Recherche

Axis 1: Models through scales

We refer to 3.1 for a presentation of the research program in this direction.

Chemotaxis models

Curvature in chemotaxis: A model for ant trail pattern formation

In 21, we propose a new model of chemotaxis motivated by ant trail pattern formation, formulated as a coupled parabolic-parabolic local PDE system, for the population density and the chemical field. The main novelty lies in the transport term of the population density, which depends on the second-order derivatives of the chemical field. This term is derived as an anticipation-reaction steering mechanism of an infinitesimally small ant as its size approaches zero. We establish global-in-time existence and uniqueness for the model, and the propagation of regularity from the initial data. Then, we build a numerical scheme and present various examples that provide hints of trail formation.

On a multi-dimensional McKean-Vlasov SDE with memorial and singular interaction associated to the parabolic-parabolic Keller-Segel model

In the article 16, we firstly prove the well-posedness of the non-linear martingale problem related to a McKean-Vlasov stochastic differential equation with singular interaction kernel in ${ℝ}^D$ for $d\ge 3$ . The particularity of our setting is that the McKean-Vlasov process we study interacts at each time with all its past time marginal laws by means of a singular space-time kernel. Secondly, we prove that our stochastic process is a probabilistic interpretation for the parabolic-parabolic Keller-Segel system in ${ℝ}^d$ . We thus obtain a well-posedness result to the latter under explicit smallness condition on the parameters of the model.

Limits of large populations with local or nonlocal interaction and heterogeneity

For the study of large populations with local interaction, Vincent Bansaye, together with Felipe Munoz and Ayman Moussa, developed approaches in a discrete space that grows simultaneously with the local size of the population 47. We introduced duality techniques to handle the convergence of the stochastic process to a cross-diffusion and the stability of the limiting PDE. Ayman Moussa and Vincent Bansaye are currently supervising a pre-thesis student, Alexandre Bertoloni, who is extending these resu lts by incorporating births and deaths with local density dependence, along with a reaction term.

Originally motivated by the morphogenesis of bacterial microcolonies, the aim of a series of articles, in a collaboration between Marie Doumic, members of the Inria project-team MUSCLEES and Marc Hoffmann, is to explore models through different scales for a spatial population of interacting, growing and dividing particles. After the modelling and simulation article 80, we studied the rigorous limits through scales of a model including growth, division and interaction.

In 26, we start from a microscopic stochastic model, write the corresponding stochastic differential equation satisfied by the empirical measure, and rigorously derive its mesoscopic (mean-field) limit. Under smoothness and symmetry assumptions for the interaction kernel, we then obtain entropy estimates, which provide us with a localization limit at the macroscopic level. Finally, we perform a thorough numerical study in order to compare the three modeling scales. An important difficulty of this work is to take into account the continuous size structure, which leads to a lack of compactness for the localisation limit.

In the article 12, we study a multi-species system, which may be seen as a discrete counterpart of the size-structured one 26. At the mesoscopic level, the system is quadratic, written under the form of transport equations with a nonlocal self-generated drift. We establish the localisation limit, that is the convergence of nonlocal to local systems, when the range of interaction tends to 0. These theoretical results are sustained by numerical simulations. The major new feature in this analysis is that we do not need diffusion to gain compactness, at odd with the existing literature. The central compactness result is provided by a full rank assumption on the interaction kernels. In turn, we prove existence of weak solutions for the resulting system, a cross-diffusion system of quadratic type.

A scenario for an evolutionary selection of ageing

Signs of ageing become apparent only late in life, after organismal development is finalized. Ageing, most notably, decreases an individual’s fitness. As such, it is most commonly perceived as a non-adaptive force of evolution and considered a by-product of natural selection. Building upon the evolutionarily conserved age-related Smurf phenotype, we propose a simple mathematical life-history trait model in which an organism is characterized by two core abilities: reproduction and homeostasis 15. Through the simulation of this model, we observe 1) the convergence of fertility’s end with the onset of senescence, 2) the relative success of ageing populations, as compared to non-ageing populations, and 3) the enhanced evolvability (i.e. the generation of genetic variability) of ageing populations. In addition, we formally demonstrate the mathematical convergence observed in 1). We thus theorize that mechanisms that link the timing of fertility and ageing have been selected and fixed over evolutionary history, which, in turn, explains why ageing populations are more evolvable and therefore more successful. Broadly speaking, our work suggests that ageing is an adaptive force of evolution.

