Multiscale molecular dynamics using the matched interface and boundary method

Calculation of solvation force in molecular dynamics simulation by deep-learning method

Electrostatic calculations are generally used in studying the thermodynamics and kinetics of biomolecules in solvent. Generally, this is performed by solving the Poisson-Boltzmann equation on a large grid system, a process known to be time consuming. In this study, we developed a deep neural network to predict the decomposed solvation free energies and forces of all atoms in a molecule. To train the network, the internal coordinates of the molecule were used as the input data, and the solvation free energies along with transformed atomic forces from the Poisson-Boltzmann equation were used as labels. Both the training and prediction tasks were accelerated on GPU. Formal tests demonstrated that our method can provide reasonable predictions for small molecules when the network is well-trained with its simulation data. This method is suitable for processing lots of snapshots of molecules in a long trajectory. Moreover, we applied this method in the molecular dynamics simulation with enhanced sampling. The calculated free energy landscape closely resembled that obtained from explicit solvent simulations.

TABI-PB 2.0: An Improved Version of the Treecode-Accelerated Boundary Integral Poisson-Boltzmann Solver

This work describes TABI-PB 2.0, an improved version of the treecode-accelerated boundary integral Poisson-Boltzmann solver. The code computes the electrostatic potential on the molecular surface of a solvated biomolecule, and further processing yields the electrostatic solvation energy. The new implementation utilizes the NanoShaper surface triangulation code, node-patch boundary integral discretization, a block preconditioner, and a fast multipole method based on barycentric Lagrange interpolation and dual tree traversal. Performance-critical portions of the code were implemented on a GPU. Numerical results for protein 1A63 and two viral capsids (Zika, H1N1) demonstrate the code's accuracy and efficiency.

Methodology-Centered Review of Molecular Modeling, Simulation, and Prediction of SARS-CoV-2

Despite tremendous efforts in the past two years, our understanding of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), virus-host interactions, immune response, virulence, transmission, and evolution is still very limited. This limitation calls for further in-depth investigation. Computational studies have become an indispensable component in combating coronavirus disease 2019 (COVID-19) due to their low cost, their efficiency, and the fact that they are free from safety and ethical constraints. Additionally, the mechanism that governs the global evolution and transmission of SARS-CoV-2 cannot be revealed from individual experiments and was discovered by integrating genotyping of massive viral sequences, biophysical modeling of protein-protein interactions, deep mutational data, deep learning, and advanced mathematics. There exists a tsunami of literature on the molecular modeling, simulations, and predictions of SARS-CoV-2 and related developments of drugs, vaccines, antibodies, and diagnostics. To provide readers with a quick update about this literature, we present a comprehensive and systematic methodology-centered review. Aspects such as molecular biophysics, bioinformatics, cheminformatics, machine learning, and mathematics are discussed. This review will be beneficial to researchers who are looking for ways to contribute to SARS-CoV-2 studies and those who are interested in the status of the field.

MLIMC: Machine Learning-Based Implicit-Solvent Monte Carlo

Monte Carlo (MC) methods are important computational tools for molecular structure optimizations and predictions. When solvent effects are explicitly considered, MC methods become very expensive due to the large degree of freedom associated with the water molecules and mobile ions. Alternatively implicit-solvent MC can largely reduce the computational cost by applying a mean field approximation to solvent effects and meanwhile maintains the atomic detail of the target molecule. The two most popular implicit-solvent models are the Poisson-Boltzmann (PB) model and the Generalized Born (GB) model in a way such that the GB model is an approximation to the PB model but is much faster in simulation time. In this work, we develop a machine learning-based implicit-solvent Monte Carlo (MLIMC) method by combining the advantages of both implicit solvent models in accuracy and efficiency. Specifically, the MLIMC method uses a fast and accurate PB-based machine learning (PBML) scheme to compute the electrostatic solvation free energy at each step. We validate our MLIMC method by using a benzene-water system and a protein-water system. We show that the proposed MLIMC method has great advantages in speed and accuracy for molecular structure optimization and prediction.

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.

DFMD: Fast and Effective DelPhiForce Steered Molecular Dynamics Approach to Model Ligand Approach Toward a Receptor: Application to Spermine Synthase Enzyme

Here we report a novel approach, the DelPhiForce Molecular Dynamics (DFMD) method, for steered molecular dynamics simulations to model receptor-ligand association involving charged species. The main purpose of developing DFMD is to simulate ligand's trajectory toward the receptor and thus to predict the "entrance" of the binding pocket and conformational changes associated with the binding. We demonstrate that the DFMD is superior compared with molecular dynamics simulations applying standard cut-offs, provides correct binding forces, allows for modeling the ligand approach at long distances and thus guides the ligand toward the correct binding spot, and it is very fast (frequently the binding is completed in <1 ns). The DFMD is applied to model the binding of two ligands to a receptor (spermine synthase) and it is demonstrated that it guides the ligands toward the corresponding pockets despite of the initial ligand's position with respect to the receptor. Predicted conformational changes and the order of ligand binding are experimentally verified.

Gaussian-Based Smooth Dielectric Function: A Surface-Free Approach for Modeling Macromolecular Binding in Solvents

Conventional modeling techniques to model macromolecular solvation and its effect on binding in the framework of Poisson-Boltzmann based implicit solvent models make use of a geometrically defined surface to depict the separation of macromolecular interior (low dielectric constant) from the solvent phase (high dielectric constant). Though this simplification saves time and computational resources without significantly compromising the accuracy of free energy calculations, it bypasses some of the key physio-chemical properties of the solute-solvent interface, e.g., the altered flexibility of water molecules and that of side chains at the interface, which results in dielectric properties different from both bulk water and macromolecular interior, respectively. Here we present a Gaussian-based smooth dielectric model, an inhomogeneous dielectric distribution model that mimics the effect of macromolecular flexibility and captures the altered properties of surface bound water molecules. Thus, the model delivers a smooth transition of dielectric properties from the macromolecular interior to the solvent phase, eliminating any unphysical surface separating the two phases. Using various examples of macromolecular binding, we demonstrate its utility and illustrate the comparison with the conventional 2-dielectric model. We also showcase some additional abilities of this model, viz. to account for the effect of electrolytes in the solution and to render the distribution profile of water across a lipid membrane.

