The Adaptive Poisson-Boltzmann Solver (APBS) software was developed to solve the equations of continuum electrostatics for large biomolecular assemblages that have provided impact in the study of a broad range of chemical, biological, and biomedical applications. APBS addresses the three key technology challenges for understanding solvation and electrostatics in biomedical applications: accurate and efficient models for biomolecular solvation and electrostatics, robust and scalable software for applying those theories to biomolecular systems, and mechanisms for sharing and analyzing biomolecular electrostatics data in the scientific community. To address new research applications and advancing computational capabilities, we have continually updated APBS and its suite of accompanying software since its release in 2001. In this article, we discuss the models and capabilities that have recently been implemented within the APBS software package including a Poisson-Boltzmann analytical and a semi-analytical solver, an optimized boundary element solver, a geometry-based geometric flow solvation model, a graph theory-based algorithm for determining pK values, and an improved web-based visualization tool for viewing electrostatics.

Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) infectivity is a major concern in coronavirus disease 2019 (COVID-19) prevention and economic reopening. However, rigorous determination of SARS-CoV-2 infectivity is very difficult owing to its continuous evolution with over 10,000 single nucleotide polymorphisms (SNP) variants in many subtypes. We employ an algebraic topology-based machine learning model to quantitatively evaluate the binding free energy changes of SARS-CoV-2 spike glycoprotein (S protein) and host angiotensinconverting enzyme 2 receptor following mutations. We reveal that the SARS-CoV-2 virus becomes more infectious. Three out of six SARS-CoV-2 subtypes have become slightly more infectious, while the other three subtypes have significantly strengthened their infectivity. We also find that SARS-CoV-2 is slightly more infectious than SARS-CoV according to computed S protein-angiotensin-converting enzyme 2 binding free energy changes. Based on a systematic evaluation of all possible 3686 future mutations on the S protein receptor-binding domain, we show that most likely future mutations will make SARS-CoV-2 more infectious. Combining sequence alignment, probability analysis, and binding free energy calculation, we predict that a few residues on the receptor-binding motif, i.e., 452, 489, 500, 501, and 505, have high chances to mutate into significantly more infectious COVID-19 strains.

This paper presents a differential geometry based model for the analysis and computation of the equilibrium property of solvation. Differential geometry theory of surfaces is utilized to define and construct smooth interfaces with good stability and differentiability for use in characterizing the solvent-solute boundaries and in generating continuous dielectric functions across the computational domain. A total free energy functional is constructed to couple polar and nonpolar contributions to the salvation process. Geometric measure theory is employed to rigorously convert a Lagrangian formulation of the surface energy into an Eulerian formulation so as to bring all energy terms into an equal footing. By minimizing the total free energy functional, we derive coupled generalized Poisson-Boltzmann equation (GPBE) and generalized geometric flow equation (GGFE) for the electrostatic potential and the construction of realistic solvent-solute boundaries, respectively. By solving the coupled GPBE and GGFE, we obtain the electrostatic potential, the solvent-solute boundary profile, and the smooth dielectric function, and thereby improve the accuracy and stability of implicit solvation calculations. We also design efficient second order numerical schemes for the solution of the GPBE and GGFE. Matrix resulted from the discretization of the GPBE is accelerated with appropriate preconditioners. An alternative direct implicit (ADI) scheme is designed to improve the stability of solving the GGFE. Two iterative approaches are designed to solve the coupled system of nonlinear partial differential equations. Extensive numerical experiments are designed to validate the present theoretical model, test computational methods, and optimize numerical algorithms. Example solvation analysis of both small compounds and proteins are carried out to further demonstrate the accuracy, stability, efficiency and robustness of the present new model and numerical approaches. Comparison is given to both experimental and theoretical results in the literature. KeywordsGeneralized Poisson-Boltzmann equation; Biomolecular surface formation and evolution; Potential driving geometric flows; Solvation free energy; Multiscale models I IntroductionAmong the various components of molecular interactions, electrostatic interactions are of special importance 6,8,41,44,55,59,71, 142 -144 , 169 ,170 because of their long range and influence on polar or charged molecules -including water, aqueous ions, and amino or nucleic acids. Electrostatic interactions are ubiquitous for any system of charged or polar molecules, such as * Please address correspondence to Guowei Wei. wei@math.msu.edu. (proteins, nucleic acids, lipid bilayers, sugars, etc.) in their aqueous environment. Electrostatic solute-solvent interactions, therefore, are of central importance in analyzing molecular structure and modeling the intramolecular and intermolecular interactions of macromolecules in simulations. There are two types of solvation models: 131, 142 , 143 , 169 explicit ...

