Differential geometry based solvation model I: Eulerian formulation

Abstract

This paper presents a differential geometry based model for the analysis and computation of the equilibrium property of solvation. Differential geometry theory of surfaces is utilized to define and construct smooth interfaces with good stability and differentiability for use in characterizing the solvent-solute boundaries and in generating continuous dielectric functions across the computational domain. A total free energy functional is constructed to couple polar and nonpolar contributions to the salvation process. Geometric measure theory is employed to rigorously convert a Lagrangian formulation of the surface energy into an Eulerian formulation so as to bring all energy terms into an equal footing. By minimizing the total free energy functional, we derive coupled generalized Poisson-Boltzmann equation (GPBE) and generalized geometric flow equation (GGFE) for the electrostatic potential and the construction of realistic solvent-solute boundaries, respectively. By solving the coupled GPBE and GGFE, we obtain the electrostatic potential, the solvent-solute boundary profile, and the smooth dielectric function, and thereby improve the accuracy and stability of implicit solvation calculations. We also design efficient second order numerical schemes for the solution of the GPBE and GGFE. Matrix resulted from the discretization of the GPBE is accelerated with appropriate preconditioners. An alternative direct implicit (ADI) scheme is designed to improve the stability of solving the GGFE. Two iterative approaches are designed to solve the coupled system of nonlinear partial differential equations. Extensive numerical experiments are designed to validate the present theoretical model, test computational methods, and optimize numerical algorithms. Example solvation analysis of both small compounds and proteins are carried out to further demonstrate the accuracy, stability, efficiency and robustness of the present new model and numerical approaches. Comparison is given to both experimental and theoretical results in the literature.

I Introduction

Among the various components of molecular interactions, electrostatic interactions are of special importance^{6,8,41,44,55,59,71,142–144,169,170} because of their long range and influence on polar or charged molecules – including water, aqueous ions, and amino or nucleic acids. Electrostatic interactions are ubiquitous for any system of charged or polar molecules, such as biomolecules (proteins, nucleic acids, lipid bilayers, sugars, etc.) in their aqueous environment. Electrostatic solute-solvent interactions, therefore, are of central importance in analyzing molecular structure and modeling the intramolecular and intermolecular interactions of macromolecules in simulations. There are two types of solvation models:^{131,142,143,169} explicit solvent models that represent the solvent in molecular or atomic detail, and implicit solvent models that essentially replace the explicit solvent with a dielectric continuum. Each type of methods has its strengths and weaknesses. Although explicit solvent models provide some of the highest levels of detail, they generally require extensive sampling to converge thermodynamic or kinetic properties of interest. On the other hand, implicit solvent models trade detail and some accuracy to eliminate costly sampling of solvent degrees of freedom. Because of their fewer degrees of freedom, implicit solvent methods have become popular for many applications in molecular simulation.^6,8,22,44,55

Solute-solvent interactions are typically described by solvation energies (or closely related quantities): the free energy of transferring the solute from a vacuum to the solvent environment of interest (e.g., water at a certain ionic strength). Solvation energies can be decomposed into polar and nonpolar contributions. It is important to realize that this polar/nonpolar decomposition is arbitrary and, although widely used, has caveats associated with the non-unique nature of the polar and nonpolar processes¹⁰² and the intrinsic coupling between these two components of solvation.^49,50

Solvation free energies can be calculated by a variety of computational methods ranging from very time-consuming quantum mechanical approaches^77,102,129 to simple phenomenological modifications of Coulomb’s law. Traditionally, explicit solvent methods with classical descriptions of intermolecular interactions have been the standard approach for obtaining very detailed descriptions of solvation.¹²⁵ However, such methods require extensive sampling of solvent degrees of freedom which creates a significant computational burden. Implicit solvent methods, as used in the present work, have become popular alternatives to more computationally-expensive approaches although they have a lower accuracy.^{7,9,41,74,131} In implicit solvent methods, an atomic-detail solute is surrounded by continuum solvent modeled by its “pre-equilibrated” effect on the solute.¹³¹

Due to the ubiquitous nature of electrostatics and the aqueous environment common to most biomolecular systems, analysis of molecular solvation and electrostatics is of significant importance to research in chemistry, biophysics, and medicine. Such analysis can be classified into two general types: quantitative analysis for thermodynamic or kinetic observables and qualitative analysis for general characteristics of biomolecular solvation. One of the primary quantitative applications of implicit solvent methods in computational biology and chemistry research has been the calculation of thermodynamic properties. Implicit solvent methods offer the advantage of “pre-equilibrating” the solvent and mobile ions, thus effectively pre-computing the solvent contribution to the configuration integral or partition function for a system.¹³¹ Such pre-equilibration is particularly evident in MM/PBSA models^{103, 122, 151, 156, 176} which combine implicit solvent approaches with molecular mechanics models to evaluate binding free energies from an ensemble of biomolecular structures. Another important and related application of implicit solvent computational methodology is the calculation and assignment of protein titration states.^{3,13,66,90–92,99,104,114,115,115,159,178} Such methods have been used to interpret experimental titration curves, decompose residue contributions to protein-protein and protein-ligand binding energetics, examine structural/functional consequences of RNA nucleotide protonation, and several other applications. Yet another application area for implicit solvent methods is in the evaluation of biomolecular kinetics where implicit solvent models are generally used to provide solvation forces for molecular Langevin dynamics, ^{97, 98, 126, 127, 155} Brownian dynamics,^{52, 61, 101, 138} or continuum diffusion ^{33, 34, 148, 149, 182} simulations.

A major qualitative use of implicit solvent methods in experimental work is the visualization and qualitative analysis of electrostatic potentials on and around biomolecular surfaces.^{6, 10, 124, 171} Visualization of electrostatic potentials was popularized by the availability of software such as Grasp¹²⁴ and is now a standard procedure for the analysis of biomolecular structures with thousands of examples in the literature, including ligand-receptor binding and drug design, protein-nucleic acid complexes, protein-protein interactions, macromolecular assembly, enzymatic mechanism analysis, etc.

The polar solvation energy is generally associated with a difference in charging free energies in vacuum and solvent. A variety of implicit solvent models are available to describe the polar solvation process;^{8, 9, 29, 41, 64, 82, 131, 137, 141, 142, 157, 165, 169, 170} however, the most widely-used methods are currently the Generalized Born^{12, 29, 43, 64, 70, 82, 109, 119, 161, 162, 188} and Poisson-Boltzmann (PB)^{6, 41, 44, 59, 71, 84, 142} models. Generalized Born methods are very fast but are only heuristic models for estimating the polar salvation energies of biomolecular structures. These methods are often used in high-throughput applications such as molecular dynamics simulations.^{12, 29, 39, 43, 55, 82, 119, 143, 162} PB methods can be formally derived from more detailed theories^{18, 72, 112} and offer a somewhat slower, but often more accurate, method for evaluating polar solvation properties.^{12, 39, 118} Additionally, PB techniques are often used to parameterize and assess the accuracy/performance of Generalized Born models.^{39, 43, 118, 118, 161, 163} Finally, unlike most generalized Born methods, PB models provide a global solution for the electrostatic potential and field within and around a biomolecule, therefore making them suited to visualization and other analysis^{20, 40, 42, 52, 61, 95, 139, 149, 166} that require global information about electrostatic properties. The PB equation^{6, 41, 44, 59, 71, 84, 142} is a nonlinear elliptic partial differential equation (PDE) which is solved for the electrostatic potential. It can be linearized for cases where the interactions between mobile ions and the solute electrostatic potential are very weak. This assumption leads to the “linearized PB equation”. The PB theory is approximate and, as a result, has several well-known limitations which can affect its accuracy.^{31, 37, 44, 54, 71, 72, 112, 135, 141, 157, 158, 169} These limitations have been reviewed in previous articles and will only be briefly summarized here. First, most continuum models assume linear and local solvent response.^{18, 54, 141, 169} However, nonlinear solvent response (usually through dielectric saturation or electrostriction), can be important in regions of strong electric field.^{54, 141, 169} Biologically-relevant examples of nonlinear solvent response have been found near highly charged ions, biomolecules, and other interfaces. Nonlocal solvent response generally involves the finite non-zero size of water and its unique hydrogen bonding with solute and other solvent molecules. Such nonlocal response can be important in describing the orientation of water at biomolecular interfaces,²⁶ differing solvation of cations and anions, and the solvation of asymmetric charge distributions. The second major limitation is the mean-field treatment of ions in PB theory.^{71, 72, 112} Mean field models assume that each ion experiences only the average influence of the other ions in solution. Such averaging precludes detailed ion-ion interactions involving steric repulsion of ions (or their solvation shells) and Coulombic interaction of ions, including repulsion and attractive pairing. The mean field assumption thereby eliminates correlations and fluctuations which can have important energetic and structural consequences for solutions of divalent and multivalent ions surrounding highly-charged molecules such as nucleic acids.^{31, 37, 135, 157, 158} As suggested by the limitations above, PB models also neglect detailed ion-solvent interactions which eliminate differences between ion species in solution and thereby prevent effects analysis of specific ion species – which can be important in biophysical modeling. However, despite these limitations, PB methods are still very important for biomolecular structural analysis, modeling, and simulation. Furthermore, these limitations are currently being addressed through new implicit solvent models^{5, 37, 112, 123, 157} and hybrid treatments^{11, 87, 110, 117, 164} which extend the applicability of PB theory while preserving some of its computational efficiency through pre-averaging solvent and ion response.

PB methods provide polar solvation energies and therefore must be complemented by nonpolar solvation models to provide a complete view of biomolecular solvent-solute interactions. Nonpolar solvation is generally associated with the insertion of the uncharged solute into solvent. There are many nonpolar solvation models available; however, recent work by Levy, Gallicchio, and others^62–64,89 as well as our own research¹⁶⁷ has demonstrated the importance of nonpolar implicit solvent models which include treatment of attractive solute-solvent dispersion terms as well as models of solvent-solvent repulsive interactions that include both area and volume contributions.¹⁶⁷

All implicit solvent models require an interface definition to indicate the separation of solute atoms from the surrounding solvent. In the context of the PB equation the solute-solvent boundary is used to define the dielectric constant and ion accessibility coefficients. For nonpolar models the solute-solvent boundary is used to define the solvent accessible domain which, in turn, defines the area and volume. The van der Waals surface, the solvent accessible surface,⁸⁶ and the molecular surface (MS)¹³⁰ are often used for this purpose. All of the physical properties of interest, including electrostatic free energies, biomolecular surface areas, molecular cavitation volumes, solvation free energies, and pK_a values are very sensitive to the interface definition.^{45, 47, 116, 152} These surface definitions have been found successful in biomolecular modeling;^{19, 38, 48, 79, 83, 94, 96, 150} however, these surfaces are simply ad hoc divisions of the solute and solvent regions of the problem domain; none of them takes into account minimization of interfacial free energies during equilibrium solvation.

The first partial differential equation (PDE) based molecular surface was constructed by Wei el al. in 2005.¹⁷⁵ Unlike the commonly used PDE based surface smoothing techniques which start with a given surface, this approach embeds the atomic information, i.e., atomic coordinates and radii, instead of a given surface, in the Eulerian formulation, and generates hypersurfaces by curvature controlled PDEs. The biomolecular surface is subsequently extracted from the hypersurface by a level-set approach.¹⁷⁵ This approach produces well defined molecular surfaces for both small molecules and large proteins. The true physical boundary of a biomolecule in solvent, as a physical concept, should be in general determined by the optimization of the free energy of the macromolecule in the aquatic environment. This issue was addressed by a variational derivation of the minimal molecular surface (MMS) that minimizes a surface free energy functional by the mean curvature flow model in 2006.^{15, 16} As in the classical theory of minimal surfaces, the mean curvature flow minimizes the surface area for a given constraint. For simple models of solute-solvent interactions, interfacial free energy minimization is often equivalent to surface area minimization, and gives rise to the MMS, which was the first biomolecular surface that has ever been constructed by the variational approach. Electrostatic solvation free energies of 26 proteins have been calculated by using MMSs.¹⁷ To our knowledge, our geometric PDE based models^{15–17, 175} are the first of their kind for biomolecular surfaces, electrostatics and solvation modeling. More recently, we have presented a general procedure for the formation and construction of biomolecular surfaces by balancing the geometric curvature effects and potential effects.¹⁴ This formalism enable the incorporation of microscopic interactions, such as van der Waals potentials, into the macroscopic curvature description. The mathematical structure of this approach was prototyped by one of the present authors in 1999.¹⁷² Most recently, differential geometry based multiscale models have been proposed to describe the dynamics and transports of chemical and biological systems, including fuel cells, ion channels, DNA packing, nanofluidic systems, and virus evolution.^{27, 173}

Geometric flows,¹⁷⁷ particularly mean curvature flows, have been of interest in applied mathematics for many years with an emphasis on image analysis, material design^{57, 68, 106, 120, 134, 136, 140, 147} and surface processing.¹⁸³ Computational techniques using the level set formalism were devised by Osher and Sethian^{132, 140} and have been applied by many others.^{25, 35, 146} The impact of the level set theory is far beyond applied mathematics. An alternative approach is to minimize the mean curvature or the energy functional of the characteristic function in the framework of the Mumford-Shah variational functional,¹¹¹ and the Euler-Lagrange formulation of surface variation.^{21, 24, 93, 121, 132, 133} In 1999, Wei introduced some of the first high-order geometric flow equations for image analysis.¹⁷² Coupled geometric flow equations were also proposed by Wei and Jia¹⁷⁴ for image edge detection and feature extraction.

