AFMPB: An Adaptive Fast Multipole Poisson-Boltzmann Solver for Calculating Electrostatics in Biomolecular Systems

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.

Linear scaling computation of forces for the domain-decomposition linear Poisson-Boltzmann method

The Linearized Poisson-Boltzmann (LPB) equation is a popular and widely accepted model for accounting solvent effects in computational (bio-) chemistry. In the present article, we derive the analytical forces using the domain-decomposition-based LPB-method with a van-der Waals or solvent-accessible surface. We present an efficient strategy to compute the forces and its implementation, allowing linear scaling of the method with respect to the number of atoms using the fast multipole method. Numerical tests illustrate the accuracy of the computation of the analytical forces and compare the efficiency with other available methods.

Comparison of the MSMS and NanoShaper molecular surface triangulation codes in the TABI Poisson-Boltzmann solver

The Poisson-Boltzmann (PB) implicit solvent model is a popular framework for studying the electrostatics of solvated biomolecules. In this model the dielectric interface between the biomolecule and solvent is often taken to be the molecular surface or solvent-excluded surface (SES), and the quality of the SES triangulation is critical in boundary element simulations of the model. This work compares the performance of the MSMS and NanoShaper surface triangulation codes for a set of 38 biomolecules. While MSMS produces triangles of exceedingly small area and large aspect ratio, the two codes yield comparable values for the SES surface area and electrostatic solvation energy, where the latter calculations were performed using the treecode-accelerated boundary integral (TABI) PB solver. However we found that NanoShaper is computationally more efficient and reliable than MSMS, especially when parameters are set to produce highly resolved triangulations.

Monte Carlo simulation and thermodynamic integration applied to protein charge transfer

We introduce a combination of Monte Carlo simulation and thermodynamic integration methods to address a model problem in free energy computations, electron transfer in proteins. The feasibility of this approach is tested using the ferredoxin protein from Clostridium acidurici. The results are compared to numerical solutions of the Poisson-Boltzmann equation and data from recent molecular dynamics simulations on charge transfer in a protein complex, the NrfHA nitrite reductase of Desulfovibrio vulgaris. Despite the conceptual and computational simplicity of the Monte Carlo approach, the data agree well with those obtained by other methods. A link to experiments is established via the cytochrome subunit of the bacterial photosynthetic reaction center of Rhodopseudomonas viridis.

Long-range electron-electron interaction and charge transfer in protein complexes: a numerical approach

With application to the nitrite reductase hexameric protein complex of Desulfovibrio vulgaris, NrfH2A4, we suggest a strategy to compute the energy landscape of electron transfer in large systems of biochemical interest. For small complexes, the energy of all electronic configurations can be scanned completely on the level of a numerical solution of the Poisson-Boltzmann equation. In contrast, larger systems have to be treated using a pair approximation, which is verified here. Effective Coulomb interactions between neighbouring sites of excess electron localization may become as large as 200 meV, and they depend in a nontrivial manner on the intersite distance. We discuss the implications of strong Coulomb interactions on the thermodynamics and kinetics of charging and decharging a protein complex. Finally, we turn to the effect of embedding the system into a biomembrane.

Robustness and Efficiency of Poisson-Boltzmann Modeling on Graphics Processing Units

Poisson-Boltzmann equation (PBE) based continuum electrostatics models have been widely used in modeling electrostatic interactions in biochemical processes, particularly in estimating protein-ligand binding affinities. Fast convergence of PBE solvers is crucial in binding affinity computations as numerous snapshots need to be processed. Efforts have been reported to develop PBE solvers on graphics processing units (GPUs) for efficient modeling of biomolecules, though only relatively simple successive over-relaxation and conjugate gradient methods were implemented. However, neither convergence nor scaling properties of the two methods are optimal for large biomolecules. On the other hand, geometric multigrid (MG) has been shown to be an optimal solver on CPUs, though no MG have been reported for biomolecular applications on GPUs. This is not a surprise as it is a more complex method and depends on simpler but limited iterative methods such as Gauss-Seidel in its core relaxation procedure. The robustness and efficiency of MG on GPUs are also unclear. Here we present an implementation and a thorough analysis of MG on GPUs. Our analysis shows that robustness is a more pronounced issue than efficiency for both MG and other tested solvers when the single precision is used for complex biomolecules. We further show how to balance robustness and efficiency utilizing MG's overall efficiency and conjugate gradient's robustness, pointing to a hybrid GPU solver with a good balance of efficiency and accuracy. The new PBE solver will significantly improve the computational throughput for a range of biomolecular applications on the GPU platforms.

