C, respectively, the first block should warm up faster
and its recorded temperature should be more positive (closer to 20 ◦C) than in the first experiment. We see that the exchange of heat
introduces variation of the recorded temperature of the first block, in a manner reflecting how we vary the initial temperature of the second
block. This is an analogy of the evolution of longitudinal polarization of protons coupled by the dipole-dipole interactions.
7.6. NOESY 85
2! As a result, we observe a cross-peak in the spectrum at the frequency Ω1 in t2 and Ω2 in t1. Proton 1 of course
influences proton 2 in the same manner. So we see another cross-peak at Ω1 in t1 and Ω2 in t2.
The described dependence of the longitudinal polarization of one nucleus on the longitudinal polarization of another
nucleus close in space is know as the nuclear Overhauser effect (NOE). Therefore, the correlated spectroscopy based
on the dipole-dipole interaction is called NOESY (NOE SpectroscopY).
Since the dipole-dipole coupling depends on the distance between the interacting magnetic moment, the exact
fraction of the z-magnetization modulated by the frequency of the neighbor depends on how far the nuclei are in
space. This a fundamentally different correlation than that observed in the COSY and TOCSY spectra. We can
summarize:
• NOESY spectrum tells us which nuclei are close in space
• COSY spectrum tells us which nuclei are connected by covalent bonds
• TOCSY spectrum tells us which nuclei belong to the same spin system
As spin systems well overlap with individual amino acids of a protein (Figure 6.1), and as certain atoms of neighbor
amino acids are close in space in all conformations (e.g., amide proton is close to α proton of the preceding residue),
a combination of TOCSY and NOESY spectra can be used to perform sequential assignment. This alternative to
the triple resonance experiments was the major approach to sequential assignment before the techniques of labeling
of proteins with 13
C and 15
N were developed. A serious drawback of the assignment based on TOCSY and NOESY
experiments is limited resolution of the 2D spectra. Therefore, proteins consisting of more than 100 amino acids are
difficult to assign using the TOCSY and NOESY spectra.
86
Lecture 8
From spectra to protein structure
8.1 Heteronuclear-edited homonuclear correlation
As mentioned in Section 7.6, the 2D spectra showing homonuclear correlation (COSY, TOCSY, NOESY) start to
be too crowded for proteins consisting of more than 100 amino acids (Figure 8.1A). Therefore, the homonuclear
experiments are combined with heternuclear correlation when applied to larger proteins. Such 3D experiments are
called heteronuclear-edited, or isotope-edited. Examples of three experiments combining TOCSY and NOESY with the
HSQC pulse sequence are shown in Figure 8.2.
The TOCSY-HSQC and NOESY-HSQC experiments (Figure 8.2A,C) are seamless combinations of pulse sequences
presented in Figures 5.9 and 7.1. Note that the z-polarizations1
are mixed in the version of the TOCSY-HSQC
experiment presented in Figure 8.2A (the mixing train is sandwiched by 90◦
pulses). The NOESY and TOCSY
sequences can be combined with HSQC also in the reversed order (starting with HSQC). In the case of 15
N-edited
versions, the advantage of the reversed order is that frequencies of all protons are detected in the direct dimension
(which offers better resolution), while only a narrow range of amide proton frequencies is measured in the indirect
dimension. However, water suppression is more difficult when starting with HSQC.
Two slices cut from an 15
N-edited NOESY-HSQC spectrum are plotted in Figure 8.1B,C. The plots document
that the 3D NOESY-HSQC experiment provides a good resolution even at a 15
N frequency corresponding to the most
crowded central region of the 1
H-15
N HSQC spectrum (Figure 8.1C).
In the experiment presented in Figure 8.2B, the TOCSY block is inserted in the middle of the HSQC sequence.
It allows us to employ isotropic mixing of 13
C nuclei connected by single bonds. Stronger coupling constants make
the polarization transfer more efficient. Note that additional simultaneous echoes need to be included before and
after the mixing train because we have to mix in-phase (not anti-phase) coherences. These INEPT steps are shorter
than 1/(4J) because two or three protons are attached to 13
C. In such cases, the in-phase and anti-phase coherence
oscillate more rapidly as functions cos(πJt) sin(πJt) and cos2
(πJt) sin(πJt), respectively. Therefore, the length of the
simultaneous echo must be shortened in order to get sufficient polarization transfer for all chemical groups. Using the
naming convention of the triple resonance experiments, the pulse sequence in Figure 8.2B is called HC(C)H-TOCSY
(the acronym HCCH-TOCSY is also used in the literature).
The TOCSY-HSQC and NOESY-HSQC experiments can be used to obtain the sequential assignment as discussed
for TOCSY and NOESY. However, the triple resonance experiments relying on much more diverse 13
C chemical shifts
provide spectra that are easier to interpret. Therefore, the TOCSY-HSQC and NOESY-HSQC spectra are usually
used as a complementary source of information.
All experiments presented in this section allow us to assign side-chain protons, as discussed in Section 8.2. HC(C)HTOCSY
is particularly useful because it provides also 13
C chemical shifts. The major application of the NOESY-HSQC
experiments is measurement of distances between atoms. In principle, the intensity of NOESY cross-peaks decreases
with the sixth power of the distance between the interacting nuclei. Therefore, peak intensities can be converted to
the interatomic distances, that serve as a very important restraint in structure calculation. More details are discussed
in Section 8.4.
1Not the y-polarizations, that were mixed in the pulse sequence presented in Figure 7.1.
87
88
A
9.0
9.0
8.5
8.5
8.0
8.0
7.5
7.5
7.0
7.0
6.5
6.5
δ2 (1H) / ppm
3.0 3.0
2.5 2.5
2.0 2.0
1.5 1.5
1.0 1.0
0.5 0.5
δ1(1H)/ppm
B
9.0
9.0
8.5
8.5
8.0
8.0
7.5
7.5
7.0
7.0
6.5
6.5
δ3 (1
H) / ppm
3.0 3.0
2.5 2.5
2.0 2.0
1.5 1.5
1.0 1.0
0.5 0.5
δ1(1H)/ppm
δ2 (15
N) : 113.721 ppm
C
9.0
9.0
8.5
8.5
8.0
8.0
7.5
7.5
7.0
7.0
6.5
6.5
δ3 (1
H) / ppm
3.0 3.0
2.5 2.5
2.0 2.0
1.5 1.5
1.0 1.0
0.5 0.5
δ1(1H)/ppm
δ2 (15
N) : 119.956 ppm
Figure 8.1: 2D NOESY spectrum (A) and two slices cut from a 3D NOESY-HSQC spectrum (B,C) of a protein consisting of 100 residues
(N-terminal domain of δ-subunit of RNA polymerase from Bacillus subtilis.
8.1. HETERONUCLEAR-EDITED HOMONUCLEAR CORRELATION 89
A
H1
N15
1
4J
1
4J
1
4J
1
4Jt1
t2
t3/ 1φ1φ 2φ
/ 3φ3φ
φ =x,x,x,x,−x,−x,−x,−x2
=y,−y,y,−y,y,−y,y,−y3φ
φ3=x,−x,x,−x,x,−x,x,−x
receiver phase: x,−x,−x,x,−x,x,x,−x
φ
φ
y
GARP
/φφ3 3
1
1
DIPSY−2rc
=x,x,−x,−x,x,x,−x,−x
=y,y,−y,−y,y,y,−y,−y
B
H1
t3
1
4J
1
4J
1
4Jt1
1
4J
C13
t2
2
/ 1φ1φ
/ 2φ2φ
CO13
t2
2
φ
φ =x,x,−x,−x,x,x,−x,−x
=y,y,−y,−y,y,y,−y,−y
2
2
GARP
y
τ τ DIPSY−2
=y,−y,y,−y,y,−y,y,−yφ
=x,−x,x,−x,x,−x,x,−xφ1
1
receiver phase: x,−x,−x,x,−x,x,x,−x
τ τ
C
H1
N15
1
4J
1
4J
1
4J
1
4Jt1
t2
t3/ 1φ1φ 2φ
/ 3φ3φ
φ =x,x,x,x,−x,−x,−x,−x2
mτ
=y,−y,y,−y,y,−y,y,−y3φ
φ3=x,−x,x,−x,x,−x,x,−x
receiver phase: x,−x,−x,x,−x,x,x,−x
φ
φ
y
GARP
/φφ3 3
1
1
=x,x,−x,−x,x,x,−x,−x
=y,y,−y,−y,y,y,−y,−y
Figure 8.2: 3D Heteronuclear-edited experiments providing homonuclear correlation: TOCSY-HSQC (A), HC(C)H-TOCSY (B), and
NOESY-HSQC (C). The isotropic mixing trains are shown as cyan boxes. The magenta pulse corrects phase distortions introduced by the
other pulse applied at the carbonyl 13C frequency. Note that the t1 and t2 incremented delays are combined with the INEPT transfer
modules in HC(C)H-TOCSY. The delay τ is set to 0.6J/(4J). Other symbols are used as explained in Figure 4.2.
90
8.2 Side-chain assignment
The triple resonance experiments, introduced in the previous lecture, allowed us to assign frequencies of peaks in spectra
to nuclei in the corresponding residues in the amino acid sequence. When the sequential assignment is completed,
it is time to think about assigning the remaining frequencies to the atoms of the side chains. As the side chains
coincide with separated spin systems (Figure 6.1), homonuclear correlation plays an important role in the side-chain
assignment. Heteronuclear correlations are also used, but only as an auxiliary tool as described in Section 8.1.
The fundamental experiment for the side-chain assignment is HC(C)H-TOCSY, described in Section 8.1 and
presented graphically in Figure 8.2B. What correlations are observed in the HC(C)H-TOCSY spectrum? All proton
frequencies, measured in two independent dimensions, are correlated because polarizations were mixed between the
delay t1 introducing modulation by proton frequency in the first indirect dimension and direct proton observation
during t3. The proton frequency offset modulating the amplitude during t1 is correlated only to the 13
C frequency
of the directly attached carbon. On the other hand, the proton frequency directly observed in t3 is correlated to 13
C
frequencies of all carbons (polarizations were mixed between the delay t2 introducing modulation by 13
C frequency in
the second indirect dimension, and acquisition of the proton signal).
The process of the side-chain assignment is summarized in Figure 8.3. The figure shows a hypothetical HC(C)HTOCSY
spectrum containing peaks of a single amino acid valine. We can look at slices of the spectrum cut at various
1
H or 13
C frequencies.
The side-chain assignment process starts with the 13
C frequencies of α-carbon and β-carbon, known from the
sequential assignment (row a in Figure 8.3). If we take a plane cut at the 13
C frequency of α-carbon (62 ppm in
our case), we see a diagonal peak (with the same proton frequencies in both dimensions of the plane) corresponding
to the α-proton, and a series of signals with the same frequency of the directly observed proton (axis F3) and with
frequencies of all other protons in the spin system in the other (indirect) dimension, labeled as F1 in the figure. The
same pattern should be found on a plane cut at the 13
Cβ
frequency (30 ppm), with the difference that the diagonal
peak corresponds to the β-proton frequency. The slices cut at 13
Cα
and 13
Cβ
give us new information, frequencies of
1
Hα
and 1
Hβ
assigned unequivocally (4.0 and 2.0 ppm, respectively), and list of frequencies of the remaining protons
(0.8 and 1.0 ppm). We know from the valine structure that this amino acid contains also two methyl groups with
protons labeled 1
Hγ1
and 1
Hγ2
, but we do not know which frequency belongs to which atom.
We can verify our assignment by cutting the spectrum in a perpendicular direction, at the frequency of 1
Hα
and
1
Hβ
(row b). Again, we see a series of peaks along the F3 dimension, but now at the 13
C frequency of the carbon
directly attached to the chosen proton.
Then, we can cut the spectrum at one of the frequencies of the methyl protons (at 1.0 or 0.8 ppm, see row c). In
each slice, we see a series of peaks along the F3 dimension at the 13
C frequency of a methyl carbon. The slices show
that the protons with the resonance frequency of 1.0 ppm are attached to the carbon resonating at 18 ppm, and the
protons with the resonance frequency of 0.8 ppm are attached to the carbon resonating at 22 ppm.
Finally, we verify the assignment in the slices cut at the methyl carbon frequencies (row d). Note that all protons
were assigned to the directly bonded carbons but we do not know which methyl group should be labeled as γ1 and
which is γ2 according to the standard nomenclature. We obtained a full nonstereospecific assignment.
The HC(C)H-TOCSY pulse experiment has some limitations. Spin systems of aliphatic and aromatic nuclei are
usually treated separately as the frequency difference between the aromatic and aliphatic nuclei is large for both
protons and carbons. We can use the HC(C)H-TOCSY pulse sequence to correlate carbons and protons of aromatic
rings, but the connectivity of each ring to the corresponding side chain must be obtained from another experiment.
Also, the HC(C)H-TOCSY spectra are relatively crowded. The analysis can be facilitated by combination of the
HC(C)-TOCSY part of the pulse sequence with the polarization transfer to the amide nitrogen and proton. We
obtain spectra similar to those recorded by the triple resonance experiments during the sequential assignment, but
with all carbon (or proton) peaks in the indirect dimension. Disadvantage of this attractive approach is relatively
low sensitivity. In spite of all improvements, the HC(C)H-TOCSY spectrum remains the most complete source of
information and is used when the fancy experiments fail to give unambiguous data.
Before we leave the area of assignments, we should mention nuclei that escaped our traps so far. We did not talk
about detection of backbone carbonyl atoms – it can be achieved easily by the HNCO triple resonance experiment
(identical to HNCA with a different carrier frequency). Side-chain amides of glutamine and asparagine appear in the
1
H-15
N HSQC spectra and can be correlated to the corresponding residues by the CBCA(CO)NH experiment. The
8.2. SIDE-CHAIN ASSIGNMENT 91
F2
F3
F1
F3 F3
F1
c
b
d
a
F2 = 62 ppm
4.0
4.0 2.0 1.0
0.8
F1 = 4.0 ppm
62
F1 = 1.0 ppm
18
F2
F2
F1
F2 = 18 ppm
1.0
F2 = 22 ppm
0.8
22
F1 = 0.8 ppm
F1 = 2.0 ppm
30
F2 = 30 ppm
2.0
Figure 8.3: Principle of side-chain assignment using the HC(C)H-TOCSY experiment.
92
Figure 8.4: Distance between protons in two residues that are close in space but far in the sequence represents a structural restraint
contributing to the description of the global protein fold.
side-chain NH group of arginine is also visible in the 1
H-15
N HSQC spectra and can be assigned from the HNCACB
spectra (the terminal NH2 groups of arginines are not observed as they exchange protons with water too quickly).
Experiments allowing assignment of other side-chain atoms are available, but they are used rarely. The assignment
obtained from the standard experiments is sufficient for most following studies, including structure determination.
8.3 NMR data for structure determination
When the resonance frequencies observed in the NMR spectra are assigned to individual atoms of our molecule, we can
start acquisition of spectra that provide parameters for structure determination. Information about three-dimensional
structure can be derived from various types of experimental data. In the following sections, we describe the process
of structure calculation based on the information derived from NOESY spectra. In sections labeled with asterisks, we
also discuss measurement of structural parameters that are employed less often.
8.4 Distances and nuclear Overhauser effect
Measurement of internuclear distances (Figure 8.4) was the first method of building three-dimensional structure of large
molecules and it remains to serve as the most important tool of NMR structure determination. We have introduced
the general idea already in Section 1.3. The distance measurements is based on evaluating the nuclear Overhauser
effect (NOE), as explained in Section 7.6. The actual pulse sequence used most frequently to internuclear obtain
distances in proteins, 13
C/15
N-edited NOESY-HSQC, was presented in Section 8.1.
When we analyze the NOESY spectra, we are interested not only in the frequencies but also in the intensities of the
cross-peaks. While the frequency identifies the interacting nuclei, the intensity measures strength of the interaction.
It is known that the interaction between magnetic dipoles quickly decreases with the their distance r (the decrease
follows the sixth power of 1/r). Based on this ”r−6
law”, only nuclei closer than approximately 0.5 nm should show
a measurable cross-peak. Also, it should be possible to measure the interatomic distances, using the e.g. following
proton pairs of known and fixed distance as a reference:
geminal protons in methylene H–C–H 0.17 nm
vicinal protons in aromatic ring H–C=C–H 0.25 nm
Hα
i and HN
i+3 protons in α-helix 0.34 nm
meta protons in aromatic ring H–C=CH–C–H 0.42 nm
Unfortunately, there are at least three reasons why the distance measurement is not as accurate as we would expect.
8.5. TORSION ANGLES AND THREE-BOND J-COUPLINGS∗
93
• The experimental error of evaluating the intensities, especially in crowded regions of spectra.
• Proton pairs are not isolated but form a network of interacting magnetic dipoles. The perturbation of magnetic
moment distribution due to direct dipole-dipole interactions with magnetic moments of the interacting partner
propagates further to the neighbors of the affected nucleus (so-called spin diffusion). The intensity of cross-peaks
is therefore not simply proportional to r−6
.
• Possible molecular motions. The observed intensity is then an average of what we would obtain for individual
frozen conformations.
The listed obstacles provide explanation why the evaluated internuclear distances alone do not define the structure
accurately. The structure determination protocol is based on combining a large set of relatively inaccurate distances
with some computational method. Among the determined distances, those between nuclei far in sequence are most
useful. Such distances are called long-range restraints.
8.5 Torsion angles and three-bond J-couplings∗
Conformation of a macromolecule is given by torsion angles. In theory, exact determination of all torsion angles
along a protein backbone would provide the global fold and addition of side-chain torsion angles would finish the
structure determination. Unfortunately, torsion angles are really local parameters, there is nothing like the longrange
NOE in the case of torsion angles. If we try to form the backbone according to the measured torsion angles,
small experimental errors quickly accumulate and the information about the overall shape of the molecule gets lost.
The structure determination strategy cannot be based on torsion angles but the torsion angles are still useful as a
complement to NOE and may help to refine the solved structure, as discussed in Section 8.7.
