An Unfitted Finite Element Poisson-Boltzmann Solver with Automatic Resolving of Curved Molecular Surface

So far, the existing Poisson-Boltzmann (PB) solvers that accurately take into account the interface jump conditions need a pregenerated body-fitted mesh (molecular surface mesh). However, qualified biomolecular surface meshing and its implementation into numerical methods remains a challenging and laborious issue, which practically hinders the progress of further developments and applications of a bunch of numerical methods in this field. In addition, even with a molecular surface mesh, it is only a low-order approximation of the original curved surface. In this article, an interface-penalty finite element method (IPFEM), which is a typical unfitted finite element method, is proposed to solve the Poisson-Boltzmann equation (PBE) without requiring the user to generate a molecular surface mesh. The Gaussian molecular surface is used to represent the molecular surface and can be automatically resolved with a high-order approximation within our method. Theoretical convergence rates of the IPFEM for the linear PB equation have been provided and are well validated on a benchmark problem with an analytical solution (we also noticed from numerical examples that the IPFEM has similar convergence rates for the nonlinear PBE). Numerical results on a set of different-sized biomolecules demonstrate that the IPFEM is numerically stable and accurate in the calculation of biomolecular electrostatic solvation energy.

Calculation of electrostatic free energy for the nonlinear Poisson-Boltzmann model based on the dimensionless potential

The Poisson-Boltzmann (PB) equation governing the electrostatic potential with a unit is often transformed to a normalized form for a dimensionless potential in numerical studies. To calculate the electrostatic free energy (EFE) of biological interests, a unit conversion has to be conducted, because the existing PB energy functionals are all described in terms of the original potential. To bypass this conversion, this paper proposes energy functionals in terms of the dimensionless potential for the first time in the literature, so that the normalized PB equation can be directly derived by using the Euler-Lagrange variational analysis. Moreover, alternative energy forms have been rigorously derived to avoid approximating the gradient of singular functions in the electrostatic stress term. A systematic study has been carried out to examine the surface integrals involved in alternative energy forms and their dependence on finite domain size and mesh step size, which leads to a recommendation on the EFE forms for efficient computation of protein systems. The calculation of the EFE in the regularization formulation, which is an analytical approach for treating singular charge sources of the PB equation, has also been studied. The proposed energy forms have been validated by considering smooth dielectric settings, such as diffuse interface and super-Gaussian, for which the EFE of the nonlinear PB model is found to be significantly different from that of the linearized PB model. All proposed energy functionals and EFE forms are designed such that the dimensionless potential can be simply plugged in to compute the EFE in the unit of kcal/mol, and they can also be applied in the classical sharp interface PB model.

Solvation Thermodynamics of Solutes in Water and Ionic Liquids Using the Multiscale Solvation-Layer Interface Condition Continuum Model

Molecular assembly processes are generally driven by thermodynamic properties in solutions. Atomistic modeling can be very helpful in designing and understanding complex systems, except that bulk solvent is very inefficient to treat explicitly as discrete molecules. In this work, we develop and assess two multiscale solvation models for computing solvation thermodynamic properties. The new SLIC/CDC model combines continuum solvent electrostatics based on the solvent layer interface condition (SLIC) with new statistical thermodynamic models for hydrogen bonding and nonpolar modes: cavity formation, dispersion interactions, combinatorial mixing (CDC). Given the structures of 500 solutes, the SLIC/CDC model predicts Gibbs energies of solvation in water with an average accuracy better than 1 kcal/mol, when compared to experimental measurements, and better than 0.8 kcal/mol, when compared to explicit-solvent molecular dynamics simulations. The individual SLIC/CDC energy mode values agree quantitatively with those computed from explicit-solvent molecular dynamics. The previously published SLIC/SASA multiscale model combines the SLIC continuum electrostatic model with the solvent-accessible surface area (SASA) nonpolar energy mode. With our new, improved parametrization method, the SLIC/SASA model now predicts Gibbs energies of solvation with better than 1.4 kcal/mol average accuracy in aqueous systems, compared to experimental and explicit-solvent molecular dynamics, and better than 1.6 kcal/mol average accuracy in ionic liquids, compared to explicit-solvent molecular dynamics. Both models predict solvation entropies, and are the first implicit-solvation models capable of predicting solvation heat capacities.

