Product-Form Poisson-Like Distributions and Complex Balanced Reaction Systems

Abstract

Stochastic reaction networks are dynamical models of biochemical reaction systems and form a particular class of continuous-time Markov chains on N n . Here we provide a fundamental characterisation that connects structural properties of a network to its dynamical features. Specifically, we define the notion of 'stochastically complex balanced systems' in terms of the network's stationary distribution and provide a characterisation of stochastically complex balanced systems, parallel to that established in the 70-80ies for deterministic reaction networks. Additionally, we establish that a network is stochastically complex balanced if and only if an associated deterministic network is complex balanced (in the deterministic sense), thereby proving a strong link between the theory of stochastic and deterministic networks. Further, we prove a stochastic version of the 'deficiency zero theorem' and show that any (not only complex balanced) deficiency zero reaction network has a product-form Poisson-like stationary distribution on all irreducible components. Finally, we provide sufficient conditions for when a product-form Poisson-like distribution on a single (or all) component(s) implies the network is complex balanced, and explore the possibility to characterise complex balanced systems in terms of product-form Poisson-like stationary distributions.

Time-dependent product-form Poisson distributions for reaction networks with higher order complexes

“…where x(t) and y(t) are the solutions to the associated deterministic model (8). We search for solutions that satisfy the DR condition by plugging (29) into the deterministic model (8)…”

“…where x(t) and y(t) are the solutions to the associated deterministic model (8), and we are assuming that y(t) > 0 for all t ≥ 0. Assuming the DR condition holds, the associated deterministic model is…”

“…Thus, suppose that there is a complex of the form z = kX for some k ≥ 2, and suppose that the solution to the forward equation (7) is given by (23) for some nontrivial process c(t). By Theorem 3.1, we may assume that the solution to the deterministic model (8) satisfies the DR condition of Definition 2.3 for the complex z. That is,…”

Dissipation in noisy chemical networks: The role of deficiency

“…So, is the condition δ = 0 necessary for a vanishing correlation EPR, for all possible values of rates? Very special systems with nonvashing deficiency which still have Poissonian steady states have been found 35 . An example is the chemical network X+Y 2X+Y, X 2X.…”

Product-Form Stationary Distributions for Deficiency Zero Networks with Non-mass Action Kinetics

“…For this class of models, and under the assumption of mass action kinetics, the fixed points of the deterministic models (Anderson 2011, 2008; Craciun 2015; Feinberg 1979, 1987; Gunawardena 2003) and the stationary distributions for the stochastic models, have been fully characterized (Anderson et al. 2010, 2015; Cappelletti and Wiuf 2016; Van Kampen 1976). In fact, it is the study of this class of networks that is largely responsible for the development of the field of chemical reaction network theory (Feinberg 1972, 1979; Gunawardena 2003; Horn 1972), a branch of applied mathematics in which the dynamical properties of the mathematical model are related to the structural properties of the interaction network.…”