# Mathematical Biology Seminar

### Michael Mascagni

Florida State University, Department of Computer Science

#### Novel Stochastic Methods in Biochemical Electrostatics

#### Monday, November 12, 2012, at 4:00pm

LCB 219

Electrostatic forces and the electrostatic properties of molecules in solution are
among the most important issues in understanding the structure and function of large
biomolecules. The use of implicit-solvent models, such as the Poisson-Boltzmann equation
(PBE), have been used with great success as a way of computationally deriving electrostatics
properties such molecules. We discuss how to solve an elliptic system of partial
differential equations (PDEs) involving the Poisson and the PBEs using path-integral
based probabilistic, Feynman-Kac, representations. This leads to a Monte Carlo method for
the solution of this system which is specified with a stochastic process, and a score
function. We use several techniques to simplify the Monte Carlo method and the stochastic
process used in the simulation, such as the walk-on-spheres (WOS) algorithm, and an
auxiliary sphere technique to handle internal boundary conditions. We then specify some
optimizations using the error (bias) and variance to balance the CPU time. We show that
our approach is as accurate as widely used deterministic codes, but has many desirable
properties that these methods do not. In addition, the currently optimized codes consume
comparable CPU times to the widely used deterministic codes. Thus, we have an very clear
example where a Monte Carlo calculation of a low-dimensional PDE is as fast or faster
than deterministic techniques at similar accuracy levels.