Torsion angles are often obtained from three-bond J-couplings. This type of interaction does not depend on the
orientation with respect to the external magnetic field, it reflects just the distribution of electrons forming chemical
bonds. One-bond J-coupling is mostly given by the bond length, two-bond J-coupling is also affected by the bond
angle. Bond lengths and angles are rather uniform and one- and two-bond J-couplings are not very interesting.2
Three-bond J-couplings depend also on the torsion angles. The dependence reflects the periodic nature of torsion
angles, with maxima corresponding to the planar arrangement of the three inspected bonds. Arithmetic and graphical
expressions of the relation between torsion angles and three-bond J-couplings are known as Karplus equation and
Karplus curve, respectively (Figure 8.5).
Measuring the J-couplings may look like a simple task: just do not use decoupling and measure distances between
individual peaks of the signal split to a doublet. Unfortunately, three-bond couplings are relatively small and cannot
be resolved in such a simple experiment. There are two general strategies of obtaining couplings smaller than the
peak-width.
The first strategy utilizes evaluation of the signal intensity rather than (unresolved) frequency. A simple example
called the spin echo difference experiment reveals the essence of the approach (Figure 8.6). We run two experiments.
The first pulse sequence contains a simultaneous echo similar to that used in INEPT. Magnetization evolves into
the anti-phase coherence during the echo and the measurable signal decreases. The other experiment represents a
control. The pulse sequence contains a refocusing echo, during which no signal is lost due to the J-coupling. Value of
the coupling constant can be easily calculated from the ratio of signals obtained in the two experiments because the
magnetization evolves under the influence of the coupling as cos(2πJτ), where 2τ is the length of the echo and J is
the measured coupling constant.
In order to obtain resolution required for proteins, a 3D experiment called HNHA is applied instead of the discussed
pair of 1D experiments. Correlation of the detected amide proton with 15
N is used to improve resolution by modulating
the signal with the 15
N frequency. The 1
H-15
N correlation is achieved using another way of transferring polarization
than INEPT, and the studied system, consisting of three nuclei, is more complex than we are used to. It is not our
goal to explain technical details of the polarization transfer in the HNHA experiment. Instead, we discuss what can
be understood by analyzing individual lines of the pulse sequence, and what is also directly related to the purpose of
the measurement.
2The one- and two-bond angles are also partially affected by the changes of electron distribution associated with torsion angle changes.
Their values are sometimes used as an auxiliary qualitative information.
94
1
φ ψ
χ
CN
CC N
O
O
0
2
4
6
8
10
12
-180 -120 -60 0 60 120 180
J-coupling/Hz
ϕ / deg.
HN(k)-Hα(k)
HN(k)-C′(k)
HN(k)-Cβ (k)
C′(k − 1)-Hα(k)
C′(k − 1)-Cβ (k)
C′(k − 1)-C′(k)
Figure 8.5: Dependence of various 3J-coupling constants on the bakcbone torsion angle ϕ.
t
Y
τ τ
X
Y
X
t
τ τ
Figure 8.6: Spin echo difference experiment. The measured J-coupling between nuclei labeled X and Y evolves during the simultaneous
echo (top). A refocusing echo (bottom) is applied to obtain a reference intensity not influenced by the measured J-coupling. In order to
perform the reference experiment at the same conditions, the 180◦ pulse applied to the coupled nucleus Y is not omitted, but moved to
the beginning of the sequence (red rectangle).
8.5. TORSION ANGLES AND THREE-BOND J-COUPLINGS∗
95
H1
N15
t3
1
4JHN
t2
1
4JHN 2
τ 1
8J
t11
8JNH 42
τ t11
8JNH 42
τ t11
8JNH 42
τ
2
τ
2
τ
2
τ
2
τ
t1
4
t1 t11
4J
t1
2
1
4J
t1
2
y
NH GARP
τ τ
2 2NH NH
Figure 8.7: HNHA experiment. The measured J-coupling between amide proton and α-proton evolves during the echoes shown in green.
We start analyzing the line labeled 15
N. We note that the blue blocks of the pulse sequence combine polarization
transfer with the incremented delay t1 in the manner of constant time evolution. This trick is described in Section 5.3
and also implemented in the pulse sequence presented in Figure 6.5. Each blue block can be decomposed into an
incremented delay t1/2 and a decremented refocusing echo, shown in red and black below the pulse sequence, respectively.
Any refocusing echo, regardless of possible incrementation or decrementation, cancels the effect of chemical
shift evolution. As a consequence, the signal is modulated by ΩNt1 (where ΩN is the frequency offset of amide 15
N).
We continue by analysing the line labeled 1
H. For the amide and alpha protons (1
HN
and 1
Hα
, respectively), the
green blocks represent two simultaneous echoes (note that all pulses affect both protons). If the delay τ/2 is equal to
1/(4J), where J is the 1
HN
-1
Hα
J-coupling constant, the echoes with the adjacent 90◦
pulses act as INEPT modules.
Apart from the correlation with 15
N magnetic moments,3
we converted the in-phase 1
HN
and 1
Hα
coherences to the
anti-phase coherences of the other proton at the beginning of t2. In such a case, we obtain signal oscillating with Ω of
1
HN
in t3 modulated by Ω of 1
Hα
in t2, and wise versa. This corresponds to cross-peaks in the 3D spectrum.
So far, we discussed the pulse sequence as if τ were set to a well known value of 1/(2J). But the purpose of the
experiment is to measure an unknown value of J, which depends on the torsion angle. And the torsion angle may
differ in different amino-acid residues. If J is smaller than 1/(2τ), a portion of the in-phase coherence is preserved and
contributes to the signal. This contribution introduces modulation by Ω of 1
HN
in t2 for the signal evolving with Ω of
1
HN
in t3 and modulation by Ω of 1
Hα
in t2 for the signal evolving with by Ω of 1
Hα
in t3. Such modulation corresponds
to the diagonal peaks in the 3D spectrum. As the polarization is transferred twice, with the sin(πJτ) dependence on
the three-bond proton-proton coupling, the intensity of the diagonal peak and cross-peak is proportional to cos2
(πJτ)
and sin2
(πJτ), respectively. Therefore, the value of J can be calculated from the ratio of the peak intensities, which
is equal to tan2
(πJτ), where τ is the known and well chosen duration of the polarization transfer.
The other strategy is abbreviated as E.COSY (Exclusively COrrelated SpectroscopY). The E.COSY experiments
include frequency measurement in the absence of decoupling, but no just by omitting the decoupling pulse train in any
pulse sequence. The pulse sequences are carefully designed so that small couplings are resolved with a help of a large,
usually one-bond, coupling. In the case of proteins, the three-bond 3
J(HN
Hα
) coupling (dependent on the backbone
torsion angle ϕ), is measured most often using a slightly modified HNCA pulse sequence called HNCA-J (Figure 8.8).
First, polarization is transferred to 13
Cα
by two INEPT blocks, exactly like in the standard HNCA experiments.
The sequence starts to differ from the standard HNCA experiment during t1. Frequency of 13
Cα
, modulating the signal
during t1, is itself modulated by the large one-bond 13
Cα
–1
Hα
J-coupling in the indirect dimension (by incrementing
the delay t1 without proton decoupling). The developed 13
C anti-phase coherence, in terms of correlation with the
magnetic moment distribution of the α-proton, can be described by the solid and dashed array introduced in Section 3.3.
The solid arrow evolves with the frequency equal to ΩC − πJ(Cα
Hα
), the dashed one with the frequency equal to
ΩC + πJ(Cα
Hα
). The following INEPT modules transfer the polarization to 15
N and finally to amide protons.4
3We can ignore that polarization is transferred to 15N when we discuss the effect of the 1HN-1Hα J-coupling, but not when we design
the pulse sequence. In particular, the phases of the 90◦ pulses must be set accordingly.
4We know that the first two INEPT modules create a 13Cα coherence that is ”twice” anti-phase in terms of correlation with the magnetic
moment distribution of amide proton and 15N, and that the last two INEPT steps restore the coherence of amide protons that is in-phase
in terms of correlation with the magnetic moment distribution of amide proton and 15N. This fact is not important for the measurement
of the J coupling constant and we ignore it for the sake of simplicity (but it must be taken into account when setting phases of the 90◦
96
N15
C13
CO13
t3
1
4JHN
1
4JHN
1
4JNC
1
2JHN
1
4JNC
1
4JHN
1
4JHN
t1
1
4JHN
1
4J
t2 1
4JNC
t2
−x
1
y
H
α
WALTZ 16
GARPNC 2 2
Figure 8.8: HNCA-J experiment. The total flip angle of the cyan pulses is 360 ◦. The green box labeled WALTZ 16 is a train of
decoupling pulses (with a similar effect as GARP), applied so that the 1HN-15N J coupling evolves only during the preceding delay with
a length corresponding to 1/(2J((HNN)).
B
C
ΩH ΩHΩH
1
J(CH)
3
CA
Ω J(CH)
1
J(CH)
Figure 8.9: Peaks in a slice cut at the 15N frequency from a conventional decoupled HNCA spectrum (A), coupled HNCA spectrum (B),
and HNCA-J spectrum (C). The small (< 10 Hz) three-bond coupling 3J(HNHα) is poorly resolved in the coupled HNCA spectrum, but
well resolved in the HNCA-J spectrum due to the large (140 Hz) 1J(CαHα) coupling.
8.6. ORIENTATION IN SPACE AND RESIDUAL DIPOLAR COUPLINGS∗
97
When the signal is acquired, the amide protons are coupled to Hα
. Therefore, an anti-phase amide proton coherence,
in terms of correlation with the magnetic moment distribution of the α-proton, evolves during t3. The pulse sequence is
designed carefully, so that it is the same distribution of the α-proton magnetic moments as during t1: the total rotation
by all proton pulses between t1 and t3 is 360◦
(shown in cyan in Figure 8.8) returning magnetic moments to their
original direction. The oscillation between in-phase and anti-phase proton coherences can be again described by a solid
and dashed arrows rotating at frequencies of ΩH −πJ(HN
Hα
) and ΩH +πJ(HN
Hα
). The point is that the evolution of
the proton solid arrow in t3 correlates with the evolution of the 13
C solid arrow in t1, and the same applies to proton
and dashed arrows (the solid arrows in t1 and t3 are the same because the 1
Hα
magnetic moment distributions are
the same, the same applies to the dashed arrows). As a result, two peaks appear in the two-dimensional spectrum
(Figure 8.9), one at ΩC −πJ(Cα
Hα
), ΩH −πJ(HN
Hα
), the other one at ΩC +πJ(Cα
Hα
), ΩH +πJ(HN
Hα
). In the direct
dimension, the peaks are displaced by the small three-bond coupling J(HN
Hα
) but this difference can be measured
accurately because the peaks are displaced in the indirect dimension by large J(Cα
Hα
).
8.6 Orientation in space and residual dipolar couplings∗
The direct interaction between two magnetic dipolar moments depends on orientation of the interacting nuclei in the
magnetic field, as discussed in Section 2.6 (see Figure 2.6). Energy of this interaction is proportional to (3 cos2
Θ − 1),
where Θ is the angle between the direction of the magnetic induction ⃗B0 and the vector connecting the nuclei. As the
molecule rotates in solution, the angle Θ varies. If the rotation is isotropic, the function (3 cos2
Θ−1) averages to zero
and no interaction can be measured. However, we can dissolve our macromolecule in a liquid crystal or place it in a
strained gel, and restrict rotations in some directions. The liquid crystals can be formed by filamentous viruses, flat
discs of phospholipid bilayers (”bicelles”), etc. The partial alignment of the molecule in the liquid crystal does not
allow complete averaging of (3 cos2
Θ−1) and some residual of the interaction can be measured as splitting in spectra.
Technically, these residual dipolar couplings cannot be distinguished from the J-couplings between the two interacting
nuclei. The value of the residual dipolar coupling can be determined by comparing two experiments, one in regular
isotropic buffer, another one in liquid crystal. Apparent coupling measured in the first, isotropic, sample (by one of
the methods introduced in the previous section) is just the J-coupling. The difference between values obtained in the
two experiments is the residual dipolar coupling.
Residual dipolar couplings are unique because they reflect average orientation of a pair of nuclei with respect to
an external coordinate frame (given by the magnetic field orienting components of the liquid crystal or direction of gel
compression). Therefore, orientations of two pairs can be compared regardless of the actual distance between these
nuclear pairs. Such type of information is very useful in the case of rod-like molecules (e.g., nucleic acid helices).
Possible bending of the molecule is clearly manifested in the relative orientation of the residues at the opposite ends
of the molecule. Such information is not available from NOESY spectra because NOE can be only measured between
nuclei close in space.
Direct geometrical interpretation of the residual dipolar couplings is possible but not straightforward. In order to
determine exact mutual orientation of distant nuclear pairs, five independent residual dipolar coupling values need
to be measured. Of course, it is not possible to measure more than one coupling for each pair of nuclei. However,
information from several nuclear pairs can be put together, if the relative orientations of these pairs is known and
fixed. For example, it is possible to determine mutual orientation of two α-helices in a protein if we treat each helix
as a rigid object of known local geometry (Figure 8.10). Most often, residual dipolar couplings are not interpreted
directly but submitted to a computational method as structural restraints.
8.7 Building a structural model
There are two possible approaches to interpretation of the NMR data. The first approach is based on human judgment.
This approach gives us full control over the interpretation but it is extremely inefficient if large amount of data need
to be analyzed and compared. The human interpretation can be successfully applied to prediction of the secondary
structure and to critical evaluation of the computer-derived models.
pulses as mentioned when discussing HNHA). We only pay attention to coherences that are anti-phase in terms of correlation with the
magnetic moment distribution of the α-proton.
98
Figure 8.10: Mutual orientation of the blue and red α-helices (green arrows) can be determined by measuring multiple residual dipolar
couplings in each helix.
We already mentioned that the human-based data interpretation is often used when the fragments of regular
secondary structure are searched for. There are several NMR parameters that can be easily interpreted in terms of
secondary structure elements:
• Rate of proton exchange with water is slowed down when the proton is involved in a hydrogen bond. Secondary
structures are stabilized by regular hydrogen-bond networks and the exchange rates can be used as rough
indicators of the presence of such network. The experiment is usually based on a quick exchange of water with
deuterium oxide and repeated measurement of quick 1
H-15
N HSQC spectra. Signals disappear as the protons
are replaced with deuterons. Easily accessible protons are replaced within minutes while the protons involved in
hydrogen bonds may survive for weeks.
• Three-bond J-couplings were discussed in Section 8.5. The J-coupling between amide and α protons can be
obtained relatively easily. Values larger than 8 Hz indicate β-sheets while values below 6 Hz are typical for
helical regions.
• Nuclear Overhauser effect is the key structural information. Some NOEs can be inspected separately as indicators
of secondary structures (Figure 8.11). In β-sheets, the distance between α protons of the k-th residue and amide
protons of (k+1)-th residue is much shorter than the distance between two amide protons in neighboring residues.
Therefore, an intense NOE between Hα
k and HN
k+1 is a strong indication of the β-sheet. On the other hand, NOEs
stronger for the HN
k –HN
k+1 pairs than for the Hα
k –HN
k+1 are typical for helical structures. Helices can be further
characterized by medium-range NOEs. Cross-peaks are usually observed for the pairs Hα
k –HN
k+3, Hα
k –HN
k+4, and
Hα
k –Hβ
k+3 in α-helices. Presence of a cross peak for the Hα
k –HN
k+2 pair but not for the Hα
k –HN
k+4 pair indicates a
310-helix.
• Chemical shift is often difficult to interpret but it works nicely in the secondary structure prediction. Chemical
shifts of carbonyl and alpha 13
C have significantly higher values in helices than in random coil, on the other
hand chemical shifts of α protons and 13
Cβ
are elevated in β-sheets. The real beauty of the chemical shifts is
that we obtain them for free – as soon as the assignment is finished, the structural information is available.
Direct interpretation of the NOE data can be also applied to determination of the β-sheet topology. The information
how the strands of β-sheets are aligned can be obtained from the cross-peaks between backbone protons. It is less
straightforward to establish the topology of helices as determination of their relative position requires analysis of
side-chain NOE cross-peaks.
8.7. BUILDING A STRUCTURAL MODEL 99
Figure 8.11: Proton-proton distances distinguishing α-helix (left) from β-sheet (right). The distance between Hα
k and HN
k+1 is shown in
magenta, the distance between HN
k and HN
k+1 is shown in green and the distances between Hα
k and HN
k+2, Hα
k and HN
k+3, Hα
k and HN
k+4 are
shown in cyan. Hydrogen, carbon, nitrogen, and oxygen atoms are shown in white, black, blue, and red, respectively.