Frontiers in biomolecular mesh generation and molecular visualization systems

With the development of biomolecular modeling and simulation, especially implicit solvent modeling, higher requirements are set for the stability, efficiency and mesh quality of molecular mesh generation software. In this review, we summarize the recent works in biomolecular mesh generation and molecular visualization. First, we introduce various definitions of molecular surface and corresponding meshing software. Second, as the mesh quality significantly influences biomolecular simulation, we investigate some remeshing methods in the fields of computer graphics and molecular modeling. Then, we show the application of biomolecular mesh in the boundary element method (BEM) and the finite element method (FEM). Finally, to conveniently visualize the numerical results based on the mesh, we present two types of molecular visualization systems.

Molecular Surface Remeshing with Local Region Refinement

Molecular surface mesh generation is a prerequisite for using the boundary element method (BEM) and finite element method (FEM) in implicit-solvent modeling. Molecular surface meshes typically have small angles, redundant vertices, and low-quality elements. In the implicit-solvent modeling of biomolecular systems it is usually required to improve the mesh quality and eliminate low-quality elements. Existing methods often fail to efficiently remove low-quality elements, especially in complex molecular meshes. In this paper, we propose a mesh refinement method that smooths the meshes, eliminates invalid regions in a cut-and-fill strategy, and improves the minimal angle. We compared our method with four different state-of-the-art methods and found that our method showed a significant improvement over state-of-the-art methods in minimal angle, aspect ratio, and other meshing quality measurements. In addition, our method showed satisfactory results in terms of the ratio of regular vertices and the preservation of area and volume.

Divalent Ion-Mediated DNA-DNA Interactions: A Comparative Study of Triplex and Duplex

Ion-mediated interaction between DNAs is essential for DNA condensation, and it is generally believed that monovalent and nonspecifically binding divalent cations cannot induce the aggregation of double-stranded (ds) DNAs. Interestingly, recent experiments found that alkaline earth metal ions such as Mg²⁺ can induce the aggregation of triple-stranded (ts) DNAs, although there is still a lack of deep understanding of the surprising findings at the microscopic level. In this work, we employed all-atom dynamic simulations to directly calculate the potentials of mean force (PMFs) between tsDNAs, between dsDNAs, and between tsDNA and dsDNA in Mg²⁺ solutions. Our calculations show that the PMF between tsDNAs is apparently attractive and becomes more strongly attractive at higher [Mg²⁺], although the PMF between dsDNAs cannot become apparently attractive even at high [Mg²⁺]. Our analyses show that Mg²⁺ internally binds into grooves and externally binds to phosphate groups for both tsDNA and dsDNA, whereas the external binding of Mg²⁺ is much stronger for tsDNA. Such stronger external binding of Mg²⁺ for tsDNA favors more apparent ion-bridging between helices than for dsDNA. Furthermore, our analyses illustrate that bridging ions, as a special part of external binding ions, are tightly and positively coupled to ion-mediated attraction between DNAs.

