Analysis of stochastically modeled biochemical processes with applications to numerical methods.

摘要

In this dissertation, we study stochastically modeled chemical reaction networks and associated simulation methods. The bulk of this dissertation focuses on three selected topics. Firstly, we present an efficient Runge-Kutta type simulation method and compare its weak error with those of other methods. In particular, we make a comparison with the usual Euler method, which is termed tau-leaping in the current context. The new method is found to be an order of magnitude more accurate than Euler's method, making it the first high-order numerical method for the models considered in this dissertation. Secondly, we study different coupling methods of stochastically modeled biochemical processes and provide an asymptotic relation between two such couplings found commonly in the literature. This work is motivated by the fact that variance reduction is a critical aspect of many computational methods, such as in finite difference schemes for the estimation of sensitivities and multi-level Monte Carlo algorithms for the estimation of expectations. Thirdly, we will prove a large population result on a class of chemical reaction networks which allow for reactions to have "interruptible" delay. The technique of the proof is similar in nature to that of Nancy Garcia's large population result on an S.I.R model with generally distributed infectious period, though this was not known at the time of writing. Finally, along with a package for the implementation of multi-level Monte Carlo for MATLAB, we present two miscellaneous results including: (i) a proof that complex balanced chemical reaction networks are non-explosive, and (ii) an application of the multi-level Monte Carlo algorithm for the purpose of sensitivity analysis, which produces the most efficient method for the approximation of sensitivities to date.