A stabilized finite volume element method for solving Poisson-Nernst-Planck equations

One difficulty in solving the Poisson-Nernst-Planck (PNP) equations used for studying the ion transport in channel proteins is the possible convection-dominant problem in the Nernst-Planck equations. In this paper, to overcome this issue, considering the general mixed boundary conditions of concentration functions on the interface, a novel stabilized finite volume element method based on the standard weak formulation to solve the steady-state PNP equations is proposed and analyzed. Numerical tests on four ion-channel proteins served as benchmark with varying boundary conditions in a certain range show that the new stabilized technique not only improves the robustness of the new PNP solver, but also makes the computed (especially the maximal) concentration values much more reasonable.

Computational modeling of protein conformational changes - Application to the opening SARS-CoV-2 spike

We present a new approach to compute and analyze the dynamical electro-geometric properties of proteins undergoing conformational changes. The molecular trajectory is obtained from Markov state models, and the electrostatic potential is calculated using the continuum Poisson-Boltzmann equation. The numerical electric potential is constructed using a parallel sharp numerical solver implemented on adaptive Octree grids. We introduce novel a posteriori error estimates to quantify the solution's accuracy on the molecular surface. To illustrate the approach, we consider the opening of the SARS-CoV-2 spike protein using the recent molecular trajectory simulated through the Folding@home initiative. We analyze our results, focusing on the characteristics of the receptor-binding domain and its vicinity. This work lays the foundation for a new class of hybrid computational approaches, producing high-fidelity dynamical computational measurements serving as a basis for protein bio-mechanism investigations.

An improved Poisson-Nernst-Planck ion channel model and numerical studies on effects of boundary conditions, membrane charges, and bulk concentrations

In this paper, an improved Poisson-Nernst-Planck ion channel (PNPic) model is presented, along with its effective finite element solver and software package for an ion channel protein in a solution of multiple ionic species. Numerical studies are then done on the effects of boundary value conditions, membrane charges, and bulk concentrations on electrostatics and ionic concentrations for an ion channel protein, a gramicidin A (gA), and five different ionic solvents with up to four species. Numerical results indicate that the cation selectivity property of gA occurs within a central portion of ion channel pore, insensitively to any change of boundary value condition, membrane charge, or bulk concentration. Moreover, a numerical scheme for computing the electric currents induced by ion transports across membrane via an ion channel pore is presented and implemented as a part of the PNPic finite element package. It is then applied to the calculation of current-voltage curves, well validating the PNPic model and finite element package by electric current experimental data.

On regularization of charge singularities in solving the Poisson-Boltzmann equation with a smooth solute-solvent boundary

Numerical treatment of singular charges is a grand challenge in solving the Poisson-Boltzmann (PB) equation for analyzing electrostatic interactions between the solute biomolecules and the surrounding solvent with ions. For diffuse interface PB models in which solute and solvent are separated by a smooth boundary, no effective algorithm for singular charges has been developed, because the fundamental solution with a space dependent dielectric function is intractable. In this work, a novel regularization formulation is proposed to capture the singularity analytically, which is the first of its kind for diffuse interface PB models. The success lies in a dual decomposition - besides decomposing the potential into Coulomb and reaction field components, the dielectric function is also split into a constant base plus space changing part. Using the constant dielectric base, the Coulomb potential is represented analytically via Green's functions. After removing the singularity, the reaction field potential satisfies a regularized PB equation with a smooth source. To validate the proposed regularization, a Gaussian convolution surface (GCS) is also introduced, which efficiently generates a diffuse interface for three-dimensional realistic biomolecules. The performance of the proposed regularization is examined by considering both analytical and GCS diffuse interfaces, and compared with the trilinear method. Moreover, the proposed GCS-regularization algorithm is validated by calculating electrostatic free energies for a set of proteins and by estimating salt affinities for seven protein complexes. The results are consistent with experimental data and estimates of sharp interface PB models.

