Section: New Results

Mathematical methods and methodological approach to biology

Mathematical analysis of biological models

Mathematical study of semi-discrete models

Participants : Frédéric Grognard, Ludovic Mailleret, Pierre Bernhard, Elsa Rousseau, Nicolas Bajeux, Bapan Ghosh.

Semi-discrete models have shown their relevance in the modeling of biological phenomena whose nature presents abrupt changes over the course of their evolution [81]. We used such models and analyzed their properties in several practical situations that are developed in Section 7.2.3, some of them requiring such a modeling to describe external perturbations of natural systems, and others to take seasonality into account. External perturbations of interacting populations occur when some individuals are introduced or removed from a natural system, which occurs frequently in pest control applications, either through the direct removal of pests, or through the introduction of biological control agents in deterministic [15] or stochastic [43], [33] fashion. Seasonality is an important property of most agricultural systems in temperate environments since the year is divided into a cropping season and a `winter' season, where the crop is absent, as in the dynamics of plant pathogens [24].

Model reduction and sensitivity analysis

Participants : Suzanne Touzeau, Jean-Luc Gouzé, Stefano Casagranda, Valentina Baldazzi.

Analysis and reduction of biochemical models. Dynamic models representing complex biological systems with numerous interactions can reach high dimensions and include complex nonlinearities. A model reduction method based on process weighing and pruning was developed and implemented on various models [67]. A global sensitivity analysis was performed to check the method robustness against parameter uncertainty and variability. A more general method robust to initial conditions has been elaborated [31]. This work is part of Stefano Casagranda's ongoing PhD thesis and is also a collaboration with Bayer (Sophia-Antipolis).

Estimation and control

Participants : Suzanne Touzeau, Natacha Go, Jean-Luc Gouzé.

Parameter identification in complex systems. In complex biological systems, especially when data are scarce, identifying the model parameters is a challenge and raises identifiability issues. So we developed a specific procedure based on sensitivity analysis, to select the parameters to be estimated, to define their ranges and to set the values of the remaining parameters [72]. We used this method to fit a within-host immunological model to a large data set of individual viremia profiles. Our aim was not to reproduce individual profiles, but to identify parameter sets compatible with the data. So we based our fitting criterion on viral indicators rather than the whole viremia dynamics and we defined realistic data-based ranges for these indicators. We used a genetic algorithm for the minimisation. This ongoing work is part of Natacha Go's post-doctorate, supported by the MIHMES project, in collaboration with the Roslin Institute, Edinburgh, UK. It benefits from the resources and support of NEF computation cluster.

Parameter identification in compartmental systems. In collaboration with F. Dayan (R&D Manager, Dassault Systèmes), we work on practical problems of identifiability of parameters in linear pharmacokinetic models.

Metabolic and genomic models

Participants : Jean-Luc Gouzé, Madalena Chaves, Olivier Bernard, Valentina Baldazzi, Stefano Casagranda, Francis Mairet, Sofia Almeida, Claudia Lopez Zazueta, Lucie Chambon, Ivan Egorov.

Hybrid models analysis

Attractor computation using interconnected Boolean networks Following the work in  [94] and  [68], we have generalized the method for computation of the asymptotic graph. In addition, we have extended this methodology for the case of Boolean networks with synchronous updates (collaboration with D. Figueiredo and M.A. Martins from the University of Aveiro, Portugal).

Periodic orbits in non monotonic negative feedback circuits We study the occurrence of periodic solutions in an $n$-dimensional class of negative feedback systems defined by smooth vector fields with a window of not necessarily monotonic activity. By circumscribing the smooth system by two piecewise linear ones, we show there exists an invariant toroidal region which contains a periodic orbit of the original smooth system [29]. This orbit is unique under some conditions on the parameters.

Piecewise linear representation of genetic regulatory networks The main goal was to develop a methodology for constructing piecewise linear and discrete models from a continuous model: given an initial partition of the state space, or grid, a piecewise constant vector field and diagram of transitions were computed based on the original ODE in the grid (M2 thesis of C. Kozia).

Continuous models analysis

A reduced model for the mammalian cell cycle This work focuses on identifying and analysing the main mechanisms underlying the cell cycle. A reduced two-dimensional model was proposed and calibrated against experimental data on cyclin B. As a validation, the model faithfully predicts the period of the cell cycle in response to an external growth factor input (experimental data on the periods is from F. Delaunay's lab). This work in collaboration with F. Delaunay (and part of the PhD thesis of Sofia Almeida) has been submitted to a journal.

Modeling the apoptotic signaling pathway The goal is to study the origins of cell-to-cell variability in response to anticancer drugs and provide a link between complex cell signatures and cell response phenotype. To do this, we have been analysing models of the apoptosis pathway to compare the effects of different sources of variability at the transcriptional, translational and receptor levels (collaboration with J. Roux, for the PhD thesis of Luis Pereira).

Transcription and translation models in bacteria. We study detailed models of transcription and translation for genes in a bacterium, in particular the model of gene expression of RNA polymerase. We also study other models of the global cellular machinery. This is part of the PhD thesis of Stefano Casagranda, and done in collaboration with Inria IBIS project-team, in particular with D. Ropers.

Reduction of metabolic networks. We develop a dynamical reduction reduction for metabolic networks through Elementary Flux Modes and Quasi Steady State Approximation. The aim is, in the spirit of [1], to obtain a system of lower dimensions, with some accumulative variables. This is part of the PhD thesis of Claudia Lopez Zazueta.

Estimation and control

Optimal allocation of resources in a bacterium. We study by techniques of optimal control the optimal allocation between metabolism and gene expression during growth of bacteria, in collaboration with Inria IBIS project-team. We showed that a good suboptimal control solution could be implemented in the cell by ppGpp (a small molecule involved in the regulation of ribosomes) [23]. We developed different versions of the problem, and consider a new problem where the aim is to optimize the production of a product (ANR project Reset).

Control of a model of synthesis of a virulence factor. In collaboration with J.-A. Sepulchre (INLN Nice), we model the production of a virulence factor by a bacterium in a continuous stirred tank reactor. The production of this enzyme is genetically regulated, and degrades a polymeric external substrate into monomers. A nonlinear control is built [74].

Slow-Fast analysis of metabolic models

Metabolic modelling generally assumes balanced growth, i.e. that there is no accumulation of intermediate compound, and that the metabolism is rapidly at quasi steady state. We go beyond this hypothesis by considering that some metabolic reactions are slow, while other are fast. Then we analyse the differential system using Tikhonov's Theorem. We compare the results obtained using the Drum approach [16], and show that Drum is a reasonable approximation, provided that growth rate stays low.