Level-Set Variational Implicit-Solvent Modeling of Biomolecules with the Coulomb-Field Approximation

Prediction of multiple dry-wet transition pathways with a mesoscale variational approach

Water fluctuates in a hydrophobic confinement, forming multiple dry and wet hydration states through evaporation and condensation. Transitions between such states are critical to both thermodynamics and kinetics of solute molecular processes, such as protein folding and protein-ligand binding and unbinding. To efficiently predict such dry-wet transition paths, we develop a hybrid approach that combines a variational implicit solvation model, a generalized string method for minimum free-energy paths, and the level-set numerical implementation. This approach is applied to three molecular systems: two hydrophobic plates, a carbon nanotube, and a synthetic host molecule Cucurbit[7]uril. Without an explicit description of individual water molecules, our mesoscale approach effectively captures multiple dry and wet hydration states, multiple dry-wet transition paths, such as those geometrically symmetric and asymmetric paths, and transition states, providing activation energy barriers between different states. Further analysis shows that energy barriers depend on mesoscopic lengths, such as the separation distance between the two plates and the cross section diameter of the nanotube, and that the electrostatic interactions strongly influence the dry-wet transitions. With the inclusion of solute atomic motion, general collective variables as reaction coordinates, and the finite-temperature string method, together with an improved treatment of continuum electrostatics, our approach can be further developed to sample an ensemble of transition paths, providing more accurate predictions of the transition kinetics.

Coupling Monte Carlo, Variational Implicit Solvation, and Binary Level-Set for Simulations of Biomolecular Binding

We develop a hybrid approach that combines the Monte Carlo (MC) method, a variational implicit-solvent model (VISM), and a binary level-set method for the simulation of biomolecular binding in an aqueous solvent. The solvation free energy for the biomolecular complex is estimated by minimizing the VISM free-energy functional of all possible solute-solvent interfaces that are used as dielectric boundaries. This functional consists of the solute volumetric, solute-solvent interfacial, solute-solvent van der Waals interaction, and electrostatic free energy. A technique of shifting the dielectric boundary is used to accurately predict the electrostatic part of the solvation free energy. Minimizing such a functional in each MC move is made possible by our new and fast binary level-set method. This method is based on the approximation of surface area by the convolution of an indicator function with a compactly supported kernel and is implemented by simple flips of numerical grid cells locally around the solute-solvent interface. We apply our approach to the p53-MDM2 system for which the two molecules are approximated by rigid bodies. Our efficient approach captures some of the poses before the final bound state. All-atom molecular dynamics simulations with most of such poses quickly reach the final bound state. Our work is a new step toward realistic simulations of biomolecular interactions. With further improvement of coarse graining and MC sampling, and combined with other models, our hybrid approach can be used to study the free-energy landscape and kinetic pathways of ligand binding to proteins.

Variational implicit-solvent predictions of the dry-wet transition pathways for ligand-receptor binding and unbinding kinetics

Ligand-receptor binding and unbinding are fundamental biomolecular processes and particularly essential to drug efficacy. Environmental water fluctuations, however, impact the corresponding thermodynamics and kinetics and thereby challenge theoretical descriptions. Here, we devise a holistic, implicit-solvent, multimethod approach to predict the (un)binding kinetics for a generic ligand-pocket model. We use the variational implicit-solvent model (VISM) to calculate the solute-solvent interfacial structures and the corresponding free energies, and combine the VISM with the string method to obtain the minimum energy paths and transition states between the various metastable ("dry" and "wet") hydration states. The resulting dry-wet transition rates are then used in a spatially dependent multistate continuous-time Markov chain Brownian dynamics simulation and the related Fokker-Planck equation calculations of the ligand stochastic motion, providing the mean first-passage times for binding and unbinding. We find the hydration transitions to significantly slow down the binding process, in semiquantitative agreement with existing explicit-water simulations, but significantly accelerate the unbinding process. Moreover, our methods allow the characterization of nonequilibrium hydration states of pocket and ligand during the ligand movement, for which we find substantial memory and hysteresis effects for binding vs. unbinding. Our study thus provides a significant step forward toward efficient, physics-based interpretation and predictions of the complex kinetics in realistic ligand-receptor systems.