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Cα
CSI
Hα
CSI
Cβ
CSI
consensus CSI
◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦)................................CSI prediction
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Cα
CSI
Hα
CSI
Cβ
CSI
.......(◦◦◦).......(◦◦◦◦)..........(◦◦◦◦◦◦◦◦......3
J prediction
◦◦-•-◦◦◦◦◦•-◦◦◦•-◦◦◦◦◦◦--◦• • ◦◦•◦◦◦◦◦◦◦◦◦ •◦◦◦•3
J(Hα
HN
)
..◦◦◦◦◦◦◦◦◦).......=>..<==.......................3
J prediction
- ◦-◦◦◦◦◦◦◦◦-•◦ -•-••◦◦••• ◦◦◦• ---•-◦◦-◦---•-3
J(Hα
HN
)
.......(◦◦)....(◦◦◦◦◦◦◦◦◦)...<=>.(◦◦◦◦◦◦◦◦)....(◦◦TALOS prediction
-----•◦◦◦◦--••◦◦◦◦◦◦◦◦◦◦◦---•••-◦◦◦◦◦◦◦◦◦◦--••◦◦◦TALOS
◦◦◦◦◦◦◦◦◦◦◦◦).....................................TALOS prediction
-◦◦◦◦◦◦-◦◦◦◦◦-----•------•------•-----------------TALOS
........(◦◦◦◦)..(◦◦◦◦◦◦◦◦◦◦◦◦◦◦.◦◦◦◦◦◦◦◦◦◦◦◦◦)..(◦NOE prediction
NN(i, i + 1)
αN(i, i + 1)
αN(i, i + 2)
αN(i, i + 3)
αN(i, i + 4)
◦◦◦◦◦◦◦◦◦◦◦◦◦◦)..............(◦◦).<====>........NOE prediction
NN(i, i + 1)
αN(i, i + 1)
αN(i, i + 2)
αN(i, i + 3)
αN(i, i + 4)
10 20 30 40 50
GIKQYSQEELKEMALVEIAHELFEEHKKPVPFQELLNEIASLLGVKKEEL
60 70 80 90
GDRIAQFYTDLNIDGRFLALSDQTWGLRSWYPYDQLDEETQLEHHHHHH
.......(◦◦◦)...(◦◦◦◦◦◦◦◦◦◦◦)....(◦◦◦◦◦◦◦◦◦◦◦)..(◦◦CSI prediction
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Cα
CSI
Hα
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Cβ
CSI
consensus CSI
◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦)................................CSI prediction
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CO CSI
Cα
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Hα
CSI
Cβ
CSI
consensus CSI
.......(◦◦◦).......(◦◦◦◦)..........(◦◦◦◦◦◦◦◦......3
J prediction
◦◦-•-◦◦◦◦◦•-◦◦◦•-◦◦◦◦◦◦--◦• • ◦◦•◦◦◦◦◦◦◦◦◦ •◦◦◦•3
J(Hα
HN
)
..◦◦◦◦◦◦◦◦◦).......=>..<==.......................3
J prediction
- ◦-◦◦◦◦◦◦◦◦-•◦ -•-••◦◦••• ◦◦◦• ---•-◦◦-◦---•-3
J(Hα
HN
)
.......(◦◦)....(◦◦◦◦◦◦◦◦◦)...<=>.(◦◦◦◦◦◦◦◦)....(◦◦TALOS prediction
-----•◦◦◦◦--••◦◦◦◦◦◦◦◦◦◦◦---•••-◦◦◦◦◦◦◦◦◦◦--••◦◦◦TALOS
◦◦◦◦◦◦◦◦◦◦◦◦).....................................TALOS prediction
-◦◦◦◦◦◦-◦◦◦◦◦-----•------•------•-----------------TALOS
........(◦◦◦◦)..(◦◦◦◦◦◦◦◦◦◦◦◦◦◦.◦◦◦◦◦◦◦◦◦◦◦◦◦)..(◦NOE prediction
NN(i, i + 1)
αN(i, i + 1)
αN(i, i + 2)
αN(i, i + 3)
αN(i, i + 4)
◦◦◦◦◦◦◦◦◦◦◦◦◦◦)..............(◦◦).<====>........NOE prediction
NN(i, i + 1)
αN(i, i + 1)
αN(i, i + 2)
αN(i, i + 3)
αN(i, i + 4)
Figure 8.12: Identification of secondary structure elements of the N-terminal domain of δ-subunit of RNA polymerase from Bacillus
subtilis based on direct interpretation of NMR data. Intensity of NOESY cross-peaks between atoms listed in the first column (in residues
shown in parentheses) is depicted as thickness of the bars under the sequence. Chemical shifts were interpreted using program TALOS
(open and filled circles correspond to predicted helical and extended conformations, respectively) and chemical shift index (CSI). The
chemical shift index indicates if the measured chemical shift deviates from the random value to a value typical for helical (shown as open
circle) or extended (shown as filled circle) structure. Helical and extended secondary structures predicted from various NMR parameters
are depicted above the sequence as ◦ ◦ ◦ ◦ and <==>, respectively.
100
A
x1
x2
-10
-5
0
5
10
15
20
Energy
B
x1
x2
-10
-5
0
5
10
15
20
Energy
C
x1
x2
-10
-5
0
5
10
15
20
Energy
Figure 8.13: Dependence of energy on two coordinates x1, x2 of the conformational space (e.g. two torsion angles) in unrestrained
molecular dynamics simulation (A), energy penalty for violating the experimental restraints, i.e., deviating from values x1, x2 derived from
NMR data (B), and dependence of energy on the coordinates x1, x2 in restrained molecular dynamics simulation (C).
The other interpretation approach is using an automated algorithm. The automated data interpretation is applied
when three-dimensional models of biomacromolecules are to be created and refined. With the help of computers, this
approach is very efficient but the calculations start to play a role of ”black-boxes”. The fact that computer simulation
was successfully finished does not prove that the obtained structural model is correct.
The most obvious way of calculating three-dimensional structure of molecules is solving trigonometric equations
defined by the measured interatomic distances. This distance geometry approach works fine in civil engineering. Its use
in NMR structure calculation is limited. The distances are not measured exactly but roughly guessed from the NOE
intensities. The obtained structures have the correct general fold and correct topology of helices and β-sheets, but
local geometry looks rather wild. The bond length and angles are far from anything that we learned in the chemistry
classes.
Molecular mechanics is the opposite approach. It is based on very good empirical knowledge of local structural
parameters such as bond lengths and angles. In structure calculation, it is typically enhanced by introducing molecular
motions. As the quantum description is computationally too expensive for large molecules, the molecular motions are
mimicked by Newton’s equations of motion. Such calculations are usually referred to as molecular dynamics. Molecular
mechanics suffers from a different disease than distance geometry methods do. It reflects local geometry very well but it
has a hard time to find the correct general fold. Methods designed to help to check broader spectrum of conformations
have been developed, e.g., simulated annealing (simulation of heating molecules to very high temperatures which gives
the molecule enough energy to overcome any energy barrier and adopt any possible conformation). Still, identification
of the correct fold is difficult. Computational methods judge the structures based on their energy and the energy
difference between various folds is very small (Figure 8.13A). Many macromolecules are even present in two or more
conformations under physiological conditions.
The natural solution of the described limitations of distance geometry and molecular dynamics is their marriage.
The combined approach, known as restrained molecular dynamics uses the NMR data to get the rough general fold
(”coarse tuning” of the structure) and the molecular dynamics to refine local geometry (”fine tuning” of the structure).
The calculations are mostly based on approximating energy with harmonic potentials (similar to the description of
the energy of an ideal spring or a pendulum). For example, bonds are defined as springs having the lowest energy for the
length equal to the known distance of the corresponding atoms (this information is obtained from experimental data or
quantum calculations of small molecules containing the given functional group). Bond angles are defined in a similar
way, planar structures are fixed by artificial potentials, torsion angles are described by periodic functions describing
energy minima for the favored conformations. Van der Waals forces are also simulated (typically in a simplified
form). The same applies to electrostatic interactions. Their physical description is simple, but their calculation is
time-consuming because the electrostatic interactions decrease only slowly with the distance of the interacting atoms.
Therefore, each charged atom is in principle influenced by all remaining charges in the molecule, and a lot of attention
has to be paid to the decision which interactions can be neglected without biasing the calculated structure.
The NMR data are also introduced through energy. Each value from NMR measurement serves as a restraint and
8.7. BUILDING A STRUCTURAL MODEL 101
A B
Figure 8.14: Backbone conformations of N-terminal domain of δ-subunit of RNA polymerase from Bacillus subtilis calculated by ten
independent runs of restrained molecular dynamics simulations (A) and the structure with the lowest energy presented as a cartoon showing
the secondary structures (B).
is given an artificial potential (Figure 8.13B). For example, a distance between two atoms does not increase the total
energy as long as it is in agreement with the intensity of the corresponding NOESY cross-peak. The range of allowed
distances is usually broad because the experimental errors are large. If the calculated distance exceeds the rather
generous boundaries, energy of the molecule starts to increase rapidly, forcing the atoms to go back closer to each
other. The art of NMR structure calculation is based on finding the right combinations of the empirical potentials
keeping correct local energy and of the artificial potentials forcing the calculated structure to agree with the NMR
data (Figure 8.13C). Usually, the calculations are repeated many times (Figure 8.14A). From the calculated structures,
those of the lowest energy and of the best mutual agreement are expected to reflect the actual conformation most
accurately (Figure 8.14B). A family of ten or twenty such structures is than deposited and referred to as ”the NMR
structure” or ”the solution structure” of the molecule. Reliability of the obtained structure greatly depends on the
quality and quantity of the obtained NMR data and on the correct use of the structure calculation protocol.
102
Lecture 9
Molecular motions and NMR relaxation
9.1 Real power of biomolecular NMR
We spent a fair amount of time to describe what needs to be done if we want to build a structural model of a large
molecule. We learned that many tricks must be used so that our effort is awarded by the ball-and-stick picture on the
screen of our computer. Now it is the time to show how far from reality the precious result of our work is. Our model
is like a skeleton, but real molecules are not dead. It is not possible to represent a molecule realistically with the
model of a single conformation. Fortunately, NMR methodology is sensitive to the motions that make the molecules
alive. Actually, among experimental methods designed to investigate molecular motions, NMR is probably the most
powerful one Let us now look how can NMR animate our molecular models.
9.2 Connection between molecular motions and NMR relaxation
In general, relaxation is a return of some observable quantity to its equilibrium value. In optical spectroscopy (infrared
to ultraviolet), the relaxation is usually very fast and dominated by spontaneous emission of a photon at the frequency
given by the energy difference between the ground state and the excited state. However, spontaneous emission,
proportional to the third power of the photon frequency (and of the energy difference), is completely inefficient
in NMR. Yet, magnetic moments perturbed by a radio wave in the NMR experiment return to their equilibrium
distribution, although very slowly. The only cause of this process is interaction of magnetic moments with magnetic
fields in their surroundings.
Setup of the NMR experiments allows us to study evolution of the total magnetization vector ⃗M in time. When we
observe certain type of nuclei, e.g., 15
N, precession frequencies of individual 15
N magnetic moments in the molecule
slightly differ because each nucleus feels a slightly different magnetic field. This would result in the loss of coherence
even if the molecules did not move. However, the lost coherence can be gained back by applying an echo if the magnetic
field variations do not change in time. Such constant variations have the same effect as changing the magnetic field
by the pulsed field gradients (see Section 4.3) or simply having the magnetic field inhomogeneous (in reality, the
homogeneity is never perfect).
If the molecule moves, the magnetic fields are not only different for each nucleus but they also change in time.
Random motions obviously cause fluctuations of the magnetic field. Relaxation is a consequence of such random
fluctuations. As the relation between stochastic molecular motions and NMR relaxation is well defined, relaxation can
be used to study dynamics of molecules. It should be emphasized that relaxation is a statistical phenomenon and its
analysis requires statistical approaches. We do not build the apparatus of the statistical description of relaxation in
this text, we only present its main features.
The qualitative understanding of the effects of fluctuating fields does not require more than knowledge of the roles
of the precisely defined external magnetic fields in the basic NMR experiment. It is useful to decompose the magnetic
field variations into components parallel to the external field ⃗B0 (variations of Bz) and perpendicular to it (variations
of Bx and By).
The variations in the z direction are simply changing the strong static magnetic field of our magnet. As the
103
104
θ
Figure 9.1: The actual magnetic field of the observed nucleus (cyan) is influenced by local magnetic fields of spin magnetic moments of
other nuclei (one is shown in green) and of orbital magnetic moments of electrons (one is shown in red). Orientations of the local magnetic
fields are given by the orientation of the whole molecule (if the molecule is rigid). The orientation of the molecule can be described e.g. by
a direction of selected chemical bonds (e.g. of the bond between the green and cyan nuclei, shown in black). A fragment of a molecule is
shown in yellow, other symbols are used as in Figures 2.6A and 2.7B.
precession frequencies of magnetic moments are directly proportional to ⃗B0 defining the z directions (⃗ω0 = −γ ⃗B0),
variations of Bz cause changes of the precession frequencies of individual magnetic moments. This influences the
observed magnetization. The magnetization flipped to the −y direction by a short pulse of a radio wave starts to
rotate in the xy plane (perpendicular to ⃗B0) with the precession frequency of individual nuclei. If the local vertical
field Bz at the observed nucleus varies in space, precession frequencies of magnetic moments of the nucleus in different
molecules in the sample also vary, as mentioned above. And if the molecular motions randomly change Bz so that
it also fluctuates in time, the effect cannot be reversed by an echo. The coherence created by the radio-wave pulse
is getting lost and the observed signal decreases. No particular frequency of molecular motions is required, they just
have to randomize Bz sufficiently quickly. The total energy of all magnetic moments is preserved, there is no change
in Mz. Therefore, there is no energy exchange with the kinetic energy of water molecules hitting our molecule of
interest, the process is adiabatic.
Fluctuations of the magnetic fields perpendicular to ⃗B0 resemble the effect of the radio waves. Magnetic induction
of the radio waves oscillates in the x direction. If the frequency of the oscillations resonate with the precession
frequency, the radio wave rotates magnetization from the vertical direction, changing its orientation from z to −y →
−z → +y → +z etc. We can assume that a molecular motion may create accidentally for a short time (between two
collisions of molecules changing the speed and direction of the molecular rotation) a horizontal local field rotating
with a frequency ⃗ωmol matching the precession frequency ⃗ω0. Such a field would rotate the magnetic moment of the
given nucleus in a similar manner as the radio wave. This is changing the z components and consequently the average
energy of the magnetic moment (the average energy is given by E = −µzB0 because fluctuating components of ⃗B are
averaged to zero). The energy difference is balanced by a change of the kinetic energy of the surrounding molecules
(mostly molecules of water). In other words, magnetic moments exchange energy with the surroundings, the process
is non-adiabatic. Collisions giving the molecule the right rotation frequency are relatively rare, so it takes a long time
to see an effect. But in several seconds, the occasional events of energy exchange bring the system to the equilibrium
distribution of magnetic moments, given by the Boltzmann law.
The so-far general discussion can be applied to the actual sources of the local fields, namely spin magnetic moments
9.3. ROTATIONAL DIFFUSION 105
of other nuclei (dipolar coupling, described in Section 2.6) and orbital magnetic moments of electrons (chemical shift,
introduced in Section 2.7). The effects of the local fields depend on
• magnitude of the interaction (interaction constant), described below by the parameter b
• orientation of the interacting magnetic moments in ⃗B0, given by the orientation of the whole molecule (see
Figure 9.1).
Consequently, the local field fluctuates if the molecular motions vary
• magnitude of the interaction
• orientation of the molecule (see Figures 2.6A–C and 2.7B,C).
As the relation between the molecular motions and relaxation is not simple, we first discuss a simplified case of a
rigid molecule, whose motions do not change the magnitude of the interaction. In other words, we are going to analyze
first the effect of random changes of the orientation of the molecule. The orientation of the whole molecule should
not be confused with the orientation of the observed magnetic moments. These orientations are almost completely
independent, except for the effects discussed in this lecture.
9.3 Rotational diffusion
We now explore the simplest fluctuation of molecular orientation, a random rotation of a rigid molecule, known as
rotational diffusion. As this is a stochastic process, we must describe it statistically.
First, we identify a geometric parameter Θ describing the actual dependence of local ⃗B on the orientation of the
molecule. In the case of a rigid spherical molecule, it is sufficient to analyze Θ = (3 cos2
θ − 1)/2, where the angle θ
is defined in Figure 2.6. The analysis (not done here) shows that relaxation depends on Θ(t) · Θ(0), where Θ(t) is the
actual value of Θ, and Θ(0) is the value of Θ at the beginning (at the time t = 0). Θ(t) · Θ(0) changes randomly and
differently for different molecules. It is therefore impossible to calculate Θ(t)·Θ(0) for a particular molecule. However,
the average value for the whole sample Θ(t) · Θ(0) can by calculated exactly using the statistical approach (solving
the diffusion equation, the second Fick’s law in spherical coordinates). The result, not derived here, is simply
Θ(t) · Θ(0) ≡ C(t) =
1
5
e−t/τ
(9.1)
for fluctuations of Bz
and
Θ(t) · Θ(0) ≡ C(t) =
3
10
e−t/τ
(9.2)
for fluctuations of Bx. The constant τ in Eqs. 9.1 and 9.2 is the correlation time and Θ(t) · Θ(0), or shortly C(t),
is the time correlation function.
An exact solution can be found for any shape of a rigid molecule, the general solution is a sum of five exponential
functions, but we do not need such accuracy in order to understand the basic principles.
9.4 Quantitative description of relaxation
The quantitative knowledge of the relation between the observed relaxation rates and the statistically described
molecular motions is necessary for the studies of molecular dynamics by NMR relaxation. We start the quantitative
description of relaxation by presenting a general equation defining the desired relation
(Relaxation rate) =
ω
(numerical factor) × (magnitude)
2
b2
× spectral density function at ω
J(ω)
. (9.3)
106
To avoid a fuzzy general discussion, we rewrite the equation for a particular case of amide 15
N in a peptide bond,
influenced by the electrons surrounding it and the amide proton attached to it. This is the molecular fragment that is
studied most often because it provides information for all non-proline residues of a protein and cheap selective labeling
by 15
N makes the measurement accurate and feasible. The following table includes three relaxation rates that are
measured most frequently, the longitudinal relaxation rate R1, the transverse relaxation rate R2, and cross-relaxation
rate Rx, calculated from steady-state NOE and R1. The subscripts N and H in the table refer to amide 15
N and amide
proton, respectively.
We we type contribution of
start observe of R chemical shift to dipolar coupling to
from Bz Bx Bz Bz Bx Bx Bx
MN,z MN,z R1 = 0 +3
4b2
eJ(ωN) +0 +0 +3
4b2
nJ(ωN)+3
2b2
nJ(ωH + ωN)+1
4b2
nJ(ωH − ωN)
MN,x MN,x R2 =1
2b2
eJ(0)+3
8b2
eJ(ωN)+1
2b2
nJ(0)+3
4b2
nJ(ωH)+3
8b2
nJ(ωN)+3
4b2
nJ(ωH + ωN)+1
8b2
nJ(ωH − ωN)
MH,z MN,z Rx = 0 +0 +0 +0 +0 +3
2b2
nJ(ωH + ωN)−1
4b2
nJ(ωH − ωN)
The symbols in the equations have the following meaning.
• R1, R2, and Rx are experimentally determined relaxation rates, input for the analysis of molecular motions.
