Introduction to Soft Matter Models
of Membranes and Proteins
by Robert Vácha
Fundamental constants:
Constant Symbol Value [SI]
Boltzmann constant k, kB 1.381 ×10−23 JK−1
Avogadro’s number Na 6.022 ×1023 mol−1
Gas constant R=k×Na 8.314 JK−1mol−1
Electron charge e 1.602 ×10−19 C
Speed of light c 2.998 ×108 ms−1
Vacuum permittivity 0 = 107
4πc2 8.854 ×10−12 C2J−1m−1
Unit symbols and relations:
Quantity SI Relations
Length m 1 m = 100 cm = 109 nm = 1010 Å
1 nm = 10 Å= 10−9 m = 10−7 cm
Energy J 1 J = 0.239 cal = 6.242×1018 eV = 2.43×1020 kT (25◦C)
1 kT (25◦C) = 0.529 kcal mol−1 = 2.478 kJ mol−1
1 kT = 4.114×10−21 J (25◦C) = 4.045×10−21 J (20◦C)
1 kcal mol−1 = 4.184 kJ mol−1 (1 cal = 4.184 J)
1 eV = 1.602×10−19 J = 23.06 kcal mol−1
1 cm −1 = 1.986×10−23 J
Surf. tension Jm−2 1 dyne cm−1 = 1 mN m−1 = 1 mJ m−2
Contents
1. Introduction to coarse-grained models . . . . . . . . . . . . 1
2. Ideal chain approximation . . . . . . . . . . . . . . . . . . . 9
3. Ideal thin membrane . . . . . . . . . . . . . . . . . . . . . . 19
4. Interaction of uncharged membrane and proteins . . . . . . 30
5. Interaction of charged membrane and proteins . . . . . . . 44
6. Towards realistic interactions of membranes and proteins . 54
7. Thermodynamic principles of self-assembly . . . . . . . . . 66
8. Langmuir adsorption . . . . . . . . . . . . . . . . . . . . . . 76
9. Transport phenomena . . . . . . . . . . . . . . . . . . . . . 84
10. Advanced Monte Carlo . . . . . . . . . . . . . . . . . . . . 95
Test questions . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Basics of thermodynamics . . . . . . . . . . . . . . . . . . . 109
Basics of statistical mechanics . . . . . . . . . . . . . . . . . 110
Derivation of Boltzmann distribution . . . . . . . . . . . . . . 112
Virial coefficients and itneractions . . . . . . . . . . . . . . . 113
Ideal chain - radius of gyration Rg . . . . . . . . . . . . . . . 114
Ideal chain - probability of the end-to-end distance . . . . . . . 116
Dispersion interactions with thin membrane . . . . . . . . . . 118
Depletion interactions in solution of ideal chains . . . . . . . . 119
Dissociation constants for amino acids . . . . . . . . . . . . . 121
Derivation of equilibrium equation between aggregates of different
sizes . . . . . . . . . . . . . . . . . . . . . . . . 121
Distribution of finite size aggregates . . . . . . . . . . . . . . 122
Fluctuation-dissipation theorem . . . . . . . . . . . . . . . . . 123
Introduction to Metropolis Monte Carlo method . . . . . . . . 125
1. Introduction to
coarse-grained models
(Coarse-graining levels: from all-atom to single particle models
and continuum approximation)
Soft matter is quite a general term, which covers a wide variety of different
systems and materials such as polymers, colloids, surfactants, proteins,
membrane, cells, tissues, and other biological matter. The common
feature of all soft matter substances is their ’softness’, which can be
characterized by an elastic bulk modulus being in the range from 1 Pa
to 100 MPa. This means that with a pressure change of 1 atmosphere,
the material volume compresses from 105 to 0.1 %. For comparison
the bulk modulus of water and steel is 2 GPa and 200 GPa respectively,
while air has one of about 0.1 MPa. These species usually form very
complex fluids with interactions on order of a few kT, which are comparable
with thermal fluctuations and entropic forces. As a result the soft
matter structures are dynamic and changes in the solution composition
and temperature can lead to significant structural changes (no need to
break chemical bonds).
Soft matter processes are mostly about diffusion and changes in
large molecules in solution. It is therefore important to understand organization
and changes at the "mesoscopic" scale rather than investigate
atomistic details. The characteristic length scale is on the order
2
of 1 - 100 nm and characteristic times are on the order of milliseconds
to seconds rather than the angstroms and femto- and picosecond times
relevant for the movement of individual atoms and small molecules.
Such large scales are a typical domain of coarse-grained simulations,
which are widely used and still being developed. Coarse-grained
model is a term used for non-atomistic models, where some of the degrees
of freedom are averaged out. Strictly speaking, atomistic models
are also coarse-grained as they neglect or average quantum effects and
wave description of particles. Nevertheless, the terminology of coarse
grained models is typically used for models, where a single particle consists
of more atoms etc. Naturally, the first question is how to coarsegrain
the system of interest, or which model to use for it. This can
sometimes be a challenging task as we only know that the right model
has to contain all the physically important details of the studied phenomena/system,
while the irrelevant degrees of freedom are averaged
and described effectively. This means that the effective interaction is a
function of temperature, pH and, salt concentration (for instance the solvent
could be described as a dielectric medium with a certain viscosity,
which are both temperature-dependent). Therefore, different phenomena
often require different coarse grained model and a whole library of
different models have been developed.
To gain a better idea of the current library of coarse-grained models
we will outline a classification system below. However, this classification
is only for a simple overview, as it is not widely recognized and is
incomplete (e.g. does not contain mean field models). We list some
well-known examples of protein and membrane models.
Classification based on the size of elementary building units:
1. Finite element models – molecules or their parts are represented
as finite particles and we can divide this group further
based on the number of heavy atoms included in one
particle
• First order – contains 2 - 5 heavy atoms; examples
3
are the MARTINI force field, Klein and Shinoda model,
Voth’s model, Smit’s model, Tube model, ELBA, PRIMO,
PaLaCe, ...
• Second order – usually includes 5 - 20 heavy atoms;
a typical example is an amino acid model (1 bead per
amino acid) for proteins, UNRES, VAMM, OPEP, Bereau
and Deserno model, SCORPION model, McCammon
model..
• Third order – coarse grain 20 - 100 heavy atoms into
one particle. A typical example is the three-bead lipid
model by Lipowsky, which was further developed by
Deserno, Frank model, HCG model, etc.
...
• Single particle – whole molecule is represented as a
single particle, HAS model for lipids and many patchy
particles for proteins.
2. Lattice models – particles or regions of the system are represented
as points on a lattice with the specific interactions
between them
• Ising model – interaction points with two possible states
• Density Functional Theory – points contribute to the
free energy of the system through their local density
(entropy) and interactions
3. Continuum models – molecules or parts of the system are
described uniformly by their macroscopic properties based
on the basis of which we can further categorise this group
• Electrostatic – solution described as a dielectric screening
of the Coulomb interactions
• Elastic – membrane as an elastic thin sheet
4
• Density
Classification based on parameterization of the model:
1. Approaches to coarse-graining
• TOP DOWN approach – the parameters of the interactions
are fitted to macroscopic values such as partition
coefficients, density, elasticity, binding constants, etc.,
which are commonly known from experiments
• BOTTOM UP approach – model parameters are fitted
to reproduce some microscopic properties such as radial
distribution functions, molecular forces, etc., these
are usually obtained from more detailed simulations.
2. Type of interaction between elementary building blocks
• Isotropic – acts between two particles independently of
their orientation - most common
• Anisotropic – orientation-dependent interaction between
two particles - interaction of dipoles or patchy particles
• Three and more body – potential depends on the positions
of more than two particles – single-particle water
model
Note that the different models can be combined in a calculation
and there are also models where both top-down and bottomup
approaches were combined. The species of interest or more
flexible part of the system are often described in more details
than the rest of the system. For example the MARTINI force field
uses 1:3-4 mapping with explicit water, which can be polarizable
or non-polarizable. In the contrast the UNRES (united residue)
model uses one particle for each amino acid, two particles for 4
backbone heavy atoms and implicit solvent. The more detailed
5
backbone representation was found to be necessary to describe
protein folding, where hydrogen bonds within the protein backbone
play an important role.
A knowledge-based approach is sometimes recognized separately
from a top-down approach. This approach is based on protein
structures from the PDB database. The statistics of molecular
contacts and their distance dependence is used in interaction
parameterization. The resulting Go, network, and nativecentric
models were used to study protein folding, fluctuations,
and protein-protein interactions. Despite the success of these
models, it has been shown that the resulting models are not transferable
to new protein structures. This casts a shadows on the
physical significance of results from these models for non-native
(untrained) structures.
Why is there a need for coarse grained models and why is
their popularity growing rapidly? Because human imagination in
model development enable us to study systems that might not
be feasible for all-atom simulations for decades. Let’s estimate
how long it will thaw for a larger system to become feasible for
all-atom simulation. Let’s say we want to simulate a system that
is 10 times larger (103
in volume) than is feasible by atomistic
simulations today. We could naively think that the computational
cost (time required) will grow with the number of interactions, i.e.
with N2
, where N is number of particles in the system (needing
to calculate the forces between all pairs of atoms). Using more
efficient algorithms we can achieve better scaling, so the computational
cost would increase with N. Still, this gives us a time that
is 103
longer. Moreover, the time required for equilibration and
sampling of such a system would also increase. We can make
a simple estimate based on the diffusion which dominates such
large systems. If we take a single particle in the center of the simulation
box and calculate the time required for its diffusion to the
edge of the box, it would take 102
longer for our larger system.
6
So we can estimate that one would need computers 105
faster
(possibly much more than that). Computer speed increases exponentially,
doubling every 1.5 years, according to Moore’s law.
This gives us 25 years, until we can perform such a simulation.
Note that due to massive parallelization, we can simulate allatom
system that have millions of atoms nowadays; however, the
time scales required to study such large systems remains challenging.
Moreover, we gain insights into key aspects of the studied
systems from the models, so we can understand the microscopic
origin of a given macroscopic phenomenon. Sometimes
the models are so simple that they may look very unrealistic, for
example ideal gas, ideal chain, etc. Nevertheless, the main property
of a good model is its predictive power, and simple models
have predicted a lot of phenomena and are still used as first ap-
proximations.
Most coarse-grained simulations of proteins and phospholipid
membranes today employ finite element models with the solvent
described as a continuum or finite elements. We will therefore
focus on those. We start with a description of the simplest models
covering conformations, entropy, and the changes in them for
proteins and membranes. This is followed by a discussion of enthalpic
contributions to the intra- and intermolecular interactions,
focusing on their strength and distance dependence, information
necessary for the development of the models using a top-down
approach. The parameterization of coarse-grained models is then
completed with description of methodology for the bottom-up approach.
We then focus on principles of self assembly, surface
association, and the motion in solutions, basic phenomena in the
field of coarse-grained matter. Last but not least some advanced
methods commonly applied in these areas are explained.
We will now look at two examples that demonstrates how much
we can learn from simplest coarse-grained models. In the early
7
fifties John G. Kirkwood made a prediction that hard spheres with
no attractive interaction can freeze. This counter-intuitive idea
was based on a theoretical approximation and was not well accepted
at the time. Only later computer simulations by Alder,
Wainwright, Wood, and Jacobson confirmed the existence of such
a gas-to-crystal transition (interestingly, it was recently shown that
the face-centered cubic phase is more stable than the hexagonal
close-packed phase). This is the first order transition and
it is purely entropic as there is no attractive interaction (dH=0).
It is common that entropically driven processes are somewhat
counter-intuitive, since most processes in our everyday experience
are dominated by enthalpy. Some examples of such processes
are described in the following chapters and one shall not
neglect the entropy effects in soft matter.
The second example we will mention now is the existence of a
liquid phase. This may sound strange, since we see liquids every
day, but from a molecular point of view it is hard to make any
predictions about material phases based purely on the interaction
potential of atoms. Let’s look at hard spheres again, but this time
with isotropic attraction. Would this lead to any other phase than
gas and solid? Think about its phase diagram. If the range of
attraction is zero, there is no liquid phase. The imaginary critical
temperature is below gas/solid coexistence and we will have no
knowledge of it. When the attraction range expands the critical
temperature (below which a liquid exists) increases, and once it
passes across the solid/gas coexistence line the liquid begins to
exist. Recent simulations and experiments suggested that this
point exist when the range of attraction is about one-third of the
particle diameter. Once we know about the liquid phase, it is easy
to explain it. Nevertheless, it demonstrates that there might be
other phases, whose critical temperature is below the coexistence
line and hence we have no knowledge of them yet. Finding such
materials or interaction potential remains an open task.
8
Sources and further readings:
1. W. G. Noid; Perspective: Coarse-grained models for biomolecular
systems; The Journal of Chemical Physics 2013, 139
(9), 090901; DOI: 10.1063/1.4818908
2. H. I. Ingólfsson, C. A. Lopez, J. J. Uusitalo, D. H. de Jong,
S. M. Gopal, X. Periole, S. J. Marrink; The power of coarse
graining in biomolecular simulations; Wiley Interdisciplinary
Reviews: Computational Molecular Science 2013, DOI: 10.1002/wcms.
2. Ideal chain approximation
(Random coil, Kuhn length, end-to-end distance)
Proteins are essential biological macromolecules with a variety
of roles in all organisms. They are built from 20 standard amino
acids attached together by peptide bonds forming a single linear
polymer. Proteins can fold in a specific structure or remain
unstructured in a conformation called random coil. In this conformation
the elementary building blocks are oriented randomly
and the overall shape is dictated by the statistical distribution of
all possible populations. This could be theoretically described by
an ideal chain approximation, a basic theory for ideal polymers.
The ideal chain is a polymer of N independent non-interacting
segments, each of length l. All the segments can freely rotate
around the joining points. This model might sound like an oversimplification,
however, there are many proteins and fully flexible
polymers that behave like this when the segment length is large
enough (it can contain several amino acids). The ideal chain is
the simplest model of polymer approximation and in a way it is
similar to the ideal gas approximation in the theory of molecular
gases. Note that there is an important similarity between an ideal
chain and random walk, which will help us in our calculations and
understanding. In both cases the direction of each segment/step
of length l is random and independent of other segments/steps.
10
An example of random walk is Brownian motion, where the motion
of a large particle in solution can be described by diffusion.
Figure 2.1: Illustration of ideal chain (random walk), where all segments
have length l and RN is a vector from beginning to the end of N-th
segment.
The key parameter in the ideal chain model is the length l of
the segment. It is an effective length, over which the protein can
be treated as rigid body. Moreover, the individual segments of this
length must be independent. Such an effective length is called the
Kuhn length and the protein parts of this length are called Kuhn
segments after the swiss chemist Werner Kuhn, who first suggested
the existence of these effective segments in polymers. In
proteins, the Kuhn length can vary from about 1 nm to 100 nm, depending
on the protein sequence, charges, secondary structure,
etc. Naturally, the Kuhn length is directly connected to the flexibility
of the protein. If we take a point in the protein and study how
much the direction at given distance d deviates from the starting
point, we will find that it exponentially decays with exp(−d/lp).
Where lp is a characteristic length called the persistence length.
For short distances d lp the protein can be treated as a rigid
rod and for large distance d lp the direction of the protein is
11
independent of its initial direction. Since the protein can be considered
to be a rigid rod for +lp and −lp from the initial point, the
Kuhn length can be calculated as:
l = 2lp
The persistence length lp can be also expressed using the bending
stiffness B as:
lp =
B
kT
,
where k is the Boltzmann constant 1.380×10−23
JK−1
. Other frequently
used constants are: Avogadro’s number Na = 6.022 ×
1023
mol−1
and ideal gas constant R = 8.31 JK−1
mol−1
. In soft
matter, kT is commonly used as units of energy, since it corresponds
to the thermal energy. Its relation to other units at 25◦
is
kT = 4.11 × 10−21
J= 2.479 kJmol−1
= 0.593 kcal mol−1
.
Now that we have defined our basic parameters, we can calculate
the end-to-end distance RN of an ideal chain as a measure
of its size. Analogously, with random walk we can start calculating
a distance how far you get after walking lost in a deep forest.
Let’s assume that we will walk for 10 hours and because we are
relatively fit, we will walk with an average speed of 4 km/h. The
total length of our track will thus be 40 km, however, if we lose
our direction after a few hundred meters, i.e l = 400 m, we will
actually not get very far from the starting point. 400 meters may
not sound like very far, but in experiments with blindfolded people,
they usually lost their initial direction within the first 100 - 200
m. Therefore, we will take this distance as an estimate for the
persistence length of our walk. As we know random walk can be
described by diffusion (if not see Chapter 9.), the distance from
our origin is given by the equation:
R2
= 2nDt, (2.2)
12
where D is a diffusion coefficient and n is the dimensionality
of the process. We can calculate the diffusion coefficient from
our speed and uninterrupted path in one direction as 2nD =
l2
/t = lv. Hence, we obtain R2
= lvt, which can be rewritten
as |R| = (lvt)1/2
= 4 km. This means that after walking 40 km
we will actually only get 4 km away from our starting point. Moreover,
it is highly likely that our path will cross itself, so we will have
the impression of walking in circles, which many people lost in a
dense mist or blizzard experience. Thus it is better to go slowly
and concentrate on your direction than walk fast. To obtain the
distance dependence on the number of segments, we simply divide
the total path into segments of length l (vt = Nl) and obtain
|RN | = lN1/2
(2.3)
Now we will calculate the same RN for a protein using the
ideal chain approximation. We start with a definition of the end-toend
distance as a sum of N segments each defined by its vector
ri of size l:
RN =
N
i=1
ri (2.4)
As all the segments are independent we can write the averaged
value as:
RN =
N
i=1
ri (2.5)
All orientations of ri have the same probability, i.e. in 1D +l is
equally likely as −l, hence the averaged value is zero. Therefore,
we end up with RN = 0. This does not mean that the size of
the end-to-end distance is zero, but that RN has no preferred orientation.
The size of the end-to-end distance has to be calculated
13
as the square root of RN
2
.
RN
2
=
N
i=1
ri
2
(2.6)
=
N
i=1
ri
N
j=1
rj (2.7)
=
N
i=1
N
j=1
rirj (2.8)
=
N
i=1
N
j=1
lilj cos θij (2.9)
=
N
i=1
N
j=1
l2
cos θij (2.10)
Since all orientations have the same probability, all angles from
0 to 180◦
are equally likely and cos θij = 0, apart from cases
when i = j for which cos θii = 1. Therefore, the sum simplifies
to
RN
2
= l2
N (2.11)
and the average end-to-end distance is:
|RN | ≡ RN
2
= lN1/2
(2.12)
Not surprisingly, we have obtained the same result as in random
walk (Eq. 2.3), where we used the diffusion equation.
If we want to compare to the measured size of proteins, we
can calculate the radius of gyration Rg. It is defined as the mean
14
squared distance between the monomers and the center of mass:
R2
g ≡
1
N
N
i=1
(|ri − RCM |)2
(2.13)
This can be related (see Appendix) to the end-to-end distance as:
< R2
g >
1
6
RN
2
(2.14)
We can also calculate the probability distribution of the endto-end
distance. After a little bit of math (shown in the Appendix)
we end up with the 1D probability:
P(RN ) =
1
√
2πNl2
exp −RN
2
/2Nl2
(2.15)
and for 3D:
P(RN ) =
3
√
2πNl2
3/2
exp −3RN
2
/2Nl2
(2.16)
The end-to-end distribution is a Gaussian function with a variance
corresponding to the mean-squared end-to-end distance. This
distribution could have been anticipated, since the central limit
theorem of statistics states that a large number of random events
have a Gaussian distribution. The assumptions used in the above
derivation are large number of segments N and the end-to-end
distance being smaller than the fully stretched chain RN Nl.
