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

The first part of this course deals with complements to probability theory: conditional expectation, Gaussian vectors. The second part presents one of the main families of discrete-time stochastic processes, Markov chains. These are sequences of dependent random variables, whose dependency relationship is relatively simple, since each variable depends only on the preceding one. They are 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 the biosciences, ranging from cutting-edge 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 the robustness and optimality of biological systems, gene regulation and the fundamental principles underlying membrane and genome organization.

The main topics will first be introduced 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 bibliographical studies, the use of existing code or the development of new code (depending on the student's experience) and will make up half of the final assessment.

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

Quantifying phenomena, their physical interpretation and physico-mathematical modeling are also fundamental aspects of the course. The course opens up to philosophy and to the whole range of themes of this Master's program, centered on the study of the physical principles of the organization and dynamics of living, complex matter.

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The importance of statistical science in the process of scientific discovery and industrial advancement is that it enables the formulation of inferences concerning phenomena of interest, to which risks of error or degrees of confidence can be associated. 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 the basis for all statistical methodologies.

This module is intended to provide a fairly comprehensive presentation of these basic principles and of the mathematical tools, results and theorems used in inferential statistics. It covers the notions of point and interval estimation, hypothesis testing and fundamental concepts such as exponential families, the maximum likelihood principle and the use of p-value.
To implement certain applications, we will present the tools adapted from R software.

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