Fyzika biopolymerů
Robert Vácha
Kamenice 5, A4 2.13
robert.vacha@mail.muni.cz
Solvatace
"2
Solvatace
IUPAC definition: solvation is an interaction of a solute with the solvent, which leads
to stabilization of the solute species in the solution. In the solvated state, a solute in a
solution is surrounded or complexed by solvent molecules. Solvated species can
often be described by coordination number, and the complex stability constants.
první solvatační vrstva - je v kontaktu s rozpuštěnou látkou a je nejvíce ovlivněna
druhá solvatační vrstva - je v kontaktu s prvni solvatační vrstvou a je ovlivněna
přítomností rozpuštěné látky méně
nejčastějším solventem je voda => solvatace = hydratace
"3
Solvatace
● solvent je nezbytný pro funkci biologických systémů, které ovlivňuje:
- přímo = aktivní účast v biologických procesech např. enymatická reakce 
- nepřímo = stabilizace biologicky aktivních konformací biomolekul 
● interakce rozpuštěná látka-voda silně ovlivňuje konformace biopolymerů
● hydrofobní efekt u protein foldingu
● solvent hraje klíčovou roli při tvorbě komplexů, rozpoznávání ligandů, interakcí
mezi DNA a proteiny
● stíní elektrostatické interakce 
Hydratační páteř na DNA
Hydratačnípáteř
oviceDNA
"4
Patametry solvatace
● solvatační číslo 
- počet molekul rozpouštědla (vod) ovlivněných rozpuštěnou molekulou
(obvykle první a druhá solvatační vrstva)
● relativní rezidenční časy  
- je-li rezidenční čas u rozpuštěné látky/ rezidenční čas v roztoku
> 1 zvýšení strukturního stupně 
< 1 narušení struktury
● Stokesův poloměr 
- efektivní hydrodynamický poloměr pohybujicí se sféry se setjnou difuzní
konstantou (obvykle zahrnuje i silněji interagující vody)
- výpočet ze Stokesova zákona:  
- porovnává se s poloměrem otáčení (gyration)
● Slip plane
solvatační číslo
● počet vod ovlivněných iontem (lze zjistit pomocí NMR)
relativní rezidenční časy
● vlastní difúze (difúze vody ve vodě)
● čas ti
(voda v blízkosti iontu), t (voda v blízkosti jiné vody), ti
/t
určuje stupeň demobilizace vody v blízkosti iontu
● ti
/t > 1 zvýšení strukturního stupně vody v blízkosti iontu
● ti
/t < 1 narušení struktury vody (chaotropní ionty)
Stokesův poloměr rH
● efektivní poloměr makroskopické sféry s hydrofilním povrchem,
může být uplatněn Stokesův zákon:
⃗F=6 πηr ⃗v
- hypotetická vzdálenost do které se solvent hýbe
s rozpuštěnou látkou - používá se při měření
elektrostatického potenciálu a odhadu náboje
"5
Radiální distribuční funkce RDF,g(r)
- representuje pravděpodobnost výskytu částice B ve
vzdálenosti r od částice A
- je to párová korelační funkce
- je normalizovaná na hustotu ideálního plynu (1 v ∞)
- lze i pro stejné částice gAA(r)
- lze v 3D i 2D
- charakterizuje dané skupenství
- není dobře definovaná v nehomogenním systému
- jde porovnat s rozptylovými experimenty
- může zachytit fázové strukturní změny
- lze z ní spočítat vazebnou konstantu
- lze spočítat jako histogram
- v periodických okrajových podmínkách omezena polovinou
boxu
Radial	distribution	function	(RDF,	g(r))
8
in	practice,	RDF:
• can	be	calculated	using	histogram	methods
• is	normalized	to	‘ideal	gas’	density,	should	be	equal	1	at	∞
• under	p.b.c.,	it	is	limited	to	half-box	size
• represents	‘probability’	of	finding	the	particle	B	at	the	
distance	r from	particle	A
• is	a	pair	correlation	function
• can	be	calculated	also	for	same	particles	gAA(r)
• can	be	calculated	in	2D	(to	analyze	lateral	preferences)
• not	well	defined	in	non-homogeneous	 systems!
