Implicit solvent models

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Abstract

Implicit solvent models for biomolecular simulations are reviewed and their underlying statistical mechanical basis is discussed. The fundamental quantity that implicit models seek to approximate is the solute potential of mean force, which determines the statistical weight of solute conformations, and which is obtained by averaging over the solvent degrees of freedom. It is possible to express the total free energy as the reversible work performed in two successive steps. First, the solute is inserted in the solvent with zero atomic partial charges; second, the atomic partial charges of the solute are switched from zero to their full values. Consequently, the total solvation free energy corresponds to a sum of non-polar and electrostatic contributions. These two contributions are often approximated by simple geometrical models (such as solvent exposed area models) and by macroscopic continuum electrostatics, respectively. One powerful route is to approximate the average solvent density distribution around the solute, i.e. the solute–solvent density correlation functions, as in statistical mechanical integral equations. Recent progress with semi-analytical approximations makes continuum electrostatics treatments very efficient. Still more efficient are fully empirical, knowledge-based models, whose relation to explicit solvent treatments is not fully resolved, however. Continuum models that treat both solute and solvent as dielectric continua are also discussed, and the relation between the solute fluctuations and its macroscopic dielectric constant(s) clarified.

Introduction

Computer simulations in which a large number of solvent molecules are treated explicitly represent one of the most detailed approaches to study the influence of solvation on complex biomolecules [1]. However, a significant computational cost is associated with the large number of solvent molecules required to model a bulk solution. In practical situations, a large fraction of the time is spent calculating a detailed trajectory of the solvent molecules, even though it is primarily the solute behavior that is of interest. Furthermore, despite their cost, computer simulations with explicit solvent molecules are not exempt from approximations. For example, difficulties arise in thermodynamic perturbation free energy calculations involving charged species when long range electrostatic interactions are truncated or summed over an infinite periodic array using Ewald techniques [2].

Partly due to these difficulties, it is desirable to develop different approaches in which the influence of the solvent is incorporated implicitly. Approximate schemes treating the solvent implicitly can provide useful quantitative estimates and remain computationally inexpensive. Such approaches avoid the statistical errors associated with averages extracted from simulations with a large number of solvent molecules. In addition, implicit solvent models can play an important role as conceptual tools for analyzing the results of simulations generated with explicit solvent molecules. A statistical mechanical formulation of implicit solvent is helpful to better understand the nature of solvation phenomena in general.

The goal of this article is to provide an overview of implicit solvent models commonly used in biomolecular simulations. A number of questions concerning the formulation and the development of implicit solvent models are addressed. The article begins with a rigorous formulation of implicit solvent from statistical mechanics. The potential of mean force (PMF) is introduced. A decomposition in terms of non-polar and electrostatic contributions is described. Approximations such as integral equations, scaled particle theory, and classical continuum electrostatics are discussed. Solvent boundary potentials for implicit/explicit mixed schemes are briefly reviewed. Continuum models of both solute and solvent are also described. Lastly, miscellaneous approximations of implicit solvation are discussed. The paper ends with a short summary.

Section snippets

Rigorous formulation of implicit solvent models

First and foremost, it is important to clarify the significance of implicit solvent models from first principles. To this effect, we consider a molecular solute immersed in a solvent in contact with a heat bath at temperature T. It is expected that the system is fluctuating over a large number of configurations. The statistical properties of the system are best characterized in terms of a probability function P(X,Y) [3],P(X,Y)=e−U(X,Y)/kBTdXdYe−U(X,Y)/kBTwhere the complete configuration of the

Relative and absolute values: reversible work

As long as the normalization condition given by Eq. (5)is satisfied, an arbitrary offset constant may be added to W(X) without affecting averages in Eq. (8). The absolute value of the PMF is thus unimportant. For convenience, it is possible to choose the value of the free energy W(X) relative to a reference system in which the solute–solvent interactions are absent. The free energy W(X) may be expressed ase−W(X)/kBT=dYe−[Uu(X)+Uvv(Y)+Uuv(X,Y)]/kBTdYe−Uvv(Y)/kBT.It is customary to write W(X)=Uu

Free energy decomposition

Intermolecular forces are dominated by short-range harsh repulsive interactions, arising from Pauli's exclusion principle, and long-range electrostatic interactions, arising from the non-uniform charge distribution. It is convenient to express the potential energy Uuv(X,Y) as a sum of non-polar and electrostatic contributions,Uuv(X,Y)=U(np)uv(X,Y)+U(elec)uv(X,Y).Such a representation of the microscopic non-bonded interactions is commonly used in most force fields for computer simulations of

Solvent boundary potentials and implicit/explicit mixed schemes

A description in which all atomic and structural details of the solvent molecules are ignored may not always be desirable. In some cases, it may be advantageous to use a mixed scheme which combines an implicit solvent model with a limited number of explicit solvent molecules. For example, one intermediate approach consists in including a small number of explicit solvent molecules in the vicinity of the solute, and representing the remaining bulk with an effective solvent boundary potential 86,

Summary

The microscopic significance of an implicit solvent potential for a solute in a fixed configuration has been clarified. A statistical mechanical formulation of the `potential of mean force' or `solvent free energy surface' provides a robust theoretical framework to express the influence of solvent rigorously. Nonetheless, going beyond such formal considerations, it is clear that any implicit model is an approximation that must be parameterized carefully to yield accurate results. Not

Acknowledgements

Financial support from FCAR (Québec) and CNRS (France) is acknowledged. B.R. is a research fellow of the Medical Research Council of Canada.

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