Accurate solution of multi-region continuum biomolecule electrostatic problems using the linearized Poisson-Boltzmann equation with curved boundary elements

We present a boundary-element method (BEM) implementation for accurately solving problems in biomolecular electrostatics using the linearized Poisson-Boltzmann equation. Motivating this implementation is the desire to create a solver capable of precisely describing the geometries and topologies prevalent in continuum models of biological molecules. This implementation is enabled by the synthesis of four technologies developed or implemented specifically for this work. First, molecular and accessible surfaces used to describe dielectric and ion-exclusion boundaries were discretized with curved boundary elements that faithfully reproduce molecular geometries. Second, we avoided explicitly forming the dense BEM matrices and instead solved the linear systems with a preconditioned iterative method (GMRES), using a matrix compression algorithm (FFTSVD) to accelerate matrix-vector multiplication. Third, robust numerical integration methods were employed to accurately evaluate singular and near-singular integrals over the curved boundary elements. Fourth, we present a general boundary-integral approach capable of modeling an arbitrary number of embedded homogeneous dielectric regions with differing dielectric constants, possible salt treatment, and point charges. A comparison of the presented BEM implementation and standard finite-difference techniques demonstrates that for certain classes of electrostatic calculations, such as determining absolute electrostatic solvation and rigid-binding free energies, the improved convergence properties of the BEM approach can have a significant impact on computed energetics. We also demonstrate that the improved accuracy offered by the curved-element BEM is important when more sophisticated techniques, such as nonrigid-binding models, are used to compute the relative electrostatic effects of molecular modifications. In addition, we show that electrostatic calculations requiring multiple solves using the same molecular geometry, such as charge optimization or component analysis, can be computed to high accuracy using the presented BEM approach, in compute times comparable to traditional finite-difference methods.     

Reduction potentials and acidity constants of Mn superoxide dismutase calculated by QM/MM free-energy methods

We used two theoretical methods to estimate reduction potentials and acidity constants in Mn superoxide dismutase (MnSOD), namely combined quantum mechanical and molecular mechanics (QM/MM) thermodynamic cycle perturbation (QTCP) and the QM/MM-PBSA approach. In the latter, QM/MM energies are combined with continuum solvation energies calculated by solving the Poisson-Boltzmann equation (PB) or by the generalised Born approach (GB) and non-polar solvation energies calculated from the solvent-exposed surface area. We show that using the QTCP method, we can obtain accurate and precise estimates of the proton-coupled reduction potential for MnSOD, 0.30±0.01 V, which compares favourably with experimental estimates of 0.26-0.40 V. However, the calculated potentials depend strongly on the DFT functional used: The B3LYP functional gives 0.6 V more positive potentials than the PBE functional. The QM/MM-PBSA approach leads to somewhat too high reduction potentials for the coupled reaction and the results depend on the solvation model used. For reactions involving a change in the net charge of the metal site, the corresponding results differ by up to 1.3 V or 24 pK(a) units, rendering the QM/MM-PBSA method useless to determine absolute potentials. However, it may still be useful to estimate relative shifts, although the QTCP method is expected to be more accurate.     

Highly accurate biomolecular electrostatics in continuum dielectric environments

Implicit solvent models based on the Poisson-Boltzmann (PB) equation are frequently used to describe the interactions of a biomolecule with its dielectric continuum environment. A novel, highly accurate Poisson-Boltzmann solver is developed based on the matched interface and boundary (MIB) method, which rigorously enforces the continuity conditions of both the electrostatic potential and its flux at the molecular surface. The MIB based PB solver attains much better convergence rates as a function of mesh size compared to conventional finite difference and finite element based PB solvers. Consequently, highly accurate electrostatic potentials and solvation energies are obtained at coarse mesh sizes. In the context of biomolecular electrostatic calculations it is demonstrated that the MIB method generates substantially more accurate solutions of the PB equation than other established methods, thus providing a new level of reference values for such models. Initial results also indicate that the MIB method can significantly improve the quality of electrostatic surface potentials of biomolecules that are frequently used in the study of biomolecular interactions based on experimental structures.     