Tailoring the Variational Implicit Solvent Method for New Challenges: Biomolecular Recognition and Assembly

Predicting solvation free energies and describing the complex water behavior that plays an important role in essentially all biological processes is a major challenge from the computational standpoint. While an atomistic, explicit description of the solvent can turn out to be too expensive in large biomolecular systems, most implicit solvent methods fail to capture “dewetting” effects and heterogeneous hydration by relying on a pre-established (i.e., guessed) solvation interface. Here we focus on the Variational Implicit Solvent Method, an implicit solvent method that adds water “plasticity” back to the picture by formulating the solvation free energy as a functional of all possible solvation interfaces. We survey VISM's applications to the problem of molecular recognition and report some of the most recent efforts to tailor VISM for more challenging scenarios, with the ultimate goal of including thermal fluctuations into the framework. The advances reported herein pave the way to make VISM a uniquely successful approach to characterize complex solvation properties in the recognition and binding of large-scale biomolecular complexes.

Implicit Solvation Using the Superposition Approximation (IS-SPA): An Implicit Treatment of the Nonpolar Component to Solvation for Simulating Molecular Aggregation

Nonpolar solute-solvent interactions are the driving force for aggregation in important chemical and biological phenomena including protein folding, peptide self-assembly, and oil-water emulsion formation. Currently, the most accurate and computationally efficient description of these processes requires an explicit treatment of all solvent and solute atoms. Previous computationally feasible implicit solvent models, such as solute surface area approaches, are unsuccessful at capturing aggregation features including both structural and energetic trends while more theoretically rigorous approaches, such as Reference Interaction Site Model (RISM), are accurate but extremely computationally demanding. Our approach, denoted Implicit Solvation using the Superposition Approximation (IS-SPA), builds on previous theory utilizing the Kirkwood superposition approximation to approximate the mean force of the solvent from solute parameters. We introduce and verify a parabolic first solvation shell truncation of atomic solvation, fitting water distributions around a molecule, and a Monte Carlo integration of the mean solvent force. These extensions allow this method to be implemented as an efficient nonpolar implicit solvent model for molecular simulation. The approximations in IS-SPA are first explored and justified for the homodimerization of an array of different sized Lennard-Jones spheres. The accuracy and transferability of the approach are demonstrated by its ability to capture the position and relative energies of the desolvation barrier and free energy minimum of alkane homodimers. The model is then shown to reproduce the phase separation and solubility of cyclohexane and water. These promising results, coupled with 2 orders of magnitude speed-up for dilute systems as compared to explicit solvent simulations, demonstrate that IS-SPA is an appealing approach to boost the time- and length-scale of molecular aggregation simulations.

"Martinizing" the Variational Implicit Solvent Method (VISM): Solvation Free Energy for Coarse-Grained Proteins

Solvation is a fundamental driving force in many biological processes including biomolecular recognition and self-assembly, not to mention protein folding, dynamics, and function. The variational implicit solvent method (VISM) is a theoretical tool currently developed and optimized to estimate solvation free energies for systems of very complex topology, such as biomolecules. VISM's theoretical framework makes it unique because it couples hydrophobic, van der Waals, and electrostatic interactions as a functional of the solvation interface. By minimizing this functional, VISM produces the solvation interface as an output of the theory. In this work, we push VISM to larger scale applications by combining it with coarse-grained solute Hamiltonians adapted from the MARTINI framework, a well-established mesoscale force field for modeling large-scale biomolecule assemblies. We show how MARTINI-VISM (^MVISM) compares with atomistic VISM (^AVISM) for a small set of proteins differing in size, shape, and charge distribution. We also demonstrate ^MVISM's suitability to study the solvation properties of an interesting encounter complex, barnase-barstar. The promising results suggest that coarse-graining the protein with the MARTINI force field is indeed a valuable step to broaden VISM's and MARTINI's applications in the near future.

