Error bounds for the linear noise approximation to stationary distributions of chemical reaction networks

Abstract: Species interacting according to chemical reactions are often modeled by a continuous time Markov chain that describes the evolution of counts of the species over time. Such Markov chains typically have a large or infinite number of states and are thus computationally difficult to analyze. Therefore, approximations exploiting the fact that the volume and molecular counts are both large are often used. The most common such approximations are the reaction rate equations (RREs), which are a deterministic model, and the linear noise approximation (LNA), which is a diffusion approximation to fluctuations about the solution of the RREs. Limit theorem results, due to Kurtz (1971), establish the validity of the RREs and of the LNA for finite times. However, such results do not justify approximating the stationary distribution of a chemical reaction network using the RREs or LNA. The validity of these approximations for the stationary distribution has only been investigated for special cases, such as when the Markov chain’s state space is bounded in concentration, or when the chemical reaction network has a special structure. Here, we use Stein’s method to derive bounds on the approximation error for the LNA applied to the stationary distribution of a chemical reaction network. Specifically, we give a non-asymptotic bound on the 1-Wasserstein distance between an appropriately scaled Markov chain and its LNA, under certain technical conditions, that decays to zero with increasing system size. Our results do not require that the Markov chain’s state space be bounded, nor do they require that the chemical reaction network have a special structure. We further show how global stability properties of an equilibrium point of the RREs are sufficient to obtain such error bounds. Our results can be used to check when the LNA is a suitable approximation of the stationary distribution of a chemical reaction network without having to perform computationally costly simulations.

algebraic geometrydynamical systemsprobability

Audience: researchers in the topic

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Seminar on the Mathematics of Reaction Networks

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This seminar series focuses on progress in mathematical theory for the study of reaction networks, mainly in biology and chemistry. The scope is broad and accommodates works arising from dynamical systems, stochastics, algebra, topology and beyond.

We aim at providing a common forum for sharing knowledge and encouraging discussion across subfields. In particular we aim at facilitating interactions between junior and established researchers. These considerations will be represented in the choice of invited speakers and we will strive to create an excellent, exciting and diverse schedule.

The seminar runs twice a month, typically on the 2nd and 4th Thursday of the month, at 17:00 Brussels time (observe that this webpage shows the schedule in your current time zone). Each session consists of two 25-minute talks followed by 5-minute questions. After the two talks, longer discussions will take place for those interested. To this end, we will use breakout rooms. For this to work well, you need to have the latest version of Zoom installed (version 5.3.0 or higher), and use the desktop client or mobile app (not supported on ChromeOS).

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