Numerical calculation of protein-ligand binding rates through solution of the Smoluchowski equation using smooth particle hydrodynamics

TL;DR

This paper introduces a Lagrangian particle-based smoothed particle hydrodynamics (SPH) method to accurately compute protein-ligand binding rates by solving the Smoluchowski equation with a novel Robin boundary condition, improving agreement with experiments.

Contribution

The paper presents a new SPH-based numerical approach with Robin boundary conditions for calculating diffusion-controlled ligand binding rates, accommodating imperfect enzyme reactions.

Findings

Robin boundary condition improves agreement with experimental rates.

Method successfully applied to ligand binding to acetylcholinesterase.

Framework adaptable to larger biomolecular systems.

Abstract

Background. The calculation of diffusion-controlled ligand binding rates is important for understanding enzyme mechanisms as well as designing enzyme inhibitors. We demonstrate the accuracy and effectiveness of a Lagrangian particle-based method, smoothed particle hydrodynamics (SPH), to study diffusion in biomolecular systems by numerically solving the time-dependent Smoluchowski equation for continuum diffusion. Results. The numerical method is first verified in simple systems and then applied to the calculation of ligand binding to an acetylcholinesterase monomer. Unlike previous studies, a reactive Robin boundary condition (BC), rather than the absolute absorbing (Dirichlet) boundary condition, is considered on the reactive boundaries. This new boundary condition treatment allows for the analysis of enzymes with "imperfect" reaction rates. Rates for inhibitor binding to mAChE are calculated at various ionic strengths and compared with experiment and other numerical methods. We find that imposition of the Robin BC improves agreement between calculated and experimental reaction rates. Conclusions. Although this initial application focuses on a single monomer system, our new method provides a framework to explore broader applications of SPH in larger-scale biomolecular complexes by taking advantage of its Lagrangian particle-based nature.