Experiments providing them will be discussed in the next lecture.
• Numerical factors account for differences between correlation functions (cf. Eqs. 9.1 and 9.2) and for the factor
of one half multiplying J(ω) in Eq. 9.10 (see below).
• be and bn are constants describing magnitude of the interaction (interaction constants), known in advance with
a relatively high precision and accuracy. For the amide 15
N nucleus,
be = −γNB0(δmax − δmin), (9.4)
bn =
µ0h
8π2
γNγH
r3
NH
, (9.5)
where δmax and δmin are the amide 15
N chemical shifts in orientations providing the highest and lowest values,
respectively (the difference is approximately −160 ppm), rNH is the distance between the amide 15
N and 1
H
nuclei, µ0 is the magnetic permeability of vacuum and h is the Planck’s constant.
• J(ω) are most interesting values, representing the actual bridge between molecular motions (described statistically
by the time correlation function C(t)) and NMR at particular frequencies. As we discuss later, the J(ω)
values provide a deep into the dynamics, but not all researches are used to looking at them. The following
section explains how is the dependence of relaxation rates on the molecular motions encoded in J(0).
9.5 Spectral density function
The effect of local field fluctuations on the relaxation can be described as C(t) cos(ωt), where the correlation function
C(t) describes the molecular motions and cos(ωt) accounts for the frequency dependence. It can be derived that the
relaxation rates depend on all values of C(t) cos(ωt) in the whole history of the measurement. The cumulative effect
of C(t) cos(ωt) can be evaluated by integrating C(t) cos(ωt) from the beginning of the measurement until the actual
time (Figure 9.2A,B):
tactual
0
C(t) cos(ωt)dt. (9.6)
9.5. SPECTRAL DENSITY FUNCTION 107
A
-0.2
-0.1
0
0.1
0.2
-80 -60 -40 -20 0 20 40 60 80
t1 t2
C(t)
t / ns
B
-0.2
-0.1
0
0.1
0.2
-80 -60 -40 -20 0 20 40 60 80
t1 t2
C(t)
C(t)cos(ωt)
t / ns
C
-0.2
-0.1
0
0.1
0.2
-80 -60 -40 -20 0 20 40 60 80
t1t2
C(-t)
C(t)cos(ωt)
t / ns
D
-0.2
-0.1
0
0.1
0.2
-80 -60 -40 -20 0 20 40 60 80
C(t)
C(t)cos(ωt)
t / ns
Figure 9.2: A, integration of C(t) describes the cumulative frequency-independent effect of fluctuations of Bz. If the upper limit of the
integration (t1) is sufficiently long so that C(t) decays close to zero, further extending the upper limit (e.g. to t2) does not change the
integral significantly. B, integration of C(t) cos(ωt) describes the cumulative effect of fluctuations of Bx resonating with the frequency ω.
C, for random fluctuations, correlation functions and integrals of C(t) cos(ωt) describing the loss of the correlation backward and forward
in time are identical. D, for sufficiently long upper integration limit in Panel B, the value of the integral tends to one half of the integral of
the sum of backward and forward C(t) cos(ωt) functions, calculated from −∞ to ∞. The values are plotted for τ = 20 ns and precession
frequency of 15N at 11.75 T (500 MHz spectrometer).
Usually, we integrate for a long time when C(t) already decayed almost to zero. In such a case, extending the
time further does not change the value of the integral any more (Figure 9.2B). The integral becomes constant, time
independent. Therefore, it does not change even if we prolong the integration time to infinity:
tlong
0
C(t) cos(ωt)dt ≈
∞
0
C(t) cos(ωt)dt. (9.7)
Moreover, we obtain the same integral if we look backwards in the time (Figure 9.2C). As the molecules move
randomly, the geometric parameters Θ a nanosecond before we start counting the time are correlated with Θ(0) exactly
like a nanosecond after t = 0. Mathematically,
0
−∞
C(t) cos(ωt)dt =
∞
0
C(t) cos(ωt)dt. (9.8)
If we add both integrals, we obtain (Figure 9.2D)
108
0
−∞
C(t) cos(ωt)dt +
∞
0
C(t) cos(ωt)dt =
∞
−∞
C(t) cos(ωt)dt = 2
∞
0
C(t) cos(ωt)dt. (9.9)
In physics, the integral from −∞ to ∞ is a well established function, labeled J(ω), and called spectral density
function. It is therefore convenient to use this already defined function, and express our cumulative effect of local field
fluctuations as
tactual
0
C(t) cos(ωt)dt ≈
1
2
∞
−∞
C(t) cos(ωt)dt. (9.10)
For the correlation function in Eq. 9.1, the integral can be calculated easily
0
−∞
1
5
e−t/τ
cos(ωt)dt =
1
5
τ
1 + ω2τ2
. (9.11)
The graph of the resulting function is the same as the shape of a peak in an NMR spectrum, a typical resonance
curve with maximum at ω = 0 (Figure 9.3). It tells us that J(ω) can be interpreted as a factor integrating the whole
history of loosing correlation and including dependence on characteristic frequencies ω. In the case of 15
N relaxation
in peptide bonds, discussed in Section 9.4, the five characteristic frequencies are 0, ωH, ωN, ωH + ωN, and ωH − ωN. In
the case of 13
C relaxation, e.g. in the C8–H8 bonds of purine bases of nucleic acids, the characteristic frequencies are
0, ωH, ωC, ωH + ωC, and ωH − ωC. The individual frequencies are discussed bellow and their effects are illustrated in
Figures 9.4 and 9.5. In order to make the illustration of the effect of molecular rotation easier to follow, the magnetic
moment of 13
C, which differs less than 15
N in its precession frequency from 1
H, is shown in Figures 9.4 and 9.5.
• ω = 0. Molecular motions at a low frequency slowly move the 13
C or 15
N nucleus in different molecules to different
regions of the local fields (the fields created by the orbital magnetic moment of the surrounding electrons and
by the spin magnetic moment ⃗µH of the attached proton). This is depicted in Figure 9.4. Mathematically, the
change of the direction of ⃗µC is given by the vector product ⃗µC × ⃗B, where ⃗B is the actual magnetic field at the
place of ⃗µC. The same applies to ⃗µN. Graphically, the result of the vector product of ⃗µC (thick green arrow in
Figure 9.4) and ⃗B (thin cyan arrows in Figure 9.4) is the red arrow in Figure 9.4. A stochastic molecular rotation
about a horizontal axis moves the observed nucleus (15
N in our discussion, 13
C in Figure 9.4) from a position
where the field of the attached proton accelerates its precession (Figure 9.4A, the red arrows always point in
the direction of the rotation of the green arrow) to positions where the effect of ⃗µH on the precession frequency
is negligible (Figure 9.4B, the red arrows oscillate roughly vertically and do not pull the green arrow in any
constant direction) or opposite (Figure 9.4C, the red arrows always point against the direction of the rotation of
the green arrow). For the sake of clarity, the size of all three arrows are similar in Figure 9.4. However, the effect
of the red arrow is very small on the time scale presented in Figure 9.4: the time needed for a visible change of
the rotation frequency of the green arrow in the direction of the red arrow is much longer, than the 0.5 ns time
difference between the frames in Figure 9.4. Moreover, the molecular motions are random, they do not rotate the
green nucleus in Figure 9.4 with a constant frequency. Different molecules reorient differently and Bz fluctuates
differently in different molecules. As a result, the coherence of the rotation of ⃗µC or ⃗µN about the axis z slowly
disappears (Section 9.2). Note that the loss of the coherence does not require any particular correlation between
the rate of the molecular motion and the precession frequency of ⃗µC or ⃗µN. The cumulative effect is given
by integrating the correlation function for the whole history, without any frequency factor (Figure 9.2A). This
missing frequency dependence may be included into the general equation as a multiplication by 1 = cos(0). Also
note that the molecular motions much slower than the precession frequency (symbolically described as ω = 0)
do not pull ⃗µC or ⃗µN out of the xy plane (the red arrows rapidly oscillate in the z direction or rotate in the xy
plane in Figure 9.4).
• ω = ωH. Molecular motions also vary Bz at 13
C or 15
N in another way, not by changing position of the 13
C
or 15
N nucleus in the field of ⃗µH, but by changing the field of ⃗µH itself. Molecular motions occasionally cause
9.5. SPECTRAL DENSITY FUNCTION 109
A
-0.2
-0.1
0
0.1
0.2
0 20 40 60 80
0 MHz
C(t)cos(ωt)
t / ns
B
-0.2
-0.1
0
0.1
0.2
0 20 40 60 80
25 MHz
C(t)cos(ωt)
t / ns
C
-0.2
-0.1
0
0.1
0.2
0 20 40 60 80
100 MHz
C(t)cos(ωt)
t / ns
D
0
1
2
3
4
5
-0.6 -0.4 -0.2 0 0.2 0.4 0.6
0 MHz
25 MHz
100 MHz
0.5J(ω)/nsrad
-1
ω / rad ns-1
Figure 9.3: Spectral density function. Integrals displayed in Panels A–C represent three values of the spectral density function J(ω) (D).
The data are plotted for τ = 20 ns and precession frequency of 13C and proton at 2.85 T (100 MHz spectrometer).
oscillations of Bx with a frequency ωmol resonating with ωH. An example of molecular rotation about z with a
frequency close to ωH is shown in Figure 9.5A. Unlike in Figure 9.4A, rotation of the position of the green arrow
(due to the molecular motion) is synchronized with the rotation of its direction (due to the precession about ⃗Bz).
As a consequence, ⃗µH (thick cyan arrow in Figure 9.5A) remains perpendicular to Bx of 13
C (thin green arrows
in Figure 9.5A). Therefore, the red arrow (showing direction of the change of the ⃗µH orientation) remains vertical
and pulls ⃗µH to the z direction as long as the molecule rotates as depicted. In general, temporary molecular
rotations about z witch frequencies resonating with ωH rotate ⃗µH from the z direction to −z and vice versa.
Such a change of ⃗µH, occurring only when ωmol matches ωH, further spreads values of Bz at 13
C in different
molecules, in addition to the frequency independent variations, discussed above. This is why the contributions
to R2 also include the term J(ωH).
• ω = ωC or ω = ωN. As discussed in Section 9.2, the fluctuations of Bx contribute to the relaxation when Bx
rotates for a short time with a frequency matching ωC or ωN (as a result of a molecular rotation shown in
Figure 9.5B). In comparison to Figure 9.5A, the roles of ⃗µH and ⃗µC are switched in Figure 9.5B. As we observe
relaxation of the transverse 13
C magnetization, we are interested directly in the changes in the orientation of ⃗µC.
Figure 9.5B shows that ⃗µC are constantly pulled up (to z) from the original direction in the xy plane, as long as
the molecule rotates as depicted. Therefore, the transverse13
C magnetization disappears. Eq. 9.11 shows that
the term J(ωC) describes resonance with the 13
C frequency ωC (for a rigid spherical molecule).
• ω = ωC +ωH or ω = ωN +ωH. Precession of ⃗µH does not resonate with the precession of the 13
C or 15
N magnetic
moment. However, if a molecule rotates for a short time about the vertical axis with an angular frequency ωmol
which is accidentally matching ωC + ωH or ωN + ωH, the precession of ⃗µH combined with the molecular rotation
110
resonates with the precession of ⃗µC or ⃗µN, respectively. It is shown in Figure 9.5C for 13
C. The green arrow,
rotating with the frequency ωC and moving along the cyan circle with the frequency ωmol = ωC + ωH, is always
approximately perpendicular to the local magnetic field of 1
H (thin cyan arrows) such that the red arrow is
always pointing up (in the z direction).
• ω = ωC − ωH or ω = ωN − ωH. The precession of ⃗µH combined with a horizontal molecular rotation at ωC − ωH
resonates with the precession of the 13
C magnetic moment, as shown in Figure 9.5D. The green arrow, rotating
with the frequency ωC and moving along the cyan circle with the frequency ωmol = ωC − ωH, is on average
perpendicular to the local magnetic field of 1
H (thin cyan arrows) such that the red arrow is mostly pointing up
(in the z direction, note that the coordinate frame is oriented differently in Figure 9.5D in order to depict the
molecular rotation about a horizontal axis).
9.5. SPECTRAL DENSITY FUNCTION 111
t=3.5ns
y
z
A
xy
z
C
t=0.0nst=4.0nst=3.0nst=2.5nst=2.0nst=1.5nst=1.0nst=0.5ns
x
B
z
yx
Figure9.4:Illustrationhowrandommolecularmotionsmuchslowerthantheprecessionfrequencyaffect⃗µCintheCHbondina500MHzspectrometer(⃗B0=11.75T).
Thefirstdiagramineachrowshowspositionsofthenuclei(13Cingreen,1Hincyan),theseconddiagramshowsorientationofthecoordinateframe,thefollowing
diagramsaresnapshotsoftheorientationsofmagneticmomentsshownasarrows(13Cingreen,1Hincyan)inamagneticfieldof1H(shownascyanforcelines).The
redarrowindicatesthedirectionofthechangeoftheorientationof⃗µC.Notethattheactualchangeisverysmall(notobservableinthepictures)intheshortdepicted
periodof4ns.Themoleculeisorientedsuchthattheinternuclearvector(showingorientationof13Crelativeto1H)ispointingapproximatelyinthezdirectioninPanel
A,tiltedbyapproximately55◦fromthezdirectioninPanelB,andtiltedbyapproximately90◦fromthezdirectioninPanelC.Themagneticfieldof1Hslightlyspeeds
uptheprecessionof13CmagneticmomentinPanelAandslowsdowntheprecessionof13CmagneticmomentinPanelC.Theeffectofthemagneticfieldof1Hon
theprecessionof13CisnegligibleinPanelB(theredarrowinPanelBoscillatesroughlyverticallyanddoesnotpull⃗µCinanydirection).Randommolecularmotions
changingtherelativeorientationof13Cand1HbetweenpositionsdepictedinPanelsA,B,andCthuscausefluctuationsofthe13Cprecessionfrequency,butdonot
drivethedistributionofthe13Cmagneticmomentstothethermalequilibrium.
112
D
t=0.0nst=4.0nst=3.5nst=3.0nst=2.5nst=2.0nst=1.5nst=1.0nst=0.5ns
ωH
zx
y
A
ωC
zx
y
B
ωH+ωC
zx
y
C
z
yx
−ωCωH
Figure9.5:Illustrationhowrandommolecularmotionsresonatingwithcharacteristicfrequenciesaffect⃗µCintheCHbondina500MHzspectrometer(⃗B0=11.75T).
Thefirstdiagramineachrowshowspositionsofthenuclei(13Cingreen,1Hincyan),theseconddiagramshowsorientationofthecoordinateframe,thefollowing
diagramsaresnapshotsoftheorientationsofmagneticmomentsshownasarrows(13Cingreen,1Hincyan)inamagneticfieldoftheneighbour(shownasforcelines).
Theredarrowindicatesthedirectionofthechangeoftheorientationoftheinfluencedmagneticmoment.Notethattheactualchangeisverysmall(notobservablein
thepictures)intheshortdepictedperiodof4ns.Themoleculeisassumedtorotatewithaconstantfrequencyduringthe4nstimeinterval.A,⃗µHorientedhorizontally,
⃗µCorientedvertically,1Habovethexyplane,13Cinthecenterofthecoordinatesystem.RotationatωHaboutahorizontalaxispulls⃗µHup(redarrowpointingupin
thegivencoordinateframe)inthemagneticfieldsof⃗µC.Thisreorientationof⃗µH(observedonlyatatimemuchlongerthanthedepictedperiodof4ns)modulatesthe
effectoftheJ-coupling.B,⃗µCorientedhorizontally,⃗µHorientedvertically,13Cabovethexyplane,1Hinthecenterofthecoordinatesystem.RotationatωCabout
ahorizontalaxispulls⃗µCup(redarrowpointingupinthegivencoordinateframe)inthemagneticfieldof⃗µH.C,⃗µCorientedhorizontally,⃗µHorientedhorizontally,
13Cclosetothezaxis,1Hinthecenterofthecoordinatesystem.RotationatωH+ωCaboutaverticalaxispulls⃗µCup(redarrowpointingupinthegivencoordinate
frame)inthemagneticfieldsof⃗µH.D,⃗µCorientedhorizontally,⃗µHorientedhorizontally,13Cintheyzplane,1Hinthecenterofthecoordinatesystem.Rotationat
ωH−ωCaboutahorizontalaxispulls⃗µCup(redarrowpointingmostlyupinthegivencoordinateframe)inthemagneticfieldsof⃗µH.Thereorientationof⃗µCinPanels
B–D(observedonlyattimemuchlongerthanthedepictedperiodof4ns)contributestothereturnofthedistributionofthe13Cmagneticmomentstothethermal
equilibrium.
Lecture 10
Measurement of relaxation rates
10.1 General principle
NMR experiments for measurements of relaxation rates usually follow a general scheme. First, it is necessary to
create a starting distribution of magnetic moments defining the type relaxation experiment we are going to run (the
first column ”we start from” in the table of equations describing relaxation rates R1, R2, Rx in Section 9.4. Such
magnetic moment distribution is called the spin state in the following text. Then the spin state is let to relax for a
variable period of time (relaxation delay). Typically, some manipulation of the spin system is also necessary during
the relaxation delay to suppress coherent evolutions or relaxation pathways which need to be avoided. Finally, the
selected spin state is transformed into an observable magnetization detected as an NMR signal which intensity reflects
the polarization of the selected spin state at the end of the relaxation delay. The experiment is usually repeated several
times, while the relaxation delay is varied. The change of the polarization of the selected spin state is recorded as a
dependence on the relaxation time, which is then analyzed to extract the required relaxation rate.
10.2 Inversion recovery experiment
The inversion recovery experiment represents probably the simplest NMR relaxation experiment. Usually, it is used
to study the relaxation of small molecules and it is also used in magnetic resonance imaging. The experiment is
shown in a graphical form in Figure 10.1. In the beginning, the spin magnetization is oriented parallel with the
external magnetic field. The experiment starts with a 180◦
pulse, which inverts the polarization of the studied nuclei.