While implicit solvent models are very efficient, they are approximate treatments of the solvent environment around a solute assuming linear and local solvent response to all solute perturbations.^{18, 46, 131, 169} As such, they can fail to accurately describe solvent behavior in situations where nonlinear or nonlocal solvent response is important to the phenomenon of interest or where the atomic details of solute-solvent interactions are needed.^{1, 2, 4, 8, 23, 26, 30, 36, 49–51, 58, 60, 75, 76, 81, 85, 88, 107, 108, 128, 145, 153, 154, 160, 167} Limitations of the PB equation have been indicated above and follow directly from these assumptions of linearity and locality in solvent-solute interactions. Similar limitations are present for the nonpolar models discussed. However, despite their shortcomings, implicit solvent models have many uses in a variety of chemical, physical, and biomedical research fields and therefore continue to be an important area of computational research.

Yet, an important problem in implicit solvent models is the lack of sufficient descriptions of polar-nonpolar coupling and solvent-solute interactions.^{4, 23, 26, 36, 49, 58, 60} This problem has recently considered by Dzubiella et al⁴⁹ who proposed an interesting free energy optimization procedure to couple polar-nonpolar interactions. Their model includes contributions from pressure, Gauss and mean curvatures, van der Waals interactions and electrostatic effects. Biomolecular surfaces were generated from this model via the level set approach³² which is similar to our earlier Eulerian geometric flow approaches of biomolecular surfaces and solvation analysis.^{15–17, 175}

The use of the PB theory in computational biology, chemistry, and physics still faces a number of challenges – many of which are related to the definition of the solute-solvent interface. First, there is significant controversy over the choice of solute-solvent interface.^{45, 47} This controversy stems, in part, from the ad hoc nature of the current popular surface definitions. Second, many popular surface definitions lack the stability and differentiability for routine use in molecular simulations due to extreme sensitivity to atomic positions, radii, etc. This sensitivity often drives the use of alternative “smoothed” solvent-solute interface definitions^{69, 78} which can introduce additional computational artifacts.¹⁵² Finally, the wide range of surface definitions has led to confusion and misuse of parameter (radii) sets developed for implicit solvent calculations with specific surface definitions.

The objective of the present work is to introduce a differential geometry based approach for the generation of smooth surfaces with good stability and differentiability for use in both polar and nonpolar modeling, thereby addressing many of these current challenges. We propose to optimize a polar and nonpolar solvation free energy functional to drive the construction of realistic solvent-solute boundaries and thereby improve the accuracy and stability of implicit solvent calculations. The differential geometry theory of surfaces and manifolds is employed to result in new coupled geometric and potential flows for the generation of a physical solvent-solute boundary and the optimization of solvation energy. Technically, the smoothness of the resulting solute-solvent boundary is ensured by coupled geometric and potential flows of parabolic type. Computational methods and algorithms are constructed and carefully validated to ensure their accuracy, stability, and efficiency. The proposed solute-solvent boundary model and associated free energy functional are tested by their applications in several common biomolecular modeling tasks.

The rest of this paper is organized as follows. Section §II is devoted to the theoretical foundation of the present differential geometry based solvation model. A variational framework is established to couple different parts of the solvation contributions. Governing equations are derived by variational principles. The solution of the governing equations leads to physical solvent-solute boundaries and accurate solvation free energies. Numerical methods and algorithms are discussed in Section §III. Schemes of the second order numerical accuracy are designed for the construction and evolution of solute characteristic function. Appropriate preconditioners are used for solving the generalized Poisson-Boltzmann equations. The coupled equations are solved by two iterative schemes. Section §IV presents validation and analysis of the proposed numerical approaches. The accuracy and convergence of various computational schemes, including the surface area formulation based on the geometric measure theory, are carefully tested to ensure their computational reliability and efficiency. The applications of the proposed theories, methods and algorithms are considered to two sets of compounds: small molecules and proteins in Section §V. Comparison is given to results in the literature. Finally, this paper ends with concluding remarks.

II Differential geometry based solvation model

In this section, a differential geometry based model of solvation is briefly described for macromolecules and their aquatic environment that are near equilibrium. More details about the differential geometry based multiscale formalism, particularly dynamics and transport aspects, can be found elsewhere.¹⁷³ For a system near equilibrium, the density of charged particles in the solvent can be approximated by the Boltzmann distribution, which considerably reduces the number of degrees of the freedom of the solvation system. Alternatively, the Nernst-Planck equations or the full set of the Navier-Stokes equations might be utilized to describe systems that are far from the equilibrium.¹⁷³

II.A Solute-solvent boundary

Let us consider a multi-domain setting of a macromolecule and solvent system. The macromolecule is described in discrete atomic detail, while the aqueous solvent is treated as a continuum. Therefore, the domain Ω ∈ ℝ³ is essentially divided into two (types of ) regions, i.e., aqueous solvent domain Ω_s and macromolecular domain Ω_m. Therefore, one has Ω = Ω_s ∪ Ω_m. However, because electron wavefunctions of the solvent and the solute overlap at the atomic scale, Ω_s and Ω_m should overlap with each other at the boundary of molecules and solvent, i.e., Ω_b = Ω_s ∩ Ω_m ≠ ⊘, where Ω_b is the region of solvent-solute boundary. Therefore, we propose a characteristic function S : ℝ³ → ℝ to characterize this overlapping solvent-solute boundary. As such, S(x) is a description function or a characteristic function of the solute domain, i.e., it is one (S = 1) inside the biomolecule and zero (S = 0) in the aquatic solvent. However, S takes a value between zero and one at the solvent-solute boundary region. Therefore, (1 − S) is a description function or a characteristic function for the solvent domain. The profiles of S and (1 − S) are depicted in Fig. 1 for a simple system. It is seen that there is a transition region at the solvent-solute boundary where the solvent and the solute regions overlap. Obviously, in our model, the evaluation of all the solvent-solute properties depends on S. Physically, S and thus the profile of solvent-solute boundary, must be determined by the energy minimization principle. Therefore, our task is to identify the energy functional that to be optimized. This task is accomplished via the differential geometry of surfaces and manifolds in the present work.

Figure 1.

The cross line of S and (1 − S) of a diatomic system described in Section IV.C

II.B Total free energy functional

The solvation process of macromolecules involves a number of interactions. As discussed in the Introduction, typically, the free energy of solvation models consists of polar and nonpolar contributions, as well as polar and nonpolar interactions.

II.B.1 Polar free energy functional

The polar part is standardly represented by electrostatic interactions, which are of special importance because of their long range and influence on polar or charged molecules including water, aqueous ions, and amino or nucleic acids. They are also some of the most important aspects that determine the physical and chemical properties of biomolecules, such as protein folding, protein-DNA binding, gene expression and regulation, etc. The widely used free energy functional of the electrostatic system was given by Sharp and Honig¹⁴² and Gilson et al.⁶⁷ However, their formulation is based on a sharp interface that separates the solvent and solute domains. In our formulation, we incorporate the function S into the polar solvation free energy functional

$$G_{\mathrm{p}}={\displaystyle {\int }_{\mathrm{\Omega }}\left\{S\left[{\rho }_m\varphi -\frac{1}{2}{\epsilon }_m{|\nabla \varphi |}^2\right]+(1-S)\left[-\frac{1}{2}{\epsilon }_s{|\nabla \varphi |}^2-k_{B}T{\displaystyle \sum _{i=1}^{N_c}{c}_i(e^{-\varphi q_i/k_{B}T}-1)}\right]\right\}d\mathbf{r}}$$ (1)

where ϕ is the electrostatic potential whose domain is the whole computational domain Ω, and ε_s and ε_m are the dielectric constants of the solvent and solute, respectively. Here ${\rho }_m={\displaystyle {\sum }_j{Q}_j\delta (\mathrm{r}-{\mathrm{x}}_j)}$ is the density of molecular charges, with Q_j being the partial charge on an atom located at x_j, q_i is the charge of ion species i, N_c is the number of ion species, k_B is the Boltzmann constant, T is the temperature, and c_i is the bulk concentration of the ith ionic species. The term associated with S is the electrostatic free energy of the solute and that with (1 − S) is the electrostatic free energy of the solvent.

The above electrostatic free energy functional is inherently multidomain in nature and the domain is divided into the solute subdomain and the solvent subdomain as indicated by S and 1 − S, respectively. These subdomains do not have to be mutually exclusive. A discrete description of the solute and a continuum description of the solvent are also employed in Eq. (1) in the framework of the implicit solvent treatment, in which the charge density of mobile ions follows the Boltzmann distribution. Moreover, it will be demonstrated that the present electrostatic free energy functional is able to reproduce the classical Poisson-Boltzmann equation when a sharp solvent-solute interface is used and S becomes a Heaviside function. Finally, we note that the terms that are quadratic in the potential gradient in Eq. (1) have negative signs. Therefore, the free energy will be optimized instead of being minimized. In this work, we have adopted the earlier sign convention in the field.^{67, 142}

II.B.2 Non-polar free energy functional

For the nonpolar contribution, we consider the following nonpolar solvation free energy functional proposed by Wagoner and Baker.¹⁶⁷

$$G_{\text{np}}=\gamma (\text{Area})+p(\text{Vol})+{\rho }_0{\displaystyle {\int }_{{\mathrm{\Omega }}_s}{U}^{\text{att}}d\mathbf{r}}$$ (2)

where Area is the surface area of the macromolecule, γ is the surface tension, Vol represents the volume occupied by the molecule of interest, p is the hydrodynamic pressure, ρ₀ is the solvent bulk density and U^att(r) is the attractive portion of the van der Waals potential at point r. The first term is the surface energy. It measures the disruption of intermolecular and/or intramolecular bonds that occurs when a surface is created. The second term is the mechanical work of creating the vacuum of a biomolecular size in the solvent. The third term represents the attractive dispersion effects near the solvent-solute interface which has been shown by Wagoner and Baker¹⁶⁷ to play a crucial role in accurate nonpolar solvation analysis.

To obtain a functional relation for S, it is necessary to rewrite nonpolar free energy formulation in terms of S(r). The enclosed volume of biomolecule can be given by

$$\text{Vol}={\displaystyle {\int }_{{\mathrm{\Omega }}_m}d\mathbf{r}={\displaystyle {\int }_{\mathrm{\Omega }}S(\mathbf{r})d\mathbf{r}.}}$$ (3)

Similarly the attractive dispersion term can be rewritten in the form

$${\rho }_0{\displaystyle {\int }_{{\mathrm{\Omega }}_s}{U}^{\text{att}}d\mathbf{r}={\rho }_0{\displaystyle {\int }_{\mathrm{\Omega }}(1-S(\mathbf{r}))U^{\text{att}}d\mathbf{r},}}$$ (4)

where we assume that the solvent bulk density ρ₀ is a constant in space.

Typically, one expresses the area of a unique surface as a surface integration over the biomolecular boundary in the Lagrangian formulation. However, this approach does not work directly in our formulation because no sharp solvent-solute boundary is assumed. In fact, the concept of the surface area cannot be defined in the same manner as in the sharp surface case. For a smooth boundary, there are infinitely many surfaces and the surface area can be defined as a weighted mean of this family of surfaces. Additionally, for practical purpose, we need an appropriate Eulerian formulation for the surface area so that we can put all energy contributions into an equal footing. Therefore, we need to convert the surface integral into a volume one. To this end, we make use of the coarea formula in the geometric measure theory.⁵³ For a scalar field B in ℝ³, with C¹ continuity condition, integrating a function f over its isolevel c in a region Ω can be done directly by a volume integral over Ω through the expression

$${\displaystyle {\int }_{ℝ}{\displaystyle {\int }_{B^{-1}\cap \mathrm{\Omega }}fd\sigma dc={\displaystyle {\int }_{\mathrm{\Omega }}\Vert \nabla B\Vert f(\mathbf{r})d\mathbf{r},}}}$$ (5)

where c denotes an isovalue of B, and B⁻¹ represents the c-isosurface, i.e., the set of points {r_c} such that B(r_c) = c. Here, the coarea formula prescribes a relationship between the sum of area integrals and a global volume integral. In our case, we introduce the concept of mean surface area of the family of isosurfaces which are subsets of point satisfying {S(r) = y}, where 0 ≤ y ≤ 1. Therefore the mean surface area can be given by a volume integral as

$$\begin{array}{cc}\text{Area}\hfill & ={\displaystyle {\int }_0^1{\displaystyle {\int }_{S^{-1}(c)\cap \mathrm{\Omega }}d\sigma dc}}\hfill \\ \hfill & ={\displaystyle {\int }_{\mathrm{\Omega }}\Vert \nabla S(\mathbf{r})\Vert d\mathbf{r}.}\hfill \end{array}$$ (6)

Note that ∇S ≠ 0 only in the region of the solvent-solute boundary. Numerical test of this formulation will be presented in Section §IV.A.

