Biham, Ofer

Title: Reaction networks with fluctuations: from inter-stellar chemistry to intra-cellular biology
Author: Ofer Biham
Affiliation: Racah Institute of Physics, Hebrew University, Jerusalem, Israel.
Abstract:
Complex reaction networks are common in physical, chemical and biological systems. In many cases, the reactive species are confined to a small volume, in which the copy number of each species is low. Relevant examples include biochemical networks in cells and reactions on dust grains in the interstellar medium. In these systems the reaction rates are often dominated by fluctuations and cannot be evaluated by rate equations. Stochastic methods such as the direct integration of the master equation or Monte Carlo simulations (e.g. using the Gillespie algorithm) are needed. However, the number of microscopic states (and the number of equations) increase exponentially with the complexity of the network. This makes the stochastic simulations infeasible for complex networks.

In this talk I will describe the effects of fluctuations on the reaction rates in interstellar chemistry networks and on bistability and oscillations in genetic regulatory networks in cells. I will also present two methods that provide a dramatic reduction in the number of equations, and enable efficient stochastic simulations of complex reaction networks. In the multiplane method [1], the reduction is achieved by breaking the network into a set of maximal fully connected sub-networks (cliques). Lower-dimensional master equations are constructed for the marginal probability distributions associated with the cliques, with suitable couplings between them. In the moment equations method [2], the reduction is achieved by writing a closed set of equations for the first and second moments, using a suitable closure condition. Applications of the two methods will be shown and their advantages and limitations in comparison with the Gillespie algorithm will be discussed.

[1] A. Lipshtat and O. Biham, Efficient simulations of gas-grain chemistry in interstellar clouds, Phys. Rev. Lett. 93, 170601 (2004).
[2] B. Barzel and O. Biham, Efficient stochastic simulations of complex reaction networks on surfaces, J. Chem. Phys 127, 144703 (2007).
[3] F. Le Petit, B. Barzel, O. Biham, E. Roueff and J. Le Bourlot, Incorporation of stochastic chemistry on dust grains in the Meudon PDR code using moment equations: I. Application to the formation of H2 and HD, Astron. Astrophys. 505, 1153 (2009).
[4] B. Barzel and O. Biham, Stochastic analysis of dimerization systems, Phys. Rev. E 80, 031117 (2009).