Kirkilionis, Markus and Sbano, Luca (2010) An averaging principle for combined interaction graphs - connectivity and appications to genetic switches. Advances in Complex Systems, Vol.13 (No.3 Sp. Iss. SI). pp. 293-326. doi:10.1142/S0219525910002669 ISSN 0219-5259.

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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 analyze a class of simplified hybrid systems. Once the dynamics is defined the mutual interaction of both ODEs and Markov chains can be analyzed 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 a 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.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics Q Science
Divisions:	Faculty of Science, Engineering and Medicine > Science > Mathematics
Journal or Publication Title:	Advances in Complex Systems
Publisher:	World Scientific Publishing Company PTE Ltd
ISSN:	0219-5259
Official Date:	June 2010
Dates:	Date Event June 2010 Published
Volume:	Vol.13
Number:	No.3 Sp. Iss. SI
Number of Pages:	34
Page Range:	pp. 293-326
DOI:	10.1142/S0219525910002669
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
Funder:	European Commission
Grant number:	12990

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