This is the preparation phase of the experiment followed with the variable relaxation delay, during which the spin
system returns to the equilibrium in which it was at the beginning of the experiment (before applying the 180◦
pulse).
Therefore, the absolute value of the polarization in the direction opposite to the initial one starts to decrease, later it
passes the zero value, switches the sign, and it continues increasing until it reaches its equilibrium value. This process
of the return to the equilibrium is interrupted by the application of a 90◦
pulse which transforms the longitudinal
polarization into an observable magnetization and it is detected. The intensity of the signal is proportional to the
size of the longitudinal polarization at the end of the chosen length of the relaxation delay. The dependence of the
intensity of the detected signal I(t) on the length of the relaxation delay t is described by Eq 10.1:
I(t) = I∞(1 − 2e−R1t
), (10.1)
where R1 is the longitudinal relaxation rate and I∞ is the equilibrium signal intensity. The example of the
dependence is shown in Figure 10.1.
10.3 Measurement of the transverse relaxation rate
A simple experiment for a measurement of the transverse relaxation is discussed in this section and its scheme is
shown in Figure 10.2. First, the transverse magnetization is created by the application of a 90◦
pulse. Then, the
relaxation delay follows. It is necessary to keep in mind that magnetization in the transverse plane is a subject of
113
114
d1 FT
td1 FT
td1 FT
td1 FT
td1 FT
td1 FT
td1 FT
t
A B C
-1
-0.5
0
0.5
1
0 0.5 1 1.5 2 2.5 3 3.5
I/AU
t / s
Figure 10.1: A, Scheme of the inversion recovery experiment designed for a measurement of the longitudinal relaxation rate of an isolated
spin. The experiment is repeated with variable length of the relaxation delay t, a series of seven experiments is presented in this example.
Narrow and wide boxes represent 90◦ and 180◦ degree pulses, respectively. (Note, that the delay d1 between the scans is not proportional
in its length to the rest of the pulse sequence. For the depicted case it should be at least as long as the longest relaxation delay in the last
scheme to achieve a state close to the equilibrium.) B, Spectrum obtained after the Fourier transformation of the obtained signal from the
series of the inversion recovery experiments (the simplest case of a single peak in the spectrum is presented). C, The dependence of the
measured intensities in spectra on the relaxation delay t of the corresponding experiment. The red line shows the theoretical dependence
described by Eq. 10.1.
10.4. MEASUREMENT OF THE STEADY STATE NUCLEAR OVERHAUSER EFFECT 115
d1 ττ
N
d1 FTττ ττ ττ
d1 FTττ ττ
d1 FTττ
d1 FTττ ττ ττ ττ
d1 FTττ ττ ττ ττ ττ 0
0.2
0.4
0.6
0.8
1
0 0.01 0.02 0.03 0.04 0.05 0.06
I/AU
t / s
A
C
D
B
Figure 10.2: A, Scheme of the simple experiment for the measurement of the transverse relaxation rate of an isolated spin. Narrow and
wide boxes represent 90◦ and 180◦ degree pulses, respectively. Central part in the bracket is a spin echo of the length 2τ and it is repeated
N times to achieve variable relaxation time as shown for a series of five experiments in Panel B. Fourier transformation of the obtained
signal from each experiment provides a signal with various intensity as shown in Panel C (the simplest case of a single peak in the spectrum
is presented). D, The dependence of the measured intensities in spectra on the relaxation delay t = 2Nτ of the corresponding experiments.
The red line shows the theoretical dependence described by Eq. 10.2.
coherent evolutions. Even the simplest case, i.e. isolated spin will be subject of effects of evolution due to chemical
shift. Therefore, unlike for the inversion recovery experiment, additional spin system manipulation is necessary to
be applied during the relaxation delay to suppress this effect. As was already discussed in Section 3.6, the effect of
coherent evolution due to the chemical shift can be suppressed by the refocusing spin echo. So, the relaxation delay is
composed by a spin-echo block which is repeated various number of times to achieve variable relaxation delays. The
transverse relaxation causes decrease of the spin coherence until the equilibrium state is reached, which is characterized
by a zero coherence and no observable magnetization. The signal can be detected after the end of the relaxation delay
without any need of further spin state manipulation (because the transverse magnetization is directly observable).
The amount of the transverse magnetization is directly proportional to the detected signal and it decays with the
increasing relaxation delay t following the exponential decay:
I(t) = I(0)e−R2t
, (10.2)
where R2 is the transverse relaxation rate and I(t) is the signal intensity the end of the relaxation delay with the
total length t = 2Nτ, where τ is the one half of the length of the used spin echo and N is number of repetitions of
the spin-echo elements. The example of the experimental data is shown in Figure 10.2.
10.4 Measurement of the steady state Nuclear Overhauser Effect
So far we discussed only the cases when we disturbed a selected spin polarization from equilibrium to create a new
spin state and we studied its own return to the equilibrium. This type of relaxation is called auto-relaxation. But it is
possible to study also the cases when we disturb one type of polarization and we observe the effect of the consequent
perturbation of the equilibrium on another spin state. This type of relaxation is called cross-relaxation, and its
116
d1
FT
d'1 FT
1
H
1
H
A: steady state
B: reference
13
C / 15
N
13
C / 15
N
Figure 10.3: Schemes of pulse sequences used to measure the cross-relaxation between proton and heteronuclear polarizations (of 13C or
15N shown in this example). Narrow boxes represent 90◦ degree pulses. A, Scheme of the simple experiment for the measurement of the
steady state NOE with the applied saturation of protons. B, Reference experiment to measure the signal intensity originating from the
equilibrium 13C or 15N polarization. Note, that delays between scans in the experiments, d′
1 and d1, have different length: d′
1 ≫ d1. A
long d′
1 must be used to let the system reach a state very similar to the equilibrium. C, Fourier transformation is used to obtain spectra
in both reference and steady state experiment (only a single peak in the spectrum is presented for the sake of simplicity). The ratio of the
intensities of the signals in the spectra originating from the steady state experiment and the reference experiment provides the information
about the cross-relaxation rate Rx.
consequence is the nuclear Overhauser effect (NOE). The experiment allowing us to evaluate NOE and to calculate
the cross-relaxation rate Rx is explained in this section for a simple spin system: 13
C-1
H pair or 15
N-1
H pair. It
is important that both nuclei have magnetic moments and are close in space. Then, each of them is affected not
only by the external magnetic field, but also by the magnetic field from the other nuclei, i.e., 13
C / 15
N spin ”feels”
magnetization of proton, and vice versa (see Section 2.6). The magnetic moments are no longer isolated but they
interact with each other via the dipole-dipole interaction (or dipolar coupling). At the beginning of the experiment,
magnetic moment distributions of both 13
C / 15
N and 1
H are polarized producing a net magnetization in the direction
aligned with the external magnetic field. This equilibrium is then disturbed by a very long irradiation by a radio wave
resonating with the precession frequency of one of the nuclei, typically proton. This irradiation may be of various form
but its results is the same as shown for solvent suppression by presaturation in Section 6.2. The irradiation rotates
the polarization out of the z direction so thata coherence is created while relaxation simultaneously has the opposite
effect dephasing the coherences and returning the polarization back to the z-direction. As a result, a steady state
is achieved characterized by an absence of proton polarization in any direction. Because the proton is not isolated,
but it interacts with the carbon, the 13
C / 15
N polarization is also affected by the change of the proton magnetic
moment distribution, and also its polarization is changed as it adapts to the established steady state (different from
the equilibrium, where the proton longitudinal polarization is present). Comparison of the equilibrium 13
C / 15
N
polarization and its longitudinal polarization upon the steady state conditions provides the information about the
cross-relaxation between the proton and 13
C / 15
N polarizations. For the quantitative analysis, it is necessary to keep
the recovery delay before the acquisition of the reference spectra (when no proton saturation is applied) long enough
so that the system is in a state very close to the equilibrium.
10.5 Measurement of the relaxation rates in biomacromolecules
The experiments discussed above in Sections 10.2–10.4 are not suited for the analysis of dynamics of more complex
systems like biomacromolecules. But the experiments used for the proteins, nucleic-acids, etc., are derived from the
above ones. Usually, dynamics of biomacromolecules is investigated by analysis of relaxation rates of 15
N or 13
C in
isolated 15
N-1
H or 13
C-1
H spin pairs which represent the simplest case of interacting magnetic moments. Moreover,
it is relatively easy to produce a sample suitable for the relaxation measurements, where the mentioned spin pairs can
be considered as isolated. Typical probes of dynamics are amide 15
N-1
H pairs from the peptide bonds of a protein
10.5. MEASUREMENT OF THE RELAXATION RATES IN BIOMACROMOLECULES 117
enriched with 15
N isotopes, 15
N-1
H spin pairs at position 1 of guanines, or at position 3 of uracils or thymines in
nucleic acid samples enriched with 15
N isotopes. 13
C-1
H spin pairs at position 8 in purine bases enriched with 13
C
isotopes may serve as another example. Although these spin pairs are not completely isolated, other spin magnetic
moments are not located very close to them and weak interactions with other nuclei do not corrupt the analysis.
Typically, the relaxation rates are determined from the analysis of peak intensities in 2D spectra, but sometimes
a measurement of relaxation rates using a more dimensional spectra is necessary to achieve the desired resolution. It
is also very useful to increase the sensitivity of the experiments by polarization transfers between proton and 15
N or
13
C in a similar way as introduced in Section 4.1. Combinations with the HSQC sequence (Section 4.2) are used most
often. Simplified versions of such experiments, emphasizing the basic principles, are discussed in this section.
Our overview of the basic experiments starts with the experiment for the measurement of longitudinal relaxation
rates, shown in Figure 10.4A. The spin state preparation step (in the red frame) includes a 90◦
pulse applied to 13
C
or 15
N followed by a gradient, aimed to destroy longitudinal polarization of 13
C or 15
N and its correlation with the
proton longitudinal polarization (so-called two spin order).
Then the INEPT step is used to transfer the polarization from protons to the attached heteronuclei (13
C or
15
N) to profit from the larger polarization of proton. Unlike in the standard HSQC experiment (and similar to
HC(C)H-TOCSY, see Figure 8.2B), a second INEPT transfer is necessary to create a pure in-phase polarization of
the heteronucleus.
The preparation step ends with a single 90◦
pulse used to transform the 13
C or 15
N in-phase coherence into the
polarization aligned with the external static magnetic field. Its absolute value is labeled Pinit in this paragraph.
Depending on the phase of the last applied pulse (+y or −y), a longitudinal polarization either parallel or anti-parallel
with the external static magnetic field is created, which is distinguished by the sign of the Pinit in the following equations
(positive for parallel orientation). Note that parallel polarization is not equal to the equilibrium polarization of 13
C
or 15
N. It is larger due to the initial polarization transfer from the attached proton, as discussed in Section 4.1.
In the following block (shown in a blue frame), the polarization P(t) decays towards the equilibrium polarization
P∞. The decay of P(t) is exponential with a longitudinal relaxation rate constant R1. For the parallel polarization at
the beginning of the relaxation delay we get
P1(t) = (Pinit − P∞)e−R1t
+ P∞, (10.3)
while if we apply the 90◦
pulse with the opposite phase we start with antiparallel polarization −Pinit and the
evolution of the polarization P2(t) towards the equilibrium P∞ follows the equation
P2(t) = −(Pinit + P∞)e−R1t
+ P∞. (10.4)
It is not convenient to evaluate more unknown variables than necessary, therefore the experiment is measured twice
in an interleaved manner and the signals originating from both polarizations P1(t) and P2(t) are subtracted, providing
a simple mono-exponential decay
P1(t) − P2(t) = 2Pinite−R1t
. (10.5)
Compared to the simple inversion-recovery experiment one should notice that the relaxation period is not so
simple but there is a proton inversion pulse applied in the middle of the relaxation period. It inverts the effect of the
cross-relaxations, so that the cross-relaxations effects before and after this pulse cancel each other.
The third part of the pulse sequence, shown in a green frame in Figure 10.4A, converts the 13
C or 15
N longitudinal
polarization modulated by the exponential decay to a measurable signal. A 90◦
pulse at the end of the relaxation
delay flips the 13
C or 15
N longitudinal polarization to the horizontal one, corresponding to the 13
C or 15
N coherence.
Its evolution during the incremented delay t1 introduces modulation of the signal by the 13
C or 15
N chemical shift.
This goal is achieved in the same way as in the HC(C)H-TOCSY, see Figure 8.2B) HSQC experiment 4.2. Finally,
the polarization is transferred back to proton using two INEPT steps (like in the HC(C)H-TOCSY experiment, cf.
Figure 8.2B). The created proton coherence is detected in the real time t2. Fourier transformation of the accumulated
signal results in a series of two dimensional spectra, with the proton frequency offset in the direct dimension, the 13
C
or 15
N frequency offset in the indirect dimension, and peak intensity decaying exponentially in the series of spectra
with the R1 rate constant.
118
The experiment for the measurement of the transverse relaxation rate (shown in Figure 10.4B) is very similar to the
experiment for the longitudinal relaxation rate, except for the relaxation delay element (shown in a blue frame) and
few differences in the central part of the pulse sequence. First, the last 90◦
pulse applied prior the relaxation period
is missing because the desired in-phase polarization is already created at the end of the second INEPT step. The
relaxation period for the transverse relaxation has similar form to that one introduced in Section 10.3. It is composed
of different numbers of (refocusing) spin echoes. It is necessary to keep in mind that a J-coupled pair of magnetic
moments evolves not only due to the chemical shift. The horizontal polarization oscillates between the in-phase and
anti-phase coherences as discussed in Section 3.3. This oscillation is refocused at the end of the spin echo, but evolves
during the echo (see Section 3.6). Therefore, the spin-echoes must be sufficiently short so that the original in-phase
coherence is refocused before it has enough time to evolve into an anti-phase coherence, which relaxes differently. If
the echo is too long, the measured relaxation rate represents a mixture of the relaxation rates of the in-phase and
anti-phase coherences. The length of the spin echo 2τ should be chosen to fulfill the condition: 2τπJ ≪ 1, where J
is the J-coupling constant. In addition, it is also necessary to suppress the cross-relaxation between the in-phase and
anti-phase coherences. Therefore, additional proton 180◦
pulses are inserted in the relaxation period, which effectively
suppresses the effect of this cross-relaxation.1
The relaxation block of the pulse sequence is followed by the incremented delay t1, when the signal is modulated by
the 13
C or 15
N frequency offset. Because we keep the 13
C or 15
N transverse magnetization throughout the relaxation
delay, the 90◦
pulse creating the in-phase magnetization in the case of the experiment for the measurement of the
longitudinal relaxation rate is missing. The rest of the pulse sequence is identical with the experiment for the measurement
of the longitudinal relaxation rate. The dependence of the detected signal on the relaxation delay reflects the
relaxation of the coherence during the relaxation delay and it follows Eq. 10.2, where the relaxation delay is calculated
slightly differently: t = 2Nτ + 8τ.
Finally, we describe the experiment for the measurement of the steady-state NOE. Its principle is the same as
discussed for its simple 1D version in the previous section 10.4. Two experiments are recorded. In the first one, the
steady state condition is created by a long irradiation of protons (Figure 10.4C). In the second one, the system is let
relax long enough to reach its equilibrium (or a state very close to it) prior to the initial 90◦
pulse (Figure 10.4D).
Therefore, the created coherence is given by the equilibrium longitudinal polarization of 13
C or 15
N. The final part of
both sequences is the same as in the experiment for the measurement of the longitudinal relaxation rate (shown in a
green frame).
1The proton 180◦ pulses are applied after the first quarter of the relaxation period and at three quarters of the relaxation period. The
proton 180◦ pulses invert the effect of the cross-relaxation, so that the sum of the effects of the cross-relaxation in the outer parts of the
relaxation period cancels with the effect in the middle part.
10.5. MEASUREMENT OF THE RELAXATION RATES IN BIOMACROMOLECULES 119
d1
1
H
13
C/15
N
G
y
t1/2t1/2
y
ττ
N/4
ττττ
N/4
ττττττττττ
N/2
d1
1
H
13
C/15
N
t1/2t1/2
y
d'1
1
H
13
C/15
N
t1/2t1/2
y
d1
1
H
13
C/15
N
G
y
±y
t/2t/2
t1/2t1/2
y
a)
b)
c)
d)
Figure 10.4: Simplified pulse sequences of experiments used to determine heteronuclear longitudinal relaxation rates (A), transverse
relaxation rates (B) and steady state NOE (C and D) in biomacromolecules. The examples are provided for 13C-1H or 15N-1H spin pairs.
The lines correspond to radio-waves applied at the precession frequency of protons and 13C or 15N, and to the magnetic field gradients
(labeled G). Narrow and wide boxes represent 90◦ and 180◦ pulses, respectively. The pulses are applied with x phase unless indicate
otherwise. The open rectangles represent decoupling trains. The delay ∆ is equal to 1/(4J) where J is the J-coupling constant. Two
experiments are depicted for the measurement of the steady state NOE: the experiment in which the steady state is created (C) by proton
saturation (low rectangle), and the reference experiment (D) in which the equilibrium 13C or 15N polarization is measured. The delays
d1 and d′
1 have different length. A long d′
1 must be used to assure achieve of a state close to the equilibrium. The red, blue, and green
frame mark the preparation phases of the experiments, the variable relaxation delays, and the detection elements of the pulse sequences,
respectively.
120
Lecture 11
Analysis of internal motions
11.1 Internal motions
So far, we discussed just the rotational diffusion of the whole molecule. It is much more exciting to study internal
motions of the molecule. A brief list of internal molecular motions with their time scales motions is presented in
Table 11.1.