Binding site elucidation and structure guided design of macrocyclic IL-17A antagonists

Interleukin-17A (IL-17A) is a principal driver of multiple inflammatory and immune disorders. Antibodies that neutralize IL-17A or its receptor (IL-17RA) deliver efficacy in autoimmune diseases, but no small-molecule IL-17A antagonists have yet progressed into clinical trials. Investigation of a series of linear peptide ligands to IL-17A and characterization of their binding site has enabled the design of novel macrocyclic ligands that are themselves potent IL-17A antagonists.

Stochastic level-set variational implicit-solvent approach to solute-solvent interfacial fluctuations

Recent years have seen the initial success of a variational implicit-solvent model (VISM), implemented with a robust level-set method, in capturing efficiently different hydration states and providing quantitatively good estimation of solvation free energies of biomolecules. The level-set minimization of the VISM solvation free-energy functional of all possible solute-solvent interfaces or dielectric boundaries predicts an equilibrium biomolecular conformation that is often close to an initial guess. In this work, we develop a theory in the form of Langevin geometrical flow to incorporate solute-solvent interfacial fluctuations into the VISM. Such fluctuations are crucial to biomolecular conformational changes and binding process. We also develop a stochastic level-set method to numerically implement such a theory. We describe the interfacial fluctuation through the "normal velocity" that is the solute-solvent interfacial force, derive the corresponding stochastic level-set equation in the sense of Stratonovich so that the surface representation is independent of the choice of implicit function, and develop numerical techniques for solving such an equation and processing the numerical data. We apply our computational method to study the dewetting transition in the system of two hydrophobic plates and a hydrophobic cavity of a synthetic host molecule cucurbit[7]uril. Numerical simulations demonstrate that our approach can describe an underlying system jumping out of a local minimum of the free-energy functional and can capture dewetting transitions of hydrophobic systems. In the case of two hydrophobic plates, we find that the wavelength of interfacial fluctuations has a strong influence to the dewetting transition. In addition, we find that the estimated energy barrier of the dewetting transition scales quadratically with the inter-plate distance, agreeing well with existing studies of molecular dynamics simulations. Our work is a first step toward the inclusion of fluctuations into the VISM and understanding the impact of interfacial fluctuations on biomolecular solvation with an implicit-solvent approach.

Identification of protein-ligand binding sites by the level-set variational implicit-solvent approach

Protein–ligand binding is a key biological process at the molecular level. The identification and characterization of small-molecule binding sites on therapeutically relevant proteins have tremendous implications for target evaluation and rational drug design. In this work, we used the recently developed level-set variational implicit-solvent model (VISM) with the Coulomb field approximation (CFA) to locate and characterize potential protein–small-molecule binding sites. We applied our method to a data set of 515 protein–ligand complexes and found that 96.9% of the cocrystallized ligands bind to the VISM-CFA-identified pockets and that 71.8% of the identified pockets are occupied by cocrystallized ligands. For 228 tight-binding protein–ligand complexes (i.e, complexes with experimental pK_d values larger than 6), 99.1% of the cocrystallized ligands are in the VISM-CFA-identified pockets. In addition, it was found that the ligand binding orientations are consistent with the hydrophilic and hydrophobic descriptions provided by VISM. Quantitative characterization of binding pockets with topological and physicochemical parameters was used to assess the “ligandability” of the pockets. The results illustrate the key interactions between ligands and receptors and can be very informative for rational drug design.

Numerical Treatment of Stokes Solvent Flow and Solute-Solvent Interfacial Dynamics for Nonpolar Molecules

We design and implement numerical methods for the incompressible Stokes solvent flow and solute-solvent interface motion for nonpolar molecules in aqueous solvent. The balance of viscous force, surface tension, and van der Waals type dispersive force leads to a traction boundary condition on the solute-solvent interface. To allow the change of solute volume, we design special numerical boundary conditions on the boundary of a computational domain through a consistency condition. We use a finite difference ghost fluid scheme to discretize the Stokes equation with such boundary conditions. The method is tested to have a second-order accuracy. We combine this ghost fluid method with the level-set method to simulate the motion of the solute-solvent interface that is governed by the solvent fluid velocity. Numerical examples show that our method can predict accurately the blow up time for a test example of curvature flow and reproduce the polymodal (e.g., dry and wet) states of hydration of some simple model molecular systems.