Finally, the electrostatic free energy functional is complemented by the nonpolar free energy functional to give the total free energy functional of solvation for biomolecules at equilibrium

$$\begin{array}{cc}{G}_{\text{total}}\hfill & ={\displaystyle {\int }_{\mathrm{\Omega }}\mathrm{\gamma }\Vert \nabla S(\mathbf{r})\Vert +pS(\mathbf{r})+{\rho }_0(1-S(\mathbf{r}))U^{\text{att}}}\hfill \\ \hfill & +S(\mathbf{r})\left[{\rho }_m\varphi -\frac{1}{2}{\epsilon }_m{\left|\nabla \varphi \right|}^2\right]\hfill \\ \hfill & +(1-S(\mathbf{r}))\left[-\frac{1}{2}{\epsilon }_s{\left|\nabla \varphi \right|}^2-k_{B}T{\displaystyle \sum _{i=1}^{N_c}{c}_i(e^{-\varphi q_i/k_{B}T}-1)}\right]d\mathbf{r}.\hfill \end{array}$$ (7)

Note that the polar and nonpolar parts are coupled via the characteristic function S, which is determined by the total energy optimization instead of the surface free energy optimization as done in our earlier work.¹⁷ The above total free energy expression provides a basis for the evaluation of the solvation free energy and a starting point for the derivation of governing equations for the solvation analysis. A similar coupling of polar and nonpolar interactions was described previously by Dzubiella and co-workers;^{49, 50} however, the implementation of non-polar interactions and the representation of continuum and discrete domains differ significantly from the present work.

II.C Governing equations

The solvation free energy functional is a functional in terms of characteristic function S and potential ϕ. The integral is taken over the whole space. From the physical point of view, there should exist an optimal function S(r) and an optimal potential ϕ at the equilibrium state which optimize the total energy. Since S and ϕ can vary independently in our formulation, to optimize G_total, it is necessary that

$$\frac{\mathrm{\delta }{G}_{\text{total}}}{\mathrm{\delta }\varphi }\Rightarrow S{\rho }_m+\nabla \cdot ([(1-S){\epsilon }_s+S{\epsilon }_m]\nabla \varphi )+(1-S){\displaystyle \sum _{i=1}^{N_c}{c}_i{q}_i{e}^{-\varphi qi/k_{B}T}=0}$$ (8)

and

$$\frac{\delta G_{\text{total}}}{\delta S}\Rightarrow -\nabla \cdot \left(\mathrm{\gamma }\frac{\nabla S}{\Vert \nabla S\Vert }\right)+p-{\rho }_0{U}^{\text{att}}+{\rho }_m\varphi -\frac{1}{2}{\epsilon }_m{|\nabla \varphi |}^2+\frac{1}{2}{\epsilon }_s{|\nabla \varphi |}^2+k_{B}T{\displaystyle \sum _{i=1}^{N_c}{c}_i(e^{-\varphi q_i/k_{B}T}-1)=0,}$$ (9)

where $\nabla \cdot \left(\gamma \frac{\nabla S}{\Vert \nabla S\Vert }\right)$ is a generalized Laplace-Beltrami operator, which is a generalization of the usual Laplacian operator to a smooth manifold.^{14, 173} In general, γ can be a function of the position γ = γ(r) to reflect surface hydrophobicity at different locations. However, it is treated as a constant in our present computation. From Eq. (8) we result in the generalized Poisson-Boltzmann equation (GPBE)

$$-\nabla \cdot (\epsilon (S)\nabla \varphi )=S{\rho }_m+(1-S){\displaystyle \sum _{i=1}^{N_c}{c}_i{q}_i{e}^{-\varphi q_i/k_{B}T}},$$ (10)

where the dielectric function is given by

$$\epsilon (S)=(1-S){\epsilon }_s+S{\epsilon }_m.$$ (11)

This expression provides a smooth dielectric profile. Figure 2 shows the cross line of the dielectric function ε(S) of a diatomic system. It is seen that there is a smooth transition region for the dielectric constant to change from ε_s to ε_m. The solution procedure of Eq. (10) can differ very much from that of the standard PB equation, due to the smooth dielectric function. Particularly, many mathematical difficulties of solving elliptic equations with discontinuous oefficients^{180, 181, 184, 186, 187} can be avoided in the present generalized Poisson-Boltzmann equation.

Figure 2.

The cross line profile of ε(S) of a diatomic system described in Section IV.C. Here, we have set ε_s = 80 and ε_m = 1.

For a weak electrostatic potential, i.e., ϕ ≪ 1, one can linearize the generalized PB equation

$$-\nabla \cdot (\epsilon (S)\nabla \varphi )+(1-S){\overline{\kappa }}^2\varphi =S{\rho }_m,$$ (12)

where κ̄ is a modified Debye-Hϋckel screening function describing the ion strength.⁷³

Furthermore, the solution of Eq. (9) leads to a “physical solvent-solute boundary” (S). As discussed in earlier work,^{14, 17, 173} the solution of this elliptic partial differential equation can be attained via a parabolic partial differential equation

$$\frac{\partial S}{\partial t}=\Vert \nabla S\Vert \left[\nabla \cdot \left(\mathrm{\gamma }\frac{\nabla S}{\Vert \nabla S\Vert }\right)+V\right],$$ (13)

where the generalized “potential” V is defined as

$$V=-p+{\rho }_0{U}^{\text{att}}-{\rho }_m\varphi +\frac{1}{2}{\epsilon }_m{\left|\nabla \varphi \right|}^2-\frac{1}{2}{\epsilon }_s{\left|\nabla \varphi \right|}^2-k_{B}T{\displaystyle \sum _{i=1}^{N_c}{c}_i(e^{-\varphi q_i/k_{B}T}-1).}$$ (14)

Note that Eq. (13) has the same differential operator as the mean curvature flow equation,¹⁷ except for the extra external source term. Therefore, it is a special case of the potential driven geometric flow equation proposed in our earlier work.¹⁴ In Eq. (13), as t → ∞, the initial profile of S evolutes into a steady state solution, which solves the original Eq. (9).

It is interesting to see that the sharp solvent-solute interface and the standard PB equation, as well as related interface conditions, can be recovered from Eq. (10). For a sharp interface, S becomes a Heaviside function, having value 1 for the solute subdomain and 0 for the solvent subdomain. As such, the smooth transition region in the dielectric function disappears and the dielectric function becomes discontinuous. Then, Eq. (10) reduces to the classical form of the Poisson-Boltzmann equation⁷³

$$\begin{array}{cc}-{\epsilon }_m{\mathrm{\nabla }}^2{\varphi }_m={\rho }_m\hfill & \forall \mathbf{r}\in {\mathrm{\Omega }}_m\hfill \\ -{\epsilon }_s{\mathrm{\nabla }}^2{\varphi }_s={\displaystyle {\sum }_j{q}_j{c}_j{e}^{(-{\varphi }_s{q}_j/k_{B}T)}}\hfill & \mathrm{\forall }\mathbf{r}\in {\mathrm{\Omega }}_s\hfill \end{array}$$

with appropriate interface conditions

$${\varphi }_s={\varphi }_m,\text{and}\hspace{1em}{\epsilon }_m\nabla {\varphi }_m\cdot \mathbf{n}={\epsilon }_s\nabla {\varphi }_s\cdot \mathbf{n}\forall \mathbf{r}\in \mathrm{\Gamma },$$ (15)

where ϕ_m and ϕ_s represent the potential in the solute domain Ω_m and solvent domain Ω_s, respectively, Γ denotes the sharp interface, and n is the normal vector of the solvent-solute sharp interface.

Note that the generalized Poisson-Boltzmann Eq. (10) and the potential driven geometric flow equation (13) are strongly coupled. Therefore, these two equations have to be solved by appropriate iterative procedures. This aspect will be discussed in Section III.C.

III Methods and algorithms

This section presents a variety of computational methods and algorithms for the solution of the generalized Poisson-Boltzmann equation and the generalized geometric flow equation.

III.A Discretization schemes of the governing equations

We design second order finite difference schemes for governing equation derived from the free energy optimization.

III.A.1 The generalized Poisson-Boltzmann equation

For the solution of the generalized PB equation, the finite difference scheme is utilized in this study. The continuous dielectric definition allows us to obtain accurate solution by using the standard second order center difference scheme. Let the pixel (i, j, k) represent the position (x_i, y_j, z_k). The discretized form of Eq. (12) is

$$\epsilon (x_i+\frac{1}{2}h,y_j,z_k)[\varphi (i+1,j,k)-\varphi (i,j,k)]+\epsilon (x_i-\frac{1}{2}h,y_j,z_k)[\varphi (i-1,j,k)-\varphi (i,j,k)]+\epsilon (x_i,y_j+\frac{1}{2}h,z_k)[\varphi (i,j+1,k)-\varphi (i,j,k)]+\epsilon (x_i,y_j-\frac{1}{2}h,z_k)[\varphi (i,j-1k)-\varphi (i,j,k)]+\epsilon (x_i,y_j,z_k+\frac{1}{2}h)[\varphi (i,j,k+1)-\varphi (i,j,k)]+\epsilon (x_i,y_j,z_k-\frac{1}{2}h)[\varphi (i,j,k-1)-\varphi (i,j,k)]-4(1-S(i,j,k)){\overline{\kappa }}^2\varphi (i,j,k)h^2=-4S(i,j,k)q(i,j,k)/h$$ (16)

where h is the grid spacing, and q(i, j, k) is the fractional charge at grid point (x_i, y_j, z_k), which is resulted from the interpolation of the charge density ρ_m. The second order interpolation (i.e., the trilinear mapping) is used to distribute charges. Thus, the discretized PB equation can be cast into the standard linear algebraic equation system of the form AX = B, where X is the solution of the interest, A is the discretization matrix and B is the source term associated with the continuum and discrete charges. The boundary condition is built up by the far field condition and practically obtained by the sum of potential contributions from individual atom charges with a decay factor from the continuum charge strength κ.⁶⁵ Initially, we have explored the use of the biconjugate gradient method as the linear solver. Matrix acceleration is discussed in a later section. The initial guess of the solution is set to 0 and the convergence tolerance is set as 10⁻⁶. It is shown in the test section that the PB solver is able to deliver the designed second-order accuracy.

III.A.2 The potential driven geometric evolution equation

To attain the solution of geometry flow equation (13), one has to determine all of the involved physical parameters first. Some parameters in the literature,^{89, 113} including the CHARMM force field, can be adopted for this purpose. However, due to the nonpolar solvation energy expression in our model, not all parameters can be adopted from the literature. Some modifications are necessary. In particular, surface tension γ should vary according to the surface definition.^{89, 113} Therefore, it is used as a fitting parameters in our model. To this end, we write

$$\begin{array}{cc}\frac{\partial S}{\partial t}\hfill & =\Vert \nabla S\Vert \left[\nabla \cdot \left(\mathrm{\gamma }\frac{\nabla S}{\Vert \nabla S\Vert }\right)+V\right],\hfill \\ \hfill & =\mathrm{\gamma }\Vert \nabla S\Vert \left[\nabla \cdot \left(\frac{\nabla S}{\Vert \nabla S\Vert }\right)+V_{\mathrm{\gamma }}\right]\hfill \end{array}$$ (17)

where $V_{\mathrm{\gamma }}=\frac{V}{\mathrm{\gamma }}$ with γ as a nonzero parameter and V being defined in Eq. (14). In addition to the Lennard Jones parameters ε_i, σ_s and σ_i, what will be determined in the generalized geometry flow equation are p/γ, ρ0/γ, ε_s/γ, ε_p/γ and ρ_m/γ. Note that since ρ_m/γ is assigned at the center of atoms, term (ρ_m/γ) ϕ has nonzero value only at atomic center so that it does not have contributions to the surface evolution when S evolves only outside the van der Waals volume.

