Analytic and numerical solutions of framework models

Framework models are stochastic models designed to incorporate information from simulations of the dynamics of biomolecules. The framework models can then calculate properties of the molecules on a much longer time scale than can be simulated directly by the molecular dynamics.; This dissertation develops various methods to solve framework models both analytically and numerically. The first method developed is the lumped state approximations (LSA). It is a method for handling boundary conditions for diffusion on an interval that simplifies the description of transitions into and out of the interval. Also discussed are mean first passage time calculations both using and not using the lumped state approximation.; The next method introduced is the generalized King-Altman method. It is a diagrammatic method for finding the steady state solutions of Markov chains that are commonly used to represent the interactions of enzyme systems in the field of enzyme kinetics. This thesis develops a generalized King-Altman method for the solution of framework models whose state diagrams includes one-dimensional segments of states. State-dependent diffusion coefficients are easily incorporated by the method. Mean first passage time calculations are also made using this method.; The last method discussed is a time-dependent single particle model with nonlocal Robin boundary conditions. There are two numerical solutions of this model, a finite element solution and Markov chain solution. An analytic mean first passage time calculation is also developed and solved. Then the model is applied to the problem of ion blockers passing over high-energy barriers. Our results are then compared with experiment. An analytic solution for the special case of a linear potential and constant diffusion coefficient of the time-dependent model is also developed.