An Averaging Principle for Combined Interaction Graphs — Connectivity and Applications to Genetic Switches

Abstract

Time-continuous dynamical systems defined on graphs are often used to model complex systems with many interacting components in a non-spatial context. In the reverse sense attaching meaningful dynamics to given 'interaction diagrams' is a central bottleneck problem in many application areas, especially in cell biology where various such diagrams with different conventions describing molecular regulation are presently in use. In most situations these diagrams can only be interpreted by the use of both discrete and continuous variables during the modelling process, corresponding to both deterministic and stochastic hybrid dynamics. The conventions in genetics are well-known, and therefore we use this field for illustration purposes. In [25] and [26] the authors showed that with the help of a multi-scale analysis stochastic systems with both continuous variables and finite state spaces can be approximated by dynamical systems whose leading order time evolution is given by a combination of ordinary differential equations (ODEs) and Markov chains. The leading order term in these dynamical systems is called average dynamics and turns out to be an adequate concept to analyse a class of simplified hybrid systems. Once the dynamics is defined the mutual interaction of both ODEs and Markov chains can be analysed through the (reverse) introduction of the so called Interaction Graph, a concept originally invented for time-continuous dynamical systems, see [5]. Here we transfer this graph concept to the average dynamics, which itself is introduced as an heuristic tool to construct models of reaction or contact networks. The graphical concepts introduced form the basis for any subsequent study of the qualitative properties of hybrid models in terms of connectivity and (feedback) loop formation. arXiv:0803.0635v3 [q-bio.MN]

Multiscale Analysis in Many-Body Systems

“…In this case, upon the condition that the dynamics of the Markov chain is faster than the one associated with the vector fields χ 's, the vector field (25) is the leading order term of the dynamics on the directed graph. To explore the features of the average dynamics the reader may consult [8].…”

“…This paper is intended as a general introduction to an approach aimed at studying systems (like a cell) whose processes take place at very different spatial, time and size scales (see [19][20][21]) involving different type of molecular "players". Some relevant applications of this approach can be found in [8][9][10]. The author would like to thank one of the referee for drawing his attention to [2,4,13] where the multiscale analysis is addressed from a more probabilistic point of view.…”

Multi-scale genetic dynamic modelling I : an algorithm to compute generators

“…Details on the derivation and expansion of the Master equation are described in the books by van Kampen (1992) and Gardiner (1990). The theory for molecular systems with finite states and infinite copy numbers has been studied in detail by Kirkilionis and Sbano in a series of articles, (Kirkilionis 2010;Kirkilionis and Sbano 2010;Sbano and Kirkilionis 2008a, b, c).…”

“…The mathematical and numerical analysis of these models needs to explore how the dynamics of the system depends on this dimensional or functional molecular choices. We expect such changes to be small if the corresponding interaction graph (see Kirkilionis and Sbano (2010) where this concept has been introduced) has only a few more discrete Markov states. But more precise results are needed via mathematical perturbation analysis, (Kirkilionis and Sbano in preparation).…”

“…The first section reviews the general modelling framework for molecular systems with finite states and infinite copy numbers, which is discussed in more detail by Kirkilionis and Sbano (Kirkilionis 2010;Sbano and Kirkilionis 2008a, b, c). The focus lies here on how deterministic equations can be derived from an underlying discrete stochastic description of the system by a continuum limit and an additional time-scaling argument.…”