Numerical interpretation of molecular surface field in dielectric modeling of solvation

Continuum solvent models, particularly those based on the Poisson-Boltzmann equation (PBE), are widely used in the studies of biomolecular structures and functions. Existing PBE developments have been mainly focused on how to obtain more accurate and/or more efficient numerical potentials and energies. However to adopt the PBE models for molecular dynamics simulations, a difficulty is how to interpret dielectric boundary forces accurately and efficiently for robust dynamics simulations. This study documents the implementation and analysis of a range of standard fitting schemes, including both one-sided and two-sided methods with both first-order and second-order Taylor expansions, to calculate molecular surface electric fields to facilitate the numerical calculation of dielectric boundary forces. These efforts prompted us to develop an efficient approximated one-dimensional method, which is to fit the surface field one dimension at a time, for biomolecular applications without much compromise in accuracy. We also developed a surface-to-atom force partition scheme given a level set representation of analytical molecular surfaces to facilitate their applications to molecular simulations. Testing of these fitting methods in the dielectric boundary force calculations shows that the second-order methods, including the one-dimensional method, consistently perform among the best in the molecular test cases. Finally, the timing analysis shows the approximated one-dimensional method is far more efficient than standard second-order methods in the PBE force calculations. © 2017 Wiley Periodicals, Inc.

DelPhiForce, a tool for electrostatic force calculations: Applications to macromolecular binding

Long-range electrostatic forces play an important role in molecular biology, particularly in macromolecular interactions. However, calculating the electrostatic forces for irregularly shaped molecules immersed in water is a difficult task. Here, we report a new tool, DelPhiForce, which is a tool in the DelPhi package that calculates and visualizes the electrostatic forces in biomolecular systems. In parallel, the DelPhi algorithm for modeling electrostatic potential at user-defined positions has been enhanced to include triquadratic and tricubic interpolation methods. The tricubic interpolation method has been tested against analytical solutions and it has been demonstrated that the corresponding errors are negligibly small at resolution 4 grids/Å. The DelPhiForce is further applied in the study of forces acting between partners of three protein-protein complexes. It has been demonstrated that electrostatic forces play a dual role by steering binding partners (so that the partners recognize their native interfaces) and exerting an electrostatic torque (if the mutual orientations of the partners are not native-like). The output of DelPhiForce is in a format that VMD can read and visualize, and provides additional options for analysis of protein-protein binding. DelPhiForce is available for download from the DelPhi webpage at http://compbio.clemson.edu/downloadDir/delphiforce.tar.gz © 2017 Wiley Periodicals, Inc.

ESES: Software for Eulerian solvent excluded surface

Solvent excluded surface (SES) is one of the most popular surface definitions in biophysics and molecular biology. In addition to its usage in biomolecular visualization, it has been widely used in implicit solvent models, in which SES is usually immersed in a Cartesian mesh. Therefore, it is important to construct SESs in the Eulerian representation for biophysical modeling and computation. This work describes a software package called Eulerian solvent excluded surface (ESES) for the generation of accurate SESs in Cartesian grids. ESES offers the description of the solvent and solute domains by specifying all the intersection points between the SES and the Cartesian grid lines. Additionally, the interface normal at each intersection point is evaluated. Furthermore, for a given biomolecule, the ESES software not only provides the whole surface area, but also partitions the surface area according to atomic types. Homology theory is utilized to detect topological features, such as loops and cavities, on the complex formed by the SES. The sizes of loops and cavities are measured based on persistent homology with an evolutionary partial differential equation-based filtration. ESES is extensively validated by surface visualization, electrostatic solvation free energy computation, surface area and volume calculations, and loop and cavity detection and their size estimation. We used the Amber PBSA test set in our electrostatic solvation energy, area, and volume validations. Our results are either calibrated by analytical values or compared with those from the MSMS software. © 2017 Wiley Periodicals, Inc.

The impact of surface area, volume, curvature, and Lennard-Jones potential to solvation modeling

This article explores the impact of surface area, volume, curvature, and Lennard-Jones (LJ) potential on solvation free energy predictions. Rigidity surfaces are utilized to generate robust analytical expressions for maximum, minimum, mean, and Gaussian curvatures of solvent-solute interfaces, and define a generalized Poisson-Boltzmann (GPB) equation with a smooth dielectric profile. Extensive correlation analysis is performed to examine the linear dependence of surface area, surface enclosed volume, maximum curvature, minimum curvature, mean curvature, and Gaussian curvature for solvation modeling. It is found that surface area and surfaces enclosed volumes are highly correlated to each other's, and poorly correlated to various curvatures for six test sets of molecules. Different curvatures are weakly correlated to each other for six test sets of molecules, but are strongly correlated to each other within each test set of molecules. Based on correlation analysis, we construct twenty six nontrivial nonpolar solvation models. Our numerical results reveal that the LJ potential plays a vital role in nonpolar solvation modeling, especially for molecules involving strong van der Waals interactions. It is found that curvatures are at least as important as surface area or surface enclosed volume in nonpolar solvation modeling. In conjugation with the GPB model, various curvature-based nonpolar solvation models are shown to offer some of the best solvation free energy predictions for a wide range of test sets. For example, root mean square errors from a model constituting surface area, volume, mean curvature, and LJ potential are less than 0.42 kcal/mol for all test sets. © 2016 Wiley Periodicals, Inc.

Multiscale method for modeling binding phenomena involving large objects: application to kinesin motor domains motion along microtubules

Many biological phenomena involve the binding of proteins to a large object. Because the electrostatic forces that guide binding act over large distances, truncating the size of the system to facilitate computational modeling frequently yields inaccurate results. Our multiscale approach implements a computational focusing method that permits computation of large systems without truncating the electrostatic potential and achieves the high resolution required for modeling macromolecular interactions, all while keeping the computational time reasonable. We tested our approach on the motility of various kinesin motor domains. We found that electrostatics help guide kinesins as they walk: N-kinesins towards the plus-end, and C-kinesins towards the minus-end of microtubules. Our methodology enables computation in similar, large systems including protein binding to DNA, viruses, and membranes.

