Math modelling major
Master 1 courses
1st Semester
Shared courses
Organisation: TD (18h)
Lecturers: Julien Lefèvre
Evaluation: projects and oral presentation
The goal of this course is to introduce students to multidisciplinary research topics through seminars and visits to research laboratories and private sector companies.
The teaching unit consists of a presentation of the main professions involved in biological modelling. It will be carried out in two ways. First of all, some seminars will be offered with speakers from outside Aix-Marseille University in the academic and industrial field. Secondly, students will benefit from an immersion in Centuri laboratories where they will discover multidisciplinary research topics. To conclude this unit, students will be asked to present a specific problem related to data processing and modelling. The scientific aspect will also have to be integrated into a reflection on the underlying professional issues, whether in the academic or private sector.
Organisation: Lectures (6h), TD (6h), TP (6h)
Lecturers: Brigitte Mossé, Elisabeth Remy
Evaluation: projects and final written exam
This UE presents the finite Boolean dynamical systems, that are mathematical tools more and more used in the field of the modelling of biological regulatory networks.
Prerequisites in logics, set theory, graph theory and Boolean finite dynamical systems are provided, as well as definitions of the different possible associated dynamics, the corresponding regulatory graphs and logical formula. The rôle of feedback circuits in the dynamics is specially emphasised.
Applications of these tools for the modelling of biological networks are presented, mainly in context of diseases. The practical lessons (TP) are done using GINsim, a free software dedicated to the logical modelling of regulatory networks.
Organisation: Lectures (18h)
Lecturers: Laurence Röder, François Muscatelli
Evaluation: final written exam
This unit is the first part of the Fundamentals of biology course. In this first part, we will give a general presentation of the role of macromolecules in cell function, particularly in the regulation of gene expression, epigenetic inheritance and speciation as a source of biological diversity during evolution.
Organisation: Lectures (14h), TP (16h)
Lecturers: Laurent Pézard, Michael Kopp, Andreas Zanzoni
Evaluation: projects and final written exam
Computational biology introduce the biological concepts needed to model complex systems, implement modelling approaches (differential, logical, stochastic or deterministic equations) to develop mathematical models of a biological system, analyze mathematical models and biological data to understand complex systems and assess the adequacy between a biological question.
The course is divided in 3 sections:
- Computational neuroscience: dynamic models of neuron function: dynamic behavioural simulation, biological aspects, computer complexity, analytical aspects
- Bioinformatics: alignment, molecular phylogeny, prediction and modelling of structural aspects of proteins, cis-regulation
- Evolution and population dynamics.
Organisation: Lectures (12h), TD (12h), TP (12h)
Lecturers: Julien Lefèvre
Evaluation: continuous exams
This course recalls the general basics of programming, implemented here with Python.
It also provides the essential elements for a modern practice of programming in the Python language: use of an IDE, version control, use of existing modules, good practice for coding. Part of this course is also devoted to some classic algorithms for sorting and manipulating current data structures as well as algorithms dedicated to bioinformatics.
Mathematics courses
Organisation: Lectures (9h), TD (9h)
Lecturers: Olivier Guès
Evaluation: continuous monitoring and final written exam
The purpose of this course is to deepen some notions of functional analysis that can be encountered in the study of mathematical models of biological problems. For instance:
- Banach spaces. The notion of norm on a vector space. Convergent sequences and notion of Banach space. Fixed point Theorem.
- Examples in finite dimension. Completeness. Notion of compactness and the link with closed and bounded subsets. Illustrations in biology : problems of optima. Behavior of recursive sequences. Fixed points of discrete dynamical systems. Stability/instability of a fixed point.
- Examples in infinite dimension. Examples of classical functional spaces. Spaces of continuous functions on a compact set, C(K,Rn), H1([0,1]), spaces of periodic functions. Linear continuous operators.
- Examples of applications of the fixed point theorem. Illustrations in Biology.
- Study of functional equations, integral equations. Examples from Biology.
- Study of density and approximation property (Dirichlet Theorem, Gibbs phenomenon, Theorems of Fejer and of Weierstrass.).
Organisation: Lectures (9h), TD (9h)
Lecturers: Christophe Gomez
Evaluation: continuous monitoring and final written exam
The purpose of this course is to revise and deepen the fundamental concepts in probability. More precisely, the following concepts will be discussed:
- Dependence, law, expectancy, conditional density
- Generating functions
- Markovian models
- Branching, Poisson, birth and death processes.
