The potential of mean force

Typical forms of the radial distribution function are shown in Fig. 38 for a liquid of hard core and of Lennard—Jones spheres (using the Percus— Yevick approximation) [447, 449] and Fig. 44 for carbon tetrachloride [452a]. Significant departures from unity are evident over considerable distances. The successive maxima and minima in g(r) correspond to essentially contact packing, but with small-scale orientational variation and to significant voids or large-scale orientational variation in the liquid structure, respectively. Such factors influence the relative location of reactants within a solvent and make the incorporation of the potential of mean force a necessity. 

As discussed in Chap. 6, Sect. 2.3, the potential of mean force should be incorporated via the diffusion and drift equation [eqns. (44) and (137)]. Emeis and Fehder [456a], Northrup and Hynes [103] and Rice 

This rate coefficient is approximately 30% less than the Smoluchoswki value. Finally, it might reasonable have been assumed that the rate of reaction of encounter pairs is not much larger than the diffusion-limited rate. Incorporating the partially reflecting boundary condition gives a steady-state rate coefficient of 

It is interesting to note that eqn. (190) is reminiscent of the steady-state Collins and Kimball rate coefficient [4] [eqn. (27)] with kact replaced by kacig R) and 4ttRD by eqn. (189). Equation (190) for the rate coefficient is significantly less than the Smoluchowski rate coefficient on two counts hydrodynamics repulsion and rate of encounter pair reaction. Had experimental studies shown that a measured rate coefficient was within a factor of two of the Smoluchowski rate coefficient, it would be tempting to invoke partial diffusion control of the reaction rate. The reduction of rate due to hydrodynamic repulsion should be included first and then the effect of moderately slow reaction rates between encounter pairs. 

Finally, it may be noted that the analysis of homogeneous reaction and of escape/recombination probabilities using the kinetic theory of liquids is rather more complex, but can incorporate all these complications in a more natural and fundamental manner. Kapral and co-workers [37, 285, 286] have made considerable progress in this direction and their work is discussed in Chap. 12. 

Northrup and Hynes [103] have remarked that the effects of the potential of mean force as well as hydrodynamic repulsion are very much more apparent in their effect on the survival (and escape) probability of a reactant pair of radicals than their effect on the rate coefficient. For instance, considering the escape probability of Fig. 20, suppose that an escape probability of 0.75 had been determined experimentally. Initial distances of separation Tq = 4i or 312 would have been deduced from the diffusion equation analysis alone or from the diffusion equation with the potential of mean force and hydrodynamic repulsion included. Again, the effect of a moderately slow rate of reaction of encounter pairs further reduces the recombination probability. Consequently, as the inherent uncertainty in the magnitudes of U r), D(r) and feact may be as much as a factor of 2, the estimation of an initial separation distance, Tq, of a radical pair from experimental measurements of escape probabilities may be in doubt by a factor of 30% or more. Careful and detailed analysis of the recombination of radical pairs has been made by Northrup and Hynes 

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The pair correlation fiinction has a simple physical interpretation as the potential of mean force between two particles separated by a distance r... 

The concept of the potential of mean force can be extended to mixtures and solutions. Consider two ions in a sea of water molecules at fixed temperature T and solvent density p. The potential of mean force w r) is the direct interaction between the ions u.j r) = plus the interaction between the ions tln-ough water... 

McMillan-Mayer theory of solutions [1,2], which essentially seeks to partition the interaction potential into tln-ee parts that due to the interaction between the solvent molecules themselves, that due to die interaction between the solvent and the solute and that due to the interaction between the solute molecules dispersed within the solvent. The main difference from the dilute fluid results presented above is that the potential energy u(r.p is replaced by the potential of mean force W(rp for two particles and, for particles of solute in the solvent, by the expression... 

The McMillan-Mayer theory allows us to develop a fomialism similar to that of a dilute interacting fluid for solute dispersed in the solvent provided that a sensible description of W can be given. At the Ihnit of dilution, when intersolute interactions can be neglected, we know that the chemical potential of a can be written as = W (a s) + IcT In where W(a s) is the potential of mean force for the interaction of a solute... 

At low solvent density, where isolated binary collisions prevail, the radial distribution fiinction g(r) is simply related to the pair potential u(r) via g ir) = exp[-n(r)//r7]. Correspondingly, at higher density one defines a fiinction w r) = -kT a[g r). It can be shown that the gradient of this fiinction is equivalent to the mean force between two particles obtamed by holding them at fixed distance r and averaging over the remaining N -2 particles of the system. Hence w r) is called the potential of mean force. Choosing the low-density system as a reference state one has the relation... 

