réseaux sociaux

Master Mathématiques et applications | Computational and Mathematical Biology – CMB

RESPONSABLE

AIM

image externe

CMB is an international Master of science dedicated to the analysis and the modelling of complex biological systems, created as a part of the graduate school of the Turing Center of living systems (CENTURI).

It is shared by the departments of computer science, mathematics and biology, and aims at gathering students with a high potential coming from different backgrounds and at benefiting from the interdisciplinary synergies between training experts in that scientific area on the rise.

CMB offers a program by and for scientific research, with many opportunities for pursuing studies towards a PhD or obtaining a job in numerous companies having a research department.

On the mathematical side, the main courses are dedicated to the main mathematical tools for modeling (deterministic or stochastics models, dynamical systems) or the statistical tools for interpreting biological data (statistical test, parametric estimation). Other courses allow in parallel to develop skills in computing and biology.

LINKS WITH RESEARCH

La formation s’appuie sur les activités et les compétences des laboratoires de recherche membres du centre Turing des systèmes vivants (CENTURI).

RECOMMENDED PREREQUISITES

As this Master is a research-focused study path, a BSc in mathematics or statistics obtained with Honors is required.

WEB SITE

PLAQUETTE DE LA FORMATION

T�l�charger la plaquette de la formation au format PDF

PROFESSIONAL SKILLS

  • Acquiring the biological concepts required to model living systems.
  • Using and mastering the computational and mathematical methods and tools to elaborate and analyse models of biological systems (Ordinary differential equations, stochastic differential equations, partial differential equation, discrete dynamical systems).
  • Using and mastering the statistical methods and tools to analyse biological experimental data or help confronting mathematical models and biological systems.
  • Choosing and implementing the methodology adapted to the modelling of a given problem / biological system.
  • Evaluating the feasibility and the efficiency of a computational choice for the analysis of a complex system.
  • Collaborating to a research activity as a part of an interdisciplinary group.

PLAQUETTE DE LA FORMATION

T�l�charger la plaquette de la formation au format PDF

Master 1 CMB mentions Mathématiques appliquées, statistique et Mathématiques et applications (60 ECTS)

Semestre 1 CMB mentions Mathématiques appliquées, statistique et Mathématiques et applications (30 ECTS)

Fundamentals in biology 1 (3 ECTS)

CONTENT

This course Fundamentals in biology is divided in 2 parts taught during the 1st and 2nd semester of CMB. In the first semester, we will give a presentation of the evolutionary theories that have founded modern biology (from Lamarck to Darwin), and a synthesis of the discoveries that have led to current concepts of molecular and cellular biology : the role of macromolecules in cell function (information transfer between DNA, RNA, proteins, regulation, etc.), heredity and cellular adaptation.

Below are some topics covered during the lectures :

  • Information, evolution causes for living organisms
  • Cellular information
  • Epigenetics – phenomenons, information, adaptation, mechanisms

PDF brochure.

VOLUME OF TEACHINGS

  • Lectures: 18 hours

Professional perspectives for biological systems modelling (3 ECTS)

CONTENT

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.

PDF brochure.

VOLUME OF TEACHINGS

  • Tutorials: 18 hours

Discrete modelling of dynamical biological systems 1 (3 ECTS)

CONTENT

This course is an introduction to the basics of finite dynamic systems and PLC networks (definitions of local functions, global function/relationship, automata, interaction graph, transition graph) as well as the main static and dynamic properties. A part of the lecture will also focus on the parallel update mode.

The lectures should provide students the skills to 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, evaluate the adequacy between a biological question, available data, and mathematical formalisms and interpret and validate a study.

PDF brochure.

VOLUME OF TEACHINGS

  • Lectures: 6 hours
  • Tutorials: 6 hours
  • Pratical works: 6 hours

Programming & Algorithms (6 ECTS)

CONTENT

3 parts : 1) Unix : - file system and basic shell commands - text utilities - redirections, pipe 2) Programming language (Python) : - basic principles of imperative programming : variable, type, assignment, operators - control flow (conditional, iterations, loops) - basic data structures : tuple, list, dict - files (text, input/output) - local and external modules : math, random, numpy, scipy, matplotlib, pandas - functions 3) Algorithms : arrays, sorting, lists, stacks, queues

PDF brochure.

VOLUME OF TEACHINGS

  • Lectures: 12 hours
  • Tutorials: 12 hours
  • Pratical works: 12 hours

Computational Biology (3 ECTS)

CONTENT

Computational biology will introduce the biological concepts necessary 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 2 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

PDF brochure.

VOLUME OF TEACHINGS

  • Lectures: 14 hours
  • Pratical works: 16 hours

Probability (3 ECTS)

CONTENT

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.