• can	be	compare	with	scattering	experiments
• can	be	used	to	indicate	structural	phase	transitions
r
∆r
Radial	distribution	function	(RDF,	g(r))
8
in	practice,	RDF:
• can	be	calculated	using	histogram	methods
• is	normalized	to	‘ideal	gas’	density,	should	be	equal	1	at	∞
• under	p.b.c.,	it	is	limited	to	half-box	size
• represents	‘probability’	of	finding	the	particle	B	at	the	
distance	r from	particle	A
• is	a	pair	correlation	function
• can	be	calculated	also	for	same	particles	gAA(r)
• can	be	calculated	in	2D	(to	analyze	lateral	preferences)
• not	well	defined	in	non-homogeneous	 systems!
• can	be	compare	with	scattering	experiments
• can	be	used	to	indicate	structural	phase	transitions
r
∆r
"6
Skupenství a RDF,g(r)
11
RDF	example	II:	DPPC	monolayer
r
∆r
• 2D	RDF	for	studying	lateral	arrangement	of	
molecules
• phase	transition	in	monolayer	can	be	analyzed	
11
RDF	example	II:	DPPC	monolayer
r
∆r
• 2D	RDF	for	studying	lateral	arrangement	of	
molecules
• phase	transition	in	monolayer	can	be	analyzed	
"7
Příklady RDF,g(r)
Experiment
RDF	example	I:	water
9
Soper 2013
RDF	water	- MD RDF	water	– neutron	scattering
• hydration	structure	analysis
• comparison	with	experiment
10
RDF	example	I:	water
• numbers	of	atoms	in	
hydration	shells
"8
Strukturní faktor a RDF,g(r)
strukturní faktor (měřitelný experimentálně, např gama rozptylem) je Fourieriva
transformace radiální distribuční funkce
"9
Vazebná konstanta a RDF,g(r)
vazebné konstanta
ligand is allowed to equilibrate across the membrane, and the
concentrations of ligand inside and outside the bag are
measured. The excess concentration of ligand inside the bag
is attributed to binding and hence is equated with the
concentration of the complex, enabling evaluation of K. The
value of K can be obtained from a full-ﬂedged binding
isotherm or, at least in principle, from a single measurement
at a well-chosen ligand concentration.
The theoretician’s challenge is to account for measured
afﬁnity data and ultimately to predict binding afﬁnities to
useful accuracy. The ﬁrst requisite for accomplishing this is
an energy model that accurately and efﬁciently provides the
energy of the system as a function of its conﬁguration.
Developing such a model is highly nontrivial and will
continue to be a subject of research in many labs, but it is not
the focus of the present study. We address instead the second
requisite, a theory or formula that says how to use an energy
model to compute a binding afﬁnity that can legitimately be
compared with experiment. Three major competing theories
are considered.
In one theory, K is evaluated as the integral of the Mayer
factor over all space (Hill, 1986) (equivalent to the second
virial coefﬁcient),
KMayer ¼ C°
Z
ðeÿbW
ÿ 1Þdr; (3)
where W is the potential of mean force between the two
molecules. Thus, Groot focuses on the compressibility of
a mixture of receptors and ligands to show that the binding
constant is the integral over all space of the receptor-ligand
correlation function, and notes that this quantity goes to
the Mayer integral in the limit where receptor-receptor and
ligand-ligand interactions are negligible (Groot, 1992). Dill
reaches the same result via analysis of the equilibrium
dialysis experiment (Stigter and Dill, 1996). An appealing
feature of the Mayer integral is that there is little difﬁculty in
deﬁning what is meant by the complex: the integral extends
over all space, yet is ﬁnite because the Mayer factor goes to
zero at long range (which applies so long as the potential of
pointed out that the equilibrium dialysis experiment could
yield a negative binding constant (van Holde, 1971). One
might say that the equilibrium dialysis experiment provides
a global assessment of the interactions—both attractive and
repulsive—of the receptor with the ligand, whereas the
signal technique provides a local assessment of the afﬁnity of
a speciﬁc region of the receptor for the ligand.
An alternative theoretical approach involves viewing the
bound complex, the free receptor, and the free ligand, as
three distinct chemical species. From this perspective,
the binding constant should be computed as the ratio of the
partition function of the complex to the product of the
partition functions of the free molecules. This approach is
widely used to compute covalent binding constants, as
exempliﬁed by the treatment of the reaction 2H 
 H2 in
many physical chemistry textbooks, and there is no obvious
reason why it should not be applicable to noncovalent
binding as well. Assuming classical statistical thermodynamics
where the spacing of quantized energy levels is
assumed to be much smaller than thermal energy, as was also
done for the other two theories of binding considered in this
article, the binding constant in this approach is simply the
integral of the Boltzmann factor for the potential of mean
force of the receptor and ligand (Chandler, 1979; Shoup and
Szabo, 1982; Jorgensen, 1989; Gilson et al., 1997):
KBoltzmann ¼ C°
Z
e
ÿbW
dr: (4)
(The Appendix reviews how this expression can be obtained
for the reaction 2H 
 H2, starting from the usual translational,
rotational, and vibrational partition functions.)
