Smooth Dielectric Function Research Articles

On delivering polar solvation free energy of proteins from energy minimized structures using a regularized super-Gaussian Poisson-Boltzmann model.

The biomolecules interact with their partners in an aqueous media; thus, their solvation energy is an important thermodynamics quantity. In previous works (J. Chem. Theory Comput. 14(2): 1020-1032), we demonstrated that the Poisson-Boltzmann (PB) approach reproduces solvation energy calculated via thermodynamic integration (TI) protocol if the structures of proteins are kept rigid. However, proteins are not rigid bodies and computing their solvation energy must account for their flexibility. Typically, in the framework of PB calculations, this is done by collecting snapshots from molecular dynamics (MD) simulations, computing their solvation energies, and averaging to obtain the ensemble-averaged solvation energy, which is computationally demanding. To reduce the computational cost, we have proposed Gaussian/super-Gaussian-based methods for the dielectric function that use the atomic packing to deliver smooth dielectric function for the entire computational space, the protein and water phase, which allows the ensemble-averaged solvation energy to be computed from a single structure. One of the technical difficulties associated with the smooth dielectric function presentation with respect to polar solvation energy is the absence of a dielectric border between the protein and water where induced charges should be positioned. This motivated the present work, where we report a super-Gaussian regularized Poisson-Boltzmann method and use it for computing the polar solvation energy from single energy minimized structures and assess its ability to reproduce the ensemble-averaged polar solvation on a dataset of 74 high-resolution monomeric proteins.

Reproducing ensemble averaged electrostatics with Super-Gaussian-based smooth dielectric function: application to electrostatic component of binding energy of protein complexes

Reproducing ensemble averaged electrostatics with Super-Gaussian-based smooth dielectric function: application to electrostatic component of binding energy of protein complexes

Gaussian-Based Smooth Dielectric Function: A Surface-Free Approach for Modeling Macromolecular Binding in Solvents.

Conventional modeling techniques to model macromolecular solvation and its effect on binding in the framework of Poisson-Boltzmann based implicit solvent models make use of a geometrically defined surface to depict the separation of macromolecular interior (low dielectric constant) from the solvent phase (high dielectric constant). Though this simplification saves time and computational resources without significantly compromising the accuracy of free energy calculations, it bypasses some of the key physio-chemical properties of the solute-solvent interface, e.g., the altered flexibility of water molecules and that of side chains at the interface, which results in dielectric properties different from both bulk water and macromolecular interior, respectively. Here we present a Gaussian-based smooth dielectric model, an inhomogeneous dielectric distribution model that mimics the effect of macromolecular flexibility and captures the altered properties of surface bound water molecules. Thus, the model delivers a smooth transition of dielectric properties from the macromolecular interior to the solvent phase, eliminating any unphysical surface separating the two phases. Using various examples of macromolecular binding, we demonstrate its utility and illustrate the comparison with the conventional 2-dielectric model. We also showcase some additional abilities of this model, viz. to account for the effect of electrolytes in the solution and to render the distribution profile of water across a lipid membrane.

Reproducing the Ensemble Average Polar Solvation Energy of a Protein from a Single Structure: Gaussian-Based Smooth Dielectric Function for Macromolecular Modeling.

Typically, the ensemble average polar component of solvation energy (ΔGpolarsolv) of a macromolecule is computed using molecular dynamics (MD) or Monte Carlo (MC) simulations to generate conformational ensemble and then single/rigid conformation solvation energy calculation is performed on each snapshot. The primary objective of this work is to demonstrate that Poisson-Boltzmann (PB)-based approach using a Gaussian-based smooth dielectric function for macromolecular modeling previously developed by us (Li et al. J. Chem. Theory Comput. 2013, 9 (4), 2126-2136) can reproduce that ensemble average (ΔGpolarsolv) of a protein from a single structure. We show that the Gaussian-based dielectric model reproduces the ensemble average ΔGpolarsolv(⟨ΔGpolarsolv⟩) from an energy-minimized structure of a protein regardless of the minimization environment (structure minimized in vacuo, implicit or explicit waters, or crystal structure); the best case, however, is when it is paired with an in vacuo-minimized structure. In other minimization environments (implicit or explicit waters or crystal structure), the traditional two-dielectric model can still be selected with which the model produces correct solvation energies. Our observations from this work reflect how the ability to appropriately mimic the motion of residues, especially the salt bridge residues, influences a dielectric model's ability to reproduce the ensemble average value of polar solvation free energy from a single in vacuo-minimized structure.

On the Modeling of Polar Component of Solvation Energy using Smooth Gaussian-Based Dielectric Function.

Traditional implicit methods for modeling electrostatics in biomolecules use a two-dielectric approach: a biomolecule is assigned low dielectric constant while the water phase is considered as a high dielectric constant medium. However, such an approach treats the biomolecule-water interface as a sharp dielectric border between two homogeneous dielectric media and does not account for inhomogeneous dielectric properties of the macromolecule as well. Recently we reported a new development, a smooth Gaussian-based dielectric function which treats the entire system, the solute and the water phase, as inhomogeneous dielectric medium (J Chem Theory Comput. 2013 Apr 9; 9(4): 2126-2136.). Here we examine various aspects of the modeling of polar solvation energy in such inhomogeneous systems in terms of the solute-water boundary and the inhomogeneity of the solute in the absence of water surrounding. The smooth Gaussian-based dielectric function is implemented in the DelPhi finite-difference program, and therefore the sensitivity of the results with respect to the grid parameters is investigated, and it is shown that the calculated polar solvation energy is almost grid independent. Furthermore, the results are compared with the standard two-media model and it is demonstrated that on average, the standard method overestimates the magnitude of the polar solvation energy by a factor 2.5. Lastly, the possibility of the solute to have local dielectric constant larger than of a bulk water is investigated in a benchmarking test against experimentally determined set of pKa's and it is speculated that side chain rearrangements could result in local dielectric constant larger than 80.

