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Closure Scheme for Chemical Master Equations - Is the Gibbs entropy maximum for stochastic reaction networks at steady state?

Presented by: 
Yiannis Kaznessis
Friday 8th April 2016 - 09:00 to 09:45
INI Seminar Room 1
Stochasticity is a defining feature of biochemical reaction networks, with molecular fluctuations influencing cell physiology. In principle, master probability equations completely govern the dynamic and steady state behavior of stochastic reaction networks. In practice, a solution had been elusive for decades, when there are second or higher order reactions. A large community of scientists has then reverted to merely sampling the probability distribution of biological networks with stochastic simulation algorithms. Consequently, master equations, for all their promise, have not inspired biological discovery.

We recently presented a closure scheme that solves chemical master equations of nonlinear reaction networks [1]. The zero-information closure (ZI-closure) scheme is founded on the observation that although higher order probability moments are not numerically negligible, they contain little information to reconstruct the master probability [2]. Higher order moments are then related to lower order ones by maximizing the entropy of the network. Using several examples, we show that moment-closure techniques may afford the quick and accurate calculation of steady-state distributions of arbitrary reaction networks.

With the ZI-closure scheme, the stability of the systems around steady states can be quantitatively assessed computing eigenvalues of the moment Jacobian [3]. This is analogous to Lyapunov’s stability analysis of deterministic dynamics and it paves the way for a stability theory and the design of controllers of stochastic reacting systems [4, 5].

In this seminar, we will present the ZI-closure scheme, the calculation of steady state probability distributions, and discuss the stability of stochastic systems.

1. Smadbeck P, Kaznessis YN. A closure scheme for chemical master equations. Proc Natl Acad Sci U S A. 2013 Aug 27;110(35):14261-5.

2. Smadbeck P, Kaznessis YN. Efficient moment matrix generation for arbitrary chemical networks, Chem Eng Sci, 20
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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons