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.     

Constructing irregular surfaces to enclose macromolecular complexes for mesoscale modeling using the discrete surface charge optimization (DISCO) algorithm

Salt-mediated electrostatics interactions play an essential role in biomolecular structures and dynamics. Because macromolecular systems modeled at atomic resolution contain thousands of solute atoms, the electrostatic computations constitute an expensive part of the force and energy calculations. Implicit solvent models are one way to simplify the model and associated calculations, but they are generally used in combination with standard atomic models for the solute. To approximate electrostatics interactions in models on the polymer level (e.g., supercoiled DNA) that are simulated over long times (e.g., milliseconds) using Brownian dynamics, Beard and Schlick have developed the DiSCO (Discrete Surface Charge Optimization) algorithm. DiSCO represents a macromolecular complex by a few hundred discrete charges on a surface enclosing the system modeled by the Debye-Hückel (screened Coulombic) approximation to the Poisson-Boltzmann equation, and treats the salt solution as continuum solvation. DiSCO can represent the nucleosome core particle (>12,000 atoms), for example, by 353 discrete surface charges distributed on the surfaces of a large disk for the nucleosome core particle and a slender cylinder for the histone tail; the charges are optimized with respect to the Poisson-Boltzmann solution for the electric field, yielding a approximately 5.5% residual. Because regular surfaces enclosing macromolecules are not sufficiently general and may be suboptimal for certain systems, we develop a general method to construct irregular models tailored to the geometry of macromolecules. We also compare charge optimization based on both the electric field and electrostatic potential refinement. Results indicate that irregular surfaces can lead to a more accurate approximation (lower residuals), and the refinement in terms of the electric field is more robust. We also show that surface smoothing for irregular models is important, that the charge optimization (by the TNPACK minimizer) is efficient and does not depend on the initial assigned values, and that the residual is acceptable when the distance to the model surface is close to, or larger than, the Debye length. We illustrate applications of DiSCO's model-building procedure to chromatin folding and supercoiled DNA bound to Hin and Fis proteins. DiSCO is generally applicable to other interesting macromolecular systems for which mesoscale models are appropriate, to yield a resolution between the all-atom representative and the polymer level.     

A semi-implicit solvent model for the simulation of peptides and proteins

We present a new model of biomolecules hydration based on macroscopic electrostatic theory, that can both describe the microscopic details of solvent-solute interactions and allow for an efficient evaluation of the electrostatic hydration free energy. This semi-implicit model considers the solvent as an ensemble of polarizable pseudoparticles whose induced dipole describe both the electronic and orientational solvent polarization. In the presented version of the model, there is no mutual dipolar interaction between the particles, and they only interact through short-ranged Lennard-Jones interactions. The model has been integrated into a molecular dynamics code, and offers the possibility to simulate efficiently the conformational evolution of biomolecules. It is able to provide estimations of the electrostatic solvation free energy within short time windows during the simulation. It has been applied to the study of two small peptides, the octaalanine and the N-terminal helix of ribonuclease A, and two proteins, the bovine pancreatic trypsin inhibitor and the B1 immunoglobin-binding domain of streptococcal protein G. Molecular dynamics simulations of these biomolecules, using a slightly modified Amber force field, provide stable and meaningful trajectories in overall agreement with experiments and all-atom simulations. Correlations with respect to Poisson-Boltzmann electrostatic solvation free energies are also presented to discuss the parameterization of the model and its consequences.     