Multiscale models of compact bone

This work is concerned about multiscale models of compact bone. We focus on the lacuna–canalicular system. The interstitial fluid and the ions in it are regarded as solvent and others are treated as solute. The system has the characteristic of solvation process as well as non-equilibrium dynamics. The differential geometry theory of surfaces is adopted. We use this theory to separate the macroscopic domain of solvent from the microscopic domain of solute. We also use it to couple continuum and discrete descriptions. The energy functionals are constructed and then the variational principle is applied to the energy functionals so as to derive desirable governing equations. We consider both long-range polar interactions and short-range nonpolar interactions. The solution of governing equations leads to the minimization of the total energy.

Linearly scaling and almost Hamiltonian dielectric continuum molecular dynamics simulations through fast multipole expansions

Hamiltonian Dielectric Solvent (HADES) is a recent method [S. Bauer et al., J. Chem. Phys. 140, 104103 (2014)] which enables atomistic Hamiltonian molecular dynamics (MD) simulations of peptides and proteins in dielectric solvent continua. Such simulations become rapidly impractical for large proteins, because the computational effort of HADES scales quadratically with the number N of atoms. If one tries to achieve linear scaling by applying a fast multipole method (FMM) to the computation of the HADES electrostatics, the Hamiltonian character (conservation of total energy, linear, and angular momenta) may get lost. Here, we show that the Hamiltonian character of HADES can be almost completely preserved, if the structure-adapted fast multipole method (SAMM) as recently redesigned by Lorenzen et al. [J. Chem. Theory Comput. 10, 3244-3259 (2014)] is suitably extended and is chosen as the FMM module. By this extension, the HADES/SAMM forces become exact gradients of the HADES/SAMM energy. Their translational and rotational invariance then guarantees (within the limits of numerical accuracy) the exact conservation of the linear and angular momenta. Also, the total energy is essentially conserved-up to residual algorithmic noise, which is caused by the periodically repeated SAMM interaction list updates. These updates entail very small temporal discontinuities of the force description, because the employed SAMM approximations represent deliberately balanced compromises between accuracy and efficiency. The energy-gradient corrected version of SAMM can also be applied, of course, to MD simulations of all-atom solvent-solute systems enclosed by periodic boundary conditions. However, as we demonstrate in passing, this choice does not offer any serious advantages.

Matched Interface and Boundary Method for Elasticity Interface Problems

Elasticity theory is an important component of continuum mechanics and has had widely spread applications in science and engineering. Material interfaces are ubiquity in nature and man-made devices, and often give rise to discontinuous coefficients in the governing elasticity equations. In this work, the matched interface and boundary (MIB) method is developed to address elasticity interface problems. Linear elasticity theory for both isotropic homogeneous and inhomogeneous media is employed. In our approach, Lamé's parameters can have jumps across the interface and are allowed to be position dependent in modeling isotropic inhomogeneous material. Both strong discontinuity, i.e., discontinuous solution, and weak discontinuity, namely, discontinuous derivatives of the solution, are considered in the present study. In the proposed method, fictitious values are utilized so that the standard central finite different schemes can be employed regardless of the interface. Interface jump conditions are enforced on the interface, which in turn, accurately determines fictitious values. We design new MIB schemes to account for complex interface geometries. In particular, the cross derivatives in the elasticity equations are difficult to handle for complex interface geometries. We propose secondary fictitious values and construct geometry based interpolation schemes to overcome this difficulty. Numerous analytical examples are used to validate the accuracy, convergence and robustness of the present MIB method for elasticity interface problems with both small and large curvatures, strong and weak discontinuities, and constant and variable coefficients. Numerical tests indicate second order accuracy in both L∞ and L2 norms.

Second order Method for Solving 3D Elasticity Equations with Complex Interfaces

Elastic materials are ubiquitous in nature and indispensable components in man-made devices and equipments. When a device or equipment involves composite or multiple elastic materials, elasticity interface problems come into play. The solution of three dimensional (3D) elasticity interface problems is significantly more difficult than that of elliptic counterparts due to the coupled vector components and cross derivatives in the governing elasticity equation. This work introduces the matched interface and boundary (MIB) method for solving 3D elasticity interface problems. The proposed MIB elasticity interface scheme utilizes fictitious values on irregular grid points near the material interface to replace function values in the discretization so that the elasticity equation can be discretized using the standard finite difference schemes as if there were no material interface. The interface jump conditions are rigorously enforced on the intersecting points between the interface and the mesh lines. Such an enforcement determines the fictitious values. A number of new techniques has been developed to construct efficient MIB elasticity interface schemes for dealing with cross derivative in coupled governing equations. The proposed method is extensively validated over both weak and strong discontinuity of the solution, both piecewise constant and position-dependent material parameters, both smooth and nonsmooth interface geometries, and both small and large contrasts in the Poisson's ratio and shear modulus across the interface. Numerical experiments indicate that the present MIB method is of second order convergence in both L∞ and L2 error norms for handling arbitrarily complex interfaces, including biomolecular surfaces. To our best knowledge, this is the first elasticity interface method that is able to deliver the second convergence for the molecular surfaces of proteins..

Multidimensional persistence in biomolecular data

Persistent homology has emerged as a popular technique for the topological simplification of big data, including biomolecular data. Multidimensional persistence bears considerable promise to bridge the gap between geometry and topology. However, its practical and robust construction has been a challenge. We introduce two families of multidimensional persistence, namely pseudomultidimensional persistence and multiscale multidimensional persistence. The former is generated via the repeated applications of persistent homology filtration to high-dimensional data, such as results from molecular dynamics or partial differential equations. The latter is constructed via isotropic and anisotropic scales that create new simiplicial complexes and associated topological spaces. The utility, robustness, and efficiency of the proposed topological methods are demonstrated via protein folding, protein flexibility analysis, the topological denoising of cryoelectron microscopy data, and the scale dependence of nanoparticles. Topological transition between partial folded and unfolded proteins has been observed in multidimensional persistence. The separation between noise topological signatures and molecular topological fingerprints is achieved by the Laplace-Beltrami flow. The multiscale multidimensional persistent homology reveals relative local features in Betti-0 invariants and the relatively global characteristics of Betti-1 and Betti-2 invariants.