From the probability distribution we can calculate the force that
keeps the end-to-end distance in equilibrium. Our force doing the
work can be calculated as:
< FW (RN ) >=
dW
dRN
(2.17)
15
The chain force is of the same size but opposite direction:
< F(RN ) >= −
dW
dRN
(2.18)
Since there is no interaction, the internal energy stays constant
and all the work done in a change of the end-to-end distance is
transferred into heat.
dU = dW + dQ = 0 (2.19)
Furthermore, when we only make small changes we can assume
that the change in our system is reversible:
dQ = TdS (2.20)
Hence, the chain force is:
< F(RN ) >= −
dW
dRN
=
dQ
dRN
= T
dS
dRN
(2.21)
Thus to calculate the force we need to know the entropy change
dependence on the end-to-end distance. The entropy is defined
as:
S = k ln(Ω) (2.22)
where k is the Boltzmann constant and Ω is the number of microstates
corresponding to the evaluated macro-state (end-to-end
distance). Because we already know the probability distribution
of the end-to-end distances, we know the number of micro-states
up to the multiplication constant C.
Ω(RN ) = CP(RN ) (2.23)
and so
S = k ln(Ω) = k ln P(RN ) + k ln(C) (2.24)
16
Substituting this into 2.21 we obtain:
< F(RN ) >= kT
d ln P(RN )
dRN
(2.25)
Finally, using the probability of the end-to-end distance (Eq. 2.16)
leads to the equation of state for an ideal chain:
< F( )RN >= −
3kT
Nl2
RN , (2.26)
where RN is the end-to-end distance. We should note that this
force is purely entropic, since there are no interactions (∆H = 0).
The linear increase in the force with the end-to-end distance is
similar to Hook’s law used for small deformations in materials. Importantly,
the force increases linearly with temperature, so it will
be harder to stretch or shrink the ideal chain at higher temperatures.
This might be counter-intuitive, but it is a well-known fact in
polymer science and we can see demonstrations of it every day.
Try to heat up a rubber band and it will become stiffer.
Be aware that at large extensions the chain is not ideal and the
equation 2.26 fails. When RN Nl an alternate model has to be
employed, leading to diverging force at full extension (Nl = RN ):
< F(RN ) >∼ −
kTN
Nl − |RN |
(2.27)
More realistic model is a worm-like chain model, where flexibility
is uniformly distributed along the chain. This continuous model
is constructed by a limit, where number of segments goes to infinity
and the segment length goes to zero, while the total length
is kept constant (L = Nl). The derivation is similar as above, but
17
summations becomes integrations:
RN
2
= ri
2
(2.28)
=
L
0
L
0
cos θijdridrj (2.29)
=
L
0
L
0
exp −
|ri − rj|
lp
dridrj (2.30)
= 2lp L − lp(1 − e−L/lp
) (2.31)
In limit of long chain L >> lp, the square end-to-end distance
converges to 2Llp, which is equal to ideal chain for segment
length l = lp/2.
In addition, the finite volume of the chain can be added as an
excluded volume, which gives the scaling of the squared end-toend
distance to be N6/5
. These models successfully predicted
the stretching of proteins and their behavior in a free or confined
environment. Note that there are more chain models such as
the bead-spring model, where successive bead particles along a
coarse-grained chain are connected by harmonic potential.
Exercise: Let’s consider proteins with no tertiary structure,
such as denatured proteins or intrinsically disordered proteins.
Calculate the ideal chain end-to-end distance and radius of gyration
of denatured Ubiquitin protein, which has 76 amino acids.
Compare the calculated radius of gyration with the experimental
value of 25.2 nm. Then calculate the excluded volume radius of
gyration of the 129-amino-acid-long protein Lysozyme and compare
it with the experimental value of 35.8 nm. Comment on your
results. The average persistence length of disordered Ubiquitin is
4.00 nm. And assume Lysozyme to be intrinsically disordered.
Sources and further readings:
18
1. A. Y. Grosberg and A. R. Khokhlov; Giant Molecules: Here,
There, and Everywhere (2nd Edition); World Scientific Publishing
Company, 2010
2. J. R. C. Van der Maarel; Introduction To Biopolymer Physics;
World Scientific Publishing Company, 2007
3. Ideal thin membrane
(Elastic stretching, bending, Helfrich’s theory, endocytosis)
The biological membrane is a vital component of all cells. It separates
the content of the cell and its organelles from surrounding
fluids. Its main function is to act as a selective filter which controls
the traffic and communication across these open systems. Our
current view of membranes dates back to 1925 when Gorter and
Grendel showed that it is a thin double layer of lipid molecules.
Proteins were added later and it was in 1972 that Singer and
Nicolson introduced the "fluid mosaic" model, where the membrane
is described as a fluid bilayer composed of a lipid mixture
with embedded and attached proteins. This model has been updated
to its current form by the addition of the complex interplay
of all the membrane components, which can lead to spontaneous
organization such as phase separation or the formation of more
rigid regions called rafts. The rafts are speculated to have various
important biological functions. The lipids in rafts are in a gel
phase, where lipid hydrocarbon tails are highly ordered. The rest
of the membrane is typically in the fluid phase, where the tails are
disordered. The lateral diffusion of lipids is on the order of 10−3
nm2
ns−1
in the fluid phase and 2-3 orders of magnitude smaller
in the gel phase.
From a macroscopic view, membranes are flexible thin sheets
20
with thicknesses of about 5 nm. The shape and mechanical response
of a membrane can be understood from the elastic theory
of an idealized plate or shell. We will explain this theory on a small
square of homogeneous material. When the square is stretched
it expands from the original area A0 to a new size, denoted A.
In the first approximation (similar to a harmonic oscillator) we can
write the energy change as:
∆E = Estretch =
1
2
K(A − A0)2
, (3.2)
where K is a constant representing the membrane resistance to
the increase in area. As a matter of convention we will define a
new constant Kstretch = KA0 and obtain:
Estretch =
1
2
Kstretch
(A − A0)2
A0
(3.3)
The convention of the stretching constant (modulus) is chosen
based on Hook’s law which states that the stretch A−A0
A0
is directly
proportional to lateral tension(stress) σ ≡ dE
dA
. The integral form
of the equation 3.3 is:
Estretch =
membrane
Kstretch
(A − A0)
A0
dA =
membrane
σdA
(3.4)
The typical value of Kstretch for a phospholipid membrane is in
range of 0.15 - 0.30 N/m. This value is similar to rubber, however,
compared to rubber a phospholipid membrane is not made
of long polymers. As a result a stretch of more than just a few
percent leads to the membrane rupture. Note, that Kstretch can
be calculated for homogenous materials from the Young’s modulus
Y as Kstretch = Y t, where t is the thickness of the stretched
sheet, assuming that it does not change during deformation.
The second type of deformation we will investigate is bending.
We start again from a square of homogeneous material, but this
21
time we bend the square along one axis as shown in Figure 3.2.
As with stretching, we will use only the first term of expansion:
Ebend =
1
2
Kbend
(V − V0)2
V0
(3.5)
To calculate the bending change in energy, we have to integrate
Figure 3.2: Side view illustration of bending of a homogeneous material
of thickness t. Bending radius R is defined by the curvature of mid plane
(dashed). At distance z from mid plane a small volume of length l and
thickness dz is displayed.
the change in an elementary volume dV over the sheet thickness.
The upper half of the material becomes stretched, while the lower
part is compressed by the bending. Since Kbend corresponds to
the deformation of volume it is equal to Young modulus Y , so:
Ebend =
1
2
Y
V
(dV − dV0)2
dV0
(3.6)
=
1
2
Y
A
t/2
−t/2
(dx dy dz − dxdydz)2
dxdydz
, (3.7)
22
where dx dy dz is the size of the elementary volume after deformation.
Because we have bending around a single axis dy =
dy, we assume that the thickness of the bend material has not
changed, dz = dz. The Eq. 3.7 then simplifies to:
Ebend =
1
2
Y
A
t/2
−t/2
(dx − dx)2
dx
dydz (3.8)
(3.9)
To evaluate the integral, we will use the substitution derived from
Figure 3.2.
We can see that the length of the elementary volume dx expands
to dx at distance z from the mid plane. Using the expression
for the angle β (ratio of arc length to the radius) we get:
dx
R + z
=
dx
R
(3.10)
dx
dx
=
R + z
R
(3.11)
Now we subtract one from both sides to get:
dx − dx
dx
=
R + z − R
R
(3.12)
dx − dx
dx
=
z
R
(3.13)
(dx − dx)2
dx2
=
z2
R2
(3.14)
(dx − dx)2
dx
=
z2
R2
dx (3.15)
23
We can substitute this into Eq. 3.9, leading to:
Ebend =
1
2
Y
A
t/2
−t/2
z2
R2
dxdydz (3.16)
=
1
2
Y A
z3
3R2
t/2
−t/2
(3.17)
=
1
24
Y
t3
R2
A, (3.18)
where A is the area of our square. Now we can compare the
bending and stretching energy. Employing Kstretch = Y t we get:
Ebend =
1
24
Kstretch
t2
R2
A (3.19)
This means that the lateral stress for bending is dE
dA
∼ t2
R2 and for
stretching it is ∼ A−A0
A0
. Thus the bending is less energy demanding
than the stretching, as we see for other materials in every day
life. Try for example to stretch or compress a sheet of paper without
bending. The energy necessary to bend the membrane to a
radius of 1 mm is roughly equal to a relative change in area of
10−11
.
Note that the bending energy is commonly expressed per unit
area:
bend =
1
24
Kbend
t3
R2
(3.20)
This tells us that the energy density depends on the square of
the curvature 1/R, as this is the only variable for a sheet of fixed
size and thickness. For convenience we define a new constant
κ ≡ t3
12
Y , which is the bending modulus for a given thickness.
The bending energy per unit area then becomes:
bend =
1
2
κ
1
R2
(3.21)
24
A typical value of κ for phospholipid membrane in range 6 - 8
10−20
J = 60 - 80 pN nm ≈ 15 - 20 kT at ambient conditions.
This is an important value which perfectly suits the needs of cells.
It is just slightly larger than the thermal energy (a few kT) which
means that the thermal energy is enough to gently bend the membrane,
but not enough to break it. Moreover, it also means that
the membrane can be deformed by proteins, their self-assembly,
or by ATP hydrolysis providing ∼ 15 kT per single molecule. We
should note that a phospholipid membrane has zero shear modulus.
The reason for this is that the membrane is fluid, so shear
strain does not cause any change in energy.
In nature, the membrane is usually bent around more than
one axis. In fact, the membrane can have various shapes and
the curvature is not the same everywhere. In order to describe
such bending correctly, we have to look at bending locally and
then integrate the local bending energy over the whole membrane
area.
The correct description of local curvature can be obtained
from differential geometry, and we will use its general results and
implications (the derivation can be found elsewhere). At each
point of the deformed surface we can define a curvature c ≡ 1/R
in each direction. Nevertheless, there are always two directions
in which the directional curvatures are extremal (see figure 3.3).
These curvatures are called the principal curvatures (c1 and c2),
which are orthogonal, but do not have to be unique. One way
to think about this is to imagine a normal vector to the surface
at given point. The normal vector defines a plane that is called
normal section plane. All vectors within the plane defines directions
on the surfaces with varying curvature. By rotating the plane
around the normal vector we can find minimum and maximum
curvatures, which are the principal curvatures.
There are two important characteristics of the curvature that
25
can be calculated from the principal curvature. The first is the
mean curvature:
H ≡
1
2
(c1 + c2) =
1
2
1
R1
+
1
R2
(3.22)
and the second is the Gaussian curvature:
K ≡ c1c2 =
1
R1R2
(3.23)
Figure 3.3: Schematic visualization of locally curved surface with two
principal curvatures.
The mean curvature H describes how much the surface is
curved, while the Gaussian curvature K tells us how the surface
can be unfolded, i.e. what shape it would have if the mean curvature
were zero. With these two characteristics, we can properly
describe the bending energy of the surface using a terms up to
the order 1/R3
. In general, the bending energy depends on a
combination of all the linear and the quadratic invariants c2
1 + c2
1,
(c1 −c2)2
, (c1 +c2)2
, and c1c2. However, they are not independent
26
and the first two expressions can be expressed as combination of
the last two. Thus, we can construct the expression for bending
energy purely from the mean and Gaussian curvature.
The following formula for bending energy density is called the
Helfrich theory/equation after its main author Wolfgang Helfrich,
but two other authors, Canham and Evans, who participated in its
development, are sometimes added. It is:
bend =
1
2
κ(c1 + c2 − c0)2
+ κgc1c2 (3.24)
=
1
2
κ(2H − c0)2
+ κgK (3.25)
with the integral form:
Ebend =
membrane
1
2
κ(c1 + c2 − c0)2
+ κgc1c2 dA, (3.26)
where κg is a bending modulus associated with Gaussian curvature.
Note that Gaussian curvature is sometimes labeled KG
because instead of the mean curvature H we can use the extrinsic
curvature K ≡ 2H. The spontaneous curvature c0 is a
curvature that the membrane would spontaneously adopt (without
any bending). A non-zero value reflects a membrane asymmetry
either due to the lipid composition or a different chemical environment
on each side of the membrane in non-equilibrium situation.
If we are interested in bending the membrane without changing
its topology we can use the Gauss-Bonnet theorem, which
states that the integral of a Gaussian curvature over the surface
can be expressed as the integral over the boundary. Therefore,
if the boundary conditions remain the same, the Gaussian curvature
contribution is constant and the changes in bending energy
are:
∆Ebend =
membrane
1
2
κ(c1 + c2 − c0)2
dA (3.27)
27
Now we can combine the stretching and bending contributions
together to obtain the total elastic energy of a membrane deformation
as:
Etot =
membrane
σ +
1
2
κ(c1 + c2 − c0)2
+ κgc1c2 dA (3.28)
If there are no other factors in the system, the total elastic energy
will be in equilibrium at its minimum (the elastic energy represents
the free energy of the system). By varying the equation 3.28 we
can obtain a general shape equation, however, the resulting differential
equation and its solutions go beyond the scope of this
chapter. Nevertheless, we should mention that this theory is very
fruitful and various interesting phenomena have been explained
with it. To name just a few, the shape of red blood and other cells,
budding of the membrane, contact angle and shape of vesicles in
contact, and the change in the fluctuations and phase transitions
of membranes in contact. One analogy to the membrane and its
shapes is appealing. It is the analogy of soap bubbles, which are
also made of a thin layer with two water/hydrophobic interfaces
at the boundary. However, it can be misleading as there are significant
differences. For instance, if one makes a hole in a soap
bubble, it collapses, while a membrane heals itself by closing the
hole.
When two membranes come close to each other, their thermal
fluctuations start to influence each other. The suppression of
these fluctuation modes gives rise to repulsive forces. There are
two types of fluctuations: undulation and protrusion. Undulation
is a fluctuation of a membrane plane and can be well understood
in elastic terms. The corresponding undulation interaction of two
membranes at distance d is equals to −3π2
k2
T2
/128κd2
. Since
the bending modulus κ is highly dependent on the membrane
composition and surrounding solution, the undulation interaction
is highly dependent on the system. Importantly, the suppression
28
of the thermal fluctuations can lead to membrane phase transition,
and so the fluid membrane can become gel-like when it is
adsorbed at or pressed against a solid interface. Protrusion is a
movement of a single phospholipid out of the membrane plane. It
is a local distortion, which corresponds to the membrane roughness.
In addition, there is water depolarization and membrane
dehydration as two membranes get very close to each other. All
of these short-range effects add together to a repulsive interaction
on the order of kT, which exponentially decreases with a decay
length of 1 - 2 nm, and it was shown in 2012 that its origin is both
enthalpic and entropic.
Exercise: We have a cell and a virus is trying to get inside
it via endocytosis (Fig. 3.4). Endocytosis is a process during
which a particle gets internalized into the cell. There are different
types of endocytosis involving different proteins and amounts
of ATP. We will be interested in endocytosis without ATP, i.e. a
thermodynamically spontaneous process. For simplicity we will
assume that the virus particle is a sphere with radius R and that
its surface is homogeneously covered by cell-attractive proteins
(ligands). These attractive proteins are interacting with membrane
receptors (membrane-embedded proteins) that are homogeneously
distributed in the membrane. Due to the assumption of
homogeneity, we can define an effective attraction (per unit area)
w between the virus and the membrane. The total attractive interaction
Eatr can then be written as:
Eatr = wA, (3.29)
where A is the contact area between the virus shell and membrane.
At the beginning the virus particle is far from the membrane.
Upon the interaction between the membrane and virus,
the elastic membrane can bend and wrap around the virus particle.
This happens only if the attraction density is large enough.
Calculate the size at which the wrapping is a spontaneous pro-
29
cess and apply the formula to the Semliki Forest virus. Compare
the threshold radius with a size of the virus, which radius is about
R = 30 nm. Note that the virus contains about 80 attractive sites,
each binding with a few kT, leading to w = 0.035 kT nm−2
. For
the membrane deformation, use a typical bending modulus κ of
20 kT and its κg was recently estimated to -15 kT.
Figure 3.4: Schematic visualization of endocytosis.
Sources and further readings:
1.
4. Interaction of uncharged
membrane and proteins
(Dispersion forces, Hamaker constant, depletion forces, Derjaguin
approximation)
When we go beyond the ideal systems, we have to consider the
interactions between molecules. In the following chapters we will
discuss these interactions, with a focus on their strength and distance
dependence. This is crucial for coarse-graining, since we
need to know which interactions are the dominant ones and what
parameterization (functional profile) should be used for them.
We start with the interaction of uncharged molecules. Do they
interact when we neglect gravitational forces? The answer is
yes they do, and a common example is noble gases, which can
become liquid at low temperatures and/or high pressures. The
first explanation was given by Fritz London, hence the forces are
sometimes called London forces. The physical origin of the interaction
is due to the polarization of the electron cloud around
an atom when another atom gets close. This is why the term
induced dipole-induced dipole forces is sometimes used. However,
the most frequent name is dispersion forces, which originates
from its similarity to the quantum mechanical theory of light
dispersion. The derivation is based on quantum mechanics (per-
31
turbation method using the multipole expansion of 1/r interaction)
leading to:
ELondon = −
3
2(4π 0)2
IiIj
Ii + Ij
αiαj
r6
, (4.2)
where Ii is the first ionization potential of the i-th atom, α is polarizability,
and r is the distance between the atoms. Note that
the total dispersion force between two molecules (particles) depends
on the polarizability of the whole molecule, which is usually
orientation-dependent.
If one of the uncharged molecules has a permanent dipole
then there is also a dipole-induced dipole interaction, which is
sometimes called Debye interaction. Similarly to London interaction,
it depends on 1/r6
:
EDebye = −
1
(4π 0)2
p2
i αj
r6
, (4.3)
where p is the permanent dipole of the molecule. When both
molecules have permanent dipoles there is also a dipole-dipole
interaction. This commonly orients molecules and is sometimes
called Keesom interactions after its discoverer. The orientationaveraged
interaction free energy is:
EKeesom = −
1
(4π 0)2
1
3kT
p2
i p2
j
r6
(4.4)
All of the above interactions add together to form the van der
Waals interaction, which is the first order approximation of the interaction
of uncharged molecules. Naturally, there are higher order
terms of multipole expansion such as quadrupole-quadrupole,
etc. but they are typically weaker and more short ranged.