Phenotypic plasticity trade-offs in an age-structured model of bacterial growth under stress

Under low concentrations of antibiotics causing DNA damage, Escherichia coli bacteria can trigger stochastically a stress response known as the SOS response. While the expression of this stress response can make individual cells transiently able to overcome antibiotic treatment, it can also delay cell division, thus impacting the whole population’s ability to grow and survive. In order to study the trade-offs that emerge from this phenomenon, we propose a bi-type age-structured population model that captures the phenotypic plasticity observed in the stress response 

. Individuals can belong to two types: either a fast-dividing but prone to death “vulnerable » type, or a slow-dividing but “tolerant » type. We study the survival probability of the population issued from a single cell as well as the population growth rate in constant and periodic environments. We show that the sensitivity of these two different notions of fitness with respect to the parameters describing the phenotypic plasticity differs between the stochastic approach (survival probability) and the deterministic approach (population growth rate). Moreover, under a more realistic configuration of periodic stress, our results indicate that optimal population growth can only be achieved through fine-tuning simultaneously both the induction of the stress response and the repair efficiency of the damage caused by the antibiotic.

Dynamics of a kinetic model describing protein transfers in a cell population

We consider a cell population structured by a positive real number which represents the number of P-glycoproteins carried by the cell. In this article, we introduce a kinetic model to describe the dynamics of the cell population, and consider an asymptotic limit of this equation: if transfers are frequent, the population can be described through a system of two coupled ordinary differential equations. The main idea of this manuscript is to combine Wasserstein distance estimates on the kinetic operator to more classical estimates on the macroscopic quantities.

Macroscopic limit from a structured population model to the Kirkpatrick-Barton model

We consider an ecology model in which the population is structured by a spatial variable and a phenotypic trait. The model combines a parabolic operator on the spatial variable with a kinetic operator on the trait variable. We combine a contraction argument based on Wasserstein estimates on the phenotypic variable with parabolic estimates controlling the spatial regularity of solutions to prove the convergence of the population size and the mean phenotypic trait to solutions of the Kirkpatrick-Barton model, which is a well-established model in evolutionary ecology.

Axis 2: qualitative analysis of structured populations

We refer to 3.2 for a presentation of the research program in this direction.

Long-time behaviours

Long-time behaviour of a multidimensional age-dependent branching process with a singular jump kernel.

In the article 

, we investigate the ergodic behaviour of a multidimensional age-dependent branching process with a singular jump kernel, motivated by studying the phenomenon of telomere shortening in cell populations. Our model tracks individuals evolving within a continuous-time framework indexed by a binary tree, characterised by age and a multidimensional trait. Branching events occur with rates dependent on age, where offspring inherit traits from their parent with random increase or decrease in some coordinates, while the most of them are left unchanged. Exponential ergodicity is obtained at the cost of an exponential normalisation, despite the fact that we have an unbounded age-dependent birth rate that may depend on the multidimensional trait, and a non-compact transition kernel. These two difficulties are respectively treated by stochastically comparing our model to Bellman-Harris processes, and by using a weak form of a Harnack inequality. We conclude this study by giving examples where the assumptions of our main result are verified.

Long-time behaviour of a degenerate stochastic system modeling the response of a population to its environmental perception.

In 23, accepted for publication in Electronic Communications in Probability, we study the asymptotics of a two-dimensional stochastic differential system with a degenerate diffusion matrix. This system describes the dynamics of a population where individuals contribute to the degradation of their environment through two differentbehaviors, responding more or less intensively to their environmental perception. We exploit the almost one-dimensional form of the dynamical system to compute explicitly the Freidlin-Wentzell action functional. This allows us to give conditions under which the small noise regime of the invariant measure is concentrated around the equilibria of the dynamical system having the smallest diffusion coefficient.

Time reversal and ancestral lineages

Ancestral lineage for interacting populations.

In 8, we consider Markov jump processes describing structured populations with interactions via density dependance. We propose a Markov construction with a distinguished individual which allows to describe the random tree and random sample at a given time via a change of probability. This spine construction involves the extension of type space of individuals to include the state of the population. The jump rates outside the spine are also modified. We apply this approach to some issues concerning evolution of populations and competition. For single type populations, we derive the diagram phase of a growth fragmentation model with competition and the growth of the size of birth and death processes with multiple births. We also describe the ancestral lineages of a uniform sample in multitype populations.