Enriched gradient recovery for interface solutions of the Poisson-Boltzmann equation

Accurate calculation of electrostatic potential and gradient on the molecular surface is highly desirable for the continuum and hybrid modeling of large scale deformation of biomolecules in solvent. In this article a new numerical method is proposed to calculate these quantities on the dielectric interface from the numerical solutions of the Poisson-Boltzmann equation. Our method reconstructs a potential field locally in the least square sense on the polynomial basis enriched with Green's functions, the latter characterize the Coulomb potential induced by charges near the position of reconstruction. This enrichment resembles the decomposition of electrostatic potential into singular Coulomb component and the regular reaction field in the Generalized Born methods. Numerical experiments demonstrate that the enrichment recovery produces drastically more accurate and stable potential gradients on molecular surfaces compared to classical recovery techniques.

Molecular Mean-Field Theory of Ionic Solutions: A Poisson-Nernst-Planck-Bikerman Model

We have developed a molecular mean-field theory—fourth-order Poisson–Nernst–Planck–Bikerman theory—for modeling ionic and water flows in biological ion channels by treating ions and water molecules of any volume and shape with interstitial voids, polarization of water, and ion-ion and ion-water correlations. The theory can also be used to study thermodynamic and electrokinetic properties of electrolyte solutions in batteries, fuel cells, nanopores, porous media including cement, geothermal brines, the oceanic system, etc. The theory can compute electric and steric energies from all atoms in a protein and all ions and water molecules in a channel pore while keeping electrolyte solutions in the extra- and intracellular baths as a continuum dielectric medium with complex properties that mimic experimental data. The theory has been verified with experiments and molecular dynamics data from the gramicidin A channel, L-type calcium channel, potassium channel, and sodium/calcium exchanger with real structures from the Protein Data Bank. It was also verified with the experimental or Monte Carlo data of electric double-layer differential capacitance and ion activities in aqueous electrolyte solutions. We give an in-depth review of the literature about the most novel properties of the theory, namely Fermi distributions of water and ions as classical particles with excluded volumes and dynamic correlations that depend on salt concentration, composition, temperature, pressure, far-field boundary conditions etc. in a complex and complicated way as reported in a wide range of experiments. The dynamic correlations are self-consistent output functions from a fourth-order differential operator that describes ion-ion and ion-water correlations, the dielectric response (permittivity) of ionic solutions, and the polarization of water molecules with a single correlation length parameter.

An efficient second-order poisson-boltzmann method

Immersed interface method (IIM) is a promising high-accuracy numerical scheme for the Poisson-Boltzmann model that has been widely used to study electrostatic interactions in biomolecules. However, the IIM suffers from instability and slow convergence for typical applications. In this study, we introduced both analytical interface and surface regulation into IIM to address these issues. The analytical interface setup leads to better accuracy and its convergence closely follows a quadratic manner as predicted by theory. The surface regulation further speeds up the convergence for nontrivial biomolecules. In addition, uncertainties of the numerical energies for tested systems are also reduced by about half. More interestingly, the analytical setup significantly improves the linear solver efficiency and stability by generating more precise and better-conditioned linear systems. Finally, we implemented the bottleneck linear system solver on GPUs to further improve the efficiency of the method, so it can be widely used for practical biomolecular applications. © 2019 Wiley Periodicals, Inc.

Poisson-Fermi modeling of ion activities in aqueous single and mixed electrolyte solutions at variable temperature

The combinatorial explosion of empirical parameters in tens of thousands presents a tremendous challenge for extended Debye-Hückel models to calculate activity coefficients of aqueous mixtures of the most important salts in chemistry. The explosion of parameters originates from the phenomenological extension of the Debye-Hückel theory that does not take steric and correlation effects of ions and water into account. By contrast, the Poisson-Fermi theory developed in recent years treats ions and water molecules as nonuniform hard spheres of any size with interstitial voids and includes ion-water and ion-ion correlations. We present a Poisson-Fermi model and numerical methods for calculating the individual or mean activity coefficient of electrolyte solutions with any arbitrary number of ionic species in a large range of salt concentrations and temperatures. For each activity-concentration curve, we show that the Poisson-Fermi model requires only three unchanging parameters at most to well fit the corresponding experimental data. The three parameters are associated with the Born radius of the solvation energy of an ion in electrolyte solution that changes with salt concentrations in a highly nonlinear manner.