A self-consistent phase-field approach to implicit solvation of charged molecules with Poisson-Boltzmann electrostatics

Dielectric boundary based implicit-solvent models provide efficient descriptions of coarse-grained effects, particularly the electrostatic effect, of aqueous solvent. Recent years have seen the initial success of a new such model, variational implicit-solvent model (VISM) [Dzubiella, Swanson, and McCammon Phys. Rev. Lett. 96, 087802 (2006) and J. Chem. Phys. 124, 084905 (2006)], in capturing multiple dry and wet hydration states, describing the subtle electrostatic effect in hydrophobic interactions, and providing qualitatively good estimates of solvation free energies. Here, we develop a phase-field VISM to the solvation of charged molecules in aqueous solvent to include more flexibility. In this approach, a stable equilibrium molecular system is described by a phase field that takes one constant value in the solute region and a different constant value in the solvent region, and smoothly changes its value on a thin transition layer representing a smeared solute-solvent interface or dielectric boundary. Such a phase field minimizes an effective solvation free-energy functional that consists of the solute-solvent interfacial energy, solute-solvent van der Waals interaction energy, and electrostatic free energy described by the Poisson-Boltzmann theory. We apply our model and methods to the solvation of single ions, two parallel plates, and protein complexes BphC and p53/MDM2 to demonstrate the capability and efficiency of our approach at different levels. With a diffuse dielectric boundary, our new approach can describe the dielectric asymmetry in the solute-solvent interfacial region. Our theory is developed based on rigorous mathematical studies and is also connected to the Lum-Chandler-Weeks theory (1999). We discuss these connections and possible extensions of our theory and methods.

Target the More Druggable Protein States in a Highly Dynamic Protein--Protein Interaction System

The proteins of the Bcl-2 family play key roles in the regulation of programmed cell death by controlling the integrity of the outer mitochondrial membrane and the initiation of the apoptosis process. We performed extensive molecular dynamics simulations to investigate the conformational flexibility of the Bcl-xL protein in both the apo and holo (with Bad peptide and ABT-737) states. The accelerated molecular dynamics method implemented in Amber 14 was used to produce broader conformational sampling of 200 ns simulations. The pocket mining method based on the variational implicit-solvent model tracks the dynamic evolution of the ligand binding site with a druggability score characterizing the maximal affinity achievable by a drug-like molecule. Major movements were observed around the α3-helical domain and the loop region connecting the α1 and α2 helices, reshaping the ligand interaction in the BH3 binding groove. Starting with the apo crystal structure, which is recognized as "closed" and undruggable, the BH3 groove transitioned between the "open" and "closed" states during equilibrium simulation. Further analysis revealed a small percentage of the trajectory frames (∼10%) with a moderate degree of druggability that mimic the ligand-bound states. The ability to attain and detect by computer simulation the most suitable conformational states for ligand binding in advance of compound synthesis and crystal structure solution is of immense value to the application and success of structure-based drug design.