However, this approach poses a problem that is particularly
noticeable in the case of noncovalent binding: the Boltzmann
factor does not go to zero as W goes to zero at long range, so
KBoltzmann is at risk of becoming inﬁnite. Thus, to apply this
formula, one must deﬁne the domain of integration, in effect
establishing the receptor-ligand distance at which the
receptor and ligand no longer form a complex. When the
potential of mean force has a deep and circumscribed energy
24 Mihailescu and Gilson
Binding
y of Maryland Biotechnology Institute, Rockville, Maryland
constant of a receptor and ligand can be written as a two-body integral
he ligand. Interestingly, however, three different theories of binding in the
udy uses theory, as well as simulations of binding experiments, to test the
ed by a signal that detects the ligand in the binding site, the most accurate
ctor, where the bound complex is deﬁned in terms of an exclusive binding
panding a ligand can increase its binding constant, is borne out by the
binding isotherms can be obtained when the region over which the signal
ne. Interestingly, the binding constant measured by equilibrium dialysis,
yield a binding constant that differs from that obtained from a signal
ral of the Mayer factor.
n is of
role in
n, and
emistry
tection,
olecular
funda-
uestion.
n force
can be
olecule
of this
ns (see
em by
of the
des the
ity that
e most
nt is to
to the
is, by
spectro-
enzyme
e of the
oretical
gnal for
because
rmined
for a given value of the signal to be interpreted in terms of the
extent of binding.) The theoretical form of the binding
isotherm is obtained by considering the association of
a receptor R and a ligand L to form the complex RL. The
equilibrium constant for this reaction, the binding constant,
may be written as
K [
gRLCRLC°
gRCRgLCL
 
eq
; (1)
where CX and gX indicate, respectively, the concentration
and activity coefﬁcient of species X, C° is the standard
concentration expressed in the same units as the other
concentrations, and the subscript eq indicates a quantity
evaluated under equilibrium conditions. It is often assumed
that the activity coefﬁcients are near 1, so these terms are
frequently not written explicitly. (Note that K is free of units
when C° is correctly included in its deﬁnition.) The binding
isotherm gives the fraction of the receptor with bound ligand,
r, as
r [
CRL
CR 1 CRL
¼
KCL
1 1 KCL
: (2)
With luck, an experimental isotherm will match this
theoretical curve-ﬁtting, in which case the binding constant
K can be extracted via curve-ﬁtting. If the experimental
isotherm diverges signiﬁcantly from the theoretical ideal,
then it is appropriate to ask whether equilibrium dimerization
23
z RDF g(r)= exp( -dW(r)/kT ), kde W(r) je PMF… profil dG
souvisí s druhým
viriálním koeficientem
pointed out that the equilibrium dialysis experiment could
yield a negative binding constant (van Holde, 1971). One
might say that the equilibrium dialysis experiment provides
a global assessment of the interactions—both attractive and
repulsive—of the receptor with the ligand, whereas the
signal technique provides a local assessment of the afﬁnity of
a speciﬁc region of the receptor for the ligand.
An alternative theoretical approach involves viewing the
bound complex, the free receptor, and the free ligand, as
three distinct chemical species. From this perspective,
the binding constant should be computed as the ratio of the
partition function of the complex to the product of the
partition functions of the free molecules. This approach is
widely used to compute covalent binding constants, as
exempliﬁed by the treatment of the reaction 2H 
 H2 in
many physical chemistry textbooks, and there is no obvious
reason why it should not be applicable to noncovalent
binding as well. Assuming classical statistical thermodynamics
where the spacing of quantized energy levels is
assumed to be much smaller than thermal energy, as was also
done for the other two theories of binding considered in this
article, the binding constant in this approach is simply the
integral of the Boltzmann factor for the potential of mean
force of the receptor and ligand (Chandler, 1979; Shoup and
Szabo, 1982; Jorgensen, 1989; Gilson et al., 1997):
KBoltzmann ¼ C°
Z
e
ÿbW
dr: (4)
(The Appendix reviews how this expression can be obtained
for the reaction 2H 
 H2, starting from the usual translational,
rotational, and vibrational partition functions.)