SDPBS Web Server for Calculation of Electrostatics of Ionic Solvated Biomolecules

The Poisson-Boltzmann equation (PBE) is one important implicit solvent continuum model for calculating electrostatics of protein in ionic solvent. We recently developed a PBE solver library, called SDPBS, that incorporates the finite element, finite difference, solution decomposition, domain decomposition, and multigrid methods. To make SDPBS more accessible to the scientific community, we present an SDPBS web server in this paper that allows clients to visualize and manipulate the molecular structure of a biomolecule, and to calculate PBE solutions in a remote and user friendly fashion. The web server is available on the website https://lsextrnprod.uwm.edu/electrostatics/.

Communication: modeling charge-sign asymmetric solvation free energies with nonlinear boundary conditions

We show that charge-sign-dependent asymmetric hydration can be modeled accurately using linear Poisson theory after replacing the standard electric-displacement boundary condition with a simple nonlinear boundary condition. Using a single multiplicative scaling factor to determine atomic radii from molecular dynamics Lennard-Jones parameters, the new model accurately reproduces MD free-energy calculations of hydration asymmetries for: (i) monatomic ions, (ii) titratable amino acids in both their protonated and unprotonated states, and (iii) the Mobley "bracelet" and "rod" test problems [D. L. Mobley, A. E. Barber II, C. J. Fennell, and K. A. Dill, "Charge asymmetries in hydration of polar solutes," J. Phys. Chem. B 112, 2405-2414 (2008)]. Remarkably, the model also justifies the use of linear response expressions for charging free energies. Our boundary-element method implementation demonstrates the ease with which other continuum-electrostatic solvers can be extended to include asymmetry.

Biological applications of classical electrostatics methods

Continuum electrostatics modeling of solvation based on the Poisson–Boltzmann (PB) equation has gained wide acceptance in biomolecular applications such as energetic analysis and structural visualization. Successful application of the PB solvent models requires careful calibration of the solvation parameters. Extensive testing and validation is also important to ensure accuracy in their applications. Limitation in the continuum modeling of solvation is also a known issue in certain biomolecular applications. Growing interest in membrane systems has further spurred developmental efforts to allow inclusion of membrane in the PB solvent models. Despite their past successes due to careful parameterization, algorithm development and parallel implementation, there is still much to be done to improve their transferability from the small molecular systems upon which they were developed and validated to complex macromolecular systems as advances in technology continue to push forward, providing ever greater computational resources to researchers to study more interesting biological systems of higher complexity.

A Stabilized Finite Element Method for Modified Poisson-Nernst-Planck Equations to Determine Ion Flow Through a Nanopore

The conventional Poisson-Nernst-Planck equations do not account for the finite size of ions explicitly. This leads to solutions featuring unrealistically high ionic concentrations in the regions subject to external potentials, in particular, near highly charged surfaces. A modified form of the Poisson-Nernst-Planck equations accounts for steric effects and results in solutions with finite ion concentrations. Here, we evaluate numerical methods for solving the modified Poisson-Nernst-Planck equations by modeling electric field-driven transport of ions through a nanopore. We describe a novel, robust finite element solver that combines the applications of the Newton's method to the nonlinear Galerkin form of the equations, augmented with stabilization terms to appropriately handle the drift-diffusion processes. To make direct comparison with particle-based simulations possible, our method is specifically designed to produce solutions under periodic boundary conditions and to conserve the number of ions in the solution domain. We test our finite element solver on a set of challenging numerical experiments that include calculations of the ion distribution in a volume confined between two charged plates, calculations of the ionic current though a nanopore subject to an external electric field, and modeling the effect of a DNA molecule on the ion concentration and nanopore current.

