Necessary Conditions for Multistationarity and Stable Periodicity

Snoussi, El Houssine

doi:10.1142/s0218339098000042

Cited by 213 publications

(177 citation statements)

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“…Since then, these conjectures have been proven in different frameworks [20,21,22,23,24,25,26,27,28,29]. As for intersecting circuits, we will argue here that beyond the impact that circuits have on the dynamical behaviour of a network, the interactions of circuits via their intersections also account significantly for certain dynamical properties of networks.…”

Section: Circuits and Intersecting Circuitsmentioning

confidence: 91%

“…A huge mathematical effort has been done to identify the sources of the attractor multiplicity, known to be closely related to the number of positive circuits underlying regulatory networks [4,15,22,23,57,75,76,77,78] as well as to understand the causes of attractor uniqueness. Both these problems are of high interest if we want to explain the number of differentiated functions (around 300 for the human being [57]) as well as the presence of some unique functions that are devoted to one tissue.…”

Section: Appendix F Biological Importance Of Attractorsmentioning

confidence: 99%

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“Immunetworks”, intersecting circuits and dynamics

Demongeot

Elena

Noual

et al. 2011

Journal of Theoretical Biology

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This paper proposes a study of biological regulatory networks based on a multi-level strategy. Given a network, the first structural level of this strategy consists in analysing the architecture of the network interactions in order to describe it. The second dynamical level consists in relating the patterns found in the architecture to the possible dynamical behaviours of the network. It is known that circuits are the patterns that play the most important part in the dynamics of a network in the sense that they are responsible for the diversity of its asymptotic behaviours. Here, we pursue further this idea and argue that beyond the influence of underlying circuits, intersections of circuits also impact significantly on the dynamics of a network and thus need to be payed special attention to. For some genetic regulation networks involved in the control of the immune system ("immunetworks"), we show that the small number of attractors can be explained by the presence, in the underlying structures of these networks, of intersecting circuits that "inter-lock".

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Section: Circuits and Intersecting Circuitsmentioning

confidence: 91%

Section: Appendix F Biological Importance Of Attractorsmentioning

confidence: 99%

“Immunetworks”, intersecting circuits and dynamics

Demongeot

Elena

Noual

et al. 2011

Journal of Theoretical Biology

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“…for oscillations, e.g. for homeostasis) in different formalisms, and in particular for ODE systems in [43,39,40,41,42] and recently in [38] for the ODE semantics of non-linear reaction systems.…”

Section: Influence Graph Associated To a Well-formed Reaction Systemmentioning

confidence: 99%

“…This result generalizes a previous result in [36,37] to reactions with inhibitors. It shows that the influence graph of a well-formed reaction system with inhibitors is essentially independent of the kinetics, can be computed in linear time in the number of reactions when the number of species per reaction is bounded, and can thus advantageously be used to perform multi-stationarity analyses by circuit analysis à la Thomas [38,39,40,41,42,43,5,4,3].…”

Section: Introductionmentioning

confidence: 99%

Inferring reaction systems from ordinary differential equations

Fages

Gay

Soliman

2015

Theoretical Computer Science

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In Mathematical Biology, many dynamical models of biochemical reaction systems are presented with Ordinary Differential Equations (ODE). Once kinetic parameter values are fixed, this simple mathematical formalism completely defines the dynamical behavior of a system of biochemical reactions and provides powerful tools for deterministic simulations, parameter sensitivity analysis, bifurcation analysis, etc. However, without requiring any information on the reaction kinetics and parameter values, various qualitative analyses can be performed using the structure of the reactions, provided the reactants, products and modifiers of each reaction are precisely defined. In order to apply these structural methods to parametric ODE models, we study a mathematical condition for expressing the consistency between the structure and the kinetics of a reaction, without restricting to Mass Action law kinetics. This condition, satisfied in particular by standard kinetic laws, entails a remarkable property of independence of the influence graph from the kinetics of the reactions. We derive from this study a heuristic algorithm which, given a system of ODEs as input, computes a system of reactions with the same ODE semantics, by inferring well-formed reactions whenever possible. We show how this strategy is capable of automatically curating the writing of ODE models in SBML, and present some statistics obtained on the model repository biomodels.net. 1 .

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“…a negative circuit) has been shown to be a necessary condition for multistationarity (resp. oscillations) in different settings [4,5,6,7,8]. There are several tools providing different kinds of analyses for either reaction models or influence graphs.…”

Section: Introductionmentioning

confidence: 99%

Formal Cell Biology in Biocham

Fages

Soliman

Formal Methods for Computational Systems Biology

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Abstract. Biologists use diagrams to represent interactions between molecular species, and on the computer, diagrammatic notations are also employed in interactive maps. These diagrams are fundamentally of two types: reaction graphs and activation/inhibition graphs. In this tutorial, we study these graphs with formal methods originating from programming theory. We consider systems of biochemical reactions with kinetic expressions, as written in the Systems Biology Markup Language (SBML), and interpreted in the Biochemical Abstract Machine (Biocham) at different levels of abstraction, by either an asynchronous boolean transition system, a continuous time Markov chain, or a system of Ordinary Differential Equations over molecular concentrations. We show that under general conditions satisfied in practice, the activation/inhibition graph is independent of the precise kinetic expressions, and is computable in linear time in the number of reactions. Then we consider the formalization of the biological properties of systems, as observed in experiments, in temporal logics. We show that these logics are expressive enough to capture semi-qualitative semi-quantitative properties of the boolean and differential semantics of reaction models, and that model-checking techniques can be used to validate a model w.r.t. its temporal specification, complete it, and search for kinetic parameter values. We illustrate this modelling method with examples on the MAPK signalling cascade, and on Kohn's map of the mammalian cell cycle.

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Necessary Conditions for Multistationarity and Stable Periodicity

Cited by 213 publications

References 0 publications

“Immunetworks”, intersecting circuits and dynamics

“Immunetworks”, intersecting circuits and dynamics

Inferring reaction systems from ordinary differential equations

Formal Cell Biology in Biocham

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