EV DW = EKeesom + EDebye + ELondon (4.5)
32
Van der Waals interactions for atoms are equal to London dispersion
forces, since atoms do not have permanent dipoles. Common
values of van der Waals interaction are on the order of a few
kT. For comparison, the strength of the interaction for hydrogen
bonds is about 10 kT and covalent bonds are another order of
magnitude stronger (on the order of 100 kT). However, numerous
van der Waals interactions allows Gecko lizard to hold on side
walls or ceilings.
A more general description of van der Waals interactions, where
all the above contributions are naturally together, is included in
Lifschitz theory. It is a quantum electrodynamics derivation of
Casimir effect for interaction in different dielectrics. According to
this theory, the polarizabilities are frequency-dependent and the
above results correspond to the zero frequency approximation.
The advantage of this approach is that we can also calculate dispersion
interactions in a continuum medium, i.e. solvent. The
particle interaction is then dependent on the excess polarizabilities
of the molecules compared to the solvent.
EV DW ∼ −
(αi − αs)(αj − αs)
r6
, (4.6)
where the subscript s stands for solvent. We can see that van
der Waals interaction is always attractive (negative) for similar
molecules. However, the interaction can become repulsive if one
molecule has a larger polarizability than the solvent and the other
molecule a lower polarizability than the solvent.
If we do not know the polarizabilities for the species of interest,
we can take advantage of the Clausius-Mosotti expression for the
refractive index n:
4
3
πρα =
n2
− 1
n2 + 2
, (4.7)
where ρ is a density. Thus, we can estimate which particles/materials
will be miscible and how much they will tend to aggregate. The
33
refractive index of water is 1.333. Most proteins as well as the inner
part of membranes have a refractive index of about 1.5, which
is similar to hydrocarbons. This is not surprising, since there are
hydrocarbons in the hydrophobic core of these objects. In fact, hydrophobic
attraction is the dominant interaction of self-assembly,
but we will discuss this later.
Proteins and membranes are not small atoms, hence if we
want to calculate their van der Waals interactions, we will have to
integrate it over their volumes. For simplicity we will assume the
interaction to be additive here.
We start with a small molecule next to a flat surface at distance
d. The surface is made of molecules with number density
ρ and each molecule interacts with energy −C/r6
. The easiest
integration is over small rings axially aligned with the surface normal
going through the molecule as depicted in Figure 4.5. All
points on the ring with radius x are the same distance from the
molecule r =
√
x2 + z2, where z is the distance from the middle
of the ring to the molecule. The number of surface molecules in
such a ring is 2πxρdxdz.
E·| = −
∞
z=d
∞
0
2πxρ
C
(x2 + z2)6/2
dxdz
= −2πρC
∞
z=d
1
4z4
dz = −
πρC
6d3
(4.8)
Importantly, this means that the van der Waals interaction between
the molecule and the surface is fairly long range ∼ 1/d3
. If
the surface is replaced with a wall of finite thickness t the integration
leads to:
E· = −
1
6
πρC
1
(d + t)3
−
1
d3
, (4.9)
which for small thicknesses t → 0 decreases with ∼ 1/d4
. For
34
more interactions with a wall of finite thickness that represents a
phospholipid membrane see Appendix.
Figure 4.5: Schematic picture of the molecule next to the wall for the
calculation of dispersion interaction integrating over the axial rings within
the wall.
Having the expression for the interaction of single molecules
with the surface, we can calculate the interaction between larger
objects and the surface by simply integrating over all the object
molecules. An example could be a large sphere of radius R interacting
with the surface as depicted in Figure 4.6. All spherical
molecules at distance z from the surface create a circular slice of
volume πx2
dz, where x is the radius of a given slice. From geometry
we can calculate the slice radius as x2
= R2
− (R − (z −
d))2
= (z − d)(2R − z + d), where d is the distance between the
35
Figure 4.6: Schematic picture of a sphere next to the wall for the calculation
of dispersion interaction using integration over the axial disks
within the sphere.
surfaces. Consequently we integrate:
E•| =
d+2R
d
ρsphπ(z − d)(2R − z + d)
πρsurf C
6z3
dz
= −
1
6
π2
ρsphρsurf C
d+2R
d
(z − d)(2R − z + d)
z3
dz
(4.10)
36
This can be simplified for close distances R d − z to:
E•| = −
1
6
π2
ρsphρsurf C
∞
d
(z − d)(2R)
z3
dz (4.11)
= −
1
6
π2
ρsphρsurf CR
∞
d
2
z2
−
2d
z3
dz (4.12)
= −
1
6
π2
ρsphρsurf CR
−2
z
+
d
z2
∞
d
(4.13)
= −
π2
ρsphρsurf CR
6d
(4.14)
Note that the interaction decreases with 1/d. We could also solve
the Eq 4.10 directly and obtain the full solution for all distances:
= −
1
6
π2
ρsphρsurf C
d2
+ 2Rd − 4dz − 4Rz
2z2
− ln(z)
d+2R
d
= −
1
6
π2
ρsphρsurf C
2R(d + R)
d(d + 2R)
− ln(1 + 2R/d)
In limiting cases we obtain a 1/d dependence for small d and 1/d3
for large distances, which confirms the above results.
Similarly, we can calculate the interaction between two surfaces,
where we integrate over squares of unit area. The energy
per unit area is then:
E|| = −
1
6
πρsurf1ρsurf2C
∞
d
1
z3
dz (4.15)
= −
πρsurf1ρsurf2C
12d2
(4.16)
We can see that it is convenient to define a new constant,
which contains all the material properties such as densities ρ and
polarizabilities (C ∼ α1α2). This constant is called the Hamaker
constant, A:
A ≡ π2
ρ1ρ2C (4.17)
37
The advantage of the Hamaker constant is that it is similar for
all materials, being in the range of 1 - 100 kT = 0.4 - 40.0 ×
10−20
J (water: 4 kT, hydrocarbon: 1 kT). The reason for this is
that the polarizabilities of molecules scale with their volume, while
their number densities scale with the inverse of volume. Hence,
A ∼ ρ1ρ2α1α2 ∼ 1
v1
1
v2
v1v2 = const. Using this constant, we can
express the interactions for species of different shape as follows:
Two atoms with distance r between their centers:
E·· = −
C
r6
(4.18)
Atom and surface separated by distance d:
E·| = −
πρC
6d3
(4.19)
Atom and membrane:
E· = −
1
6
πρC
1
(d + t)3
−
1
d3
(4.20)
Atom and sphere:
E·• = −
2
3
πρC
R(3d3
+ 6dR + 2R2
)
d3(d + 2R)3
(4.21)
Sphere and surface:
E•| = −
A
6
2R(d + R)
d(d + 2R)
− ln(1 + 2R/d) (4.22)
Sphere and surface d R:
E•| = −
AR
6d
(4.23)
38
Two spheres:
E•• = −
A
6
2R1R2
d(d + 2(R1 + R2))
+
2R1R2
(d + 2R1)(d + 2R2)
+ ln
d(d + 2R1 + 2R2)
(d + 2R1)(d + 2R2)
(4.24)
Two spheres d R1, R2:
E•• = −
R1R2
R1 + R2
A
6d
(4.25)
Sphere and membrane d R:
E• =
1
6
AR
1
d + t
−
1
d
(4.26)
Two surfaces (per unit area):
E|| = −
A
12πd2
(4.27)
Two membranes (per unit area):
E =
A
12π
2
(d + t)2
−
1
(d + 2t)2
−
1
d2
(4.28)
Cylinder of unit length parallel to the surface:
E◦| = −
A
√
R
12
√
2d3/2
(4.29)
Two parallel cylinders of unit length:
E◦◦ = −
R1R2
R1 + R2
A
12
√
2d3/2
(4.30)
39
Two cylinders perpendicular to each other:
E/\ = − R1R2
A
6d
(4.31)
Now we can estimate the dispersion interactions between proteins
and membranes. However, we should keep in mind that we
used two important assumptions. The first was the additivity of
pair interactions and the second was no retardation of dispersion.
In reality the interactions are not additive (e.g. aromatic rings) and
electromagnetic forces do not propagate instantaneously, which
can decrease the range of the interaction from r−6
to roughly r−7
.
This retardation become important when the separation distance
d is large - on the order of c/f, where c is the speed of light and
f is atomic dipole fluctuations. In practice, it plays a role for separations
larger than 5 nm.
The interaction between curved surfaces can be calculated
from the interaction of flat surfaces using the Derjaguin approximation.
This is a very useful approximation, which is valid for
short-ranged interactions without any assumption about the shape
(profile) of the interaction. It states that the interaction for curved
objects can be integrated over the two facing surfaces if their separation
is smaller than their curvature.
Etot(d) =
∞
d
w(z)dA, (4.32)
where d is the separation of objects, A is the cross-section area
and w(z) is the interaction per unit area between two flat surfaces
separated by distance z. For rotationally symmetric objects, we
can use cylindrical coordinates to get:
Etot(d) = 2π
∞
d
w(z(r))rdr (4.33)
We can demonstrate this with a simple case of two spheres of
radii R1 and R2 separated by distance d. We need to integrate the
40
interaction over the two facing surfaces, i.e. over 2πxdx, where x
is the radius of the ring in the cross-section of the spheres. The
total interaction is:
Etot =
∞
d
2πxw(z)dx, (4.34)
where w(z) is the interaction density between the two flat surfaces
separated by distance z. In order to be able to integrate
we have to relate the distance between the two cross-sections
z = d+z1 +z2 and the radius of the rings x. Using geometry and
Figure 4.7: Schematic picture of two spheres for Derjaguin approxima-
tion.
the fact that R1 z1 we simplify the relation (R1−z1)2
+x2
= R2
1
to z1 = x2
2R1
. Similarly, we obtain the expression for z2. Combining
these two we can express the distance between the rings as
z = d + x2
2R1
+ x2
2R2
. Thus dz = ( 1
R1
+ 1
R2
)xdx and the total
interaction energy can be expressed as:
Etot =
∞
d
2π
R1R2
R1 + R2
w(z)dz, (4.35)
41
From here we can directly see that the force between the two
curved objects will be:
Ftot = −
∂Etot
∂d
= 2π
R1R2
R1 + R2
w(z) (4.36)
This relation has proved to be fruitful for comparing of experiments
and theory, because it is much easier to do such a calculation
for two flat surfaces, while in experiments it is easier to measure
the force between two spheres by Atomic Force Microscopy
or optical tweezers.
To test the relation we can use the above expression for dispersion
interactions. The interaction between two spheres E••
evaluated from the interaction between two surfaces E|| is:
E•• =
∞
d
2π
R1R2
R1 + R2
E||dz (4.37)
= 2π
R1R2
R1 + R2
∞
d
−
A
12πz2
dz (4.38)
= −
R1R2
R1 + R2
A
6d
(4.39)
If our objects of interest are in solution, which proteins and
membranes usually are, there is an additional interaction called
depletion interaction. The origin of the interaction is in the osmotic
pressure of the solvent in bulk and the depleted volume between
close objects (see Figure 4.8). Thus the depletion interaction has
an entropic character.
We demonstrate this interaction on two flat surfaces of unit
area with zero interaction. The surfaces are immersed in solvent
made of small non-interacting solvent spheres of radius r. Clearly
at separations smaller than 2r, there is no solvent between the
surfaces and so the force is constant in this region. The force can
be calculated from osmotic pressure, i.e. the pressure of an ideal
42
Figure 4.8: Illustrative figure of depletion interaction.
gas pid = n
V
RT = ρkT, times unit area:
Edepletion = −
d
2r
−ρkTdz = ρkT(d − 2r) (4.40)
At separations larger than r the force is zero, so the energy of
separating the surfaces remains constant. This idea was proposed
by Asakura and Oosawa for unexpected attraction forces
between colloid particles in polymer solutions in 1954. In the appendix,
you can find a more advanced model for a solution of ideal
chains.
Exercise: Calculate the van der Waals interaction energy between
two lysozyme proteins and lysozyme protein and membrane
as a function of distance and plot the dependence. Assume
that lysozyme has a spherical shape with a radius of about 20 nm
and the membrane core has thickness 4 nm. The Hamaker constant
for lysozyme interacting with itself in water is 2.87 x 10−20
J,
43
while for a typical hydrocarbon oil it is 0.54 x 10−20
J.
Sources and further readings:
1. V. Adrian Parsegian; Van der Waals Forces A Handbook for
Biologists, Chemists, Engineers, and Physicists; Cambridge
University Press, 2005
2. Jacob N. Israelachvili; Intermolecular and Surface Forces
(3rd Edition); Academic Press, 2011
3. Ken A. Dill and Sarina Bromberg; Molecular Driving Forces:
Statistical Thermodynamics in Biology, Chemistry, Physics,
and Nanoscience (2nd Edition); Garland Science, 2010
5. Interaction of charged
membrane and proteins
(Poisson Boltzmann, Debye length, Strong coupling limit, Hofmeister
series)
In addition to the non-charged interactions described above, there
could be a Coulomb interaction in the system of interest. Coulomb
interaction between charged objects is a typical pairwise additive
interaction, which is fairly long ranged (∼ 1/r) and usually
stronger than dispersion. Its formula is simply Coulomb’s law:
FCoulomb(r) =
1
4π
q1q2
r2
= q1E2 (5.2)
VCoulomb(r) =
1
4π
q1q2
r
= q1ψ2 (5.3)
where F is the force, E is the electric field, V is the potential energy,
and ψ is the electrostatic potential. is a dielectric constant
of the given medium and q are the charges of the interaction objects
1 and 2 denoted in the subscript. Charges in solutions usually
originate, in the dissociation of salts, acidic or basic groups,
ionization, or by adsorption of the already dissociated molecules
on the object. This means that there is usually a nearby co-charge
to every charge present in soft matter systems.
45
To investigate the interactions, we start with a homogeneously
charged object and mobile counter-ions in the surrounding solution.
The charged object creates an electrostatic potential ψ,
which influences the distribution of counter-ions . Their relationship
is given by the Poisson equation:
ψ = −
qρ
, (5.4)
where q is the charge of a single counter-ion, ρ is the number
density of counter-ions, and is the Laplace operator. We are
interested in the equilibrium situation, where the chemical potential
(free energy per mol) µ is constant over the whole system:
µ = qψ + kT ln(ρ) = const. (5.5)
This equation provides us with a Boltzmann distribution of counterions,
which can be written as:
ρ = ρ0 exp −
qψ
kT
(5.6)
where ρ0 is a constant representing a density at ψ = 0, in our
case at infinity distance. By combining Eq. 5.4 and Eq. 5.6 we
obtain the Poisson-Boltzmann equation:
ψ = −
qρ0
exp −
qψ
kT
(5.7)
Proteins and membranes are usually not only surrounded by counterions
but also by additional salts. The total number density is then
a sum of individual densities:
ρ =
i
ρ0i exp −
qiψ
kT
(5.8)
where qi is the charge of the individual species, which is usually
expressed as a multiplication of elementary charge e and valency
46
zi: qi ≡ ezi. Therefore, the general Poisson-Boltzmann equation
can be rewritten as:
ψ = −
i
eziρ0i
exp −
eziψ
kT
(5.9)
This is a nonlinear second-order differential equation, and apart
from a few simple examples it does not have an analytical solution.
It is usually solved numerically and/or by approximations.
The exact solution is determined by the boundary conditions of
the system, which is typically system charge neutrality and/or
known electrostatic potential at infinity.
If we know electrostatic potential at certain point (from presence
of electrode or solution of the Poisson-Boltzmann equation)
we can calculate the density distribution of ions at that point.
Derivation starts by deriving the Boltzmann distribution:
dρ
dx
= −
qρ0
kT
exp −
qψ
kT
dψ
dx
(5.10)
and by substituting this result into the Poisson-Boltzmann equa-
tion:
dρ
dx
=
kT
d2
ψ
dx2
dψ
dx
(5.11)
=
2kT
d
dx
dψ
dx
2
(5.12)
By integration from infinity we obtain:
ρ = ρ0 +
2kT
dψ
dx
2
(5.13)
A more general expression, where more ions in solution are involved,
is:
i
ρi =
i
ρ0i +
2kT
dψ
dx
2
i
ρ0i +
E2
2kT
(5.14)
47
Note that here is ion density instead of individual ions, hence the
Poisson-Boltzmann theory is a mean field theory using averaged
density values.
The most common approximations to Poisson-Boltzmann equation
is its ’linearization’, in which the exponential function is approximated
by a first order expansion ex
1 + x leading to the
Debye-Hückel approximation:
ψ =
i
e2
z2
i ρ0i
kT
ψ (5.15)
We also used an electro neutrality condition i eziρ0i = 0, so the
first term after approximation vanishes. The Eq. 5.15 is a linear
differential equation, which is much easier to solve, however, it is
only valid for eziψ kT. In practice, this usually holds below
100 mM salt concentration. The solution to this equation is an
exponential function in 1D:
ψ(x) = ψ(0) exp −
x
λ
(5.16)
where λ ≡ kT
i e2z2
i ρ0i
is a characteristic distance called the Debye
screening length and it represents the distance at which the
compensating charge is distributed around the object. The Debye
length can be written using Ionic strength I as λ ≡ kT
e2NaI
,
where I = 1
2 i ciz2
i and Na is Avogadro’s number. Naturally, it
decreases with increasing concentrations of ions. For a solution
of monovalent equimolar salt (1:1), e.g. NaCl, the Debye length
is 0.3 nm for 1M solution, 1.0 nm for 0.1 M solution, and 10 nm
for 1 mM solution (roughly 0.304/
√
c nm). For divalent salts such
as MgSO4 it is half of the above values. As a result the charge of
the membrane and proteins is well screened below 1 nm at physiological
conditions (200 mM salt concentration). For convenience
we define the Bjerrum length λb, which is the distance, at which
48
Figure 5.9: Schematic profiles of ion density distributions next to a
charged wall.
two elementary charges interact with the same strength as the
thermal energy, λb = e2
4π kT
. (λb =56 nm in vacuum and λb = 0.71
nm in pure water)
The above solution of exponential decay is very important and
frequently used. It is also the main idea of the Gouy-Chapman
theory, in which large objects in solution are surrounded by a
electric double layer. The first layer is thin and is formed by dissociated
or adsorbed ionic groups. The second layer is diffuse
and it is formed by counter-ions, whose distribution exponentially
decays with distance from the object surface. At large distances,
49
i.e. distances larger than the Debye screening length, the objects
can be viewed as surrounded by a double layer.
Using the electric double layer assumption together with the
Derjaguin approximation and weak overlap assumption, we can
derive the following expressions for interaction energies between
the species at distance d.
Two small molecules of radius R << d:
E·· =
Z1Z2e2
4π
exp(−κd + 2κR)
d(1 + 2κR)
(5.17)
Two spheres (R >> d):
E•• =
R1R2
R1 + R2
Z exp(−κd) (5.18)
Sphere and surface:
E•| = RZ exp(−κd) (5.19)
Two surfaces (per unit area):
E|| =
κ
2π
Z exp(−κd) (5.20)
where κ ≡ 1
λ
is the inverse Debye length and Z is an interaction
constant with a similar role to the Hamaker Constant A for van
der Waals interactions. Z is defined by the surface potential ψ0 of
an isolated surface and related to surface charge density:
Z = 64π
kT
e
2
tanh2 zeψ0
4kT
(5.21)
The hyperbolic tangent is often written separately as:
γ = tanh(zeψ0/4kT), which is at ambient conditions and ψ0 in
mV: γ = tanh(zψ0/103).