The discretization scheme used here for the solution of the generalized geometry flow equation (17) is similar to what we designed previously.^{14, 17} It can be rewritten in the form

$$\begin{array}{cc}\frac{\partial S}{\partial t\prime }\hfill & =\frac{(S_x^2+S_y^2)S_{\mathit{\text{zz}}}+(S_x^2+S_z^2)S_{\mathit{\text{yy}}}+(S_y^2+S_z^2)S_{\mathit{\text{xx}}}}{S_x^2+S_y^2+S_z^2}\hfill \\ \hfill & -\frac{2S_x{S}_y{S}_{\mathit{\text{xy}}}+2S_x{S}_z{S}_{\mathit{\text{xz}}}+2S_z{S}_y{S}_{\mathit{\text{yz}}}}{S_x^2+S_y^2+S_z^2}+\sqrt{S_x^2+S_y^2+S_z^2}{V}_{\mathrm{\gamma }},\hfill \end{array}$$ (18)

where t′ = tγ. To obtain the discretized form, we introduce the following notations. We consider a discrete time t_n := nτ where n is a non-negative integer and τ is the time stepping size. We denote $S_{\mathit{\text{ijk}}}^n$ to be the discretized form of S(x_i, y_j, z_k, t_n). An explicit scheme of the generalized geometry flow equation is given by

$$S_{\mathit{\text{ijk}}}^{n+1}-S_{\mathit{\text{ijk}}}^n:=[{\upsilon }_x{\delta }_x^2+{\upsilon }_y{\delta }_y^2+{\upsilon }_z{\delta }_z^2]S_{\mathit{\text{ijk}}}^n+\tau f_{\mathit{\text{ijk}}}^n,$$ (19)

where

$$\begin{array}{c}\hfill f_{\mathit{\text{ijk}}}^n={\left\{-2\frac{S_x{S}_y{S}_{\mathit{\text{xy}}}+S_x{S}_z{S}_{\mathit{\text{xz}}}+S_z{S}_y{S}_{\mathit{\text{yz}}}}{S_x^2+S_y^2+S_z^2}+\sqrt{S_x^2+S_y^2+S_z^2}{V}_{\gamma }\right\}}_{\mathit{\text{ijk}}}^n\hfill \\ \hfill {\upsilon }_x=\tau {\left\{\frac{(S_y^2+S_z^2)}{S_x^2+S_y^2+S_z^2}\right\}}_{\mathit{\text{ijk}}}^n,\hfill \\ \hfill {\upsilon }_y=\tau {\left\{\frac{(S_x^2+S_z^2)}{S_x^2+S_y^2+S_z^2}\right\}}_{\mathit{\text{ijk}}}^n,\hfill \\ \hfill {\upsilon }_z=\tau {\left\{\frac{(S_x^2+S_y^2)}{S_x^2+S_y^2+S_z^2}\right\}}_{\text{ijk}}^n,\hfill \end{array}$$

where

$$\begin{array}{c}\hfill {\delta }_x^2{S}_{\mathit{\text{ijk}}}^n=(S_{(i-1)\mathit{\text{jk}}}^n-2S_{\mathit{\text{ijk}}}^n+S_{(i+1)\mathit{\text{jk}}}^n)/h^2\hfill \\ \hfill {\delta }_y^2{S}_{\mathit{\text{ijk}}}^n=(S_{i(j-1)k}^n-2S_{\mathit{\text{ijk}}}^n+S_{i(j+1)k}^n)/h^2\hfill \\ \hfill {\delta }_z^2{S}_{\mathit{\text{ijk}}}^n=(S_{\mathit{\text{ij}}(k-1)}^n-2S_{\mathit{\text{ijk}}}^n+S_{\mathit{\text{ij}}(k+1)}^n)/h^2\hfill \\ \hfill {\{S_x\}}_{\mathit{\text{ijk}}}^n=(S_{(i+1)\mathit{\text{jk}}}^n-S_{(i-1)\mathit{\text{jk}}}^n)/2h\hfill \\ \hfill {\{S_y\}}_{\mathit{\text{ijk}}}^n=(S_{i(j+1)k}^n-S_{i(j-1)k}^n)/2h\hfill \\ \hfill {\{S_z\}}_{\mathit{\text{ijk}}}^n=(S_{\mathit{\text{ij}}(k+1)}^n-S_{\mathit{\text{ij}}(k-1)}^n)/2h\hfill \\ \hfill {\{S_{\mathit{\text{xy}}}\}}_{\mathit{\text{ijk}}}^n=(S_{(i+1)(j+1)k}^n+S_{(i-1)(j-1)k}^n-S_{(i+1)(j-1)k}^n-S_{(i-1)(j+1)k}^n)/4h^2\hfill \\ \hfill \{S_{\mathit{\text{xz}}}\}{S}_{\mathit{\text{ijk}}}^n=(S_{(i+1)j(k+1)}^n+S_{(i-1)j(k-1)}^n-S_{(i+1)j(k-1)}^n-S_{(i-1)j(k+1)}^n)/4h^2\hfill \end{array}$$

and

$${\{S_{\mathit{\text{yz}}}\}}_{\mathit{\text{ijk}}}^n=(S_{i(j+1)(k+1)}^n+S_{i(j-1)(k-1)}^n-S_{i(j+1)(k-1)}^n-S_{i(j-1)(k+1)}^n)/4h^2.$$

For the initial value of S, we consider

$$S(x,y,z,0)=\{\begin{array}{cc}1,\hfill & (x,y,z)\in D\hfill \\ 0,\hfill & \text{otherwise}\hfill \end{array}$$ (20)

where we define the domain enclosed by the solvent accessible surface to be $D={\cup }_{i=1}^{N_a}\{\mathbf{r}:\left|\mathbf{r}-{\mathbf{r}}_{\mathbf{i}}\right|<r_i+r_p\}$, with r_p being the probe radius. Here N_a denotes the total number of the atoms for a given biomolecular system. Let the atom centers be r_i = (x_i, y_i, z_i), i = 1, …, N_a, and r_i represents the radius of the ith atom. To protect the van der Waals surface and make the computation more efficient, we only update the values of S(x, y, z, t) at the points in between the van der Waals surface and the solvent accessible surface; i.e.,(x,y,z) ∈ ${\cup }_{i=1}^{N_a}\{\mathbf{r}:r_i<\left|\mathbf{r}-{\mathbf{r}}_i\right|<(r_i+r_p)\}$. Numerically, to avoid possible zeros in the denominator of Eq. (18) we add a very small number, such as 10⁻⁷ into the square root expression, which does not affect the result at all. The forward Euler method is initially used for time integration, and the explicit second order central difference scheme is performed for the spatial discretization. Moreover, the numerical algorithms based on semi-implicit scheme can be applied to relax the step size limitation and to accelerate the geometry flow solver. This aspect is studied in the next section.

III.B Acceleration procedures

The computational efficiency of the solution process is an important issue and can be a bottleneck for further applications of the present model. Particularly, when this model is applied to molecular dynamic simulation, the generalized PB equation and geometry flow equation are to be solved up to millions of times. Therefore, any nontrivial improvement in computational efficiency will make the present model more feasible to many practical applications in chemical and biological systems.

III.B.1 Precondition of the PB solver

The linear algebraic system of the discretized PB equation can be solved by two major approaches: direct methods and iterative methods. Large memory requirement prohibits direct methods to be used in the matrix resulted from large chemical and biological systems. Widely used iterative methods, including Gauss-Seidel and successive over-relaxation (SOR), work well for the generalized PB equation, but typically converge slowly. Conjugate Gradient method is quite efficient for symmetric and positively definite matrices. However, the sparse matrix A resulted from Eq. (16) is seven-fold banded but non-symmetric because the dielectric distribution function is not a constant and varies in the transition region. The biconjugate gradient (BiCG) method can be a good choice for non-symmetric systems and has been adopted in a variety of our MIB schemes,^{180, 181, 184, 186, 187} but attentions are still to be paid in regard to the convergence issue. We have studied the application of pre-conditioners in two linear solver libraries, the SLATEC (http://people.sc.fsu.edu/~burkardt/f_src/slatec/slatec.html) and the PETSc (http://www.mcs.anl.gov/petsc/petsc-as/) to the solution of the PB equation.²⁸ It turned out that combination of stabilized biconjugate gradient method (BiCG) and the blocked Jacobi preconditioner (BJAC) from the PETSc and the combination of the orthomin method (OM) and the incomplete LU factorization preconditioner (ILU) from the SLATEC performed better compared to other tested solvers, preconditoners and their combinations.²⁸ In this study, we focus on the combination of the ILU and the OM from the SLATEC, which is easy to incorporate into the present iteration procedure and provides a stand-alone package, while the PETSc needs to be pre-installed before being used. In Section §IV, we further demonstrate the improvement by the combination of pre-conditioners and the iterative linear solvers.

III.B.2 Initial guess of the generalized PB solution

A good initial guess is always desired for the speedup of the iterative PB solver. Normally, the initial guess can be simply set to 0 because it is complicated and computationally expensive to find good ones. However, in our iteration procedure, it is found that the electrostatic potential distribution does not change dramatically from the prior calculation due to the small adjustment in dielectric from the prior step. Therefore, it is beneficial to take the prior potential as a good guess for solving the linear system. It turns out that the generalized PB solver converges faster when the electrostatic potential from the previous iteration is used as an input.

III.B.3 Convergence criteria in the generalized PB solver

The convergence criterion directly influences the accuracy and CPU cost of the solution of the generalized PB equation. The smaller convergence criterion, the more accurate the solution of linear system is. However, the smaller convergence criterion requires more iterations and longer CPU time in the iterative solution process. Therefore, it is worthwhile to find a criterion which is a good compromise between the accuracy and the efficiency. Typically, a value of 10⁻⁶ is used in many chemical and biological applications. Later on we will numerically investigate the effect of convergence criterion on the electrostatic solvation energy, mean surface area and mean volume which are used to compute the total salvation free energy. With the 10⁻⁸ as a standard, we will examine the efficiency and the accuracy for several relaxed convergence criteria, such as 10⁻⁶, 10⁻⁴, 10⁻³, 10⁻² and 10⁻¹.

III.B.4 Numerical schemes for the generalized geometric flow equation

The explicit scheme described in Section §III.A for the generalized geometric flow equation has the advantage of saving much memory. Nevertheless, it is not very efficient because a very small time step size i.e., time stepping, is required to guarantee the stability of the algorithm. This motivates us to find a faster semi-implicit algorithm. An alternative direction implicit (ADI) scheme reported in our earlier work¹⁴ was the fastest scheme among the tested ones under typical accuracy requirement in the mean curvature flow. One of the reasons is that the traditional ADI method, widely used in linear diffusion equations, is unconditionally stable. It allows a much larger time stepping than does the explicit scheme in guaranteeing the accuracy and avoiding the stability concern. Another reason is that the capability of applying the fast O(N) Thomas algorithm to solving the tridiagonal linear system in the ADI further speeds up the computation. Considering the similarity of the current differential operator and the mean curvature flow, we adopt the splitting algorithm based ADI scheme to speed up our generalized geometric flow solver. To this end, we modify Eq. (19) as

$$\left(1-\frac{{\mathrm{\upsilon }}_x}{2}{\mathrm{\delta }}_x^2-\frac{{\mathrm{\upsilon }}_y}{2}{\mathrm{\delta }}_y^2-\frac{{\mathrm{\upsilon }}_z}{2}{\mathrm{\delta }}_z^2\right)S_{\mathit{\text{ijk}}}^{n+1}=\left(1+\frac{{\mathrm{\upsilon }}_x}{2}{\mathrm{\delta }}_x^2+\frac{{\mathrm{\upsilon }}_y}{2}{\mathrm{\delta }}_y^2+\frac{{\mathrm{\upsilon }}_z}{2}{\mathrm{\delta }}_z^2\right)S_{\mathit{\text{ijk}}}^n+\tau f(S_{\mathit{\text{ijk}}}^n).$$ (21)

It follows that

$$\left(1-\frac{A_x}{2}\right)\left(1-\frac{A_y}{2}\right)\left(1-\frac{A_z}{2}\right)S_{\mathit{\text{ijk}}}^{n+1}=\left[\left(1+\frac{A_x}{2}\right)\left(1+\frac{A_y}{2}\right)\left(1+\frac{A_z}{2}\right)-\frac{A_x{A}_y{A}_z}{4}\right]S_{\mathit{\text{ijk}}}^n+\tau f\left(S_{\mathrm{\text{ijk}}}^n\right),$$ (22)

where

$$A_x={\upsilon }_x{\delta }_x^2,A_y={\upsilon }_y{\delta }_y^2,A_z={\upsilon }_z{\delta }_z^2.$$ (23)

Here υ_x, υ_y, υ_z, ${\delta }_x^2,{\delta }_y^2$ and ${\delta }_z^2$ are defined in Section §III.A. The following multi-step implementation can be carried out. Step 1:

$$\left(1-\frac{A_x}{2}\right)S_{\mathit{\text{ijk}}}^{n+\frac{1}{3}}=\left(1+\frac{A_x}{2}+A_y+A_z\right)S_{\mathit{\text{ijk}}}^n+\tau f(S_{\mathit{\text{ijk}}}^n)$$ (24)

Step 2:

$$\left(1-\frac{A_y}{2}\right)S_{\mathit{\text{ijk}}}^{n+\frac{2}{3}}=S_{\mathit{\text{ijk}}}^{n+\frac{1}{3}}-\frac{A_y}{2}{S}_{\mathit{\text{ijk}}}^n$$ (25)

Step 3:

$$\left(1-\frac{A_z}{2}\right)S_{\mathit{\text{ijk}}}^{n+1}=S_{\mathit{\text{ijk}}}^{n+\frac{2}{3}}-\frac{A_z}{2}{S}_{\mathit{\text{ijk}}}^n{.}_{}^{}$$ (26)

This ADI algorithm is used to integrate our generalized geometric flow equation (10).

III.C Dynamical coupling of the generalized Poisson Boltzmann and geometry flow equations

As described in Section §II, the present differential geometry based solvation model prescribes a procedure to set up the total free energy functional of the solvation. By the variational principle, we obtain generalized coupled PB equation (10) and potential driven geometric flow equation (10). The solution of these coupled nonlinear equations provides a “physical” dielectric profile ε(S) and the electrostatic potential ϕ and thereby enables the calculation of the solvation free energy. The solution of the potential driven geometric flow equation (10) requires the knowledge of ϕ, while the solution of the generalized PB equation (10) requires the input of S and ε (S). Therefore, in principle, the geometric flow equation needs to be solved simultaneously with the generalized PB equation until a self-consistency is reached. In this study, we explore two self-consistent iteration procedures.

