Some Results on the Structural Properties and the Solution of the Chemical Master Equation

The Chemical Master Equation (CME) is a well known tool for studying (bio)chemical processes involving few copies of the species involved, because it is a framework able to capture random behaviors that are neglected by deterministic approaches based on the concentration dynamics. In this work, we investigate some structural properties of CMEs and their solutions, with a particular focus on the efficient computation of the stationary distribution. We introduce a generalized notion of one-step process, which results in a sparse dynamical matrix describing the collection of the scalar CMEs, also showing a recursive block-tridiagonal structure. Further properties are inferred by means of a graph-theoretical interpretation of the reaction network. Examples are included to illustrate the notions and to show the effectiveness of the proposed approach.