2006/10/31 by Mukhtar Ullah, Olaf Wolkenhauer
Biochemistry, Genetics and Molecular Biology · Mathematics · #Applied mathematics #Bioinformatics and Genomic Networks #Combinatorics #Computer science #Differential equation #Evolution and Genetic Dynamics #Fokker–Planck equation #Gene Regulatory Network Analysis #Joint probability distribution #Jump #Jump process #Markov chain #Markov process #Master equation #Mathematical analysis #Mathematics #Physics #Probabilistic logic #Probability density function #Probability distribution #Statistical physics #Statistics #Stochastic process #Tree (set theory) #q-bio.QM
paper · pdf · doi:10.1049/iet-syb:20070017
published as IET Syst Biol. 1 (2007) 247-254 · 18 pages, 2 figures
arxiv created 2007/07/11 · openalex publication_date 2007/07/11 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.