PDF brochure.

VOLUME OF TEACHINGS

  • Lectures: 9 hours
  • Tutorials: 9 hours

Continuous dynamical systems, linear algebra and modelling (6 ECTS)

CONTENT

The purpose of this course is twofolds. The first part of the course will concern linear algebra tools that can be usefull in the studies of biological systems. Cite for instance the notions of

  • Iterative 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.

PDF brochure.

VOLUME OF TEACHINGS

  • Lectures: 12 hours
  • Tutorials: 12 hours
  • Pratical works: 12 hours

Functional analysis (3 ECTS)

CONTENT

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.)

PDF brochure.

VOLUME OF TEACHINGS

  • Lectures: 9 hours
  • Tutorials: 9 hours

Semestre 2 CMB mentions Mathématiques appliquées, statistique et Mathématiques et applications (30 ECTS)

Fundamentals of biology 2 (4 ECTS)

CONTENT

The course is in conitnuity with the course Fundamentals of biology 1. 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 ;
  • Nervous system

Some 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

PDF brochure.

VOLUME OF TEACHINGS

  • Lectures: 12 hours
  • Tutorials: 12 hours

Statistical and computational tools for biological modelling (6 ECTS)

Graph theory and algorithms 1

CONTENT

This introductory course focuses on graphs as mathematical objects and some of its uses to solve applications to biological networks. After intruducing different classes of graphs and their properties, the following points will be developped :

  • Planar graphs, graphs on a surface, Euler characteristic
  • Interval graphs, perfect graphs

PDF brochure.

VOLUME OF TEACHINGS

  • Lectures: 6 hours
  • Tutorials: 6 hours
  • Pratical works: 6 hours

Statistics for biology

CONTENT

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, …)

PDF brochure.

VOLUME OF TEACHINGS

  • Lectures: 6 hours
  • Tutorials: 6 hours
  • Pratical works: 6 hours

Advanced statistics (3 ECTS)

CONTENT

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

PDF brochure.

VOLUME OF TEACHINGS

  • Lectures: 6 hours
  • Tutorials: 6 hours
  • Pratical works: 6 hours

Hilbert and Fourier analysis (3 ECTS)

CONTENT

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

PDF brochure.

VOLUME OF TEACHINGS

  • Lectures: 6 hours
  • Tutorials: 6 hours
  • Pratical works: 6 hours

Research project and scientific communication (6 ECTS)

CONTENT

At the end of the courses Professional perspectives for biological systems modelling , and Fundamentals of biology 1,, students will choose a scientific article at the interface of several disciplines on which they will work in groups. They will have to present in a memory and an oral presentation, to explain the biological context and the related basic concepts, to explain the methods used to interpret the biological data, to synthesize the results obtained in the article.

PDF brochure.

CenTuri Seminars (2 ECTS)

CONTENT

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.

PDF brochure.

VOLUME OF TEACHINGS

  • Lectures: 12 hours

Option: mathematics or statistics (6 ECTS)

EDP et analyse numérique (6 ECTS)

Unavailable contents.

VOLUME OF TEACHINGS

  • Lectures: 24 hours
  • Tutorials: 36 hours

2 options statistics S2 (6 ECTS)

Option: statistics (6 ECTS)

Statistique décisionnelle (3 ECTS)

Unavailable contents.

VOLUME OF TEACHINGS

  • Lectures: 15 hours
  • Tutorials: 15 hours

Signaux à temps discret, modèles déterministes et aléatoires (3 ECTS)

Unavailable contents.

VOLUME OF TEACHINGS

  • Lectures: 15 hours
  • Tutorials: 15 hours

Chaînes de Markov, martingales (3 ECTS)

Unavailable contents.

VOLUME OF TEACHINGS

  • Lectures: 15 hours
  • Tutorials: 15 hours

Séries chronologiques (3 ECTS)

Unavailable contents.

VOLUME OF TEACHINGS

  • Lectures: 10 hours
  • Tutorials: 6 hours
  • Pratical works: 8 hours

Master 2 mentions Mathématiques appliquées, statistique et Mathématiques et applications (60 ECTS)

Semestre 3 CMB mentions Mathématiques appliquées, statistique et Mathématiques et applications (30 ECTS)

Biological interaction networks (3 ECTS)

Unavailable contents.

VOLUME OF TEACHINGS

  • Lectures: 10 hours
  • Tutorials: 8 hours
  • Pratical works: 12 hours

Statistical inference and big data in biology (3 ECTS)

CONTENT

This course is an introduction to inferential statistics. It will be illustrated with biogical examples. The course will be composed by three parts

  • Multiple tests
  • Classification
  • Time series analysis

PDF brochure.

