Mathematical Population Genetics and Evolution of Bacterial Cooperation

Contents
Introduction
1. Prelude
1.1 Model approaches: from stochastic to deterministic models
1.1.1 Survival of one individual
1.1.2 Several individuals
1.1.3 Many individuals
1.1.4 Infinite population size
2. Neutral Evolution
2.1 Deterministic setting: Hardy–Weinberg model
2.2 Finite population size
2.2.1 Finite population size: Wright–Fisher
2.2.2 Moran model
2.2.3 Ecological versus evolutionary time
2.2.4 Invasion analysis for the Moran model
2.2.5 Coalescence and Ewens’s Sampling Formula in the Moran model
2.2.5.1 Ancestrial Process
2.2.5.2 Neutral mutations and the infinite allele model
2.2.5.3 Interpretation of data
2.2.5.4 Pairwise differences
2.2.5.5 Segregating Sites
2.2.5.6 Site-Frequency Spectrum
2.2.5.7 Ewens’s Sampling Formula
2.2.6 Excursion: Hubbells unified theory and the ESF
2.2.6.1 Etiennes refined sampling formula
2.2.6.2 Effect of fragmentation of rain forest on bat communities
2.3 Neutral evolution on graphs
2.3.1 Probability for fixation
2.3.2 Identity by descent
2.3.3 Island model
2.4 Neutral evolution on the lattice: the Voter Model
2.4.1 Mathematical toolbox: Particle processes
2.4.1.1 Random measures and random variables in Aг
2.4.1.2 Construction and elementary properties of particle processes
2.4.2 The Voter Model
2.5 Diffusion limit
2.5.1 Wright–Fisher model: from the time-discrete case to the SDE
2.5.2 Moran model — time-continuous case
2.5.3 Expected time to fixation
2.5.4 Variable population size and site-frequency spectrum
2.5.4.1 Constant population
2.5.5 Quiescent states: Seedbank model
2.5.5.1 Seedbanks as bet-hedging strategies
2.5.5.2 Consequences of seedbanks on the SFS
2.5.5.3 Formal analysis: small delay approximation
2.5.6 Moran with logistic population dynamics
2.5.6.1 Strategy for the analysis
2.5.6.2 Singular Approximation of the Fokker–Planck equation
3. Frequency-Independent Selection
3.1 Strong selection
3.1.1 Hardy–Weinberg model with selection
3.1.1.1 Directional selection
3.1.1.2 Balancing selection
3.1.1.3 Bistability: Disruptive selection
3.1.2 Strong selection in haploid, finite populations
3.1.2.1 Directional selection and clonal expansion
3.2 Weak selection
3.2.1 Weak selection and the Moran model
3.2.2 Diffusion limit and the SFS
3.2.3 Weak selection and seedbanks
3.2.3.1 Site-frequency spectra
4. Frequency-Dependent Selection
4.1 Hardy–Weinberg model and sickle-cell anemia
4.1.1 The Kermack–McKendrick model
4.1.2 Evolutionary host-pathogen interaction in sickle-cell anemia
4.2 Game Theory
4.2.1 Two player games and Nash equilibria
4.2.2 Pareto dominance
4.2.3 Cooperation and two-person games
4.2.3.1 Mutualistic cooperative games
4.2.3.2 The altruistic cooperative games
4.3 Replicator equation
4.3.1 Properties of the replicator equation
4.3.1.1 Two strategy-games
4.3.1.2 Replicator equation and Nash equilibria
4.3.1.3 Replicator equation and Lotka–Volterra systems
4.3.2 Cooperative Games
4.3.2.1 Mutualistic cooperative Games
4.3.2.2 Altruistic cooperative Games
4.4 Moran model with frequency-dependent selection
4.4.1 Fixation probability for a Markov Chain
4.4.2 Cooperation as weak effect
4.4.3 Probability for extinction
4.5 From the Moran model to the replicator equation
4.5.1 Interlude: Diffusion limit for a Markov process
4.5.2 Diffusion limit and connection with replicator dynamics
5. Multilevel Evolution and Price Equation
5.1 Interlude: Selfish genes, Haldane’s idea, and Hamilton’s rule
5.2 A first look on multilevel evolution
5.2.1 Invasion analysis
5.2.2 Simpson’s paradox and multilevel evolution
5.2.3 Interpretation of the relatedness r
5.3 Price equation
5.3.1 Notation
5.3.2 The Price equation
5.3.3 Fisher’s fundamental theorem of selection
5.3.4 Price equation, Prisoner’s Dilemma, and Hamilton’s rule
5.3.5 Multilevel evolution, kin and group selection
5.4 Relatedness and games on graphs
5.4.1 Game dynamics on a graph
5.4.2 Inclusive fitness and relatedness
5.4.3 Evolution on graphs — the full fitness approach
5.4.3.1 Back to the neutral case — relatedness
5.4.3.2 Hamilton’s rule, once again
5.4.3.3 Death-Birth dynamics
5.4.3.4 Birth-Death dynamics
6. Adaptive Dynamics
6.1 Evolution of virulence
6.2 The Prisoner’s Dilemma and adaptive dynamics
6.2.1 Additive model
6.2.2 Local interactions
7. Bacterial Cooperation
7.1 Proximate and ultimate cooperation
7.1.1 What is social behavior?
7.1.2 Basic forms of social interactions
7.1.3 Mechanisms improving the stability of cooperation
7.1.3.1 Cooperation and multilevel evolution
7.1.3.2 Stabilizing mechanisms
7.2 Division of labour and phaenotypic heterogeneity
7.2.1 Damage and plasmid segregation
7.2.1.1 Damage segregation
7.2.1.2 Unequal plasmid segregation
7.2.2 Switching environment (no sensing)
7.2.3 Switching environment (sensing)
7.3 Cooperation and quiescence/persister states
7.4 Kin recognition
7.4.1 Kin recognition and punishment
7.4.2 Quorum sensing, plasticity and bacterial cooperation
7.5 Cooperative and altruistic genes in plasmids
7.5.1 Plasmids and cooperation in large, homogeneous populations
7.5.2 Plasmids also mutate
7.5.3 Cooperating plasmids and multilevel evolution
7.6 Cooperation as hitchhiker on selfish straegies
7.6.1 Colicin A — multilevel setting
7.6.2 Colicin B — hitchhiking cooperation
7.7 Cooperative behavior between species
7.7.1 Mutualism in the homogeneous population
7.7.2 Generalized Hamilton’s rule
7.7.3 Cooperator recognition and punishment
7.8 Red Queen and Red King: Time scales in co-evolution
7.8.1 Red Queen
7.8.2 Red King
7.8.2.1 Randomly mixing population
7.8.2.2 Symbionts living within hosts
7.9 Cooperation and evolutionary suicide
Concluding Remarks
Appendix A Mathematical Tools
A.1 Some elementary terms from dynamical systems
A.1.1 ɷ-limit sets and Lyapunov functions
A.1.2 Long term behavior of linear dynamical systems
A.1.3 Singular perturbation theory in a nutshell
A.2 Basic stochastic terms
A.2.1 Random variables: definitions
A.2.1.1 Moments
A.2.1.2 Sum of random variables
A.2.1.3 Marginal distribution
A.2.1.4 Generating function
A.2.2 Random variables: conditional events and more
A.2.2.1 Conditioned probability
A.2.2.2 Independence
A.2.2.3 Law of total probability
A.2.2.4 Law of iterated expectations
A.2.2.5 Bayes Theorem
A.2.3 Random variables: important examples
A.2.3.1 Bernoulli Experiment
A.2.3.2 Binomial distribution
A.2.3.3 Poisson distribution
A.2.3.4 Geometric distribution
A.2.3.5 Multinomial distribution
A.2.3.6 Exponential distribution
A.2.3.7 Normal distribution
A.2.4 Asymptotic behavior, Chebyshev’s inequality
A.2.4.1 Approximations for Binomial and Poisson distributions
A.2.4.2 Law of large numbers, central limit theorem
A.2.4.3 Chebyshev’s inequality
A.3 Basic facts about stochastic processes
A.3.1 Markov Chain
A.3.2 Markov Process
A.3.2.1 General case
A.3.2.2 Waiting times
A.3.2.3 Irreducibility and invariant measure
A.3.2.4 Embedded jump chain
A.3.2.5 Absorbing states
A.3.3 Poisson Process
A.3.4 Stochastic differential equations — a brief primer
A.3.4.1 Diffusion limit
A.3.4.2 Transport equation and SDE
A.3.5 Stochastic coupling of processes
A.4 Martingale stopping theorem
A.5 Polyas theorem about random walks
A.6 Kakutanis Fixed point theorem
Appendix B Further Notes
B.1 The proof of Ewens’s Sampling Formula via induction
Appendix C Glossary
Bibliography
Index