Table 11.1: Typical time scales of internal molecular motions
Time scale type of motion
10−15
s Bond-length fluctuations (stretching vibrations)
10−14
–10−13
s Bond angle and torsion angle fluctuations
10−12
–10−11
s Helix bending, β-sheet twisting, simultaneous local torsion angle flips,
conformational changes in disordered regions
10−10
–10−7
s Local conformational changes
(helix growing, sheet breathing, local folding events)
10−6
–10−3
s Fast changes of conformation, fast chemical reactions
10−2
–100
s Interconversion of conformational states, slow chemical reactions,
folding of small proteins (with no prolines and disulfide bridges)
> 100
s Slow folding events (proline isomerization, S–S bond formation),
exchange of amide and imino protons with water
We now explore the effect of internal motions on the NMR relaxation, starting with motions that affect only
orientation of the interacting magnetic moments but do not change magnitude of the interaction. In the NMR
experiments, we monitor motions of vectors connecting nuclei interaction via the dipole-dipole interaction, or of
vectors defining orientation of the electronic environment surrounding the observed nucleus.1
We now attempt to describe the internal motions changing orientation of the interacting magnetic moments under
some simplifying circumstances. First, we assume that only one type of motion affects the amide bond. Second, we
assume that the motion is random and can be described in a fashion similar to the rotational diffusion. Therefore,
we expect an exponential decay for the correlation function of the internal motion. We cannot use Eq. 9.1 directly.
Eq. 9.1 describes unrestricted random rotation, which is an unrealistic assumption for a motion of a chemical bond in
a well-structured molecule. To fix the problem, a so-called order parameter can be introduced to describe motional
restriction. The single exponential term in Eq. 9.1 is then replaced by the expression
CI(t) = (1 − S2
)e−RIt
+ S2
, (11.1)
where S2
represents the order parameter and RI is a constant describing the rate of loosing the correlation, equal
to the inverse value of the correlation time describing the fast motions. Note that CI(t) decays exponentially from
unity (for t = 0) to S2
(for t → ∞). Replacing S2
with zero gives us back the correlation function from Eq. 9.1. We
1Technically, the electron distribution influencing observed frequencies of nuclei are described by chemical shift tensors.
121
122
see that S2
= 0 represents the unrestricted limit. On the other hand, setting S2
= 1 makes CI(t) equal to one for any
t. It shows that S2
= 1 describes complete rigidity, i.e., lack of any internal motion.
The rotational diffusion is always present in solution. So, we need to combine Eqs. 9.1 and 11.1. To do it, we need
the last assumption. We expect that the internal motions and overall rotational diffusion are independent and that
the whole molecule still behaves as a sphere. Then, we can write
C(t) = CO(t)CI(t) =
1
5
S2
e−RIt
+
1
5
(1 − S2
)e−(RI+RO)t
, (11.2)
where RO = 1/τ from Eq. 9.1 and the subscripts O and I distinguish overall and internal motions, respectively. It
is useful to note that the effect of the internal motions cannot be distinguished from the overall rotational diffusion if
the internal motions are slower. It can be checked easily by setting RI ≪ RO. Then,
C(t) = CO(t)CI(t) =
1
5
S2
e−ROt
+
1
5
(1 − S2
)e−(RI+RO)t
→
1
5
S2
e−ROt
+
1
5
(1 − S2
)e−ROt
=
1
5
e−ROt
. (11.3)
Fourier transformation of C(t) gives the spectral density function
J(ω) = S2
2
5 RO
R2
O + ω2
+ (1 − S2
)
2
5 (RO + RI)
(RO + RI)2 + ω2
, (11.4)
which can be used in the equation defining the relaxation rates. Note that we obtain a sum of two functions
resembling the description of NMR peaks. Such expressions are known as the Lorentzian functions.
Further complexity is introduced if we need more than one type of internal motions to describe the relaxation or
if the overall shape of the molecule is far for spherical.
11.2 Motional parameters from NMR measurements
The measured values of R1, R2, and Rx (Sections 10.2–10.5) depend on five values of the spectral density function,
taken at five different frequencies. If we study relaxation of 15
N, these frequencies are zero, ωN, ωN + ωH, ωH, and
ωN − ωH. Analysis of the form of the spectral density function introduced in the Eq. 11.4 shows that it is a linear
combination of Lorentzian functions. An example of its profile is shown in Figure 11.1. Two facts can be noticed.
First, the magnetogyric ratio of proton is approximately 10× larger than the magnetogyric ratio of 15
N, so the last
three frequencies do not differ too much. Second, the dependence does not decay rapidly at frequencies close to ωH
in cases of biomolecules studied by high-field NMR. Therefore, the last three values of spectral density function at
frequencies ωN + ωH, ωH, and ωN − ωH differ very slightly and they can be replaced by a single ”high frequency” value
of the spectral density function. We see that the measured three relaxation rates are just linear combinations of three
values of the spectral density function. It is easy to calculate the spectral density function values from the relaxation
rates. We know that the spectral density functions map the rate of molecular motions. The described analysis of the
relaxation data, known as reduced spectral density mapping, represents a straightforward way to the information about
molecular motions.
If we want to obtain more intuitive description of motions, we can fit the measured relaxation rates to the corresponding
equations and obtain directly the correlation rates (or times) and the order parameters. Of course, the
number of fitted parameters cannot be larger than the number of measured relaxation rates (can be three, although
it is recommended to repeat the measurements at several different magnetic fields to improve the analysis). As a consequence,
only one type of internal motions can be fully described by its order parameter (defining range of motion)
and correlation rate (defining the rate of the motion) if the relaxation rates were acquired only at a single field. Note
that such description of motions is obtained for each residue which amide peak is resolved in the 1
H-15
N correlation
spectrum. The third available parameter may be the rate of overall rotational diffusion, but this parameter is the
same for all residues and it is usually obtained in an independent way.2
The third parameter can be used to partially
describe an additional type of motion. The analysis based on fitting correlation rates and order parameters is known
as the model-free approach. The name reflects the fact that we do not need a mechanistic description of the motion
to learn the average range and rate.
2The average of R2/R1 ratios provides a good estimate.
11.3. CHEMICAL EXCHANGE 123
0.001
0.01
0.1
1
10
100
0 1 2 3 4 5
J(ω)/nsrad-1
ω / ns rad
-1
Figure 11.1: Example of the spectral density function with highlighted values of the spectral density values at frequencies that contribute
to the relaxation rates of the 15N-1H pair at 14.1 T (600 MHz spectrometer).
11.3 Chemical exchange
We now extend our description of molecular motions to structural changes affecting also the magnitude of the interaction.
The values of bn and be vary if the distance between interacting nuclei, or the electron distribution around the
observed nucleus change. The causes of such variations can be changes in conformation but also in chemical configuration,
including intermolecular interactions and chemical reactions. In NMR spectroscopy, the structural changes
are called conformational exchange or chemical exchange.
We have shown in Eq. 11.3 that the rate constants describing fast relaxation are not sensitive to much slower
changes in orientation of vectors describing mutual orientations of interacting magnetic moments. However, slow
exchange processes changing magnitude of the interaction have impact on the shape of the NMR spectral lines and
affect the evolution of the magnetization components parallel and perpendicular to the external magnetic field.
The rate equations describing the chemical exchange are similar to the equations describing evolution of the magnetization
components due to the fast relaxation. We just need to treat the magnetization components separately for
each state of the molecule, and to add the chemical rate constants to the relaxation rates and precession frequencies in
the equations. The description can be also applied to bimolecular processes such as ligand binding, proton dissociation,
or enzyme catalysis.
11.4 Effect of chemical exchange on the transverse magnetization
Effects of the chemical exchange on the evolution of the magnetization components perpendicular to the external
magnetic field can be studied in a similar fashion as the R2 relaxation. Direct evaluation of the line-width in the
spectrum provides a quick estimate of the exchange contribution to the loss of coherence. Spectra can be simulated
for various values of the rate constants and compared to the experimental data. However, the line shape is in reality
also influenced by inhomogeneities of the magnetic field. Therefore, the determined relaxation rate may be affected
by a systematic error.
More sophisticated methods known as relaxation dispersion analysis are used if we want to fully exploit the
information hidden in the spectra. There are several methods known as relaxation dispersion experiments allowing
us to vary the chemical exchange contribution to apparent R2 by changing experimental conditions. Here, we briefly
introduce a method called by he abbreviation composed of the first letters of its authors: CPMG (Carr-Purcell-
124
Meiboom-Gill). This experiment is derived from the experiment for the measurement of the transverse relaxation
rate 10.3. During the description of the experiment for the measurement of the transverse relaxation rate, it was
demonstrated that its relaxation delay must be composed from spin-echo elements. The choice of the length of the
single spin echo elements is not important as long as we are studying an isolated spin which is not undergoing a
chemical exchange. In this case the chemical shift will be same during the whole spin echo and the coherence will be
refocused regardless of the length of the delay τ (half of the spin echo). But, the length of the τ becomes important if
the studied molecules undergo an exchange affecting the chemical shift of the observed nucleus. Then, the evolutions
in the left and right part of the spin echo are different for different molecules depending on when the molecule switched
the state during the spin echo. An example of such a case is given in the Figure 11.2C. Consequently, an additional loss
of coherence is observed and the measured relaxation rate is elevated by this effect. Clearly, if the difference between
the chemical shifts corresponding to the exchanging states is larger, the loss of the signal due to its dephasing is more
rapid, because the magnetic moments in different states reorient more quickly. The example given in Figure 11.2
shows a case when the numbers of the studied molecule in both states are equal. However, it is often not the case and
the ratio of the molecules in different states is another important factor. It is obvious that if most of the molecules are
present in one state and only a relatively small fraction of them is in the other one, the loss of coherence is diminished.
The effect of the loss of coherence can be suppressed if the refocusing pulses are applied faster as it is shown in
Figure 11.2D. If we apply pulses sufficiently frequently, we are able to suppress the dephasing effect of the exchange
as it is illustrated in Figure 11.2E). Obviously, the slower the exchange is, the less frequent application of refocusing
pulses is sufficient to suppress the effects of the exchange. Therefore, the analysis of the dependence of the relaxation
rate on the frequency of application of the refocusing pulses during the relaxation delay allows one to extract the
information about the rate of the exchange.
The example of the dependence of the transverse relaxation rate on the increasing frequency of the application of
the refocusing pulses during the relaxation delay is shown in Figure 11.3. You may notice a decreasing profile, which is
becoming flat for higher frequencies, when the refocusing pulses are applied fast enough to suppress dephasing of the
signal due to the exchange. Analysis of such a dependence provides information about the rate of the exchange (a kinetic
parameter), equilibrium populations of individual states undergoing the exchange (a thermodynamic parameter), and
chemical shifts of both (or all) interchanging states (encoded structural information).
Finally, it is important to recall that the experiments for studies of motions of biomolecules (Section 10.5) are
slightly more complicated because they are based on the analysis of 13
C of 15
N nuclei in the 13
C-1
H or 15
N-1
H
pairs. Therefore, additional effects like cross-relaxation and effects due the J-coupling between these nuclei must be
considered. We can for example recall that during the spin echo the in-phase coherence evolves into the anti-phase
coherence and back. Consequently, the relaxation reflects the weighted average of the relaxation rates of these two
coherences. Because these two rates are not same, the measured relaxation rate depends on the length of the spinecho
element even without the presence of the chemical exchange and this effect would interfere with the measured
relaxation dispersion curve, which makes the analysis difficult. Therefore, this effect must be either suppressed or
compensated.
The former approach (suppression) is represented by the experiment when continuous proton decoupling is applied
during the whole relaxation delay (i) ensuring that the in-phase coherence do not evolve into the anti-phase coherence
at all and (ii) suppressing the cross-relaxation effects. The pulse sequence (Figure 11.4A) is very similar to the pulse
sequence for the measurement of the transverse relaxation rate in biomolecules (Figure ??B).
The latter approach (compensation) requires also modification of the preparation phase of the pulse sequence.
There is only one INEPT step present in order to create the anti-phase 15
N coherence. So, it resembles rather the
beginning of the HSQC experiment introduced in Section 4.2. The following relaxation delay is split into two parts
and a simultaneous echo followed by a 180◦
pulse is inserted in the middle. This element exchanges the in-phase and
anti-phase coherences in the middle of the relaxation delay. The initially created anti-phase coherence evolves into an
in-phase coherence and back during each spin-echo in the first part. Then the anti-phase coherence is transformed into
the in-phase coherence, which evolves into the anti-phase coherence and back during each spin-echo in the second part
of the relaxation delay. Such an approach effectively averages the time when the in-phase and anti-phase coherences
are present, regardless of the chosen length of the spin echo. The proton 180◦
pulse, following the central simultaneous
echo, is necessary to suppress the effects of the cross relaxation.3
3A combination of switching between the anti-phase and in-phase coherences by the simultaneous echo with inverting of the crossrelaxation
effects by the proton 180◦ pulse is counter-productive for the suppression of the cross relaxation effects. In order to suppress
11.4. EFFECT OF CHEMICAL EXCHANGE ON THE TRANSVERSE MAGNETIZATION 125
τ1 τ2 τ3 τ4 τ5 τ6 τ7 τ8 τ9 τ10 τ11 τ12zero frequency
(on resonance)
20 degree per step
τ1 τ2 τ3 τ4 τ5 τ6+ + + + + τ7 τ8 τ9 τ10 τ11 τ12
+ + + + +
A B
180
τ1 τ2 τ3 τ4 τ5 τ6+ + + + + τ7 τ8 τ9 τ10 τ11 τ12
+ + + + +
C
180 180
τ1 τ2 τ3+ + τ4 τ5 τ6 τ7 τ8 τ9+ + + + + τ10 τ11 τ12
+ +
τ1 τ2 τ3+ + τ4 τ5 τ6 τ7 τ8 τ9+ + + + + τ10 τ11 τ12
+ +
D
180 180180
τ1 τ2+ τ3 τ4 τ5 τ6+ + + τ7 τ8 τ9 τ10+ + + τ11 τ12
+
τ1 τ2+ τ3 τ4 τ5 τ6+ + + τ7 τ8 τ9 τ10+ + + τ11 τ12
+
E
Figure 11.2: Illustration of the effect of various lengths of the spin echo elements on the coherence loss due to an exchange. A, Definition
of the system used to explain the function of the CPMG experiment. Open and filled circles represent two states of the molecule undergoing
the exchange. B, Variants of the exchange events. The studied system is represented by eight molecules distinguished by colors. Each of
them undergoes a unique variant of the exchange events. The relaxation delay is divided into 12 steps τ1–τ12. The rate of the exchange is
limited so that the molecule has to keep its state for at least 4 time steps before it can change the state again. C–E, three variants of the
splitting of the relaxation delay by spin echo elements with increasing number of spin echoes. The colored arrows display the orientation
of the magnetisation in the transverse plane for each molecule. The arrows allow us to follow the effect of the exchange and refocusing
pulses on each molecule. The final sum of the signals increases with the increasing frequency of application of the refocusing pulses.
126
6.7
6.8
6.9
7.0
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
0 200 400 600 800 1000
R2/s
-1
ν / s
-1
Figure 11.3: Example of the experimental CPMG relaxation dispersion profile. The data eshibit a decay of a transverse relaxation rate
with increasing frequency of the application of the refocusing pulses during the relaxation delay. The displayed frequency ν is calculated
as 1/(4τ), where τ is the half of the used spin echo element. The red line correspond to the theoretical dependence simulated for the
parameters defining the studied exchange.
There is no difference in sensitivity to the chemical exchange between the two methods described above because
both in-phase and anti-phase coherences are affected by the chemical exchange. However, in the first case the effect
of the chemical exchange increases the pure in-phase relaxation rate, while in the second case it does the same to the
average of in-phase and anti-phase relaxation rates.
The last experiment discussed in this section dedicated to the study of slow exchange in biomolecules is called
Chemical Exchange Saturation Transfer (CEST). It is designed to investigate the exchange with so highly skewed
populations between states so that minor state(s) is not visible in the spectra. It resembles the experiment for the
measurement of the longitudinal relaxation rate (Section 10.5), because the 13
C / 15
N longitudinal polarization is also
created prior the beginning of the relaxation delay but the relaxation delay is different. The pulse sequence is shown in
Figure 11.4C. A very weak continuous irradiation is applied during the relaxation delay, which has similar a effect as
selective presaturation discussed in Section 6.2. It effectively suppresses any signal which is present at the frequency of
the irradiating radio wave. The experiment is typically repeated with a fixed length of the irradiation but its frequency
is changed step by step throughout the region where the expected chemical shift of the minor state(s) is expected. If
the frequency of the irradiating radio wave gets close to the precession frequency of the minor state, magnetic moments
of the minor state aadopt distribution of with no polarization. The loss of polarization is transferred via chemical
exchange to magnetic moments in the major state. Therefore, intensity of the observable peak (of the major state)
decreases. A dependence of the signal intensities on the frequency of the irradiation reveals the position of the minor
states in the spectra (typically, an additional spectrum is acquired with the relaxation delay completely omitted to
obtain reference peak intensities and the peak intensities relative to this reference is plotted). The dependence is
characterized by a major minimum, when the irradiation resonates with the visible peak in the spectrum and other
minima showing the positions of frequencies corresponding to the other states as it is shown in Figure 11.5.
them, we shall not do both at once, therefore the second proton inversion pulse is applied at the end of the simultaneous echo to reverse
the effect of the proton inversion pulse in the middle of the echo on the cross-relaxation.
11.4. EFFECT OF CHEMICAL EXCHANGE ON THE TRANSVERSE MAGNETIZATION 127
d1
1
H
13
C / 15
N
G
y
±y
t
t1/2 t1/2
y
A
d1
1
H
13
C / 15
N
G
y
t1/2 t1/2
y
ττ
N
B
d1
1
H
13
C / 15
N
G
y
t1/2 t1/2
y
ττ
N/2 N/2
ττ
C
Figure 11.4: Schemes of simplified experiments used to the study chemical exchange in biomolecules. A, CPMG experiment keeping
the in-phase coherence during the relaxation delay. B, relaxation compensated CPMG experiment with the inter-exchange of in-phase an
anti-phase coherences during the relaxation delay. C, CEST experiment. The examples are provided for 13C-1H or 15N-1H spin pairs.
