M2 Evolution of Genomes, Populations and Species: Data and Models (EvoGEM)

This programming course is intended for students from any background.

Concepts covered include navigating and managing files under Unix, executing commands on the command line, and automating repetitive tasks. In Python, fundamental syntax, control structures (conditions and loops) and common data structures such as lists, tuples and dictionaries will be taught, as well as basics of programming in R, focusing on data analysis and manipulation, the use of packages, and the creation of reproducible scripts.

This course introduces students to the basics of computer programming through the use of three essential environments: Unix, Python and R. This course aims to develop a practical understanding of programming tools and to foster autonomy in solving common computing problems.

Location: Sorbonne Université, Campus Jussieu (Paris 5)

Expected mathematical proficiency after completing an M1 in Biology and a liking for and interest in mathematical formalism.

Linear Algebra: Notions of matrices and vectors, matrix calculations, usual matrix properties, inversion, diagonalization.

Probability Basics: Random variables and common probability distributions, expected value and variance, convergence in distribution, central limit theorem, and law of large numbers. Properties of random vectors and matrices, Markov chains.

Statistics: Inference, likelihood, linear models, multivariate analyses (PCA, DFA, …), introduction to Bayesian approaches.

Ordinary Differential Equations: Review of derivatives and integrals, concept of differential equations, solving a simple linear equation, system of differential equations, notion of dynamic equilibrium in systems.

This teaching unit aims to consolidate/revise basic knowledge in mathematics and statistics. The objective is to recall the use of mathematical formalisms, especially matrix notation, review fundamental elements of probability and statistical inference, introduce Bayesian approaches, and revise basic concepts in the treatment of ordinary differential equations.

Location: Sorbonne Université, Campus Jussieu (Paris 5)

There is no pre-requisite for this course.

Genetic basics – 2 h
Reproduction & heredity – 2h
Metabolism & Developpement- 2h
From genes to genomes – 3h
Regulation of gene expression – 2h
Systematic & biodiversity – 1h30
Conference – 2h

This biology refresher course is intended for students from Mathematics and Computer Science backgrounds. It aims to acquire and/or consolidate basic knowledge in cellular and molecular biology, developmental biology, genetics, and genomics. We also illustrate, through examples, how data derived from ‘omics’ technologies can address fundamental questions in evolutionary biology, as well as current challenges related to health and threats to biodiversity.

Location: Sorbonne Université, Campus Jussieu (Paris 5)

To propose a general framework for work in evolutionary biology, starting from a historical and epistemological approach and using examples from contemporary work to address the major issues of the 21st century. To understand how theories and models shape the constitution of data sets and, conversely, how data challenges established theories and models.

Location: Museum National d'Histoire Naturelle (Paris 5)

– Biology basic knowledge.
– Genetics & mathematics fundamentals.

– Evolution of genetic polymorphism at one locus in a neutral Wright-Fisher model.
– Consequences of diploidy and influence of mating systems.
– Multilocus polymorphism and recombination.
– Bayesian statistics in population genetics
– Polymorphism in meta-populations and speciation.
– Evolution of polymorphism under selection.
– Introduction to molecular evolution.

This course will address the fundamental processes that are at work in evolution through the lens of population genetics and molecular evolution. It will devote time to statistical inferences based on the patterns of polymorphisms in sequence data. The intended participants are students motivated by the understanding of evolutionary processes and not entirely reluctant to theoretical models.

Location: Ecole Normale Supérieure PSL and/or Collège de France (Paris 5)

To be able to follow this course, there must be an understanding of Mendelian genetics and population genetics. Familiarity with linear models, probability theory, matrix algebra, R and/or is highly recommended.

Heritability, additive genetic variance, infinitesimal model, relatedness, dominance, epistasis, linear mixed models, animal model, selection gradients and intensity, G matrix, M matrix, adaptive landscapes, genome-wide association analysis, genomic prediction, polygenic scores, polygenic adaptation

This course offers an in-depth exploration of quantitative genetics, building on foundational knowledge in biology, mathematics, and genetics. It focuses on the genetic basis of complex traits influenced by many genes. Students will be introduced to core quantitative genetic models, including the partitioning of phenotypic variance and the concept of heritability. Key statistical approaches will be covered, such as the use of animal models to estimate variance components, and theoretical frameworks like the Price-Robertson identity to understand selection and evolutionary responses. The course will also delve into advanced topics including genomic selection, genome-wide association studies (GWAS), and multivariate evolution through the study of the G-matrix, M-matrix, and adaptive landscapes. Emphasis will be placed on the application of quantitative genetic tools in interpreting experimental evolution and investigating adaptation in natural populations, particularly in the context of global environmental change. A significant portion of the second week will be dedicated to hands-on mathematical and computational modeling of quantitative trait evolution, guided through tutored projects.