Accurate, robust, and reliable calculations of Poisson-Boltzmann binding energies

Poisson-Boltzmann (PB) model is one of the most popular implicit solvent models in biophysical modeling and computation. The ability of providing accurate and reliable PB estimation of electrostatic solvation free energy, ΔGel, and binding free energy, ΔΔGel, is important to computational biophysics and biochemistry. In this work, we investigate the grid dependence of our PB solver (MIBPB) with solvent excluded surfaces for estimating both electrostatic solvation free energies and electrostatic binding free energies. It is found that the relative absolute error of ΔGel obtained at the grid spacing of 1.0 Å compared to ΔGel at 0.2 Å averaged over 153 molecules is less than 0.2%. Our results indicate that the use of grid spacing 0.6 Å ensures accuracy and reliability in ΔΔGel calculation. In fact, the grid spacing of 1.1 Å appears to deliver adequate accuracy for high throughput screening. © 2017 Wiley Periodicals, Inc.

PB-AM: An open-source, fully analytical linear poisson-boltzmann solver

We present the open source distributed software package Poisson-Boltzmann Analytical Method (PB-AM), a fully analytical solution to the linearized PB equation, for molecules represented as non-overlapping spherical cavities. The PB-AM software package includes the generation of outputs files appropriate for visualization using visual molecular dynamics, a Brownian dynamics scheme that uses periodic boundary conditions to simulate dynamics, the ability to specify docking criteria, and offers two different kinetics schemes to evaluate biomolecular association rate constants. Given that PB-AM defines mutual polarization completely and accurately, it can be refactored as a many-body expansion to explore 2- and 3-body polarization. Additionally, the software has been integrated into the Adaptive Poisson-Boltzmann Solver (APBS) software package to make it more accessible to a larger group of scientists, educators, and students that are more familiar with the APBS framework. © 2016 Wiley Periodicals, Inc.

Salt contribution to the flexibility of single-stranded nucleic acid offinite length

Nucleic acids are negatively charged macromolecules and their structure properties are strongly coupled to metal ions in solutions. In this article, the salt effects on the flexibility of single-stranded (ss) nucleic acid chain ranging from 12 to 120 nucleotides are investigated systematically by the coarse-grained Monte Carlo simulations where the salt ions are considered explicitly and the ss chain is modeled with the virtual-bond structural model. Our calculations show that, the increase of ion concentration causes the structural collapse of ss chain and multivalent ions are much more efficient in causing such collapse, and both trivalent/small divalent ions can induce more compact state than a random relaxation state. We found that monovalent, divalent, and trivalent ions can all overcharge ss chain, and the dominating source for such overcharging changes from ion-exclusion-volume effect to ion Coulomb correlations. In addition, the predicted Na(+) and Mg(2+)-dependent persistence length l(p)'s of ss nucleic acid are in accordance with the available experimental data, and through systematic calculations, we obtained the empirical formulas for l(p) as a function of [Na(+)], [Mg(2+)] and chain length.

Multivalent ion-mediated nucleic acid helix-helix interactions: RNA versus DNA

Ion-mediated interaction is critical to the structure and stability of nucleic acids. Recent experiments suggest that the multivalent ion-induced aggregation of double-stranded (ds) RNAs and DNAs may strongly depend on the topological nature of helices, while there is still lack of an understanding on the relevant ion-mediated interactions at atomistic level. In this work, we have directly calculated the potentials of mean force (PMF) between two dsRNAs and between two dsDNAs in Co(NH3)6 (3+) (Co-Hex) solutions by the atomistic molecular dynamics simulations. Our calculations show that at low [Co-Hex], the PMFs between B-DNAs and between A-RNAs are both (strongly) repulsive. However, at high [Co-Hex], the PMF between B-DNAs is strongly attractive, while those between A-RNAs and between A-DNAs are still (weakly) repulsive. The microscopic analyses show that for A-form helices, Co-Hex would become 'internal binding' into the deep major groove and consequently cannot form the evident ion-bridge between adjacent helices, while for B-form helices without deep grooves, Co-Hex would exhibit 'external binding' to strongly bridge adjacent helices. In addition, our further calculations show that, the PMF between A-RNAs could become strongly attractive either at very high [Co-Hex] or when the bottom of deep major groove is fixed with a layer of water.

