Switching and Diffusion Models for Gene Regulation Networks

We analyze a hierarchy of three regimes for modeling gene regulation. The most complete model is a continuous time, discrete state space, Markov jump process. An intermediate “switch plus diffusion” model takes the form of a stochastic differential equation driven by an independent continuous time Markov switch. In the third “switch plus ODE” model the switch remains, but the diffusion is removed. The latter two models allow for multiscale simulation where, for the sake of computational efficiency, system components are treated differently according to their abundance. The switch plus ODE regime was proposed by Paszek [Bull. Math. Biol., 69 (2007), pp. 1567–1601], who analyzed the steady state behavior, showing that the mean was preserved, but the variance only approximated that of the full model. Here, we show that the tools of stochastic calculus can be used to analyze first and second moments for all time. A technical issue to be addressed is that the state space for the discrete-valued switch is infin...

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