Abstract: The time evolution of chemically reacting molecules is sometimes modeled using a stochastic formulation, which takes into account the inherent randomness of molecular motion. This formulation is especially useful for complex reactions inside living cells, where small populations of key reactants can set the stage for significant stochastic effects. In this talk, we show how Stochastic Hybrid Systems can be used to construct stochastic models for chemical reactions.
Hybrid systems combine continuous-time dynamics with discrete modes of operation. The states of such system usually have two distinct components: one that evolves continuously, typically according to a differential equation; and another one that only changes through instantaneous jumps. To model chemical reactions, we actually need Stochastic Hybrid Systems (SHSs) where transitions between discrete modes are triggered by stochastic events, much like transitions between states of a continuous-time Markov chains. However, the rate at which transitions occur is allowed to depend on both the continuous and the discrete states of the SHS.
Several tools are available to analyze SHSs. Among these, we discuss the use of the extended generator, infinite-dimensional moment dynamics, and finite-dimensional truncations by moment closure. The application of these tools is illustrated in the context of modeling the evolution of populations of molecules undergoing a system of chemical reactions.