Exploring accurate Poisson-Boltzmann methods for biomolecular simulations

Accurate and efficient treatment of electrostatics is a crucial step in computational analyses of biomolecular structures and dynamics. In this study, we have explored a second-order finite-difference numerical method to solve the widely used Poisson-Boltzmann equation for electrostatic analyses of realistic bio-molecules. The so-called immersed interface method was first validated and found to be consistent with the classical weighted harmonic averaging method for a diversified set of test biomolecules. The numerical accuracy and convergence behaviors of the new method were next analyzed in its computation of numerical reaction field grid potentials, energies, and atomic solvation forces. Overall similar convergence behaviors were observed as those by the classical method. Interestingly, the new method was found to deliver more accurate and better-converged grid potentials than the classical method on or nearby the molecular surface, though the numerical advantage of the new method is reduced when grid potentials are extrapolated to the molecular surface. Our exploratory study indicates the need for further improving interpolation/extrapolation schemes in addition to the developments of higher-order numerical methods that have attracted most attention in the field.

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.

Biomolecular surface construction by PDE transform

This work proposes a new framework for the surface generation based on the partial differential equation (PDE) transform. The PDE transform has recently been introduced as a general approach for the mode decomposition of images, signals, and data. It relies on the use of arbitrarily high-order PDEs to achieve the time-frequency localization, control the spectral distribution, and regulate the spatial resolution. The present work provides a new variational derivation of high-order PDE transforms. The fast Fourier transform is utilized to accomplish the PDE transform so as to avoid stringent stability constraints in solving high-order PDEs. As a consequence, the time integration of high-order PDEs can be done efficiently with the fast Fourier transform. The present approach is validated with a variety of test examples in two-dimensional and three-dimensional settings. We explore the impact of the PDE transform parameters, such as the PDE order and propagation time, on the quality of resulting surfaces. Additionally, we utilize a set of 10 proteins to compare the computational efficiency of the present surface generation method and a standard approach in Cartesian meshes. Moreover, we analyze the present method by examining some benchmark indicators of biomolecular surface, that is, surface area, surface-enclosed volume, solvation free energy, and surface electrostatic potential. A test set of 13 protein molecules is used in the present investigation. The electrostatic analysis is carried out via the Poisson-Boltzmann equation model. To further demonstrate the utility of the present PDE transform-based surface method, we solve the Poisson-Nernst-Planck equations with a PDE transform surface of a protein. Second-order convergence is observed for the electrostatic potential and concentrations. Finally, to test the capability and efficiency of the present PDE transform-based surface generation method, we apply it to the construction of an excessively large biomolecule, a virus surface capsid. Virus surface morphologies of different resolutions are attained by adjusting the propagation time. Therefore, the present PDE transform provides a multiresolution analysis in the surface visualization. Extensive numerical experiment and comparison with an established surface model indicate that the present PDE transform is a robust, stable, and efficient approach for biomolecular surface generation in Cartesian meshes.

ADAPTIVE FINITE ELEMENT MODELING TECHNIQUES FOR THE POISSON-BOLTZMANN EQUATION

We consider the design of an effective and reliable adaptive finite element method (AFEM) for the nonlinear Poisson-Boltzmann equation (PBE). We first examine the two-term regularization technique for the continuous problem recently proposed by Chen, Holst, and Xu based on the removal of the singular electrostatic potential inside biomolecules; this technique made possible the development of the first complete solution and approximation theory for the Poisson-Boltzmann equation, the first provably convergent discretization, and also allowed for the development of a provably convergent AFEM. However, in practical implementation, this two-term regularization exhibits numerical instability. Therefore, we examine a variation of this regularization technique which can be shown to be less susceptible to such instability. We establish a priori estimates and other basic results for the continuous regularized problem, as well as for Galerkin finite element approximations. We show that the new approach produces regularized continuous and discrete problems with the same mathematical advantages of the original regularization. We then design an AFEM scheme for the new regularized problem, and show that the resulting AFEM scheme is accurate and reliable, by proving a contraction result for the error. This result, which is one of the first results of this type for nonlinear elliptic problems, is based on using continuous and discrete a priori L(∞) estimates to establish quasi-orthogonality. To provide a high-quality geometric model as input to the AFEM algorithm, we also describe a class of feature-preserving adaptive mesh generation algorithms designed specifically for constructing meshes of biomolecular structures, based on the intrinsic local structure tensor of the molecular surface. All of the algorithms described in the article are implemented in the Finite Element Toolkit (FETK), developed and maintained at UCSD. The stability advantages of the new regularization scheme are demonstrated with FETK through comparisons with the original regularization approach for a model problem. The convergence and accuracy of the overall AFEM algorithm is also illustrated by numerical approximation of electrostatic solvation energy for an insulin protein.