Parameterization of the Hamiltonian Dielectric Solvent (HADES) Reaction-Field Method for the Solvation Free Energies of Amino Acid Side-Chain Analogs

Optimization of the Hamiltonian dielectric solvent (HADES) method for biomolecular simulations in a dielectric continuum is presented with the goal of calculating accurate absolute solvation free energies while retaining the model's accuracy in predicting conformational free-energy differences. The solvation free energies of neutral and polar amino acid side-chain analogs calculated by using HADES, which may optionally include nonpolar contributions, were optimized against experimental data to reach a chemical accuracy of about 0.5 kcal mol(-1). The new parameters were evaluated for charged side-chain analogs. The HADES results were compared with explicit-solvent, generalized Born, Poisson-Boltzmann, and QM-based methods. The potentials of mean force (PMFs) between pairs of side-chain analogs obtained by using HADES and explicit-solvent simulations were used to evaluate the effects of the improved parameters optimized for solvation free energies on intermolecular potentials.

MIB Galerkin method for elliptic interface problems

Material interfaces are omnipresent in the real-world structures and devices. Mathematical modeling of material interfaces often leads to elliptic partial differential equations (PDEs) with discontinuous coefficients and singular sources, which are commonly called elliptic interface problems. The development of high-order numerical schemes for elliptic interface problems has become a well defined field in applied and computational mathematics and attracted much attention in the past decades. Despite of significant advances, challenges remain in the construction of high-order schemes for nonsmooth interfaces, i.e., interfaces with geometric singularities, such as tips, cusps and sharp edges. The challenge of geometric singularities is amplified when they are associated with low solution regularities, e.g., tip-geometry effects in many fields. The present work introduces a matched interface and boundary (MIB) Galerkin method for solving two-dimensional (2D) elliptic PDEs with complex interfaces, geometric singularities and low solution regularities. The Cartesian grid based triangular elements are employed to avoid the time consuming mesh generation procedure. Consequently, the interface cuts through elements. To ensure the continuity of classic basis functions across the interface, two sets of overlapping elements, called MIB elements, are defined near the interface. As a result, differentiation can be computed near the interface as if there is no interface. Interpolation functions are constructed on MIB element spaces to smoothly extend function values across the interface. A set of lowest order interface jump conditions is enforced on the interface, which in turn, determines the interpolation functions. The performance of the proposed MIB Galerkin finite element method is validated by numerical experiments with a wide range of interface geometries, geometric singularities, low regularity solutions and grid resolutions. Extensive numerical studies confirm the designed second order convergence of the MIB Galerkin method in the L∞ and L2 errors. Some of the best results are obtained in the present work when the interface is C1 or Lipschitz continuous and the solution is C2 continuous.

A Galerkin formulation of the MIB method for three dimensional elliptic interface problems

We develop a three dimensional (3D) Galerkin formulation of the matched interface and boundary (MIB) method for solving elliptic partial differential equations (PDEs) with discontinuous coefficients, i.e., the elliptic interface problem. The present approach builds up two sets of elements respectively on two extended subdomains which both include the interface. As a result, two sets of elements overlap each other near the interface. Fictitious solutions are defined on the overlapping part of the elements, so that the differentiation operations of the original PDEs can be discretized as if there was no interface. The extra coefficients of polynomial basis functions, which furnish the overlapping elements and solve the fictitious solutions, are determined by interface jump conditions. Consequently, the interface jump conditions are rigorously enforced on the interface. The present method utilizes Cartesian meshes to avoid the mesh generation in conventional finite element methods (FEMs). We implement the proposed MIB Galerkin method with three different elements, namely, rectangular prism element, five-tetrahedron element and six-tetrahedron element, which tile the Cartesian mesh without introducing any new node. The accuracy, stability and robustness of the proposed 3D MIB Galerkin are extensively validated over three types of elliptic interface problems. In the first type, interfaces are analytically defined by level set functions. These interfaces are relatively simple but admit geometric singularities. In the second type, interfaces are defined by protein surfaces, which are truly arbitrarily complex. The last type of interfaces originates from multiprotein complexes, such as molecular motors. Near second order accuracy has been confirmed for all of these problems. To our knowledge, it is the first time for an FEM to show a near second order convergence in solving the Poisson equation with realistic protein surfaces. Additionally, the present work offers the first known near second order accurate method for C1 continuous or H2 continuous solutions associated with a Lipschitz continuous interface in a 3D setting.

Persistent homology analysis of protein structure, flexibility, and folding

Proteins are the most important biomolecules for living organisms. The understanding of protein structure, function, dynamics, and transport is one of the most challenging tasks in biological science. In the present work, persistent homology is, for the first time, introduced for extracting molecular topological fingerprints (MTFs) based on the persistence of molecular topological invariants. MTFs are utilized for protein characterization, identification, and classification. The method of slicing is proposed to track the geometric origin of protein topological invariants. Both all-atom and coarse-grained representations of MTFs are constructed. A new cutoff-like filtration is proposed to shed light on the optimal cutoff distance in elastic network models. On the basis of the correlation between protein compactness, rigidity, and connectivity, we propose an accumulated bar length generated from persistent topological invariants for the quantitative modeling of protein flexibility. To this end, a correlation matrix-based filtration is developed. This approach gives rise to an accurate prediction of the optimal characteristic distance used in protein B-factor analysis. Finally, MTFs are employed to characterize protein topological evolution during protein folding and quantitatively predict the protein folding stability. An excellent consistence between our persistent homology prediction and molecular dynamics simulation is found. This work reveals the topology-function relationship of proteins.

