[Paper Review] Time-dependent product-form Poisson distributions for reaction networks with higher order complexes

David F. Anderson, David Schnoerr+1 more|arXiv (Cornell University)|Apr 25, 2019

Gene Regulatory Network AnalysisBiochemistry, Genetics and Molecular Biology21 references3 citations

TL;DR

This paper establishes a necessary and sufficient condition—dynamical and restricted complex balancing (DR)—for the time-dependent distribution of a stochastic reaction network with higher-order complexes to remain a product of Poisson distributions for all time, provided the initial distribution is a product of Poissons. The result extends classical product-form stationary distributions to the transient regime via a time-varying mean vector solving the deterministic mass-action equations.

ABSTRACT

It is well known that stochastically modeled reaction networks that are complex balanced admit a stationary distribution that is a product of Poisson distributions. In this paper, we consider the following related question: supposing that the initial distribution of a stochastically modeled reaction network is a product of Poissons, under what conditions will the distribution remain a product of Poissons for all time? By drawing inspiration from Crispin Gardiner's "Poisson representation" for the solution to the chemical master equation, we provide a necessary and sufficient condition for such a product-form distribution to hold for all time. Interestingly, the condition is a dynamical "complex-balancing" for only those complexes that have multiplicity greater than or equal to two (i.e. the higher order complexes that yield non-linear terms to the dynamics). We term this new condition the "dynamical and restricted complex balance" condition (DR for short).

Motivation & Objective

• To identify conditions under which the time-dependent distribution of a stochastic reaction network remains a product of Poissons when initialized as such.
• To extend the classical product-form stationary distribution result (product of Poissons) to the transient, time-dependent regime.
• To characterize the dynamics of the mean vector c(t) that preserves the product-form structure.
• To generalize Gardiner's Poisson representation to networks with higher-order complexes.
• To establish a necessary and sufficient condition for the preservation of product-form Poisson distributions beyond the stationary case.

Proposed method

• Introduces the dynamical and restricted (DR) complex balance condition, which applies only to complexes of order two or higher.
• Uses Gardiner's Poisson representation to derive the time-dependent solution of the chemical master equation.
• Derives a necessary and sufficient condition for the time-dependent distribution to remain a product of Poissons by equating coefficients in the Kolmogorov forward equation.
• Employs polynomial independence arguments to prove that the DR condition is both necessary and sufficient.
• Shows that the time-varying mean vector c(t) satisfies the deterministic mass-action ODEs, ensuring the product-form structure is preserved.
• Establishes equivalence between the DR condition and the preservation of the product-form via coefficient matching in the master equation.

Experimental results

Research questions

• RQ1 Under what conditions does a stochastically modeled reaction network with higher-order complexes preserve a product-form Poisson distribution for all time if initialized as such?
• RQ2 Can the time-dependent distribution of a reaction network remain a product of Poissons even when the network includes non-linear (higher-order) complexes?
• RQ3 What is the necessary and sufficient condition on the network structure and kinetics that ensures the product-form is preserved dynamically, not just in steady state?
• RQ4 How does the time-varying mean vector c(t) relate to the deterministic mass-action dynamics in preserving the product-form?
• RQ5 Is there a generalization of Gardiner’s Poisson representation to networks with higher-order complexes that preserves the product-form structure?

Key findings

• The necessary and sufficient condition for the time-dependent distribution to remain a product of Poissons is the dynamical and restricted (DR) complex balance condition, which applies only to complexes of order two or higher.
• The time-varying mean vector c(t) that preserves the product-form is the solution to the deterministic mass-action ODEs with initial condition c(0) = c̃.
• The DR condition ensures that the coefficient matching in the Kolmogorov forward equation holds for all x, thereby preserving the product-form structure.
• The functions f_i(x) = g_{x,c}(z_i) associated with binary and higher-order complexes are linearly independent, which is key to proving the necessity of the DR condition.
• For networks with only zeroth- and first-order complexes, the DR condition reduces to the classical complex balance, recovering Gardiner’s known result.
• The proof relies on polynomial degree analysis to show that the vanishing of the coefficient sum implies the DR condition must hold for all higher-order complexes.