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, 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 [67]. 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, through the introduction of artificial habitats for the predatores or through the introduction of biological control agents in deterministic [13] or stochastic fashion. This was the main topic of Nicolas Bajeux's PhD thesis [11].

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 study of the durability of plant resistance to root-knot nematodes [28], [41].

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 [57] and implemented on various models [39]. 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. This work was part of Stefano Casagranda's PhD thesis [12] 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 ABC-like method, less computationally expensive than standard Bayesian fitting procedures such as ABC [6]. 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 several parameter sets compatible with the data and reflecting the variability among individuals. So we based our fitting criterion on viral indicators rather than the whole viremia dynamics [44]. This work was part of Natacha Go's post-doctorate, supported by the MIHMES project, in collaboration with the Roslin Institute, Edinburgh, UK. It benefited from the resources and support of NEF computation cluster.

Parameter identification in compartmental systems. In collaboration with F. Dayan (Exactcure), we work on practical problems of identifiability of parameters in linear pharmacokinetic models. This was the subject of the internship of Laurent Dragoni.

Metabolic and genomic models

Participants : Jean-Luc Gouzé, Madalena Chaves, Olivier Bernard, Valentina Baldazzi, Stefano Casagranda, Francis Mairet, Ivan Egorov, Sofia Almeida, Claudia Lopez Zazueta, Lucie Chambon, Luis Gomes Pereira, Eleni Firippi, Ignacio Lopez Munoz.

Hybrid models analysis

Applying differential dynamic logic to biological networks. In [26] we have explored the framework of differential dynamic logic for the analysis of hybrid systems and, in particular, piecewise linear models of biological networks (collaboration with D. Figueiredo and M.A. Martins from the University of Aveiro, Portugal).

Attractor computation using interconnected Boolean networks. Following the work in [10] and  [58], 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 [9]. We prove that this orbit is unique under some conditions on the parameters.

Continuous models analysis

Reduced models for the mammalian cell cycle and clock. In the context of project ANR ICycle, we have focused on identifying and analysing the main mechanisms underlying the cell cycle and the circadian clock in mammalian cells. A reduced two-dimensional model of the cell cycle is described [38]; 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 is in collaboration with F. Delaunay and part of the PhD thesis of Sofia Almeida.

Interconnection of reduced models of the mammalian cell cycle and clock. Also in the context of project ANR ICycle, we have studied several possibilities for the interconnection between these two mammalian oscillators, using the reduced model already described in [38] and two different possible oscillatory circuits of low dimension. This work is part of the Master's thesis of Eleni Firippi.

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 [45]. 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 [17]. We also study other models of the global cellular machinery, and growth models ([22]). This is part of the PhD thesis of Stefano Casagranda, and done in collaboration with Inria IBIS project-team, in particular with D. Ropers.

Analysis and reduction of a model of sugar metabolism in peach fruit. Predicting genotype-to-phenotype relationships under contrasting environments is a big challenge for plant biology and breeding. A model of sugar metabolism in peach fruit has been recently developed and applied to 10 peach varieties [25]. The aim of this ongoing work is to reduce model's size and complexity to allow for calibration on a whole progeny of 106 genotypes and for further application to virtual breeding (collaboration with B. Quilot-Turion and Mohamed Memmah (INRA Avignon) and part of the PhD thesis of Hussein Kanso).

Analysis of an integrated cell division-endoreduplication and expansion model. The development of a new organ depends on cell-cyle progression and cell expansion, but the interaction and coordination between these processes is still unclear. An integrated model of fruit development has been developed and used to investigate the regulation of cell expansion capabilities. To this aim, different control schemes are tested by means of specific model variants and simulation results compared to observed data in tomato [14].

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) [5]. We developed different versions of the problem [43], [36], and consider a new problem where the aim is to optimize the production of a product (ANR projects Reset and Maximic).

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 [32].

Large scale metabolic modeling

Metabolic modeling 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 have proposed a new approach called DRUM where this hypothesis is relaxed by splitting the metabolic network into subnetworks and assuming that some compounds can accumulate between the subnetworks [2], [49]. This approach was successfully applied to several cases where the strong variations in light or nutrient resources induce a strong accumulation in the microalgal cells which could not be represented by the state of the art approaches [48]. More recently we have expended this approach to the modeling of diauxic growth for heterotrophic or mixotrophic microalgae [15].

Slow-Fast analysis of metabolic models

Metabolic modeling 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 [2], and show that Drum is a reasonable approximation, provided that growth rate stays low. This is part of the PhD thesis of Claudia Lopez Zazueta.