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Section: New Results

Aggregation kinetics

Participants : Tom Banks, Thibault Bourgeron, Marc Hoffmann, Marie Doumic-Jauffret, Nathalie Krell, Benoît Perthame, Stéphanie Prigent, Human Rezaei, Nathalie Robert, Léon Matar Tine [Univ. Lyon and Dracula Inria team] , Jorge Zubelli [IMPA, Rio de Janeiro] .

Time Asymptotics for Fragmentation Equations

Fragmentation and growth-fragmentation equations is a family of problems with varied and wide applications. This paper is devoted to description of the long time time asymptotics of two critical cases of these equations, when the division rate is constant and the growth rate is linear or zero. The study of these cases may be reduced to the study of the following fragmentation equation:

t u ( t , x ) + u ( t , x ) = x k 0 ( x y ) u ( t , y ) d y .

Using the Mellin transform of the equation, we determine the long time behavior of the solutions. Our results show in particular the strong dependence of this asymptotic behavior with respect to the initial data.

Estimating the division rate in a size-structured population.

The problem which was considered in [5] consists in estimating the division rate from the stable size distribution of the population, which is easily measured, but non-smooth. We propose a method based on the Mellin transform for growth-fragmentation equations with self-similar kernels. We build a sequence of functions which converges to the density of the population in division, simultaneously in several weighted L2 spaces, as the measurement error goes to 0. This improves previous results for self-similar kernels  (Perthame and Zubelli, Inv. Prob., 2007) and allows us to understand the partial results for general fragmentation kernels  (Doumic and Tine, J. Math. Biol., 2012). Numerical simulations confirm the theoretical results. Moreover, our numerical method is tested on real biological data, arising from a bacteria growth and fission experiment.

What governs bacterial growth? The “sizer” vs the “timer” model

We applied the previously seen inverse problem methodology [5] to a fundamental biological problem: what governs the bacterial growth?

Many organisms coordinate cell growth and division through size control mechanisms: cells must reach a critical size to trigger a cell cycle event. Bacterial division is often assumed to be controlled in this way, but experimental evidence to support this assumption is still lacking. Theoretical arguments show that size control is required to maintain size homeostasis in the case of exponential growth of individual cells. Nevertheless, if the growth law deviates slightly from exponential for very small cells, homeostasis can be maintained with a simple `timer' triggering division. Therefore, deciding whether division control in bacteria relies on a `timer' or `sizer' mechanism requires quantitative comparisons between models and data.

The timer and sizer hypotheses find a natural expression in models based on partial differential equations. Here we test these models with recent data on single-cell growth of Escherichia coli. We demonstrate that a size-independent timer mechanism for division control, though theoretically possible, is quantitatively incompatible with the data and extremely sensitive to slight variations in the growth law. In contrast, a sizer model is robust and fits the data well. In addition, we tested the effect of variability in individual growth rates and noise in septum positioning and found that size control is robust to this phenotypic noise.

Confrontations between cell cycle models and data usually suffer from a lack of high-quality data and suitable statistical estimation techniques. In the study [16] we had overcome these limitations by using high precision measurements of tens of thousands of single bacterial cells combined with recent statistical inference methods to estimate the division rate within the models. We therefore provided the first precise quantitative assessment of different cell cycle models.

Size distribution of amyloid fibrils. Mathematical models and experimental data.

More than twenty types of proteins can adopt misfolded conformations, which can co-aggregate into amyloid fibrils, and are related to pathologies such as Alzheimer's disease. In [15] , we surveyed mathematical models for aggregation chain reactions, and discussed the ability to use them to understand amyloid distributions. Numerous reactions have been proposed to play a role in their aggregation kinetics, though the relative importance of each reaction in vivo is unclear: these include activation steps, with nucleation compared to initiation, disaggregation steps, with depolymerization compared to fragmentation, and additional processes such as filament coalescence or secondary nucleation. We have statistically analysed the shape of the size distribution of prion fibrils, with the specific example of truncated data due to the experimental technique (electron microscopy). A model of polymerization and depolymerization succeeds in explaining this distribution. It is a very plausible scheme though, as evidenced in the review of other mathematical models, other types of reactions could also give rise to the same type of distributions.

To clarify how these fibrils are able to incorporate additional units, prion fibril aggregation and disaggregation kinetics were experimentally studied using Static Light Scattering (SLS) [45] . Values that are functions of i2ci (for i>0) with ci being the concentration of fibrils of size i, were then measured as a function of time. An initial model, adapted from the Becker-Döring system that considers all fibrils to react similarly is not able to reproduce the observed in vitro behaviour. Our second model involves an additional compartment of fibrils unable to incorporate more prion units. This model leads to kinetic coefficients which are biologically plausible and correctly simulates the first experimental steps for prion aggregation.

In the formation of large clusters out of small particles, the initializing step is called the nucleation, and consists in the spontaneous reaction of agents which aggregate into small and stable polymers called nucleus. After this early step, the polymers are involved into a bunch of reactions such as polymerization, fragmentation and coalescence. Since there may be several orders of magnitude between the size of a particle and the size of an aggregate, building efficient numerical schemes to capture accurately the kinetics of the reaction is a delicate step of key importance. In [29] , we propose a conservative scheme, based on finite volume methods on an adaptive grid, which is able to render out the early steps of the reaction as well as the later chain reactions.