Parameterization for molecular Gaussian surface and a comparison study of surface mesh generation

The molecular Gaussian surface has been frequently used in the field of molecular modeling and simulation. Typically, the Gaussian surface is defined using two controlling parameters; the decay rate and isovalue. Currently, there is a lack of studies in which a systematic approach in the determination of optimal parameterization according to the geometric features has been done. In this paper, surface area, volume enclosed by the surface and Hausdorff distance are used as three criteria for the parameterization to make the Gaussian surface approximate the solvent excluded surface (SES) well. For each of these three criteria, a search of the parameter space is carried out in order to determine the optimal parameter values. The resulted parameters are close to each other and result in similar calculated molecular properties. Approximation of the VDW surface is also done by analyzing the explicit expressions of the Gaussian surface and VDW surface, which analysis and parameters can be similarly applied to the solvent accessible surface (SAS) due to its geometric similarity to the VDW surface. Once the optimal parameters are obtained, we compare the performance of our Gaussian surface generation software TMSmesh with other commonly used software programs, focusing primarily on mesh quality and fidelity. Additionally, the Poisson-Boltzmann solvation energies based on the surface meshes generated by TMSmesh and those generated by other software programs are calculated and compared for a set of molecules with different sizes. The results of these comparisons validate both the accuracy and the applicability of the parameterized Gaussian surface.

RNA folding: structure prediction, folding kinetics and ion electrostatics

Beyond the "traditional" functions such as gene storage, transport and protein synthesis, recent discoveries reveal that RNAs have important "new" biological functions including the RNA silence and gene regulation of riboswitch. Such functions of noncoding RNAs are strongly coupled to the RNA structures and proper structure change, which naturally leads to the RNA folding problem including structure prediction and folding kinetics. Due to the polyanionic nature of RNAs, RNA folding structure, stability and kinetics are strongly coupled to the ion condition of solution. The main focus of this chapter is to review the recent progress in the three major aspects in RNA folding problem: structure prediction, folding kinetics and ion electrostatics. This chapter will introduce both the recent experimental and theoretical progress, while emphasize the theoretical modelling on the three aspects in RNA folding.

Calculating the Bimolecular Rate of Protein-Protein Association with Interacting Crowders

We have recently introduced a method termed Poisson-Boltzmann semianalytical method (PB-SAM) for solving the linearized Poisson-Boltzmann equation for large numbers of arbitrarily shaped dielectric cavities with controlled precision. In this work we extend the applicability of the PB-SAM approach by deriving force and torque expressions that fully account for mutual polarization in both the zero- and first-order derivatives of the surface charges, that can now be embedded into a Brownian dynamics scheme to look at electrostatic-driven mesoscale assembly and kinetics. We demonstrate the capabilities of the PB-SAM approach by simulating the protein concentration effects on the bimolecular rate of association of barnase and barstar, under periodic boundary conditions and evaluated through mean first passage times. We apply PB-SAM to the pseudo-first-order reaction rate conditions in which either barnase or barstar are in great excess relative to the other protein (124:1). This can be considered a specific case in which the PB-SAM approach can be applied to crowding conditions in which crowders are not inert but can form interactions with other molecules.

Progress in developing Poisson-Boltzmann equation solvers

This review outlines the recent progress made in developing more accurate and efficient solutions to model electrostatics in systems comprised of bio-macromolecules and nano-objects, the last one referring to objects that do not have biological function themselves but nowadays are frequently used in biophysical and medical approaches in conjunction with bio-macromolecules. The problem of modeling macromolecular electrostatics is reviewed from two different angles: as a mathematical task provided the specific definition of the system to be modeled and as a physical problem aiming to better capture the phenomena occurring in the real experiments. In addition, specific attention is paid to methods to extend the capabilities of the existing solvers to model large systems toward applications of calculations of the electrostatic potential and energies in molecular motors, mitochondria complex, photosynthetic machinery and systems involving large nano-objects.