III.C.1 Approach I

The iteration process can be carried out by breaking up the process into an iterative sequence of two steps as follows: Starting with an initial guess of characteristic function S, one finds out the temporary electrostatic potential ϕ by solving the generalized PB equation with a given initial S. Once the electrostatic potential is obtained, the electrostatic energy can be calculated. The second step is to solve the time-dependent generalized geometric flow equation for S with a prior calculated potential ϕ. In this step, the time integration can usually reach a quasi-steady state. With the updated quasi-steady S, one can come back to the first step for the next cycle until the solvation free energy converges to within a predetermined criteria. However, in practice, simply re-inserting S into the PB solver may diverge. Because the quasi-steady S may vary dramatically during the iteration. Note that all changes in S are concentrated around the solvent-solute boundary, as the final solution of the potential driven geometric flow equation reflects the balance between the intrinsic curvature energy and the external potential terms. A large change in S near the solvent-solute boundary in turn leads to much adjustment in the electrostatic potential which differs much in the solute and solvent regions. To avoid this problem, we adopt a simple relaxation algorithm: the characteristic function S used for the PB solver is a linear combination of the previous one S_old and the newly generated one S_new

$$S=\mathrm{\alpha }{S}_{\text{new}}+(1-\mathrm{\alpha })S_{\text{old}},0<\alpha <1.$$ (27)

It turns out that the convergence of the generalized PB equation is guaranteed if α is small enough. In this work, α can be taken in the range from 0.1 to 0.7. The choice of α is explored later. Note that this approach may fail sometimes when the generalized geometric flow equation blows up due to a large variation in the temporary electrostatic potential. We therefore utilize a similar procedure for the electrostatic potential used in the evolution of the generalized geometric flow equation

$$\varphi =\alpha \prime {\varphi }_{\text{new}}+(1-\alpha \prime ){\varphi }_{\text{old}},0<\alpha \prime <1,$$ (28)

where ϕ_old and ϕ_new are previous and newly resolved electrostatic potentials, respectively. This treatment can avoid the blow-up of the generalized geometric flow soluation.

III.C.2 Approach II

To reduce the dramatic changes in S and ϕ, we can consider a straightforward way in which solving generalized PB equation follows each time-step integration of the generalized geometry flow equation. However, this treatment makes the whole iterative procedure computationally over expensive as many more PB solution processes are required. Additionally, it may not be necessary since the change in the S from one time step to another one is so small that the change in the corresponding potential distribution should be very small. Indeed, it is practical to update electrostatic potential after a number of time steps (i.e., 10 to 100 steps) in the generalized geometry flow equation integration rather than every time step. We call the number of time integration between two ϕ updatings the number of intermittency N_step. This approach effectively speeds up the whole process. Additionally, the relaxation algorithm given in Eqs. (27) and (28) can also be used here to guarantee the convergence. Moreover, in this approach, one better starts the iterative process by solving the S from Eq. (13) without the electrostatic potential term. So that the later iteration procedure can focus on the impact of electrostatic potential to the generation of the solvent-solute boundary. This treatment reduces the total iteration number and save the computational time significantly.

In fact, there is a relationship between Approach I and Approach II. When the number of the time integrations becomes larger and larger, Approach II returns to Approach I. In Section IV, we systematically study the difference between these two approaches. This can be done by comparing the impacts of different approaches on the resulting total solvation free energy, surface area and volume of the solute molecule. It is found that these two approaches lead to the same results. This, to some degree, indicates the reliability and validity of the proposed iteration procedures.

III.D Evaluation of the solvation free energy

Once the electrostatic potential ϕ and the characteristic function S are obtained, the solvation free energy is given by

$$\mathrm{\Delta }G=G-G_0$$ (29)

where G₀ is the energy calculated from the homogeneous environment with ε_s = ε_m = 1 without the nonpolar energy part. Therefore, we have

$$\mathrm{\Delta }G=G_{\mathrm{p}}+G_{\text{np}}-G_0.$$ (30)

The expressions of G_p and G_np are taken from Eq. (7). Here G_p − G₀ can be considered as the electrostatic solvation free energy. In all calculations presented here except for salt effect calculation, mobile ions will be set to zero corresponding to a solution without salt. Therefore we have

$$G_{\mathrm{p}}={\displaystyle {\int }_{\mathrm{\Omega }}S(r){\rho }_m\phi d\mathbf{r}}-\frac{1}{2}{\displaystyle {\int }_{\mathrm{\Omega }}\epsilon (S(r)){\left|\nabla \varphi \right|}^{2}d\mathbf{r}}=\frac{1}{2}{\displaystyle {\int }_{\mathrm{\Omega }}S(r){\rho }_m\varphi d\mathbf{r}.}$$ (31)

Discretizing the integral yields

$$G_{\mathrm{p}}=\frac{1}{2}{\displaystyle \sum _{i=1}^{N_m}Q({\mathbf{r}}_i)\varphi ({\mathbf{r}}_i),}$$ (32)

where Q(r_i) is the ith partial charge at r_i in the biomolecule, and N_m is the total number of partial charges. Now the electrostatic solvation free energy can be calculated as

$$\mathrm{\Delta }{G}_{\mathrm{p}}=G_{\mathrm{p}}-G_0=\frac{1}{2}{\displaystyle \sum _{i=1}^{N_m}Q({\mathbf{r}}_i)(\varphi ({\mathbf{r}}_i)-{\varphi }_0({\mathbf{r}}_i)),}$$ (33)

where ϕ and ϕ₀ are electrostatic potentials in the presence of the solvent and the vacuum, respectively. Here ϕ is computed from the generalized Poisson equation (10) using the continuous dielectric distribution

$$-\mathrm{\nabla }\cdot (\epsilon (S)\mathrm{\nabla }\varphi (\mathbf{r}))=S_{\rho m}$$ (34)

where ε(S) and ρ_m are the same as the ones in Eq. (10). The homogeneous solution ϕ₀ is computed with ε(S) = ε_m in the whole domain. The nonpolar part, G_np, is computed exactly by Eq. (2).

IV Numerical test and validation

This section provides systematic validations for the computational algorithms and schemes proposed in the last two sections. We first examine the behavior of the coarea formula, then continue testing through various equation solvers, and finally check the impact of the potential term in our generalized geometric flow equation.

IV.A The behavior of the coarea formula

As described earlier, the coarea formula plays an important role in describing the mean surface area of an infinite family of smooth solvent-solute boundaries by a volume integral. This Eulerian formulation puts the free energy of the surface area and other free energies in an equal footing. Usually, the isovalue of the surface area in the coarea formula can be any positive real number. But for the present derivation, it is limited to be between 0 and 1 because S is defined as a characteristic function of the solute. Here, we numerically explore the behavior of the coarea formula in a bounded open set. To this end, we design some test cases as follows: Let B be a smooth function with a specific expression according to the geometry in the coarea formula, we set

$$\text{mean surface area of}\{\mathrm{x}|0<B(\mathrm{x})<1\}={\displaystyle {\int }_0^1{\displaystyle {\int }_{B^{-1}{\cap }^{\mathrm{\Omega }}}d\sigma dc}={\displaystyle {\int }_{\mathrm{\Omega }}\Vert \mathrm{\nabla }B\Vert d\mathrm{x}}}$$ (35)

where Ω is a bounded open set. Therefore, the mean surface area has the same value as the volume of open set Ω∩{x|0 < B(x) < 1}. Computationally, integrating over the norm of the gradient of B gives rise to the corresponding mean surface area. The volume integral of a density function f is just simply approximated by

$${\displaystyle {\int }_{\mathrm{\Omega }}f(x,y,z)d\mathbf{r}\approx {\displaystyle \sum _{(i,j,k)\in J}f(x_i,y_j,z_k)h^3}}$$ (36)

where the summation is over J, the set of points inside Ω, and (x_i, y_j, z_k) is the coordinates of grid points (i, j, k). Table (1) lists the numerical results and exact values of the surface areas for the following cases

1. A unit sphere: $\mathrm{\Omega }=\{(x,y,z)|x^2+y^2+z^2\le 1\}$ and $B=\sqrt{x^2+y^2+z^2}$.

2. A cylinder: $\mathrm{\Omega }=\{(x,y,z)|x^2+y^2<1,-4\le z\le 4\}$ and $B=\sqrt{x^2+y^2}$.

3. A ellipsoid: $\mathrm{\Omega }=\{(x,y,z)|{(x/a)}^2+{(y/b)}^2+{(z/c)}^2\le 1\}$ and $B=\sqrt{{(x/a)}^2+{(y/b)}^2+{(z/c)}^2}$, where a = 20/7, b = 25/14, and c = 25/14.

Table 1.

Areas computed from the coarea formula for bounded open sets

	grid spacing
case	0.5	0.25	0.1	0.05	0.025	exact value
Sphere	4.00	4.00	4.15	4.17	4.18	4.189
Cylinder	22.50	23.25	24.49	24.84	25.01	25.133
Ellipsoid	37.75	37.97	38.10	38.17	38.16	38.163

It is evident that the numerical result converges to the exact value. The errors from the cylinder are slightly larger than those from the sphere and ellipsoid because the cylinder has non-smooth edges. However, the errors are small for all cases. Therefore, we conclude that the mean value of the areas of the family of smooth solvent-solute surfaces indeed converges to the area of the corresponding sharp surface. Thus, the present definition of the mean surface area of an infinite family of smooth surfaces is an important generalization of the classic concept of the area of a sharp surface.

IV.B Accuracy of the generalized PB solver

In this section, we investigate the accuracy of the proposed numerical solvers. The generalized geometric flow equation (19) has the same differential operator as the mean curvature flow¹⁷ except for the extra source terms. Previously, we have numerically proved that the explicit Euler algorithm delivers the reliability and convergence of the solution of geometric flow equations, and the finite central different scheme is of second order convergence in space.¹⁴ Here, we focus on the test of the accuracy of the generalized PB solver with the proposed dielectric function ε(S). Although the discretization form of the second-order finite difference PB expression has been used for other continuous dielectric definitions,⁷⁸ the accuracy of this approach has not been examined. Moreover, it is worthwhile to validate the generalized PB solver due to its different settings of dielectric function, i.e., the ε(S) profiles generated by the geometric flow equation. For this purpose, we construct a benchmark test of a simple one-ball system. We examine the convergence order and the accuracy of the finite difference scheme in solving the generalized PB equation. In particular, we consider a modified Poisson equation with a designed dielectric definition given by

$$\epsilon (\mathbf{r})={\epsilon }_{1}Su(\mathbf{r})+{\epsilon }_2(1-Su(\mathbf{r}))$$ (37)

where r = (x, y, z), ε₁ and ε₂ are two constants to be determined, and

$$Su(\mathbf{r})=\{\begin{array}{cc}1\hfill & \text{if}\sqrt{x^2+y^2+z^2}<a;\hfill \\ -2{\left(\frac{b-\sqrt{x^2+y^2+z^2}}{b-a}\right)}^3+3{\left(\frac{b-\sqrt{x^2+y^2+z^2}}{b-a}\right)}^2\hfill & \text{if}a\le \sqrt{x^2+y^2+z^2}\le b;,\hfill \\ 0\hfill & \text{if}b<\sqrt{x^2+y^2+z^2};\hfill \end{array}$$ (38)

where a < b. Note that through the definition of ε and Su(r), this designed case has the same features of the dielectric definition as that in our model. The exact solution is designed to be

$${\varphi }_0(\mathbf{r})=\text{cos}(x)\text{cos}(y)\text{cos}(z).$$ (39)

Then the modified equation becomes

$$\nabla \cdot (\epsilon \nabla {\varphi }_0)={\epsilon }_x{\nabla }_x{\varphi }_0+{\epsilon }_y{\nabla }_y{\varphi }_0+{\epsilon }_z{\nabla }_z{\varphi }_0+\epsilon {\nabla }^2{\varphi }_0$$ (40)

where

$$\begin{array}{c}\hfill {\nabla }^2{\varphi }_0(x,y,z)=-3\text{cos}(x)\text{cos}(y)\text{cos}(z)\hfill \\ \hfill {\nabla }_x{\varphi }_0=-\text{sin}(x)\text{cos}(y)\text{cos}(z)\hfill \\ \hfill {\nabla }_y{\varphi }_0=-\text{sin}(y)\text{cos}(x)\text{cos}(z)\hfill \end{array}$$

and

$${\nabla }_z{\varphi }_0=-\text{sin}(z)\text{cos}(x)\text{cos}(y).$$

For ε, if $a\le \sqrt{x^2+y^2+z^2}\le b$, we have

$$\begin{array}{c}{\epsilon }_x(\mathbf{r})=6({\epsilon }_2-{\epsilon }_1)\left(\frac{b-\sqrt{x^2+y^2+z^2}}{b-a}\right)\left(\frac{\sqrt{x^2+y^2+z^2}-a}{b-a}\right)\left(\frac{x}{(b-a)\sqrt{x^2+y^2+z^2}}\right)\hfill \\ {\epsilon }_y(\mathbf{r})=6({\epsilon }_2-{\epsilon }_1)\left(\frac{b-\sqrt{x^2+y^2+z^2}}{b-a}\right)\left(\frac{\sqrt{x^2+y^2+z^2}-a}{b-a}\right)\left(\frac{y}{(b-a)\sqrt{x^2+y^2+z^2}}\right)\hfill \end{array}$$

and

$${\epsilon }_z(\mathbf{r})=6({\epsilon }_2-{\epsilon }_1)\left(\frac{b-\sqrt{x^2+y^2+z^2}}{b-a}\right)\left(\frac{\sqrt{x^2+y^2+z^2}-a}{b-a}\right)\left(\frac{z}{(b-a)\sqrt{x^2+y^2+z^2}}\right).$$

Otherwise, ε_x = ε_y = ε_z = 0. Table (2) lists the computed errors under different mesh sizes with a = 1 and b = 3. The standard absolute norm error measurement L_∞ is employed. Here ε₁ is fixed to be 1 and ε₂ is taken to be 80 or 10. The second order convergence in space is observed for the scheme. Furthermore, it is found that a large ε₂ may slightly deteriorate the convergence.

