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.     

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.     

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.     

An efficient simulation technique for electrostatic free energies with applications to azurin

We present an efficient technique for Monte Carlo simulation of electrostatic free energy changes in biomolecular systems. It is a development of a recent method for the study of the influence of electrostatic interactions on the ion binding properties and redox potentials of biomolecules. The electrolyte solution is described by the primitive model, in which ions are treated as hard charged spheres and the solvent is replaced by a structureless continuum. The protein is kept fixed in the center of a spherical simulation cell, and the dielectric constant has the solvent value throughout the cell. By a multiparticle perturbation approach, it is possible to obtain a number of free energy changes within one simulation only. The usefulness of the method is illustrated with a study of the copper binding electron-transport protein azurin (from Alcaligenes denitrificans). The change in acidity of the histidine residues upon changing the redox state of the copper ion is calculated. The theoretical predictions agree well with available experiments. © 1995 by John Wiley & Sons, Inc.     

An efficient second-order poisson–boltzmann method

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

A Fast and Robust Poisson-Boltzmann Solver Based on Adaptive Cartesian Grids

An adaptive Cartesian grid (ACG) concept is presented for the fast and robust numerical solution of the 3D Poisson-Boltzmann Equation (PBE) governing the electrostatic interactions of large-scale biomolecules and highly charged multi-biomolecular assemblies such as ribosomes and viruses. The ACG offers numerous advantages over competing grid topologies such as regular 3D lattices and unstructured grids. For very large biological molecules and multi-biomolecule assemblies, the total number of grid-points is several orders of magnitude less than that required in a conventional lattice grid used in the current PBE solvers thus allowing the end user to obtain accurate and stable nonlinear PBE solutions on a desktop computer. Compared to tetrahedral-based unstructured grids, ACG offers a simpler hierarchical grid structure, which is naturally suited to multigrid, relieves indirect addressing requirements and uses fewer neighboring nodes in the finite difference stencils. Construction of the ACG and determination of the dielectric/ionic maps are straightforward, fast and require minimal user intervention. Charge singularities are eliminated by reformulating the problem to produce the reaction field potential in the molecular interior and the total electrostatic potential in the exterior ionic solvent region. This approach minimizes grid-dependency and alleviates the need for fine grid spacing near atomic charge sites. The technical portion of this paper contains three parts. First, the ACG and its construction for general biomolecular geometries are described. Next, a discrete approximation to the PBE upon this mesh is derived. Finally, the overall solution procedure and multigrid implementation are summarized. Results obtained with the ACG-based PBE solver are presented for: (i) a low dielectric spherical cavity, containing interior point charges, embedded in a high dielectric ionic solvent - analytical solutions are available for this case, thus allowing rigorous assessment of the solution accuracy; (ii) a pair of low dielectric charged spheres embedded in a ionic solvent to compute electrostatic interaction free energies as a function of the distance between sphere centers; (iii) surface potentials of proteins, nucleic acids and their larger-scale assemblies such as ribosomes; and (iv) electrostatic solvation free energies and their salt sensitivities - obtained with both linear and nonlinear Poisson-Boltzmann equation - for a large set of proteins. These latter results along with timings can serve as benchmarks for comparing the performance of different PBE solvers.     

Electrostatic potential around charged finite rodlike macromolecules: nonlinear Poisson-Boltzmann theory

In this paper, we show that the far field electrostatic potential created by a highly charged finite size cylinder within the nonlinear Poisson-Boltzmann (PB) theory, is remarkably close to the potential created within the linearized PB approximation by the same object at a well-chosen fixed potential. Comparing the nonlinear electrostatic potential with its linear counterpart associated to a fixed potential boundary condition (called the effective surface potential), we deduce the effective charge of the highly charged cylinder. Values of the effective surface potential are provided as a function of the bare surface charge and Debye length of the ionic solution. This allows to compute the anisotropic electrostatic interaction energy of two distant finite rods.     

A new outer boundary formulation and energy corrections for the nonlinear Poisson-Boltzmann equation

