Family tree of Markov models in systems biology

Abstract

Motivated by applications in systems biology, a probabilistic framework based on Markov processes is proposed to represent intracellular processes. The formal relationships between different stochastic models referred to in the systems biology literature are reviewed. As part of this review, a novel derivation of the differential Chapman-Kolmogorov equation for a general multidimensional Markov process made up of both continuous and jump processes, is presented. First, the definition of a time-derivative for a probability density is focused, but placing no restrictions on the probability distribution, in particular, it is not assumed to be to be confined to a region that has a surface (on which the probability is zero). In this derivation, the master equation gives the jump part of the Markov process and the Fokker-Planck equation gives the continuous part. As a result, a 'family tree' for stochastic models in systems biology is sketched, providing explicit derivations of their formal relationship and clarifying assumptions involved.