Table 2.

Errors and convergence orders for the generalized PB solver (ε₁ = 1)

	ε2 = 80		ε2 = 10
spacing	L∞	order	L∞	order
1	0.22		0.13
0.5	8.13*10-2	1.65	3.18*10-2	2.02
0.25	2.06*10-2	1.99	7.97*10-3	2.00
0.125	5.44*10-3	1.94	1.98*10-3	2.01

IV.C Convergence of boundary profile and dielectric function

In the present model, the characteristic function S defines the solvent-solute boundary. Consequently, it can significantly affect the solution of the generalized Poisson-Boltzmann equation, the surface area and volume, and thus, plays a key role in the solvation free energy calculation. To illustrate the evolution and the convergence of the generalized geometric flow equation and corresponding S function, we solve Eq. (13) without the electrostatic potential term in this test. However, the electrostatic solvation free energy at a given time can still be calculated. The expression of attractive interaction $U_i^{\text{att}}$ needs to be given explicitly in order to solve the geometric flow equation. Here we consider the following 6–12 Lennard-Jones (L-J) pair potential to model $U_i^{\text{att}}$:

$$U_i^{\text{att}}(\mathbf{r})={\epsilon }_{\mathbf{i}}\left[{\left(\frac{{\sigma }_{\mathbf{i}}+{\sigma }_{\mathbf{s}}}{\Vert \mathbf{r}-{\mathbf{r}}_{\mathbf{i}}\Vert }\right)}^{12}-2{\left(\frac{{\sigma }_{\mathbf{i}}+{\sigma }_{\mathbf{s}}}{\Vert \mathbf{r}-{\mathbf{r}}_{\mathbf{i}}\Vert }\right)}^6\right]$$ (41)

where ε_i is the well-depth parameter, and σ_i and σ_s are solute atomic and solvent radii, respectively. Here r and r_i are positions. The L-J potential can be divided into attractive U^att and repulsive U^rep in different ways. It can be a “6–12” decomposition as follows:

$$\begin{array}{cc}{U}_i^{\text{att},6/12}(\mathbf{r})\hfill & =-2{\epsilon }_i{\left(\frac{{\sigma }_i+{\sigma }_s}{\Vert \mathbf{r}-{\mathbf{r}}_i\Vert }\right)}^6\hfill \\ U_i^{\text{rep},6/12}(\mathbf{r})\hfill & ={\epsilon }_i{\left(\frac{{\sigma }_i+{\sigma }_s}{\Vert \mathbf{r}-{\mathbf{r}}_i\Vert }\right)}^{12}.\hfill \end{array}$$ (42)

Or it can also be a “WCA” decomposition based on the original WCA theory⁸⁹

$$U_i^{\text{att},\text{WCA}}(\mathbf{r})=\{\begin{array}{cc}-{\epsilon }_i(\mathbf{r})\hfill & 0<\Vert \mathbf{r}-{\mathbf{r}}_i\Vert <{\sigma }_i+{\sigma }_s\hfill \\ U_i^{\text{LJ}}(\mathbf{r})\hfill & \Vert \mathbf{r}-{\mathbf{r}}_i\Vert \ge {\sigma }_i+{\sigma }_s,\hfill \end{array}$$ (43)

$$U_i^{\text{rep},\text{WCA}}(\mathbf{r})=\{\begin{array}{cc}{U}_i^{\text{LJ}}(\mathbf{r})+{\epsilon }_i(\mathbf{r})\hfill & 0<\Vert \mathbf{r}-{\mathbf{r}}_i\Vert <{\sigma }_i+{\sigma }_s\hfill \\ 0\hfill & \Vert \mathbf{r}-{\mathbf{r}}_i\Vert \ge {\sigma }_i+{\sigma }_s.\hfill \end{array}$$ (44)

All the calculations in this work are carried out by using the WCA decomposition.

To illustrate the S profile and evolution, we consider a diatomic system with the van der Waals radius 2.2Å and coordinates (x, y, z) = (−3, 0, 0) and (1.4, 0, 0). The spacing and time stepping are chosen as h = 0.25Å and τ = h²/4.5, respectively. The solvent probe radius is set to 2Å, which is used for the initial value setting and constraint construction. In fact a much small solvent probe radius can ensure the correct surface topology.¹⁷ The computational domain is set to [−8.70, 7.05] × [−5.7, 5.55] × [−5.7, 5.55]. Thus, the size of computational system is 64 × 46 × 46. The L-J parameters are set as follows: σ_i is taken from atomic radius and σ_s is chosen to be 0.65Å. We set well depth ε_i = 0.039 kcal/mol and bulk density coefficient ρ₀/γ = 2, where, γ = 1/15kcal/(molŲ). To compute the electrostatic solvation free energy during the evolution of solvent-solute boundary, 1 unit charge is set on the center of each atom. We choose the dielectric constants ε_m = 1 and ε_s = 80, respectively. We set pressure coefficient p/γ = 0.2. A different γ is used for real systems.

The evolution process of diatomic solvent-solute boundary is depicted through a group of cross section profiles of the S function in Fig. 3 where the values of S from a set of points of (x, y, 0.05) are described. The cross sections start with a relatively fat-shaped interface which reflects the solvent accessible density. Here, S = 1 inside the molecular domain and S = 0 in the solvent domain. Then the solvent-solute boundary is driven inward by intrinsic geometric curvature effect and external potential. At the same time, there appears a transition region between the solvent and the solute. Finally a convergence is reached with a balance among intrinsic geometric curvature effect, different potentials and enforced constraints. To have a clear idea about the distribution feature of the S function, we draw a cross line from the cross section graph at T = 5 along x = −0.75 which are shown in Fig. 1, where the functions of S and 1 − S are described together. It can be seen that the S function in the transition region is rather smooth. Once the S function is determined, the dielectric function ε(S) is calculated according to Eq. (11). Here the dielectric function ε(S) corresponding to the S function in Fig (1) is also exhibited in Fig. 2. It has a pattern similar to 1 − S but with different values. It is important to note that the dielectric function ε(S) is also very smooth at the solute-solvent boundary. That is why the classical finite difference scheme can be applied to solve the generalized PB equation without reducing the accuracy of the solution.

Figure 3.

The evolutionary profiles of the S function at cross section (x, y, 0.05) in a diatomic system plotted from six intermediate states.

Once the solution of the generalized PB equation is computed, the electrostatic solvation free energy can be calculated. Therefore, the time history of the free energy along with the evolution of solvent-solute boundary can be recorded. To illustrate the convergence pattern of the solvation free energy, we compute the electrostatic solvation free energies in intermediate states during the time evolution. The results are shown in Fig. 4. In order to show evolution histories of the surface area, volume and solvation free energy together in one plot, we plot two linear functions F(Volume) and J(Area) which have the same time dependence as the volume and the surface area, respectively. Here T denotes the time span and $N=\frac{T}{\tau }$ represents the number of computational steps in the generalized geometric flow solver. It is found that the solvation free energy decreases with respect to the time evolution, which is consistent with our theoretical formulation. It is observed that the solution of our model converges to a steady state in terms of volume (ų), area (Ų) and electrostatic solvation free energy (kcal/mol). Moreover, to obtain the results at the steady state, N = 200 or T = 3.5 is large enough to be taken as the stopping time in our geometry flow solver for this system. Normally, it takes a longer evolution time for a large system to set down to the steady state. The total integration time could be considerably shortened had a small probe radius been used.¹⁷

Figure 4.

The time evolution histories of the electrostatic solvation free energy, F(Volume) and J(Area), where F(Volume) = volume/5 – 180 and J(Area) = (surface area)/5 – 200.

IV.D Consistency of iteration procedures

If the electrostatic potential effect is taken into account during the solvent-solute boundary evolution, the iteration procedure has to be used to update the electrostatic potential repeatedly. As described earlier, there are two possible iterative approaches which can be explored to solve the coupling system, in which the simple relaxation algorithm guarantees the convergence of the algorithm. The question is whether these two approaches lead to the same outcome.

To study the consistency between these two approaches, the above mentioned diatomic system is employed as well as a small molecule called glycerol triacetate (Gly) from a set of 17 test compounds whose detailed information is given in Section §V. The self-consistent iteration is performed until the electrostatic solvation free energy converges to within 0.01 kcal/mol. The electrostatic solvation free energy(kcal/mol), surface area (Ų) and volume

(ų) resulted from these two different methods are compared. The results are shown in Table 3. Here we take α = 0.5 for both methods. The electrostatic potential ϕ is updated in every 15 steps of the generalized geometric flow integrations in Approach II. It is evident that the results from these two approaches are almost the same. Therefore, they can be alternatives for each other at least in small molecular systems. But for large protein systems, as we mentioned, it is better to use the second approach to avoid the possible blowup in the generalized geometry flow caused by unpredictable large changes in the temporary electrostatic potential. Thus, in the following tests and applications, the second method is applied except specified.

Table 3.

Comparison between two iteration approaches

	2 atoms		Gly
	Approach I	Approach II	Approach I	Approach II
Energy	−231.18	−231.18	−12.44	−12.44
Surface area	128.67	128.67	271.91	272.02
Volume	100.72	100.83	287.85	287.93

In Approach II, the relaxation factor α and the number of intermittency N_step need to be determined. We are interested in knowing whether the relaxation factor α plays a role in the final result. Similarly, it is important to know whether the N_step makes a difference in the converged result. We address these issues by examining the effects of α and N_step on the electrostatic solvation free energy, surface area and enclosed volume. The above mentioned diatomic system is used here again.

It is known that a stable α value is between 0 and 1 but can not be very close to 1. We consider a number of α values in the diatomic system while keeping other settings fixed. Table 4 shows the electrostatic solvation free energy, surface area and volume for α = 0.1, 0.2, 0.5 and 0.7. It is found that the procedure diverges when α ≥ 0.8. However, convergence is achieved as long as the relaxation factor α is small enough. Once the convergence is achieved there is no much difference in the final outcome. We therefore take α = 0.5 in the following tests and applications.

Table 4.

Effect of relaxation factor α on final results

α	0.1	0.2	0.5	0.7	0.8
Energy	−231.26	−231.18	−231.18	−231.18	divergence
Surface area	100.73	100.94	100.83	100.71
Volume	128.65	128.62	128.67	128.71

To study the effect of the number of intermittency, we take N_step = 5, 10, 15, 40 and 100, while fixing α = 0.5 and other settings. The results are listed in Table 5. It is seen that all values obtained from different number of intermittency are very close to each other. However, a numerically too large or too small N_step is not preferable. If N_step is too large, Approach II goes back to the first one. If N_step is too small, the iterative process may stop too early because the perturbation in each iteration is so weak that the convergence criteria is satisfied unexpectedly sometimes. In addition, small step number makes the whole process computationally expensive.

Table 5.

Effect of the number of intermittency in Approach II

Nstep	100	40	15	10	5
Energy	−231.18	−231.19	−231.18	−231.17	−231.11
Surface area	100.83	100.83	100.83	100.89	101.07
Volume	128.63	128.63	128.67	128.48	128.63

IV.E Efficiency of the accelerated iteration procedure

We study the efficiency of the accelerated self-consistent iteration in this section. At the beginning we analyze the CPU time usage based on an original combination of methods: Biconjugate Gradient (BiCG) method for the generalized PB solver and a widely used explicit scheme for the generalized geometric flow (GGF) solver. In addition, as commonly used in the linear system of the PB equation, we take 10⁻⁶ as the initial convergence criteria and set the first guess of the electrostatic potential in each generalized PB run to be 0. The above mentioned diatomic and Gly systems will be utilized through this efficiency test. Table 6 lists the breakup of time spending in the different parts of the self-consistent iteration procedure. It is seen that for these two systems the major computation cost lies in the routines of the generalized PB solver and the generalized geometric flow solver (more than 90%). Therefore, the total time will be dramatically reduced when efficient accelerations are achieved in the generalized PB solver and generalized geometric flow solver. Note that all of the computations are performed on a SGI Altix 350 workstation with a 1.4 GHz Itanium processor and 4 GB memory. Additionally, we explore the improvement made to the generalized PB solver, the generalized geometric flow solver, and consequently to the total time cost. First, we combine an appropriate preconditioner with the iterative solver of the linear system. Additionally, we make use of the prior electrostatic potential as a first guess for the next PB run. Moreover, we obtain the approximations through the relaxation of the convergence criteria of the linear solver. Finally, we employ the ADI scheme to integrate the generalized geometric flow equation.

