Numerical Simulation Methods Of Stochastic Hybrid Chemical Reaction System

Chemical reaction network is an important tool to describe life activities. For a chemical reaction system of a single cell, random fluctuations can be observed experimen-tally due to the sparse number of molecular entities. This means that the deterministic ordinary differential equation model is no longer valid, and it is necessary to build a stochastic process model. Meanwhile, the numerical simulation of a stochastic chemical reaction network becomes a problem worthy of discussion.The stochastic chemical reaction network considered in this thesis is a hybrid system: the reaction probability rate constants are dependent on some factor variables, while the time-evolution of these factor variables is affected by the population variable of the network; the population variable and the factor variables of probability rate constants form a piecewise deterministic Markov process (PDMP) and become the jump component and the continuous component of the PDMP. Stochastic Hodgkin-Huxely neuron model is a typical PDMP model, where the evolution differential equation of the membrane potential of the neuron is coupled with a jump process that represents the stochastic open and closing of ion channels embedded in the membrane.It is challenging to sample a sample path of such a hybrid system exactly. The difficulty is sampling the inter-jump time, which is essentially a hitting time problem with a random threshold. We propose an approximation numerical method by interpolating the conditional log survival function the inter-jump time piecewise linearly. Meanwhile, the convergence of the approximated sample trajectories to the exact solution are proven, and error estimations are provided. Furthermore, other equivalent numerical methods and their corresponding approximation methods are discussed, and the molecule state tracking method for a monomolecular reaction network is given. Finally, we prospect a generalized stochastic hybrid chemical reaction network — coupled diffusion model, and provide its exact simulation method.