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Section: New Results

Time-series and asymptotic dynamics

Participants : Anne Siegel, Jacques Nicolas, Jérémie Bourdon, Jean Coquet, Victorien Delannée, Vincent Picard, Nathalie Théret.

Identification of logical models for signaling pathways: towards a systems biology loop. Logical models of signaling pathways are a promising way of building effective in silico functional models of a cell. The automated learning of Boolean logic models describing signaling pathways can be achieved by training to phosphoproteomics data. This data is unavoidably subject to noise. As a result, the learning process leads to a family of feasible logical networks rather than a single model. This family is composed of logic models proposing different internal wirings for the system, implying that the logical predictions from this family may suffer a significant level of variability leading to uncertainty. In our work, combinatorial optimization methods based on recent logic programming paradigm allow to enumerate, and discriminate the family of logical models explaining data. Together, these approaches enable a robust understanding of the system response. The results are implemented in the caspo software [Jacques Nicolas, Anne Siegel] [22] , [23]

Boolean Network Identification from Multiplex Time Series Data. The ASP-based learning algorithm developed in the team to train logical models of signaling networks focuses on the comparison of two time-points and assumes that the system has reached an early steady state. We have generalized such a learning procedure in order to discriminate Boolean networks according to their transient dynamics. To that goal, we exhibit a necessary condition that must be satisfied by a Boolean network dynamics to be consistent with a discretized time series trace. This approach was included in the ASP-based framework designed for the caspo software. We ended up with a global learning algorithm and compared it to learning approaches based on static data. [Anne Siegel] [31]

Representation of symbolic dynamical systems generated by a substitution. Iterated morphisms are combinatorial processes which are related to several classes of dynamical systems appearing in several fields of computer sciences and mathematics: numeration, ergodic theory, discrete geometry. They may be associated to fractal sets called "Rauzy fractals" whose topological properties are linked to the properties of the underlying dynamical system. We have introduced a generic algorithm framework to check such topological properties within a complete family of iterated morphism. This makes efficient the verification of conjectures on several families of substitutions related to multi-dimensional continued fraction algorithms. [Anne Siegel] [32] , [25] , [14]

Multivariate Normal Approximation for the Stochastic Simulation Algorithm: Limit Theorem and Applications. We present a central limit theorem for the Gillespie stochastic trajectories when the living system has reached a steady-state, that is when the internal bio-molecules concentrations are assumed to be at equilibrium. It appears that the stochastic behavior in steady-state is entirely characterized by the stoichiometry matrix of the system and a single vector of reaction probabilities. We propose several applications of this result such as deriving multivariate confidence regions for the time course of the system and a constraints-based approach which extends the flux balance analysis framework to the stochastic case. [Jérémie Bourdon, Vincent Picard, Anne Siegel] [20] , [12]

A Logic for Checking the Probabilistic Steady-State Properties of Reaction Networks. Designing probabilistic reaction models and determining their stochastic kinetic parameters are major issues in systems biology. In order to assist in the construction of reaction network models, we introduce a logic that allows one to express asymptotic properties about the steady-state stochastic dynamics of a reaction network. Basically, the formulas can express properties on expectancies, variances and co-variances. We demonstrate that deciding the satisfiability of a formula is NP-hard. [Jérémie Bourdon, Vincent Picard, Anne Siegel] [28] , [12]