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

Abstract

In many applications, for example when computing statistics of fast subsystems in a multiscale setting, we wish to find the stationary distributions of systems of continuous-time Markov chains. Here we present a class of models that appears naturally in certain averaging approaches whose stationary distributions can be computed explicitly. In particular, we study continuous-time Markov chain models for biochemical interaction systems with non-mass action kinetics whose network satisfies a certain constraint. Analogous with previous related results, the distributions can be written in product form.

On balance relations for irreversible chemical reaction networks

“…One of the most notable results in chemical reaction network theory is the deficiency zero theorem [2], which gives general conditions under which the reaction rate equations have a nontrivial stable equilibrium. More recently, this result has been extended to the chemical master equation, where it has been shown that, under the same conditions, the stochastic model has a product form stationary density [11,12]. The networks to which these theorems apply are known as complex balanced [13], and can be thought of as a generalization of the class of detailed balance networks.…”

Mixture distributions in a stochastic gene expression model with delayed feedback

“…The underlying probability distributions are typically defined as solutions to a specific master equation (Paulsson, 2005;Veerman et al, 2018;Albert, 2019). Explicit solutions to the master equation, especially at steady state, can be found for models with few components Zhou and Liu, 2015) and/or with special structural properties (Kumar et al, 2015;Anderson and Cotter, 2016). Generally, however, explicit solutions are unavailable or intractable and one resorts to stochastic simulation or seeks a numerical solution to a finite truncation of the master equation (Munsky and Khammash, 2006;Borri et al, 2016;Gupta et al, 2017).…”

Buffering gene expression noise by microRNA based feedforward regulation

“…Unfortunately, exact characterisations of the copy-number distributions in a reaction system are often unavailable. Systems operating at a complex-balanced equilibrium are a notable exception in that they admit tractable product-form distributions [21][22][23][24]. Steady-state distributions have also been characterised in terms of generating functions in a growing collection of simple models that are not complex-balanced [25][26][27][28][29][30].…”

“…Section 3 contains the derivation of the LNA results and cross-validates them with large-scale kinetic Monte Carlo simulations. Section 4 focuses on a special case in which the model admits a product-form distribution predicted by the Chemical Reaction Network Theory [23,24]. Section 5 introduces a QSS approximation and shows that it can outperform the LNA in a specific parametric regime.…”