Table 6.

CPU time analysis from original schemes

	2 atoms		Gly
	Time(s)	%	Time(s)	%
Total	23.95		58.4
GF	4.95	21	11.31	19
PB	18.25	76	45.24	77
Other	0.75	3	2.03	4

First of all, we do the following improvement: Take the prior potential solution as the first guess of each run of the linear solver, then replace BiCG scheme with a combination of the preconditioner and the iterative solver (ILU/OM), while keeping other settings unchanged.

Table 7 gives the total computational costs of the generalized PB solver in the diatomic system as well as the total iteration numbers which are inside the parenthesis. The third column lists the time spending for original schemes, the fourth one makes use of prior potential as a first guess and the fifth one records the time spending from the usage of the preconditioner and the new first guess setting. It is seen that the gain of speedup is related to the size of system: The larger size is the system, the more acceleration is achieved. For a 127× 92× 92 system, combination of the above two implementations can obtain a speedup up to a factor of 4, while a single adjustment does not give much impressive improvement. It is also found that although the total iteration number reduces dramatically by adding the preconditioner, the total computational cost is reduced with a much smaller factor. The reason is that the PB solver with a preconditioner takes more time in each step.

Table 7.

Speedup from adjustment of initial guess and preconditioner in PB solver

h	size	BiCG	BiCGi	ILU/OM	Speedup
1	17× 12× 12	0.0557(252)	0.0322(152)	0.0371(50)	1.50
0.5	32× 23× 23	0.775(419)	0.467(248)	0.420(82)	1.85
0.25	64× 46× 46	17.676(841)	10.410(490)	6.947 (166)	2.54
0.125	127× 92× 92	525.74(2771)	263.11(1371)	130.76 (410)	4.02

Next, we study the impact of the convergence criteria to the electrostatic solvation free energy of the diatomic system. Table 8 summarizes the calculated electrostatic solvation free energies and total time cost of the PB solver under different convergence criteria. It indicates that 10⁻⁴ is good enough to deliver accurate results. In fact, 10⁻² is still fine but 10⁻¹ is clearly unacceptable. In this study, we take 10⁻⁴ as the convergence criteria of the linear system in the following calculations except specified. Because it is able to save much time compared to 10⁻⁶ while at the same time maintains the accuracy to a satisfied level. In practical application, one might use 10⁻². A further reduction in computational time is possible if one sets the probe radius to r_p = 0.25r_vdW, where r_vdW is the van der Waals radius.¹⁷

Table 8.

Influence of convergence criteria on electrostatic solvation free energy and computational time for the diatomic system

Criteria	10−8	10−6	10−4	10−3	10−2	10−1
Energy (kcal/mol)	−231.17	−231.17	−231.19	−231.28	−231.07	−239.80
PB Time (s)	10.44	6.95	4.17	3.09	2.05	0.90

Finally, we implement the ADI scheme in the generalized geometric flow solver. Thus we can use a much larger time increment than that used in an explicit scheme without the stability concern. For example, if grid spacing is h = 0.25, the time step size can be taken as large as 0.2 for the ADI scheme to be a good balance between accuracy and efficiency, while it has to be less than 0.02 in the explicit scheme. The acceleration of the generalized geometric flow solver can be found in Table 9, which is obtained by applying all of the speedup strategies we have discussed to the diatomic and Gly systems. This table shows all the time spending for major routines in the iterative process before and after the acceleration. It indicates that speedup in the PB solver can reach a factor of 4 or 5. However, the speedup in the total time is not as impressive as in the PB solver. It is about a factor of 3 in the Gly system and about a factor of 2 in the diatomic system. The reason is that the acceleration in the generalized geometric flow equation through the ADI can not have the same speedup factor as that of the PB solver. The electrostatic solvation free energies are also given in the table for a comparison before and after the speedup. Little difference in energies is observed due to varying schemes and the approximation.

Table 9.

Comparison of CPU time (s) in the iteration procedures with and without accelerations

	2 atoms			Gly
	without accel.	with accel.	Speedup	without accel.	with accel.	Speedup
Total	23.95	8.87	2.70	58.40	16.28	3.59
GF	4.95	3.67	1.35	11.13	5.57	2.00
PB	18.25	4.45	4.10	45.24	8.71	5.19
Other	0.75	0.75		2.03	2.00
Energy	−231.17	−231.18		−12.44	−12.44

IV.F Impact of potentials in the geometric flow equation

The potential source terms in the generalized geometric flow equation include pressure, long-ranged attractive dispersion interaction and electrostatic potential. The solution (S) of the generalized geometric flow reflects a balance between the intrinsic geometric curvature effect and several external potentials at the equilibrium. In this section, we illustrate the impact of involved potentials to the characteristic function S, and consequently to the solvation free energy. Although in our model there is no explicit surface definition to be demonstrated, the impacts of these potentials can be reflected by volume, area and the electrostatic solvation free energy. In particular, if the flow is driven inward by a potential, the volume should become smaller, and an outward driving makes the volume larger. These are true at least during the early stage of the solvent-solute boundary evolution. In fact, they are not true for the system near the equilibrium. The present study is carried out through two proteins (PDB ID 1ajj and 1fca) from protein data bank (PDB). Their detailed coordinates and parameters are given in the application section of 22 proteins. Without any potential term, the mean curvature flow equation is driven purely by intrinsic geometric curvature effect, which leads to the minimal molecular surface (MMS).¹⁷ With the MMS as a reference, each time we use one additional potential term in Eq. (17) to produce a new characteristic function S which will be used in the PB solver to calculate the electrostatic potential. In other words, starting with the MMS, we attain the different characteristic functions S with either pressure, attractive non-polar potential, or electrostatic potential separately. Only when electrostatic potential term is taken into account, is it needed to run the self-consistent iteration process for the solution of the coupled system. Table 10 gives the calculated volume and electrostatic solvation free energy under each potential term. We also calculate the solvation free energy when all the potentials are turned on. It is seen that all the potentials involved here drive the flow inward so that there are more solvent components between or around two spherical solutes when just an individual potential is turned on. This is consistent with the experimental observations¹⁶⁸ and our previous studies.¹⁴

Table 10.

Effects of potentials on the solvent-solute boundary

	1ajj		1fca
	Volume	Energy	Volume	Energy
MMS	6601.9	−975.6	9345.7	−1082.2
Pressure	6195.1	−1032.1	8866.6	−1123.2
Attractive	5533.8	−1139.3	8107.2	−1192.8
Electrostatic	6585.8	−1061.4	9329.3	−1112.4
Total potential	5381.6	−1165.6	7886.8	−1211.9

V Applications

We consider two types of problems in this section. First, we apply our new approach to a set of 17 small molecules. Then, some protein examples are studied. The Dirichlet boundary condition is used for both the generalized Poisson-Boltzmann equation and the generalized geometric flow equation as in our previous calculations^{17, 65, 179, 185}

V.A Set of 17 test molecules

We apply our optimized surface model (OSM) of solvation to compute the solvation free energies of a set of 17 small compounds. This test set was studied by Nicholls et al¹¹³ using a number of approaches, including quantum mechanics, PB theory etc. An important aspect about this test set is that experimental data are available. Therefore, solvation free energies predicted from our new model can be compared with both experimental values and other numerical results. Moreover, these compounds are considered as a challenging test set for computational methods because the existence of polyfunctional or interacting polar groups, which lead to strong solvent-solute interactions.

In our calculation, we set the dielectric constants ε_m = 1 and ε_s = 80. We use γ as fitting parameter, and its initial value is set to γ = 1/15 kcal/(molŲ) to compute other γ-dependent parameters, see Eq. (17). We choose ρ₀/γ = 2 by comparing the bulk density 0.033Å⁻³ and the possible γ value. For micro-molecular systems, pressure p can be very small and sometimes is neglected in the calculation.³² But here we still take it into account and set p/γ to 0.2. Note that in the numerical simulation, all ratio parameters here are treated as dimensionless. For L-J parameters, σ_s is chosen to be 0.65Å as a good fitting solvent radius and σ_i is the solute atomic radii.¹⁶⁷ Note that due to the continuum representation of solvent in our model, the 6–12 Lennard Jones potential formula (41) differs from the standard version — the distance used in our formula is no longer the distance between the centers of solute atoms and the centers of solvent atoms but the distance between a specific position in the solvent area and the centers of solute atoms. This should make the setting of well depth ε_i different from the ones taken from AMBER or OPLS force fields. However, the performance of the L-J potential should be similar, i.e, the value of the L-J potential in the solvent caused by a solute atom only depends on the distance from the center of the atom. It implies that the value of L-J potential caused by a solute atom should be a constant on the van der Waals surface of the atom. In other words, ${\epsilon }_i\left[{\left(\frac{{\sigma }_i+{\sigma }_s}{\Vert \mathbf{r}-{\mathbf{r}}_i\Vert }\right)}^{12}-2{\left(\frac{{\sigma }_i+{\sigma }_s}{\Vert \mathbf{r}-{\mathbf{r}}_i\Vert }\right)}^6\right]=D_i$ if r is on the vdW surface of the atom. Here the constant D_i should have different values for various types of atoms. For simplicity we use a uniform constant D to determine the value of ε_i given σ_s and σ_i. In the present calculation, we pick 1.0 for D and the WCA expression is chosen as the attractive van der Waals potential. We choose grid spacing h = 0.25Å and time stepping τ = h²/4.5. Here, γ (kcal/(molŲ)) serves as a fitting parameter, and its final value is 0.0065 kcal/(molŲ).

Structure and charge information of the 17 compounds are adopted from those of Nicholls et al¹¹³ and can be obtained from the supporting information of their paper. In particular, charges are taken from the OpenEye-AM1-BCC v1 parameters.⁸⁰ Atomic coordinates and radii are based on their new parametrization called ZAP-9 in which certain types of radii are adjusted by them from Bondi radii to improve the agreement with experimental free energy. With these structures and charges parameters, the root mean square error (RMS) obtained in their paper is 1.71± 0.05 kcal/mol via the explicit solvent model. And the smallest RMS error of their single - conformer Poisson-Boltzmann approach is 1.87±0.03 kcal/mol.¹¹³ Such a large RMS error indicates the challenge of this test set. Usually, different surface definitions in implicit solvent models should have their own optimal radii set. In particular, a continuous dielectric definition based model is supposed to have radii set with larger values than those of a discontinuous dielectric model. Otherwise, the calculated free energy does not give a good fitting to experimental data.¹⁵² This also occurs in the present model. Therefore, we multiply the radii from ZAP-9 by a common factor 1.1. It turns out this treatment leads to a good agreement with experimental data in terms of electrostatic solvation free energies and total solvation free energies. The results are summarized in Table 11, which gives a comparison between calculated and experimental values of solvation free energies of 17 compounds. RMS error of the present model is 1.76 kcal/mol which is similar to that of Nicholls et. al, i.e., 1.87 kcal/mol. This RMS error is competitive to that of the explicit solvent approach (1.71±0.05 kcal/mol) under the same charge and structure parameters set.¹¹³ This may be credited to the more satisfactory nonpolar terms and the enforcement of the potential driven geometric flow. Here, as expected, major errors are from the calculation of benzamides which are between 3.5 and 4.0 kcal/mol, see Fig. 5. Without these benzamide compounds, the RMS error drops from 1.76 kcal/mol to 1.24 kcal/mol. This problem with benzamides is likely due to radius adjustment for the carbonyl oxygens and tertiary nitrogens in ZAP 9 under the OpenEye-AM1-BCC v1 charges.¹¹³ In other words, these large errors from benzamides can not be avoided if both OpenEye-AM1-BCC v1 charge and corresponding optimized ZAP 9 radii are used in PB approaches. Based on these considerations, one possible approach for improvement is to create a new charge set more appropriate for the PB approach with the same ZAP radii. It may be realized by introducing quantum mechanical corrections to our model to take care of charge density. However, this aspect is out of the scope of the present paper and will be investigated in our future work.

Table 11.

