M1 - Modeling Biological and Environmental Systems - Mention MATHEMATIQUES - IDIL

The first part of this course concerns complements of probability theory: conditional expectation, Gaussian vectors. The second part presents one of the main families of stochastic processes in discrete time : Markov chains. These are sequences of dependent random variables, whose dependence relation is relatively simple since each variable depends only on the previous one. It is also a very powerful modeling tool. We will study the main properties of these processes, as well as their behavior in long time and the estimation of their parameters.

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This EU aims to provide a broad overview of emerging quantitative interdisciplinary fields in bioscience, ranging from advanced experimental techniques in microscopy and synthetic biology, to systems approaches.

In an innovative way, these methodological aspects will be presented in the context of biological and biophysical concepts such as robustness and optimality of biological systems, gene regulation and the fundamental principles underlying membrane and genome organization.

The main topics will be introduced first with traditional lectures and will be developed through individual or team projects where students will learn to apply specific techniques through examples, and see how these can be used to explore specific biological questions. These projects will involve literature reviews, use of existing code, or development of new code (depending on the student's experience) and will constitute half of the final assessment.

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The course aims to provide a general introduction to cellular and molecular biology and to put into context the use of modern physics, through its quantitative methods and approaches, to describe biological systems and their complexity from the molecular to the cellular and tissue scales.

A fundamental point addressed is also the quantification of phenomena, their physical interpretation and their physico-mathematical modeling. The course opens to the philosophy and to the set of themes of this master's course centered on the study of the physical principles of the organization and the dynamics of the living and complex matter.

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The importance of statistical science in the process of scientific discovery and industrial advancement is that it allows the formulation of inferences about phenomena of interest to which one can associate risks of error or degrees of confidence. The calculation of these risks of error is based on probability theory, but the principles and methods for associating these risks with inferences constitute a theoretical corpus that serves as a basis for all statistical methodologies.

This module is intended to be a fairly complete presentation of these basic principles and of the tools, results and mathematical theorems used in inferential statistics. It develops the notions of point and interval estimation, hypothesis testing and fundamental concepts such as exponential families and the principle of maximum likelihood and the use of p-value.
For the implementation of certain applications, the adapted tools of the R software will be presented.

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