The nonlinear Poisson-Boltzmann equation (PBE) has been successfully used for the prediction of numerous electrostatic properties of highly charged biopolyelectrolytes immersed in aqueous salt solutions. While numerous numerical solvers for the 3D PBE have been developed, the formulation of the outer boundary treatments used in these methods has only been loosely addressed, especially in the nonlinear case. The de facto standard in current nonlinear PBE implementations is to either set the potential at the outer boundaries to zero or estimate it using the (linear) Debye-Hückel (DH) approximation. However, an assessment of how these outer boundary treatments affect the overall solution accuracy does not appear to have been previously made. As will be demonstrated here, both approximations can, under certain conditions, produce completely erroneous estimates of the potential and energy salt dependencies. A related concern for calculations carried out on grids of finite extent (e.g., all current finite difference and finite element implementations) is the contribution to the energy and salt dependence from the exterior region outside the computational grid. This too is shown to be significant, especially at low salt concentration where essentially all of the contributions to the excess osmotic pressure and ion stress energies originate from this exterior region. In this paper the authors introduce a new outer boundary treatment that is valid for both the linear and nonlinear PBE. The authors also formulate energy corrections to account for contributions from outside the computational domain. Finally, the authors also consider the effects of general ion exclusion layers upon biomolecular electrostatics. It is shown that while these layers tend to increase the surface electrostatic potential, under physiological salt conditions and high net charges their effect on the excess osmotic pressure term, which is a measure of the salt dependence of the total electrostatic free energy, is weak. To facilitate presentation and allow very fine resolutions and/or large computational domains to be considered, attention is restricted to the 1D spherically symmetric nonlinear PBE. Though geometrically limited, the modeling principles nevertheless extend to general PBE solvers as discussed in the Appendix. The 1D model can also be used to benchmark and validate the salt effect prediction capabilities of existing PBE solvers.     

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.     

Flexible polyelectrolyte simulations at the Poisson-Boltzmann level: a comparison of the kink-jump and multigrid configurational-bias Monte Carlo methods

We present a new approach for simulating the motions of flexible polyelectrolyte chains based on the continuous kink-jump Monte Carlo technique coupled to a lattice field theory based calculation of the Poisson-Boltzmann (PB) electrostatic free energy "on the fly." This approach is compared to the configurational-bias Monte Carlo technique, in which the chains are grown on a lattice and the PB equation is solved for each configuration with a linear scaling multigrid method to obtain the many-body free energy. The two approaches are used to calculate end-to-end distances of charged polymer chains in solutions with varying ionic strengths and give similar numerical results. The configurational-bias Monte Carlo/multigrid PB method is found to be more efficient, while the kink-jump Monte Carlo method shows potential utility for simulating nonequilibrium polyelectrolyte dynamics.     

MIBPB: A software package for electrostatic analysis

The Poisson–Boltzmann equation (PBE) is an established model for the electrostatic analysis of biomolecules. The development of advanced computational techniques for the solution of the PBE has been an important topic in the past two decades. This article presents a matched interface and boundary (MIB)‐based PBE software package, the MIBPB solver, for electrostatic analysis. The MIBPB has a unique feature that it is the first interface technique‐based PBE solver that rigorously enforces the solution and flux continuity conditions at the dielectric interface between the biomolecule and the solvent. For protein molecular surfaces, which may possess troublesome geometrical singularities, the MIB scheme makes the MIBPB by far the only existing PBE solver that is able to deliver the second‐order convergence, that is, the accuracy increases four times when the mesh size is halved. The MIBPB method is also equipped with a Dirichlet‐to‐Neumann mapping technique that builds a Green's function approach to analytically resolve the singular charge distribution in biomolecules in order to obtain reliable solutions at meshes as coarse as 1 Å — whereas it usually takes other traditional PB solvers 0.25 Å to reach similar level of reliability. This work further accelerates the rate of convergence of linear equation systems resulting from the MIBPB by using the Krylov subspace (KS) techniques. Condition numbers of the MIBPB matrices are significantly reduced by using appropriate KS solver and preconditioner combinations. Both linear and nonlinear PBE solvers in the MIBPB package are tested by protein–solvent solvation energy calculations and analysis of salt effects on protein–protein binding energies, respectively. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2011     

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.     

Chain stiffness, salt valency, and concentration influences on titration curves of polyelectrolytes: Monte Carlo simulations

Monte Carlo simulations have been used to study two different models of a weak linear polyelectrolyte surrounded by explicit counterions and salt particles: (i) a rigid rod and (ii) a flexible chain. We focused on the influence of the pH, chain stiffness, salt concentration, and valency on the polyelectrolyte titration process and conformational properties. It is shown that chain acid-base properties and conformational properties are strongly modified when multivalent salt concentration variation ranges below the charge equivalence. Increasing chain stiffness allows to minimize intramolecular electrostatic monomer interactions hence improving the deprotonation process. The presence of di and trivalent salt cations clearly promotes the chain degree of ionization but has only a limited effect at very low salt concentration ranges. Moreover, folded structures of fully charged chains are only observed when multivalent salt at a concentration equal or above charge equivalence is considered. Long-range electrostatic potential is found to influence the distribution of charges along and around the polyelectrolyte backbones hence resulting in a higher degree of ionization and a lower attraction of counterions and salt particles at the chain extremities.     

