Block-tridiagonal state-space realization of Chemical Master Equations: a tool to compute explicit solutions

Chemical Master Equations (CMEs) provide a comprehensive way to model the probabilistic behavior in biochemical networks. Despite their widespread diffusion in systems biology, the explicit computation of their solution is often avoided in favor of purely statistic Monte Carlo methods, due to the dramatically high dimension of the CME system. In this work, we investigate some structural properties of CMEs and their solutions, focusing on the efficient computation of the stationary distribution. We introduce a generalized notion of one-step process, which results in a sparse dynamic matrix describing the collection of the scalar CMEs, showing a recursive block-tridiagonal structure as well. Further properties are inferred by means of a graph-theoretical interpretation of the reaction network. We exploit this structure by proposing different methods, including a dedicated LU decomposition, to compute the explicit solution. Examples are included to illustrate the introduced concepts and to show the effectiveness of the proposed approach.