Two lines correspond to two channels used to apply radio-wave pulses to proton and to 13C or 15N. The third lines labeled G represents
the pulsed field gradients. Narrow and wide filled boxes represent 90◦ and 180◦ degree pulses, respectively. The pulses are applied with
the x phase unless indicated otherwise. The lower open rectangle applied during the acquisition is heteronuclear decoupling. The delay ∆
is equal to 1/4J where J is J-coupling constant. The lower filled rectangle applied in the CEST experiment during the relaxation delay is
the CEST irradiation. The lower open rectangle applied during the relaxation delays in Panels A and C is proton decoupling. The red,
blue, and green boxes depict the preparation phase of the experiments, the variable relaxation delay, and the detection element of the pulse
sequences, respectively.
128
I/I0
112114116118120122124126128
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
δ (15
N) / ppm
Figure 11.5: Illustration of CEST experimental data displayed as a dependence ratio between peak intensity in the CEST experiment
and reference experiment on the frequency of the irradiation during the CEST relaxation delay. The major and minor minima in the profile
show the frequencies of the highly populated state and the minor state, respectively.
11.5 Effect of chemical exchange on the longitudinal magnetization
Chemical exchange also affects the z component of magnetization. Experiments similar to the steady-state NOE
measurement can be run. Although the results of such saturation transfer experiments are similar to the effect of
NOE, the physical mechanism is completely different. Signal intensities are changed by NOE because two protons are
close in space and their magnetic dipolar moments interact directly. In the saturation transfer experiments, on the other
hand, we observe two forms of the same proton. If the form (e.g., conformation) of the molecule changed during the
experiment, the proton is measured at the frequency corresponding to the new form but the signal intensity is affected
by the magnetic moment distribution in the original form. Saturation transfer experiments are easy to evaluate,
but applicable just to the two-site exchange (for technical reasons). Two-dimensional versions of this experiment,
resembling NOESY, are also available.
The same principles apply to the inversion transfer experiment. Like in the R1 measurement, longitudinal polarization
of a nucleus in one form are selectively inverted and the signal intensity of both forms is monitored. Evaluation
of the inversion transfer requires multi-exponential fitting but more information can be obtained compared to the
saturation transfer experiments.
Lecture 12
Intermolecular interactions
12.1 Studies of intermolecular interactions
NMR is recognized as a useful tool for binding studies of biomolecules. It allows confirmation of interactions, identification
of binding interface and it provides information about binding constant, stoichiometry, and rate of binding
events. Methods of fast and cheap identification of binding partners in mixtures of potential ligands have been also
developed.
12.2 Titration
We start our discussion of NMR binding studies by describing titration of a biomolecule by its binding partner.
This method can be used not only to investigate a simple binding event described in the example below, but it
is applicable also to studies of complex binding mechanisms like multistep binding, cooperativity, induced fit or
conformational selection, etc. Typically, the sample of the biomolecule is prepared with some isotope enrichment so
that heteronuclear experiments can be used. The 15
N enrichment of proteins is usually sufficient for titration studies,
unless we are working with a protein which 15
N-1
H HSQC is characterized by a severe signal overlap. The ligand
is usually not isotopically enriched and we investigate exclusively the changes in the protein spectrum induced by
the binding. The protein spectrum (most often 15
N-1
H HSQC) is acquired repeatedly, the first spectrum is recorded
without any ligand present in the solvent, and then the measurement is repeated after each titration step, i.e., with
an increasing content of the ligand. In such manner, we obtain a series of NMR spectra starting from that of the free
protein up to the spectrum of the protein fully saturated by the ligand. The titration step must be chosen so that it
is possible to follow the changes in the spectra.
Direct comparison of the first spectrum (free protein) and the last spectrum (highest concentration of ligand)
allows us to identify binding regions of the protein if the protein assignment is known. If both the assignment and the
structure of the protein is known, it is possible to map the changes in frequencies to the structure in order to identify
the binding site(s). As multidimensional spectra with different nuclei detected in the individual axis are typically
measured, the chemical shift changes of different nuclei must be combined, taking into the account the typical range
of frequencies of the nuclei of interest. The resulting value is the chemical shift perturbation parameter (CSP). For the
experiment applied most often, i.e. for 15
N-1
H HSQC, CSP is calculated as
CSP =
1
2
(∆δ2
H + (α∆δN)2), (12.1)
where the scaling factor α is typically chosen based on the observed ranges of chemical shifts of both nuclei. For
example, if the range of amide 15
N shift is between 131 and 109 ppm and the range of amide protons is between 10.0
and 6.9 ppm, the α factor can be calculated as α = (10.0 − 6.9)/(131 − 109) = 0.14. However, one has to keep in
mind that each protein has slightly different ranges of chemical shifts. It should be also stressed out that there is no
theoretical justification of the assumption that the relative sensitivity of chemical shifts of individual nuclei to the
ligand binding is given by the ranges of the chemical shifts.
129
130
It is necessary to keep in mind that high values of CSP are not a direct evidence of a binding to that area of the
protein. Such simplification can lead to a misinterpretation of data. The chemical shifts may be disturbed for example
also by allosteric effects far away from the binding site.
The detailed analysis of the spectra of the whole titration series provides additional information about the binding
constant and stoichiometry and kinetic of the binding. Changes in the spectra during the titration depends on the
relative size of the rate of the exchange (sum of binding and dissociation rate) and the difference between chemical
shifts in the free and bound state. When the exchange rate is much larger than the difference of the chemical shifts
(kex = kassoc + kdissoc ≫ ∆ω), the binding is in the fast exchange regime (usually low-affinity binding). On the other
hand, if kex =≪ ∆ω, the binding is in the slow exchange regime (usually strong binding).
The fast exchange regime can be recognized by a subsequent shift of the peak between the initial position (free
protein) towards the final position corresponding to the bound form. On the other hand, a progressive loss of the
signal intensity at the initial free-state position and increase of signal intensity at the position of the bound state are
characteristic for the slow exchange regime. The peak linewidths are not dramatically increased during the titration
in the fast or slow exchange regimes. But, the peaks are getting significantly broadened (eventually even beyond the
detection limit) in the cases when the exchange rate is similar to the difference of the chemical shifts. This case is
called intermediate exchange and it provides a bit more complex picture than the slow or fast exchange regime. In
the beginning of the titration, the peak starts to shift towards the position of the bound form, but it becomes also
broadened and therefore its intensity drops. It might be completely undetectable during the middle phase of the
titration, but it gets sharper in the final part of the titration when there is an extent of the ligand and the peak
position gets close to its target position. All three cases are illustrated in the Figure 12.1. Note, that if a multiple
dimensional spectrum is measured with different nuclei detected in the individual dimensions the discussed regime can
be different for each of the dimensions.
In order to quantify the changes in the spectra, different parameters are used in the slow and fast exchange regimes.
If the exchange is fast, a gradual increase of CSP can be followed (Figure 12.3). In the case of a simple equilibrium
between ligand-free and ligand-bound form of the protein, CSP divided by its maximum value is equal to the fraction
of the protein in its bound form:
y =
[PL]
cP
=
cP − [P]
cP
=
CSP
CSP[L]≫[P]
, (12.2)
where cP, [P], [L], and [PL] are the concentrations of total protein, free protein, free ligand, and of the complex,
respectively. This allows us to determine the dissociation constant Kd directly from the titration curve (dependence
of y on the total concentration of the ligand in each titration step cL). The shape of the titration curve can be derived
as follows.
Kd =
[P][L]
[PL]
=
(cP − [PL])(cL − [PL])
[PL]
(12.3)
Kd
[PL]
=
cP − [PL]
[PL]
cL − [PL]
[PL]
=
cL
cP
cP
[PL]
2
− 1 +
cL
cP
cP
[PL]
+ 1 (12.4)
cP
[PL]
2
− 1 +
Kd + cP
cL
cP
[PL]
+
cP
cL
=
1
y
2
− 1 +
Kd + cP
cL
1
y
+
cP
cL
= 0 (12.5)
CSP[L]≫[P]
CSP
=
1
y
=
1
2

1 +
Kd + cP
cL
± 1 +
Kd + cP
cL
2
− 4
cP
cL

 (12.6)
CSP
CSP[L]≫[P]
= y =
2
1 + Kd+cP
cL
± 1 + Kd+cP
cL
2
− 4cP
cL
. (12.7)
The dependence is plotted in Figure 12.2 for high affinity (red), intermediate affinity (green), and low affinity
(blue). Titrating the proteins in six steps (i.e., recording the spectrum of a free protein and of a protein with one to
12.2. TITRATION 131
δ(15
N)/ppm
123
122
121
120
118
119
δ (1
H) / ppm
8.0 7.9 7.8
step 1
δ (1
H) / ppm
8.0 7.9 7.8
step 2
δ (1
H) / ppm
8.0 7.9 7.8
step 3
δ (1
H) / ppm
8.0 7.9 7.8
step 4
δ (1
H) / ppm
8.0 7.9 7.8
step 5
δ(15
N)/ppm
123
122
121
120
118
119
δ (1
H) / ppm
8.0 7.9 7.8
step 1
δ (1
H) / ppm
8.0 7.9 7.8
step 2
δ (1
H) / ppm
8.0 7.9 7.8
step 3
δ (1
H) / ppm
8.0 7.9 7.8
step 4
δ (1
H) / ppm
8.0 7.9 7.8
step 5
δ(15
N)/ppm
123
122
121
120
118
119
δ (1
H) / ppm
8.0 7.9 7.8
step 1
δ (1
H) / ppm
8.0 7.9 7.8
step 2
δ (1
H) / ppm
8.0 7.9 7.8
step 3
δ (1
H) / ppm
8.0 7.9 7.8
step 6
δ (1
H) / ppm
8.0 7.9 7.8
step 5
δ (1
H) / ppm
8.0 7.9 7.8
step 4
δ (1
H) / ppm
8.0 7.9 7.8
step 6
δ (1
H) / ppm
8.0 7.9 7.8
step 6
A)
C)
B)
Figure 12.1: Changes observed in the spectra of a protein upon its titration with a ligand. Titration is divided into six steps in this
example. The red and black peaks stand for the case of the free protein and protein fully saturated with its ligand. Orange, green, cyan,
and blue peaks then represent spectra with increasing ligand concentration, respectively. The intensities of the peaks are depicted by the
number of contours. Fast (A), intermediate (B), and slow (C) exchange regimes of the binding (with respect to the 15N chemical shift) are
displayed. Note that the regime is fast for proton dimensions in all cases as in this example the binding do not change the proton chemical
shift.
132
0.0
0.2
0.4
0.6
0.8
1.0
0 1 2 3 4 5
y=[PL]/cP
cL/cP
Figure 12.2: Titration curves for three values of the dissociation constant Kd: Kd = 0.01cP (red), Kd = 0.1cP (green), and Kd = cP
(blue). The limit for Kd → 0 (irreversible binding) is shown in black. Total concentrations of the added ligand are plotted relative to the
total protein concentration.
five aliquots of ligand corresponding to cL ranging from cP to 5 × cP) is likely to provide reliable value of Kd for the
intermediate-affinity binding (green). In this case, Kd is the only unknown parameter in fitting the experimental data
to Eq. 12.7. If the affinity is too low (blue curve), it may be difficult to record a spectrum of protein saturated with
the ligand and CSP[L]≫[P] must treated as another fitted parameter in fitting the experimental data to Eq. 12.7. In
the biologically more interesting case of high affinity (red curve), fitting Kd would be difficult because the curve is
close to the limit of tight binding (black lines), when the dependence does not change with decreasing Kd any more.
In the case of the slow exchange, peaks do not move in the spectra and SCP cannot be evaluated. In principle,
the intensities (most accurately volumes) of peaks of the free and bound form are in the ratio [P] : [PL]. However,
this assumption is true only if the relaxation rates of free protein and of the complex are identical. If the relaxation
rates differ, the correct ratio can be obtained only for areas of peaks in one-dimensional experiments where detection
immediately follows the 90◦
excitation pulse. Furthermore, the slow exchange is typical for dissociation constants
much lower than concentrations used in NMR experiments (cf. the red curve in Figure 12.2). In such cases, the
dependence of y on cL described by Eq. 12.7 approaches its linear limit (shown in black in Figure 12.2), independent
of Kd (because Kd + cP ≈ cP). Therefore, an accurate determination of Kd may not be possible. On the other hand,
the linear dependence allows us to precisely determine stoichiometry of binding (equal to one in the example presented
in Figure 12.2).
The intermediate exchange represents a case which is most challenging but also richest in information. A careful
choice of the ligand:protein ratio, so that the peak broadening due to the exchange occurs but the peak intensity can
be still measured reliably (e.g., step 5 in Figure 12.1B), allows us to determine not only Kd, but also the exchange rate
kex, using the methods described in Section 11.4. As kex = kassoc + kdissoc and Kd = kdissoc/kassoc, the rate constants
kdissoc and kassoc can be calculated.
12.3 Saturation transfer difference
The saturation transfer difference experiment (STD) is very popular in industry, particularly in the pharmaceutical
industry where it is used as a relatively quick, cheap and effective drug screening tool. The experiment is based on
comparison of proton spectra, therefore it is not necessary to introduce any isotope labeling of the biomolecule or of
the small molecules which are screened for binding to the biomacromolecule. In addition, the price is reduced by the
12.3. SATURATION TRANSFER DIFFERENCE 133δ(15
N)/ppm
123
122
121
120
118
119
δ (1
H) / ppm
8.0 7.9 7.8 7.7 7.6 7.5 7.4 7.3 7.2 7.1 7.0 6.9 6.8
Δδ1 (1
H)
Δδ2 (1
H)
Δδ3 (1
H)
Δδ3 (15
N)
Δδ1(15
N)
Δδ2(15
N)
Figure 12.3: Chemical shift perturbation (CSP). An overlay of a region of hypothetical 1H-15N HSQC spectra is shown. The red peaks
correspond to the spectrum of a free protein. The black peaks are from a spectrum acquired during titration with the binding ligand.
Black arrows represent vectors composed of the chemical shift changes of individual nuclei. Coordinates of the vector are used to calculate
CSP for each residue.
134
fact that the protein sample concentration at the level of 10 µM is sufficient.
The saturation transfer experiment and its functionality is shown in the figure ??. Two 1D proton spectra of the
mixture of the protein and potential ligands are recorded. At the beginning of the first one, a weak and selective
irradiation is applied at protein resonance frequencies to saturate exclusively them. Therefore the signal positions
of the ligand must be known to avoid accidental saturation of ligand by this irradiation. The second spectrum is
measured with the frequency of the irradiating radio wave far off-resonance to avoid any saturation. The two resulting
spectra are subtracted from each other. Considering the very small concentration of the protein, there are no protein
signals directly visible. But, if the ligand interacts with the protein at a site close to the proton which was saturated
(i.e., with frequency close to the frequency at which the irradiating radio wave was applied), then the saturation
influence distribution of magnetic moments that are close in space, including those of the ligand. This is observed
as a decrease of the ligand signals compared to their intensity in the reference spectrum. Therefore, the subtraction
of the spectrum measured as a reference from the spectrum measured with the saturation reveals only signals of the
interacting molecules.
12.4 Isotope-edited and isotope-filtered NOE experiment
We close the chapter dedicated to the interaction studies by an experiment allowing us to calculate structures of
intermolecular complexes based on the Nuclear Overhauser effect (NOE). The experiment is applicable to complexes
of macromolecules with small ligands, but it is often used in studies of interactions of two large biomolecules.
As discussed in Section 7.6, NOE transfers the polarization between nuclei that are close in space, but not necessarily
connected via chemical bonds. Therefore, NOE can be used for investigation of interactions when surfaces of two
biomolecules get close to each other. In these cases, the intermolecular NOE cross peaks are observed and provide a
direct evidence of the interaction. However, the pulse sequences presented in Figures 7.1C and 8.2C would provide
correlations of all protons close in space, including those in the same molecule (intra-molecular cross-peaks). The
number of intra-molecular cross-peaks is significantly larger than the few inter-molecular peaks we are interested in.
Identification of the inter-molecular peaks in spectra of complexes is possible, but challenging.
To simplify the spectra, one interaction partner is prepared with 13
C or 15
N isotope enrichment while the other
is unlabeled. The labeled molecule is usually protein as labeling of nucleic acids, oligosaccharides and small organic
molecules is more expensive and often technically difficult. The spectra are then measured using a pulse sequence
derived from the NOE experiment described earlier 7.6. Either isotope-filtering or isotope-editing elements are inserted
in the sequence. Both element are based on the same phenomena and are shown in Figure 12.5. Below, we describe
the 13
C isotope-filtering or isotope-editing element, but exactly the same idea can be used for 15
N filtering/editing.
Supposing the proton polarization present in the beginning of the sequence, the elements start with a proton
excitation pulse followed by a simultaneous echo with the delays optimized for a transfer of a proton in-phase coherence
to the proton anti-phase coherence, which actually happens only if the proton is bound to 13
C in our case. The natural
abundance of these isotopes is negligible and we can assume that they are absent completely in the unlabeled molecule.
Therefore, the effect of the same pulse element results only in refocusing the proton chemical shift as the spin echo
element always does. Then, the sign of the anti-phase coherence can be altered by application of a 180◦
pulse applied
to 13
C, while the application of such a pulse has no effect on the proton in-phase coherence (which was created by the
element for the protons not coupled to the heteronucleus). Finally, the proton anti-phase coherence is transferred back
to the proton in-phase coherence by the same series of pulses and delays as was used in the first half of the sequence.
The proton in-phase coherence can be transformed to the proton longitudinal polarization by the second of the proton
90◦
pulses. In unlabeled molecules, the proton in-phase coherence is also refocused by the second spin echo and the
proton polarization is restored by the second proton 90◦
pulse as well. So, the result is the same in both labeled and
unlabeled molecules. However, the sign of polarization can be altered only for the case of a proton coupled to the
13
C nucleus. Therefore, we can perform the experiment twice with and without the 180◦
pulse applied at the 13
C
frequency and either sum or subtract both spectra. In the first case, which is called isotope filtering, the signal arising
from protons not coupled to the 13
C are observed, while the signals coming from the protons coupled to the 13
C are
effectively eliminated. The subtraction will work in the opposite manner. This is called isotope editing.
The isotope editing and filtering can be inserted both before and after the mixing element of the NOE experiment.
