Oscillatory processes during the aggregation and the fragmentation of the amyloid fibrils.

Abstract : The objective of this thesis is to study the physical process of protein aggregation and fragmentation. More specifically, oscillatory kinetic phenomena are identified during this process and are the subject of an in-depth analysis. The process of protein aggregation and fragmentation is intimately linked to the contraction and development of a vast class of incurable neurodegenerative diseases, amyloid diseases. Our work focuses on prion diseases, a subcategory of amyloid diseases, caused by the misfolding of protein conformation and the aggregation of these misfolded proteins into fibrils. First, our study focuses on experimental static light scattering data (SLS). The signals obtained correspond approximately to the second moment of the size distribution of amyloid fibrils in vitro and therefore provide information on the evolution of the size distribution over time. We note damped oscillations at specific locations on the signals. These oscillations highlight the presence of complex, underlying kinetic phenomena during protein aggregation/ fragmentation processes. The analysis of SLS signals leads us to build a parametric characterization of oscillations in the frequency domain (Fourier domain). We propose a numerical procedure to obtain these parameters. Then, we build a statistical test of hypotheses. We thus obtain a p-value that allows us to quantitatively assert the presence of oscillations in the experimental signals. In a second step, we introduce and mathematically analyze a kinetic model of proteins capable of generating oscillations. The model is a variant of the polymerization/depolymerization system and considers two species of monomers: a pathological monomer that polymerizes and a healthy monomer that depolymerizes. Unlike traditional models, depolymerization is catalytic and non-linear and an exchange phenomenon occurs between the two species of monomers and polymers. The model combines a Lotka-Volterra system for monomers with a growth/fragmentation system: Becker-Döring in the discrete size setting, Lifshitz-Slyozov in the continuous size setting. In the discrete size model, the oscillations are damped and under certain conditions we prove the exponential convergence towards a stationary state. While in the continuous model, the system oscillates perpetually or converges to a Dirac depending on the shape of the reaction coefficients. By complexifying the model, in particular by integrating other species of polymers and kinetic reactions, it is possible to achieve a realistic modelling of protein kinetic processes. The mathematical study of these models leads to new interesting problems, improves and clarifies the understanding of the underlying physical phenomena.
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Mathieu Mezache. Oscillatory processes during the aggregation and the fragmentation of the amyloid fibrils.. Mathematics [math]. Sorbonne Université - Laboratoire Jacques-Louis Lions; Inria Paris, 2019. English. ⟨tel-02435325⟩

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