Variational multiscale models for charge transport

This work presents a few variational multiscale models for charge transport in complex physical, chemical and biological systems and engineering devices, such as fuel cells, solar cells, battery cells, nanofluidics, transistors and ion channels. An essential ingredient of the present models, introduced in an earlier paper (Bulletin of Mathematical Biology, 72, 1562-1622, 2010), is the use of differential geometry theory of surfaces as a natural means to geometrically separate the macroscopic domain from the microscopic domain, meanwhile, dynamically couple discrete and continuum descriptions. Our main strategy is to construct the total energy functional of a charge transport system to encompass the polar and nonpolar free energies of solvation, and chemical potential related energy. By using the Euler-Lagrange variation, coupled Laplace-Beltrami and Poisson-Nernst-Planck (LB-PNP) equations are derived. The solution of the LB-PNP equations leads to the minimization of the total free energy, and explicit profiles of electrostatic potential and densities of charge species. To further reduce the computational complexity, the Boltzmann distribution obtained from the Poisson-Boltzmann (PB) equation is utilized to represent the densities of certain charge species so as to avoid the computationally expensive solution of some Nernst-Planck (NP) equations. Consequently, the coupled Laplace-Beltrami and Poisson-Boltzmann-Nernst-Planck (LB-PBNP) equations are proposed for charge transport in heterogeneous systems. A major emphasis of the present formulation is the consistency between equilibrium LB-PB theory and non-equilibrium LB-PNP theory at equilibrium. Another major emphasis is the capability of the reduced LB-PBNP model to fully recover the prediction of the LB-PNP model at non-equilibrium settings. To account for the fluid impact on the charge transport, we derive coupled Laplace-Beltrami, Poisson-Nernst-Planck and Navier-Stokes equations from the variational principle for chemo-electro-fluid systems. A number of computational algorithms is developed to implement the proposed new variational multiscale models in an efficient manner. A set of ten protein molecules and a realistic ion channel, Gramicidin A, are employed to confirm the consistency and verify the capability. Extensive numerical experiment is designed to validate the proposed variational multiscale models. A good quantitative agreement between our model prediction and the experimental measurement of current-voltage curves is observed for the Gramicidin A channel transport. This paper also provides a brief review of the field.

Poisson-Boltzmann-Nernst-Planck model

The Poisson-Nernst-Planck (PNP) model is based on a mean-field approximation of ion interactions and continuum descriptions of concentration and electrostatic potential. It provides qualitative explanation and increasingly quantitative predictions of experimental measurements for the ion transport problems in many areas such as semiconductor devices, nanofluidic systems, and biological systems, despite many limitations. While the PNP model gives a good prediction of the ion transport phenomenon for chemical, physical, and biological systems, the number of equations to be solved and the number of diffusion coefficient profiles to be determined for the calculation directly depend on the number of ion species in the system, since each ion species corresponds to one Nernst-Planck equation and one position-dependent diffusion coefficient profile. In a complex system with multiple ion species, the PNP can be computationally expensive and parameter demanding, as experimental measurements of diffusion coefficient profiles are generally quite limited for most confined regions such as ion channels, nanostructures and nanopores. We propose an alternative model to reduce number of Nernst-Planck equations to be solved in complex chemical and biological systems with multiple ion species by substituting Nernst-Planck equations with Boltzmann distributions of ion concentrations. As such, we solve the coupled Poisson-Boltzmann and Nernst-Planck (PBNP) equations, instead of the PNP equations. The proposed PBNP equations are derived from a total energy functional by using the variational principle. We design a number of computational techniques, including the Dirichlet to Neumann mapping, the matched interface and boundary, and relaxation based iterative procedure, to ensure efficient solution of the proposed PBNP equations. Two protein molecules, cytochrome c551 and Gramicidin A, are employed to validate the proposed model under a wide range of bulk ion concentrations and external voltages. Extensive numerical experiments show that there is an excellent consistency between the results predicted from the present PBNP model and those obtained from the PNP model in terms of the electrostatic potentials, ion concentration profiles, and current-voltage (I-V) curves. The present PBNP model is further validated by a comparison with experimental measurements of I-V curves under various ion bulk concentrations. Numerical experiments indicate that the proposed PBNP model is more efficient than the original PNP model in terms of simulation time.

