Section: New Results

Aggregation Kinetics

Participants : Aurora Armiento, Tom Banks [CRSC, NCSU, Raleigh, USA] , Thibault Bourgeron, José Antonio Carrillo [Imperial College, London, United Kingdom] , Marie Doumic, Miguel Escobedo [Universidad del País Vasco, Bilbao, Spain] , Sarah Eugène, Marc Hoffmann [Ceremade, Université Paris-Dauphine] , François James [MAPMO, Université d'Orléans] , Nathalie Krell [Université de Rennes 1] , Carola Kruse, Frédéric Lagoutière [Département de mathématiques d'Orsay] , Philippe Moireau [Inria Paris Saclay, M3DISIM project-team] , Benoît Perthame, Stéphanie Prigent, Human Rezaei [VIM, INRA Jouy-en-Josas] , Lydia Robert [Laboratoire Jean Perrin, UPMC] , Philippe Robert [Inria Paris, RAP project-team] , Maria Teresa Teixeira [IBCP, Paris] , Nicolas Vauchelet, Min Tang [Jiaotong University, Shanghai] , Zhou Xu [IBCP, Paris] , Wei-Feng Xue [University of Kent, United Kingdom] .

Heterogeneity as an intrinsic feature in biological dynamics

Combining deterministic and probabilistic approaches, we investigated in two different applications - namely senescence and protein aggregation - the impact of heterogeneity on dynamical features of the considered populations.

Yeast Senescence and Telomere replication In eukaryotes, the absence of telomerase results in telomere shortening, eventually leading to replicative senescence, an arrested state that prevents further cell divisions. While replicative senescence is mainly controlled by telomere length, the heterogeneity of its onset is not well understood. Insights on this key question may have consequences both for cancer and aging issues.

In collaboration with T. Teixeira and Z. Xue from IBCP, we proposed a mathematical model based on the molecular mechanisms of telomere replication and shortening to decipher the causes of this heterogeneity [7] . Using simulations fitted on experimental data obtained from individual lineages of senescent Saccharomyces cerevisiae cells, we decompose the sources of senescence heterogeneity into interclonal and intraclonal components, and show that the latter is based on the asymmetry of the telomere replication mechanism. We also evidence telomere rank-switching events with distinct frequencies in short-lived versus long-lived lineages, revealing that telomere shortening dynamics display important variations. Thus, the intrinsic heterogeneity of replicative senescence and its consequences find their roots in the asymmetric structure of telomeres.

These promising first results lead us to an ongoing collaboration, and hopefully will allow still more insight on complex mechanisms not yet modelled mathematically.

Variability in nucleated polymerisation

The kinetics of amyloid assembly show an exponential growth phase preceded by a lag phase, variable in duration as seen in bulk experiments and experiments that mimic the small volumes of cells. To investigate the origins and the properties of the observed variability in the lag phase of amyloid assembly currently not accounted for by deterministic nucleation dependent mechanisms, we formulated a new stochastic minimal model that is capable of describing the characteristics of amyloid growth curves despite its simplicity [44] . We then solved the stochastic differential equations of our model and gave a mathematical proof of a central limit theorem for the sample growth trajectories of the nucleated aggregation process. These results give an asymptotic description for our simple model, from which closed-form analytical results capable of describing and predicting the variability of nucleated amyloid assembly were derived. We also demonstrated the application of our results to inform experiments in a convenient and clear way. Our model offers a new perspective and paves the way for a new and efficient approach on extracting vital information regarding the key initial events of amyloid formation.

Inverse Problems and Data Assimilation Applied to Protein Aggregation and other settings

As mathematical models become more complex with multiple states and many parameters to be estimated using experimental data, there is a need for critical analysis in model validation related to the reliability of parameter estimates obtained in model fitting. This leads to a fundamental question: how much information with respect to model validation can be expected in a given data set or collection of data sets?