50
For small surface potentials (below 25 mV), this can be simplified
using the surface charge density of particles σ = κψ0 as:
Two spheres:
E•• =
2πσ2
κ2
R1R2
R1 + R2
exp(−κd) (5.22)
Sphere and surface:
E•| =
2πσ2
R
κ2
exp(−κd) (5.23)
Two surfaces (per unit area):
E|| =
2σ2
κ
exp(−κd) (5.24)
While the above expressions are very helpful for estimates
of the interaction strength between membranes and proteins, we
have to keep in mind the approximations we made. For example,
at close distances the ion-correlation effects will start to be important.
The diffuse counter-ion layer is a highly polarizable layer,
and its fluctuations will give rise to attractive forces between the
particles ( van der Waals forces have a similar origin). Indeed,
it has been shown that ion-correlations can lead to an attraction
between walls or spheres of the same charge.
Moreover, once the separation between the particles approaches
the size of ions, the Poisson Boltzmann description becomes inaccurate,
as the ions form a two-dimensional layer. This regime
is called the Strong coupling regime and was investigated only
recently with significant help from computer simulations. Simulations
were not only able to test Strong coupling, but able to
investigate the gap between the Strong coupling theory and the
Poisson Boltzmann description.
In addition, the ions are only characterized by their charges
in the Poisson Boltzmann equation, so all the finite size effects
51
of the ions are missed. Unfortunately, there are plenty of examples
where the ion size is crucial. For instance, the vital biological
difference between sodium and potassium, two very common alkali
metals of very similar size. While sodium can pass through
ion pumps in the membrane, potassium cannot. This leads to an
ionic imbalance with potassium inside the cells and with sodium
outside, which is used for the transport of larger molecules and
correct functioning of proteins (there is an approximately 150 mM
concentration of sodium outside and 5 mM inside the cells, while
potassium has a 5 mM concentration outside and 150 mM inside
the cell). Thus Poisson Boltzmann fails to explain Hofmeister series
(also sometimes called Lyotropic series). These are series for
cations and anions of salts which have been formed based on the
salt effect on the salting out of egg white proteins. However, since
their discovery in 1888 by Franz Hofmeister, they were found to
be useful in many protein phenomena where ions follow the same
trend. The explanation for this is found in the pairing between
salt ions and proteins, which is related to their hydration free energy
(roughly corresponding to their size), which determines the
strength of the pairing. Once the size of ion is large enough, ions
can also pair with uncharged groups and adhere there. Note that
some modified versions of the Poisson-Boltzmann theory, which
take the finite size of ions into account, were developed recently
solved numerically.
Be aware that the charge distribution on proteins is not static
and a presence of other charged proteins, membrane, or external
field can induce protonation or deprotonation of amino acids. The
new charge distribution can be described in induced multipoles
(charge, dipole, quadrupole, etc.), which size is dependent on the
protein capacitance, C. The capacitance is defined as a change
of the mean charge upon a small change of electric field:
C = −
∂ < q >
∂ψ
=
< q2
> − < q >2
e2
(5.25)
52
The induced interaction of two molecules can then be written as:
E = −
e2
2kT(4π )2
C1C2
R2
−
1
2kT(4π )2
Q2
2C1 + Q2
1C2
R2
−... (5.26)
where Q is the mean charge of the molecule.
For a single amino acid, the capacitance can be related to
change of pH:
C = −
1
ln 10
∂ < q >
∂ pH
(5.27)
The largest capacitance is thus at pKa, where the molecular charge
changes the fastest with the pH change. Histidine has the closest
pKa to neutral pH and is thus the most relevant amino acid to
consider in biological applications (see appendix for pKa table of
free standing amino acids). For proteins and larger molecules the
interaction usually needs to be calculated numerically, since the
pKa of the residue is influenced by presence of charged residues
in vicinity.
Exercise: Calculate the electrostatic repulsion of two Lysozyme
proteins in blood at pH 7.4, where lysozyme has a charge of +7.
Assume the radius of Lysozyme to be 20 nm.
Exercise: What pKa has histidine on the surface of Lysozyme
and close to a negatively charged membrane at physiological conditions
and neutral pH? Consider standard histidine pKa = 6.0
and membrane surface charge density to be -0.3 e/nm2
. Calculate
also apparent equilibrium constant pKapp that is defined with
respect to bulk concentration of protons, [H+
]∞.
pKapp(x) = − log
[X−
][H+
]∞
[HX]
(5.28)
Sources and further readings:
53
1. Jacob N. Israelachvili; Intermolecular and Surface Forces
(3rd Edition);Academic Press, 2011
2. Ken A. Dill and Sarina Bromberg; Molecular Driving Forces:
Statistical Thermodynamics in Biology, Chemistry, Physics,
and Nanoscience (2nd Edition); Garland Science, 2010
6. Towards realistic
interactions of membranes
and proteins
(DLVO, hydrophobic effect, parametrization of potentials )
By combining the electrostatic and dispersion interactions we obtain
the DLVO potential, which describes the interaction between
charged proteins and membranes at a coarse-grained level. Originally,
the DLVO potential was developed for charged colloids by
Derjaguin and Landau and, independently, by Verweij and Overbeek
(the name DLVO is constructed from the first letters of the
surnames of these authors). The interaction profiles are found to
be in very good agreement with the measured profiles between
two colloidal particles by Atomic Force Microscopy (AFM).
Typically, the dispersion interaction is dominant at short distances,
while the electrostatic interaction is the determining factor
over large separations. The DLVO interaction between two highly
charged particles (of the same sign) is dominated by electrostatic
repulsion with a single van der Waals minimum at contact. When
the particle charges are not very large or the solution contains
enough electrolytes, a second minimum at a larger separation
occurs (see Figure 6.10). The two minima are separated by a bar-
55
rier which is pH- and electrolyte- dependent. If the barrier is much
higher than kT, the solution becomes kinetically stable with only
the secondary minimum populated. Upon an increase in concentration,
the height of the barrier decreases, which leads to a slow
aggregation. Once the concentration reaches a critical concentration,
the barrier height diminishes to zero and particles undergo a
rapid aggregation. The usual position of the barrier is for separations
in the range of 1 to 4 nm. Note that for protein crystallization
a slow aggregation is desirable, because rapid aggregation often
leads to a random orientation.
Figure 6.10: Schematic profile of DLVO potential displayed together with
its van der Walls and electrostatic contributions.
One of the successes of DLVO potential is its explanation of
56
the empirical Schulze-Hardy rule, which states that the critical coagulation
concentration varies with the inverse sixth power of the
colloid charge. We will start the derivation from DLVO potential
for two spherical particles with the same radius R:
E•• = 64πkTRρ∞
γ2
κ2
exp(−κd) −
AR
12d
(6.2)
where γ = tanh(zeψ0/4kT). At critical concentration the barrier
is zero, hence E••(db) = 0 leading to:
ρ∞ =
Aκ2
768πkTγ2db
exp(κdb) (6.3)
The position of the barrier db can be calculated from the maximum
definition dE••(db)/dd = 0 and using the above equation leading
to db = 1/κ. Substituting the position of the barrier in the above
gives:
ρ∞ =
Aκ3
768πkTγ2
exp(1) (6.4)
Inserting a definition of κ2
= z2e2ρ∞
kT
, we obtain:
ρ∞ =
7682
π2 3
k5
T5
γ4
exp(2)
A2e6z6
(6.5)
For high potentials (ψ0 > 100mV) the γ = tanh(zeψ0/4kT) 1
and so the critical concentration approximates to ρ ∼ 1/z6
which
is the Schulze-Hardy rule. However, for low constant potential
γ = tanh(zeψ0/4kT) zeψ0/4kT, leading to the modified rule
ρ ∼ ψ4
0/z2
.
DLVO theory also successfully in explains the stability of protein
solutions with various electrolytes. An example of such a
solution is milk, which is a stable solution of charged proteins
that can become unstable upon the addition of vinegar or salt,
57
leading to protein aggregation. However, there are many wellknown
examples where DLVO fails. The main limitation is due
to the Poisson-Boltzmann approximations. Poisson-Boltzmann is
a mean-field continuum theory, which does not include ion size
effects and ion-ion correlation effects.
It has been shown that combining van der Waals attraction
and electrostatic repulsion results in a rich phase diagram which
could also include finite size aggregates such as clusters or fibers.
The reason for this is that for few particles, the short-range van
der Waals could overcompensate for the weak charge repulsion.
When more particles are aggregated then the total charge of the
cluster significantly increases, leading to a strong repulsion of additional
particles, however, the attractive van der Waals attraction
increases slightly with increased aggregate size. If we want to
consider more complicated objects than spheres, the problem of
calculating the phase diagram becomes quite difficult, and using
computer simulations becomes beneficial.
To complete the interactions of membranes and proteins we
have to discuss the hydrophobic effect, which we have not yet
covered. It is one of the most important driving forces for protein
folding and molecular self-assembly. The hydrophobic parts of
molecules are not exposed to solution and become mostly buried
in the assembled structures. The residual hydrophobic amino
acids at the protein surfaces are responsible for the hydrophobic
interaction between proteins.
The origin of the hydrophobic interaction is connected to hydrogen
bonding in water, which is responsible for many water abnormalities.
When water is in contact with hydrophobic molecule(s),
its hydrogen bond network is disrupted. Indeed, at large interfaces,
such as a water/oil interface, the interface water molecules
have less hydrogen bonds and the hydrophobic effect is enthalpic
in origin. However, the complete picture is not so simple. Smaller
58
hydrophobic molecules only slightly disrupt the hydrogen bonds.
When the curvature is large enough, the number of hydrogen
bonds remains roughly constant, but the number of possible arrangements
of these hydrogen bonds decreases. The smaller
number of micro states results in an entropic character of the
hydrophobic effect for small molecules. An extreme example of
this behavior are clathrates, the relatively stable cages around
small hydrophobic molecules formed at high pressures. The radius
crossover from enthalpic to entropic is around 1 nm.
The hydrophobic free energy, dG, can be calculated from the
water-accessible surface area of the nonpolar part of the molecule
(dG = γdA). The water accessible surface can be constructed
by rolling a sphere of radius 0.14 nm, the approximate size of
a water molecule, over the surface of the studied molecule. For
large hydrophobic particles (larger than 1 nm in radius) the proportional
constant γ is a macroscopic surface tension, e.g. for
a water/oil interface it is 4.7 kcal/mol/nm2
= 7.9 kT/nm2
= 33
dynes/cm. For smaller molecules such as alkanes it was shown
that the effective surface tension is about 4 kT/nm2
. For mid
range molecules/curvatures one shall extrapolate between these
two values.
The strength and distance dependence of the above interactions
provide us with solid grounds for estimating and evaluating
the most important interactions in the system of interest. Based
on this chemo-physical intuition, we can develop a coarse-grained
model via a top-down approach. In detailed top-down models, the
interaction sites are related to specific chemical groups. The potential
is then typically fitted to known thermodynamic properties
such as partition coefficients, density, permittivity, binding constants,
persistence length, etc. Large-scale top-down models do
not necessarily describe any particular system, but rather capture
phenomenological features such as the shape and distribution of
interaction sites. The behavior of such models is typically inves-
59
tigated by the systematic variation of model parameters. The information
obtained can help to predict the results of experiments
under different conditions as well as help to interpret the actual
results of those experiments.
We finish the top-down approach with a brief summary of the
typical strength and distance dependence of important interactions
between small molecules (for dipoles we use thermally averaged
interactions):
Interactions Range Energy /kT (kJ/mol)
Covalent bond 1 - 2 A 40 - 350 (100 - 900)
Hydrogen bond 2.5 - 3.5 A 2 - 12 (5 - 30)
Charge charge r−1
16 - 30 (40 - 70)
Charge dipole r−4
4 - 12 (10 - 30)
Dipole dipole r−6
1 - 3 (2 - 8)
van der Waals r−6
0.5 - 2 (1 - 5)
Bottom-up
The bottom-up approach is based on mapping M, a mathematical
connection (M : r → R) between detailed (typically
all-atom) configuration r and the coarse-grained configuration R,
which is related to the accuracy, efficiency, and transferability of
the coarse-grained model.
Ri = Mi(r) (6.6)
The mapping is usually a linear combination of the Cartesian coordinates
of a number of atoms to each coarse-grained building
block. The degrees of freedom that will be averaged out in the
mapping should be: 1. not essential to the process or property of
interest; 2. numerous, so the computational gain of the coarsegrained
model is significant; 3. largely uncorrelated with the essential
degrees of freedom or the essential degrees of freedom
need to be also represented in an effective way. Typically these
60
include high frequency motions and uncorrelated slow motions;
however, it might not be trivial to determine which slow motions
are uncorrelated. Note that the more coarse-grained the model,
the more specific (restricted) it is to the system it was parameterized
for. This non-transferability of the model is a price we often
pay for its speed. For example, if a protein is parameterized in an
implicit water model, it will not be valid for a system in a mixed
solvent such as water/ethanol. The reason for this is given by the
surface tension, which is different in both systems and hence the
hydrophobic parts of the protein should interact differently to take
this into account.
The mapping defines how we split the molecule of interest
into coarse-grained blocks. For small molecules, we usually think
about the desired number of blocks and then geometrically split
the molecule into parts of equal size. For non-homogenous large
molecules (proteins of 100 nm) it is beneficial to perform a normal
mode analysis, motion analysis, or experiments to identify the
rigid and flexible parts of the molecule. Rigid parts could be replaced
by one or few blocks to represent their shape correctly. In
contrast, the flexible parts should be divided into a larger number
of blocks that could properly describe its motion and fit its elastic
parameters such as persistence length. Importantly, if the chosen
model does not describe the molecule properly, a refined model
might be necessary.
Once we have our building blocks, we can start to parameterize
their interactions. Statistical mechanics provides us with a
formal framework to determine the interactions. This is based on
the fact that the probability of each coarse-grained configuration
should be equal to the sum of all corresponding probabilities in a
detailed model.
e−
W (R)
kT = dre−
u(r)
kT
N
i=1
δ (Mi(r) − Ri) (6.7)
61
Where u is a potential in the detailed model and Mi is a mapping
of the i-th configuration. W(R) is a general many-body potential
at the coarse-grained level, which is a free energy profiles or in
other words the potentials of mean force averaged over detailed
model configurations. The solution of this equation is called the
inverse problem of statistical mechanics and Henderson theorem
states that there is a unique solution, i.e. a one-to-one correspondence
between the potential and the distribution function. Unfortunately,
W(R) is impossible to calculate for most systems and
its many-body character makes it very inconvenient in practical
use. The challenge of the bottom-up approach is thus to find an
approximation to W(R), which is computationally efficient and yet
accurate. We typically limit ourselves to isotropic two-body potentials,
for which the closest fit to W(R) is not unique and depends
on the fitting procedure. Naturally, the approximation is also related
to the transferability of the obtained potentials to different
thermodynamic states than the ones for which it was parameter-
ized.
Note that even though the integral determines the mass of the
given interaction site, which reproduces the equilibrium momenta
distribution, it does not imply that the dynamic properties are reproduced
in the coarse-grained model.
The simplest bottom-up method is called the Boltzmann Inversion
or Direct Boltzmann Inversion. It simply converts the distribution
of populated configurations from a detailed model to a
coarse-grained potential U:
U(R) = −kT ln
p(r)
J(r)
, (6.8)
where p is probability and J is the corresponding Jacobian factor.
In many cases this is simplified to spherically isotropic potentials
calculated from radial distribution functions ρ:
U(R) = −kT ln ρ(r) (6.9)
62
There are two common ways to obtain the necessary data from
the detailed model. The first is to do one simulation with the whole
system of interest, while the second one uses dilute simulations of
individual building blocks. This method does not include the effect
of the environment and the presence of other coarse-grained sites
in its vicinity, which is very important in dense systems.
The correlation among the different coarse-grained sites is
captured by the Iterative Boltzmann Inversion method. As the
name suggests this method is iterative, where a coarse-grained
simulation is carried out in each step. The coarse-grained potential
is modified based on the difference between the probabilities
from the detailed and coarse-grained simulation.
U(R)new = U(R)old + kT ln
P(R|Uold)
p(r)
(6.10)
The Iterative Boltzmann Inversion thus aims to reproduce pair histograms
from the detailed model as accurately as possible.
Inverse Monte Carlo is also an iterative scheme, but in addition
to the radial distribution functions, their cross correlations
distributions are also used. This includes the fact that modifying
the potential at one distance can affect the radial distribution function
at other distances. In general, Inverse Monte Carlo should be
faster and more likely to converge (not get trapped in local minima)
for multi-component systems than the Iterative Boltzmann
Inversion.
Note that some of the radial distributions functions can be provided
from experiments or there can be additional restrictions during
the fitting, such as pressure or compressibility values. Such
coarse-grained models would be a hybrid between the bottom-up
and top-down approaches.
An alternative variational approach is Force Matching (FM),
where the goal is to provide a potential that would fit the mean
63
forces calculated with the more detailed model. Mathematically,
one minimizes a function:
χ2
(F) =
1
3N
N
s=1
|Fs(M(r)) − fs(r)|2
, (6.11)
where fs(r) is the net force on a coarse-grained site s in configuration
r. Fs(M(r)) is a net force on the same interaction site using
the coarse-grained model in the mapped configuration M(r).
Brackets correspond to the average over a canonical ensemble.
The FM thus focuses on the dynamic rather than the structural
properties. In the extended ensemble version of FM higher order
distribution functions (correlations) are also necessary, and
the optimized potentials have an increased transferability. This
method is called Multiscale coarse-graining. The ability of the FM
model to reproduce the radial distribution functions has been suggested
as a means of checking the completeness of the coarse
grained model, showing that the mapping is fine enough. Similarly,
net forces could be used as a check for the structure based
methods.
All the above methods can be understood in terms of minimizing
the information loss during the coarse-graining process. The
information lost during coarse-graining is:
Φ(R) = ln
pr(r)δ(R − M(r))dr
PR(R)
, (6.12)
where P is the probability of configuration R in the coarse-grained
configuration space and p is the probability of configuration r in a
detailed configuration space. The relative entropy associated with
this information loss is the negative log likelihood of the coarsegrained
model relative to the detailed one.
Srel = pR(R)Φ(R)dR (6.13)
64
The relative entropy is positive or zero and it vanishes when the
coarse-grained model perfectly reproduces the sampling of the
configuration space from the detailed model. The task of coarse
graining is then to find the interaction potentials which minimize
Srel for all configurations λ.
δSrel
δU
=
δSrel
δ λ U(Rλ)
= 0 (6.14)
If we limit ourselves to pair potentials, this could be reformulated
as a shown set of linear equations, whose solution leads to the
desired coarse-grained potential. The numerical solution then
corresponds to the methods described above. The relative entropy
may also be helpful in optimization of the mapping.
Iterative Boltzmann Inversion and Inverse Monte Carlo minimize
the average of Φ(R) and involve a nonlinear optimization
problem that is solved by iterative simulations. Force matching
and Multiscale coarse-graining methods have the advantage of
minimizing |Φ(R)|2
, which leads to a linear optimization problem
that can be solved directly.