However, this approach poses a problem that is particularly
noticeable in the case of noncovalent binding: the Boltzmann
factor does not go to zero as W goes to zero at long range, so
KBoltzmann is at risk of becoming inﬁnite. Thus, to apply this
formula, one must deﬁne the domain of integration, in effect
establishing the receptor-ligand distance at which the
receptor and ligand no longer form a complex. When the
potential of mean force has a deep and circumscribed energy
Mihailescu and Gilson
the ligand is considered ‘‘bound’’ when its attraction to the
receptor exceeds thermal energy (Luo and Sharp, 2002).
However, neither of these suggestions has been validated by
comprehensive theory or by comparison with simulated
binding experiments. It may also be tempting to argue that
the very idea of a ligand-receptor complex is an artiﬁcial
construct. Thus, the Boltzmann integral has been incorporated
into theories of ion-pairing to explain deviations from
Debye-Hu¨ckel theory (Bjerrum, 1926; Prue, 1969; Justice
and Justice, 1976). In this context, the distance at which
two ions cease being a ‘‘pair’’ can be chosen based upon
theoretical convenience (Justice and Justice, 1976). However,
for pairwise noncovalent binding, the deﬁnition of the
bound complex cannot be arbitrary because even a weak
binding interaction can generate a perfectly reasonable
isotherm that ﬁts the chemical equilibrium model and thus
yields a single distinct value of the binding constant. Thus, it
would appear, as previously pointed out (Groot, 1992), that
nature knows how to deﬁne the complex, even if we do not.
A third theoretical approach, pioneered by Andersen
(1973) and further developed by Hoye and Olaussen (1980),
and Wertheim (1984), explicitly accounts for a solution of
ligands and receptors, rather than limiting attention to a single
ligand and receptor as in the two theories discussed above.
Central to this theory is an exclusive, or saturating, energy
model; that is, one where a receptor and ligand do not attract
other receptors or ligands once they have paired off.
Exclusivity is essential for dimerization: if each receptor
could bind multiple ligands and each ligand could bind
multiple receptors, then one would see polymerization or
even a phase change, rather than dimerization. In what will
here be called the Andersen theory, cluster expansions are
used to show that an exclusive interaction potential leads to
formation of ligand-receptor dimers, and that the concentration
of dimers is related to the concentrations of free ligands
and free receptors by a binding equilibrium. The binding
constant is computed by separating the interaction potential
into two parts: a short-ranged, repulsive part WR and a softer,
longer-ranged, attractive part WA. The binding constant is
then given by
KAndersen ¼ C°
Z
e
ÿbWR
ðe
ÿbWA
ÿ 1Þdr: (5)
(Hoye and Olaussen use this same approach, but their
formula for K has the form of the Mayer integral because
their RL interaction potential includes no steric contribution.)
This formula appears to have two practical advantages. First,
because the term eÿbWA
ÿ 1 goes to zero at long range, there
seems to be no need for the geometric deﬁnition of the
complex that is required to obtain a ﬁnite value of KBoltzmann.
Second, the term eÿbWR
brings the integrand to zero in the
because it is not always clear how W is to be separated into
WR and WA. It also is not clear how to handle an attractive
potential with a long-ranged component that extends beyond
the zone in which binding is exclusive. Finally, this theory,
like the Mayer theory, predicts that the binding constant goes
to zero as the depth of the attractive energy well goes to zero,
even though it is clearly possible to observe ‘‘complexes,’’
as deﬁned by a spectroscopic signal, even if the attractive
potential goes to zero. Accordingly, Jackson and co-workers
note that the number of ligand-receptor complexes from this
theory will not correspond exactly to the number obtained by
a count of ligand-receptor pairs that are within bonding
distance (Jackson et al., 1988).
In summary, pairwise noncovalent binding is more subtle
than it initially might appear, and there is still no generally
accepted theory for this fundamental phenomenon. From
a practical standpoint, although the differences among the
three theories diminish for small, tight-binding molecules,
there are receptor-ligand systems that bind weakly enough
for the theories to differ signiﬁcantly, so the question of
which theory to use is important if one wishes to develop
quantitative models of weak binding. This article therefore
seeks to further elucidate the theoretical basis of pairwise
noncovalent binding. The central approach is to compare
theory with simulations designed to mimic actual experimental
measurements. Two types of experiments are considered:
1) spectroscopic detection of binding to generate an
isotherm that is ﬁtted to a theoretical isotherm, and 2) equilibrium
dialysis. To our knowledge, this article represents the
ﬁrst direct comparison of the three theories discussed above.