Stochastic dynamics simulation of alanine dipeptide: Including solvation interaction determined by boundary element method

A merger of the Poisson–Boltzmann equation and stochastic dynamics simulation is examined using illustrative calculations of alanine dipeptide. The boundary element method (BEM) is used to calculate the hydration forces acting on the solute molecule based on the surroundings. Computational efficiency is achieved by the use of a simple hydration model and a coarse boundary element. Nonetheless, the conformational distribution obtained from this new method is reasonable compared with other theoretical and computational results. Detailed analysis has been accomplished in terms of the hydration interactions and solvation energies. The results indicate that the new simulation method provides an obvious improvement over the conventional stochastic dynamics simulation technique. The further improvement of the hydration model and future application to large molecules are also discussed. © 1997 John Wiley & Sons, Inc. J Comput Chem 18 : 1440–1449, 1997     

A new set of atomic radii for accurate estimation of solvation free energy by Poisson–Boltzmann solvent model

The Poisson–Boltzmann implicit solvent (PB) is widely used to estimate the solvation free energies of biomolecules in molecular simulations. An optimized set of atomic radii (PB radii) is an important parameter for PB calculations, which determines the distribution of dielectric constants around the solute. We here present new PB radii for the AMBER protein force field to accurately reproduce the solvation free energies obtained from explicit solvent simulations. The presented PB radii were optimized using results from explicit solvent simulations of the large systems. In addition, we discriminated PB radii for N‐ and C‐terminal residues from those for nonterminal residues. The performances using our PB radii showed high accuracy for the estimation of solvation free energies at the level of the molecular fragment. The obtained PB radii are effective for the detailed analysis of the solvation effects of biomolecules. © 2014 The Authors Journal of Computational Chemistry Published by Wiley Periodicals, Inc.     

Incorporating variable dielectric environments into the generalized Born model

A generalized Born (GB) model is proposed that approximates the electrostatic part of macromolecular solvation free energy over the entire range of the solvent and solute dielectric constants. The model contains no fitting parameters, and is derived by matching a general form of the GB Green function with the exact Green's function of the Poisson equation for a random charge distribution inside a perfect sphere. The sphere is assumed to be filled uniformly with dielectric medium epsilon(in), and is surrounded by infinite solvent of constant dielectric epsilon(out). This model is as computationally efficient as the conventional GB model based on the widely used functional form due to Still et al. [J. Am. Chem. Soc. 112, 6127 (1990)], but captures the essential physics of the dielectric response for all values of epsilon(in) and epsilon(out). This model is tested against the exact solution on a perfect sphere, and against the numerical Poisson-Boltzmann (PB) treatment on a set of macromolecules representing various structural classes. It shows reasonable agreement with both the exact and the numerical solutions of the PB equation (where available) considered as reference, and is more accurate than the conventional GB model over the entire range of dielectric values.     

Calculation of the Maxwell stress tensor and the Poisson-Boltzmann force on a solvated molecular surface using hypersingular boundary integrals

The electrostatic interaction among molecules solvated in ionic solution is governed by the Poisson-Boltzmann equation (PBE). Here the hypersingular integral technique is used in a boundary element method (BEM) for the three-dimensional (3D) linear PBE to calculate the Maxwell stress tensor on the solvated molecular surface, and then the PB forces and torques can be obtained from the stress tensor. Compared with the variational method (also in a BEM frame) that we proposed recently, this method provides an even more efficient way to calculate the full intermolecular electrostatic interaction force, especially for macromolecular systems. Thus, it may be more suitable for the application of Brownian dynamics methods to study the dynamics of protein/protein docking as well as the assembly of large 3D architectures involving many diffusing subunits. The method has been tested on two simple cases to demonstrate its reliability and efficiency, and also compared with our previous variational method used in BEM.     