VOLUME OF TEACHINGS

  • Lectures: 6 hours
  • Tutorials: 6 hours
  • Pratical works: 6 hours

Optimization and numerical calculus (3 ECTS)

CONTENT

The purpose of this course is to investigate the fundamental notions in the optimization theory with or without constraints. The theory will be illustrated in the case of regression problems. The outline of the course is the following :

  • The classical results on optimization problems with or without constraints
  • Application to the problems of approximation
  • Examples of nonlinear regression

PDF brochure.

VOLUME OF TEACHINGS

  • Lectures: 6 hours
  • Tutorials: 6 hours
  • Pratical works: 6 hours

Interdisciplinary project in modelling (6 ECTS)

CONTENT

Following the module Research project and scientific communication, 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.

PDF brochure.

CenTuri Seminars 3 (3 ECTS)

CONTENT

Following the module Centuri seminars 2, the students will attend all the Centuri seminars of this semester. For two of them, they will be asked to broaden their knowledges on the subject and present an oral and written synthese. The work will be done in collaboration.

PDF brochure.

VOLUME OF TEACHINGS

  • Lectures: 10 hours
  • Tutorials: 10 hours
  • Pratical works: 10 hours

Advanced biology (2 options among 3) (6 ECTS)

Advanced Biology : Developmental Biology (3 ECTS)

Unavailable contents.

VOLUME OF TEACHINGS

  • Lectures: 6 hours
  • Tutorials: 6 hours
  • Pratical works: 6 hours

Advanced Biology : Immunology (3 ECTS)

Unavailable contents.

VOLUME OF TEACHINGS

  • Lectures: 6 hours
  • Tutorials: 6 hours
  • Pratical works: 6 hours

Advanced biology : Neurobiology (3 ECTS)

Unavailable contents.

VOLUME OF TEACHINGS

  • Lectures: 6 hours
  • Tutorials: 6 hours
  • Pratical works: 6 hours

S3 Options (6 ECTS)

Introduction à la logique modale (3 ECTS)

Unavailable contents.

VOLUME OF TEACHINGS

  • Lectures: 20 hours

Modèles de calcul naturel - MOCANA (3 ECTS)

Unavailable contents.

VOLUME OF TEACHINGS

  • Lectures: 20 hours

Modèles à variables latentes (3 ECTS)

Unavailable contents.

VOLUME OF TEACHINGS

  • Lectures: 12 hours
  • Tutorials: 12 hours

Algorithmes stochastiques (3 ECTS)

Unavailable contents.

VOLUME OF TEACHINGS

  • Lectures: 12 hours
  • Tutorials: 12 hours

Signal et big data, compressed sensing (3 ECTS)

Unavailable contents.

VOLUME OF TEACHINGS

  • Lectures: 12 hours
  • Tutorials: 12 hours

Data science: Big Data (3 ECTS)

Big data

Unavailable contents.

VOLUME OF TEACHINGS

  • Lectures: 12 hours
  • Tutorials: 18 hours

Data science: Sparse representation of signals, images, graphs (3 ECTS)

Représentations parcimonieuses de signaux et images

Unavailable contents.

VOLUME OF TEACHINGS

  • Lectures: 12 hours
  • Tutorials: 18 hours

Méthodes numériques probabilistes (3 ECTS)

Unavailable contents.

VOLUME OF TEACHINGS

  • Lectures: 12 hours
  • Tutorials: 12 hours

Méthodes d'estimation paramétrique (3 ECTS)

Unavailable contents.

VOLUME OF TEACHINGS

  • Lectures: 12 hours
  • Tutorials: 9 hours

Calcul scientifique (3 ECTS)

Unavailable contents.

VOLUME OF TEACHINGS

  • Lectures: 8 hours
  • Tutorials: 15 hours

Calcul scientifique avancé (3 ECTS)

Unavailable contents.

VOLUME OF TEACHINGS

  • Lectures: 8 hours
  • Tutorials: 15 hours

Semestre 4 CMB Computational and Mathematical Biology (30 ECTS)

Master thesis (30 ECTS)

CONTENT

Internship in research laboratories or in companies.

PDF brochure.

FURTHER INFORMATION

Secretariat :

  • Sonia KERBACHE (sonia.kerbache@univ-amu.fr) tél. : 04 91 82 90 80, Aix-Marseille Université, Campus Luminy. Bâtiment TPR2 ET GRAND H. 163 av. de Luminy. case 901. 13009 Marseille.

HOW TO APPLY AND REGISTER

https://formations.univ-amu.fr/xml/PRSMA5AF-en.xml