(254f) Stochastic Analysis of Chemical Reaction Networks Using Moment Closure Techniques

Random noise has long been known to contribute significantly to the dynamics of chemical reaction networks in systems that contain species with low copy numbers (e.g. DNA in a bacterium). These systems are common in biology and are typically referred to as being far from the thermodynamic limit. The fact that random noise or fluctuations are vital to understanding the behavior of such systems necessitates the use of stochastic simulation algorithms. While it is getting progressively easier to run the large kinetic Monte Carlo (kMC) simulations necessary for the simulation of biological systems, many issues remain. When compared to deterministic models, for example, it is not easy to obtain steady-state results or perform prior analysis for stochastic models using kMC simulations.

Decades ago it was recognized that while individual kMC trajectories are non-deterministic, the underlying distribution and its statistics (or moments, the mean, variance, etc.) are indeed deterministic. The equations that govern the dynamics of these moments are rather simple to obtain as well. The major issue is that for any chemical network with 2^nd order reactions the set of equations is open. The mean dynamics depend on the variance, the variance on the skewness, etc. In such cases a moment-closure technique is needed: a way to relate higher-order moments to lower-order moments so that the set of equations can be solved. Several moment-closure schemes have been developed over the years, both analytical (cumulant closure) and numerical (our Zero-Information Closure), but stochastic analysis using moment-closure, for the most part, has not been attempted.

Using several examples, most prominently the Brusselator oscillatory system, we show that moment-closure techniques may afford the ability to use deterministic analytical tools for analysis of stochastic simulations. In particular, steady-state distributions are obtained quickly, the resulting linearized Jacobian is determined, and an eigenproblem is solved to determine both eigenvalues and moment vectors. While the accuracy of such analysis is under investigation, the possibility of using moment-closure techniques to quickly identify interesting steady-state and dynamic behavior in stochastic simulation is tantalizing.

Many moment-closure techniques have been developed and explored in recent years, but relatively little work has been put into exploring the analytical consequences of such work. By changing the focus of stochastic simulation from the non-deterministic trajectories of kMC algorithms to the deterministic statistics of the ensemble distribution a whole panel of new analytical techniques becomes available. For small-scale networks in particular such analysis may be particularly useful in determining when interesting behavior occurs or model reduction is possible.