AQUASOL: An efficient solver for the dipolar Poisson-Boltzmann-Langevin equation

The Poisson-Boltzmann (PB) formalism is among the most popular approaches to modeling the solvation of molecules. It assumes a continuum model for water, leading to a dielectric permittivity that only depends on position in space. In contrast, the dipolar Poisson-Boltzmann-Langevin (DPBL) formalism represents the solvent as a collection of orientable dipoles with nonuniform concentration; this leads to a nonlinear permittivity function that depends both on the position and on the local electric field at that position. The differences in the assumptions underlying these two models lead to significant differences in the equations they generate. The PB equation is a second order, elliptic, nonlinear partial differential equation (PDE). Its response coefficients correspond to the dielectric permittivity and are therefore constant within each subdomain of the system considered (i.e., inside and outside of the molecules considered). While the DPBL equation is also a second order, elliptic, nonlinear PDE, its response coefficients are nonlinear functions of the electrostatic potential. Many solvers have been developed for the PB equation; to our knowledge, none of these can be directly applied to the DPBL equation. The methods they use may adapt to the difference; their implementations however are PBE specific. We adapted the PBE solver originally developed by Holst and Saied [J. Comput. Chem. 16, 337 (1995)] to the problem of solving the DPBL equation. This solver uses a truncated Newton method with a multigrid preconditioner. Numerical evidences suggest that it converges for the DPBL equation and that the convergence is superlinear. It is found however to be slow and greedy in memory requirement for problems commonly encountered in computational biology and computational chemistry. To circumvent these problems, we propose two variants, a quasi-Newton solver based on a simplified, inexact Jacobian and an iterative self-consistent solver that is based directly on the PBE solver. While both methods are not guaranteed to converge, numerical evidences suggest that they do and that their convergence is also superlinear. Both variants are significantly faster than the solver based on the exact Jacobian, with a much smaller memory footprint. All three methods have been implemented in a new code named AQUASOL, which is freely available.

Treatment of charge singularities in implicit solvent models

This paper presents a novel method for solving the Poisson-Boltzmann (PB) equation based on a rigorous treatment of geometric singularities of the dielectric interface and a Green's function formulation of charge singularities. Geometric singularities, such as cusps and self-intersecting surfaces, in the dielectric interfaces are bottleneck in developing highly accurate PB solvers. Based on an advanced mathematical technique, the matched interface and boundary (MIB) method, we have recently developed a PB solver by rigorously enforcing the flux continuity conditions at the solvent-molecule interface where geometric singularities may occur. The resulting PB solver, denoted as MIBPB-II, is able to deliver second order accuracy for the molecular surfaces of proteins. However, when the mesh size approaches half of the van der Waals radius, the MIBPB-II cannot maintain its accuracy because the grid points that carry the interface information overlap with those that carry distributed singular charges. In the present Green's function formalism, the charge singularities are transformed into interface flux jump conditions, which are treated on an equal footing as the geometric singularities in our MIB framework. The resulting method, denoted as MIBPB-III, is able to provide highly accurate electrostatic potentials at a mesh as coarse as 1.2 A for proteins. Consequently, at a given level of accuracy, the MIBPB-III is about three times faster than the APBS, a recent multigrid PB solver. The MIBPB-III has been extensively validated by using analytically solvable problems, molecular surfaces of polyatomic systems, and 24 proteins. It provides reliable benchmark numerical solutions for the PB equation.