The article is organized as follows. The Theory section
presents a novel combinatorial theory of binding which
shows that exclusive binary associations lead directly to the
standard binding isotherms associated with Eq. 1 and
indicates that, when binding is measured via a signal, the
binding constant is an integral of the Boltzmann factor
(KBoltzmann), whereas when binding is measured by equilibrium
dialysis, the binding constant is the integral of the
Mayer factor (KMayer). Methods describes Monte Carlo
simulations used to test the theories of binding discussed
above, and Results and Discussion compares the simulation
results with theory.
THEORY
The theory of Andersen (1973), Hoye and Olaussen (1980), and Wertheim
(1984) hinges on a recognition of the importance, for pairwise binding, of
exclusivity in the interaction between the receptor and the ligand. Pairwise
exclusivity is a requirement for the formation of dimers, as opposed to higher
order multimers or even a phase transition, as concentration increases.
Exclusivity also provides an intuitively satisfying explanation of the fact that
the binding constant can be written as an integral involving only one ligand
Binding Theory 25
integral přes Mayerovu f-funkci
"10
Experimentální metody
● rentgenová difrakce
- rozptyl na elektronech (el. obal atomu) = citlovější na těžší atomy
- elektron. hustota se průměruje přes čas a velké množství struktur
- v krystalu přímá evidence přítomnosti vody v interakci s biomolekulou
 
● neutronová difrakce
- rozptyl na jádrech = citlivá na vodíky, vhodná ke studiu vody
● SAXS, SANS
- distrubuce velikostí
● NMR
- strukturní i dynamické informace o vodě v blízkosti biomolekuly
- NOE: sledování solventu v přímé interakci s danou biomolekulou, omezené
časové rozlišení 
"11
Experimentální metody
● optická spektroskopie
- femtosekundová fluorescenční spektroskopie - pík je citlivý na dipól. moment
sondy, který závisí na polarizaci solventu (množství vod a jejich reorientace)
možnost vysokého časového rozlišení s prostorovým rozlišením  
- nelineární spektroskopie (VSFG, HFG) - citlivá na nehomogení prostědí =
signál z rozhraní
- infračervená spektroskopie - citlivá na tvorbu H-vazeb, umožňuje studovat
specifické interakce solut-solvent, kvalitativní informace
 
● frekvenční závislost permitivity - síla interakce (omezení reorientace)
"12
Implicitní solvatace
- molekuly rozpouštědla jsou nahrazeny spojitým médiem o vlastnostech
odpovídající rozpouštědlu
- umožňuje rychlé a jednoduché výpočty - interakce biopolymerů, jejich konformace
nebo určení solvatační energie/rozpustnosti
- SASA (solvent accessible
surface area) hlavně se
používá pro odhad
hydrofobní interakce
"13
Solvatační energie
Bornova solvatační energie (1920)
- volná energie na vložení náboje do dané kavity v roztoku (elektrostatická energie/
práce potřebná na přenesení náboje z vakua do daného média)
Zobecněný Bornův model - zahrnující zjednodušené řešení Poisson-Boltzmanovy
rovnice (a=𝛼=poloměr atomů..problematická definice)
Kavitační energie - energie potřebná na vytvoření kavity v roztoku
implicitní model = není první solvatační vrstva ….
"14
Typy solventůPolymer size depends on intra-molecular interactions (solvent quality).
Increase monomer size by factor of 108 b ~ 1cm.
Poor solvent Theta solvent
R = bN1/3
Long-range repulsion
Consider N = 1010
R = bN1/2
Good solvent
R = bN3/5
R ~ L = bN
~ 20 m
~ 1 km
~ 10 km
~ 105 km
Astronomical Variations of Polymer Size
ideal-like
globule
swollen
extended
"15
Solventy a fázový diagram
Polymer Solutions
φ
T
2-phase
Tc
φ` φ``
φc
0
poor solvent
Theta solvent
0v 3
=
−
= b
T
T θ
Chains are nearly ideal NbR =
Overlap concentration
NR
Nb 1
3
3
*
=≈θφ
good solvent
θ
φ < φ∗
dilute θ
φ∗
φ = φ∗ φ∗<φ<<1
semidilute
θ−solvent
Chain is ideal if it is smaller than
thermal blob
v
4
b
T ≈ξ
Boundaries of dilute θ-regime
N
b
b
T
T 3
3
v =
−
=
θ
±≈
N
T
1
1θTemperatures at which chains begin
to either swell or collapse
v
"16
Mayerova funkce
Real Chains: Monomer Interactions
U(r)
r
Effective interactions potential between two
monomers in a solution of other molecules.
