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Inverse problems and data assimilation methods applied to protein polymerisation

Abstract : The aim of this PhD thesis is to set up a mathematical strategy to investigate the physical process of protein aggregation. The study of this largely unknown process is particularly important since it has been identified as a key feature of a wide class of incurable diseases, called amyloid diseases. Prion diseases belong to this class and are caused by the aggregation of a misfolded configuration of the prion protein. Our work contributes to the research on prion diseases, by focusing on two kinds of aggregates : oligomers and fibrils. Oligomers, which are suspected of being the most toxic aggregates, are studied in the first part of this thesis. We base our work on the analysis of two types of experimental data. On the one hand, we consider Static Light Scattering (SLS) data, which can be interpreted biologically as the measurement of the average oligomer size and mathematically as the second moment of aggregate concentration. On the other hand, we consider oligomer size distribution data collected at several instants by using Size Exclusion Chromatography (SEC). Our study leads to the important conclusion that at least two different types of oligomers are present. Moreover, we provide a description of the interaction between these oligomers by proposing, for the first time, a two-species model. Our model is composed of a set of ODEs with the kinetic rates as parameters. The qualitative description provided by this model has been coupled to the information contained in the noisy experimental SLS data in a data assimilation framework. By means of the extended Kalman filter method, we solve a non-linear inverse problem, thereby estimating the kinetic coefficients associated to the experimental data. To validate this model we have compared our estimation to the experimental SEC data, observing a very good agreement between the two. Our oligomer species characterisation may lead to new strategies to design a first targeted treatment for prion diseases. The methodology applied to the study of oligomers can be seen as a first step in the analysis of fibrils. Due to the physical properties of these aggregates, fewer and less precise experiments can be performed and so a mathematical approach can provide a valuable contribution to their study. Our contribution is to propose a general strategy to estimate the initial condition of a fibril system. Inspired by the Lifshitz-Slyozov theory, we describe this system by a transport equation coupled with an integral equation. The estimation is performed making use of some empirical observations on the system. We consider the general case of observing a moment of order n. It is indeed possible to measure the first moment by Thioflavine T fluorescence or the second moment by SLS. We provide a theoretical and numerical solution of the initial condition estimation problem in the linear case of a depolymerising system. In particular, for constant depolymerisation rates, we propose a kernel regularisation strategy, that provides a first characterisation of the estimation. In the variable depolymerisation rates, we outline the variational data assimilation method 4d-Var. This method is more general and can be easily adapted to treat different problems. This inverse problem is particularly interesting since it can also be applied in other fields such as the cell cycle or dust formation.
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Contributor : Aurora Armiento Connect in order to contact the contributor
Submitted on : Thursday, January 26, 2017 - 5:21:38 PM
Last modification on : Tuesday, January 11, 2022 - 11:16:04 AM
Long-term archiving on: : Thursday, April 27, 2017 - 2:35:40 PM


  • HAL Id : tel-01447286, version 1


Aurora Armiento. Inverse problems and data assimilation methods applied to protein polymerisation. Optimization and Control [math.OC]. Université Paris 7 - Diderot, 2017. English. ⟨tel-01447286⟩



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