Recent progress in adapting Poisson–Boltzmann methods to molecular simulations

Electrostatic solvation modeling based upon the Poisson–Boltzmann equation is widely used in studies of biomolecular structures and functions. This manuscript provides a thorough review of published efforts to adapt the numerical Poisson–Boltzmann methods to molecular simulations so that these methods can be extended to biomolecular studies involving conformational fluctuation and/or dynamics. We first review the fundamental works on how to define the electrostatic free energy and the Maxwell stress tensor. These topics are followed by three different strategies in developing algorithms to compute electrostatic forces and how to improve their numerical performance. Finally procedures are also presented in detail on how to discretize these algorithms for numerical calculations. Given the pioneer works reviewed here, further developmental efforts will be on how to balance efficiency and accuracy in these theoretical sound approaches — two important issues in applying any numerical algorithms for routine biomolecular applications. Even if not reviewed here, more advanced numerical solvers are certainly necessary to achieve higher accuracy than the widely used classical methods to improve the overall performance of the numerical Poisson–Boltzmann methods.

A fast alternating direction implicit algorithm for geometric flow equations in biomolecular surface generation

In this paper, a new alternating direction implicit (ADI) method is introduced to solve potential driven geometric flow PDEs for biomolecular surface generation. For such PDEs, an extra factor is usually added to stabilize the explicit time integration. However, two existing implicit ADI schemes are also based on the scaled form, which involves nonlinear cross derivative terms that have to be evaluated explicitly. This affects the stability and accuracy of these ADI schemes. To overcome these difficulties, we propose a new ADI algorithm based on the unscaled form so that cross derivatives are not involved. Central finite differences are employed to discretize the nonhomogenous diffusion process of the geometric flow. The proposed ADI algorithm is validated through benchmark examples with analytical solutions, reference solutions, or literature results. Moreover, quantitative indicators of a biomolecular surface, including surface area, surface-enclosed volume, and solvation free energy, are analyzed for various proteins. The proposed ADI method is found to be unconditionally stable and more accurate than the existing ADI schemes in all tests. This enables the use of a large time increment in the steady state simulation so that the proposed ADI algorithm is very efficient for biomolecular surface generation.

Electrostatics of proteins in dielectric solvent continua. I. An accurate and efficient reaction field description

We present a reaction field (RF) method which accurately solves the Poisson equation for proteins embedded in dielectric solvent continua at a computational effort comparable to that of an electrostatics calculation with polarizable molecular mechanics (MM) force fields. The method combines an approach originally suggested by Egwolf and Tavan [J. Chem. Phys. 118, 2039 (2003)] with concepts generalizing the Born solution [Z. Phys. 1, 45 (1920)] for a solvated ion. First, we derive an exact representation according to which the sources of the RF potential and energy are inducible atomic anti-polarization densities and atomic shielding charge distributions. Modeling these atomic densities by Gaussians leads to an approximate representation. Here, the strengths of the Gaussian shielding charge distributions are directly given in terms of the static partial charges as defined, e.g., by standard MM force fields for the various atom types, whereas the strengths of the Gaussian anti-polarization densities are calculated by a self-consistency iteration. The atomic volumes are also described by Gaussians. To account for covalently overlapping atoms, their effective volumes are calculated by another self-consistency procedure, which guarantees that the dielectric function ε(r) is close to one everywhere inside the protein. The Gaussian widths σ(i) of the atoms i are parameters of the RF approximation. The remarkable accuracy of the method is demonstrated by comparison with Kirkwood's analytical solution for a spherical protein [J. Chem. Phys. 2, 351 (1934)] and with computationally expensive grid-based numerical solutions for simple model systems in dielectric continua including a di-peptide (Ac-Ala-NHMe) as modeled by a standard MM force field. The latter example shows how weakly the RF conformational free energy landscape depends on the parameters σ(i). A summarizing discussion highlights the achievements of the new theory and of its approximate solution particularly by comparison with so-called generalized Born methods. A follow-up paper describes how the method enables Hamiltonian, efficient, and accurate MM molecular dynamics simulations of proteins in dielectric solvent continua.

Multiscale geometric modeling of macromolecules I: Cartesian representation

This paper focuses on the geometric modeling and computational algorithm development of biomolecular structures from two data sources: Protein Data Bank (PDB) and Electron Microscopy Data Bank (EMDB) in the Eulerian (or Cartesian) representation. Molecular surface (MS) contains non-smooth geometric singularities, such as cusps, tips and self-intersecting facets, which often lead to computational instabilities in molecular simulations, and violate the physical principle of surface free energy minimization. Variational multiscale surface definitions are proposed based on geometric flows and solvation analysis of biomolecular systems. Our approach leads to geometric and potential driven Laplace-Beltrami flows for biomolecular surface evolution and formation. The resulting surfaces are free of geometric singularities and minimize the total free energy of the biomolecular system. High order partial differential equation (PDE)-based nonlinear filters are employed for EMDB data processing. We show the efficacy of this approach in feature-preserving noise reduction. After the construction of protein multiresolution surfaces, we explore the analysis and characterization of surface morphology by using a variety of curvature definitions. Apart from the classical Gaussian curvature and mean curvature, maximum curvature, minimum curvature, shape index, and curvedness are also applied to macromolecular surface analysis for the first time. Our curvature analysis is uniquely coupled to the analysis of electrostatic surface potential, which is a by-product of our variational multiscale solvation models. As an expository investigation, we particularly emphasize the numerical algorithms and computational protocols for practical applications of the above multiscale geometric models. Such information may otherwise be scattered over the vast literature on this topic. Based on the curvature and electrostatic analysis from our multiresolution surfaces, we introduce a new concept, the polarized curvature, for the prediction of protein binding sites.

