Section: New Results

Mathematical methods and methodological approach to biology

Mathematical analysis of biogical models

Mathematical study of ecological models

Participants : Frédéric Grognard, Ludovic Mailleret, Suzanne Touzeau, Clotilde Djuikem, Israël Tankam Chedjou.

Semi-discrete models. Semi-discrete models have shown their relevance in the modeling of biological phenomena whose nature presents abrupt changes over the course of their evolution [41]. We used such models and analyzed their properties in several practical situations, some of them requiring such a modeling to describe external perturbations of natural systems such as harvest, and others to take seasonality into account. We developed such models in the context of the analysis of the effect of stochasticity and Allee effects on the introduction of populations [14], seasonality in the dynamics of coffee leaf rust [59] and of banana and plantain burrowing nematodes [67], as well as for the protection of plant resistance against root-knot nematodes [66].

Models in plant epidemiology. We developed and analysed dynamical models describing plant-parasite interactions, in order to better understand, predict and control the evolution of damages in crops. We considered several pathosystems, further described in Section 7.2.3, describing and controlling the impact on plants of fungi [59], [39], viruses [36], nematodes [67], [66], and pests [60].

Estimation and control

Participants : Frédéric Grognard, Ludovic Mailleret, Suzanne Touzeau, Yves Fotso Fotso, Samuel Nilusmas, Israël Tankam Chedjou.

Parameter identification in complex systems. In complex biological systems, identifying model parameters is a challenge that raises identifiability issues. To fit a within-host immunological model to a large data set of individual viremia profiles, we developed an Approximate Bayesian Computation (ABC)-like method that yielded several parameter sets compatible with the data and reflecting the variability among individuals [25]. This work benefited from the resources and support of NEF computation cluster.

Optimal control and optimisation. We developed several approaches to control the evolution of crop pests. To reduce crop losses due to plant-parasitic nematodes, we optimised (i) rotation strategies between resistant and susceptible cultivars of horticultural crops [76], or (ii) fallow periods between plantain cropping seasons [67]. These optimisation problems were solved on a finite time horizon. They benefited from the resources and support of NEF computation cluster.

We also solved an optimal control problem to limit the damages due to coffee berry borers [60]. It consisted in designing the most cost-efficient application of a biopesticide over time. Using Pontryagin's maximum principle, we determined the existence and structure of the solution. The problem was solved numerically using BOCOP (https://www.bocop.org/).

Analysis of multistability and periodic behavior with hybrid models

Participants : Madalena Chaves, Eleni Firippi.

Probabilistic dynamics tool for hybrid models In a collaboration with D. Figueiredo and M.A. Martins from the University of Aveiro, Portugal (project PHC Pessoa), a tool was developed for simulating weighted reactive models [55]. These are essentially discrete models with dynamics described by state transition graphs: each transition has a given weight and the graph has the capacity to alter its accessibility relations.

M. Chaves and M.A. Martins jointly edited a book with selected papers from the Symposium on Molecular Logic and Computational Synthetic Biology [70], gathering work on different formalisms and applications of hybrid models.

Coupling and synchronization of piecewise linear systems This work studies the coupling of $N$ identical positive feedback loops described by piecewise linear differential equations. Under diffusive coupling, and for different conditions on the coupling parameters, the $N$ systems may synchronize or, alternatively, generate a set ofnew steady states that form a specific pattern [49]. An unexpected result is the existence of a special relationship between the number of components $N$ and the maximal concentration-to-activity threshold ratio ($V_1/\left({\gamma }_1{\theta }_1\right)$). This relationship implies that, for very specific parameter sets, the $N$ compartments cannot be guaranteed to synchronize.

Dynamics of complex feedback architectures

Participants : Madalena Chaves, Jean-Luc Gouzé.

To analyze the closed-loop dynamics of metabolic pathways under gene regulation, we propose a method to construct a state transition graph for a given regulatory architecture consisting of a pathway of arbitrary length, with any number of genetic regulators, and under any combination of positive and negative feedback loops [19]. Using this formalism, we analyze a “metabolator”-like mechanism (a pathway with two metabolites and three enzymes) and prove the existence of two co-existing oscillatory behaviors: damped oscillations towards a fixed point or sustained oscillations along a periodic orbit [20].

Metabolic and genomic models

Participants : Jean-Luc Gouzé, Olivier Bernard, Valentina Baldazzi, Lucie Chambon, Carlos Martinez Von Dossow, Agustin Yabo, Alex Dos Reis de Souza, Walid Djema, Sofya Maslovskaya.

Analysis and reduction of a model of sugar metabolism in peach fruit. Predicting genotype-to-phenotype relationships 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 [80]. A reduction pipeline combining several reduction strategies has been developed to reduce both model size and nonlinearity and allow for further application to virtual breeding (collaboration with B. Quilot-Turion and Mohamed Memmah (INRA Avignon) as part of the PhD thesis of Hussein Kanso) [64]. A paper is currently under revision for Mathematical Biosciences.

