Explicit ion, implicit water solvation for molecular dynamics of nucleic acids and highly charged molecules

An explicit ion, implicit water solvent model for molecular dynamics was developed and tested with DNA and RNA simulations. The implicit water model uses the finite difference Poisson (FDP) model with the smooth permittivity method implemented in the OpenEye ZAP libraries. Explicit counter-ions, co-ions, and nucleic acid were treated with a Langevin dynamics molecular dynamics algorithm. Ion electrostatics is treated within the FDP model when close to the solute, and by the Coulombic model when far from the solute. The two zone model reduces computation time, but retains an accurate treatment of the ion atmosphere electrostatics near the solute. Ion compositions can be set to reproduce specific ionic strengths. The entire ion/water treatment is interfaced with the molecular dynamics package CHARMM. Using the CHARMM-ZAPI software combination, the implicit solvent model was tested on A and B form duplex DNA, and tetraloop RNA, producing stable simulations with structures remaining close to experiment. The model also reproduced the A to B duplex DNA transition. The effect of ionic strength, and the structure of the counterion atmosphere around B form duplex DNA were also examined.     

Boundary element solution of the linear Poisson-Boltzmann equation and a multipole method for the rapid calculation of forces on macromolecules in solution

The Poisson-Boltzmann equation is widely used to describe the electrostatic potential of molecules in an ionic solution that is treated as a continuous dielectric medium. The linearized form of this equation, applicable to many biologic macromolecules, may be solved using the boundary element method. A single-layer formulation of the boundary element method, which yields simpler integral equations than the direct formulations previously discussed in the literature, is given. It is shown that the electrostatic force and torque on a molecule may be calculated using its boundary element representation and also the polarization charge for two rigid molecules may be rapidly calculated using a noniterative scheme. An algorithm based on a fast adaptive multipole method is introduced to further increase the speed of the calculation. This method is particularly suited for Brownian dynamics or molecular dynamics simulations of large molecules, in which the electrostatic forces must be calculated for many different relative positions and orientations of the molecules. It has been implemented as a set of programs in C++, which are used to study the accuracy and speed of this method for two actin monomers.     

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.     

An improved pairwise decomposable finite-difference Poisson-Boltzmann method for computational protein design

Our goal is to develop accurate electrostatic models that can be implemented in current computational protein design protocols. To this end, we improve upon a previously reported pairwise decomposable, finite difference Poisson-Boltzmann (FDPB) model for protein design (Marshall et al., Protein Sci 2005, 14, 1293). The improvement involves placing generic sidechains at positions with unknown amino acid identity and explicitly capturing two-body perturbations to the dielectric environment. We compare the original and improved FDPB methods to standard FDPB calculations in which the dielectric environment is completely determined by protein atoms. The generic sidechain approach yields a two to threefold increase in accuracy per residue or residue pair over the original pairwise FDPB implementation, with no additional computational cost. Distance dependent dielectric and solvent-exclusion models were also compared with standard FDPB energies. The accuracy of the new pairwise FDPB method is shown to be superior to these models, even after reparameterization of the solvent-exclusion model.     

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.     

New analytic approximation to the standard molecular volume definition and its application to generalized Born calculations

In a recent article (Lee, M. S.; Salsbury, F. R. Jr.; Brooks, C. L., III. J Chem Phys 2002, 116, 10606), we demonstrated that generalized Born (GB) theory provides a good approximation to Poisson electrostatic solvation energy calculations if one uses the same definitions of molecular volume for each. In this work, we present a new and improved analytic method for reproducing the Lee-Richards molecular volume, which is the most common volume definition for Poisson calculations. Overall, 1% errors are achieved for absolute solvation energies of a large set of proteins and relative solvation energies of protein conformations. We also introduce an accurate SASA approximation that uses the same machinery employed by our GB method and requires a small addition of computational cost. The combined methodology is shown to yield an efficient and accurate implicit solvent representation for simulations of biopolymers.     

Molecular mechanics and dynamics of biomolecules using a solvent continuum model

An easy implementation of molecular mechanics and molecular dynamics simulation using a continuum solvent model is presented that is particularly suitable for biomolecular simulations. The computation of solvation forces is made using the linear Poisson-Boltzmann equation (polar contribution) and the solvent-accessible surface area approach (nonpolar contribution). The feasibility of the methodology is demonstrated on a small protein and a small DNA hairpin. Although the parameters employed in this model must be refined to gain reliability, the performance of the method, with a standard choice of parameters, is comparable with results obtained by explicit water simulations. Copyright 2001 John Wiley & Sons, Inc. J Comput Chem 22: 1830-1842, 2001     

Hybrid boundary element and finite difference method for solving the nonlinear Poisson-Boltzmann equation

A hybrid approach for solving the nonlinear Poisson-Boltzmann equation (PBE) is presented. Under this approach, the electrostatic potential is separated into (1) a linear component satisfying the linear PBE and solved using a fast boundary element method and (2) a correction term accounting for nonlinear effects and optionally, the presence of an ion-exclusion layer. Because the correction potential contains no singularities (in particular, it is smooth at charge sites) it can be accurately and efficiently solved using a finite difference method. The motivation for and formulation of such a decomposition are presented together with the numerical method for calculating the linear and correction potentials. For comparison, we also develop an integral equation representation of the solution to the nonlinear PBE. When implemented upon regular lattice grids, the hybrid scheme is found to outperform the integral equation method when treating nonlinear PBE problems. Results are presented for a spherical cavity containing a central charge, where the objective is to compare computed 1D nonlinear PBE solutions against ones obtained with alternate numerical solution methods. This is followed by examination of the electrostatic properties of nucleic acid structures.     

