### Topics of Taiwan Short Course on Applied Mathematics in Biophysics - Dec. 2010

Bo Li's part - 4 lectures on 2 topics.

Lecture 1: Continuum Electrostatics in BiomolecularModeling I. The Poisson-Boltzmann Equation
Lecture 2: Continuum Electrostatics in Biomolecular Modeling II. Dielectric Boundary Forces: A Shape Derivative Approach

Abstract (for both I and II): Electrostatic interactions generate strong forces that influence crucially the conformation and dynamics of a biomolecular system. Continuum electrostatics is an efficient modeling of such forces. The Poisson-Boltzmann (PB) equation is one of the most commonly used such continuum models. In these lectures, we introduce the PB equation, describe its analytic solutions in special cases, and give a derivation of the equation using a variational approach. We also present some new mathematical results of the PB equation. We then discuss how to derive the dielectric boundary forces as the shape derivatives of the electrostatic free energy. Such forces are used in the level-set numerical simulations of equilibrium biomolecular conformations in the variational solvation approach.

Lecture 3: Variational Implicit Solvation I. The Theory
Lecture 4: Variational Implicit Solvation II. The Level-Set Implementation

Abstract (for both I and II): In these lectures, we introduce a novel variational approach to the molecular solvation with a continuum solvent. In this approach, an effective free-energy functional of all possible solute-solvent interfaces is minimized to determine equilibrium conformations and minimum solvation free energies. The functional consists of volume and surface energies of solutes, solute-solvent dispersive interactions, and electrostatic contributions. The electrostatic free energy is obtained by solving the PB equation or by the Coulomb-field or Yukawa-field approximation. Solute molecular mechanics can be coupled with the variational solvation. We also describe a robust level-set method that we have developed to track numerically such equilibrium solute-solvent interfaces. Special techniques are designed to treat the Gaussian curvature arising from the Tolman correction of surface tension. We present extensive numerical results with comparison with molecular dynamics simulations to demonstrate the success of this new approach in capturing the hydrophobic interaction and drying-and-wetting fluctuation between multiple equilibrium states. These properties are in general difficult to describe by most of the existing implicit-solvent models in which ad hoc solute-solvent interfaces are pre-defined and different parts of the free energy are decoupled.

Hong Qian's Part - 4 lectures for 2 to 3 topics.

Lecture 1: Nonlinear Stochastic Dynamics in Mesoscopic Biochemical Systems I: The Chemical Master Equation (CME) and Gillespie Algorithm
Lecture 2: Nonlinear Stochastic Dynamics in Mesoscopic Biochemical Systems II: The CME, Stationary Solutions, and Multi-Scale Dynamics

Abstract (for both I and II): Treating a single biological cell as a homogeneous biochemical reaction system in a small volume, we introduce an analytic theory based on a multi-dimensional birth-and-death process for the nonlinear, stochastic biochemical dynamics. We study the stochastic trajectories as well as Kolmogorov forward equation for the probability distribution. In the infinite system's size limit, the system of nonlinear, ordinary differential equations is derived. Near each steady state, a Gaussian process will be obtained. The stationary solution to the CME is discussed, and its implications to dynamics is presented. Using several simple examples, we show how a multi-scale dynamics emerge from the stochastic dynamics.

Lecture 3: Diffusion Processes and Fokker Planck Equation Models in Biophysics

Abstract: Diffusion process is one of the most widely used stochastic model for continuous dynamic variables in physics. We introduce the Fokker-Planck equations and several applications in biophysics: Kramers problem, Smoluchowski theory, etc.

Lecture 4: Diffusion Approximation of CME and Keizer's Paradox

Abstract: In the large system's size limit, can one approximate the CME with a diffusion processes that is globally valid? We show this attractive idea is unattainable due to the existence of multiple time scales in nonlinear systems. We solve a simple 1-dimensional model to illustrate the issue, known as Keizer's paradox, and its mathematical basis in the finite time (non-uniform) convergence of the V tending infinity.