Exercise: Calculate how much you have to lower the pH of
milk to make a yoghurt (you can verify your result experimentally
using milk and vinegar). Milk is a protein suspension stable
at neutral pH. The most abundant milk protein, casein, forms
micelles 100 nm in size and with a density 6 times lower than
lysozyme. The zeta potential of those micelles was measured to
be -8 mV at neutral pH. The Debye screening length of milk is
about 1 nm. Assume that the slip plane where the zeta potential
is measured is 2.5 nm from the micelle, as it is fluffy. For lower pH
levels, use these values of zeta potentials (-2.5 mV for pH 5, -1.2
mV for pH 4.8, -0.5 mV for pH 4.6, 0.0 mV for pH 4.5, and +0.5
mV for pH 4.4).
Exercise: Calculate the hydrophobic interaction and its distance
dependence between two hydrophobic spherical particles
65
and two rods in a parallel orientation. Could this correspond to
globular protein and α-helix?
Sources and further readings:
1. W. G. Noid; Perspective: Coarse-grained models for biomolecular
systems; The Journal of Chemical Physics 2013, 139
(9), 090901; DOI: 10.1063/1.4818908
2. Jacob N. Israelachvili; Intermolecular and Surface Forces
(3rd Edition); Academic Press, 2011
3. Ken A. Dill and Sarina Bromberg; Molecular Driving Forces:
Statistical Thermodynamics in Biology, Chemistry, Physics,
and Nanoscience (2nd Edition); Garland Science, 2010
7. Thermodynamic
principles of self-assembly
(Critical micelle concentration, size distribution aggregates,
shape argument )
In this chapter we will look at the thermodynamics of the selfassembly
of molecules and particles, a vital cellular process. This
usually means the association of hydrophobic or amphiphilic (partly
hydrophobic and partly hydrophilic) species in aqueous solutions.
The resulting molecular aggregates can have various geometries
and sizes depending on the shape and type of interactions of the
molecules of interest. The interactions between molecules consist
of van der Waals interactions, hydrogen bonds, and electrostatic
interactions. There are no covalent bonds, which means
that the strength of the interactions are on the orders of kT or
dozens of kT, therefore the association can be modified by the
temperature and composition of the solvent.
The size distribution of aggregates is determined by thermodynamics.
At equilibrium, chemical potential has to be the same
over the entire system. If we have aggregates of various sizes,
their chemical potential can be written as:
µN = µo
N + RT ln(aN ), (7.2)
67
where aN is the activity (also called relative activity) of the aggregate
composed of N molecules. The chemical potential of a
single molecule in such an aggregate is:
µ1N = µo
1N +
RT
N
ln
a1N
N
, (7.3)
where a1N = NaN is the activity of a single molecule in the Naggregate
and µo
1N = µo
N /N is the standard chemical potential of
a single molecule. Thus, at equilibrium we obtain the relationship
between different aggregates:
µ1N = µ1M (7.4)
µo
1N +
RT
N
ln
a1N
N
= µo
1M +
RT
M
ln
a1M
M
(7.5)
For an alternative derivation of this equation, see the appendix.
We can obtain the relation of the N-aggregate to monomers we
can get by simply using M = 1:
µo
1 + RT ln (a1) = µo
1N +
RT
N
ln
a1N
N
(7.6)
and the mole fraction of molecules in the N-aggregate can thus
be expressed as:
x1N = N x1 exp −
µo
1N − µo
1
RT
N
(7.7)
assuming the activity coefficient to be 1 (ai = γixi). Remember
that the total molar fraction of solutes x can never exceed 1:
x =
∞
N=1
x1N ≤ 1 (7.8)
Similarly, the concentration of molecules in the N-aggregate can
be written as:
c1N /co
= N c1/co
exp −
µo
1N − µo
1
RT
N
(7.9)
68
where co
is the standard concentration. Note that deriving from
molar fraction (cN = xN ∗ i ci) one would obtain above expression
with total concentration instead of the standard concentra-
tion.
The system equilibrium can also be written as a set of chemical
reactions. For the equilibrium between monomers and aggregates
of size N we can write:
NA ⇔ AN (7.10)
The equilibrium constant is then:
K =
cN /co
(c1/co)N
=
c1N (co
)N−1
NcN
1
(7.11)
= exp −
∆G
kT
= exp −
N(µo
1N − µo
1)
RT
(7.12)
From this we again obtain the equation 7.9.
The equation 7.9 obtained above has important implications,
as the total mole fraction can be expressed as a polynomial of x1
with small coefficients. As a result the total concentration is dominated
by a linear term for small concentrations. This is caused
by the large translational entropy that monomers would lose upon
association. Monomer concentration thus increases linearly with
increasing total concentration (limc→0 c1 = c). Naturally, this
does not continue forever. When the loss of monomer translation
entropy is comparable to the association energy, the solution
saturates and all additional molecules join the aggregates. This
threshold concentration is called the Critical Micelle Concentration
(CMC) or Critical Aggregation Concentration (CAC). The idealized
profile is shown in Figure 7.11).
We can express this behavior from Eq. 7.7. Let’s consider a
system where molecules prefer to aggregate in micelles of size
69
N. We can then write:
x1N ≤ 1 (7.13)
N x1 exp −
µo
1N − µo
1
RT
N
≤ 1 (7.14)
x1 ≤
1
N1/N
exp
µo
1N − µo
1
RT
(7.15)
The right side then corresponds to the CMC of the given system.
Figure 7.11: Schematic profiles molar fractions.
The dependence of µo
1N on N determines the aggregate size
distribution. In general, there could be several peaks, however,
70
this is not the case in simple systems as standard chemical potential
can be approximated by:
∆µo
1N = ∆µo
1∞ + A
g
NB/d
(7.16)
where g is the interaction free energy of two neighboring particles,
d is a dimensionality factor, and ∆ means the chemical potential
difference from the monomer.A, B are specific constants for the
given system. The validity of Eq. 7.16 can demonstrated in the
simple case of linear aggregates, where only the nearest neighbors
would interact. The total chemical potential of aggregate
N∆µo
1N is equal to the number of contacts times the molar free
energy of the contact, g. So for all N we can write:
N∆µo
1N = −(N − 1)g (7.17)
∆µo
1N = −(1 − 1/N)g (7.18)
This can be reformulated using the limit (∆µo
1∞ = limN→∞ −(1−
1/N)g) to:
∆µo
1N = ∆µo
1∞ +
g
N
(7.19)
Similarly, we can estimate the number of contacts using square
arrangement in 2D. The size of the square, a is equal to N1/2
. All
middle particles have 4 neighbors, side particles have 3 neighbors,
with the exception of the 4 corner particles which have only
2 neighbors. The free energy of the aggregate can then be estimated
from the contacts as:
4g×(a−2)(a−2)+3g×4(a−2)+2g×4 = 4ga2
−4ga (7.20)
In terms of the number of particles, this is 4gN − 4gN1/2
. The
total free energy of the system can be written as:
N∆µo
1N = −(4gN − 4gN1/2
) (7.21)
∆µo
1N = −4g + 4g
1
N1/2
(7.22)
71
Using the limit for an infinite aggregate, we get ∆µo
1∞ = −4g, so
we obtain:
∆µo
1N = ∆µo
1∞ + 4
g
N1/2
(7.23)
The factor 4 is given by the square arrangement, and if we used
a circular one it would be different. In general, the number of
interactions is given by the number of neighbors times N minus
the number of boundaries, which scale as N1/d
.
Using the standard chemical potential of the monomer, Eq.
7.16 can be reformulated as:
µo
1N − µo
1 = (µo
1∞ − µo
1) + A
g
NB/d
= (µo
1∞ − µo
1) 1 −
A
NB/d
(7.24)
since we know that g = −∆µo
1∞ from the derivations above.
Substituting Eq. 7.24 into Eq. 7.7 we obtain the expression
for aggregate distribution:
x1N = N x1 exp −
µo
1∞ − µo
1
RT
1 −
A
NB/d
N
(7.25)
= NxN
1 exp −
µo
1∞ − µo
1
RT
N − AN1−B/d)
(7.26)
For convenience we define a new constant C ≡ exp −
µo
1∞−µo
1
RT
and rewrite the above formula.
x1N = N [x1C]N
C−AN1−B/d
(7.27)
Above the CMC, we can approximate this as:
x1N = NC−AN1−B/d
(7.28)
Because C > 1, then for B/d < 1, the x1N decreases for larger
N and there will be no large aggregates. On the other hand
72
B/d > 1 leads to strong aggregation, where x1N increases with
N. This results in phase separation (infinite aggregate). B/d = 1
is a threshold value, which can lead to finite size aggregates with
distribution x1N N exp − N√
xC
well above the CMC (see appendix).
With this distribution, the majority of the molecules are
in the aggregate Nmax =
√
xC and the distribution of aggregates
is quite broad, therefore, the solution is polydisperse. However,
keep in mind that these results are only valid for simple systems
such as spheres with isotropic interaction, where Eq. 7.16 is valid.
In real systems the interactions are anisotropic and micelles of finite
size appear for geometrical reasons.
The effect of the molecule shape on aggregate structure and
size can be nicely explained on well-studied surfactants with hydrocarbon
tails. The core of an aggregated micelle is composed
of hydrocarbon tails surrounded by hydrophilic headgroups, which
are in contact with the solution. The size of the hydrophobic core
is dictated by the volume of surfactant hydrocarbon tails vt, while
the area in contact with water is dictated by the area of the headgroup
a0. The last parameter describing the surfactant molecule
is the critical length of the hydrocarbon tails lc. It is defined as a
length beyond which the hydrocarbons cannot be considered to
be an incompressible fluid and is slightly smaller than the length
of fully extended tails. For saturated hydrocarbons, we can use
the estimate (in A)
lc ≤ 1.54 + 1.265n (7.29)
where n is the number of hydrocarbons in the chain. Similarly, the
estimate for volume (in A3
) is:
vt 27 + 26.9n (7.30)
Both lc and vt grow linearly with the length of the tail and their
ratio is thus constant vt/lc 21 A2
which is the cross-sectional
area of the hydrocarbon chain.
73
Using the vt, lc and a0 parameters we can calculate the packing
parameter p:
p =
vt
lca0
(7.31)
This parameter is characteristic for the shape of surfactant and
determines the morphology of the aggregate (Israelachvili 1976).
Let us start with a spherical micelle of radius R. The number of
molecules in the micelle can be calculated from the volume or the
surface as:
N =
4
3
πR3
/vt = 4πR2
/a0 (7.32)
thus, for radius we obtain:
R = 3
vt
a0
(7.33)
At the same time, we know that the radius has to satisfy R ≤ lc.
As a result we obtain the formula:
1
3
≥
vt
lca0
= p (7.34)
Thus the packing parameter for a spherical micelle is p ≤ 1
3
meaning that the molecules have a conical shape. A well-known
example of such a molecule is sodium dodecyl sulfate.
Similarly, cylindrical micelles (hexagonal phase) are formed
with surfactants with a packing parameter 1
3
≤ p ≤ 1
2
and with the
shape of a truncated cone. An example is the cationic lipid with a
single alkyl chain CTAB - hexadecyl trimethylammonium bromide.
Surfactants with the shape of even more truncated cones with
packing parameter 1
2
≤ p ≤ 1 self-assemble into cubic phase or
vesicles (examples are lipids with two hydrocarbon chains, such
as phosphatidylcholines). Bilayers are composed of cylindrical
surfactants with p 1 represented by phosphatidyl ethanolamine.
Surfactants with p ≥ 1 have the shape of a wedge (inverted cone)
74
p 1/3 1/2 1 2 3
shape
cone
truncated
cylinder
truncated
wedge
cone wedge
phase
micellar hexagonal lamellar
inverted inverted
hexagonal micellar
Table 7.1: Summarizing table of related packing parameter, surfactant
shape, and formed phase.
which is truncated up to a value of 3. Wedge-shaped surfactants
form inverted phases. Vesicles or inverted cubic phase are
formed by surfactants with packing parameter 1 < p ≤ 2. For
2 < p ≤ 3, surfactants such as ceramide aggregate into an inverted
hexagonal phase (inverse cylindrical micelles) and for surfactants
with 3 > p the equilibrium structure is inverse micelles.
These results are summarized in Table 7.1.
The exact packing parameter and phase formed depends not
only on the surfactant headgroup and length of the hydrocarbon
tail, but also on the composition of the solution, number of unsaturated
bonds in the tails and the temperature. The composition
of the solution mostly affects the headgroup area a0, which could
be decreased by increasing the salt concentration or increased
by charging the headgroup via a change in pH. Unsaturated or
branched tails increase the vt/lc ratio, which could also be in-
75
creased by the addition of hydrophobic molecules into the solution.
Temperature effects are more complex, as they change both
the headgroup area and tail volume at the same time.
Naturally, mixtures of surfactants behave somewhere in between
the pure compounds and the phase diagram can be quite
rich and complex. This is utilized in living cells, which can synthesize
various lipid types and thus control the composition and
properties of the cellular membranes.
We can only analytically describe aggregation for very simple
systems, thus more realistic systems are studied in experiments
or computer simulations. Nevertheless, it is good to keep in mind
the basic thermodynamic principles of aggregation.
Sources and further readings:
1. Intermolecular and Surface Forces (3rd Edition); Jacob N.
Israelachvili; Academic Press, 2011
8. Langmuir adsorption
(Adsorption and Grand canonical ensemble)
In the previous chapter, we studied the aggregation of molecules
in bulk. In this chapter we continue with the association of molecules
with surfaces, which can be a model for protein adsorption at a
membrane or small molecule adsorption at a large protein with
many interaction sites. The most used and well known model
is called Langmuir adsorption (see Figure 8.12). It considers a
surface with M binding sites. Each site can either be empty or
occupied by one adsorbed molecule. The adsorbed molecule interacts
with the site with a potential energy of size . Moreover,
the model assumes that the adsorbed molecules do not interact
with each other.
Let us suppose that N molecules are adsorbed to M sites.
The total energy interaction energy is E(N) = −N and the
number of possible realizations of such a state is:
Ω(N) =
M
N
=
M!
N!(M − N)!
(8.2)
As a result, the canonical partition function is:
Q(N, M, T) =
i
e−
Ei
kT =
M!
N!(M − N)!
e
N
RT , (8.3)
77
Figure 8.12: Langmuir model of adsorption.
where i denotes a microstate (one realization) with energy Ei.
The chemical potential is thus:
µ =
∂F
∂n
= −
∂kT ln(Q)
∂n
= −
∂RT ln(Q)
∂N
(8.4)
= −RT
∂
∂N
ln
M!
N!(M − N)!
+
RT
(8.5)
For large M and N we can use the Stirling approximation ln(n!) ≈
n ln(n) − n and its derivation
d ln(n!)
dn
≈ ln(n) + 1 − 1 = ln(n) to
get:
µ = −RT [0 − ln(N) − ln(M − N)(−1)] − (8.6)
= − + RT ln
N
M − N
(8.7)
To compare with experiments we have to calculate a fractional
coverage defined as xc ≡ N
M
. The equation 8.7 can be rewritten
78
as:
µ = − + RT ln
xc
1 − xc
(8.8)
xc
1 − xc
= exp
µ +
RT
(8.9)
xc =
1
1 + exp −µ+
RT
(8.10)
More commonly expressed in the form:
xc =
exp µ+
RT
1 + exp µ+
RT
(8.11)
A typical profile that we obtain with this relationship depends is
shown in Figure 8.13.
When the concentration of adsorbed molecules is high, the
molecules commonly start interacting in some way (could be both
direct or indirect), which leads to a deviation from the Langmuir
model. Note that in the derivation above one can use kT instead
of RT, since kT is pure unit of energy and RT is energy per mol.
Now we derive the same equation in the Grand Canonical ensemble,
where the number of particles can change. The system
has a constant chemical potential, volume, and temperature
(µV T). The Grand canonical partition function is:
Ξ ≡
N
Q(N, V, T)e
µN
RT (8.12)
Thus using Eq. 8.3 we obtain for our system:
Ξ =
M
N=0
M!
N!(M − N)!
e
N( +µ)
RT (8.13)
79
Figure 8.13: Typical profile of Langmuir adsorption.
We use binomial theorem (1 + x)M
= M
N=0
M!
N!(M−N)!
xN
to simplify
the above expression to:
Ξ = 1 + e
+µ
RT
M
(8.14)
The average number of molecules in the system (adsorbed molecules)
can be calculated from the expression:
< N >≡
N
N ∗ p(N) (8.15)
where p(N) is the probability of the state with N adsorbed molecules.
80
Therefore, we obtain:
< N > = N NQ(N, V, T)e
µN
RT
N Q(N, V, T)e
µN
RT
=
∂ ln Ξ
∂ µ
RT
(8.16)
=
M exp +µ
RT
1 + exp +µ
RT
(8.17)
and for fraction coverage xc we obtain the same as above:
xc =
exp µ+
RT
1 + exp µ+
RT
(8.18)
The derivation in the Grand canonical ensemble is easier and
more elegant. It demonstrates that the Grand Canonical ensemble
can be used to great benefit, especially in open and more
complex systems.
Note that the Langmuir adsorption model can also be derived
phenomenologically from the reaction description. We start from
considering the reaction NF + MF N, where NF represents
a free molecule, MF represents a free binding site, and N is the
adsorbed molecule in the binding site. The equilibrium constant
can be expressed as:
K =
aN
aNF
aMF
(8.19)
and the fraction coverage as:
xc =
N
M
=
N
N + MF
=
1
1 + MF
N
(8.20)
The activity coefficient is 1 in Langmuir model, so we can reformulate
the fraction coverage as:
xc =
1
1 +
aMF
aN
=
1
1 + 1
KaNF
(8.21)
81
Since we also know that the activity is related to the chemical
potential (aNF
= exp(− µ
RT
)) and the energy of this reaction
is related to the equilibrium constant (K = exp(−RT
)), we can
express the fraction coverage as:
xc =
1
1 + exp −µ+
RT
(8.22)
This is commonly expressed in the form:
xc =
exp µ+
RT
1 + exp µ+
RT
(8.23)
If we want to study adsorption in computer simulations, we
can either perform a simulation in the canonical ensemble with
a system large enough to represent infinite bulk, or we can employ
a Grand Canonical ensemble simulation. Simulation with the
Grand Canonical ensemble only needs to simulate the surface,
therefore the system is much smaller and faster to calculate. The
bulk reservoir with constant temperature and pressure (chemical
potential) is replaced by a fluctuating number of particles in the
system. To perform such simulations, we have to implement insertion
and deletion moves. During an insertion move we take the
molecule from a hypothetical tempered reservoir and place it at a
random position in the system. The insertion is accepted with the
probability:
p(N → N+1) = min 1,
V
Λ3(N + 1)
exp
µ − E(N + 1) + E(N)
kT
(8.24)
In a deletion move we randomly select a molecule and remove it
with the probability:
p(N → N−1) = min 1,
Λ3
N
V
exp
−µ − E(N − 1) + E(N)]
kT
,
(8.25)
82
where E is energy of the system with a given number of molecules,
µ stands for chemical potential, and Λ is the de Broglie wavelength
Λ ≡ h√
2πmkT
. The exponential pre-factor corresponds to
the changed number of possible realizations of the system, and
it originates from the non-interacting part of the partition function
V N
Λ3N N!
