Diffusion equations for populations evolution

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Table of contents

1 modeling evolutionary constraints at different scales 
1.1 Some philosophy (of science)
1.2 Two examples of constraints in evolution
1.3 Statistical mechanics offers a theoretical framework to study evolution
1.4 Thesis organization
i immune systems constrain the evolutionary paths of viruses 
2 pathogens against immune systems, an arms race across timescales
2.1 Background and motivation
2.2 Technical tools: stochastic processes and numerical simulations
2.2.1 Markov processes
2.2.2 Fokker-Plank and Langevin equations
2.2.3 Numerical simulations of stochastic processes
2.3 Conceptual tools: theoretical models of evolution and epidemiology
2.3.1 Diffusion equations for populations evolution
2.3.2 From genotypes to phenotypes to fitness: cross-reactivity in recognition space
2.3.3 Evolution in structured and fluctuating fitness landscapes
2.3.4 Traveling wave theory of adaptation
2.3.5 Epidemiological models
3 multi-lineage evolution in viral populations driven by host immune systems 
3.1 Abstract
3.2 Introduction
3.3 Methods
3.3.1 The model
3.3.2 Initial conditions and parameter fine-tuning
3.3.3 Detailed mutation model
3.4 Results
3.4.1 Modes of antigenic evolution
3.4.2 Stability
3.4.3 Phase diagram of evolutionary regimes
3.4.4 Incidence rate
3.4.5 Speed of adaptation and intra-lineage diversity
3.4.6 Antigenic persistence
3.4.7 Dimension of phenotypic space
3.4.8 Robustness to details of intra-host dynamics and population size control
3.5 Discussion
4 viruses phenotypic diffusion: escaping the immune systems chase 
4.1 Introduction
4.1.1 From the microscopic model to Langevin equations
4.1.2 Simplified description
4.1.3 Deterministic fixed points
4.2 Phenomenological model in phenotypic space
4.2.1 Fitness function
4.2.2 System’s scales
4.3 Numerical simulations
4.3.1 Implementation
4.3.2 Observables estimation — clustering analysis
4.3.3 Preliminary numerical results
4.4 Wave solution
4.4.1 Regulation of population size
4.4.2 Traveling wave scaling in phenotypic space
4.5 Adding other dimensions to the linear wave
4.5.1 Shape of viral dispersion
4.5.2 Lineage trajectory diffusivity in antigenic space
4.6 Conclusions and near future directions
ii infer evolutionary constraints at finer scales: proteins, evolution and statistical physics 
5 statistical physics for protein sequences 
5.1 Background and motivation
5.2 Statistical mechanics, inference and protein sequences
5.2.1 Canonical ensemble
5.2.2 Maximum Likelihood
5.2.3 Maximum Entropy principle and inverse Potts problem
5.3 Parameters and optimization
5.3.1 Boltzmann learning
5.3.2 Gauge invariance and regularization
5.4 General applications of DCA
5.5 Repeat proteins families
5.5.1 Repeat proteins
5.5.2 Global ensemble features of repeat proteins sequence space
5.5.3 Making sense of empirical patterns: repeats evolutionary model
6 size and structure of the sequence space of repeat proteins 
6.1 Abstract
6.2 Introduction
6.3 Results
6.3.1 Statistical models of repeat-protein families
6.3.2 Statistical energy vs unfolding energy
6.3.3 Equivalence between two definitions of entropies
6.3.4 Entropy of repeat protein families
6.3.5 Effect of interaction range
6.3.6 Multi-basin structure of the energy landscape
6.3.7 Distance between repeat families
6.4 Discussion
7 evolutionary model for repeat arrays 
7.1 Introduction
7.2 Model
7.2.1 Parameters inference
7.3 Results
7.4 Exploring mechanisms behind duplications and deletions
7.4.1 Multi-repeat duplications and deletions
7.4.2 Similarity dependent duplications and deletions
7.4.3 Asymmetric similarity dependence between duplications and deletions
7.5 The road ahead
7.5.1 Duplications bursts model
7.6 Conclusions
iii conclusions and future perspectives 
8 concluding remarks 
8.1 Discussion and conclusion
8.2 Future perspectives
8.2.1 Viral-immune coevolution
8.2.2 Protein evolution
a multi-lineage evolution in viral populations driven by host immune systems: supplementary information 
a.1 Simulation details
a.1.1 Initialization
a.1.2 Control of the number of infected hosts
a.2 Detailed mutation model
a.3 Analysis of simulations
a.3.1 Lineage identification
a.3.2 Turn rate estimation
a.3.3 Phylogenetic tree analysis
b size and structure of the sequence space of repeat proteins: supplementary information 
b.1 Methods
b.1.1 Data curation
b.1.2 Model fitting
b.1.3 Models with different sets of constraints
b.1.4 Entropy estimation
b.1.5 Entropy error
b.1.6 Calculating the basins of attraction of the energy landscape
b.1.7 Kullback-Leibler divergence
c evolutionary model for repeat arrays – supplementary information 
c.1 Dataset
c.2 Quasi-equilibrium
c.3 Numerical simulations
c.4 Parameters learning
c.5 Energy gauge for contacts prediction
c.6 Similarity dependent dupdel rates
c.6.1 Asymmetric duplications and deletions
c.7 Duplication bursts rates from model definition
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