exp(-U/kT)
0
1
2
1 2 3 4
r/b
Relative probability of finding
two monomers at distance r
-1
-0.5
0
0.5
1
1.5
1 2 3 4
r/b
f
Mayer f-function
1
)(
exp)( −−=
kT
rU
rf
Excluded volume
∫−= rdrf 3
)(v
Mayer f-function
U(r)
r
Effective interactions potential between two
monomers in a solution of other molecules.
exp(-U/kT)
0
1
2
1 2 3 4
r/b
Relative probability of finding
two monomers at distance r
-1
-0.5
0
0.5
1
1.5
1 2 3 4
r/b
f
Mayer f-function
1
)(
exp)( −−=
kT
rU
rf
Excluded volume
∫−= rdrf 3
)(v
Mayer f-function
Mayerova f-funkce
Real Chains: Monomer Interactions
U(r)
r
Effective interactions potential between two
monomers in a solution of other molecules.
exp(-U/kT)
0
1
2
1 2 3 4
r/b
Relative probability of finding
two monomers at distance r
-1
-0.5
0
0.5
1
1.5
1 2 3 4
r/b
f
Mayer f-function
1
)(
exp)( −−=
kT
rU
rf
Excluded volume
∫−= rdrf 3
)(v
Mayer f-function
Real Chains: Monomer Interactions
U(r)
r
Effective interactions potential between two
monomers in a solution of other molecules.
exp(-U/kT)
0
1
2
1 2 3 4
r/b
Relative probability of finding
two monomers at distance r
-1
-0.5
0
0.5
1
1.5
1 2 3 4
r/b
f
Mayer f-function
1
)(
exp)( −−=
kT
rU
rf
Excluded volume
∫−= rdrf 3
)(v
Mayer f-function
aproximace vyloučeneho objemu
(excluded volume)
pravděpodobnost nalezení částic ve vzdálenosti r
(nenormalizovaná)
Potenciál
Real Chains: Monomer Interactions
U(r)
r
Effective interactions potential between two
monomers in a solution of other molecules.
exp(-U/kT)
0
1
2
1 2 3 4
r/b
Relative probability of finding
two monomers at distance r
-1
-0.5
0
0.5
1
1.5
1 2 3 4
r/b
f
Mayer f-function
1
)(
exp)( −−=
kT
rU
rf
Excluded volume
∫−= rdrf 3
)(v
Mayer f-function
"17
Classification of Solvents
Athermal solvents 3
v b≈high T limit
Good solvents
Theta solvents
Poor solvents
0v =
3
v0 b<<
0v <
f rrepulsion dominates
attraction balances repulsion
f r
attraction dominates f r
Typically repulsion dominates at higher temperatures
while attraction dominates at lower temperatures.
∫−= rdrf 3
)(v
f rb
1
)(
exp)( −−=
kT
rU
rf
Dobrý a špatný solvent
Mayerova f-funkce
excluded volume
"18
folding funnel - sbalování proteinů do přirozeného stavu
Protein folding / sbalování proteinů
"19
Hydrofobní kolaps
je hlavně entropické
"20
Molten globule
- univerzální intermediát při popisu sbalování a rozbalování proteinu
- hydrofobní residua hlavně uvnitř a hydrofilní residua venku
- některá residua již v přirozeném kontaktu, “skoro” natinví
konformace, sekundární struktura často blízká nativní formě proteinu
- malé uspořádání bočních řetězců, méně kompaktní než nativní
protein
"21
Více modelů
"22
Levinthalův paradox
- pokud by pro každé reziduum existovaly 2 možné konformace, pak pro řetězec
se 100 rezidui existuje 2100 alternativních struktur, a protože přechod z jedné
konformace do druhé nemůže být rychlejší než 1 ps, prohledávání prostoru
potenciální energie by trvalo nejméně ~2100 ps (~1010 let)
Otázka: Jak se dokáže protein sbalit do nativní formy během krátké doby (s-min)?