Ion-mediated RNA structural collapse: effect of spatial confinement

RNAs are negatively charged molecules that reside in cellular environments with macromolecular crowding. Macromolecular confinement can influence the ion effects in RNA folding. In this work, using the recently developed tightly bound ion model for ion fluctuation and correlation, we investigate the effect of confinement on ion-mediated RNA structural collapse for a simple model system. We find that for both Na(+) and Mg(2+), the ion efficiencies in mediating structural collapse/folding are significantly enhanced by the structural confinement. This enhancement of ion efficiency is attributed to the decreased electrostatic free-energy difference between the compact conformation ensemble and the (restricted) extended conformation ensemble due to the spatial restriction.

Protein Nano-Object Integrator (ProNOI) for generating atomic style objects for molecular modeling

Background

With the progress of nanotechnology, one frequently has to model biological macromolecules simultaneously with nano-objects. However, the atomic structures of the nano objects are typically not available or they are solid state entities. Because of that, the researchers have to investigate such nano systems by generating models of the nano objects in a manner that the existing software be able to carry the simulations. In addition, it should allow generating composite objects with complex shape by combining basic geometrical figures and embedding biological macromolecules within the system.

Results

Here we report the Protein Nano-Object Integrator (ProNOI) which allows for generating atomic-style geometrical objects with user desired shape and dimensions. Unlimited number of objects can be created and combined with biological macromolecules in Protein Data Bank (PDB) format file. Once the objects are generated, the users can use sliders to manipulate their shape, dimension and absolute position. In addition, the software offers the option to charge the objects with either specified surface or volumetric charge density and to model them with user-desired dielectric constants. According to the user preference, the biological macromolecule atoms can be assigned charges and radii according to four different force fields: Amber, Charmm, OPLS and PARSE. The biological macromolecules and the atomic-style objects are exported as a position, charge and radius (PQR) file, or if a default dielectric constant distribution is not selected, it is exported as a position, charge, radius and epsilon (PQRE) file. As illustration of the capabilities of the ProNOI, we created a composite object in a shape of a robot, aptly named the Clemson Robot, whose parts are charged with various volumetric charge densities and holds the barnase-barstar protein complex in its hand.

Conclusions

The Protein Nano-Object Integrator (ProNOI) is a convenient tool for generating atomic-style nano shapes in conjunction with biological macromolecule(s). Charges and radii on the macromolecule atoms and the atoms in the shapes are assigned according to the user’s preferences allowing various scenarios of modeling. The default output file is in PQR (PQRE) format which is readable by almost any software available in biophysical field. It can be downloaded from: http://compbio.clemson.edu/downloadDir/ProNO_integrator.tar.gz

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.

Progress in the prediction of pKa values in proteins

The pK(a) -cooperative aims to provide a forum for experimental and theoretical researchers interested in protein pK(a) values and protein electrostatics in general. The first round of the pK(a) -cooperative, which challenged computational labs to carry out blind predictions against pK(a) s experimentally determined in the laboratory of Bertrand Garcia-Moreno, was completed and results discussed at the Telluride meeting (July 6-10, 2009). This article serves as an introduction to the reports submitted by the blind prediction participants that will be published in a special issue of PROTEINS: Structure, Function and Bioinformatics. Here, we briefly outline existing approaches for pK(a) calculations, emphasizing methods that were used by the participants in calculating the blind pK(a) values in the first round of the cooperative. We then point out some of the difficulties encountered by the participating groups in making their blind predictions, and finally try to provide some insights for future developments aimed at improving the accuracy of pK(a) calculations.