DIFFUSED SOLUTE-SOLVENT INTERFACE WITH POISSON-BOLTZMANN ELECTROSTATICS: FREE-ENERGY VARIATION AND SHARP-INTERFACE LIMIT

A phase-field free-energy functional for the solvation of charged molecules (e.g., proteins) in aqueous solvent (i.e., water or salted water) is constructed. The functional consists of the solute volumetric and solute-solvent interfacial energies, the solute-solvent van der Waals interaction energy, and the continuum electrostatic free energy described by the Poisson-Boltzmann theory. All these are expressed in terms of phase fields that, for low free-energy conformations, are close to one value in the solute phase and another in the solvent phase. A key property of the model is that the phase-field interpolation of dielectric coefficient has the vanishing derivative at both solute and solvent phases. The first variation of such an effective free-energy functional is derived. Matched asymptotic analysis is carried out for the resulting relaxation dynamics of the diffused solute-solvent interface. It is shown that the sharp-interface limit is exactly the variational implicit-solvent model that has successfully captured capillary evaporation in hydrophobic confinement and corresponding multiple equilibrium states of underlying biomolecular systems as found in experiment and molecular dynamics simulations. Our phase-field approach and analysis can be used to possibly couple the description of interfacial fluctuations for efficient numerical computations of biomolecular interactions.

LS-VISM: A software package for analysis of biomolecular solvation

We introduce a software package for the analysis of biomolecular solvation. The package collects computer codes that implement numerical methods for a variational implicit-solvent model (VISM). The input of the package includes the atomic data of biomolecules under consideration and the macroscopic parameters such as solute-solvent surface tension, bulk solvent density and ionic concentrations, and the dielectric coefficients. The output includes estimated solvation free energies and optimal macroscopic solute-solvent interfaces that are obtained by minimizing the VISM solvation free-energy functional among all possible solute-solvent interfaces enclosing the solute atoms. We review the VISM with various descriptions of electrostatics. We also review our numerical methods that consist mainly of the level-set method for relaxing the VISM free-energy functional and a compact coupling interface method for the dielectric Poisson-Boltzmann equation. Such numerical methods and algorithms constitute the central modules of the software package. We detail the structure of the package, format of input and output files, workflow of the codes, and the postprocessing of output data. Our demo application to a host-guest system illustrates how to use the package to perform solvation analysis for biomolecules, including ligand-receptor binding systems. The package is simple and flexible with respect to minimum adjustable parameters and a wide range of applications. Future extensions of the package use can include the efficient identification of ligand binding pockets on protein surfaces.

STABILITY OF A CYLINDRICAL SOLUTE-SOLVENT INTERFACE: EFFECT OF GEOMETRY, ELECTROSTATICS, AND HYDRODYNAMICS

The solute-solvent interface that separates biological molecules from their surrounding aqueous solvent characterizes the conformation and dynamics of such molecules. In this work, we construct a solvent fluid dielectric boundary model for the solvation of charged molecules and apply it to study the stability of a model cylindrical solute-solvent interface. The motion of the solute-solvent interface is defined to be the same as that of solvent fluid at the interface. The solvent fluid is assumed to be incompressible and is described by the Stokes equation. The solute is modeled simply by the ideal-gas law. All the viscous force, hydrostatic pressure, solute-solvent van der Waals interaction, surface tension, and electrostatic force are balanced at the solute-solvent interface. We model the electrostatics by Poisson's equation in which the solute-solvent interface is treated as a dielectric boundary that separates the low-dielectric solute from the high-dielectric solvent. For a cylindrical geometry, we find multiple cylindrically shaped equilibrium interfaces that describe polymodal (e.g., dry and wet) states of hydration of an underlying molecular system. These steady-state solutions exhibit bifurcation behavior with respect to the charge density. For their linearized systems, we use the projection method to solve the fluid equation and find the dispersion relation. Our asymptotic analysis shows that, for large wavenumbers, the decay rate is proportional to wavenumber with the proportionality half of the ratio of surface tension to solvent viscosity, indicating that the solvent viscosity does affect the stability of a solute-solvent interface. Consequences of our analysis in the context of biomolecular interactions are discussed.

