Implicit Solvent Models for Simulations

A statistical thermodynamic development is given of a new implicit solvent model that avoids the traditional system size limitations of computer simulation of macromolecular solutions with periodic boundary conditions. This implicit solvent model is based upon the quasi-chemical approach, distinct from the common integral equation trunk of the theory of liquid solutions. The idea is geometrically to define molecular-scale regions attached to the solute macromolecule of interest. It is then shown that the quasi-chemical approach corresponds to calculation of a partition function for an ensemble analogous to, but not the same as, the grand canonical ensemble for the solvent in that proximal volume. The distinctions include: (a) the defined proximal volume — the volume of the system that is treated explicitly — resides on the solute; (b) the solute conformational fluctuations are prescribed by statistical thermodynamics and the proximal volume can fluctuate if the solute conformation fluctuates; and (c) the interactions of the system with more distant, extra-system solution species are treated by approximate physical theories such as dielectric continuum theories. The theory makes a definite connection to statistical thermodynamic properties of the solution and fully dictates volume fluctuations, which can be awkward in more ambitious approaches. It is argued that with the close solvent neighbors treated explicitly the thermodynamic results become less sensitive to the inevitable approximations in the implicit solvent theories of the more distant interactions. The physical content of this theory is the hypothesis that a small set of solvent molecules are decisive for these solvation problems. A detailed derivation of the quasi-chemical theory escorts the development of this proposal. The numerical application of the quasi-chemical treatment to Li ion hydration in liquid water is used to motivate and exemplify the quasi-chemical theory. Those results underscore the fact that the quasi-chemical approach refines the path for utilization of ion-water cluster results for the statistical thermodynamics of solutions. INTRODUCTION Direct simulation of macromolecules in aqueous solutions typically requires consideration of a mass of solution large compared to the mass of the macromolecular solute. Frequently, the bulk of the solution is of secondary interest. The extravagant allocation of computational resources for direct treatment of macromolecular solutions limits the scientific problems that may be tackled. Thus, implicit solvent models that eliminate the direct presence of the solvent in favor of an approximate description of the solvation effects have received universal and extended interest [1–53]. These issues are specifically relevant to this workshop on electrostatic interactions in solution for two reasons. First, electrostatic interactions exacerbate the difficulties of solute size mentioned above. If all interactions were short-ranged, most practitioners would be satisfied with the traditional approach, adopting periodic boundary conditions and empirically examining the system size dependence of their results by performing calculations on successively larger systems. Second, the additional computational requirements to treat genuinely chemical phenomena in solution by in situ electronic structure calculations again limits the problems that can be addressed. In fact, the principal physical concepts involved in electronic structure calculations for chemical problems in liquid water universally involve electrostatic interactions of long range. The important example of metal ion chemistry in proteins combines these points. Although a variety of implicit solvent models have been tried by now in numerous selected applications, they are still limited in fundamental aspects. A helpful recent discussion of important limitations was given by Juffer and Berendsen [20]. They emphasize that implicit solvent models should be significantly simpler than explicit models in view of the approximations and ad hoc features that are accepted. Solution chemistry problems that require direct treatment of electronic degrees of freedom are one kind of problem that must be made significantly simpler to permit a broader computational attack. A striking example is the current ‘ab initio’ molecular dynamics calculations of aqueous solutions of simple ions. They do treat electronic structure issues within simulations but have been typically limited to total system sizes of 16 – 32 water molecules in periodic boundary conditions [54], small systems by current standards with classical simulation models. These small sizes do limit the conclusions that might be drawn; the structure and dynamics of the second hydration shell of a Li ion in liquid water, the example discussed below, undoubtedly requires calculation on systems larger than 32 water molecules. Nevertheless, much can be learned from the study of such small systems, particularly when chemical effects of the interactions of a solute with proximal solvent molecules are the issues of greatest importance [55]. The idea for the developments presented here is aggressively to adopt the JufferBerendsen suggestion that implicit solvent models must be significantly simpler than explicit models and apply that philosophy to the statistical thermodynamic treatment of aqueous solutions. To that end, we acknowledge that some water molecules play a specific, almost chemical, role in these hydration phenomena. We then work out the theory that permits inclusion of a small number of such molecules explicitly. The required theory is a descendent of the quasi-chemical approximations of Guggenheim [56], Bethe [57], and Kikuchi [58]. It is a significant simplification of direct simulation calculations. Roughly described, that theory organizes and justifies treatment of a handful of water molecules essentially as ligands of the macromolecule of interest, letting the more distant solution environment be treated by simple physical approximations such as the popular dielectric continuum models. The plan of this presentation is first to introduce an example, the hydration of the Li ion, that permits a convenient discussion of the quasi-chemical theory. That example illustrates the quasi-chemical pattern for the theory, exemplifies the basic molecular information required to construct quasi-chemical predictions, and offers a simplified derivation based upon a thermodynamic model. Following that we give an extended theoretical development of the quasi-chemical organization of calculations of solvation free energies and then use those theoretical results to suggest an explicit-implicit solvent model for statistical thermodynamic calculations of solutions. FIGURE 1. The minimum energy structure of a Li ion with six water molecules. The inscribed ball identifies the inner shell occupied by the four nearest water molecules. The two further water molecules are in an outer shell. This illustrates the definition of the bonding or inner shell region. EXAMPLE: THE LI ION IN WATER The hydration of atomic ions provides a conceptually simple context in which to consider the quasi-chemical approaches developed below. The previous study of the hydration of ferric ion, Fe(aq), provides one such example [59]. Here we motivate the formal developments of the theory by considering the hydration of the Li ion in dilute aqueous solution and we will discuss the principal theoretical structures in this context first. We anticipate some of the discussion below by noting that Li(aq) proves to be a difficult case in some important respects. Thus, we expect to return to this example in later work. Indeed, the goal of the theoretical developments initiated here is of a sufficiently constructive nature that all pieces of the puzzle needn’t become available at the same time! Quasi-chemical Structure for the Hydration Free Energy The quasi-chemical theory suggests expressing the chemical potential of the lithium ion species, μLi+ , in terms of ideal and non-ideal, or interaction, contributions: