Abstract
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.
Highlights
One of the long-standing challenges in molecular biophysics is the development of accurate, yet simple models for the influence of biological fluids on biological molecules such as proteins and DNA
These simulations come at two prices: first, molecular dynamics (MD) simulations can require many hundreds of compute hours, most of which are spent on the thousands of water molecules whose individual behaviors are not of primary relevance; second, practitioners must understand numerous subtleties about simulation protocols and the parameters associated with the physical models
To support the development and testing of nonlocal electrostatic models for biomolecule solvation, we present here the nonlocal-model analogue of Kirkwood’s result: namely, an analytical approach for the electrostatic solvation free energy of an arbitrary charge distribution in a spherical solute embedded in a solvent modeled as a Lorentz nonlocal dielectric
Summary
One of the long-standing challenges in molecular biophysics is the development of accurate, yet simple models for the influence of biological fluids (aqueous solutions composed of water and dissolved ions) on biological molecules such as proteins and DNA. Green’s theorem and double reciprocity can be used to transform the coupled PDE system into a purely boundary-integral-equation (BIE) representation of the nonlocal model[26,31] In principle, both the local-formulation PDE problem and the purely BIE method are solved problems numerically, in the sense that asymptotically optimal (linear-scaling) numerical algorithms exist[27,32,33,34,35,36]. To support the development and testing of nonlocal electrostatic models for biomolecule solvation, we present here the nonlocal-model analogue of Kirkwood’s result: namely, an analytical approach for the electrostatic solvation free energy of an arbitrary charge distribution in a spherical solute embedded in a solvent modeled as a Lorentz nonlocal dielectric.
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