Variational Implicit Solvation with Poisson-Boltzmann Theory

We incorporate the Poisson-Boltzmann (PB) theory of electrostatics into our variational implicit-solvent model (VISM) for the solvation of charged molecules in an aqueous solvent. In order to numerically relax the VISM free-energy functional by our level-set method, we develop highly accurate methods for solving the dielectric PB equation and for computing the dielectric boundary force. We also apply our VISM-PB theory to analyze the solvent potentials of mean force and the effect of charges on the hydrophobic hydration for some selected molecular systems. These include some single ions, two charged particles, two charged plates, and the host-guest system Cucurbit[7]uril and Bicyclo[2.2.2]octane. Our computational results show that VISM with PB theory can capture well the sensitive response of capillary evaporation to the charge in hydrophobic confinement and the polymodal hydration behavior and can provide accurate estimates of binding affinity of the host-guest system. We finally discuss several issues for further improvement of VISM.

Heterogeneous Hydration of p53/MDM2 Complex

Water-mediated interactions play critical roles in biomolecular recognition processes. Explicit solvent molecular dynamics (MD) simulations and the variational implicit-solvent model (VISM) are used to study those hydration properties during binding for the biologically important p53/MDM2 complex. Unlike simple model solutes, in such a realistic and heterogeneous solute–solvent system with both geometrical and chemical complexity, the local water distribution sensitively depends on nearby amino acid properties and the geometric shape of the protein. We show that the VISM can accurately describe the locations of high and low density solvation shells identified by the MD simulations and can explain them by a local coupling balance of solvent–solute interaction potentials and curvature. In particular, capillary transitions between local dry and wet hydration states in the binding pocket are captured for interdomain distance between 4 to 6 Å, right at the onset of binding. The underlying physical connection between geometry and polarity is illustrated and quantified. Our study offers a microscopic and physical insight into the heterogeneous hydration behavior of the biologically highly relevant p53/MDM2 system and demonstrates the fundamental importance of hydrophobic effects for biological binding processes. We hope our study can help to establish new design rules for drugs and medical substances.

Variational Implicit-Solvent Modeling of Host-Guest Binding: A Case Study on Cucurbit[7]uril|

The synthetic host cucurbit[7]uril (CB[7]) binds aromatic guests or metal complexes with ultrahigh affinity compared with that typically displayed in protein–ligand binding. Due to its small size, CB[7] serves as an ideal receptor–ligand system for developing computational methods for molecular recognition. Here, we apply the recently developed variational implicit-solvent model (VISM), numerically evaluated by the level-set method, to study hydration effects in the high-affinity binding of the B2 bicyclo[2.2.2]octane derivative to CB[7]. For the unbound host, we find that the host cavity favors the hydrated state over the dry state due to electrostatic effects. For the guest binding, we find reasonable agreement to experimental binding affinities. Dissection of the individual VISM free-energy contributions shows that the major driving forces are water-mediated hydrophobic interactions and the intrinsic (vacuum) host–guest van der Waals interactions. These findings are in line with recent experiments and molecular dynamics simulations with explicit solvent. It is expected that the level-set VISM, with further refinement on the electrostatic descriptions, can efficiently predict molecular binding and recognition in a wide range of future applications.

Phase-field approach to implicit solvation of biomolecules with Coulomb-field approximation

A phase-field variational implicit-solvent approach is developed for the solvation of charged molecules. The starting point of such an approach is the representation of a solute-solvent interface by a phase field that takes one value in the solute region and another in the solvent region, with a smooth transition from one to the other on a small transition layer. The minimization of an effective free-energy functional of all possible phase fields determines the equilibrium conformations and free energies of an underlying molecular system. All the surface energy, the solute-solvent van der Waals interaction, and the electrostatic interaction are coupled together self-consistently through a phase field. The surface energy results from the minimization of a double-well potential and the gradient of a field. The electrostatic interaction is described by the Coulomb-field approximation. Accurate and efficient methods are designed and implemented to numerically relax an underlying charged molecular system. Applications to single ions, a two-plate system, and a two-domain protein reveal that the new theory and methods can capture capillary evaporation in hydrophobic confinement and corresponding multiple equilibrium states as found in molecular dynamics simulations. Comparisons of the phase-field and the original sharp-interface variational approaches are discussed.