Multiscale Multiphysics and Multidomain Models I: Basic Theory

This work extends our earlier two-domain formulation of a differential geometry based multiscale paradigm into a multidomain theory, which endows us the ability to simultaneously accommodate multiphysical descriptions of aqueous chemical, physical and biological systems, such as fuel cells, solar cells, nanofluidics, ion channels, viruses, RNA polymerases, molecular motors and large macromolecular complexes. The essential idea is to make use of the differential geometry theory of surfaces as a natural means to geometrically separate the macroscopic domain of solvent from the microscopic domain of solute, and dynamically couple continuum and discrete descriptions. Our main strategy is to construct energy functionals to put on an equal footing of multiphysics, including polar (i.e., electrostatic) solvation, nonpolar solvation, chemical potential, quantum mechanics, fluid mechanics, molecular mechanics, coarse grained dynamics and elastic dynamics. The variational principle is applied to the energy functionals to derive desirable governing equations, such as multidomain Laplace-Beltrami (LB) equations for macromolecular morphologies, multidomain Poisson-Boltzmann (PB) equation or Poisson equation for electrostatic potential, generalized Nernst-Planck (NP) equations for the dynamics of charged solvent species, generalized Navier-Stokes (NS) equation for fluid dynamics, generalized Newton's equations for molecular dynamics (MD) or coarse-grained dynamics and equation of motion for elastic dynamics. Unlike the classical PB equation, our PB equation is an integral-differential equation due to solvent-solute interactions. To illustrate the proposed formalism, we have explicitly constructed three models, a multidomain solvation model, a multidomain charge transport model and a multidomain chemo-electro-fluid-MD-elastic model. Each solute domain is equipped with distinct surface tension, pressure, dielectric function, and charge density distribution. In addition to long-range Coulombic interactions, various non-electrostatic solvent-solute interactions are considered in the present modeling. We demonstrate the consistency between the non-equilibrium charge transport model and the equilibrium solvation model by showing the systematical reduction of the former to the latter at equilibrium. This paper also offers a brief review of the field.

Multiscale multiphysics and multidomain models--flexibility and rigidity

The emerging complexity of large macromolecules has led to challenges in their full scale theoretical description and computer simulation. Multiscale multiphysics and multidomain models have been introduced to reduce the number of degrees of freedom while maintaining modeling accuracy and achieving computational efficiency. A total energy functional is constructed to put energies for polar and nonpolar solvation, chemical potential, fluid flow, molecular mechanics, and elastic dynamics on an equal footing. The variational principle is utilized to derive coupled governing equations for the above mentioned multiphysical descriptions. Among these governing equations is the Poisson-Boltzmann equation which describes continuum electrostatics with atomic charges. The present work introduces the theory of continuum elasticity with atomic rigidity (CEWAR). The essence of CEWAR is to formulate the shear modulus as a continuous function of atomic rigidity. As a result, the dynamics complexity of a macromolecular system is separated from its static complexity so that the more time-consuming dynamics is handled with continuum elasticity theory, while the less time-consuming static analysis is pursued with atomic approaches. We propose a simple method, flexibility-rigidity index (FRI), to analyze macromolecular flexibility and rigidity in atomic detail. The construction of FRI relies on the fundamental assumption that protein functions, such as flexibility, rigidity, and energy, are entirely determined by the structure of the protein and its environment, although the structure is in turn determined by all the interactions. As such, the FRI measures the topological connectivity of protein atoms or residues and characterizes the geometric compactness of the protein structure. As a consequence, the FRI does not resort to the interaction Hamiltonian and bypasses matrix diagonalization, which underpins most other flexibility analysis methods. FRI's computational complexity is of O(N(2)) at most, where N is the number of atoms or residues, in contrast to O(N(3)) for Hamiltonian based methods. We demonstrate that the proposed FRI gives rise to accurate prediction of protein B-Factor for a set of 263 proteins. We show that a parameter free FRI is able to achieve about 95% accuracy of the parameter optimized FRI. An interpolation algorithm is developed to construct continuous atomic flexibility functions for visualization and use with CEWAR.

Multiscale geometric modeling of macromolecules II: Lagrangian representation

Geometric modeling of biomolecules plays an essential role in the conceptualization of biolmolecular structure, function, dynamics, and transport. Qualitatively, geometric modeling offers a basis for molecular visualization, which is crucial for the understanding of molecular structure and interactions. Quantitatively, geometric modeling bridges the gap between molecular information, such as that from X-ray, NMR, and cryo-electron microscopy, and theoretical/mathematical models, such as molecular dynamics, the Poisson-Boltzmann equation, and the Nernst-Planck equation. In this work, we present a family of variational multiscale geometric models for macromolecular systems. Our models are able to combine multiresolution geometric modeling with multiscale electrostatic modeling in a unified variational framework. We discuss a suite of techniques for molecular surface generation, molecular surface meshing, molecular volumetric meshing, and the estimation of Hadwiger's functionals. Emphasis is given to the multiresolution representations of biomolecules and the associated multiscale electrostatic analyses as well as multiresolution curvature characterizations. The resulting fine resolution representations of a biomolecular system enable the detailed analysis of solvent-solute interaction, and ion channel dynamics, whereas our coarse resolution representations highlight the compatibility of protein-ligand bindings and possibility of protein-protein interactions.

Numerical Poisson-Boltzmann Model for Continuum Membrane Systems

Membrane protein systems are important computational research topics due to their roles in rational drug design. In this study, we developed a continuum membrane model utilizing a level set formulation under the numerical Poisson-Boltzmann framework within the AMBER molecular mechanics suite for applications such as protein-ligand binding affinity and docking pose predictions. Two numerical solvers were adapted for periodic systems to alleviate possible edge effects. Validation on systems ranging from organic molecules to membrane proteins up to 200 residues, demonstrated good numerical properties. This lays foundations for sophisticated models with variable dielectric treatments and second-order accurate modeling of solvation interactions.

