Finite element analysis of drug electrostatic diffusion: inhibition rate studies in N1 neuraminidase

Abstract

This article describes a numerical solution of the steady-state Poisson-Boltzmann-Smoluchowski (PBS) and Poisson-Nernst-Planck (PNP) equations to study diffusion in biomolecular systems. Specifically, finite element methods have been developed to calculate electrostatic interactions and ligand binding rate constants for large biomolecules. The resulting software has been validated and applied to the wild-type and several mutated avian influenza neurominidase crystal structures. The calculated rates show very good agreement with recent experimental studies. Furthermore, these finite element methods require significantly fewer computational resources than existing particle-based Brownian dynamics methods and are robust for complicated geometries. The key finding of biological importance is that the electrostatic steering plays the important role in the drug binding process of the neurominidase.

Figure 1

The illustration of the problem domains for Poisson-Boltzmann and Smoluchowski Equations. The Poisson-Boltzmann equation is solved on Ω, while the Smoluchowski equation is solved only on Ω_s. Γ represents the molecular surface, while Γ_a and Γ_r correspond the reactive and nonreactive boundaries.

Figure 2

(a) the external diffusion domain; (b) the non reactive molecular surface (cyan), the active site (red) and the outer sphere (pink).

Figure 3

(a) the calculated and analytical potential in the diffusion domain; (b) the relative error for calculated potential in the diffusion domain.

Figure 4

The concentration distribution calculated for Na⁺, Cl⁻ and +1e charged drug via the PBS and PNP models, respectively.

Figure 5

The local free energy and entropy densities in the PBS and PNP models.