title: General transient solution of the one-step master equation in one dimension.
abstract: Exact analytical solutions of the master equation are limited to special cases and exact numerical methods are inefficient. Even the generic one-dimensional, one-step master equation has evaded exact solution, aside from the steady-state case. This type of master equation describes the dynamics of a continuous-time Markov process whose range consists of positive integers and whose transitions are allowed only between adjacent sites. The solution of any master equation can be written as the exponential of a (typically huge) matrix, which requires the calculation of the eigenvalues and eigenvectors of the matrix. Here we propose a linear algebraic method for simplifying this exponential for the general one-dimensional, one-step process. In particular, we prove that the calculation of the eigenvectors is actually not necessary for the computation of exponential, thereby we dramatically cut the time of this calculation. We apply our new methodology to examples from birth-death processes and biochemical networks. We show that the computational time is significantly reduced compared to existing methods.
authors.affiliation: School of Biological Sciences, University of Edinburgh, Edinburgh EH9 3JR, United Kingdom.
authors.forename: Stephen
authors.initials: S
authors.lastname: Smith
authors.affiliation: Department of Mathematics, Imperial College, London SW7 2AZ, United Kingdom.
authors.forename: Vahid
authors.initials: V
authors.lastname: Shahrezaei
mesh_terms: D001419:Bacteria / Q000254:growth & development; D019281:Dimerization; D004798:Enzymes / Q000378:metabolism; D016014:Linear Models; D008390:Markov Chains; D008956:Models, Chemical; D008962:Models, Theoretical; D010766:Phosphorylation
chemical_list: D004798:Enzymes