Implementation and testing of stable, fast implicit solvation in molecular dynamics using the smooth-permittivity finite difference Poisson-Boltzmann method

A fast stable finite difference Poisson-Boltzmann (FDPB) model for implicit solvation in molecular dynamics simulations was developed using the smooth permittivity FDPB method implemented in the OpenEye ZAP libraries. This was interfaced with two widely used molecular dynamics packages, AMBER and CHARMM. Using the CHARMM-ZAP software combination, the implicit solvent model was tested on eight proteins differing in size, structure, and cofactors: calmodulin, horseradish peroxidase (with and without substrate analogue bound), lipid carrier protein, flavodoxin, ubiquitin, cytochrome c, and a de novo designed 3-helix bundle. The stability and accuracy of the implicit solvent simulations was assessed by examining root-mean-squared deviations from crystal structure. This measure was compared with that of a standard explicit water solvent model. In addition we compared experimental and calculated NMR order parameters to obtain a residue level assessment of the accuracy of MD-ZAP for simulating dynamic quantities. Overall, the agreement of the implicit solvent model with experiment was as good as that of explicit water simulations. The implicit solvent method was up to eight times faster than the explicit water simulations, and approximately four times slower than a vacuum simulation (i.e., with no solvent treatment).     

Accurate prediction of protonation state as a prerequisite for reliable MM-PB(GB)SA binding free energy calculations of HIV-1 protease inhibitors

Binding free energies were calculated for the inhibitors lopinavir, ritonavir, saquinavir, indinavir, amprenavir, and nelfinavir bound to HIV-1 protease. An MMPB/SA-type analysis was applied to conformational samples from 3 ns explicit solvent molecular dynamics simulations of the enzyme-inhibitor complexes. Binding affinities and the sampled conformations of the inhibitor and enzyme were compared between different HIV-1 protease protonation states to find the most likely protonation state of the enzyme in the complex with each of the inhibitors. The resulting set of protonation states leads to good agreement between calculated and experimental binding affinities. Results from the MMPB/SA analysis are compared with an explicit/implicit hybrid scheme and with MMGB/SA methods. It is found that the inclusion of explicit water molecules may offer a slight advantage in reproducing absolute binding free energies while the use of the Generalized Born approximation significantly affects the accuracy of the calculated binding affinities.     

IMiCMO: a new integrated ab initio multicenter molecular orbitals method for molecular dynamics calculations in solvent cluster systems

A new computational scheme integrating ab initio multicenter molecular orbitals for determining forces of individual atoms in large cluster systems is presented. This method can be used to treat systems of many molecules, such as solvents by quantum mechanics. The geometry parameters obtained for three models of water clusters by the present method are compared with those obtained by the full ab initio MO method. The results agree very well. The scheme proposed in this article also intended for use in modeling cluster systems using parallel algorithms. © 2001 John Wiley & Sons, Inc. J Comput Chem 22: 1107–1112, 2001     

FEW: A workflow tool for free energy calculations of ligand binding

In the later stages of drug design projects, accurately predicting relative binding affinities of chemically similar compounds to a biomolecular target is of utmost importance for making decisions based on the ranking of such compounds. So far, the extensive application of binding free energy approaches has been hampered by the complex and time‐consuming setup of such calculations. We introduce the free energy workflow (FEW) tool that facilitates setup and execution of binding free energy calculations with the AMBER suite for multiple ligands. FEW allows performing free energy calculations according to the implicit solvent molecular mechanics (MM‐PB(GB)SA), the linear interaction energy, and the thermodynamic integration approaches. We describe the tool's architecture and functionality and demonstrate in a show case study on Factor Xa inhibitors that the time needed for the preparation and analysis of free energy calculations is considerably reduced with FEW compared to a fully manual procedure. © 2013 Wiley Periodicals, Inc.     

Linear-scaling soft-core scheme for alchemical free energy calculations

Alchemical free energy calculations involving the removal or insertion of atoms into condensed phase systems generally make use of soft-core scaling of nonbonded interactions, designed to circumvent numerical instabilities that arise from weakly interacting "hard" atoms in close proximity. Current methods model soft-core atoms by introducing a nonlinear dependence between the shape of the interaction potential and the strength of the interaction. In this article, we propose a soft-core method that avoids introducing such a nonlinear dependence, through the application of a smooth flattening of the potential energy only in a region that is energetically accessible under normal conditions. We discuss the benefits that this entails and explore a selection of applications, including enhanced methods for the estimation of free energy differences and for the automated optimization of the placement of intermediate states in multistage alchemical calculations.     

