Computational methods for biomolecular electrostatics

Abstract

An understanding of intermolecular interactions is essential for insight into how cells develop, operate, communicate, and control their activities. Such interactions include several components: contributions from linear, angular, and torsional forces in covalent bonds, van der waals forces, as well as electrostatics. Among the various components of molecular interactions, electrostatics are of special importance because of their long range and their influence on polar or charged molecules, including water, aqueous ions, and amino or nucleic acids, which are some of the primary components of living systems. Electrostatics, therefore, play important roles in determining the structure, motion, and function of a wide range of biological molecules. This chapter presents a brief overview of electrostatic interactions in cellular systems, with a particular focus on how computational tools can be used to investigate these types of interactions.

Figure 1

A schematic comparison of implicit and explicit solvent models: (a) in the implicit solvent model, a low dielectric solute is surrounded by a continuum of high dielectric solvent; (b) in the explicit solvent model, solvent is represented by discrete water molecules.

Figure 2

A thermodynamic cycle illustrating the biomolecular solvation process. The steps are (1) un-charging the biomolecule in vacuum; (2) transferring the uncharged biomolecule from vacuum to solvent; (3) charging the biomolecule back to its normal value in solvent. The nonpolar solvation free energy is the free energy change in step (2). The polar solvation free energy is the sum of the free energy changes in steps (1) and (3).

Figure 3

Description of the terms in the Poisson-Boltzmann equation: (a) the dielectric permittivity coefficient ε(x⃑) is much smaller inside the biomolecule than outside the biomolecule with a rapid change in value across the solvent-accessible biomolecular surface; (b) the ion-accessibility parameter κ²(x⃑) is proportional to the bulk ionic strength outside the ion-accessible biomolecular surface; (c) the biomolecular charge distribution is defined as the collection of point charges located at the center of each atom.

Figure 4

Examples of the visualization of the balanol electrostatic potential in the binding site of protein Kinase A as calculated by APBS (Baker et al., 2001) and visualized with VMD (Humphrey et al., 1996).

Figure 5

Schematic of a polar solvation free energy calculation; in the initial state, the dielectric coefficient is a constant throughout the entire system and equal to the solute dielectric coefficient; in the final state, the dielectric coefficient is inhomogeneous and smaller in the solute than in the bulk solvent.

Figure 6

Thermodynamic cycle illustrating the standard procedure for calculating the electrostatic contribution to the binding free energy of a complex with rigid-body. The steps are (1) transfer the isolated molecule from a inhomogeneous dielectric into a homogeneous dielectric, the free energy change is $-(\mathrm{\Delta }{G}_{\mathit{mol}1}^{\mathit{solv}}+\mathrm{\Delta }{G}_{\mathit{mol}2}^{\mathit{solv}})$; (2) form the complex from isolated molecules in a homogeneous dielectric, the free energy change is $\mathrm{\Delta }{G}_{\mathit{binding}}^{\mathit{coul}}$, (3) transfer the complex from the homogeneous dielectric into the inhomogeneous dielectric, the free energy change is $\mathrm{\Delta }{G}_{\mathit{complex}}^{\mathit{solv}}$.