Nonuniform charge scaling (NUCS): a practical approximation of solvent electrostatic screening in proteins

In molecular mechanics calculations, electrostatic interactions between chemical groups are usually represented by a Coulomb potential between the partial atomic charges of the groups. In aqueous solution these interactions are modified by the polarizable solvent. Although the electrostatic effects of the polarized solvent on the protein are well described by the Poisson--Boltzmann equation, its numerical solution is computationally expensive for large molecules such as proteins. The procedure of nonuniform charge scaling (NUCS) is a pragmatic approach to implicit solvation that approximates the solvent screening effect by individually scaling the partial charges on the explicit atoms of the macromolecule so as to reproduce electrostatic interaction energies obtained from an initial Poisson--Boltzmann analysis. Once the screening factors have been determined for a protein the scaled charges can be easily used in any molecular mechanics program that implements a Coulomb term. The approach is particularly suitable for minimization-based simulations, such as normal mode analysis, certain conformational reaction path or ligand binding techniques for which bulk solvent cannot be included explicitly, and for combined quantum mechanical/molecular mechanical calculations when the interface to more elaborate continuum solvent models is lacking. The method is illustrated using reaction path calculations of the Tyr 35 ring flip in the bovine pancreatic trypsin inhibitor.     

Quantum dynamics in continuum for proton transport--generalized correlation

As a key process of many biological reactions such as biological energy transduction or human sensory systems, proton transport has attracted much research attention in biological, biophysical, and mathematical fields. A quantum dynamics in continuum framework has been proposed to study proton permeation through membrane proteins in our earlier work and the present work focuses on the generalized correlation of protons with their environment. Being complementary to electrostatic potentials, generalized correlations consist of proton-proton, proton-ion, proton-protein, and proton-water interactions. In our approach, protons are treated as quantum particles while other components of generalized correlations are described classically and in different levels of approximations upon simulation feasibility and difficulty. Specifically, the membrane protein is modeled as a group of discrete atoms, while ion densities are approximated by Boltzmann distributions, and water molecules are represented as a dielectric continuum. These proton-environment interactions are formulated as convolutions between number densities of species and their corresponding interaction kernels, in which parameters are obtained from experimental data. In the present formulation, generalized correlations are important components in the total Hamiltonian of protons, and thus is seamlessly embedded in the multiscale/multiphysics total variational model of the system. It takes care of non-electrostatic interactions, including the finite size effect, the geometry confinement induced channel barriers, dehydration and hydrogen bond effects, etc. The variational principle or the Euler-Lagrange equation is utilized to minimize the total energy functional, which includes the total Hamiltonian of protons, and obtain a new version of generalized Laplace-Beltrami equation, generalized Poisson-Boltzmann equation and generalized Kohn-Sham equation. A set of numerical algorithms, such as the matched interface and boundary method, the Dirichlet to Neumann mapping, Gummel iteration, and Krylov space techniques, is employed to improve the accuracy, efficiency, and robustness of model simulations. Finally, comparisons between the present model predictions and experimental data of current-voltage curves, as well as current-concentration curves of the Gramicidin A channel, verify our new model.     

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.     

A new quantum method for electrostatic solvation energy of protein

A new method that incorporates the conductorlike polarizable continuum model (CPCM) with the recently developed molecular fractionation with conjugate caps (MFCC) approach is developed for ab initio calculation of electrostatic solvation energy of protein. The application of the MFCC method makes it practical to apply CPCM to calculate electrostatic solvation energy of protein or other macromolecules in solution. In this MFCC-CPCM method, calculation of protein solvation is divided into calculations of individual solvation energies of fragments (residues) embedded in a common cavity defined with respect to the entire protein. Besides computational efficiency, the current approach also provides additional information about contribution to protein solvation from specific fragments. Numerical studies are carried out to calculate solvation energies for a variety of peptides including alpha helices and beta sheets. Excellent agreement between the MFCC-CPCM result and those from the standard full system CPCM calculation is obtained. Finally, the MFCC-CPCM calculation is applied to several real proteins and the results are compared to classical molecular mechanics Poisson-Boltzmann (MM/PB) and quantum Divid-and-Conque Poisson-Boltzmann (D&C-PB) calculations. Large wave function distortion energy (solute polarization energy) is obtained from the quantum calculation which is missing in the classical calculation. The present study demonstrates that the MFCC-CPCM method is readily applicable to studying solvation of proteins.     