Parallel Generalized Born Implicit Solvent Calculations with NAMD

Accurate electrostatic descriptions of aqueous solvent are critical for simulation studies of bio-molecules, but the computational cost of explicit treatment of solvent is very high. A computationally more feasible alternative is a generalized Born implicit solvent description which models polar solvent as a dielectric continuum. Unfortunately, the attainable simulation speedup does not transfer to the massive parallel computers often employed for simulation of large structures. Longer cutoff distances, spatially heterogenous distribution of atoms and the necessary three-fold iteration over atom-pairs in each timestep combine to challenge efficient parallel performance of generalized Born implicit solvent algorithms. Here we report how NAMD, a parallel molecular dynamics program, meets the challenge through a unique parallelization strategy. NAMD now permits efficient simulation of large systems whose slow conformational motions benefit most from implicit solvent descriptions due to the inherent low viscosity. NAMD's implicit solvent performance is benchmarked and then illustrated in simulating the ratcheting Escherichia coli ribosome involving ~250,000 atoms.

Fast Analytical Methods for Macroscopic Electrostatic Models in Biomolecular Simulations

We review recent developments of fast analytical methods for macroscopic electrostatic calculations in biological applications, including the Poisson-Boltzmann (PB) and the generalized Born models for electrostatic solvation energy. The focus is on analytical approaches for hybrid solvation models, especially the image charge method for a spherical cavity, and also the generalized Born theory as an approximation to the PB model. This review places much emphasis on the mathematical details behind these methods.

Salt contribution to RNA tertiary structure folding stability

Accurate quantification of the ionic contribution to RNA folding stability could greatly enhance our ability to understand and predict RNA functions. Recently, motivated by the potential importance of ion correlation and fluctuation in RNA folding, we developed the tightly bound ion (TBI) model. Extensive experimental tests showed that the TBI model can lead to better treatment of multivalent ions than the Poisson-Boltzmann equation. In this study, we use the model to quantify the contribution of salt (Na(+) and Mg(2+)) to the RNA tertiary structure folding free energy. Folding of the RNA tertiary structure often involves intermediates. We focus on the folding transition from an intermediate state to the native state, and compute the electrostatic folding free energy of the RNA. Based on systematic calculations for a variety of RNA molecules, we derive a set of formulas for the electrostatic free energy for tertiary structural folding as a function of the sequence length and compactness of the RNA and the Na(+) and Mg(2+) concentrations. Extensive comparisons with experimental data suggest that our model and the extracted empirical formulas are quite reliable.

Treecode-based generalized Born method

We have developed a treecode-based O(N log N) algorithm for the generalized Born (GB) implicit solvation model. Our treecode-based GB (tGB) is based on the GBr6 [J. Phys. Chem. B 111, 3055 (2007)], an analytical GB method with a pairwise descreening approximation for the R6 volume integral expression. The algorithm is composed of a cutoff scheme for the effective Born radii calculation, and a treecode implementation of the GB charge-charge pair interactions. Test results demonstrate that the tGB algorithm can reproduce the vdW surface based Poisson solvation energy with an average relative error less than 0.6% while providing an almost linear-scaling calculation for a representative set of 25 proteins with different sizes (from 2815 atoms to 65456 atoms). For a typical system of 10k atoms, the tGB calculation is three times faster than the direct summation as implemented in the original GBr6 model. Thus, our tGB method provides an efficient way for performing implicit solvent GB simulations of larger biomolecular systems at longer time scales.

On the role of electrostatics in protein-protein interactions

The role of electrostatics in protein-protein interactions and binding is reviewed in this paper. A brief outline of the computational modeling, in the framework of continuum electrostatics, is presented and the basic electrostatic effects occurring upon the formation of the complex are discussed. The effect of the salt concentration and pH of the water phase on protein-protein binding free energy is demonstrated which indicates that the increase of the salt concentration tends to weaken the binding, an observation that is attributed to the optimization of the charge-charge interactions across the interface. It is pointed out that the pH-optimum (pH of optimal binding affinity) varies among the protein-protein complexes, and perhaps is a result of their adaptation to particular subcellular compartments. The similarities and differences between hetero- and homo-complexes are outlined and discussed with respect to the binding mode and charge complementarity.