Hybrid mathematical formalism for analysis of biological regulatory networks

International audienceBiological regulatory networks often give rise to new mathematical problems for which classical methods do not apply. Challenging problems include model assembly or model reduction, network inference, parameter estimation, and control of a system towards a desired state. More recently, questions related to the coupling of basic biochemical components, their regulation and control have arised in the context of ``synthetic biology'', a discipline which aims at designing and constructing cellular networks from individual components. The difficulties in the analysis and control of biological models are typically related to insufficient knowledge of the complex network of interactions and the limitations in experimental techniques for the measurement and control of the biochemical species involved.Hybrid modeling frameworks, such as piecewise affine systems or an appropriate combination of discrete and continuous techniques, can be very helpful to deal with these problems, as they facilitate the development of intuitive algorithms, computationally amenable, but still mathematically rigorous, methods of analysis. For example, piecewise affine models generate a partition of the state space into a finite number of regular regions, and the dynamics of the system can then be represented by a discrete ``state transition graph''. This is an object that provides a useful link between detailed quantitative models and the qualitative dynamics in general discrete models. To explore the link between continuous and discrete models, we will discuss the computation of probabilities of transition between two discrete states (or two regular regions) in the graph, in terms of the parameters of the continuous dynamics. This can be applied for parameter estimation and analysis of the possible dynamical behaviors of the system. We will also discuss the construction of qualitative control strategies, by restricting the inputs to take only a finite number of vales and to be constant in each regular region. These ideas will be applied to the qualitative control of a bistable switch and to models of cellular growth of bacteria E.coli, to illustrate how a combination of different mathematical formalisms leads to gaining quantitative knowledge and predictive power for biological systems