Jaydeep P. Bardhan

Jaydeep P. Bardhan, Matthew G. Knepley, Peter R. Brune

In this paper, we present an analytical solution to 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 their eigenfunctions are the surface spherical harmonics. To introduce this uncommon approach for analytical calculations in separable geometries, we rederive Kirkwood's classic results for a protein surrounded concentrically by a pure-water ion-exclusion layer and then a dilute electrolyte (modeled with the linearized Poisson--Boltzmann equation). Our main result, however, is an analytical method for calculating the reaction potential in a protein embedded in a nonlocal-dielectric solvent, the Lorentz model studied by Dogonadze and Kornyshev. The analytical method enables biophysicists to study the new nonlocal theory in a simple, computationally fast way; an open-source MATLAB implementation is included as supplemental information.

Amirhossein Molavi Tabrizi, Matthew G. Knepley, Jaydeep P. Bardhan

In this paper we extend the familiar continuum electrostatic model with a perturbation to the usual macroscopic boundary condition. The perturbation is based on the mean spherical approximation (MSA), to derive a multiscale hydration-shell boundary condition (HSBC). We show that the HSBC/MSA model reproduces MSA predictions for Born ions in a variety of polar solvents, including both protic and aprotic solvents. Importantly, the HSBC/MSA model predicts not only solvation free energies accurately but also solvation entropies, which standard continuum electrostatic models fail to predict. The HSBC/MSA model depends only on the normal electric field at the dielectric boundary, similar to our recent development of an HSBC model for charge-sign hydration asymmetry, and the reformulation of the MSA as a boundary condition enables its straightforward application to complex molecules such as proteins.

Matthew G. Knepley, Jaydeep P. Bardhan

The structure and function of biological molecules are strongly influenced by the water and dissolved ions that surround them. This aqueous solution (solvent) exerts significant electrostatic forces in response to the biomolecule's ubiquitous atomic charges and polar chemical groups. In this work, we investigate a simple approach to numerical calculation of this model using boundary-integral equation (BIE) methods and boundary-element methods (BEM). Traditional BEM discretizes the protein--solvent boundary into a set of boundary elements, or panels, and the approximate solution is defined as a weighted combination of basis functions with compact support. The resulting BEM matrix then requires integrating singular or near singular functions, which can be slow and challenging to compute. Here we investigate the accuracy and convergence of a simpler representation, namely modeling the unknown surface charge distribution as a set of discrete point charges on the surface. We find that at low resolution, point-based BEM is more accurate than panel-based methods, due to the fact that the protein surface is sampled directly, and can be of significant value for numerous important calculations that require only moderate accuracy, such as the preliminary stages of rational drug design and protein engineering.

Jaydeep P. Bardhan, Matthew G. Knepley

Electrostatic forces play many important roles in molecular biology, but are hard to model due to the complicated interactions between biomolecules and the surrounding solvent, a fluid composed of water and dissolved ions. Continuum model have been surprisingly successful for simple biological questions, but fail for important problems such as understanding the effects of protein mutations. In this paper we highlight the advantages of boundary-integral methods for these problems, and our use of boundary integrals to design and test more accurate theories. Examples include a multiscale model based on nonlocal continuum theory, and a nonlinear boundary condition that captures atomic-scale effects at biomolecular surfaces.

Thomas S. Klotz, Jaydeep P. Bardhan, Matthew G. Knepley

Ellipsoidal harmonics are a useful generalization of spherical harmonics but present additional numerical challenges. One such challenge is in computing ellipsoidal normalization constants which require approximating a singular integral. In this paper, we present results for approximating normalization constants using a well-known decomposition and applying tanh-sinh quadrature to the resulting integrals. Tanh-sinh has been shown to be an effective quadrature scheme for a certain subset of singular integrands. To support our numerical results, we prove that the decomposed integrands lie in the space of functions where tanh-sinh is optimal and compare our results to a variety of similar change-of-variable quadratures.