Abstract
We present a heuristic derivation of Gaussian approximations for stochastic chemical reaction systems with distributed delay. In particular, we derive the corresponding chemical Langevin equation. Due to the non-Markovian character of the underlying dynamics, these equations are integro-differential equations, and the noise in the Gaussian approximation is coloured. Following on from the chemical Langevin equation, a further reduction leads to the linear-noise approximation. We apply the formalism to a delay variant of the celebrated Brusselator model, and show how it can be used to characterise noise-driven quasi-cycles, as well as noise-triggered spiking. We find surprisingly intricate dependence of the typical frequency of quasi-cycles on the delay period.
Highlights
Chemical reaction systems are modelled by sets of differential equations, known as rate equations
In particular we describe how the Gaussian approximation is obtained for Markovian systems, and how the corresponding chemical Langevin equation (CLE) is derived
In summary we have presented an intuitive heuristic derivation for the Gaussian approximation of discrete-particle dynamics with distributed delay
Summary
Chemical reaction systems are modelled by sets of differential equations, known as rate equations. In a previous piece of work [22] we proposed the use of a method known from condensed matter physics [23] to describe the time evolution of discrete reaction systems with distributed delays This technique, the socalled Martin-Siggia-Rose-Janssen-de-Dominicis generating functional [24] takes a path-based view. It considers the space of all possible time courses of the system and formulates the probability for a given path to occur as a dynamic generating functional or path integral, effectively representing the Fourier transform of the probability measure in the space of all dynamic paths This is a powerful formulation applicable to a wide class of delay systems, and it can be used to derive effective Gaussian approximations, in particular an equivalent of the CLE for delay systems. In the Appendix we briefly describe how the well-known modified next-reaction method is modified to accommodate distributed delays, and we provide further supplementary details of our analytical calculations
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