Comparison of free energies (kcal/mol) for 17 compounds

Compound	Gnp	ΔGp	ΔG	Exptl	Error
glycerol triacetate	2.27	−12.44	−10.16	−8.84	−1.32
benzyl bromide	1.40	−4.89	−3.49	−2.38	−1.11
benzyl chloride	1.35	−5.02	−3.68	−1.93	−1.75
m-bis(trifluoromethyl)benzene	2.22	−3.22	−1.00	1.07	−2.07
N,N-dimethyl-p-methoxybenzamide	1.96	−9.20	−7.24	−11.01	3.77
N,N-4-trimethylbenzamide	1.86	−7.67	−5.81	−9.76	3.95
bis-2-chloroethyl ether	1.45	−4.22	−2.77	−4.23	1.46
1,1-diacetoxyethane	1.65	−8.24	−6.59	−4.97	−1.62
1,1-diethoxyethane	1.52	−4.40	−2.88	−3.28	0.40
1,4-dioxane	1.01	−5.65	−4.64	−5.05	0.41
diethyl propanedioate	1.82	−7.85	−6.03	−6.00	−0.03
dimethoxymethane	1.03	−4.52	−3.50	−2.93	−0.57
ethylene glycol diacetate	1.59	−8.43	−6.84	−6.34	0.50
1,2-diethoxyethane	1.55	−4.31	−2.76	−3.54	0.78
diethyl sulfide	1.22	−2.39	−1.17	−1.43	0.26
phenyl formate	1.37	−7.84	−6.48	−4.08	−2.40
imidazole	0.82	−11.27	−10.45	−9.81	−0.64

Figure 5.

Correlation between experimental data and the present optimized surface model (OSM)(also results from Nicholls’) in electrostatic solvation free energies of 17 compounds.

V.B Solvation free energy of proteins

Validation by using a set of 17 molecules has shown that proposed differential geometry based solvation model works well for the energy prediction of small compounds. Since small molecules are accessible to more accurate computational means, such as quantum mechanical calculations, one of the main purposes of developing the present optimized surface model (OSM) is to attack relatively large macromolecules. To this end, we consider a test set of proteins employed by Mei et al.¹⁰⁵ For this set, the total number of residues ranges from 21 to 275. The initial structures of all proteins are taken from the protein data bank (PDB). The hydrogen atoms, which are typically missing from the X-ray data, are added to the structures to obtain full all-atom models with optimized hydrogen bondings. Partial charges at atomic sites and atomic van der Waals radii in angstroms are assigned from the CHARMM27 force field.¹⁰⁰ All of these operations, i.e, the transformation from PDB files to PQR files, can be easily done with a software PDB2PQR. Parameters of the present calculation are set in the same way as those for 17 compounds except for N_step = 2. Similar to the treatment of the 17-compound set, the radii from the CHARMM force field need to be multiplied by a common factor of 1.1. Our results are summarized in Table 12.

Table 12.

Comparison of electrostatic solvation free energies (kcal/mol) obtained from the MFCC-CPCM, the present model (OSM) and MIBPB.

			ΔGp(kcal/mol)
Protein	PDBID	No.of residues	MIBPB-III	MFCC-CPCM105	OSM
RP71955	1RPB	21	−184.68	−267.60	−192.23
Amyloid	1AMC	28	−861.65	−886.01(−798.72)	−852.68
Crambin	1CBN	46	−303.80	−361.52	−304.84
BPTI	1BPI	58	−1301.9	−1332.71(−1263.52)	−1281.19
Calbindin	1CDN	75	−2188.96	−2259.62	−2195.42
Ubiquitin	1UBQ	76	−1170.61	−997.02	−1148.81
Lysozyme	2BLX	129	−1913.40	−1887.71	−1898.07
Subtilisin	1SBT	275	−1896.5	−2062.2	−2001.4

For a comparison, The results of Mei et al are listed in Table 12 as well. Their results obtained from the molecular fractionation with conjugate caps and conductor-like polarizable continuum model (MFCC-CPCM). This is an approximate quantum approach that divides the macromolecule into fragments, such that the quantum calculations at HF/6-31 G level and B3LYP/6-31 G level can be applied. The solvation effect is estimated vis the polarizable continuum method with the classic molecular surface.¹⁰⁵ It is seen from the table that there are relatively large deviations, up to 28%, between results obtained by the present OSM and those of the MFCC-CPCM. These derivations might due to the different methodologies, computational environments and structures. In fact, the results from two different quantum basis sets have up to 10% deviation for protein Amyloid. Another deviation between results of two quantum basis sets is about 5% for the protein BPTI.

As the largest deviation between the results from the proposed OSM and that of the MFCC-CPCM is quite large as shown in Fig. 6, we consider another independent approach, the MIBPB,^{65, 179, 185} to evaluate the present method. A specific MIBPB code, the MIBPB-III which has the treatment of geometric and charge singularities,⁶⁵ is employed in our calculations. MIBPB-III has been intensively calibrated in the past and is the only known second accurate method for solving the Poisson-Boltzmann equation with both molecular surfaces and partial charges represented by the Dirac delta functions. To deliver such an accuracy, the MIBPB-III has built in the MIB scheme^{181, 187} and Dirichlet to Neumann mapping.⁶⁵ Similar to the present approach, the structural data of MIBPB-III is also prepared with the PDB2PQR software. As such, we can eliminate the difference due to the different treatment of initial data. However, the MIPPB utilizes the classic PB equation and the molecular surface, while the present method has a generalized PB equation, and an optimized smooth surface. It is seen from Table 12 that solvation energy results from the present OSM and from the MIBPB have an excellent agreement on most proteins except for Subtilisin. For this protein, the difference of energies from two methods is about 5%.

Figure 6.

Correlation between the MFCC-CPSM¹⁰⁵ and the present optimized surface model (OSM) in electrostatic solvation free energies of 8 proteins.

V.C Twenty two proteins

Encouraged by the good consistency of the proposed method and the MIBPB-III, we further compare these approaches by a larger set of protein molecules — twenty two proteins that have been frequently used in previous studies.^{14, 56, 65, 179} The implementation of these two methods is the same as that described in the last section.

Table 13 shows the results from the present continuous dielectric model, denoted as “Radii1” in the table, and those by MIBPB-III. It turns out that electrostatic solvation energies obtained via our minimization process are very close to those based on the MIBPB-III. This can also be seen through Fig. 7 which shows that the results between them are quite linearly correlated. The correlation coefficient is 0.999.

Table 13.

Electrostatic solvation free energies for 22 proteins

		ΔGp(kcal/mol)
PDB-ID	No.of atoms	MIBPB-III	Radii1	Radii0
1ajj	519	−1137.2	−1178.5	−1362.6
1bbl	576	−986.8	−965.94	−1158.7
1bor	832	−853.7	−871.4	−1066.5
1fca	729	−1200.1	−1200.6	−1340.9
1frd	1478	−2852.2	−2844.8	−3173.4
1fxd	824	−3299.8	−3291.9	−3496.9
1hpt	858	−811.6	−808.2	−1039.1
1mbg	903	−1346.1	−1328.2	−1535.4
1neq	1187	−1730.1	−1713.9	−2049.3
1ptq	795	−873.1	−866.2	−1064.5
1r69	997	−1089.5	−1072.7	−1294.0
1sh1	702	−753.3	−771.8	−973.8
1svr	1435	−1711.2	−1704.6	−2073.7
1uxc	809	−1138.7	−1125.7	−1350.9
1vii	596	−901.5	−892.0	−1052.1
2erl	573	−948.8	−935.8	−1067.3
2pde	667	−820.9	−843.0	−1049.3
451c	1216	−1024.6	−1020.6	−1291.8
1a2s	1272	−1913.5	−1900.3	−2155.0
1a7m	2809	−2155.5	−2179.8	−2666.1
1a63	2065	−2373.5	−2380.5	−2912.0
1vjw	828	−1237.9	−1226.6	−1411.4

Figure 7.

Correlation between MIBPB-III and the present model (OSM) in electrostatic solvation free energies of 22 proteins

It is still interesting to understand how important it is to use a slightly enlarged radius in smooth surface models.¹⁵² To this end, we carry out the present calculations by using the original CHARMM22 van der Waals radii, denoted as “Radii0”. This result is also listed in Table 13. It is seen that results from the original CHARMM22 van der Waals radii can have over 20% deviations from those of “Radii1”. This helps to come to a conclusion that for continuous dielectric models, it is necessary to enlarge atomic radii obtained from widely used force fields. Otherwise, the results will be inconsistent with those of other analysis. This is in agreement with the observation in the literature.¹⁵² The necessity of using larger radii is also shown clearly in Fig 8 by the differences of electrostatic solvation free energies obtained from the MIBPB-III and the present calculations with original radii (Radii0) or enlarged radii (Radii1).

Figure 8.

Differences between electrostatic solvation free energies obtained from the MIBPB and the present model with original radii (Radii0) or enlarged radii (Radii1).

Additionally, it is useful to demonstrate that the electrostatic potential function computed in the present OSM can be illustrated at arbitrary isosurface of the characteristic function S. This is done by first computing a sharp surface at a given S value, then projecting the ϕ value on the isosurface of a given S value. Figure 9 shows three plots of the electrostatic potentials at S = 0.25, 0.5 and 0.75. A comparison of these potentials indicates the fast/slow electrostatic potential changing regions in the solvent-solute boundary. These regions are also interactive regions in the protein-protein or protein-ligand interactions.

Figure 9.

Surface potential display of one protein(PDBID: 1frd) at different isosurfaces

Finally, it remains an important issue to further improve the computational efficiency, although systematical efforts have been made in this work to reduce CPU cost. Since the coupled generalized PB and geometry flow equations are needed to evolve self-consistently to reach the steady state, it takes more CPU time for the present method to calculate the total free energy than some existing approaches that compute the polar and nonpolar energies separately.

VI Concluding remarks

This paper presents a novel differential geometry based solvation model. A crucial concept in the present model is the characteristic function or the description function of solute molecules which is one inside the solute domain and zero inside the solvent. Near the solvent-solute boundary, the characteristic function gradually changes from one to zero over a region of transition. The exact position and width of the transition region are determined by a variational framework, which is formulated based on the total solvation free energy. As a key ingredient of the present framework, the total energy encompasses coupled polar and nonpolar contributions. The polar solvation free energy functional is described by the electrostatic theory at equilibrium, while the nonpolar solvation free energy functional consists of surface energy, mechanical work and attractive solvent-solute interactions. Both the polar and nonpolar solvation free energies are coupled through the characteristic function S. This approach is similar in spirit to the works of Sharp and Honig,¹⁴² Gilson et al.,⁶⁷ Dzubiella et al.^{49, 50} and some of our own work;^{15, 17} however, the present work is unique in its implementation of non-polar interactions and differential geometry based representation of continuum and discrete domains. The differential geometry aspects are built upon our geometric flow formulations and variational minimization of surface free energy.^{14, 15, 17, 172, 174, 175} In the present work, the characteristic function of the solute is naturally described by differential geometry theory of surfaces and manifolds. Geometric measure theory is utilized to convert the Lagrangian formulation of the surface into appropriate Eulerian formulation. By variation of the total solvation free energy functional with respect to the characteristic function and electrostatic potential, a generalized geometric flow equation for the electrostatic potential and a generalized Poisson-Boltzmann equation for the characteristic function are obtained. Unlike the standard Poisson-Boltzmann equation, the generalized Poisson-Boltzmann admits a smooth dielectric profile governed by the generalized geometric flow equation, which provides a physical description of the true solvent-solute dielectric boundary, according to the variational principle. The generalized geometric flow equation balances the intrinsic geometric curvature effect and external potential due to mechanical work, solvent-solute interactions, and the electrostatic potential. The solution of the generalized geometric flow equation and the generalized Poisson-Boltzmann equation leads to quantities for the direct evaluation of the solvation free energy.

Computational methods and algorithms are designed and developed for the solution of the generalized Poisson-Boltzmann equation and generalized geometric flow equations. Specifically, a second-order finite difference scheme is designed to solve the generalized Poisson-Boltzmann equation. The effect of the appropriate preconditioner to the basic Poisson-Boltzmann solver is explored. Both a simple-minded Euler method and an appropriate alternative direction implicit (ADI) scheme are constructed to solve the nonlinear generalized geometric flow equation. The ADI scheme allows a relatively large time stepping and provides better efficiency. Finally, two iterative approaches are designed and tested for the solution of the coupled Poisson-Boltzmann equation and the generalized geometric flow equation. All computational methods have been extensively validated.

Numerical applications of the present differential geometry based solvation paradigm are considered to a set of 17 small compounds and a set of proteins. The 17 small compounds consist of very different types of molecules, ranging from polar to nonpolar. The strong solvent-solute interactions are presented in many of these compounds. The present model takes a blind approach to these compounds, i.e., the same set of parameters are used for all the compounds without parameter fitting or adjustment according to an individual molecule. Our new model outperforms the standard Poisson-Boltzmann method and is compared well with the quantum mechanical method employed by Nicholls et al.¹¹³ To further demonstrate the utility and usefulness of the present differential geometry based solvation model, we considered a group of protein molecules in our study. Similar to the treatment of 17 small compounds, the same set parameters are applied to all the proteins to compute electrostatic free energies and/or solvation free energies. Results from present model compare well with those of quantum mechanical calculations based on the polarizable continuum method.¹⁰⁵ The present predictions have also been compared with those obtained with the standard Poisson-Boltzmann theory utilizing sharp interfaces.⁶⁵ Results from these two theories shown a very good consistency. However, our study on small compounds, particularly on the benzamide derivatives, indicates the need to refine the charge assignment for certain conjugated polar groups. Therefore, the incorporation of quantum mechanical description in the present theory is necessary. A differential geometry based multiscale solvation paradigm including continuum, discrete and quantum descriptions is under our consideration.