A new numerical method for nonlocal electrostatics in biomolecular simulations

Journal of the Mechanical Behavior of Materials

2013

In the interest of developing a bridge between researchers modeling materials and those modeling biological molecules, we survey recent progress in developing nonlocal-dielectric continuum models for studying the behavior of proteins and nucleic acids. As in other areas of science, continuum models are essential tools when atomistic simulations (e.g. molecular dynamics) are too expensive. Because biological molecules are essentially all nanoscale systems, the standard continuum model, involving local dielectric response, has basically always been dubious at best. The advanced continuum theories discussed here aim to remedy these shortcomings by adding features such as nonlocal dielectric response, and nonlinearities resulting from dielectric saturation. We begin by describing the central role of electrostatic interactions in biology at the molecular scale, and motivate the development of computationally tractable continuum models using applications in science and engineering. For context, we highlight some of the most important challenges that remain and survey the diverse theoretical formalisms for their treatment, highlighting the rigorous statistical mechanics that support the use and improvement of continuum models. We then address the development and implementation of nonlocal dielectric models, an approach pioneered by Dogonadze, Kornyshev, and their collaborators almost forty years ago. The simplest of these models is just a scalar form of gradient elasticity, and here we use ideas from gradient-based modeling to extend the electrostatic model to include additional length scales. The paper concludes with a discussion of open questions for model development, highlighting the many opportunities for the materials community to leverage its physical, mathematical, and computational expertise to help solve one of the most challenging questions in molecular biology and biophysics.

Section: Discussionmentioning

confidence: 99%

Section: Nonlocal Dielectric Modelsmentioning

confidence: 99%

Gradient models in molecular biophysics: progress, challenges, opportunities

Journal of the Mechanical Behavior of Materials

2013

“…The theory's successful prediction of monovalent and divalent monatomic ion solvation free energies, 12 and the intriguing indication here that nonlocal models with realistic parameters predict pK a shifts matching those from empirically verified effective models, 33, 34 support the notion that nonlocal models merit serious investigation and testing. The most important next step, already underway, is the development of efficient algorithms and software to enable the study of atomistic representations of molecular solute-solvent interfaces 22,42,43 and parameterization. In addition, even relatively simple molecular dynamics calculations should be able to furnish important new insights into nonlocal models' apparent qualitative improvement as a means to understand electrostatic interactions in biomolecules, particularly the length-scale parameter λ.…”

Section: Discussionmentioning

confidence: 99%

Nonlocal continuum electrostatic theory predicts surprisingly small energetic penalties for charge burial in proteins

The Journal of Chemical Physics

2011

We study the energetics of burying charges, ion pairs, and ionizable groups in a simple protein model using nonlocal continuum electrostatics. Our primary finding is that the nonlocal response leads to markedly reduced solvent screening, comparable to the use of application-specific protein dielectric constants. Employing the same parameters as used in other nonlocal studies, we find that for a sphere of radius 13.4 Å containing a single +1e charge, the nonlocal solvation free energy varies less than 18 kcal/mol as the charge moves from the surface to the center, whereas the difference in the local Poisson model is ∼35 kcal/mol. Because an ion pair (salt bridge) generates a comparatively more rapidly varying Coulomb potential, energetics for salt bridges are even more significantly reduced in the nonlocal model. By varying the central parameter in nonlocal theory, which is an effective length scale associated with correlations between solvent molecules, nonlocal-model energetics can be varied from the standard local results to essentially zero; however, the existence of the reduction in charge-burial penalties is quite robust to variations in the protein dielectric constant and the correlation length. Finally, as a simple exploratory test of the implications of nonlocal response, we calculate glutamate pK(a) shifts and find that using standard protein parameters (ε(protein) = 2-4), nonlocal results match local-model predictions with much higher dielectric constants. Nonlocality may, therefore, be one factor in resolving discrepancies between measured protein dielectric constants and the model parameters often used to match titration experiments. Nonlocal models may hold significant promise to deepen our understanding of macromolecular electrostatics without substantially increasing computational complexity.

“…To accelerate studies of the promising Lorentz nonlocal model 17,26,27,65 , we have derived the exact analytical solution for a spherical solute containing an arbitrary charge distribution. Our approach uses Hildebrandt’s boundary-integral equation (BIE) formulation 26 and the analytically known eigendecompositions of the associated boundary-integral operators.…”

Section: Discussionmentioning

confidence: 99%

Nonlocal Electrostatics in Spherical Geometries Using Eigenfunction Expansions of Boundary-Integral Operators

Computational and Mathematical Biophysics

Knepley

Brune

2015

In this paper, we present an exact, infinite-series solution to Lorentz nonlocal continuum electrostatics for an arbitrary charge distribution in a spherical solute. Our approach relies on two key steps: (1) re-formulating the PDE problem using boundary-integral equations, and (2) diagonalizing the boundary-integral operators using the fact that their eigenfunctions are the surface spherical harmonics. To introduce this uncommon approach for calculations in separable geometries, we first re-derive Kirkwood’s classic results for a protein surrounded concentrically by a pure-water ion-exclusion (Stern) layer and then a dilute electrolyte, which is modeled with the linearized Poisson–Boltzmann equation. The eigenfunction-expansion approach provides a computationally efficient way to test some implications of nonlocal models, including estimating the reasonable range of the nonlocal length-scale parameter λ. Our results suggest that nonlocal solvent response may help to reduce the need for very high dielectric constants in calculating pH-dependent protein behavior, though more sophisticated nonlocal models are needed to resolve this question in full. An open-source MATLAB implementation of our approach is freely available online.