The dynamics of chemical reaction networks often takes place on widely differing time scales--from the order of nanoseconds to the order of several days. This is particularly true for gene regulatory networks, which are modeled by chemical kinetics. Multiple time scales in mathematical models often lead to serious computational difficulties, such as numerical stiffness in the case of differential equations or excessively redundant Monte Carlo simulations in the case of stochastic processes. We present a model reduction method for study of stochastic chemical kinetic systems that takes advantage of multiple time scales. The method applies to finite projections of the chemical master equation and allows for effective time scale separation of the system dynamics. We implement this method in a novel numerical algorithm that exploits the time scale separation to achieve model order reductions while enabling error checking and control. We illustrate the efficiency of our method in several examples motivated by recent developments in gene regulatory networks.

Abstract-Recent experiments have shown that a small number of fiber lasers can spontaneously form coherent states when suitably coupled. The observed synchrony persisted for a long time without any active control. In this paper, we develop a dynamical model for fiber laser arrays that is valid in the high gain regime. In the limiting case of a single laser analysis and simulations are presented that agree with physical expectations. Using simulations to examine array behavior we report results that are in qualitative agreement with laboratory observations.

We explore the transition to in-phase synchronization in globally coupled oscillator arrays, and compare results for van der Pol arrays with Josephson junction arrays. Our approach yields in each case an analytically tractable iterative map; the resulting stability formulas are simple because the expansion procedure identifies natural parameter groups. A third example, an array of Duffing-van der Pol oscillators, is found to be of the same fundamental type as the van der Pol arrays, but the Josephson arrays are fundamentally different owing to the absence of self-resonant interactions.

Synchronization of coupled fiber lasers has been reported in recent experiments [Bruesselbach, Opt. Lett. 30, 1339 (2005); Minden, Proc. SPIE 5335, 89 (2004)]. While these results may lead to dramatic advances in laser technology, the mechanism by which these lasers synchronize is not understood. We analyze a recently proposed [Rogers, IEEE J. Quantum Electron. 41, 767 (2005)] iterated map model of fiber laser arrays to explore this phenomenon. In particular, we look at synchronous solutions of the maps when the gain fields are constant. Determining the stability of these solutions is analytically tractable for a number of different coupling schemes. We find that in the most symmetric physical configurations the most symmetric solution is either unstable or stable over insufficient parameter range to be practical. In contrast, a lower symmetry configuration yields surprisingly robust coherence. This coherence persists beyond the pumping threshold for which the gain fields become time dependent.

Abstract-Considerable recent experimental evidence suggests that significant stochastic fluctuations are present in gene regulatory networks. The investigation of stochastic properties in genetic systems involves the formulation of a mathematical representation of molecular noise and devising efficient computational algorithms for computing the relevant statistics of the modeled processes. However, the complexity of gene regulatory networks poses serious computational difficulties and makes any quantitative prediction a difficult task. Monte Carlo based approaches are typically used in study of complex stochastic systems, but they often suffer from long computation times, slow convergence, and offer little analytic insight. The recently proposed Finite State Projection (FSP) approach provides an analytical alternative that avoids many of the shortcomings of Monte Carlo methods, but thus far it has only been demonstrated for a certain class of problems. In this paper we show that the applicability of the finite projection approach can be enhanced by taking advantage of tools from the fields of modern control theory and dynamical systems. In particular, we present an approach that utilizes singular perturbation theory in conjunction with the Finite State Projection approach to improve the computation time and facilitate model reduction. We demonstrate the effectiveness of the resulting slow manifold FSP algorithm on a simple example arising in the cellular heat shock response mechanism.