Location: Université Paris-Saclay, site IDEEV (Gif-sur-Yvette)

Biology: basic knowledge of evolutionary processes (mutation, selection, genetics) and of biodiversity description (taxon, species, population).
Probability: Markov chains, likelihood

• Database and alignments
• Phylogenetic inference : distances, maximum parsimony, maximum likelihood, bayesian inference, dating a phylogeny
• Phylogenomics and comparative genomics
• Post-tree analysis : species delimitation, ancestral state reconstruction and trait evolution, correlations, detecting selection, phylodynamics, diversification models
• Beyond trees : reticulated evolution, hybridization and introgression

The aim of this module is to cover advanced notions in phylogenetics, from simple tree reconstruction algorithms to more complex phylogenomic methods. A great deal of the module will also cover post-phylogenetic analyses, such as trait evolution, diversification models or introgression detection.

Location: Université Paris-Cité, site Grands Moulins (Paris 13)

Math 0 and MAEE (Stochastic Models in Ecology and in Evolution)

Lecture 1 (AL): Cannings model, coalescent, infinite-site model, infinite-allele model, Ewens sampling formula, Chinese restaurant process.

Lecture 2 (SR): Multivariate methods: dimension reduction (PCA), clustering (hierarchical, k-means, mixture models).

Lecture 3 (AL): Brownian motion, diffusion theory, stochastic differential equations, infinitesimal generator, hitting probabilities, Feller diffusion, Wright-Fisher diffusion.

Lecture 4 (AL): Coupling gene genealogies: ancestral recombination graph, sequential Markov coalescent.

Lecture 5 (TMH): Mixed models : Random effects, ReML inference, hypothesis testing for variance components and random effect prediction (BLUP).

Lecture 6 (TMH): GWAS : Oligogenic and polygenic models, random effect model accounting for relatedness, multiple testing.

Lecture 7 (SR): State space models: hidden Markov models (HMM), STRUCTURE, sequence evolution models.

Lecture 8 (AL): coalescent of birth-death processes, Luria-Delbrück distribution, 1/k² SFS.

Lecture 9 (AL): Trait evolution on a tree, phylogenetic covariance matrix.

The aim of this module is to deepen the mathematical concepts and tools covered in the other modules

Classes within schedule of other UEs.

Lecturers:

Amaury Lambert (AL), Stéphane Robin (SR), Tristan Mary-Huard (TMH)

Schedule:

Lectures are spread over the semester. Six lectures out of 9 are scheduled in the middle of a fortnight dedicated to one of the following modules: GPS (Genomes, Populations, Species), GQ (Quantitative Genetics), ACP (Comparative Phylogenetic Analysis), TP (Tutored Projects). When this is the case, the content of the lecture is in tight relation with the surrounding module (except TP), and the classroom is the one used for this module

• Prerequisites:
• A student enrolling in the course should know:
• what a matrix is and the elementary operations that apply to it (multiplication, diagonalization, inversion);
• what a discrete and continuous random variable is, and the related concepts (law, density, distribution function, expectation, variance);
• Knowledge of deterministic models of evolution, notably based on ordinary differential equations, is a plus;
• For those who are interested in the course but for whom these notions are unknown or remote, don't panic: you can acquire this knowledge by first taking the DYST module.\

Contents:

Introduction to classical probabilistic dynamic models in ecology and evolution, i.e. Markov processes:

• in discrete time and space (Markov chains);
• in continuous time and discrete space (Poisson processes, birth and death);
• in discrete time and continuous space (Gaussian random walk);
• in discrete time and continuous space (Brownian motion and stochastic differential equations);

Goals:

At the end of the module, the student will be able to:

• Define classical probabilistic dynamical models (random walks, birth-and-death processes, coalescent, Brownian motion);
• Deduce Markov chain characteristics (ergodicity, stationarity, periodicity) in discrete time and space from its transition law;
• Numerically implement a simulation code (exact or approximate) of a Markov process to study its properties empirically;\

Bio0/Maths0/Info0 and EVO modules

The aim of this cross-disciplinary module is to introduce students to working with data and models by means of a tutored project

Location: Museum National d'Histoire Naturelle, Paris 5