Stochastic Models of Resource Allocation in Chemical Reaction Networks
This paper analyses of a stochastic model of a chemical reaction network with three types of chemical species ${\cal R}$, ${\cal M}$ and ${\cal U}$ that interact to transform a flow of external resources, the chemical species ${\cal Q}$, to produce a product, the chemical species ${\cal P}_r$. A regulation mechanism involving the sequestration of the chemical species ${\cal R}$ when the flow of resources is too low is investigated. The original motivation of the study is of analyzing the qualitative properties of a key regulation mechanism of gene expression in biological cells, the {\em stringent response}. A scaling analysis of a Markov process in $\N^5$ representing the state of the chemical reaction network is achieved. It is shown that, depending on the parameters of the model, there are, quite surprisingly, three possible asymptotic regimes. To each of them corresponds a stochastic averaging principle with a fast process expressed in terms of a network of $M/M/\infty$ queues. One of these regimes, the optimal sequestration regime, does not seem to have been identified up to now. Under this regime, the input flow of resources is low but the state of the network is still acceptable in terms of unused macro-molecules, showing the remarkable efficiency of this regulation mechanism. The technical proofs of the main convergence results rely on a combination of coupling arguments, technical estimates of the solutions of SDEs, of sample paths of fast processes in particular, and the stability properties of some dynamical systems in $\R^2$.