Calculation of solvation free energy from quantum mechanical charge density and continuum dielectric theory

We have combined ultrasoft pseudopotential density functional theory utilizing plane wave basis with a Poisson-Boltzmann/solvent-accessible surface area (PB/SA) model to calculate the solvation free energy of small neutral organic compounds in water. The solute charge density obtained from density functional theory was directly used in solving the Poisson-Boltzmann equation to obtain the reaction field. The polarized electronic wave function of the solute in the solvent was solved by including the reaction field in the density functional Hamiltonian. The quantum mechanical and Poisson-Boltzmann equations were solved self-consistently until the charge density and reaction field converged. Using the solute charge density directly instead of a point-charge representation permitted asymmetric distortion and spreading out of the electron cloud. Because the electron density could leave the van der Waals surface to penetrate into the high-dielectric solvent, the reaction field generated by this density was generally smaller than that obtained by using the point-charge representation. In applying this model to calculate the solvation free energy of 31 small neutral organic molecules spanning a range of 25 kcal/mol, we obtained a root-mean-square error of only 1.3 kcal/mol if we allowed one adjustable parameter to shift the calculated solvation free energy.     

Application of MDGRAPE-3, a special purpose board for molecular dynamics simulations, to periodic biomolecular systems

We describe the application of a special purpose board for molecular dynamics simulations, named MDGRAPE-3, to the problem of simulating periodic bio-molecular systems. MDGRAPE-3 is the latest board in a series of hardware accelerators designed to calculate the nonbonding long-range interactions much more rapidly than normal processors. So far, MDGRAPEs were mainly applied to isolated systems, where very many nonbonded interactions were calculated without any distance cutoff. However, in order to regulate the density and pressure during simulations of membrane embedded protein systems, one has to evaluate interactions under periodic boundary conditions. For this purpose, we implemented the Particle-Mesh Ewald (PME) method, and its approximation with distance cutoffs and charge neutrality as proposed by Wolf et al., using MDGRAPE-3. When the two methods were applied to simulations of two periodic biomolecular systems, a single MDGRAPE-3 achieved 30-40 times faster computation times than a single conventional processor did in the both cases. Both methods are shown to have the same molecular structures and dynamics of the systems.     

Adapting Poisson-Boltzmann to the self-consistent mean field theory: application to protein side-chain modeling

We present an extension of the self-consistent mean field theory for protein side-chain modeling in which solvation effects are included based on the Poisson-Boltzmann (PB) theory. In this approach, the protein is represented with multiple copies of its side chains. Each copy is assigned a weight that is refined iteratively based on the mean field energy generated by the rest of the protein, until self-consistency is reached. At each cycle, the variational free energy of the multi-copy system is computed; this free energy includes the internal energy of the protein that accounts for vdW and electrostatics interactions and a solvation free energy term that is computed using the PB equation. The method converges in only a few cycles and takes only minutes of central processing unit time on a commodity personal computer. The predicted conformation of each residue is then set to be its copy with the highest weight after convergence. We have tested this method on a database of hundred highly refined NMR structures to circumvent the problems of crystal packing inherent to x-ray structures. The use of the PB-derived solvation free energy significantly improves prediction accuracy for surface side chains. For example, the prediction accuracies for χ(1) for surface cysteine, serine, and threonine residues improve from 68%, 35%, and 43% to 80%, 53%, and 57%, respectively. A comparison with other side-chain prediction algorithms demonstrates that our approach is consistently better in predicting the conformations of exposed side chains.     