Examination of several density functionals in numerical Kohn–Sham calculations for atoms

We studied several exchange‐only and exchange–correlation energy density functionals in numerical, i.e., basis‐set‐free, nonrelativistic Kohn–Sham calculations for closed‐shell ¹S states of atoms and atomic ions with N electrons, where 2≤N≤120. Accurate total energies are presented to serve as reference data for algebraic approaches, as do the numerical Hartree–Fock results, which are also provided. Gradient‐corrected exchange‐only functionals considerably improve the total energies obtained from the usual local density approximation, when compared to the Hartree–Fock results. Such an improvement due to gradient corrections is not seen in general for highest orbital energies, neither for exchange‐only results (to be compared with Hartree–Fock results), nor for exchange–correlation results (to be compared with experimental ionization energies). © 2001 John Wiley & Sons, Inc. Int J Quant Chem 82: 227–241, 2001     

IBIsCO: a molecular dynamics simulation package for coarse-grained simulation

IBIsCO is a parallel molecular dynamics simulation package developed specially for coarse-grained simulations with numerical potentials derived by the iterative Boltzmann inversion (IBI) method (Reith et al., J Comput Chem 2003, 24, 1624). In addition to common features of molecular dynamics programs, the techniques of dissipative particle dynamics (Groot and Warren, J Chem Phys 1997, 107, 4423) and Lowe-Andersen dynamics (Lowe, Europhys Lett 1999, 47, 145) are implemented, which can be used both as thermostats and as sources of friction to compensate the loss of degrees of freedom by coarse-graining. The reverse nonequilibrium molecular dynamics simulation method (Müller-Plathe, Phys Rev E 1999, 59, 4894) for the calculation of viscosities is also implemented. Details of the algorithms, functionalities, implementation, user interfaces, and file formats are described. The code is parallelized using PE_MPI on PowerPC architecture. The execution time scales satisfactorily with the number of processors.     

Efficient polarized basis sets for evaluation of static and dynamic molecular electric properties

New medium size Gaussian‐type basis set R‐ORP for evaluation of static and dynamic electric properties in molecular systems is presented. It is obtained in a close resemblance to the original ORP basis set, from the source basis set through addition of two first‐order polarization functions whose exponent values are optimized with respect to the finite field restricted open‐shell Hartree–Fock (ROHF) atomic polarizabilities. As the source set the VTZ basis set of Ahlrichs and coworkers, augmented with additional diffuse functions and contracted to the form [6s/3s] for hydrogen and [11s7p/4s3p] for carbon through fluorine, is chosen. The resulting basis set is of the form [6s2p/3s2p] for hydrogen and [11s7p2d/4s3p2d] for other atoms. Presented basis set is next tested in the CCSD static and dynamic molecular polarizability and hyperpolarizability calculations for a set of ten and four test molecules, respectively, for which very accurate reference data exist. Additionally, the recently developed ORP basis set is employed in the calculations to examine the limits of its applicability. Results are compared to the literature data obtained in both, large and diffuse, as well as reduced‐size basis sets. In the case of polarizability calculations, the aug‐pc‐1 and R‐ORP are the optimal choices among the investigated smaller basis sets, with the overall performance of the aug‐pc‐1 set being better. Among the larger sets, the ORP performs better in the case of average polarizability, while the RMSE values for polarizability anisotropy are practically identical for d‐aug‐cc‐pVDZ and ORP sets. Finally, the R‐ORP and ORP basis sets compete other small bases in the evaluation of the first hyperpolarizability in investigated systems. © 2016 Wiley Periodicals, Inc.     

Calculating protein–ligand binding affinities with MMPBSA: Method and error analysis

Molecular Mechanics Poisson–Boltzmann Surface Area (MMPBSA) methods have become widely adopted in estimating protein–ligand binding affinities due to their efficiency and high correlation with experiment. Here different computational alternatives were investigated to assess their impact to the agreement of MMPBSA calculations with experiment. Seven receptor families with both high‐quality crystal structures and binding affinities were selected. First the performance of nonpolar solvation models was studied and it was found that the modern approach that separately models hydrophobic and dispersion interactions dramatically reduces RMSD's of computed relative binding affinities. The numerical setup of the Poisson–Boltzmann methods was analyzed next. The data shows that the impact of grid spacing to the quality of MMPBSA calculations is small: the numerical error at the grid spacing of 0.5 Å is already small enough to be negligible. The impact of different atomic radius sets and different molecular surface definitions was further analyzed and weak influences were found on the agreement with experiment. The influence of solute dielectric constant was also analyzed: a higher dielectric constant generally improves the overall agreement with experiment, especially for highly charged binding pockets. The data also showed that the converged simulations caused slight reduction in the agreement with experiment. Finally the direction of estimating absolute binding free energies was briefly explored. Upon correction of the binding‐induced rearrangement free energy and the binding entropy lost, the errors in absolute binding affinities were also reduced dramatically when the modern nonpolar solvent model was used, although further developments were apparently necessary to further improve the MMPBSA methods. © 2016 Wiley Periodicals, Inc.