Computational methods for stochastic biological systems

Abstract

The virtual completion of the genome project and prodigious amounts of work by different biologists throughout the world have elucidated many of the components of biological systems. The genes (and hence proteins) are largely known, and the tools of molecular biology allows one to manufacture and express them, so as to understand their function. Given this increased understanding of components, the next step in understanding complex biology will be understanding systems, which will almost certainly involve formal, detailed, and quantitative models. One of the great challenges of modeling biological systems is that they tend to “break the math.” Biological systems have small numbers of molecules, operate far from equilibrium, change shape and size, etc. This thesis develops mathematical and computational tools for biological systems with few molecules. Such systems are particularly problematic because the usual macroscopic view of chemistry, in which concentrations of molecules vary continuously, continually, and deterministically, does not work. Rather, one needs to use the mesoscopic view of chemistry: molecules undergo discrete reaction events, and the timing of these events is probabilistic. There are many standard numerical computational techniques for the macroscopic view, but far fewer for the mesoscopic view. This thesis develops (1) an efficient, exact stochastic simulation algorithm, to generate trajectories of mesoscopic biological systems, (2) a sensitivity analysis algorithm, to quantify how a model's predictions depend on the exact values of parameters (e.g., rate constants) used, and (3) a parameter estimation algorithm, to estimate the values of model parameters from observed trajectories.