Evaluation of Hydration Free Energy by Level-Set Variational Implicit-Solvent Model with Coulomb-Field Approximation

In this article, we systematically apply a novel implicit-solvent model, the variational implicit-solvent model (VISM) together with the Coulomb-Field Approximation (CFA), to calculate the hydration free energy of a large set of small organic molecules. Because these molecules have been studied in detail by molecular dynamics simulations and other implicit-solvent models, they provide a good benchmark for evaluating the performance of VISM-CFA. With all-atom Amber force field parameters, VISM-CFA is able to reproduce well not only the experimental and MD simulated total hydration free energy but also the polar and nonpolar contributions individually. The correlation between VISM-CFA and experiments is R² = 0.763 for the total hydration free energy, with a root-mean-square deviation (RMSD) of 1.83 kcal/mol, and the correlation to results from TIP3P explicit water MD simulations is R² = 0.839 with a RMSD = 1.36 kcal/mol. In addition, we demonstrate that VISM captures dewetting phenomena in the p53/MDM2 complex and hydrophobic characteristics in the system. This work demonstrates that the level-set VISM-CFA can be used to study the energetic behavior of realistic molecular systems with complicated geometries in solvation, protein–ligand binding, protein–protein association, and protein folding processes.

Ionic Interactions Are Everywhere

Ionic solutions are dominated by interactions because they must be electrically neutral, but classical theory assumes no interactions. Biological solutions are rather like seawater, concentrated enough so that the diameter of ions also produces important interactions. In my view, the theory of complex fluids is needed to deal with the interacting reality of biological solutions.

Variational Implicit Solvation with Solute Molecular Mechanics: From Diffuse-Interface to Sharp-Interface Models

Central in a variational implicit-solvent description of biomolecular solvation is an effective free-energy functional of the solute atomic positions and the solute-solvent interface (i.e., the dielectric boundary). The free-energy functional couples together the solute molecular mechanical interaction energy, the solute-solvent interfacial energy, the solute-solvent van der Waals interaction energy, and the electrostatic energy. In recent years, the sharp-interface version of the variational implicit-solvent model has been developed and used for numerical computations of molecular solvation. In this work, we propose a diffuse-interface version of the variational implicit-solvent model with solute molecular mechanics. We also analyze both the sharp-interface and diffuse-interface models. We prove the existence of free-energy minimizers and obtain their bounds. We also prove the convergence of the diffuse-interface model to the sharp-interface model in the sense of Γ-convergence. We further discuss properties of sharp-interface free-energy minimizers, the boundary conditions and the coupling of the Poisson-Boltzmann equation in the diffuse-interface model, and the convergence of forces from diffuse-interface to sharp-interface descriptions. Our analysis relies on the previous works on the problem of minimizing surface areas and on our observations on the coupling between solute molecular mechanical interactions with the continuum solvent. Our studies justify rigorously the self consistency of the proposed diffuse-interface variational models of implicit solvation.

Motion of a Cylindrical Dielectric Boundary

The interplay between geometry and electrostatics contributes significantly to hydrophobic interactions of biomolecules in an aqueous solution. With an implicit solvent, such a system can be described macroscopically by the dielectric boundary that separates the high-dielectric solvent from low-dielectric solutes. This work concerns the motion of a model cylindrical dielectric boundary as the steepest descent of a free-energy functional that consists of both the surface and electrostatic energies. The effective dielectric boundary force is defined and an explicit formula of the force is obtained. It is found that such a force always points from the solvent region to solute region. In the case that the interior of a cylinder is of a lower dielectric, the motion of the dielectric boundary is initially driven dominantly by the surface force but is then driven inward quickly to the cylindrical axis by both the surface and electrostatic forces. In the case that the interior of a cylinder is of a higher dielectric, the competition between the geometrical and electrostatic contributions leads to the existence of equilibrium boundaries that are circular cylinders. Linear stability analysis is presented to show that such an equilibrium is only stable for a perturbation with a wavenumber larger than a critical value. Numerical simulations are reported for both of the cases, confirming the analysis on the role of each component of the driving force. Implications of the mathematical findings to the understanding of charged molecular systems are discussed.