Calculating solvation energies by means of a fluctuating charge model combined with continuum solvent model

Continuum solvent models have shown to be very efficient for calculating solvation energy of biomolecules in solution. However, in order to produce accurate results, besides atomic radii or volumes, an appropriate set of partial charges of the molecule is needed. Here, a set of partial charges produced by a fluctuating charge model-the atom-bond electronegativity equalization method model (ABEEMσπ) fused into molecular mechanics is used to fit for the analytical continuum electrostatics model of generalized-Born calculations. Because the partial atomic charges provided by the ABEEMσπ model can well reflect the polarization effect of the solute induced by the continuum solvent in solution, accurate and rapid calculations of the solvation energies have been performed for series of compounds involving 105 small neutral molecules, twenty kinds of dipeptides and several protein fragments. The solvation energies of small neutral molecules computed with the combination of the GB model with the fluctuating charge protocol (ABEEMσπ∕GB) show remarkable agreement with the experimental results, with a correlation coefficient of 0.97, a slope of 0.95, and a bias of 0.34 kcal∕mol. Furthermore, for twenty kinds of dipeptides and several protein fragments, the results obtained from the analytical ABEEMσπ∕GB model calculations correlate well with those from ab initio and Poisson-Boltzmann calculations. The remarkable agreement between the solvation energies computed with the ABEEMσπ∕GB model and PB model provides strong motivation for the use of ABEEMσπ∕GB solvent model in the simulation of biochemical systems.     

Treatment of geometric singularities in implicit solvent models

Geometric singularities, such as cusps and self-intersecting surfaces, are major obstacles to the accuracy, convergence, and stability of the numerical solution of the Poisson-Boltzmann (PB) equation. In earlier work, an interface technique based PB solver was developed using the matched interface and boundary (MIB) method, which explicitly enforces the flux jump condition at the solvent-solute interfaces and leads to highly accurate biomolecular electrostatics in continuum electric environments. However, such a PB solver, denoted as MIBPB-I, cannot maintain the designed second order convergence whenever there are geometric singularities, such as cusps and self-intersecting surfaces. Moreover, the matrix of the MIBPB-I is not optimally symmetrical, resulting in the convergence difficulty. The present work presents a new interface method based PB solver, denoted as MIBPB-II, to address the aforementioned problems. The present MIBPB-II solver is systematical and robust in treating geometric singularities and delivers second order convergence for arbitrarily complex molecular surfaces of proteins. A new procedure is introduced to make the MIBPB-II matrix optimally symmetrical and diagonally dominant. The MIBPB-II solver is extensively validated by the molecular surfaces of few-atom systems and a set of 24 proteins. Converged electrostatic potentials and solvation free energies are obtained at a coarse grid spacing of 0.5 A and are considerably more accurate than those obtained by the PBEQ and the APBS at finer grid spacings.     

Prediction of hydration free energies for the SAMPL4 diverse set of compounds using molecular dynamics simulations with the OPLS-AA force field

All-atom molecular dynamics computer simulations were used to blindly predict the hydration free energies of a range of small molecules as part of the SAMPL4 challenge. Compounds were parametrized on the basis of the OPLS-AA force field using three different protocols for deriving partial charges: (1) using existing OPLS-AA atom types and charges with minor adjustments of partial charges on equivalent connecting atoms and derivation of new parameters for a number of distinct chemical groups (N-alkyl imidazole, nitrate) that were not present in the published force field; (2) calculation of quantum mechanical charges via geometry optimization, followed by electrostatic potential (ESP) fitting, using Jaguar at the LMP2/cc-pVTZ(-F) level; and (3) via geometry optimization and CHelpG charges (Gaussian09 at the HF/6-31G* level), followed by two-stage RESP fitting. The absolute hydration free energy was computed by an established protocol including alchemical free energy perturbation with thermodynamic integration. The use of standard OPLS-AA charges (protocol 1) with a number of newly parametrized charges and the use of histidine derived parameters for imidazole yielded an overall root mean square deviation of the prediction from the experimental data of 1.75 kcal/mol. The precision of our results appears to be mainly limited by relatively poor reproducibility of the Lennard-Jones contribution towards the solvation free energy, for which we observed large variability that could be traced to a strong dependence on the initial system conditions.     