Analysis of an integrated cell division-endoreduplication and expansion model. The development of a new organ depends on cell-cycle progression and cell expansion, but the interaction and coordination between these processes is still unclear [27]. An integrated model of fruit development has been developed and used to test different interaction schemes, by comparing simulation results to the observed cell size distribution in tomato fruit [15].

Modeling cell growth and resource allocation. In the framework of the Maximic project (collaboration with IBIS team) and as a follow up of our previous work [82], we investigated the impact of energy metabolism on cell's strategy for resource allocation. Preliminary results show that the inclusion of energy costs leads to the emergence of a trade-off between growth rate and yield, as experimentally observed in many bacterial cells .

The allocation of cellular resources can strongly influence not only the rate of cell growth but also the resulting cell size [78]. To better investigate the connection between proteome allocation and cell volume, the original model by Giordano et al. [82] has been connected to a biophysical model of cell growth, explicitly describing cell volume increase as a function of cell's internal pressure and mechanical properties. The resulting model will be used to investigate the mechanisms (control of osmotic pressure or wall mechanics) behind cell size control under different environmental constrains [84].

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 and MCTAO project-teams. We developed different versions of the problem, and consider a new problem where the aim is to optimize the production of a product [68], [40], [50], (ANR project Maximic, PhD thesis of A. Yabo). We also study variations of the model, for example in the chemostat [57]. The precise mathematical analysis of the optimal behavior (turnpike property) is under investigation.

A synthetic community of bacteria. In the framework of IPL Cosy, we study the coexistence of two strains of bacteria E. Coli in a bioreactor. The strains have been modified synthetically to achieve some goals. The aim is to obtain a better productivity in the consortium than in a single strain, by control technics. The description of models is in revision for Plos Comp. Biol.

In collaboration with team VALSE (Lille), we also studied several problems of estimation and robust stabilization related to IPL Cosy, for two bacterial species in a bioreactor [53], [54].

Control of a model of synthesis of a virulence factor. In collaboration with J.-A. Sepulchre (UCA), we modeled 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 [37]. We also studied the problem of periodic inputs for maximization of some yield [97].

Hybrid control of genetic networks. We designed control strategies based on the measurement and control of a unique gene within positive or negative loops of genetic networks, in order to stabilize the system around its unstable fixed point. The quantized nature of genetic measurements and the new synthetic control approaches available in biology encourage the use of piecewise constant control laws. A specific partitioning of the state space and the study of successive repulsive regions allow to show global convergence and global stability for the resulting system [18]. Several other control strategies are studied [47], [48], [46]. This is part of the PhD thesis of L. Chambon.

Biochemical and signaling models

Participants : Madalena Chaves, Eleni Firippi, Sofia Almeida, Marielle Péré, Luis Gomes Pereira, Jérémie Roux.

Analysis and coupling of biological oscillators

Modeling, analysis and coupling of the mammalian cell cycle and clock A transcriptional model of the mammalian circadian clock was developed in [13] and its parameters calibrated against experimental data from F. Delaunay's lab. A cell cycle model was also previously developed by us [77]. The interactions between the two oscillators are investigated under uni- or bi-directional coupling schemes [44]. Numerical simulations replicate the oscillators' period-lock response and recover observed clock to cell cycle period ratios such as 1:1, 3:2 and 5:4 (as observed in experiments, F. Delaunay's lab). This work is in collaboration with F. Delaunay (ANR ICycle) and part of the PhD thesis of Sofia Almeida.

Period-control in a coupled system of two genetic oscillators In the context of ANR project ICycle, we consider two reduced models that mimic the dynamics of the cell cycle and clock oscillators and study the effect of each oscillator on the coupled system, from a synthetic biology perspective [56]. The first observation is that oscillator A is more likely to be the controller of the coupled system period when the dynamis of oscillator B becomes stable due to the coupling strength. Another interesting observation is that the coupled system exhibits oscillatory dynamics over an increased region of the parameter space. This work is part of the PhD thesis of Eleni Firippi (ANR ICycle).

Modeling the apoptotic signaling pathway

A detailed model of the death receptor layer In a collaboration with J. Roux and within project Imodrez, he 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. In a first approach, we constructed a detailed model to represent the death receptor-ligand binding and subsequent signaling cascade  [11]. This model was used to study the effect of intrinsic and extrinsic noise sources, and suggested the need to expand a set of reactions on the model, to account for the observed cell heterogeneity (this was part of the PhD thesis of Luis Pereira).

A basic model to explore the effect of a positive feedback loop Analysis of the detailed apoptosis receptor model uncovered a set of reactions for which the introduction of a positive feedback loop from caspase 8 was able to significantly increase the range of variability in the model in response to extrinsic noise. To better understand this mechanism and the role of positive loop in cell response variability, we are constructing a reduced model representing only the basic components: death ligand and receptor, caspase 8 and two intermediate complexes. This is part of the work of the PhD student Marielle Péré.