High-order fractional partial differential equation transform for molecular surface construction

Fractional derivative or fractional calculus plays a significant role in theoretical modeling of scientific and engineering problems. However, only relatively low order fractional derivatives are used at present. In general, it is not obvious what role a high fractional derivative can play and how to make use of arbitrarily high-order fractional derivatives. This work introduces arbitrarily high-order fractional partial differential equations (PDEs) to describe fractional hyperdiffusions. The fractional PDEs are constructed via fractional variational principle. A fast fractional Fourier transform (FFFT) is proposed to numerically integrate the high-order fractional PDEs so as to avoid stringent stability constraints in solving high-order evolution PDEs. The proposed high-order fractional PDEs are applied to the surface generation of proteins. We first validate the proposed method with a variety of test examples in two and three-dimensional settings. The impact of high-order fractional derivatives to surface analysis is examined. We also construct fractional PDE transform based on arbitrarily high-order fractional PDEs. We demonstrate that the use of arbitrarily high-order derivatives gives rise to time-frequency localization, the control of the spectral distribution, and the regulation of the spatial resolution in the fractional PDE transform. Consequently, the fractional PDE transform enables the mode decomposition of images, signals, and surfaces. The effect of the propagation time on the quality of resulting molecular surfaces is also studied. Computational efficiency of the present surface generation method is compared with the MSMS approach in Cartesian representation. We further validate the present method by examining some benchmark indicators of macromolecular surfaces, i.e., surface area, surface enclosed volume, surface electrostatic potential and solvation free energy. Extensive numerical experiments and comparison with an established surface model indicate that the proposed high-order fractional PDEs are robust, stable and efficient for biomolecular surface generation.

High-order fractional partial differential equation transform for molecular surface construction

Fractional derivative or fractional calculus plays a significant role in theoretical modeling of scientific and engineering problems. However, only relatively low order fractional derivatives are used at present. In general, it is not obvious what role a high fractional derivative can play and how to make use of arbitrarily high-order fractional derivatives. This work introduces arbitrarily high-order fractional partial differential equations (PDEs) to describe fractional hyperdiffusions. The fractional PDEs are constructed via fractional variational principle. A fast fractional Fourier transform (FFFT) is proposed to numerically integrate the high-order fractional PDEs so as to avoid stringent stability constraints in solving high-order evolution PDEs. The proposed high-order fractional PDEs are applied to the surface generation of proteins. We first validate the proposed method with a variety of test examples in two and three-dimensional settings. The impact of high-order fractional derivatives to surface analysis is examined. We also construct fractional PDE transform based on arbitrarily high-order fractional PDEs. We demonstrate that the use of arbitrarily high-order derivatives gives rise to time-frequency localization, the control of the spectral distribution, and the regulation of the spatial resolution in the fractional PDE transform. Consequently, the fractional PDE transform enables the mode decomposition of images, signals, and surfaces. The effect of the propagation time on the quality of resulting molecular surfaces is also studied. Computational efficiency of the present surface generation method is compared with the MSMS approach in Cartesian representation. We further validate the present method by examining some benchmark indicators of macromolecular surfaces, i.e., surface area, surface enclosed volume, surface electrostatic potential and solvation free energy. Extensive numerical experiments and comparison with an established surface model indicate that the proposed high-order fractional PDEs are robust, stable and efficient for biomolecular surface generation.

Exploring a charge-central strategy in the solution of Poisson's equation for biomolecular applications

Continuum solvent treatments based on the Poisson-Boltzmann equation have been widely accepted for energetic analysis of biomolecular systems. In these approaches, the molecular solute is treated as a low dielectric region and the solvent is treated as a high dielectric continuum. The existence of a sharp dielectric jump at the solute-solvent interface poses a challenge to model the solvation energetics accurately with such a simple mathematical model. In this study, we explored and evaluated a strategy based on the "induced surface charge" to eliminate the dielectric jump within the finite-difference discretization scheme. In addition to the use of the induced surface charges in solving the equation, the second-order accurate immersed interface method is also incorporated to discretize the equation. The resultant linear system is solved with the GMRES algorithm to explicitly impose the flux conservation condition across the solvent-solute interface. The new strategy was evaluated on both analytical and realistic biomolecular systems. The numerical tests demonstrate the feasibility of utilizing induced surface charge in the finite-difference solution of the Poisson-Boltzmann equation. The analysis data further show that the strategy is consistent with theory and the classical finite-difference method on the tested systems. Limitations of the current implementations and further improvements are also analyzed and discussed to fully bring out its potential of achieving higher numerical accuracy.

Biomolecular electrostatics and solvation: a computational perspective

An understanding of molecular interactions is essential for insight into biological systems at the molecular scale. Among the various components of molecular interactions, electrostatics are of special importance because of their long-range nature and their influence on polar or charged molecules, including water, aqueous ions, proteins, nucleic acids, carbohydrates, and membrane lipids. In particular, robust models of electrostatic interactions are essential for understanding the solvation properties of biomolecules and the effects of solvation upon biomolecular folding, binding, enzyme catalysis, and dynamics. Electrostatics, therefore, are of central importance to understanding biomolecular structure and modeling interactions within and among biological molecules. This review discusses the solvation of biomolecules with a computational biophysics view toward describing the phenomenon. While our main focus lies on the computational aspect of the models, we provide an overview of the basic elements of biomolecular solvation (e.g. solvent structure, polarization, ion binding, and non-polar behavior) in order to provide a background to understand the different types of solvation models.

Variational approach for nonpolar solvation analysis

Solvation analysis is one of the most important tasks in chemical and biological modeling. Implicit solvent models are some of the most popular approaches. However, commonly used implicit solvent models rely on unphysical definitions of solvent-solute boundaries. Based on differential geometry, the present work defines the solvent-solute boundary via the variation of the nonpolar solvation free energy. The solvation free energy functional of the system is constructed based on a continuum description of the solvent and the discrete description of the solute, which are dynamically coupled by the solvent-solute boundaries via van der Waals interactions. The first variation of the energy functional gives rise to the governing Laplace-Beltrami equation. The present model predictions of the nonpolar solvation energies are in an excellent agreement with experimental data, which supports the validity of the proposed nonpolar solvation model.

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.