In the biological context of amyloid formation, the question is to quantify to which extent a given model may be appropriately fitted and selected for, given relatively sparse data. Estimating reaction rates and size distributions of protein polymers is an important step towards understanding the mechanisms of protein misfolding and aggregation, a key feature for amyloid diseases. Specifically, experimental measurements often consist in the time-dynamics of a moment of the population (i.e., for instance the total polymerised mass, as in Thioflavine T measurements, or the second moment measured by Static Light Scattering).

In a first study [4] , in collaboration with H.T. Banks and H. Rezaei, we illustrated the use of tools (asymptotic theories of standard error quantification using appropriate statistical models, bootstrapping, model comparison techniques) in addition to sensitivity that may be employed to determine the information content in data sets. We do this in the context of recent models  [87] for nucleated polymerisation in proteins, about which very little is known regarding the underlying mechanisms; thus the methodology we developed may be of great help to experimentalists.

In another study [39] , related to a different biological setting (the frog olfactive tract), we use a method based on the Mellin transform, as in   [64] , to solve a spectral inverse problem arising from the modeling of the transduction of an odor into an electrical signal. The problem is to find the spatial distribution of CNG ion channels along the cilium of a frog, which allow a depolarizing influx of sodium ions, which initiate the electrical signal. This problem comes down to solving a Fredholm integral equation. We prove observability and continuity inequalities by estimating the Mellin transform of the kernel of this integral equation. We perform numerical computations using experimental data.

To get more insight into the estimation of reaction rates and size distributions of protein polymers, we are now developing an approach based on a data assimilation strategy. In this purpose, A. Armiento's Ph.D is focused on setting this framework problem when the experimental measurements consist in the time-dynamics of a moment of the population (i.e. for instance the total polymerised mass, as in Thioflavine T measurements, or the second moment measured by Static Light Scattering). In [37] we proposed a general methodology, and we solved the problem theoretically and numerically in the case of a depolymerising system. We then applied our method to experimental data of degrading oligomers, and conclude that smaller aggregates of ovPrP protein should be more stable than larger ones. This has an important biological implication, since it is commonly admitted that small oligomers constitute the most cytotoxic species during prion misfolding process.

Time asymptotics for growth-fragmentation equations

The long-term dynamics of fragmentation and growth-fragmentation equations has been for long an important research field for BANG then MAMBA research team. Thanks to these common efforts, these equations are now well understood. However, there remain some interesting open questions. In particular, if the generic long-time behaviour for the linear equation is known - given by a (generally exponential) trend towards a steady exponential growth described by the positive eigenvector linked to the dominant eigenvalue, see  [84] for most recent results - critical cases are not yet fully understood.

With Miguel Escobedo, we focused on an important critical case, when the fragmentation is constant and the growth rate is either null or linear [43] . Using the Mellin transform of the equation, we determine the long time behaviour of the solutions and the speed of convergence, which may be either exponential or at most polynomial according to the subdomain of $(t,x)\in {\mathrm{I}\mathrm{R}}_{+}^2$ which is considered. Our results show in particular the strong dependence of this asymptotic behaviour with respect to the initial data, in contrast to the generic results. Following our study, J. Bertoin and A. Watson proposed a complementary probabilistic analysis of related models  [60] . These results exemplify the continuing need for further analysis of these interesting equations.

Cell aggregation by chemotaxis

We follow our investigation on the kinetic model describing the chemotactic motion of bacteria. When taxis dominates the unbiased movements, the kinetic system is approximated by the aggregation equation. The study of such equation is challenging since blow-up in finite-time of solutions occurs. We have defined the notion of measure-valued solution [8] and we have proposed and studied a numerical scheme to simulate these solutions[18] .

In another approach, more accuracy can be obtained with the kinetic model by adding an internal variable describing the methylation level of the internal receptors of bacteria. In [55] we have investigated the link between these kinetic models with an internal variable and the one without internal variable.