Electrostatics and Classical Density Functional Theory for Biological Systems

Electrostatic interactions play an essential role in the structure and function of biomolecules (proteins, DNA, cell membranes, etc.). One of the most challenging aspects for understanding these interactions is the fact that biologically active molecules are almost always in solution---that is, they are surrounded by water molecules and dissolved ions. These solvent molecules add many thousands or even millions more degrees of freedom to any theoretical study, many of which are of only secondary importance for investigations of interest. Therefore, we model the solvent using a discontinuous dielectric coefficient, and solve the Poisson equation directly using a boundary-integral formulation and the boundary-element method, BEM, to determine the induced charge distribution on the molecular surface which accounts for the change in polarization charge across the dielectric boundary.

I am collaborating with Jay Bardhan to develop a comprehensive computational platform for solvation modeling, and we are developing both analytical and numerical tools as part of this effort. We are refining the BIBEE approximation algorithm for electrostatics problems with discontinuous dielectric coefficient in complex geometries. BIBEE approximates the operator arising from a boundary integral discretization of the Poisson equation, giving rigorous upper bounds and effective lower bounds for the solvation energy. We have also recently proposed a models for charge asymmetry and dielectric saturation. In addition, we are developing open source software for both finite element and boundary element solution of these problems.

I am collaborating with Dirk Gillespie and Robert Eisenberg at Rush University Medical Center on the modeling of biological ion channels using classical Density Functional Theory of fluids. Since its inception 30 years ago, classical density functional theory (DFT) of fluids has developed into a fast and accurate theoretical tool to understand the fundamental physics of inhomogeneous fluids. To determine the structure of a fluid, DFT minimizes a free energy functional of the fluid density by solving the Euler-Lagrange equations for the inhomogeneous density profiles of all the particle species When solving on a computer, the density can be discretized (called free minimization) or a parameterized function form such as Guassians can be assumed (called parameterized minimization). This approach has been used to model freezing, electrolytes, colloids, and charged polymers in confining geometries and at liquid-vapor interfaces. DFT is different from direct particle simulations where the trajectories of many particles are followed over long times to compute averaged quantities of interest (e.g., density profiles). DFT computes these ensemble-averaged quantities directly. However, developing an accurate DFT is difficult and not straightforward. In fact, new, more accurate DFTs are still being developed for such fundamental systems as hard-sphere fluids, electrolytes, and polymers.

Our group has already applied one dimensional DFT to biological problems involving ion channel permeation, successfully matching and predicting experimental data. We have extended this to 3D using efficient numerical schemes for functional evaluation and nonlinear system solution.