"New-version-fast-multipole-method" accelerated electrostatic interactions in biomolecular systems

Abstract

In this paper, we present an efficient and accurate numerical algorithm for calculating the electrostatic interactions in biomolecular systems. In our scheme, a boundary integral equation (BIE) approach is applied to discretize the linearized Poisson-Boltzmann (PB) equation. The resulting integral formulas are well conditioned for single molecule cases as well as for systems with more than one macromolecule, and are solved efficiently using Krylov subspace based iterative methods such as generalized minimal residual (GMRES) or bi-conjugate gradients stabilized (BiCGStab) methods. In each iteration, the convolution type matrix-vector multiplications are accelerated by a new version of the fast multipole method (FMM). The implemented algorithm is asymptotically optimal O(N) both in CPU time and memory usage with optimized prefactors. Our approach enhances the present computational ability to treat electrostatics of large scale systems in protein-protein interactions and nano particle assembly processes. Applications including calculating the electrostatics of the nicotinic acetylcholine receptor (nAChR) and interactions between protein Sso7d and DNA are presented.

FIG. 1

Series expansion approximations of the function $\frac{1}{r}$. a) For any point R(R, θ, φ)located outside of a sphere S_a of radius a, the potential generated by N charges located inside of S_a with spherical coordinates ρ(ρ_i, α_i, β_i), respectively, can be described using multipole expansions; b) in the opposite case, for any point R(R, θ, φ) located inside of S_a, the potential generated by N charges located outside of S_a with spherical coordinates ρ(ρ_i, α_i, β_i), respectively, can be described using local expansions.

FIG. 2

Schematic showing the hierarchical divided boxes for recording the neighbor boxes and interaction list in the new version FMM. The neighbor boxes (up to 27 including itself in three dimensions) of the target box b are darkly shaded, while its interaction list (up to 189 boxes in three dimensions) is indicated in yellow. The remaining far-field boxes are indicated in light blue. Also shown are the source points ρ_i and evaluation point R (field). In BEM implementation, the source particles are located at the centers of the surface triangular elements.

FIG. 3

Schematic showing the location of the evaluation point R(r⃗_p, n⃗₀) (R_p) and a BE location ρ_t.

FIG. 4

A typical surface triangulated mesh of a protein (Acetylcholinsterase).

FIG. 5

The CPU time (a) and memory usage (b) of our fast BEM-PB algorithm as compared to those from the direct calculation in one GMRES iteration step.

FIG. 6

The surface potential map of nAChR from different views. The increasing potential from negative to positive value is represented by changing the color from red to blue.

FIG. 7

Electrostatics of Sso7d-DNA. (a) The surface potential map of Sso7d and DNA at separation of 10 Å. (b) The electrostatic interaction energies as functions of separation along the center-to-center unbinding direction. U_BEM is the full electrostatic interaction energy determined by our BEM, and U_Coulomb is the sum of all the atomic pair screened Coulomb interactions between Sso7d and DNA. The curves connect the calculation points (denoted by the diamond and star symbols) consecutively by fitting with cubic splines.