Spectra obtained with two isotope filtering or isotope editing elements contain only in intra-molecular NOE signals of
12.4. ISOTOPE-EDITED AND ISOTOPE-FILTERED NOE EXPERIMENT 135
A)
1
H
1
H
1
H
1
H
1
H
1
H1
H
1
H
1
H
1
H
1
H
1
H
1
H
1
H
1
H
1
H
1
H
1
H
saturation
1
H
1
H
1
H
1
H
1
H
1
H
1
H
1
H
1
H
1
H
1
H
1
H
1
H
1
H
1
H
1
H
1
H
136
1
H
13
C / 15
N
1
H
13
C/15
N
Figure 12.5: Isotope filter/edit pulse sequence element. The narrow and wide boxes represent the 90◦ and 180◦ pulse, respectively. All
pulses are applied with the x-phase. Delays ∆ stand for 1/(4J), where J is the J-coupling constant. The red pulse applied to 13C / 15N
nucleus is either present (left) or omitted (right). Isotope-filtered and isotope-edited signals are obtained by adding and subtracting the
spectra obtained using the individual elements, respectively.
12.4. ISOTOPE-EDITED AND ISOTOPE-FILTERED NOE EXPERIMENT 137
the unlabeled or labeled interaction partner. Applying the isotope filtering element before the NOE mixing time and
the isotope editing element after the NOE mixing results in a spectrum containing only the desired inter-molecular
NOE cross-peaks created by transferring the proton polarization of the unlabeled molecule to protons of the labeled
molecule. Finally, the isotope editing element before the NOE mixing time and the isotope filtering after the NOE
mixing result in inter-molecular NOE cross-peaks due to the proton polarization transfer from the labeled molecule
to the unlabeled one. Figure 12.6 illustrates the inter-molecular and intra-molecular NOE contants and their selective
detection.
138
12
C
12
C
12
C
1
H
1
H
1
H
13
C
13
C
13
C
13
C
1
H
1
H
1
H
1
H
12
C1
H
13
C
1
H
13
C
1
H
12
C
1
H
12
C
1
H
12
C
1
H
Figure 12.6: Contacts detected by the isotope filtered/edited NOE experiment. The light yellow molecule was prepared with 13C isotope
enrichment, unlike the light magenta molecule. The blue arrows mark intramolecular contacts within the 13C labeled molecule that can be
selectively detected by inserting the isotope editing element before and after the NOE mixing time. The green arrows mark intramolecular
contacts within the 13C unlabeled molecule and can be selectively detected by inserting the isotope filtering element before and after the
NOE mixing time. The red arrows show inter-molecular NOE contacts selectively detected if the isotope filtering element is placed before
the NOE mixing time and the isotope editing element is placed after it, or vice versa.
Lecture 13
Nucleic acids
13.1 Chemical composition
Chemically, nucleic acids are esters of orthophosphoric acids. This should be kept in mind when considering their
chemical properties. For example, they are negatively charged and require a presence of a counter-ion, which is usually
either Na+
or K+
. The other part of the ester is a nucleoside, which itself consists of a sugar and a base bound together
by the glycosidic bond (Figure 13.1). The sugars involved in nucleic acids are either ribose in ribonucleic acids (RNA)
or 2-deoxy ribose in deoxyribonucleic acids (DNA). This rather small chemical difference however leads to significantly
different structural features that are also reflected in NMR spectra. The most common naturally occurring bases in
DNA are Adenine, Guanine, Cytosine and Thymine. RNAs contain Uracil instead of Thymine. The glycosidic bond
is formed between the sugar atom C1′
(pronounce ”one prime”) and a nitrogen atom of the base. The symbol prime
(′
) is used to distinguish the atoms of the sugar part from the base. Note the numbering of the base atoms. Following
the principles of chemical nomenclature, the atoms are numbered so that the heavier atoms (N in our case) get the
lowest possible numbers and the larger ring is numbered before the smaller ring. As a result of these rules, the nitrogen
atom involved in the glycosidic bond is N1 in Pyrimidine bases (Cytosine and Thymine/Uracil), while in Purine bases
(Adenine, Guanine) it is the atom N9.
The base combined with the sugar forms a nucleoside, as mentioned above. The nucleoside together with the
phosphate form nucleotide (mononucleotide). The phosphate binds the nucleoside either at the position 5′
or at the
position 3′
. The resulting nucleotides are called, for example, cytidine 5′
-monophospahate (shortened to CMP or pC)
or 2-deoxyadenosin 3′
-monophosphate (dAMP, dAp). Note there is an oxygen atom between the sugar carbon (C5′
or C3′
) and phosphorus atoms. By attaching further nucleotides we can form di-, tri-, oligo- (oligo = several), and
polynucleotides. Obviously, the chain has to end somewhere. If the is no further nucleotide attached at the 5′
sugar,
we call it the 5′
end. If the chain does not continue after the 3′
phosphate, it is called the 3′
end. By a convention, the
nucleotide sequences are written in the direction from the 5′
end to the 3′
end. For example, the well-known DickersonDrew
dodecamer (dodeca = 12) has the sequence d(CGCGAATTCGCG), where d indicates that it is a DNA. Essential
for the biological function of nucleic acids is the ability of the bases to form hydrogen bonds. It is the imino and amino
hydrogen atoms who play the most important role in the forming of the bonds. Cytosine, Adenine, and Guanine
have the amino group and Thymine/Uracil and Guanine have the imino hydrogen. Guanine is thus the only of the
bases that have both imino and amino groups. The best known base-pairing patterns are the Watson-Creek base pairs
CG and AT. Also important are the Hoogsteen base pairs for which the involvement of Purine N7 is typical. NMR
spectroscopy is an excellent tool for studying whether the base pairs are formed and evaluating which type of the base
pairs they are.
13.2 Torsion angles
If we have four atoms A, B, C, and D, and look along the BC bond, then the torsion angle along the BC bond is
between the vectors AB and CD (Figure 13.2). If the torsion angle is close to 0 ◦
, we talk about syn or cis orientation,
if the torsion angle is close to 180 ◦
, the orientation is called trans or anti. For the orientations transient from syn
139
140
C8C4
C2
C5
O2
C6
O4
N1
H6
N3
C5
C4
C2
H8
N9
N7
C4
C2’
N3
C1’
O4’
H5’’
O5’
P
O
H6
H1’
H5
O3’
H42
N1
H41
N3H3
N4
C6
O2
C5
C2
−O
O2’H2’
C3’
H5
C3’
H5’
H4’
H3’
H2’
C4’
C5’
H2’
C2’
H3’H4’
3
)7H(7C
H5’
O− H6
O4
O2
C2
C4C5
C6 N3H3
N1
O3’ H2’’
H1’
O
P
O5’
H5’’
O4’
C1’C4’
C5’
C6
H22
H21N3
C5
C4
C2
N2
O6
N1H1
H8
N9
N7
C8 N1
C6
H61
H62
N6
H2
G
U
C
A
T
Figure 13.1: Top, chemical structures of DNA (depicted for Thymidine) and RNA (depicted for Uridine). Bottom, bases found in both
DNA a RNA (Cytosine, Guanine, and Adenine)
13.3. HELICAL PARAMETERS 141
toanti, terms synperiplanar (sp), synclinal (sc), anticlinal (ac) and antiperiplanar (ap) can be used.
In nucleic acids, the torsion angles significant for characterizing the structure are the torsion angles along the
sugar-phosphate backbone (labeled α, β, γ, δ, ϵ, ζ), within the sugar ring (labeled νO to ν4), and the glycosidic torsion
angle χ (Figure 13.3).
It should be noted that the sugar ring is not planar. If we choose a plane defined by three atoms, the remaining
two can be above or below the plane. As the carbon atoms in the (deoxy)ribose carry bulky substituents, some of
the conformations are preferable for steric reasons. The sugar ring conformations are frequently referred to as sugar
pucker. If we use the C1′
-O4′
-C4′
plane as the reference, then the atoms occurring on the same side of the plane as
the C5′
are called endo and the atoms on the other side are called exo. The two most frequently occurring sugar ring
conformations in the nucleic acids are envelope conformations (i.e. conformations with 4 atoms in the plane and only
one outside the plane) called C2′
-endo (prevalent in DNA) and C3′
-endo (prevalent in RNA).
Additionally, the 5 torsion angles within the ring are not independent. It turns out that the ribose conformation
can be characterized by just two parameters. The parameters are called Pseudorotation angle (phase) P (Figure 13.4)
and the pucker amplitude νmax, defined as follows
tan P =
(ν4 + ν1) − (ν3 + νO)
2ν2 · (sin 36◦ + sin 72◦)
, (13.1)
νmax = ν2/ cos(P). (13.2)
Expressed using the Pseudorotation angle P, the C3′
-endo conformation corresponds to P = 18 ◦
and the C2′
-endo
conformation corresponds to P = 172 ◦
. When depicted on a sphere, the positions resemble North and South on the
globe, therefore the C3′
-endo and C2′
-endo are also referred to as North (N) and South (S) conformations, respectively.
The torsion angle χ around the C1′
–N glycosidic bond is defined by atoms O4′
–C1′
–N1–C2 in pyrimidines and O4′
–
C1′
–N9–C4 in purines. The trans conformation is sterically preferable as the bulky bases need space. However, if the
chain changes the direction like in the loops or quadruplexes, a cis conformation becomes inevitable at some residue.
Most frequently, the cis conformation occurs at guanosines.
13.3 Helical parameters
In nucleic acids, several terms are used to characterize the position of the base with respect to the helix axis or the
distances and angles between the bases or base pairs.
Specifically, the axis-base or axis-base pair relations can be characterized by X-displacement, Y-displacement,
Inclination and Tip. The distance and shift relation within the base-pair can be described by Shear, Stretch and
Stagger, while the intra-base pair angles are described by Buckle, Propeller twist, and Opening. For the base-pair
distance and shift relations we have Shift, Slide, and Rise and the inter-base pair angles are characterized as Tilt, Roll,
and Twist. The meaning of these terms is best described by diagrams in Figures 13.5 and 13.6.
13.4 A and B double helix
There are many tertiary structures into which nucleic acids can fold, for example hairpins, triplexes, quadruplexes,
i-motifs, to name a few. However, the most typical structures for nucleic acids are A and B helices. The A-type helix
is more typical for RNA while DNA usually forms the B-type helix. The difference between these two types of helices
becomes most obvious when we look along the helical axis. In the A helix, the base-pairs are ‘wrapped’ around the
axis forming a hole in the middle with no atoms in the place of the axis. In the B-helix, the axis goes right through the
atoms forming the hydrogen bonds between the bases. In the A-helix, the base pairs are conspicuously inclined with
respect to the helical axis by about 21 ◦
, while in the B-type helix the base-pairs are almost perfectly perpendicular
to the axis. The A-helix with 11 base-pairs per a single turn is also a little bit more twisted that the B-helix with 10
base-pairs per a turn.
142
D
A
D
A
D
A
D
AA
B
C
+θ
−θ
+θ
−θ
B
B B
B
−90 ◦
+120 ◦
+150 ◦
+sp−sp
+sc
+sc
+ac
+ac
+ap
−sc
−sc
−ac
−ac
−ap
cis
trans
±180 ◦
−150 ◦
−120 ◦
gauche(−)
−60 ◦
−30 ◦
gauche(+)
A
B C
D
A
B
0 ◦
+30 ◦
+60 ◦
+90 ◦
Figure 13.2: Torsion angles in a general sequence of four atoms A, B, C, and D. A, two views of the atoms with a positive value of the
torsion angle about the B–C bond. B, two views of the atoms with a negative value of the torsion angle about the B–C bond. C, various
descriptions of torsion angles.
13.4. A AND B DOUBLE HELIX 143
C5’
C4’ C1’
O4’
H5’’
O5’
P
H1’
O3’
−O
H5’
H4’H3’
C2’
H2’
H2’’ or O2’H2’
C3’
O
α
Base
ν2
ν1ν3
ν4 νΟ
δ
ζ
γ
β
ε
χ
Figure 13.3: Torsion angles in nucleic acids.
N
C1’C3’ O4’
C2’C5’
C4’
C5’
C4’
C3’
O4’ C1’
N
C2’
N
0
0 N
N
N
N
N
N
N
NN
N
N
N
N
N
N
N
N
N
N
0
0
0
0
0
0
0
0
0
90
36
54
72108
126
144
270
252
342
324
306
288
234
216
198
180
162
18
++
+
+
+
+
+
+
++
+
+
+
+ +
+
++
++
+
++
+
+
+
+
++
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
C2’−exo
C4’−exo
O4’−endo
C1’−exo
C3’−exo
C4’−endo
O4’−exo
C1’−endo
C2’−endo
C3’−endo
T2
3
T2
3
EO EO
E2
E3
T3
4
T2
1
T1
2
T3
4
E4E1
E4
E1
T4
O
TO
1
1
O
TT4
O
+
+
E3
E2
+
+
+
+
+
+
+
+
+
+
+
+
+
Figure 13.4: Values of P displayed using the pseudorotation wheel. All envelope and twisted conformations are marked (starting form
the center by one-letter symbol, value of P, verbal description, and depicted schematically (circles symbolize atoms in the furanose ring,
plus and minus signs mark atom s in the endo and exo positions, respectively, and signs of the angles νi are displayed next to the bonds).
Conformations C2′-endo- and C3′-endo are shown in blue and red, respectively.
144
z
xy
Stagger (Sz) Shear (Sx)
Shift (Dx)
y-displacement (dy) x-displacement (dx)
Stretch (Sy)
Rise (Dz) Slide (Dy)
Figure 13.5: Distance and shift relations between the nucleic acid bases and base pairs.
z
x
y
Tip (θ) Inclination (η)
Buckle (κ)
Tilt (τ)
Propeller twist (ω)Opening (σ)
Twist (Ω) Roll (ρ)
Figure 13.6: Angular relations between the nucleic acid bases and base pairs.
13.4. A AND B DOUBLE HELIX 145
Figure 13.7: Stick models of B- and A-type double helix (left and right, respectively).
146
Table 13.1: Properties of isotopes available for studying nucleic acids.
Isotope 10−7
γ ν at 11.74 T Natural Abundance Sensitivity
(I = 1/2) rad s−1
T−1
MHz % relativea
absoluteb
1
H 26.75 500.0 99.98 1.00 1.00
13
C 6.73 125.7 1.1 1.6 ×10−2
1.8 ×10−4
15
N −2.71 50.7 0.4 1.0 ×10−3
3.8 ×10−6
31
P 10.83 202.4 100 6.6 ×10−2
6.6 ×10−2
a
Relative sensitivity at constant field for equal number of nuclei.
b
Product of relative sensitivity and natural abundance.
13.5 Atoms in nucleic acids
Out of five elements constituting nucleic acids (H, C, N, O, P), four have isotopes suitable for NMR spectroscopy, i.e.
1
H, 13
C, 15
N, and 31
P. While all naturally occurring phosphorus is 31
P and overwhelming majority of hydrogen is 1
H,
in the case of carbon and nitrogen, we have to rely on low natural abundance isotopes 13
C and 15
N (Table 13.1). This
means that for samples with natural isotopic abundance we have to rely primarily on proton spectra and only simple
heteronuclear correlation experiments (HSQC, HMQC) are possible. Even though 31
P has 100% natural abundance
and a high frequency, there is another complication. 31
P has a large value of chemical shift anisotropy that causes fast
relaxation leading to broad lines in 1D 31
P spectra and poor sensitivity in correlation experiments. The relaxation
rate increases with the magnetic field. Therefore, moderate magnetic fields (not higher than 11.75 T corresponding to
500 MHz proton resonance frequency) are preferable for measuring 31
P spectra.
13.6 Exchangeable an non-exchangeable protons
The protons bound to oxygen and nitrogen atoms are in water solution in a fast exchange regime with the solvent.
The exchange rate of the protons in -OH groups is usually too fast for the protons to be detected at all. The nitrogen
bound protons in imino and amino groups are observable in most cases. It often helps to lower the temperature to
slow down the exchange rate. Also, the methods of measurement that saturate the water signal should be avoided as
the nuclei with the saturated spin levels from the solvent that enter the nucleic acid molecules by the exchange process
do not produce any signal.
Note the low proton density in the bases. Except for the Guanine, there are only two signals from non-exchangeable
protons (the protons in thymine methyl group are equivalent) and one signal from either imino or amino group. Guanine
has only one non-exchangeable proton and both the imino and amino groups (Figure 13.8).
On the other hand, the spin system in the sugar part of the molecule are much more complicated. It is especially
the case of RNA where the carbons C2′
, C3′
, and C4′
all carry a proton and -OH group, and consequently, the chemical
shifts from their protons are very similar, falling in the range of 3.5–5.0 ppm. In DNA, the absence of the -OH group
at the position 2′
makes a significant difference. The chemical shifts of the methylene group protons are in the range
of 1.8–3.0 ppm and the resolution of the sugar protons in DNA is thus much better. The signals of H2′
,H2′′
protons
together with the methyl proton of Thymine between 1 and 2 ppm makes the DNA spectrum clearly different from
the RNA spectrum (Figure 13.9).
13.6. EXCHANGEABLE AN NON-EXCHANGEABLE PROTONS 147
N1
C6
C5 C4
C2
O2
N4
N3
N1
C6
C5 C4
C2
O2
O4
N3−H3
N1
C6
C5 C4
C2
O2
O4
N3−H3
C8
N7
N9
C2
C4
C5
N3
N6
C6
N1
C8
N7
N9
O6
N2
C2
C4
C5
N3
C6
H42
H41
H62
H61
H22
H21
N1−H1
H8 H8
H2
H5
H6
C7
H6
H5
H6
H( 7) 3
A X3
C T U
AG
A A + A
AX AX
Figure 13.8: Spin systems in nucleic acid bases (blue). The exchangeable protons are shown in red.
HN 2
HN
3
HC
RNA
δ / ppm
H5H2,
H6
H8
H2’H1’
15 10 5 0
H3’ H4’,H5’
Figure 13.9: Proton chemical shifts in nucleic acids. The exchangeable protons are shown in red.