This paper introduces a discrete singular convolution algorithm for solving the Fokker–Planck equation. Singular kernels of the Hilbert-type and the delta type are presented for numerical computations. Various sequences of approximations to the singular kernels are discussed. A numerical algorithm is proposed to incorporate the approximation kernels for physical applications. Three standard problems, the Lorentz Fokker–Planck equation, the bistable model and the Henon–Heiles system, are utilized to test the accuracy, reliability, and speed of convergency of the present approach. All results are in excellent agreement with those of previous methods in the field.

Proteins are the most important biomolecules for living organisms. The understanding of protein structure, function, dynamics and transport is one of 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. Based on 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.

Protein structural fluctuation, typically measured by Debye-Waller factors, or B-factors, is a manifestation of protein flexibility, which strongly correlates to protein function. The flexibility-rigidity index (FRI) is a newly proposed method for the construction of atomic rigidity functions required in the theory of continuum elasticity with atomic rigidity, which is a new multiscale formalism for describing excessively large biomolecular systems. The FRI method analyzes protein rigidity and flexibility and is capable of predicting protein B-factors without resorting to matrix diagonalization. A fundamental assumption used in the FRI is that protein structures are uniquely determined by various internal and external interactions, while the protein functions, such as stability and flexibility, are solely determined by the structure. As such, one can predict protein flexibility without resorting to the protein interaction Hamiltonian. Consequently, bypassing the matrix diagonalization, the original FRI has a computational complexity of O(N 2 ). This work introduces a fast FRI (fFRI) algorithm for the flexibility analysis of large macromolecules. The proposed fFRI further reduces the computational complexity to O(N ). Additionally, we propose anisotropic FRI (aFRI) algorithms for the analysis of protein collective dynamics. The aFRI algorithms permit adaptive Hessian matrices, from a completely global 3N × 3N matrix to completely local 3 × 3 matrices. These 3 × 3 matrices, despite being calculated locally, also contain non-local correlation information. Eigenvectors obtained from the proposed aFRI algorithms are able to demonstrate collective motions. Moreover, we investigate the performance of FRI by employing four families of radial basis correlation functions. Both parameter optimized and parameter-free FRI methods are explored. Furthermore, we compare the accuracy and efficiency of FRI with some established approaches to flexibility analysis, namely, normal mode analysis and Gaussian network model (GNM). The accuracy of the FRI method is tested using four sets of proteins, three sets of relatively small-, medium-, and large-sized structures and an extended set of 365 proteins. A fifth set of proteins is used to compare the efficiency of the FRI, fFRI, aFRI, and GNM methods. Intensive validation and comparison indicate that the FRI, particularly the fFRI, is orders of magnitude more efficient and about 10% more accurate overall than some of the most popular methods in the field. The proposed fFRI is able to predict B-factors for α-carbons of the HIV virus capsid (313 236 residues) in less than 30 seconds on a single processor using only one core. Finally, we demonstrate the application of FRI and aFRI to protein domain analysis. © 2014 AIP Publishing LLC. [http://dx

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 PoissonBoltzmann equation which describes continuum electrostatics with atomic charges. The present work introduces the theory of continuum elasticity with atomic rigidity (CEWAR). The essence of CE-WAR 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.

Effective, sensitive, and reliable diagnostic reagents are of paramount importance for combating the ongoing coronavirus disease 2019 (COVID-19) pandemic when there is neither a preventive vaccine nor a specific drug available for severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). It will cause a large number of false-positive and false-negative tests if currently used diagnostic reagents are undermined. Based on genotyping of 31,421 SARS-CoV-2 genome samples collected up to July 23, 2020, we reveal that essentially all of the current COVID-19 diagnostic targets have undergone mutations. We further show that SARS-CoV-2 has the most mutations on the targets of various nucleocapsid (N) gene primers and probes, which have been widely used around the world to diagnose COVID-19. To understand whether SARS-CoV-2 genes have mutated unevenly, we have computed the mutation rate and mutation h -index of all SARS-CoV-2 genes, indicating that the N gene is one of the most non-conservative genes in the SARS-CoV-2 genome. We show that due to human immune response induced APOBEC mRNA (C > T) editing, diagnostic targets should also be selected to avoid cytidines. Our findings might enable optimally selecting the conservative SARS-CoV-2 genes and proteins for the design and development of COVID-19 diagnostic reagents, prophylactic vaccines, and therapeutic medicines. Availability Interactive real-time online Mutation Tracker.