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. enzymatická 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 Parametry 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 stejnou 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 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 mean force decays with the receptor-ligand distance rRL more sharply than rRL ÿ3 ). On the other hand, this theory yields the somewhat odd result that KMayer falls as the steric bulk of the receptor and ligand increase, because the Mayer factor equals ÿ1 in the region where the ligand overlaps the 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 well, any definition of the complex that encompasses the energy well and does not extend much further will give reasonable results (Gilson et al., 1997). However, for a weak and/or long-ranged interaction potential, the definition of the complex matters and it is difficult to decide where to draw 24 Mihailescu and Gilson B2 = 2 0 (e W/kT 1)r2 dr "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 Příklad Na obrázku je radiální distribuční funkce kapaliny o průměrné hustotě ρ, 0,0213 A−3 1. Jaká je fyzikální interpretace ρg(r)? 2. Je více částic v první nebo ve druhé solvatační vrstvě? 3. Odhadněte poloměr u atomů/molekul kapaliny. 4. Odhadněte počet nejbližších sousedů. "13 Příklad 2 Uvažujme kapalinu, která se skládá z částic, které interagují pouze odpudivou částí Lennard-Jonesova (LJ) potenciálu, u(r) = A / r12. Levá strana obrázku ukazuje, g(r), mezi dvěma částicemi v takové kapalině. 1. Která křivka odpovídá nejvyšší hustotě částic? 2. Pro kompletní křivku na levém grafu načrtněte odpovídající potenciál střední síly, w(r). 3. Vysvětlete, proč se částice v kapalině navzájem přitahují navzdory skutečnosti, že párový potenciál u(r) je vždy odpudivý. 4. Pravá strana obrázku ukazuje g(r) při teplotě T0, kde částice interagují s plným Lennard-Jonesovým potenciálem. Načrtněte hodnotu g (r) při mírně nižší teplotě, T = 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 "34 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 "35 Příklad Vypočítejte, jak dlouho bude trvat 30 kDa proteinu difundovat přes E. Coli a HeLa buňku. Protein může být aproximován koulí s poloměrem 2 nm. E. Coli má průměr 1 𝜇m, zatímco HeLa buňka má průměr asi 20 𝜇m. Předpokládejme, že buněčné stěny nemají žádný vliv na difuzi.(Viskozita vody je η = 10−3 N s m−2) "36 Příklad 2 Jak dlouho (v řádu) trvá, aby protein s poloměrem 1 nm dodifundoval z mozku do paže? Předpokládejme, že celý pohyb je uvnitř jedné osy s viskozitou vody 10−3 Pa s a že celková délka je asi 1 m. "37 Příklad - Proteinové interakce Obrázek ukazuje volnou energii interakce, w(r) mezi dvěma nabitými identickými proteiny ve dvou různých koncentracích soli. Když je protein-proteinová separace, r, větší než 50 A, lze předpokládat, že w(r) bude následovat potenciál Debye-Huckel potenciál pro interakci dvou nábojů v soli. 1. Vypočtěte délku Debyeho screeningu, D = 1 / κ, pro 0,016 M roztok NaCl. 2. Použijte w(r) při koncentraci 0,016 M soli (plná křivka) pro stanovení celkového náboje Z proteinu. 3. Je čárkovaná čára na obrázku pro vyšší nebo nižší koncentraci soli než plná čára? 4. Proč jsou w(r) strmé a odpuzující při krátkých separacích protein-protein? 5. Odhadněte poloměr proteinu. ✏r = 80, the salt is sodium chloride and you may neglect the size of the proteins. Note: In Hill’s Chapter 18 “✏” should be replaced with “4⇡✏0✏r” in order to use SI-units. 1. Calculate the Debye screening length, D = 1/, for a 0.016 M NaCl solution. 2. Use w(r) at 0.016 M salt concentration (full drawn curve) to estimate the total charge, Z, of the protein. 3. Is the dashed line in Figure 16 for a higher or lower salt concentration than the full drawn line? Motivate. 4. Why are the w(r)’s in Figure 1 steep and repulsive at short protein-protein separations? 5. Estimate the radius of the protein. 30 40 50 60 70 r in Ångstroms 0 0.5 1 1.5 2 2.5 3 w(r)inkT 0.016 mol/l ? mol/l Figure 16: Protein-protein potential of mean force at two di↵erent salt concentrations. r is the mass center separation between the proteins. 2The entropy of water molecules, that is. 54