Converging free energy estimates: MM-PB(GB)SA studies on the protein-protein complex Ras-Raf

Estimating protein-protein interaction energies is a very challenging task for current simulation protocols. Here, absolute binding free energies are reported for the complex H-Ras/C-Raf1 using the MM-PB(GB)SA approach, testing the internal consistency and model dependence of the results. Averaging gas-phase energies (MM), solvation free energies as determined by Generalized Born models (GB/SA), and entropic contributions calculated by normal mode analysis for snapshots obtained from 10 ns explicit-solvent molecular dynamics in general results in an overestimation of the binding affinity when a solvent-accessible surface area-dependent model is used to estimate the nonpolar solvation contribution. Applying the sum of a cavity solvation free energy and explicitly modeled solute-solvent van der Waals interaction energies instead provides less negative estimates for the nonpolar solvation contribution. When the polar contribution to the solvation free energy is determined by solving the Poisson-Boltzmann equation (PB) instead, the calculated binding affinity strongly depends on the atomic radii set chosen. For three GB models investigated, different absolute deviations from PB energies were found for the unbound proteins and the complex. As an alternative to normal-mode calculations, quasiharmonic analyses have been performed to estimate entropic contributions due to changes of solute flexibility upon binding. However, such entropy estimates do not converge after 10 ns of simulation time, indicating that sampling issues may limit the applicability of this approach. Finally, binding free energies estimated from snapshots of the unbound proteins extracted from the complex trajectory result in an underestimate of binding affinity. This points to the need to exercise caution in applying the computationally cheaper "one-trajectory-alternative" to systems where there may be significant changes in flexibility and structure due to binding. The best estimate for the binding free energy of Ras-Raf obtained in this study of -8.3 kcal mol(-1) is in good agreement with the experimental result of -9.6 kcal mol(-1), however, further probing the transferability of the applied protocol that led to this result is necessary.     

基于原子表面的蛋白质水合自由能预测模型

提出了一种计算蛋白质水合自由能的简化模型（SAWSA 2）.模型把蛋白质分子中的原子分为20种不同的原子类型,通过每类原子的溶剂可及化表面以及相应的溶剂化参数,就可以得到分子的水合自由能.不同原子类型的溶剂化参数通过110个蛋白质分子水合自由能拟合得到,水合自由能的标准值采用了基于求解Possion-Boltzmann方程（PB）以及分子表面计算（SA） 相结合的方法.采用得到的模型,预测了20个蛋白质分子的水合自由能,预测值的相对值和绝对值都能和PB/SA的计算值很好地吻合,大大优于两种已报导的水合自由能模型.     

GBr(6): a parameterization-free, accurate, analytical generalized born method

The Poisson-Boltzmann (PB) equation is widely used for modeling electrostatic effects and solvation for macromolecules. The generalized Born (GB) model has been developed to mimic PB results at substantial lower computational cost. Here, we report an analytical GB method that reproduces PB results with high accuracy. The analytical approach builds on previous work of Gallicchio and Levy (J. Comput. Chem. 2004, 25, 479), and incorporates an improvement, proposed by Grycuk (J. Chem. Phys. 2003, 119, 4817), of the Coulomb-field approximation used in most GB methods. Tested against PB results, our GB method has an average unsigned relative error of only 0.6% for a representative set of 55 proteins and of 0.4% and 0.3%, respectively, for folded and unfolded conformations of cytochrome b562 sampled in molecular dynamics simulations. The dependencies of the electrostatic solvation free energy on solute and solvent dielectric constants and on salt concentration are fully accounted for in our method.     

Finite volume solution of the modified Poisson-Boltzmann equation for two colloidal particles

The double layer forces between spherical colloidal particles, according to the Poisson-Boltzmann (PB) equation, have been accurately calculated in the literature. The classical PB equation takes into account only the electrostatic interactions, which play a significant role in colloid science. However, there are at, and above, biological salt concentrations other non-electrostatic ion specific forces acting that are ignored in such modelling. In this paper, the electrostatic potential profile and the concentration profile of co-ions and counterions near charged surfaces are calculated. These results are obtained by solving the classical PB equation and a modified PB equation in bispherical coordinates, taking into account the van der Waals dispersion interactions between the ions and both surfaces. Once the electrostatic potential is known we calculate the double layer force between two charged spheres. This is the first paper that solves the modified PB equation in bispherical coordinates. It is also the first time that the finite volume method is used to solve the PB equation in bispherical coordinates. This method divides the calculation domain into a certain number of sub-domains, where the physical law of conservation is valid, and can be readily implemented. The finite volume method is implemented for several geometries and when it is applied to solve PB equations presents low computational cost. The proposed method was validated by comparing the numerical results for the classical PB calculations with previous results reported in the literature. New numerical results using the modified PB equation successfully predicted the ion specificity commonly observed experimentally.     