Branching diffusion processes and spectral properties of Feynman Kac semigroup

The article 24 is motivated by the study of the long time behavior of linear functionals of birth and death diffusion processes as well as the time reversal of the spinal process by means of spectral properties of the associated Feynman-Kac semigroup. We generalize for this non Markovian semigroup the theory of quasi-stationary distribution (q.s.d.) and $Q$ -process. We consider a situation where the underlying diffusion process doesn’t come down rapidly from infinity but the compactness properties follow from the divergence of the death rate at infinity. We establish the complete spectral decomposition for the Feynman-Kac semigroup. An interesting consequence is the identification of the law of the reversal time spinal process issued from q.s.d. with the $Q$ -process of the Feynman-Kac semigroup.

Evolutionary dynamics – stochastic and deterministic mutation models

Sharp approximation and hitting times for stochastic invasion processes We are interested in the invasion phase for stochastic processes with interactions. A single mutant with positive fitness arrives in a large resident population at equilibrium. By a now classical approach, the first stage of the invasion is well approximated by a branching process. The macroscopic phase, when the mutant population is of the same order as the resident population, is described by the limiting dynamical system. We capture the intermediate mesoscopic phase for the invasive population and obtain sharp approximations. It allows us to describe the fluctuations of the hitting times of thresholds, which inherit a large variance from the first stage. We apply our results to two models which are first motivations. In particular, we quantify the hitting times of critical values in cancer emergence and epidemics 7.

Convergence of a discrete selection-mutation model with exponentially decaying mutation kernel to a Hamilton-Jacobi equation

In the preprint 32, Anouar Jeddi (Ph.D student supervised by Sylvie Méléard and Sepideh Mirrahimi) derives a Hamilton-Jacobi equation with obstacle from a discrete linear integro-differential model in population dynamics, with exponentially decaying mutation kernel. The fact that the kernel has exponential decay leads to a modification of the classical Hamilton-Jacobi equation obtained previously from continuous models as in Barles-Mirrahimi-Perthame. We consider a population composed of individuals characterized by a quantitative trait, subject to selection and mutation. In the regime of large population, small mutations and large time, we prove that the WKB transformation of the density converges to the unique viscosity solution of a Hamilton-Jacobi equation with obstacle.

The article 13 has been published in Evolution Letters. In this article, we have analysed the evolutionary dynamics of pathogens spreading in a heterogeneous host population where selection varies periodically in space. We study both the transient dynamics taking place at the front of the epidemic and the long-term evolution far behind the front. In particular, we identify the conditions where a generalist pathogen carrying multiple adaptations can outrace a coalition of specialist pathogens. We also show that finite host populations promote the spread of generalist pathogens because demographic stochasticity enhances the extinction of locally maladapted pathogens.

The article 123 is submitted to the Journal of Theoretical Biology. Shallow lakes ecosystems may experience abrupt shifts (ie tipping points) from one state to a contrasting degraded alternative state as a result of gradual environmental changes. It is crucial to elucidate how eco-evolutionary feedbacks affect abrupt ecological transitions in shallow lakes. We explore the eco-evolutionary dynamics of submerged and floating macrophytes in a shallow lake ecosystem under asymmetric competition for nutrients and light. We show how rapid trait evolution can result in complex dynamics including evolutionary oscillations, extensive diversification and evolutionary suicide. Overall, this study shows that evolution can have strong effects in the ecological dynamics of bistable ecosystems.

Exponential convergence to a steady-state for a population genetics model with sexual reproduction and selection.

We are interested in the dynamics of a population structured by a phenotypic trait. Individuals reproduce sexually, which is represented by a non-linear integral operator. This operator is combined to a multiplicative operator representing selection. When the strength of selection is small, we show that the dynamics of the population is governed by a simple macroscopic differential equation, and that solutions converge exponentially to steady-states that are locally unique. The analysis is based on Wasserstein distance inequalities using a uniform lower bound on distributions. These inequalities are coupled to tail estimates to show the stability of the steady-states.

Is there an advantage of displaying heterogeneity in a population where the individuals grow and divide by fission? This is a wide-ranging question, for which a universal answer cannot be easily provided. In 28, we aim at providing a quantitative answer in the specific context of growth rate heterogeneity by comparing the fitness of homogeneous versus heterogeneous populations. We focus on a size-structured population, where an individual’s growth rate is chosen at its birth through heredity and/or random mutations. We use the long-term behaviour to define the Malthus parameter of such a population, and compare it to the ones of averaged homogeneous populations. We obtain analytical formulae in two paradigmatic cases: first, constant rates for growth and division, second, linear growth rates and uniform fragmentation. Surprisingly, these two cases happen to display similar analytical formulae linking effective and individual fitness. They allow us to investigate quantitatively the crossed influence of heredity and heterogeneity, and revisit previous results stating that heterogeneity is beneficial in the case of strong heredity.