.
In more detail, we can follow the detailed balance condition
of the Monte Carlo method (for the basics of the Monte Carlo
method, including the detailed balance condition, see appendix).
The Grand canonical partition function is:
Ξ =
∞
N=0
V N
Λ3N N!
e
µN
RT e−
E(rN )
kT drN
(8.26)
and the probability of a configuration with N particles is
P(N) ∼
V N
Λ3N N!
e
µN
RT e−
E(rN )
kT (8.27)
thus if we are randomly selecting particles we can calculate the
transition probability ratio to be:
p(N → N + 1)
p(N + 1 → N)
=
P(N + 1)
P(N)
(8.28)
=
V
Λ3(N + 1)
e
[µ−E(N+1)+E(N)]
kT (8.29)
This probability ratio is satisfied by the above probabilities of single
moves.
In most cases we know the pressure of the system rather than
its chemical potential. In order to obtain the chemical potential for
simulation we could perform simulations without the surface and
calculate the pressure for a given chemical potential until we find
the one we want. Or we could do a simulation in the NPT ensemble
with the desired pressure and calculate the corresponding
83
chemical potential. Alternatively, we could perform the simulation
at standard pressure and use the following equation.
µ = µo
+ RT ln a = µo
+ RT ln φ
p
po
, (8.30)
where φ is the fugacity coefficient.
For optimal performance of simulations in the Grand Canonical
ensemble the probability of insertions and deletions should
not be very low.
Exercise: Show that with Langmuir adsorption you can obtain
the typical sigmoidal titration curve with a midpoint at pKa. Assume
that you titrate with strong acid/base, so you can focus on
dissociation curve.
Sources and further readings:
1. Daan Frenkel and Berend Smit; Understanding Molecular
Simulation: From Algorithms to Applications (2nd Edition);
Academic Press, 2001
9. Transport phenomena
(Diffusion, Navier Stokes, Stokes-Einstein equation, DPD )
In previous chapters we discussed the thermodynamic properties
and structures of systems in equilibrium. In this chapter we focus
on the kinetics of proteins. As mentioned in the first chapter, diffusion
dominates the processes in soft matter. Proteins and big
molecules in solution undergo a large number of collisions with
the surrounding molecules of a solution, leading to their Brownian
motion. This motion is named after Robert Brown, who studied
the motion of a pollen grain in aqueous solution. However,
the molecular explanation and derivation was done by Einstein
(1905) and Smoluchowski (1906).
The derivation starts from macroscopic measurements. If we
put a lot of grains in one part of the solution, we will observe
their diffusion to regions with low concentration. This motion is
described by Fick’s 1st law:
j = −D
∂c
∂x
(9.2)
which states that the flux j is linearly proportional to the concentration
gradient with the proportionality constant D, called the diffusion
coefficient. From the conservation of mass law, the flux is
85
the time derivative of concentration c:
∂j
∂x
= −
∂c
∂t
(9.3)
Putting these together, we obtain Fick’s 2nd law:
∂c
∂t
= D
∂2
c
∂x2
(9.4)
If we assume constant amount of material with N =
∞
−∞
cdx and
that we started to follow the particles from time zero (c(0, 0) = N),
the solution of the differential equation is:
c(x, t) =
N
√
4πDt
exp −
x2
4Dt
(9.5)
Naturally, the mean particle position is < x >= 0 as the particles
Figure 9.14: Concentration distribution change via diffusion.
diffuse in all directions with the same diffusion coefficient. The
second moment, mean square displacement, is:
< x2
>= 2Dt (9.6)
86
The macroscopic change in concentration profile could thus be related
to the microscopic motion of particles. We can do the same
derivation in n dimensions to obtain the more general expression:
< |r|2
>= 2nDt (9.7)
Before we look at how to estimate the diffusion coefficient
for molecules, let’s have a look at the molecular origin of Fick’s
law. In other words, how do particles know which direction to
go? And how do we get time-irreversible behavior from Newton’s
time-reversible laws (a drop of ink in water becomes dispersed
over time but not the other way round)?
Consider two boxes with different amounts of ink molecules,
where molecules can jump from one box to another with probability
p and stay with probability s. The average number of particles
that will jump from a box with N molecules is
< nj >=
N
i=0
iP(i, N), (9.8)
where P(i, N) is the probability of i molecules jumping from N.
As we know that each molecule will either jump or stay, we can
calculate the probabilities;
P(i, N) =
N!
i!(N − i)!
pi
sN−i
(9.9)
This is a binomial distribution where a fraction factor originates
from the number of combinations we can choose i jumping molecules.
The sum can then be rewritten with a derivative as:
< nj >=
N
i=0
p ∗
∂P(i, N)
∂p
= p
∂ N
i=0 P(i, N)
∂p
(9.10)
87
Now we will use the fact that binomial distribution sum to (p+s)N
.
Hence
< nj >= p
∂(p + s)N
∂p
= pN(p + s)N−1
= Np (9.11)
where we used the assumption that each molecule either jumps
or stay, i.e. p + s = 1. This means that the flux of molecules
from the first box to the second one per area A and time dt can
be calculated as:
< j >=
< n1 > − < n2 >
A∆t
=
p(N1 − N2)
Adt
=
p∆N
Adt
(9.12)
To make our result more general we will move to boxes of infinitesimally
small volume, where ∆N = −∆cdV = −∆cAdx.
The minus sign comes from the opposite direction of the definitions
of flux and concentration ∆j = j1 − j2 while ∆c = c2 − c1.
< j >= −
pdx∆c
dt
= −
pdx2
dt
∆c
dx
= −D
∂c
∂x
(9.13)
The flow of particles along the concentration gradient (Fick’s law)
thus originates from the simple fact that more molecules will randomly
move in that direction than in the opposite. Similarly to
the mixing of two gases, molecules do not ’know’ where to move.
They move independently towards states with more possible rep-
resentations.
To obtain an estimation of the diffusion coefficient, we start
88
with its connection to the velocity autocorrelation function:
< x2
> =
t
0
vdt
2
(9.14)
=
t
0
t
0
< v(t )v(t ) > dt dt (9.15)
=
t
0
t
0
< v(t )v(t ) > dt dt
+
t
0
t
t
< v(t )v(t ) > dt dt (9.16)
Using the substitutions z = t − t and y = t − t in the second
integral leads to:
< x2
> =
t
0
t
0
< v(t )v(t ) > dt dt
+
t
0
y
0
< v(y)v(z) > dzdy (9.17)
= 2
t
0
t
0
< v(t )v(t ) > dt dt (9.18)
From this we can evaluate the diffusion coefficient:
D = lim
t→∞
∂ < x2
>
2∂t
(9.19)
D =
∞
0
< v(t − t )v(0) > d(t − t ) (9.20)
This equation 9.20 is called the Green-Kubo relation.
The motion of individual particles with weight m in diffusion is
described by the Langevine equation
m
d2
x
dt2
= fc − γ
dx
dt
+ fs (9.21)
89
where fc is a conservative force originating in the inter-particle
potential. γ stands for the friction coefficient, related to the fluid
viscosity, and fs is a stochastic random force representing collisions
with the surrounding fluid molecules (typically represented
by white noise). This equation is an essential part of Langevin dynamics.
Molecular friction is related to macroscopic dissipation,
which is related to stochastic force by Fluctuation-dissipation theorem.
When satisfied friction and stochastic force at as thermostat
with correct thermal fluctuations.
When the Langevine equation is combined with the GreenKubo
relation, we can derive an equation for the diffusion coefficient
from the motion of a single particle. The first step is to
multiply the Langevine equation by vx(0) and average it over time
or ensemble. For a non-interacting particle we get:
0 = mvx(0)
dvx
dt
+ γvx(0)vx(t) − vx(0)fs(t) (9.22)
0 = m < vx(0)
dvx
dt
> +γ < vx(0)vx(t) >
− < vx(0)fs(t) > (9.23)
0 = m < vx(0)
dvx
dt
> +γ < vx(0)vx(t) > (9.24)
The solution of the above equation is:
< vx(0)vx(t) >=< v2
x(0) > e− γ
m
t
(9.25)
The average size of velocity < v2
x(0) > is equal to kT/m from the
Maxwell velocity distribution or Equipartition theorem. Hence,
< vx(0)vx(t) >=
kT
m
e− γ
m
t
(9.26)
The second step is to use this solution in the Green-Kubo expres-
90
sion:
D =
∞
0
kT
m
e− γ
m
t
dt (9.27)
=
kT
m
m
γ
(9.28)
=
kT
γ
(9.29)
To finish the calculation we need to know the friction coefficient,
which is not easy to obtain, as it is determined by hydrodynamics.
Below we show the main laws of hydrodynamics and the calculation
of the friction coefficient for a spherical particle.
Hydrodynamic flow of the solution is described by the NavierStokes
equation, which is a nonlinear partial differential equation.
For incompressible liquids it is:
ρ
∂v
∂t
+ v · v = − p + η 2
v + f (9.30)
where ρ is the density, v stands for the velocity vector. p represents
the hydrostatic pressure, η is the viscosity, and f is the
external force that act on the studied elementary volume dV . The
equation is derived from Newton’s second law d(mv)/dt = F applied
to a small volume of fluid. On the left side the total derivative
(dmv/dt = ∂mv/∂t + v · mv) is simplified using a continuity
equation and the assumption of fluid incompressibility. The force
on the right side of Newton’s law is split into the stress acting on a
cubic volume and the force f acting homogeneously on the given
volume. The stress tensor is then further decomposed to diagonal
terms (pressure) and off-diagonal terms (shear), which give rise
to the second term on the right side of the Navier-Stokes equation.
Note that the Navier-Stokes equation does not take into the
account the molecular structure of the fluid, which is described as
a homogeneous continuum.
91
There is no general solution to the Navier-Stokes equation,
with its nonlinear character being the main obstacle. However,
in soft matter the typical motion is so slow that it is in the low
Reynolds number regime (no turbulence). Thus, we can neglect
the term v · v and the equation 9.30 becomes linear:
ρ
∂v
∂t
= − p + η 2
v + f (9.31)
and much easier to solve. Of course the exact solution still depends
on the boundary conditions, but for several simple cases
this could be solved analytically.
For example for a hard sphere moving at steady state in solution
the Navier-Stokes equation simplifies to 0 = − p + η 2
v.
This could be solved in cylindrical coordinates leading to the transfer
velocity from the particle to the solution vt ∼ vs(R/r), where
R is the sphere’s radius, vs is the sphere’s velocity, and r is the
distance from the sphere. From the transverse velocity dependence,
we can calculate a friction force to derive the well-known
Stokes’ law:
Ff = −6πηRvs (9.32)
This result might be counter-intuitive, since at turbulent flow the
friction force depends on R2
v2
s . However, in laminar flow it depends
on Rvs, which demonstrates the importance of correctly
describing the hydrodynamics. Importantly, the velocity of the solution
decays from the sphere with 1/r, which means that the
hydrodynamic interaction between two particles can be a long
ranged one.
From Stoke’s law, we have the friction coefficient γ = 6πηR
now and we can insert it into the Eq. 9.29 to end up with the
Einstein-Stokes relation:
D =
kT
γ
=
kT
6πηR
(9.33)
92
which describes the diffusion coefficient of spherical particles in
a liquid at laminar flow. This can be used to estimate the diffusion
coefficient for roughly spherical proteins. However, when the
molecule has a more complicated shape or non-homogeneous
interactions, the Navier-Stokes equation cannot be solved analytically
and we will need to do a computer simulation to calculate
the diffusion coefficient. For completeness, water viscosity at ambient
conditions is about 10−3
Pa·s. Note that the diffusion coefficient
is a function of temperature, but not of the concentration of
solutes.
Based on the Einstein-Stokes relation, we can get an idea of
the typical time scales in soft matter. Proteins approximated by a
sphere will diffuse the distance of their radius in the following time
(combining Eq. 9.6 and Eq. 9.29):
R2
= 2nDt = 6
kT
6πηR
t (9.34)
t =
πηR3
kT
(9.35)
As a result we get times on the order of µs for proteins of radius
10 nm and milliseconds for proteins with a radius of 100 nm.
Therefore the relevant timescales are not easily accessible by
all-atom simulations. Fortunately, there are coarse-grained techniques
which can reach such long time scales and some of them
even include hydrodynamics and through which we can calculate
the diffusion coefficient.
Probably the most widely used coarse-grained technique that
includes solvent hydrodynamics is Dissipative Particle Dynamics
(DPD). However, note that there are other methods such as
the lattice Boltzmann method, multi-particle collision dynamics,
fluid particle dynamics, or fluctuating hydrodynamics. In the DPD
method, several solvent molecules are coarse-grained into one
particle, which has similar effective friction and fluctuation as an
93
all-atom solvent. As in standard Molecular Dynamics (MD), the
time is discretized and the movement in each step is done based
on the forces acting on the particles. The difference is that in DPD,
the total force contains not only the conservative forces Fc(rij)
(e.g. forces originating from inter particle potentials), but also dissipative
Fd(rij, vij) and stochastic forces Fs(rij):
mi
d2
ri
dt2
= Fi =
i=j
Fc(rij) + Fd(rij, vij) + Fs(rij) (9.36)
Dissipative force depends on both interparticle distance rij and
velocity vij :
Fd(rij, vij) = −γωd(|rij|) vij ·
rij
|rij|
rij
|rij|
(9.37)
where γ is the friction coefficient and ωd represents the variation
of the friction with distance. The ωd distance dependence is usually
limited by a cut-off distance, after which the friction is zero.
Below the cut off rc it could be constant or decaying function such
as (1 − rij/rc). The stochastic force has the form:
Fs(rij) = −σωs(|rij|)g
rij
|rij|
(9.38)
where σ is the magnitude of a random pair force and ωs is again
the distance variation. g is a random number from a Gaussian
distribution with unit variance. This formula guarantees that the
stochastic force between two particles is antisymmetric Fs(rij) =
−Fs(rji), which is needed for the conservation of momentum.
Moreover, antisymmetry saves computer time, since only one has
to calculate one value for each pair. This also distinguishes DPD
from Brownian dynamics, in which each particle experiences a
random force independently. Thus, Brownian dynamics does not
94
conserve the local momentum, which is a crucial property for correct
hydrodynamics.
Dissipative and stochastic forces act together as a thermostat,
keeping the system at the desired temperature in the canonical
ensemble (NV T). However, to recover the canonical ensemble
there is a necessary condition:
ωd = ω2
s (9.39)
Exercise: Calculate the maximum displacement for a Monte
Carlo displacement move. Assume an acceptance probability to
be 0.2. The target diffusion coefficient is 10−5
cm2
s−1
. 1D case
is trivial, so consider 3D moves, where particles is displaced with
the same probability in all directions (and distances within maximum
displacement).
Sources and further readings:
1. Daan Frenkel and Berend Smit; Understanding Molecular
Simulation: From Algorithms to Applications (2nd Edition);
Academic Press, 2001
10. Advanced Monte Carlo
(NPT, Parallel tempering, Cluster moves, Configuration bias,
Widom insertion, Wang-Landau method, Kinetic Monte Carlo,
Dynamic Monte Carlo)
In this chapter we describe some of the methods that go beyond
the simple Monte Carlo (MC) method with Metropolis sampling
of the configuration space in the canonical ensemble that is described
in the appendix. We will only cover a few advanced methods
that are most commonly used. There is a large number of different
advanced methods and new ones that are still being developed.
For example in 2011 a Hybridization move was developed
that leads to the formation and breakage of new bond constraints,
as well as a Non-equilibrium candidate method which enhances
sampling by moving particles out of equilibrium and subsequent
relaxation of the system.
A large proportion of the advanced methods take advantage
of non-physical moves in order to enhance sampling of the equilibrium
properties. This naturally leads to loss of the dynamics,
for which Molecular dynamics (MD) is frequently used. However,
there are also MC methods which can provide the dynamics of the
system such as Kinetic and Dynamic Monte Carlo. On the other
hand, if we are only interested in the equilibrium properties, MD
can be used and understood as one of the possible and usually
96
not the most efficient method to sample the configuration space.
Note that the implementation of the complicated and highly
specialized moves is usually system-dependent. This is why there
is not a universal MC package with all the methods, yet there have
been several attempts to develop packages that are reasonably
general.
We start with a simple simulation of an NPT ensemble (isobaric
isothermal). Similarly to Molecular dynamics, the desired
pressure of the system and the fluctuations are provided by changes
to the simulation box size. Here we will discuss an isotropic
change, however, an anisotropic change of the box is very similar.
In contrast to Molecular dynamics, there are no forces calculated
in Monte Carlo, instead the change to the box is chosen randomly
and the pressure is maintained via the acceptance rule.
In a canonical ensemble we used the acceptance criterion
exp(−∆E
kT
), because the canonical partition function is:
Q(N, V, T) =
V N
Λ3N N!
e− E
kT drN
(10.2)
and the MC method samples this integral. In the same way we will
use an MC scheme to sample the integral of the partition function
of the isobaric ensemble, which is:
Z(N, p, T) =
p
Λ3N N!kT
V N
e−pV
kT e− E
kT drN
dV (10.3)
In other words the density of states (the probability that a system
of N particles will have volume V at pressure p) is:
P(N, V ) = V N
e−pV
kT e− E
kT (10.4)
= exp(−
E + pV − NkT ln(V )
kT
) (10.5)
97
and the probability criterion for accepting or rejecting the volume
change is:
prob(old → new) = min 1, exp −
Enew − Eold
kT
+
p(Vnew − Vold) − NkT ln(Vnew/Vold)
kT
(10.6)
We use prob for probability here to avoid the confusion with pressure
p. Usually it is more efficient to make random changes in
the logarithm of volume (ln(V )) than in the volume itself. The
acceptance criterion then has to be modified to:
prob(old → new) = min 1, exp −
Enew − Eold
kT
+
p(Vnew − Vold) − (N + 1)kT ln(Vnew/Vold)
kT
(10.7)
since V N
dV = V N+1
d(ln(V )).
Parallel tempering (also called replica exchange) is a method
used for systems with many local minima. In order to avoid the
system being ’stuck’ in one minimum, a simultaneous run of many
simulations is used, each at a different temperature to overcome
barriers. The configurations between the different runs are swapped
based on the MC swap move and the probability of accepting
such a move is dependent on the temperature difference (the
smaller temperature difference, the more likely the swap is accepted).
The acceptance criterion for a swap move between systems
i,j in NV T ensembles could be derived from the detailed
98
balance condition:
p(i(Ti)j(Tj) → i(Tj)j(Ti))
p(i(Tj)j(Ti) → i(Ti)j(Tj))
=
Pi(Tj)Pj(Ti)
Pi(Ti)Pj(Tj)
(10.8)
=
e
−
Ei
kTj e
−
Ej
kTi
e
−
Ei
kTi e
−
Ej
kTj
(10.9)
= e
1
kTi
− 1
kTj
(Ej−Ei)
(10.10)
Hence the acceptance criterion is:
p((i(Ti)j(Tj) → i(Tj)j(Ti)) = min 1, e
Tj−Ti
kTiT j
(Uj−Ui)
(10.11)
Temperature is not the only property that can be varied and similar
methods have been developed for simulations of systems with
different densities, pH, salt concentrations, etc. The method is
general and the exchanges can be done between simulations with
different Hamiltonian.
In cluster moves,the position or configuration of several molecules
are changed at once. We typically attempt to move particles,
where separate movements would result in a low acceptance.