Formulation of a new and simple nonuniform size-modified Poisson-Boltzmann description

The nonlinear Poisson-Boltzmann equation (PBE) governing biomolecular electrostatics neglects ion size and ion correlation effects, and recent research activity has focused on accounting for these effects to achieve better physical modeling realism. Here, attention is focused on the comparatively simpler challenge of addressing ion size effects within a continuum-based solvent modeling framework. Prior works by Borukhov et al. (Phys. Rev. Lett. 1997, 79, 435; Electrochim. Acta 2000, 46, 221) have examined the case of uniform ion size in considerable detail. Generalizations to accommodate different species ion sizes have been performed by Li (Nonlinearity 2009, 22, 811; SIAM J. Math. Anal. 2009, 40, 2536) and Zhou et al. (Phys. Rev. E 2011, 84, 021901) using a variational principle, Chu et al. (Biophys. J. 2007, 93, 3202) using a lattice gas model, and Tresset (Phys. Rev. E 2008, 78, 061506) using a generalized Poisson-Fermi distribution. The current work provides an alternative derivation using simple statistical mechanics principles that place the ion size effects and energy distributions on a consistent statistical footing. The resulting expressions differ from the prior nonuniform ion-size developments. However, all treatments reduce to the same form in the cases of uniform ion-size and zero ion size (the PBE). Because of their importance to molecular modeling and salt-dependent behavior, expressions for the salt sensitivities and ionic forces are also derived using the nonuniform ion size description. Emphasis in this article is on formulation and numerically robust evaluation; results are presented for a simple sphere and a previously considered DNA structure for comparison and validation. More extensive application to biomolecular systems is deferred to a subsequent article.

Adaptively deformed mesh based interface method for elliptic equations with discontinuous coefficients

Mesh deformation methods are a versatile strategy for solving partial differential equations (PDEs) with a vast variety of practical applications. However, these methods break down for elliptic PDEs with discontinuous coefficients, namely, elliptic interface problems. For this class of problems, the additional interface jump conditions are required to maintain the well-posedness of the governing equation. Consequently, in order to achieve high accuracy and high order convergence, additional numerical algorithms are required to enforce the interface jump conditions in solving elliptic interface problems. The present work introduces an interface technique based adaptively deformed mesh strategy for resolving elliptic interface problems. We take the advantages of the high accuracy, flexibility and robustness of the matched interface and boundary (MIB) method to construct an adaptively deformed mesh based interface method for elliptic equations with discontinuous coefficients. The proposed method generates deformed meshes in the physical domain and solves the transformed governed equations in the computational domain, which maintains regular Cartesian meshes. The mesh deformation is realized by a mesh transformation PDE, which controls the mesh redistribution by a source term. The source term consists of a monitor function, which builds in mesh contraction rules. Both interface geometry based deformed meshes and solution gradient based deformed meshes are constructed to reduce the L(∞) and L(2) errors in solving elliptic interface problems. The proposed adaptively deformed mesh based interface method is extensively validated by many numerical experiments. Numerical results indicate that the adaptively deformed mesh based interface method outperforms the original MIB method for dealing with elliptic interface problems.

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.

Quantum dynamics in continuum for proton transport II: Variational solvent-solute interface

Proton transport plays an important role in biological energy transduction and sensory systems. Therefore, it has attracted much attention in biological science and biomedical engineering in the past few decades. The present work proposes a multiscale/multiphysics model for the understanding of the molecular mechanism of proton transport in transmembrane proteins involving continuum, atomic, and quantum descriptions, assisted with the evolution, formation, and visualization of membrane channel surfaces. We describe proton dynamics quantum mechanically via a new density functional theory based on the Boltzmann statistics, while implicitly model numerous solvent molecules as a dielectric continuum to reduce the number of degrees of freedom. The density of all other ions in the solvent is assumed to obey the Boltzmann distribution in a dynamic manner. The impact of protein molecular structure and its charge polarization on the proton transport is considered explicitly at the atomic scale. A variational solute-solvent interface is designed to separate the explicit molecule and implicit solvent regions. We formulate a total free-energy functional to put proton kinetic and potential energies, the free energy of all other ions, and the polar and nonpolar energies of the whole system on an equal footing. The variational principle is employed to derive coupled governing equations for the proton transport system. Generalized Laplace-Beltrami equation, generalized Poisson-Boltzmann equation, and generalized Kohn-Sham equation are obtained from the present variational framework. The variational solvent-solute interface is generated and visualized to facilitate the multiscale discrete/continuum/quantum descriptions. Theoretical formulations for the proton density and conductance are constructed based on fundamental laws of physics. A number of mathematical algorithms, including the Dirichlet-to-Neumann mapping, matched interface and boundary method, Gummel iteration, and Krylov space techniques are utilized to implement the proposed model in a computationally efficient manner. The gramicidin A channel is used to validate the performance of the proposed proton transport model and demonstrate the efficiency of the proposed mathematical algorithms. The proton channel conductances are studied over a number of applied voltages and reference concentrations. A comparison with experimental data verifies the present model predictions and confirms the proposed model.

MIB method for elliptic equations with multi-material interfaces

Elliptic partial differential equations (PDEs) are widely used to model real-world problems. Due to the heterogeneous characteristics of many naturally occurring materials and man-made structures, devices, and equipments, one frequently needs to solve elliptic PDEs with discontinuous coefficients and singular sources. The development of high-order elliptic interface schemes has been an active research field for decades. However, challenges remain in the construction of high-order schemes and particularly, for nonsmooth interfaces, i.e., interfaces with geometric singularities. The challenge of geometric singularities is amplified when they are originated from two or more material interfaces joining together or crossing each other. High-order methods for elliptic equations with multi-material interfaces have not been reported in the literature to our knowledge. The present work develops matched interface and boundary (MIB) method based schemes for solving two-dimensional (2D) elliptic PDEs with geometric singularities of multi-material interfaces. A number of new MIB schemes are constructed to account for all possible topological variations due to two-material interfaces. The geometric singularities of three-material interfaces are significantly more difficult to handle. Three new MIB schemes are designed to handle a variety of geometric situations and topological variations, although not all of them. The performance of the proposed new MIB schemes is validated by numerical experiments with a wide range of coefficient contrasts, geometric singularities, and solution types. Extensive numerical studies confirm the designed second order accuracy of the MIB method for multi-material interfaces, including a case where the derivative of the solution diverges.