Asymptotic analysis of multiscale approximations to reaction networks

Abstract

A reaction network is a chemical system involving multiple reactions and chemical species. Stochastic models of such networks treat the system as a continuous time Markov chain on the number of molecules of each species with reactions as possible transitions of the chain. In many cases of biological interest some of the chemical species in the network are present in much greater abundance than others and reaction rate constants can vary over several orders of magnitude. We consider approaches to approximation of such models that take the multiscale nature of the system into account. Our primary example is a model of a cell's viral infection for which we apply a combination of averaging and law of large number arguments to show that the ``slow'' component of the model can be approximated by a deterministic equation and to characterize the asymptotic distribution of the ``fast'' components. The main goal is to illustrate techniques that can be used to reduce the dimensionality of much more complex models.Comment: Published at http://dx.doi.org/10.1214/105051606000000420 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Continuous Time Markov Chain Models for Chemical Reaction Networks

“…These large numbers may still differ by several orders of magnitude, so normalizing all "large" quantities in the same way may still be inappropriate. Consequently, methods are developed in [4], [26], and [27] for deriving simplified models in which different species numbers are normalized in different ways appropriate to their numbers in the system.…”

“…See Section 3 of [4] and Section 6.3 of [26] for examples. It is possible to obtain diffusion processes as limits, but these are not typical for reaction networks.…”

Blending randomness in closed queueing network models

“…Also, let P = [p src he I M ×M ] e,h=1,...,E that is a row-stochastic matrix of order E 2 M 2 with zero diagonal. 4 . Expression (31) can now be rewritten as…”

“…In our model, instead, the transitions due to the random environment do not scale with increasing population levels, but are fixed. Although approaches for multi-scale stochastic processes are available [4], they cannot be applied to our model because they require a clear separation between fast and slow processes. In general, however our model of random environment is quite general and allows transitions between stages to be in the same time scale as the network's service rates.…”

Incorporating postleap checks in tau-leaping

“…4,5,6,7 Note that there are M + 1 distinct time frames in (2). The first time frame is the actual, or absolute time, t. However, each Poisson process Y k brings its own "internal" time frame.…”