On the Dielectric “Constant” of Proteins: Smooth Dielectric Function for Macromolecular Modeling and Its Implementation in DelPhi

Implicit methods for modeling protein electrostatics require dielectric properties of the system to be known, in particular, the value of the dielectric constant of protein. While numerous values of the internal protein dielectric constant were reported in the literature, still there is no consensus of what the optimal value is. Perhaps this is due to the fact that the protein dielectric constant is not a “constant” but is a complex function reflecting the properties of the protein’s structure and sequence. Here, we report an implementation of a Gaussian-based approach to deliver the dielectric constant distribution throughout the protein and surrounding water phase by utilizing the 3D structure of the corresponding macromolecule. In contrast to previous reports, we construct a smooth dielectric function throughout the space of the system to be modeled rather than just constructing a “Gaussian surface” or smoothing molecule–water boundary. Analysis on a large set of proteins shows that (a) the average dielectric constant inside the protein is relatively low, about 6–7, and reaches a value of about 20–30 at the protein’s surface, and (b) high average local dielectric constant values are associated with charged residues while low dielectric constant values are automatically assigned to the regions occupied by hydrophobic residues. In terms of energetics, a benchmarking test was carried out against the experimental pKa’s of 89 residues in staphylococcal nuclease (SNase) and showed that it results in a much better RMSD (= 1.77 pK) than the corresponding calculations done with a homogeneous high dielectric constant with an optimal value of 10 (RMSD = 2.43 pK).

Differential geometry based solvation model I: Eulerian formulation

This paper presents a differential geometry based model for the analysis and computation of the equilibrium property of solvation. Differential geometry theory of surfaces is utilized to define and construct smooth interfaces with good stability and differentiability for use in characterizing the solvent–solute boundaries and in generating continuous dielectric functions across the computational domain. A total free energy functional is constructed to couple polar and nonpolar contributions to the solvation process. Geometric measure theory is employed to rigorously convert a Lagrangian formulation of the surface energy into an Eulerian formulation so as to bring all energy terms into an equal footing. By optimizing the total free energy functional, we derive coupled generalized Poisson–Boltzmann equation (GPBE) and generalized geometric flow equation (GGFE) for the electrostatic potential and the construction of realistic solvent–solute boundaries, respectively. By solving the coupled GPBE and GGFE, we obtain the electrostatic potential, the solvent–solute boundary profile, and the smooth dielectric function, and thereby improve the accuracy and stability of implicit solvation calculations. We also design efficient second-order numerical schemes for the solution of the GPBE and GGFE. Matrix resulted from the discretization of the GPBE is accelerated with appropriate preconditioners. An alternative direct implicit (ADI) scheme is designed to improve the stability of solving the GGFE. Two iterative approaches are designed to solve the coupled system of nonlinear partial differential equations. Extensive numerical experiments are designed to validate the present theoretical model, test computational methods, and optimize numerical algorithms. Example solvation analysis of both small compounds and proteins are carried out to further demonstrate the accuracy, stability, efficiency and robustness of the present new model and numerical approaches. Comparison is given to both experimental and theoretical results in the literature.

Integrated Continuum Dielectric Approaches to treat Molecular Polarizability and the Condensed Phase: Refractive Index and Implicit Solvation.

The idea of using a dielectric continuum inside a molecule to accurately model molecular polarizability is extended to include a larger spectrum of bioorganic molecules and the condensed phase. Atomic polarization radii and an internal dielectric (εin) were fitted to reproduce ab initio B3LYP/aug-cc-pVTZ polarizability tensors taken from a data set of 707 molecules. The average unsigned error on the isotropic polarizability and anisotropy are 2.6% and 5.2%, respectively. It is shown that usual Poisson−Boltzmann contact radii and a low internal dielectric are not appropriate and require major revision. To account for the anisotropy of polarizability, the internal dielectric (εin) constant needs to be larger than 6.0. Reinterpreting the theoretical link between εin and the experimental refractive index (n), this study shows, with a set of 23 organic molecules spanning the entire range of n, that even with εin = 24 the obtained refractive indices can correlate well with experiment (slope of 1.00, intercept of 0.05, and R = 0.95). The novel methodology used here to calculate a macroscopic-like refractive index shows that the application of the EPIC parametrization to condensed phase leads to suitable behavior. Although the primary goal in developing EPIC was to include polarizability in explicit solvent calculations, we also extend the model to work with implicit solvent. This requires the use of a 3-zone smooth dielectric function to transition from the polarization dielectric inside the molecules to the dielectric continuum of the solvent. The parametrization and validation of this model are performed against 485 experimental free energies of hydration. Using 8 solvent cavity atomic radii and a single surface tension an average unsigned error of 1.1 kal/mol and a correlation coefficient of 0.9 are obtained, validating the use of the EPIC model in the condensed phase.