Self‐consistent field for fragmented quantum mechanical model of large molecular systems

Fragment‐based linear scaling quantum chemistry methods are a promising tool for the accurate simulation of chemical and biomolecular systems. Because of the coupled inter‐fragment electrostatic interactions, a dual‐layer iterative scheme is often employed to compute the fragment electronic structure and the total energy. In the dual‐layer scheme, the self‐consistent field (SCF) of the electronic structure of a fragment must be solved first, then followed by the updating of the inter‐fragment electrostatic interactions. The two steps are sequentially carried out and repeated; as such a significant total number of fragment SCF iterations is required to converge the total energy and becomes the computational bottleneck in many fragment quantum chemistry methods. To reduce the number of fragment SCF iterations and speed up the convergence of the total energy, we develop here a new SCF scheme in which the inter‐fragment interactions can be updated concurrently without converging the fragment electronic structure. By constructing the global, block‐wise Fock matrix and density matrix, we prove that the commutation between the two global matrices guarantees the commutation of the corresponding matrices in each fragment. Therefore, many highly efficient numerical techniques such as the direct inversion of the iterative subspace method can be employed to converge simultaneously the electronic structure of all fragments, reducing significantly the computational cost. Numerical examples for water clusters of different sizes suggest that the method shall be very useful in improving the scalability of fragment quantum chemistry methods. © 2015 Wiley Periodicals, Inc.     

Assessment of linear finite‐difference Poisson–Boltzmann solvers

CPU time and memory usage are two vital issues that any numerical solvers for the Poisson–Boltzmann equation have to face in biomolecular applications. In this study, we systematically analyzed the CPU time and memory usage of five commonly used finite‐difference solvers with a large and diversified set of biomolecular structures. Our comparative analysis shows that modified incomplete Cholesky conjugate gradient and geometric multigrid are the most efficient in the diversified test set. For the two efficient solvers, our test shows that their CPU times increase approximately linearly with the numbers of grids. Their CPU times also increase almost linearly with the negative logarithm of the convergence criterion at very similar rate. Our comparison further shows that geometric multigrid performs better in the large set of tested biomolecules. However, modified incomplete Cholesky conjugate gradient is superior to geometric multigrid in molecular dynamics simulations of tested molecules. We also investigated other significant components in numerical solutions of the Poisson–Boltzmann equation. It turns out that the time‐limiting step is the free boundary condition setup for the linear systems for the selected proteins if the electrostatic focusing is not used. Thus, development of future numerical solvers for the Poisson–Boltzmann equation should balance all aspects of the numerical procedures in realistic biomolecular applications. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2010     

Efficient lookup table using a linear function of inverse distance squared

The major bottleneck in molecular dynamics (MD) simulations of biomolecules exist in the calculation of pairwise nonbonded interactions like Lennard‐Jones and long‐range electrostatic interactions. Particle‐mesh Ewald (PME) method is able to evaluate long‐range electrostatic interactions accurately and quickly during MD simulation. However, the evaluation of energy and gradient includes time‐consuming inverse square roots and complementary error functions. To avoid such time‐consuming operations while keeping accuracy, we propose a new lookup table for short‐range interaction in PME by defining energy and gradient as a linear function of inverse distance squared. In our lookup table approach, densities of table points are inversely proportional to squared pair distances, enabling accurate evaluation of energy and gradient at small pair distances. Regardless of the inverse operation here, the new lookup table scheme allows fast pairwise nonbonded calculations owing to efficient usage of cache memory. © 2013 Wiley Periodicals, Inc.     

Specific ion effects in solutions of globular proteins: comparison between analytical models and simulation

Monte Carlo simulations have been performed for ion distributions outside a single globular macroion and for a pair of macroions, in different salt solutions. The model that we use includes both electrostatic and van der Waals interactions between ions and between ions and macroions. Simulation results are compared with the predictions of the Ornstein-Zernike equation with the hypernetted chain closure approximation and the nonlinear Poisson-Boltzmann equation, both augmented by pertinent van der Waals terms. Ion distributions from analytical approximations are generally very close to the simulation results. This demonstrates that properties that are related to ion distributions in the double layer outside a single interface can to a good approximation be obtained from the Poisson-Boltzmann equation. We also present simulation and integral equation results for the mean force between two globular macroions (with properties corresponding to those of hen-egg-white lysozyme protein at pH 4.3) in different salt solutions. The mean force and potential of mean force between the macroions become more attractive upon increasing the polarizability of the counterions (anions), in qualitative agreement with experiments. We finally show that the deduced second virial coefficients agree quite well with experimental results.     

Electrostatic solvation free energy of amino acid side chain analogs: implications for the validity of electrostatic linear response in water

Electrostatic free energies of solvation for 15 neutral amino acid side chain analogs are computed. We compare three methods of varying computational complexity and accuracy for three force fields: free energy simulations, Poisson-Boltzmann (PB), and linear response approximation (LRA) using AMBER, CHARMM, and OPLS-AA force fields. We find that deviations from simulation start at low charges for solutes. The approximate PB and LRA produce an overestimation of electrostatic solvation free energies for most of molecules studied here. These deviations are remarkably systematic. The variations among force fields are almost as large as the variations found among methods. Our study confirms that success of the approximate methods for electrostatic solvation free energies comes from their ability to evaluate free energy differences accurately.