Efficient solution technique for solving the Poisson-Boltzmann equation

The Poisson-Boltzmann (PB) equation has been extensively used to analyze the energetics and structure of proteins and other significant biomolecules immersed in electrolyte media. A new highly efficient approach for solving PB-type equations that allows for the modeling of many-atoms structures such as encountered in cell biology, virology, and nanotechnology is presented. We accomplish these efficiencies by reformulating the elliptic PB equation as the long-time solution of an advection-diffusion equation. An efficient modified, memory optimized, alternating direction implicit scheme is used to integrate the reformulated PB equation. Our approach is demonstrated on protein composites (a polio virus capsid protomer and a pentamer). The approach has great potential for the analysis of supramillion atoms immersed in a host electrolyte.     

Combining the polarizable Drude force field with a continuum electrostatic Poisson–Boltzmann implicit solvation model

In this work, we have combined the polarizable force field based on the classical Drude oscillator with a continuum Poisson–Boltzmann/solvent‐accessible surface area (PB/SASA) model. In practice, the positions of the Drude particles experiencing the solvent reaction field arising from the fixed charges and induced polarization of the solute must be optimized in a self‐consistent manner. Here, we parameterized the model to reproduce experimental solvation free energies of a set of small molecules. The model reproduces well‐experimental solvation free energies of 70 molecules, yielding a root mean square difference of 0.8 kcal/mol versus 2.5 kcal/mol for the CHARMM36 additive force field. The polarization work associated with the solute transfer from the gas‐phase to the polar solvent, a term neglected in the framework of additive force fields, was found to make a large contribution to the total solvation free energy, comparable to the polar solute–solvent solvation contribution. The Drude PB/SASA also reproduces well the electronic polarization from the explicit solvent simulations of a small protein, BPTI. Model validation was based on comparisons with the experimental relative binding free energies of 371 single alanine mutations. With the Drude PB/SASA model the root mean square deviation between the predicted and experimental relative binding free energies is 3.35 kcal/mol, lower than 5.11 kcal/mol computed with the CHARMM36 additive force field. Overall, the results indicate that the main limitation of the Drude PB/SASA model is the inability of the SASA term to accurately capture non‐polar solvation effects. © 2018 Wiley Periodicals, Inc.     

Generalized born model with a simple smoothing function

Based on recent developments in generalized Born (GB) theory that employ rapid volume integration schemes (M. S. Lee, F. R. Salabury, Jr., and C. L. Brooks III, J Chem Phys 2002, 116, 10606) we have recast the calculation of the self-electrostatic solvation energy to utilize a simple smoothing function at the dielectric boundary. The present GB model is formulated in this manner to provide consistency with the Poisson-Boltzmann (PB) theory previously developed to yield numerically stable electrostatic solvation forces based on finite-difference methods (W. Im, D. Beglov, and B. Roux, Comp Phys Commun 1998, 111, 59). Our comparisons show that the present GB model is indeed an efficient and accurate approach to reproduce corresponding PB solvation energies and forces. With only two adjustable parameters--a(0) to modulate the Coulomb field term, and a(1) to include a correction term beyond Coulomb field--the PB solvation energies are reproduced within 1% error on average for a variety of proteins. Detailed analysis shows that the PB energy can be reproduced within 2% absolute error with a confidence of about 95%. In addition, the solvent-exposed surface area of a biomolecule, as commonly used in calculations of the nonpolar solvation energy, can be calculated accurately and efficiently using the simple smoothing function and the volume integration method. Our implicit solvent GB calculations are about 4.5 times slower than the corresponding vacuum calculations. Using the simple smoothing function makes the present GB model roughly three times faster than GB models, which attempt to mimic the Lee-Richards molecular volume.     