Therefore the most common criteria for cluster definition is distance
proximity or strong interaction. The acceptance rules have
to be carefully devised to meet the detail balance condition, as
cluster moves could include the joining or separation of existing
clusters. The simplest way to deal with this issue is to automatically
reject any move where the number of clusters changes. The
acceptance criterion is then simply:
p(old → new) = min 1,
new
ij∈C pij
new
i∈Cj /∈C(1 − pij)
old
ij∈C pij
old
i∈Cj /∈C(1 − pij)
exp −
Enew − Eold
kT
(10.12)
99
where C stands for the cluster and pij is the probability of two
particles i and j to belonging to the same cluster. By multiplying
the probabilities for all particles we get the total probability of a
given cluster and the rest of the system. If the probability of the
cluster before and after is the same, the criterion simplifies to:
p(old → new) = min 1,
new
i∈Cj /∈C(1 − pij)
old
i∈Cj /∈C(1 − pij)
exp −
Enew − Eold
kT
(10.13)
The probabilities pij could be defined for instance using a simple
distance cutoff rc resulting in: pij = 1 for rij < rc and pij = 0 for
rij ≥ rc.
The Configuration Bias MC (CBMC) method is used to bias
our selection of configurations to those that are probable. We
will demonstrate this method on insertions or deletions of large
chain molecules such as proteins and polymers. The basic idea
is based on the growth or removal of part of the molecule (for example
amino acids) depending on its Boltzmann weight. It originates
from the Rosenbluth method, which is used for polymers in
a lattice and the acceptance of the move is given by the ratio of
Rosenbluth weights. In growth, a randomly chosen position of the
next chain segment would very likely end up with a rejection due
to the overlap with another segment. Thus, we randomly generate
several trial positions and choose one according to its Boltzmann
weight. This means that we bias the growth to the positions that
are most likely. The probability of the i-th trial position is given by:
Pi =
exp(− Ei
kT
)
l
j=1 exp(−
Ej
kT
)
, (10.14)
where l is the number of trial positions and E is the potential
energy of the system with the given trial position. However, as
100
we introduce bias we have to also modify the acceptance probability,
so the detail balance remains conserved (otherwise we
would generate conformations with a Rosenbluth distribution and
not Boltzmann distribution). To correct the acceptance probability
we have to calculate the Rosenbluth weight w(i) for the i-th
segment:
w(i) =
l
j=1
exp −
Ej(i)
kT
(10.15)
The Rosenbluth weight for the whole new chain of n segments is:
Wnew =
n
n=1
w(i) =
n
i=1
l
j=1
exp −
Ej(i)
kT
(10.16)
Now we take into account the conformations of the chains in the
current system, which are in the Boltzmann conformation. We
randomly select one of the existing chains in the system and regrow
it with l trial positions of each segment as with a new chain,
however, the first position i = 1 is not random, but from the existing
chain e. The Rosenbluth weight for the i-th segment is:
wold(i) = exp −
Ue(i)
kT
+
l
j=2
exp −
Ej(i)
kT
(10.17)
The Rosenbluth weight for the whole "old" existing chain of n segments
is:
Wold =
n
i=1
exp −
Ee(i)
kT
+
l
j=2
exp −
Ej(i)
kT
(10.18)
The acceptance criterion of the move is:
p(old → new)
= Wnew
Wold
if Wnew < Wold
= 1 if Wnew ≥ Wold
(10.19)
101
There are modifications to the Configuration Bias Monte Carlo,
which were developed to enhance sampling. One example is the
Prune-Enrichment method, in which the partial configurations are
discarded ’pruned’ if their probability is low, whereas very likely
early configurations are copied ’enriched’ to improve sampling.
Another example is the Recoil-growth method, where chains leading
to a dead end are shortened and regrown in a new conformation.
For chain molecules, Reptation moves are also frequently
used. In this move one end of the chain is marked as the head,
then as the head moves the body of the chain follows the movement
like a snake.
In general, if we have a bias for generating trial configurations
and the bias depends on the potential energy of such a configuration
(α(old → new) = f(Enew)), then the acceptance criterion
in the canonical ensemble is:
p(old → new)
= f(Eold)
f(Enew)
e−
Enew−Eold
kT if e−
Enew−Eold
kT < f(Enew)
f(Eold)
= 1 if e−
Enew−Eold
kT ≥ f(Enew)
f(Eold)
(10.20)
Now we will look at a few methods used for Free energy cal-
culations.
Widom insertion or simply particle insertion is a method to calculate
the free energy of a particle inserted into the system. For
a large system, the free energy per particle µ can be calculated
as:
µ = H(N + 1, V, T) − H(N, V, T) (10.21)
From statistical mechanics we get:
µ = −kT ln Q(N + 1, V, T) + kT ln Q(N, V, T)(10.22)
= −kT ln
Q(N + 1, V, T)
Q(N, V, T)
(10.23)
102
where Q is the canonical partition function
Q(N, V, T) =
V N
Λ3N N!
e− E
kT drN
(10.24)
After substitution we obtain:
µ = −kT ln
Λ3
V (N + 1)
− kT ln
e−
EN+1
kT drN+1
e−
EN
kT drN
= µid + µex (10.25)
The first term is an ideal gas contribution, while the second term
is the excess free energy. The integral can be calculated as an
average over the particles in the system:
µex = −kT ln e−
EN+1−EN
kT
N
drN+1
(10.26)
The ensemble average over N-particles < .. >N can be sampled
by the Monte Carlo method by simulating the system in the
canonical ensemble with N particles, where we randomly insert
additional particle and calculate the energy difference EN+1−EN .
Note that the insertions are only virtual and system continues to
evolve with N particles.
In an isobaric ensemble we can follow a similar derivation to
obtain:
µ = −kT ln
kT
pΛ3
− kT ln
pV
(N + 1)kT
e−
EN+1−EN
kT drN+1
N
(10.27)
In very dense systems, Widom insertion is not efficient, as
the configurations with low energy (inserting in a hole) are purely
sampled. It is then advantageous to combine insertions with deletions.
Two systems are simulated: the first one has N particles
103
and an additional particle is virtually inserted; the second simulation
consists of N + 1 particles and virtual removals of random
particles are carried out. From each system, we obtain the probability
distribution p of ∆E = EN+1 − EN . If there is an overlap
of these two distributions, we can calculate the free energy of the
insertion as in two systems with N particles:
pN+1(∆E) =
e−
EN+1
kT δ(EN+1 − EN − ∆E)drN+1
e−
EN+1
kT drN+1
=
e−
EN +∆E
kT δ(EN+1 − EN − ∆E)drN+1
e−
EN+1
kT drN+1
=
e−
EN +∆E
kT δ(EN+1 − EN − ∆E)drN+1
e−
EN+1
kT drN+1
e−
EN
kT drN
e−
EN
kT drN
= e−∆E
kT
e−
EN
kT δ(EN+1 − EN − ∆E)drN+1
e−
EN
kT drN
e−
EN
kT drN
e−
EN+1
kT drN+1
= e−∆E
kT pN (∆E)e−µex
kT (10.28)
This leads to:
µex = ∆E − kT ln
pN (∆E)
pN+1(∆E)
= ∆E + kT ln (pN+1(∆E)) − kT ln (pN (∆E))
(10.29)
This method is called the Overlapping distribution method and is
more general than the above case. Its difference from MD could
be related to its thermodynamics integration.
The Wang-Landau (WL) method calculates the density of states
in any system that can be characterized by a cost function. With
particle simulations, the cost function is an energy function along
104
a reaction coordinate. The reaction coordinate can be a general
combination of coordinates such as particle distance, distance
from the center of mass, axes, volume, etc. and this coordinate
is sampled (usually from min to max value in certain bin size) and
the penalty potential along this coordinate is adaptively modified.
In the end we obtain a flat sampling along the reaction coordinate
and the cost function is related to the free energy. As such the
WL method is similar to Metadynamics in MD.
Let g be a penalty function along the reaction coordinate and
h the histogram. At the beginning both functions have zero values
in all their bins. In each move during MC, the histogram of the
current bin is increased by 1 and the penalty function decreased
by constant modification factor f. (Note that in the original WL
method ln(g) and ln f were used instead of g and f as it was
made to directly correspond to the density of states, not their logarithm).
The acceptance criterion for the canonical ensemble is:
p(old → new) = min 1, exp −
Enew − Eold + gnew − gold
kT
(10.30)
The sampling continues until the histogram reaches an accurate
estimate of the density of states with the given modification factor
f. A common criterion is that the difference between the maximum
and minimum value of the histogram is smaller than a certain
threshold. The original algorithm proposed that each histogram
value should be within 20 % of the mean value. When
such a criterion is satisfied, the modification factor f is decreased
and the histogram is reseted to 0. Originally, f was decreased
by simply multiplying by 1/2. Later, it was suggested to use a
multiplication proportional to 1/t in order to improve convergence
(t is the simulation time). The procedure continues until the f is
decreased below the previously selected value (typically 10−7
).
The g is then related to the free energy of the system along the
reaction coordinate.
105
To capture dynamics, the Kinetic Monte Carlo or Dynamic
Monte Carlo are frequently used. Naturally, no unphysical moves
are allowed in these techniques and we need to know some of the
dynamic parameters of the system a priori. In Dynamic Monte
Carlo, we typically know the diffusion coefficients for all of the
moves employed, as all the moves are coupled to the same time
constant. The method was shown to lead to the same results as
Brownian dynamics when the moves and their probabilities are
calculated from the infinite dilution limit.
In Kinetic Monte Carlo, the rates of all the moves responsible
for the studied process are known and the trial moves are
attempted according to these rates. Naturally, it is crucial to have
the uncorrelated rates ri correct. At the beginning of each step,
we calculate the cumulative rate function Ri = i
j=1 rj for i =
1, 2, ..N, where N is the number of possible processes at a given
time. Consequently, we generate a random number from 0 to RN
and the m process (move) is carried out based on which Rm is selected.
At the end of each step, the simulation time is updated by
∆t = −1/RN ln x, where x is a new random number x ∈ (0, 1 >.
This algorithm is also called the residence-time algorithm or the
Bortz-Kalos-Lebowitz algorithm. A more efficient modification of
this algorithm has been suggested by grouping the same kind of
transitions.
The final method we will mention is Green’s Function Reaction
Dynamics, which is an efficient method for simulating diffusiondependent
systems. The many-body problem of the system is
split into one- and two-body problems, which can be solved analytically
using Green’s Functions. For every particle and pair of
particles we calculate the time they would interact. The events are
then chronologically simulated, and after each interaction simulation
the event list is updated.
106
Sources and further readings:
1. Daan Frenkel and Berend Smit; Understanding Molecular
Simulation: From Algorithms to Applications (2nd Edition);
Academic Press, 2001
Test questions
1. What assumptions are used for segments of ideal chain model?
2. What is the end-to-end distance in the ideal chain model (random walk)?
3. What is the sum of individual segments in the ideal chain model
N
i=1 ri?
4. How does the equilibrium size of the ideal chain depend on tempera-
ture?
5. How does the vesicle total bending energy depend on its radius?
6. What is the Mean curvature?
7. What is the Gaussian curvature?
8. What is included in the van der Waals interactions?
9. What is the distance dependence of dispersion interaction between two
atoms?
10. What name has material constant for dispersion interaction?
11. What is the distance dependence of dispersion interaction for large
spherical particles?
12. What origin is the depletion interaction?
13. Whose approximation can be used to calculate a short-range interaction
of objects with various shapes?
14. What approximations are used in the Poisson-Boltzmann theory?
15. What is the Debye screening length at physiological concentrations (nm)?
16. What name has the double layer theory of screened charged surfaces?
17. What interactions are included in the DLVO theory?
18. What is a rough range of strength of hydrogen bonds (kT)?
19. What is a rough range of strength of van der Waals interaction (kT)?
20. What is valid for hydrophobic interaction (rc is crossover radius)?
21. What is used in bottom-up approach?
22. Coarse grained models are:
23. What is true well below CMC (ci is concentration of i-mer aggregate)?
108
24. What is the packing parameter?
25. What is in the Langmuir model?
26. What is in the number of of ways to select N out of M indistinguishable
particles?
27. What formula represents the diffusion coefficient (n is number of degrees
of motion)?
28. What is the name of formula for calculation of diffusion coefficient using
velocity autocorrelation function?
29. How long (in order of magnitude) does it take for a protein of radius 1
nm to diffuse from a brain to an arm? Assume that the whole motion
is within one axon with viscosity of water 10−3
Pa·s and that the total
length is about 1 m.
Appendices
Basics of thermodynamics
Thermodynamics is a phenomenological science providing us with a series of
general relations that are valid for any substance. The main relations on which
thermodynamics is based are known as the laws of thermodynamics, which
can be formulated in various ways.
The First Law of thermodynamics states that the energy, of a closed system,
E is conserved and can be converted into two forms: heat, ’Q’, and work,
’W’.
dE = Q + W (A.2)
The Second Law says that heat can never spontaneously flow from a cold
reservoir to a warmer reservoir. In other words it is impossible to make an
engine which completely converts heat from a single heat bath into work. Yet
another formulation is that any spontaneous process cannot decrease entropy,
where the entropy change dS is for a reversible change given by the formula:
dS =
δQrev
T
(A.3)
Entropy is an extensive quantity, so the entropy of the whole system equals to
the sum of the entropy of the individual parts.
The Third law defines temperature and states that the entropy of a perfect
crystal at absolute zero is equal to zero. Sideal(T = 0K) = 0
The zeroth law of thermodynamics is sometimes automatically assumed,
but it should be added for completeness. It postulates that if two systems are
in thermal equilibrium with a third system, they must be in thermal equilibrium
with each other. TA = TC, TB = TC => TA = TB
110
Energy E can be related to enthalpy H = E + pV , Helmholtz free energy
F = E − TS, and free energy G = E − TS + pV . Each of these quantities
becomes a thermodynamic potential which systems minimize in different ensembles,
i.e. under different constant conditions.
NV E : dE = TdS − pdV +
i
µiNi
NpE : dH = TdS + V dp +
i
µiNi
NV T : dF = −SdT − pdV +
i
µiNi
NpT : dG = −SdT + V dp +
i
µiNi
Maxwell relations (connect useful but non-measurable quantities to measurable
ones):
∂T
∂V S,N
= −
∂p
∂S V,N
∂V
∂T p,N
= −
∂S
∂p T,N
∂T
∂p S,N
=
∂V
∂S p,N
∂p
∂T V,N
=
∂S
∂V T,N
∂T
∂N V,S
=
∂µ
∂S V,N
Gibbs-Duhem relation:
−SdT + V dp =
i
Nidµi
Basics of statistical mechanics
Statistical mechanics provides us with a connection between the microscopic
details of the system and its thermodynamic quantities. We use the term mi-
111
croscopic details to mean all microscopic states in which the system can exist.
This information is contained in a partition function, which evaluates the number
of states with the same probability under the given conditions. Naturally,
each ensemble has a different partition function. In a micro-canonical ensemble,
we assume that all states with the same energy are equally likely (the
ergodic hypothesis). The partition function Ω is then simply the number of
microstates at given energy E, volume V , and number of particles N.
In a canonical ensemble the probability of each state is given by the Boltzmann
factor exp( Ei
kT ) thus the partition function is Q = i exp( Ei
kT ). This can
be written in a more general way using the Hamiltonian of the system, H. In a
continuous space of interacting particles, we can write Q = exp( H
kT )dpdr.
Moreover, if the Hamiltonian is composed of kinetic energy and interaction
potential energy, the momenta can be integrated out. In a 3D system of N
particles, this leads to:
Q =
1
Λ3N N!
exp −
U
kT
dr, (A.4)
where U is the potential energy of the system in a given configuration r. The
integral prefactor originates from the number of permutations among indistinguishable
particles, N!, the number of degenerate states under the given conditions,
1, and the de Broglie wavelength Λ = h√
2πmkT
. The fact that the particles
are indistinguishable does not have to have its origin in quantum mechanics
(i.e. all electrons are the same), but in the number of (in)distinguishable
macro states with these particles.
Using the general statistical formula < x >= i xpi
i pi
one can then calculate
the various properties of the system using the partition function.
Summary of useful relations:
NVE NVT NpT µVT
microcanonical canonical isothermal-isobaric grandcanonical
Ω Q = e
E
kT Z = Qe
−pV
kT dV Ξ =
∞
N=0
Qe
µN
kT
S = k ln Ω F = −kT ln Q G = −kT ln Z pV = kT ln Ξ
1
T = ∂S
∂E E = ∂F/T
∂1/T H = ∂G/T
∂1/T N = ∂pV
∂µ
p = T ∂S
∂V p = − ∂F
∂V V = ∂G
∂p E = T ∂pV
∂T
µ = −T ∂S
∂N µ = ∂F
∂N µ = ∂G
∂N
112
The example is an ideal gas (non-interacting point particles), where the
canonical partition function simplifies to Q = V N
Λ3N N!
. Thus, F = −kT ln V N
Λ3N N!
.
The pressure of the ideal gas is p = − ∂F
∂V N,T
= NkT
V which leads to the
equation of ideal gas equation.
Derivation of Boltzmann distribution
In a canonical (NV T) ensemble, the Helmholtz free energy is at a minimum
dF = dU − TdS = 0, (A.5)
which we can use to calculate the probability of individual states i with energy
Ei. The internal energy is given by
U =< E >=
i
piEi, (A.6)
and the derivative is
dU =
i
(Eidpi + pidEi) (A.7)
The energy of any particular state is normally dependent on V or N but not
on S or T. Using this fundamental principle from quantum mechanics and the
fact that temperature is constant in a canonical ensemble, we obtain:
dU =
i
Eidpi (A.8)
The entropy is defined as
S = −k
i
pi ln pi, (A.9)
so the derivative is
dS = −k
i
(1 + ln pi)dpi (A.10)
We can now substitute Eq. A.8 and A.10 into Eq. A.5:
dF =
i
[Ei + kT(1 + ln pi)]dpi = 0 (A.11)
113
The solution to this equation is:
pi = e−
Ei
kT (A.12)
After normalization, we obtain the Boltzmann distribution:
pi =
e−
Ei
kT
i e−
Ei
kT
=
e−
Ei
kT
Q
, (A.13)
where Q is a canonical partition function.
Virial coefficients and iteractions
Equation of state of very dilute gas (solution) is given by ideal gas equation:
pV = nRT, (A.14)
which can be rewritten as:
p
kT
=
N
V
= ρ (A.15)
At higher concentrations the real gas deviates from ideal gas and the deviation
can be expressed in density expansion of equation of state:
p
kT
= ρ + B2(T)ρ2
+ B3(T)ρ3
+ ... (A.16)
This is known as virial expansion and Bn(T) are called virial coefficients. The
coefficients can be viewed as corrections to ideal gas and their importance
increases with increasing density. They are specifically related to n-body intermolecular
interactions: B2 is related to two body interactions, B3 is related to
three body interactions, etc.