- Nativní forma proteinu je určena kineticky spíše než termodynamicky a jde
cestou hledání snadno dosažitelného lokálního minima, než hledání globálního
minima volné energie.
Kinetika : sbalování nesmí obsahovat příliš vysoké energetické bariéry a nemít
mnoho mezikroků
Termodynamika : za normálních podmínek je přirozený stav jen o několik kcal/mol
stabilnější než nesbalený
Požadavky kinetiky i termodynamiky mohou být splněny současně: předpokládá
se, že v biologických procesech našly uplatnění právě ty proteiny, které se takto
formovat dokáží.
"23
Amyloidy - nesprávné sbalování
"24
Chaperons
"25
Sbalování proteinů - modelování
Go models
All-atom
"26
Denaturace
nejčastěji náhodné klubko (random coil)
"27
DenaturaceDenaturace proteinů
Fázový diagram konformačních stavů
v lysozymu. Čárkovaně přechodové
zóny.
stav nativní stav
"28
Helix-coil transition
- peptidy, proteiny, DNA, RNA
- je to modelový zjednodušený
systém pro sbalovaní proteinů
- dvou stavový model, každé
residuum je buď v helixu nebo
coilu (ising model)
- nukleace a propagace
sbalování
- kooperativní process
- dva popisy: Zimm-Bragg a
Lifson-Roig (první bere vliv
okolních residuí a druhy
zahrnuje trojici residuí)
"29
Crowding
- efekt makromolekulárního zaplnění popisující změnu vlastností molekul v roztoku,
pokud jsou přítomny ve vysoké koncentrace (koncentrace proteinů v cytosolu 300
- 400 mg / ml, v čočce až 500 mg / ml)
- vliv na sbalování a konformace proteinů
- mění associační/dissociační konstanty = afinity
- větší molekuly ovlivněny více než malé
"30
Fickovy zákony
1. zákon
Transport phenomena
usion, Navier Stokes, Stokes-Einstein equation, DPD )
evious chapters we discussed the thermodynamic properties and strucof
systems in equilibrium. In this chapter we focus on the kinetics of
eins. As mentioned in the ﬁrst chapter, diffusion dominates the proes
in soft matter. Proteins and big molecules in solution undergo a
number of collisions with the surrounding molecules of a solution,
ng to their Brownian motion. This motion is named after Robert Brown,
studied the motion of a pollen grain in aqueous solution. However,
molecular explanation and derivation was done by Einstein (1905) and
luchowski (1906).
he derivation starts from macroscopic measurements. If we put a lot of
s in one part of the solution, we will observe their diffusion to regions
low concentration. This motion is described by Fick’s 1st law:
j = D
@c
@x
(9.2)
h states that the ﬂux j is linearly proportional to the concentration grawith
the proportionality constant D, called the diffusion coefﬁcient.
m the conservation of mass law, the ﬂux is the time derivative of conation
c:
@j
@x
=
@c
@t
(9.3)
ng these together, we obtain Fick’s 2nd law:
@c
@t
= D
@2
c
@x2
(9.4)
assume constant amount of material with N =
R 1
1
cdx and that we
ed to follow the particles from time zero (c(0, 0) = N), the solution of
zákon zachování hmotnosti
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 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 =
R 1
1
cdx and that we
started to follow the particles from time zero (c(0, 0) = N), the solution of
2. zákon
Transport phenomena
fusion, Navier Stokes, Stokes-Einstein equation, DPD )
revious chapters we discussed the thermodynamic properties and strucs
of systems in equilibrium. In this chapter we focus on the kinetics of
eins. As mentioned in the ﬁrst chapter, diffusion dominates the proses
in soft matter. Proteins and big molecules in solution undergo a
e number of collisions with the surrounding molecules of a solution,
ding to their Brownian motion. This motion is named after Robert Brown,
o studied the motion of a pollen grain in aqueous solution. However,
molecular explanation and derivation was done by Einstein (1905) and
oluchowski (1906).
