Stochastic Models of Gene Expression

This thesis addresses the problem of describing the time evolution of a genetic chemical system using a stochastic approach: the Master equation approach, which is the consequence of the fundamental hypothesis of the stochastic formulation of chemical kinetics. The Master equation can be used to compute the time dependent probability distribution of the state of a chemical system. The Master equation is built on a Markov model foundation. We derive the Master equation under the assumption that there exits a Markov process satisfying three particular postulates. A mathematical formulism called Petri net which represents a chemical system is introduced. The Petri net and a positive integer k together determine a State graph. We also explore the properties of the solution of the Master equation associated with the State graph. We give a construction of a Markov process on an infinite product space and prove that the constructed Markov process satisfies the three postulates, hence satisfies the Master equation. We also give the time evolution outcomes of the stochastic simulations of a genetic auto-regulatory system using computer software, which are consistent with the long time behavior property of the solution of the Master equation.