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.     

Effect of a low-dielectric interior on DNA electrostatic response to twisting and bending

We study the effects of a low-dielectric core of rod-like macromolecules on their electrostatic persistence lengths. We use the exact solution of the linear Poisson-Boltzmann equation for the potential of a charge on the surface of a low-dielectric cylinder. We apply the results to the B-DNA molecule, modeled as a double helical array of discrete charges wound on the surface of a low-dielectric rod. For this charge geometry, we calculate the change in the electrostatic twist persistence as compared to DNA with a water-permeable core. We also discuss possible effects of the low-dielectric molecular core on DNA bending persistence.     

A quantitative evaluation of the electrostatic theory for ion pair chromatography

Summary A quantitative model for ion pair chromatography based on the electrostatic theory is described. The model is based on the solution of the linearised Poisson-Boltzmann equation in a cylinder. The obtained equations are compared with experimental data from a number of different systems. The agreement between theory and experiments is satisfactorily. Systematic deviations due to the use of the linearised equation and ion correlation effects are discussed.     

Solvation dynamics in acetonitrile: a study incorporating solute electronic response and nuclear relaxation

The solvent reorganization process after electronic excitation of a polar solute in a polar solvent such as acetonitrile is related mainly to the time evolution of the solute-solvent electrostatic interaction. Modern laser-based techniques have sufficient time resolution to follow this decay in real time, providing information to be confirmed and interpreted by theories and models. We present here a study aimed at the investigation of the different steps involved in the process taking place after a vertical S(0) --> S(1) excitation of a large size chromophore, coumarin 153 (C153), in acetonitrile, from both the solute and the solvent points of view. To do this, we use accurate quantum mechanical calculations for the solute properties within the polarizable continuum model (PCM) and classical molecular dynamics (MD) simulations, both equilibrium and nonequilibrium, for C153 in the presence of the solvent. The geometry of the solute is allowed to change in order to study the role of internal motions in the time-dependent solvation process. The solvent response function has been obtained from the simulation data and compared to experiment, while the comparison between equilibrium and nonequilibrium MD results for the solvation response confirms the validity of the linear response approximation in the C153-acetonitrile system. The MD trajectories have also been used to monitor the structure of the solvation shell and to determine its change in response to the change in the solute partial charges.     

Solvent effects on chemical processes. 3. Surface tension of binary aqueous organic solvents

The surface tension of binary solvents is modelled by analogy to solvation effects arising from solvent-solute interactions. Competitive exchange equilibria are postulated between solvent component I (water) and solvent component 2 (organic cosolvent) for solute, which in this case is air; the solvation shell is thus the surface phase. A quantitative relationship is given between surface tension and mole fractions x₁ and x₂, the model parameters being exchange equilibrium constants K₁ and K₂. The equation is analyzed, it is applied to literature surface tension data, and it is compared with an earlier model from this laboratory. Curve-fits are very good, and the parameters appear to possess physical significance.     

Dielectric Boundary Forces in Numerical Poisson-Boltzmann Methods: Theory and Numerical Strategies

Continuum modeling of electrostatic interactions based upon the numerical solutions of the Poisson-Boltzmann equation has been widely adopted in biomolecular applications. To extend their applications to molecular dynamics and energy minimization, robust and efficient methodologies to compute solvation forces must be developed. In this study, we have first reviewed the theory for the computation of dielectric boundary forces based on the definition of the Maxwell stress tensor. This is followed by a new formulation of the dielectric boundary force suitable for the finite-difference Poisson-Boltzmann methods. We have validated the new formulation with idealized analytical systems and realistic molecular systems.