ASYMPTOTIC ANALYSIS OF MULTISCALE APPROXIMATIONS TO REACTION NETWORKS

By Karen Ball, Thomas G. Kurtz, Lea Popovic, Greg A Rempala. 2005

243 citations recorded by the source corpus.

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.

Subfields: Biomedical Engineering, Computational Theory and Mathematics, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/13ddfed21e735a204f6ee223fd2f0e6c3020faee

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Citations in the corpus

Diffusion approximation for fully coupled stochastic differential equations (2020, background): “ applications in many real world dynamical systems including planetray motion, climate models (see e.g. [30, 36]), geophysical ﬂuid ﬂows (see e.g. [19]), intracellular biochemical reactions (see e.g. [5]), etc. We refer the interested readers to the books [31, 45] for a more comprehensive overview. ”
A modified next reaction method for simulating chemical systems with time dependent propensities and delays. (2007, methodology): “needing one random number per step. 3 Representation using Poisson processes We now explicitly represent biochemical systems using Poisson processes with integrated propensity functions (see [13],[7],[2]). Using such a representation allows us to consider the system as a whole as opposed to solely c”
Separation of time-scales and model reduction for stochastic reaction networks. (2010, methodology): “The present paper gives a systematic development of many of the ideas introduced in Ball et al. (2006). First, recognizing that the variation in time-scales is due both to variation in species number and to variation in rate constants, we normalize species numbers and rate constants by powers of a f”
Convergence of stochastic gene networks to hybrid piecewise deterministic processes (2011, background): “uousvariables(Haseltine and Rawlings (2002); Alfonsi et al. (2005); Alfonsi et al. (2004)). Similar approaches reducing fast reactions can be justiﬁed by diﬀusion approximations for Markov processes (Ball et al. (2006)). A diﬀerent hybrid approach is to distinguish between molecular species accordin”
Continuous Time Markov Chain Models for Chemical Reaction Networks (2011, methodology): “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.”