Adrien Richard

Modeling and Analysis of Gene Regulatory Networks

This chapter describes basic principles for modeling genetic regulatory networks, using three different classes of formalisms: discrete, hybrid, and continuous differential systems. A short review of the mathematical tools for each formalism is presented. Based on several simple examples, which are worked out in detail, this chapter illustrates the study and analysis of the networks’ dynamics, their temporal evolution and asymptotic behaviors.

Formal Methods for Modeling Biological Regulatory Networks

This chapter presents how the formal methods can be used to analyse biological regulatory networks, which are at the core of all biological phenomena as, for example, cell differentiation or temperature control. The dynamics of such a system, i.e. its semantics, is often described by an ordinary differential equation system, but has also been abstracted into a discrete formalism due to R. Thomas. This second description is well adapted to stateof-the-art measurement techniques in biology, which often provide qualitative and coarse-grained descriptions of biological regulatory networks. This formalism permits us to design a formal framework for analysing the dynamics of biological systems. The verification tools, as model checking, can then be used not only to verify if the modelling is coherent with known biological properties, but also to help biologists in the modelling process. Actually, for a given biological regulatory network, a large class of semantics can be automatically built and model checking allows the selection of the semantics, which are coherent with the biological requirement, i.e. the temporal specification. This modelling process is illustrated with the well studied genetic regulatory network controlling immunity in bacteriophage lambda.

Analysing formal models of genetic regulatory networks with delays

In this paper, we propose a refinement of the modelling of biological regulatory networks based on the discrete approach of Rene Thomas. We refine and automatise the use of delays of activation/inhibition in order to specify which variable is more quickly affected by a change of its regulators. The formalism of linear hybrid automata is well suited to allow such refinement. We then use HyTech for two purposes: to find automatically all paths from a specified initial state to another one; to synthesise constraints on the delay parameters in order to follow any specific path.

On circuit functionality in boolean networks.

It has been proved, for several classes of continuous and discrete dynamical systems, that the presence of a positive (resp. negative) circuit in the interaction graph of a system is a necessary condition for the presence of multiple stable states (resp. a cyclic attractor). A positive (resp. negative) circuit is said to be functional when it “generates” several stable states (resp. a cyclic attractor). However, there are no definite mathematical frameworks translating the underlying meaning of “generates.” Focusing on Boolean networks, we recall and propose some definitions concerning the notion of functionality along with associated mathematical results.

Static Analysis of Boolean Networks Based on Interaction Graphs: A Survey

Boolean networks are discrete dynamical systems extensively used to model biological regulatory networks. The dynamical analysis of these networks suffers from the combinatorial explosion of the state space, which grows exponentially with the number n of components. To face this problem, a classical approach consists in deducing from the interaction graph of the network, which only contains n vertices, some information on the dynamics of the network. In this paper, we present results in this topic, mainly by focusing on the influence of positive and negative feedbacks.

Boolean Models of Biosurfactants Production in Pseudomonas fluorescens

Cyclolipopeptides (CLPs) are biosurfactants produced by numerous Pseudomonas fluorescens strains. CLP production is known to be regulated at least by the GacA/GacS two-component pathway, but the full regulatory network is yet largely unknown. In the clinical strain MFN1032, CLP production is abolished by a mutation in the phospholipase C gene () and not restored by complementation. Their production is also subject to phenotypic variation. We used a modelling approach with Boolean networks, which takes into account all these observations concerning CLP production without any assumption on the topology of the considered network. Intensive computation yielded numerous models that satisfy these properties. All models minimizing the number of components point to a bistability in CLP production, which requires the presence of a yet unknown key self-inducible regulator. Furthermore, all suggest that a set of yet unexplained phenotypic variants might also be due to this epigenetic switch. The simplest of these Boolean networks was used to propose a biological regulatory network for CLP production. This modelling approach has allowed a possible regulation to be unravelled and an unusual behaviour of CLP production in P. fluorescens to be explained.

Fixed Points of Boolean Networks, Guessing Graphs, and Coding Theory

In this paper, we are interested in the number of fixed points of functions

On the Convergence of Boolean Automata Networks without Negative Cycles

f:A^n\to A^n

On the link between strongly connected iteration graphs and chaotic Boolean discrete-time dynamical systems

over a finite alphabet

Reduction and Fixed Points of Boolean Networks and Linear Network Coding Solvability

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