General solution of the chemical master equation and modality of marginal distributions for hierarchic first-order reaction networks

J Math Biol. 2018 Aug;77(2):377-419. doi: 10.1007/s00285-018-1205-2. Epub 2018 Jan 20.


Multimodality is a phenomenon which complicates the analysis of statistical data based exclusively on mean and variance. Here, we present criteria for multimodality in hierarchic first-order reaction networks, consisting of catalytic and splitting reactions. Those networks are characterized by independent and dependent subnetworks. First, we prove the general solvability of the Chemical Master Equation (CME) for this type of reaction network and thereby extend the class of solvable CME's. Our general solution is analytical in the sense that it allows for a detailed analysis of its statistical properties. Given Poisson/deterministic initial conditions, we then prove the independent species to be Poisson/binomially distributed, while the dependent species exhibit generalized Poisson/Khatri Type B distributions. Generalized Poisson/Khatri Type B distributions are multimodal for an appropriate choice of parameters. We illustrate our criteria for multimodality by several basic models, as well as the well-known two-stage transcription-translation network and Bateman's model from nuclear physics. For both examples, multimodality was previously not reported.

Keywords: Bursting process; Chemical master equation; Compound process; Conditional multimodality; Generalized binomial distribution; Generalized poisson distribution; Hierarchic reaction network; Hierarchically linear ODE system; Nuclear decay chain; Transcription–translation model.

Publication types

  • Research Support, Non-U.S. Gov't

MeSH terms

  • Biocatalysis
  • Biochemical Phenomena
  • Computer Simulation
  • Kinetics
  • Linear Models
  • Mathematical Concepts
  • Metabolic Networks and Pathways
  • Models, Biological*
  • Models, Chemical
  • Poisson Distribution
  • Probability
  • Protein Biosynthesis
  • Transcription, Genetic