The simple difhision model of the cage effect again can be improved by taking effects of the local solvent structure, i.e. hydrodynamic repulsion, into account in the same way as discussed above for bimolecular reactions. The consequence is that the potential of mean force tends to favour escape at larger distances > 1,5R) more than it enliances caging at small distances, leading to larger overall photodissociation quantum yields [H6, 117]. 

Having separated the dynamical from equilibrium (or, more accurately, quasi-equilibrium) effects, one can readily discover the origin of the activation free energy and define the concept of the potential of mean force by analysis of the expression for the TST rate constant, k in (A3.8.3). The latter can be written as [7]... 

Haynes G R and Voth G A 1993 The dependence of the potential of mean force on the solvent friction consequences for condensed phase activated rate theories J. Chem. Phys. 99 8005... 

Colloidal particles can be seen as large, model atoms . In what follows we assume that particles with a typical radius <3 = lOO nm are studied, about lO times as large as atoms. Usually, the solvent is considered to be a homogeneous medium, characterized by bulk properties such as the density p and dielectric constant t. A full statistical mechanical description of the system would involve all colloid and solvent degrees of freedom, which tend to be intractable. Instead, the potential of mean force, V, is used, in which the interactions between colloidal particles are averaged over... 

However, in many applications the essential space cannot be reduced to only one degree of freedom, and the statistics of the force fluctuation or of the spatial distribution may appear to be too poor to allow for an accurate determination of a multidimensional potential of mean force. An example is the potential of mean force between two ions in aqueous solution the momentaneous forces are two orders of magnitude larger than their average which means that an error of 1% in the average requires a simulation length of 10 times the correlation time of the fluctuating force. This is in practice prohibitive. The errors do not result from incorrect force fields, but they are of a statistical nature even an exact force field would not suffice. 

Besides yielding qualitative information, these biologically and pharmaceutically motivated applications of SMD can also yield quantitative information about the binding potential of the ligand-receptor complex. A first advance in the reconstruction of the thermodynamic potential from SMD data by discounting irreversible work was made by Balsera et al. (1997) as outlined in Sect. Reconstruction of the potential of mean force below. 

In this section we describe the behavior of a ligand subjected to three types of external forces a constant force, forces exerted by a moving stiff harmonic spring, and forces exerted by a soft harmonic spring. We then present a method of reconstruction of the potential of mean force from SMD force measurements employing a stiff spring (Izrailev et al., 1997 Balsera ct al., 1997). 

The potential of mean force is a useful analytical tool that results in an effective potential that reflects the average effect of all the other degrees of freedom on the dynamic variable of interest. Equation (2) indicates that given a potential function it is possible to calculate the probabihty for all states of the system (the Boltzmann relationship). The potential of mean force procedure works in the reverse direction. Given an observed distribution of values (from the trajectory), the corresponding effective potential function can be derived. The first step in this procedure is to organize the observed values of the dynamic variable, A, into a distribution function p(A). From this distribution the effective potential or potential of mean force, W(A), is calculated from the Boltzmann relation ... 

In this chapter we provide an introductory overview of the imphcit solvent models commonly used in biomolecular simulations. A number of questions concerning the formulation and development of imphcit solvent models are addressed. In Section II, we begin by providing a rigorous fonmilation of imphcit solvent from statistical mechanics. In addition, the fundamental concept of the potential of mean force (PMF) is introduced. In Section III, a decomposition of the PMF in terms of nonpolar and electrostatic contributions is elaborated. Owing to its importance in biophysics. Section IV is devoted entirely to classical continuum electrostatics. For the sake of completeness, other computational... 

II. BASIC FORMULATION OF IMPLICIT SOLVENT A. The Potential of Mean Force... 

The function W(X) is called the potential of mean force (PMF). The fundamental concept of the PMF was first introduced by Kirkwood [4] to describe the average structure of liquids. It is a simple matter to show that the gradient of W(X) in Cartesian coordinates is related to the average force. 

BM Pettitt, M Karplus. The potential of mean force surface for the alanine dipeptide in aqueous solution A theoretical approach. Chem Phys Lett 121 194-201, 1985. 

In Section IV.B the energy of an ion was calculated by a simple version of the quasichemical approximation. The same procedure can be used to calculate the potential of mean force 0pinfW of an ion [18], which is the average potential that the ion experiences as a function of its position x in the direction perpendicular to the surface. This consists of two... 

In the two bulk phases the potential of mean force is constant, but it may vary near the interface. The difference in the bulk values of the chemical part is the free energy of transfer of the ion, which in our model is —2mu (we assume u < 0). Let us consider the situation in which the ion-transfer reaction is in equilibrium, and the concentration of the transferring ion is the same in both phases the system is then at the standard equilibrium potential 0oo- In Ihis case the potential of mean force is the same in the bulk of both phases the chemical and the electrostatic parts must balance ... 

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