The epidemiological footprint of contact structures in models with two levels of mixing

In human contact networks, individuals often belong to several contact structures, such as a home and a workplace or school. This social organization is relevant to study in an epidemic context, as it is the subject of control measures such as telecommuting or school closures. However, the influence of these structures on the epidemic is not yet well understood. We are therefore investigating a model with two levels of mixing, namely a uniformly mixing global level, and a local level divided into two layers of contacts within households and workplaces, respectively. We are seeking to develop reduced models that closely approximate these epidemic dynamics, while being more manageable for numerical and/or theoretical analysis. In

, a simulation study compares several telecommuting strategies, and shows that there are more effective strategies than the most naive strategy of allowing only a number of workers in the workplace proportional to the size of the workplace. Next, we highlight two indicators of the epidemic impact of the location size distribution, namely the variance and the exponential growth rate observed at the start of an epidemic. In addition, we calibrate a uniformly mixing epidemic model to provide a good approximation of the home-work model, as shown by the numerical exploration. This reduced model thus provides an easily parameterizable and numerically satisfactory approximation. Finally, in 100, a sensitivity study enables us to understand the impact of model parameters on the performance of this reduced model, by quantifying the impact of epidemic parameters on its ability to predict key features of the epidemic.

To provide a mechanistic explanation of sustained then damped oscillations observed in a depolymerisation experiment, a bi-monomeric variant of the seminal Becker-Döring system has been proposed in (Doumic, Fellner, Mezache, Rezaei, J. of Theor. Biol., 2019). When all reaction rates are constant, the equations are the following:

$$\begin{array}{cc}\hfill \frac{dv}{dt}& =–vw+v\sum _{j=2}^{\infty }{c}_j,\frac{dw}{dt}=vw–w\sum _{j=1}^{\infty }{c}_j,\hfill \\ \hfill \frac{d{c}_j}{dt}& =J_{j–1}–J_j,j\ge 1,J_j=w{c}_j–v{c}_{j+1},j\ge 1,J_0=0,\hfill \end{array}$$

where $u$ and $w$ are two distinct unit species, and $c_i$ represents the concentration of clusters containing $i$ units. We study in detail the mechanisms leading to such oscillations and characterise the different phases of the dynamics, from the initial high-amplitude oscillations to the progressive damping leading to the convergence towards the unique positive stationary solution. We give quantitative approximations for the main quantities of interest: period of the oscillations, size of the damping (corresponding to a loss of energy), number of oscillations characterising each phase. We illustrate these results by numerical simulation, in line with the theoretical results, and provide numerical methods to solve the system.

Axis 3: Model-data comparison

We refer to 3.3 for a presentation of the research program in this direction.

Telomere shortening, a unifying model

Progressive shortening of telomeres ultimately causes replicative senescence and is linked with aging and tumor suppression. Studying the intricate link between telomere shortening and senescence at the molecular level and its population-scale effects over time is challenging with current approaches but crucial for understanding behavior at the organ or tissue level. In the article 

, accepted for publication in Nature Communications, we developed a mathematical model for telomere shortening and the onset of replicative senescence using data from Saccharomyces cerevisiae without telomerase. Our model tracks individual cell states, their telomere length dynamics, and lifespan over time, revealing selection forces within a population. We discovered that both cell genealogy and global telomere length distribution are key to determine the population proliferation capacity. We also discovered that cell growth defects unrelated to telomeres also affect subsequent proliferation and may act as confounding variables in replicative senescence assays. Overall, while there is a deterministic limit for the shortest telomere length, the stochastic occurrence of non-terminal arrests drive cells into a totally different regime, which may promote genome instability and senescence escape. Our results offer a comprehensive framework for investigating the implications of telomere length on human diseases. Our model has also been used further in another experimental device, where one telomere is cut at a given very short length thanks to CrisPr-Cas9 technique 20. Alongside this modelling and simulation approach, Jules Olayé’s Ph.D work focused on several interesting problems raised by this collaboration, see Sections 8.2.1, 8.2.6 and 34,

.