Organisation: Lectures (12h), TD (12h), TP (12h)
Lecturers: Florence Hubert
Evaluation: projects and final written exam
The purpose of this course is twofold. The first part of the course will concern linear algebra tools that can be useful in the studies of biological systems. Cite for instance the notions of
- Interative methods, rate of convergence of vectorial sequences
- Matrix reduction, Power method and Perron-Frobenius theorem
- Linear regression; analysis of variance, Principal Component Analysis; Singular Values Decomposition
The second part of the course will be devoted to revision and deepening of the notion and properties of differential equations and systems of differential equations which underlie the main continuous models used in biology (dynamics of populations or cells, biochemical processes, etc.). We will address both qualitative (existence, global existence, equilibria, stability of the equilibria, long-time behavior) and quantitative (positivity, parameter dependency) properties of the considered models.
In parallel to this theoretical study, numerical approximation will be studied and implemented during the computer sessions. Practicals will consist in using Python specialised libraries as scipy.integrate in order to visualise trajectories and systems behaviours.
2nd Semester
Shared courses
Organisation: Lectures (12h)
Lecturers: Bianca Habermann, Alphée Michelot, Laurence Röder
Evaluation: project and continuous monitoring
Scientific seminars constitute a good way to broaden your scientific horizon. In this regard, MSc students will frequently attend CENTURI seminars. At the end of the semester, students will be asked to write a summary of two seminars they have attended.
Organisation: Lectures (12h), TD (12h)
Lecturers: Laurence Röder, Françoise Muscatelli
Evaluation: final written exam
The second part of this module will show how these molecular mechanisms underlie the development and functioning of tissues and organisms. It will be structured around four areas: intergenerational transmission of traits; organism development; immune system and nervous system.
Examples topics covered during the lectures:
- Information and organization: intergenerational transmission (cells, organisms Information, evolution causes for living organisms)
- Organisms’ development
- Information and organization of the immune system
- Information and organization of the nervous system
Organisation: Lectures (6h), TD (6h), TP (6h)
Lecturers: Annie Broglio
Evaluation: projects and final written exam
This course aims at providing students with a practical approach of the analysis of biological data with R, based on the concepts acquired in the course “Probabilities and statistics for modelling 1”. The associated mathematical foundations will be developed in the course “Advanced statistics”. The following notions will be investigated:
- Sampling and estimation (moments, robust estimators, confidence intervals)
- Fitting
- Additional distributions
- Hypothesis testing (mean comparison, goodness of fit, …)
The course will be based on the analysis of biological datasets with the R programming language.
Lecturers: Florence Hubert, Laurence Röder
Evaluation: project
Following the module [PROJ1], the students will do a short internship in laboratory. They will have to propose a modelling or data processing problem at the math-info-bio interface. They will be asked to synthesize their results in a dissertation and an oral presentation.
Organisation: Lectures (6h), TD (6h), TP (6h)
Lecturers: Jean-Marc Freiermuth
Evaluation: projects and final written exam
This course will tackle advanced notions in statistics such as:
- Statistical inference (fundamental concepts, estimators, intervals and tests, quadratic error, bias and variance)confidence
- Likelihood (Fisher information, likelihood ratio test)
- Exponential family
- Convergence
- Multivariate Gaussian distributions
Prerequisite for CMB-B:
- [STAT1] Probabilities and statistics for modelling 1
- [STAT2] Statistics for biology
The R software will be used in the practicals.
The R software will be used in the practicals.
Mathematics courses
Organisation: Lectures (6h), TD (6h), TP (6h)
Lecturers: Julien Olivier
Evaluation: project and final written exam
Prerequisites:
- Continuous dynamical systems, linear algebra and modelling
- Functional analysis
The purpose of this course is to revise and deepen the fundamental concepts of Fourier and Hilbert analysis and to illustrate the different notions through applications in biology. The following concepts will be investigated:
- Fourier series
- The classical theorems Dirichlet, Fejer, Plancherel Parseval
- Regularity and decrease of Fourier coefficients
- Fast Fourier Transform
- Application to the representation of biological signals in neuroscience
- Fourier Transform
- Reminders of the main results
- Applications to elliptic equations in unbounded domain arising in biology
- Hilbert spaces
- Hilbertian bases, examples
- Galerkin approximation
- Application to elliptic equations in bounded domain arising in biology
Practicals in Python will illustrate the different notions.
Partial differential equations and numerical analysis
Organisation: Lectures (12h), TD (9h), TP (9h)
Lecturers: Florence Hubert
Evaluation: project and final written exam
Prerequisites:
-
- [MODALG] Continuous dynamical systems, linear algebra and modelling
- [FUNA] Functional analysis
- [HFO] Hilbert and Fourier analysis
The purpose of this course is to introduce students to partial differential equations and their numerical approximation using only elementary techniques (Fourier series, differential equations, integration).
The starting point will be PDE and the qualitative properties of these equations.
The question of numerical approximation will also be developed through finite difference schemes around the concepts of convergence, order, stability and consistency. Practicals in Python will illustrate both the qualitative properties of equations and the concepts related to numerical approximation.
Pedagogical Approach:
Lectures will provide a summary of basic concepts, which will be applied in practicals.