Paper Title: Moment closure approximations for studying stochasticity
in biological systems

Paper Type:  Poster

Authors:
Abhyudai Singh abhi@engineering.ucsb.edu
Joao Hespanha hespanha@ece.ucsb.edu

Abstract:

The time evolution of a spatially homogeneous mixture of chemically
reacting molecules is often modeled using a stochastic formulation,
and is motivated by complex reactions inside living cells, where small
populations of key reactants can set the stage for significant
stochastic effects.

In the stochastic formulation, the time evolution of the system is
described by a single equation for a grand probability function, where
time and species populations appear as independent variables, called
the Master equation. However, this equation can only be solved for
relatively few, highly idealized cases and generally Monte Carlo
simulation techniques are used which are also known as the Stochastic
Simulation Algorithm (SSA). Since one is often interested in only the
first and second order moments for the number of molecules of the
different species involved, much effort can be saved by applying
approximate methods to produce these low-order moments, without
actually having to solve for the probability density function. In this
paper, an approximate method for estimating lower-order moments is
introduced using moment closure techniques.

We show that the evolution of the populations of several species
involved in a set of chemical reactions can be modeled by a Stochastic
Hybrid System (SHS). Then, the time evolution of the moments of the
population is obtained using results from the SHS literature. It has
been shown that for such SHSs, if one creates an infinite vector
containing all the statistical moments of the populations, the
dynamics of this vector are governed by an infinite-dimensional linear
ordinary differential equation (ODE). As this infinite-dimensional ODE
cannot be solved analytically we approximate it by a
finite-dimensional nonlinear ODE, the state of which typically
contains the lower-order moments of interest. This technique is
commonly referred to as moment closure. Using the recently introduced
technique of matching time derivatives between the finite and the
infinite-dimensional ODE at some initial time, explicit formulas to
construct these finite dimensional ODE approximations are
obtained. The striking feature of these formulas are that are
independent of the reaction parameters (reaction rates and
stoichiometry).To illustrate the applicability of our results, various
examples of chemical reacting system which have been inspired from
gene regulatory networks and other biological examples are presented.