Performance comparison of generalized born and Poisson methods in the calculation of electrostatic solvation energies for protein structures

This study compares generalized Born (GB) and Poisson (PB) methods for calculating electrostatic solvation energies of proteins. A large set of GB and PB implementations from our own laboratories as well as others is applied to a series of protein structure test sets for evaluating the performance of these methods. The test sets cover a significant range of native protein structures of varying size, fold topology, and amino acid composition as well as nonnative extended and misfolded structures that may be found during structure prediction and folding/unfolding studies. We find that the methods tested here span a wide range from highly accurate and computationally demanding PB-based methods to somewhat less accurate but more affordable GB-based approaches and a few fast, approximate PB solvers. Compared with PB solvation energies, the latest, most accurate GB implementations were found to achieve errors of 1% for relative solvation energies between different proteins and 0.4% between different conformations of the same protein. This compares to accurate PB solvers that produce results with deviations of less than 0.25% between each other for both native and nonnative structures. The performance of the best GB methods is discussed in more detail for the application for force field-based minimizations or molecular dynamics simulations.     

Monte Carlo-based linear Poisson-Boltzmann approach makes accurate salt-dependent solvation free energy predictions possible

The prediction of salt-mediated electrostatic effects with high accuracy is highly desirable since many biological processes where biomolecules such as peptides and proteins are key players can be modulated by adjusting the salt concentration of the cellular milieu. With this goal in mind, we present a novel implicit-solvent based linear Poisson-Boltzmann (PB) solver that provides very accurate nonspecific salt-dependent electrostatic properties of biomolecular systems. To solve the linear PB equation by the Monte Carlo method, we use information from the simulation of random walks in the physical space. Due to inherent properties of the statistical simulation method, we are able to account for subtle geometric features in the biomolecular model, treat continuity and outer boundary conditions and interior point charges exactly, and compute electrostatic properties at different salt concentrations in a single PB calculation. These features of the Monte Carlo-based linear PB formulation make it possible to predict the salt-dependent electrostatic properties of biomolecules with very high accuracy. To illustrate the efficiency of our approach, we compute the salt-dependent electrostatic solvation free energies of arginine-rich RNA-binding peptides and compare these Monte Carlo-based PB predictions with computational results obtained using the more mature deterministic numerical methods.     

Electrostatics of ligand binding: parametrization of the generalized Born model and comparison with the Poisson-Boltzmann approach

An accurate and fast evaluation of the electrostatics in ligand-protein interactions is crucial for computer-aided drug design. The pairwise generalized Born (GB) model, a fast analytical method originally developed for studying the solvation of organic molecules, has been widely applied to macromolecular systems, including ligand-protein complexes. However, this model involves several empirical scaling parameters, which have been optimized for the solvation of organic molecules, peptides, and nucleic acids but not for energetics of ligand binding. Studies have shown that a good solvation energy does not guarantee a correct model of solvent-mediated interactions. Thus, in this study, we have used the Poisson-Boltzmann (PB) approach as a reference to optimize the GB model for studies of ligand-protein interactions. Specifically, we have employed the pairwise descreening approximation proposed by Hawkins et al.(1) for GB calculations and DelPhi for PB calculations. The AMBER all-atom force field parameters have been used in this work. Seventeen protein-ligand complexes have been used as a training database, and a set of atomic descreening parameters has been selected with which the pairwise GB model and the PB model yield comparable results on atomic Born radii, the electrostatic component of free energies of ligand binding, and desolvation energies of the ligands and proteins. The energetics of the 15 test complexes calculated with the GB model using this set of parameters also agrees well with the energetics calculated with the PB method. This is the first time that the GB model has been parametrized and thoroughly compared with the PB model for the electrostatics of ligand binding.