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).