Classical Myelo-Proliferative Neoplasms emergence and development in patients based on real-life incidence and mathematical modeling

Mathematical modeling offers the opportunity to test hypothesis concerning Myeloproliferative emergence and development. In 19, we tested different mathematical models based on a training cohort ( $n=264$ patients) (Registre de la côte d’Or) to determine the emergence and evolution times before JAK2V617F classical Myeloproliferative disorders (respectively Polycythemia Vera and Essential Thrombocytemia) are diagnosed. We dissected the time before diagnosis as two main periods: the time from embryonic development for the JAK2V617F mutation to occur, not disappear and enter in proliferation, and a second time corresponding to the expansion of the clonal population until diagnosis. We demonstrate using progressively complexified models that the rate of active mutation occurrence is not constant and doesn’t just rely on individual variability, but rather increases with age and takes a median time of $63.1+/–13$ years. A contrario, the expansion time can be considered as constant: 8.8 years once the mutation has emerged. Results were validated in an external cohort (national FIMBANK Cohort, $n=1248$ patients). Analyzing JAK2V617F Essential Thrombocytema versus Polycythemia Vera, we noticed that the first period of time (rate of active homozygous mutation occurrence) for PV takes approximatively 1.5 years more than for ET to develop when the expansion time was quasi-similar. In conclusion, our multi-step approach and the ultimate time-dependent model of MPN emergence and development demonstrates that the emergence of a JAK2V617F mutation should be linked to an aging mechanism, and indicates a 8-9 years period of time to develop a full MPN.

Quantitative effects of the stress response to DNA damage in the cell size control of Escherichia coli

In Escherichia coli the response to DNA damage shows strong cell-to-cell-heterogenity. This results in a random delay in cell division and asymmetrical binary fission of single cells, which can compromise the size homeostasis of the population. To quantify the effect of the heterogeneous response to genotoxic stress (called SOS response in E. coli) on the growth of the bacterial population, we propose a flexible time-continuous parametric model of individual-based population dynamics 22. We construct a stochastic model based on the « adder » size-control mechanism, extended to incorporate the dynamics of the SOS response and its effect on cell division. The model is fitted to individual lineage data obtained in a ‘mother machine’ microfluidic device. We show that the heterogeneity of the SOS response can bias the observed division rate. In particular, we show that the adder division rate is decreased by SOS induction and that this perturbative effect is stronger in fast-growing conditions.

A unifying mathematical approach for the coordination of DNA replication and cell division in E. coli

While significant efforts have been made to model bacterial cell division, few models have incorporated DNA replication into the control of this process. To date, models that attempt to capture the coordination between replication and division cycles are based on fundamentally different assumptions, and yet no study provide a rigorous quantitative comparaison. As a result, key questions regarding how replication quantitatively controls cell division remain unresolved. To address this, we have developed a robust mathematical framework to compare models of coordination of replication and division cycles proposed in the literature. Through theoretical analysis, we identified necessary and sufficient conditions for these models to exhibit physiological behaviour in cell cycles, and tested whether these conditions accords with experimental data. Additionally, a comprehensive statistical analysis allowed us to assess the models’ ability to reproduce the joint distributions of cell cycles related quantities. This in-depth analysis led to the development of a novel model for the coordination of replication and division cycles, which provides an accurate fit to the data across a wider range of growth conditions than previous models.

Exploitation of microfluidic data

In 35, we describe fluctuations in population sizes of Bellman-Harris processes to estimate lifetime distributions from temporal population size tracking. This involves determining two fluctuation regimes, leveraging recent results for Crump-Mode-Jagers processes, and applying these findings to microfluidic data.

Asymptotic inverse problems for fragmentation and depolymerisation models

Given a phenomenon described by a self-similar fragmentation equation, how to infer the fragmentation kernel from experimental measurements of the solution ? To answer this question at the basis of our work, a formal asymptotic expansion suggested us that using short-time observations and initial data close to a Dirac measure should be a well-adapted strategy. We prove error estimates in Total Variation and Bounded Lipshitz norms; this gives a quantitative meaning to what a ”short” time observation is. Our analysis is complemented by a numerical investigation 11.

In another study, we focused on depolymerisation reactions, which constitute frequent experiments, for instance in biochemistry for the study of amyloid fibrils. The quantities experimentally observed are related to the time dynamics of a quantity averaged over all polymer sizes, such as the total polymerised mass or the mean size of particles. The question analysed here is to link this measurement to the initial size distribution. To do so, we first derive, from the initial reaction system two asymptotic models: at first order, a backward transport equation, and at second order, an advection-diffusion/Fokker-Planck equation complemented with a mixed boundary condition at x = 0. We estimate their distance to the original system solution. We then turn to the inverse problem, i.e., how to estimate the initial size distribution from the time measurement of an average quantity, given by a moment of the solution. This question has been already studied for the first order asymptotic model, and we analyse here the second order asymptotic. Thanks to Carleman inequalities and to log-convexity estimates, we prove observability results and error estimates for a Tikhonov regularization. We then develop a Kalman-based observer approach, and implement it on simulated observations. Despite its severely ill-posed character, the second order approach appears numerically more accurate than the first-order one 27.