The derivation starts from macroscopic measurements. If we put a lot of
ns in one part of the solution, we will observe their diffusion to regions
h low concentration. This motion is described by Fick’s 1st law:
j = D
@c
@x
(9.2)
ch states that the ﬂux j is linearly proportional to the concentration grant
with the proportionality constant D, called the diffusion coefﬁcient.
m the conservation of mass law, the ﬂux is the time derivative of contration
c:
@j
@x
=
@c
@t
(9.3)
ting these together, we obtain Fick’s 2nd law:
@c
@t
= D
@2
c
@x2
(9.4)
e assume constant amount of material with N =
R 1
1
cdx and that we
ted to follow the particles from time zero (c(0, 0) = N), the solution of
84
the differential equation is:
c(x, t) =
N
p
4⇡Dt
exp
✓
x2
4Dt
◆
(9.5)
Naturally, the mean particle position is < x >= 0 as the particles diffuse
Figure 9.16: Concentration distribution change via diffusion.
in all directions with the same diffusion coefﬁcient. The second moment,
mean square displacement, is:
< x2
>= 2Dt (9.6)
84
the differential equation is:
c(x, t) =
N
p
4⇡Dt
exp
✓
x2
4Dt
◆
(9.5)
Naturally, the mean particle position is < x >= 0 as the particles diffuse
fferential equation is:
c(x, t) =
N
p
4⇡Dt
exp
✓
x2
4Dt
◆
(9.5)
rally, the mean particle position is < x >= 0 as the particles diffuse
Figure 9.16: Concentration distribution change via diffusion.
directions with the same diffusion coefﬁcient. The second moment,
square displacement, is:
< x2
>= 2Dt (9.6)
macroscopic change in concentration proﬁle could thus be related to
icroscopic motion of particles. We can do the same derivation in n
nsions to obtain the more general expression:
< |~r|2
>= 2nDt (9.7)
efore we look at how to estimate the diffusion coefﬁcient for molecules,
ave a look at the molecular origin of Fick’s law. In other words, how do
les know which direction to go? And how do we get time-irreversible
"31
Difuzní koeficient
Green-Kubo
= 2
0 0
< v(t0
)v(t00
) > dt00
dt0
(9.18)
From this we can evaluate the diffusion coefﬁcient:
D = lim
t!1
@ < x2
>
2@t
(9.19)
D =
Z 1
0
< v(t t00
)v(0) > d(t t00
) (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
dx
dt
+ fs (9.21)
nd much easier to solve. Of course the exact solution still depends on
he boundary conditions, but for several simple cases this could be solved
nalytically.
For example for a hard sphere moving at steady state in solution the
avier-Stokes equation simpliﬁes to 0 = rp + ⌘r2
v. This could be
olved in cylindrical coordinates leading to the transfer velocity from the
article to the solution vt ⇠ vs(R/r), where R is the sphere’s radius, vs
the sphere’s velocity, and r is the distance from the sphere. From the
ansverse velocity dependence, we can calculate a friction force to derive
he well-known Stokes’ law:
Ff = 6⇡⌘Rvs (9.32)
Stokesův zákon
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 = 6⇡⌘R now and
we can insert it into the Eq. 9.29 to end up with the Einstein-Stokes relation:
D =
kT
=
kT
6⇡⌘R
(9.33)
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
6⇡⌘R
t (9.34)
t =
⇡⌘R3
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,
Einstein-Stokesův zákon
90
Molecule Medium Diffusion coefﬁcient µm2/s
H+ water 7000
H2O, O2, CO2 water 2000
Protein (30 kDa), tRNA (20 kDa) water 100
Protein (30 kDa) cytoplasm 10 - 30
Protein (70 -250 kDa) cytoplasm 0.4 - 2
Protein (70 -140 kDa) membrane 0.03 - 0.2
Table 9.3: Examples of diffusion coefﬁcients for various molecules in biological
environment.
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 coarsegrained
into one particle, which has similar effective friction and ﬂuctuation
as an 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 =
X
i6=j
h
~Fc( ~rij) + ~Fd( ~rij, ~vij) + ~Fs( ~rij)
i
(9.36)
Dissipative force depends on both interparticle distance ~rij and velocity ~vij
:
~Fd( ~rij, ~vij) = !d(| ~rij|)

~vij ·
~rij
| ~rij|
~rij
| ~rij|
(9.37)
where is the friction coefﬁcient and !d represents the variation of the
friction with distance. The !d 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) = !s(| ~rij|)g
~rij
| ~rij|
(9.38)
where is the magnitude of a random pair force and !s 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
"32
Statistical	ensembles
9
• thermodynamic	 statistical	ensembles	describe	macroscopic	
conditions
• NVE – microcanonical
• NVT – canonical	(other	names:	isothermal,	Helmholtz	 canonical)
• /VT	– grand-canonical
• NPE	– isobaric
• NPT – isobaric-isothermal	(other	name:	Gibbs	canonical)
NVE NVT
T
NPT
T