From statistical mechanics it can be shown that:
B2 = V
1
2
−
Q2
Q2
1
(A.17)
B3 = V 2 2Q2
Q2
1
2Q2
Q2
1
− 1 −
1
3
6Q3
Q3
1
− 1 (A.18)
(A.19)
114
where Qn is a canonical partition function of system with n particles. Using
the pairwise additivity of interactions we can write:
Q1 =
V
dr1 =
V
Λ3
(A.20)
Q2 =
1
Λ62 V
exp −
U12
kT
dr1dr2 (A.21)
Q3 =
1
Λ93! V
exp −
U12 + U13 + U23
kT
dr1dr2dr3 (A.22)
(A.23)
The second virial coefficient for monoatomic classical gas can be related to
u(r), a potential of mean force (spherically averaged free energy) between
particles, as:
B2(T) = −
1
2V
∞
0
e
u(r)
kT − 1 4πr2
dr (A.24)
Ideal chain - radius of gyration Rg
We start from the definition of a radius of gyration
R2
g ≡
1
N
N
i=1
(|ri − RCM |)2
(A.25)
where RCM is the center of mass defined as:
RCM ≡
1
N
N
i=1
(ri) (A.26)
To derive the relation of the radius of gyration to the end-to-end distance, we
will use the following expression where we will use the following definition of
center of mass:
115
N
i,j=1
(|ri − rj|)2
=
N
i,j=1
[(ri − RCM ) − (rj − RCM )]2
=
N
i,j=1
(ri − RCM )2
−2
N
i,j=1
(ri − RCM ) · (rj − RCM )
+
N
i,j=1
(rj − RCM )2
= N
N
i=1
(ri − RCM )2
+ 0 + N
N
j=1
(rj − RCM )2
= 2N
N
i=1
(|ri − RCM |)2
(A.27)
Therefore, the radius of gyration can be rewritten as:
R2
g =
1
2N2
N
i,j=1
(|ri − rj|)2
(A.28)
The ri − rj vector is a end-to-end vector of the chain from the i-th to j-th
monomer. Employing the average end-to-end distance, we obtain:
< R2
g > =
1
2N2
N
i,j=1
l2
|i − j| (A.29)
=
l2
2N2
N
i=1
2
i
j=1
|i − j| (A.30)
116
Now we will do the summation using
a
k=1 k = a(a + 1)/2 and
a
k=1 k2
=
(2a3
+ 3a2
+ a)/6
< R2
g > =
l2
2N2
N
i=1
2 i ∗ i −
i(i + 1)
2
(A.31)
=
l2
2N2
N
i=1
(i2
− i) (A.32)
=
l2
2N2
2N3
+ 3N2
+ N
6
−
N2
+ N
2
(A.33)
=
1
6
l2
N −
l2
N
(A.34)
=
1
6
RN
2
−
l2
6N
(A.35)
The radius of gyration is equal to one sixth of the end-to-end distance for very
large N.
Ideal chain - probability of the end-to-end distance
We will calculate a probability distribution of the end-to-end distance RN of
an ideal chain made of N Kuhn segments, each of length l. For simplicity we
start with a 1D chain, where segments can be only oriented left or right. The
end-to-end distance can be expressed as RN = (NR − NL)l, where Ni is
the number of segments oriented right (i = R) or left (i = L). The number of
realizations of such a RN is given by the combinatorial expression:
Ω(NR, NL) =
N!
NR!NL!
(A.36)
Since we know that the total number of segments stays constant, we can substitute
NL = N − NR
Ω(NR) =
N!
NR!(N − NR)!
(A.37)
With this substitution RN = (NR − N + NR)l = (2NR − N)l. To simplify the
above expression we will use x = NR − NL, so RN = xl and
Ω(x) =
N!
(N+x
2 )!(N−x
2 )!
(A.38)
117
The probability of distance RN is the number of these micro-states divided
by the total number of states, which is 2N
, since all segments can be oriented
left or right. However, in Eq. A.38 we are biased for such a x where both
N + x and N − x are even numbers. By doing so we increased (doubled)
our probability, since the probability of odd combinations is zero. To correct for
this we have to add a factor of 1/2. Other possibility would be to obtain the
probability without this factor and then normalize the probability to 1.
2P(x) =
1
2N
N!
(N+x
2 )!(N−x
2 )!
(A.39)
We will now use Stirling approximation ln x! ≈ x ln x − x + ln
√
2πx in its
exponential form x! = xx
e−x
√
2πx
2P(x) =
1
2N
NN
e−N+ N+x
2 + N−x
2
(N+x
2 )( N+x
2 )(N−x
2 )( N−x
2 )
2πN
2π N+x
2 2π N−x
2
=
(N
2 )N
e0
(N+x
2 )( N+x
2 )(N−x
2 )( N−x
2 )
2N
π(N2 − x2)
=
(N
2 )N
(N2−x2
4 )( N
2 )(N+x
2 )
x
2 (N−x
2 )− x
2
2N
π(N2 − x2)
=
(N
2 )N
(N
2 )N (1 − x2
N2 )N/2(N+x
N−x )x/2
2N
π(N2 − x2)
= 1 −
x2
N2
− N
2
1 + x
N
1 − x
N
− x
2
2
πN(1 − ( x
N )2)
(A.40)
Assuming that the number of segments is huge NR N or in our substitution
x N and using the exponential expansion for large N (exp(t) ≈ (1+t/N)N
118
), we can simplify the formula to:
2P(x) = exp
x2
2N
−
x2
2N
−
x2
2N
2
πN
(A.41)
= exp −
x2
2N
2
πN
(A.42)
P(x) =
1
√
2πN
exp −
x2
2N
(A.43)
(A.44)
We can double check that the distribution is normalized. By substituting x
back, we obtain the end-to-end distance probability:
P(RN ) =
1
√
2πNl2
exp −
R2
N
2Nl2
(A.45)
Similarly, we can obtain the distribution in three dimensions.
P(RN ) =
3
√
2πNl2
3/2
exp −3R2
N /2Nl2
(A.46)
Dispersion interactions with thin membrane
The dispersion interaction with a thin membrane is calculated similarly to the
interaction with an infinite wall, only the upper integration limit is not infinite, but
the membrane thickness t.
Therefore, the interaction of a small molecule with the membrane is:
E· = −
d+t
z=d
∞
0
2πxρ
C
(x2 + z2)3
dxdz (A.47)
= πρC
d+t
d
1
2(x2 + z2)2
∞
0
dz (A.48)
= −πρC
d+t
d
1
2z4
dz (A.49)
= −
1
6
πρC(
1
(d + t)3
−
1
d3
) (A.50)
119
The interaction distance dependence is ∼ 1/d4
for small membrane thickness
t → 0. The distance scaling is thus weaker than for interaction with an infinite
wall.
The interaction of a sphere and membrane can be calculated using the
above result as simply integration over all the molecules within the sphere:
E• =
d+2R
d
ρsphπ(z − d)(2R − z + d)E·||dz (A.51)
As for the wall derivation, we will use the close distances approximation R
d − z leading to:
E• =
CR
3
π2
ρsphρmem
∞
d
z − d
(z + t)3
−
z − d
z3
dz (A.52)
The first term in the integral is calculated by employing the substitution s =
z + t. The second term is the same as for the wall.
E• =
1
3
π2
ρsphρmemCR −
1
z + t
+
d + t
2(z + t)2
+
1
z
−
d
2z2
∞
d
=
1
6
π2
ρsphρmemCR
1
d + t
−
1
d
(A.53)
Again, for small membrane thickness t → 0, the interaction distance dependence
is ∼ 1/d2
, which is again 1/d weaker (more short range) than the interaction
with an infinite wall.
The last interaction we will calculate is the interaction between two membranes,
where the energy per unit area is:
E =
1
6
πρmem1ρmem2C
d+t
d
1
(z + t)3
−
1
z3
dz (A.54)
=
πρmem1ρmem2C
12
2
(d + t)2
−
1
(d + 2t)2
−
1
d2
(A.55)
The interaction energy at the limit of small membrane thickness is ∼ 1/d4
.
Depletion interactions in solution of ideal chains
Let’s have a system consisting of hard spheres in a solution of ideal chains.
A solution of ideal chains behaves like an ideal gas with additional inner con-
120
formational freedom. The number of chain conformations is restricted in the
vicinity of a hard surface. To simplify, we can use the Asakura and Oosawa
assumption that each chain can be approximated as a hard sphere of radius
equal to the radius of gyration Rg of the given ideal chain (Rg = l2
6 (N −1/N)).
The depleted volume of the single hard sphere will then be a sphere with radius
R + Rg. If two spheres are be separated by distance d 2Rg, their
depleted volumes will overlap. The change in depleted volume is responsible
for the depletion force, which is pushing the particles together. In other words,
the force exists due to the osmotic pressure of ideal chains, which is approximated
as an ideal gas pid = nRT = ρkT (being constant for a given density).
The force is equal to the osmotic pressure times the area A(z), which is the
area of the largest axial circle within the overlap.
Edepletion = −
2Rg−d
0
−ρkTA(z)dz = −ρkTVo(d), (A.56)
where Vo(d) is the overlap volume made of two spherical caps with total vol-
ume:
Vo =
π
12
(6R + 4Rg + d)(2Rg − d)2
(A.57)
The depletion energy is thus:
Edepletion = −
π
12
(6R + 4Rg + d)(2Rg − d)2
ρkT (A.58)
We can understand the depletion energy in terms of the entropy gain of an
ideal gas whose volume increases by the overlap volume. The Asakura and
Oosawa approximation was numerically confirmed to be quite good.
Note that the depletion interaction is not pairwise additive. The overlap
of three spheres is generally not the sum of the pair overlaps, since there
could be a region where all three spheres overlap. If we used the sum of the
pair overlaps, then the three-spheres overlap would be calculated twice, so
the three-body correction to the sum is positive (the force is repulsive). This
non-additivity is important when Rg/R is large.
121
Dissociation constants for amino acids
pKa is a dissociation constant for amino acids that can be protonated or deprotonated,
defined as:
pKa =
[X−/0
][H+
]
[HX0/−]
= e−
µo(HX0/−)−µo(X−/0)−µo(H+)
kT (A.59)
Amino Acid pKa
Asp (D) 3.9
Glu (E) 4.3
His (H) 6.08
Cys (C) 8.28
Tyr (Y) 10.1
Lys (K) 10.5
Arg (R) 12.0
Derivation of equilibrium equation between aggregates
of different sizes
Let’s consider a chemical equation, where two molecules form a dimer:
A + A A2 (A.60)
The equilibrium constant of this reaction is:
K =
cA2 co2
coc2
A
= exp −
Go
A2
− 2Go
A
kT
(A.61)
The dimer concentration cA2
can be transformed into the concentration of single
A molecules in the dimer as cAinA2 = 2cA2 . Similarly, the free energy
of the dimer can be split into the single molecule contributions as ∆Go
A2
=
2∆Go
AinA2
. The above equilibrium constant equation can be thus rewritten:
1
2 cAinA2
co2
coc2
A
= exp −2
Go
AinA2
− Go
A
kT
(A.62)
122
We can use the definition of chemical potential µ as the free energy per mole
and activity a to get:
1
2 aAinA2
a2
A
= exp −2
µo
AinA2
− µo
A
RT
(A.63)
where R is a gas constant (R = NAk). Taking the logarithm of the equation,
we obtain:
ln
1
2
aAinA2
− ln a2
A = −2
µo
AinA2
− µo
A
RT
−RT ln
1
2
aAinA2
+ 2RT ln(aA) = 2(µo
AinA2
− µo
A)
µo
A + RT ln(aA) = µo
AinA2
+
1
2
RT ln
1
2
aAinA2
(A.64)
In the same way we can derive the above equation for larger aggregates:
trimer, tetramer, .. N-mer:
NA AN (A.65)
µo
A + RT ln(aA) = µo
AinAN
+
RT
N
ln
1
N
aAinAN
(A.66)
Since this formula holds for any size of N, we can write the final equilibrium
equation between aggregates of different sizes M and N as:
µo
AinAN
+
RT
N
ln
1
N
aAinAN
= µo
AinAM
+
RT
M
ln
1
M
aAinAM
(A.67)
Distribution of finite size aggregates
The molar fraction of molecules in an N-aggregate Eq. 7.27 for the threshold
value d = 1 simplifies to:
x1N = N [x1B]
N
B−1
(A.68)
We can calculate, which aggregate has the most molecules by finding the extreme
∂x1N
∂N = 0. In order to calculate this, it is useful to express the mol
fraction of monomers x1 in terms of the total molar fraction x.
x =
∞
N=1
x1N =
∞
N=1
N [x1B]
N
B−1
=
x1
(1 − x1B)2
(A.69)
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where we used the sum
∞
N=1 NaN
= a
(1−a)2 . Solving this quadratic equation
we obtain:
x1 =
2xB + 1 −
√
4xB + 1
2xB
(A.70)
The molar fraction of monomers in the N-aggregate can be rewritten as:
x1N = N
2xB + 1 −
√
4xB + 1
2x
N
B−1
(A.71)
For large concentrations (well above CMC): x >> CMC exp
µo
1N −µo
1
RT =
B−1
, we can simplify the formula and use the approximation for large N:
x1N = N 1 − 1/
√
xB
N
B−1
N exp −
N
√
xB
(A.72)
The distribution of aggregates thus has the shape of ∼ Ne−N
. The most
molecules are in the aggregate Nmax:
∂x1N
∂N
= exp −
N
√
xB
+ N exp −
N
√
xB
−
N
√
xB
= 0 (A.73)
Nmax =
√
xB (A.74)
Fluctuation-dissipation theorem
The fluctuation-dissipation theorem relates macroscopic dissipation of energy
(friction or resistive force) and system fluctuations in equilibrium. With linear
response theory it clarifies the system response to the applied perturbations
via its equilibrium fluctuations.
Brownian motion (Einstein 1905): Let’s have a potential V (x) that makes
particles flow with drift velocity
vd = −µ
dV
dx
= µF, (A.75)
where µ is mobility of particles that relates thermal drift velocity to applied
force. The net flow is then:
j = −D
∂c
∂x
+ vdc = −D
∂c
∂x
− µ
dV
dx
(A.76)
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In equilibrium the flow will vanish. The solution is then c ∼ exp(−µV
D ). At
constant temperature the concentration has to be proportional to Boltzmann
factor c ∼ exp(− V
kT ), so we obtain:
µ
D
=
1
kT
(A.77)
The Einstein result is then (using Green-Kubo expression):
µ =
D
kT
=
1
γ
(A.78)
or µ =
D
kT
=
1
kT
∞
0
< v(t)v(0) > dt (A.79)
Thermal noise and dissipation:
We start from Langevine equation and find relation between friction and
stochastic force.
m
d2
x
dt2
= fc − γ
dx
dt
+ fs (A.80)
Let’s assume that the stochastic force fs is a white noise (no correlations in
time or space). This means:
< fs(t) > = 0 (A.81)
< fs(t)fs(t ) > = 2Aδ(t − t ) (A.82)
< fs(t)v(0) > = 0 (A.83)
where A is amplitude of the force and it is related to the strength of collisions.
To find solution of Langevine equation we will assume fc = 0 no conservative
forces on the nanoparticle.
m
dv
dt
= −γv + fs (A.84)
The solution is then:
mv(t) = mv(0)e− γ
m t
+
t
0
fs(t )e− γ
m (t−t )
dt (A.85)
We can combine the solution with another available information for velocity, the
125
equipartition theorem < v2
>= kT
m . Therefore we have to calculate < v2
>:
< m2
v2
(t) > = < mv(0)e− γ
m t
+
t
0
fs(t )e− γ
m (t−t )
dt
2
> (A.86)
= m2
v2
(0)e−2 γ
m t
+ 0+ <
t
0
fs(t )e− γ
m (t−t )
dt
2
>(A.87)
The middle term vanishes because < fs(t)v(0) >= 0. Let’s solve the square
integral separately:
<
t
0
fs(t )e− γ
m (t−t )
dt
t
0
fs(t )e− γ
m (t−t )
dt > (A.88)
=
t
0
t
0
e− γ
m (t−t )
e− γ
m (t−t )
2Aδ(t − t )dt dt (A.89)
=
Am
γ
(1 − e−2 γ
m t
) (A.90)
Thus for square velocity we get:
< m2
v2
(t) >= m2
v2
(0)e−2 γ
m t
+
Am
γ
(1 − e−2 γ
m t
) (A.91)
In the limit of infinite time we can relate to equipartition theorem:
lim
t→∞
< v2
(t) >=
A
γm
=
kT
m
(A.92)
Therefore, amplitude is related to friction as:
A = kTγ (A.93)
More general solution would be in n dimensions:
γ =
1
nkT
∞
0
< fs(t)fs(0) > dt (A.94)
Introduction to Metropolis Monte Carlo method
The Monte Carlo method is a stochastic method used to evaluate multidimensional
integrals, solve differential equations, etc. In the context of molecular
126
simulations, it is used to sample the configuration space of a studied system,
leading to the evaluation of the properties of interest. For this purpose, a random
generation of configurations can be improved by using Metropolis methods,
where the configurations are generated with a probability proportional to
the Boltzmann factor exp(−E/kT), thus obtaining the relevant conformations
of the system at the given temperature.
The sampling starts from an initial configuration with energy Eold. A new
configuration with energy Enew is generated by a move (for instance a small
displacement) and this new configuration is accepted or rejected based on
the transition probability. There are several options for calculating the transition
probability depending on the particular move, but in general the transition
probability has to keep the system in equilibrium once it is reached. In other
words, once the system is in equilibrium the number of visits to particular configurations
is proportional to the configuration probability. We usually use the
much stronger condition of detailed balance, where the averaged number of
moves from the old state to the new one is equal to the average number of
reverse moves. This can be formulated as:
P(old)a(old → new) = P(new)a(new → old) (A.95)
where a is the transition probability and P the configuration probability. The
ratio of transition probabilities is then:
a(old → new)
a(new → old)
=
P(new)
P(old)
(A.96)
Each transition probability a consists of the probability of generating the given
move α and the probability of accepting such a move p. In most of the common
methods α(old → new) = α(new → old) and hence we can write the
acceptance ratio:
p(old → new)
p(new → old)
=
P(new)
P(old)
(A.97)
There are other formulas that satisfy this relation, but the standard Metropolis
condition is:
p(old → new)
= P (new)
P (old) if P(new) < P(old)
= 1 if P(new) ≥ P(old)
(A.98)
We will demonstrate this method on a simple system. Assume that we
have a system in a NV T ensemble with two particles A, B moving on the
127
x-axis with simple a displacement move. The displacement is ±1 with the
same probability in both directions. The configuration probability is then P ∼
exp(−E/kT) and the probabilities of the displacement are:
p(old → new) = e−
Enew−Eold
kT if Enew > Eold
= 1 if Enew ≤ Eold
(A.99)
This can be also written as:
p(old → new) = min[1, e−
Enew−Eold
kT ] (A.100)
The simulation algorithm is as follows:
1. randomly select the particle (A or B)
2. randomly select direction (+ or −)
3. do a displacement of size 1 and calculate the energy of the new state
4. accept the new configuration with probability p
We can see that the detailed balance is satisfied. In other words, the probability
of moving particle A from position 1 to 2 is the same as moving it from position
2 to 1.
Note that full configuration probability in a canonical ensemble is
P(N) =
V N
Λ3N N!
e−
E(rN )
kT (A.101)
since the canonical partition function is
Q =
V N
Λ3N N!
e
µN
RT e−
E(rN )
kT drN
(A.102)
Another note is about a system where we move particles left and right in
the sequence (A,B,A,B,A,B,...). The detailed balance is not satisfied, yet we
can obtain the correct sampling. However, in complex systems the situation
is more complicated, and therefore, the detailed balance condition is usually
used.