The Germinal Center Reaction

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Table of contents

1 The biology of affinity maturation, open questions and the contribution of models 
1.1 Introduction and chapter outline
1.2 The biology of Affinity Maturation
1.2.1 Structure and function of Antibodies
1.2.2 The Germinal Center Reaction
1.2.3 Differentiation into Memory and Plasma Cells
1.2.4 Vaccination
1.3 Open questions and the role of models
1.3.1 The role of models in understanding AM
1.3.2 The effect of Antigen dosage
1.3.3 GC selection and mechanisms of affinity discrimination
1.3.4 Maturation in the presence of complex Ag, selection permissiveness and homogenization
1.3.5 Immunizing against mutable pathogens
2 Effects of ag dosage: modeling and data analysis 
2.1 Introduction and chapter outline
2.2 Foreword: Bayesian inference
2.3 The experimental dataset
2.3.1 Experimental technique
2.3.2 Immunization schemes
2.3.3 Qualitative observations: the influence of Ag dosage and injection delay
2.4 Stochastic model of AM to predict affinity distributions
2.4.1 Antigen dynamics
2.4.2 GC affinity maturation
2.4.3 GC reinitialization
2.4.4 Elicited Ab-SCs
2.4.5 Three model variants
2.4.6 Values of model parameters
2.4.7 Example of model evolution at high and low Ag dosage
2.5 Model deterministic limit
2.5.1 Limit of big population size
2.5.2 Deterministic model reproduces average of stochastic simulations
2.6 Inferring model parameters
2.6.1 The likelihood function
2.6.2 Numerical likelihood maximization through Parallel Tempering
2.6.3 Comparison between model variants
2.6.4 Consistency check through artificial data generation
2.6.5 Inferred model reproduces data
2.7 Analysis of deterministic model offers insight on the effect of Ag dosage
2.7.1 Asymptotic travelling wave behavior under constant Ag concentration
2.7.2 Eigenvalue equation
2.7.3 Ag concentration determines different maturation regimes
2.8 Inference as a tool to investigate AM mechanisms
2.8.1 Degree of permissiveness in GC selection
2.8.2 Maturation with and without loss of clonality
2.8.3 Maturation as combination of beneficial mutations and selection of high-affinity precursors
2.8.4 The relative contribution of Ag-binding and competitive selection
2.8.5 Fractions of PCs and MCs amongst Ab-SCs
2.9 Conclusion
2.9.1 Summary and significance
2.9.2 Model limitations and discussion
2.9.3 Outlooks
3 Stochastic effects in maturation model: survival, lineages, competition
3.1 Introduction and chapter outline
3.2 Simplified model for Affinity Maturation under a population bottleneck
3.2.1 model definition
3.2.2 Qualitative model behavior: bottleneck and lineages
3.2.3 Limit of big population size
3.2.4 Continuous time description
3.3 Maturation speed and growth rate
3.3.1 Traveling-wave asymptotic solution
3.3.2 Dependence of growth rate and maturation speed on model parameters
3.4 Probability of survival to bottleneck
3.4.1 Lineage extinction probability and extinction time in a population bottleneck
3.4.2 Lineage size at extinction
3.4.3 Probability of bottleneck survival for the full population
3.5 Most-likely evolutionary trajectory
3.5.1 Path integral formulation
3.5.2 Method of characteristic trajectories
3.5.3 Action and trajectories in a simplified case: no competition and no silent mutations
3.5.4 Evolutionary trajectories with and without competition
3.6 Conclusion and perspectives
4 Perspectives – microscopic b-t cell interactions 
4.1 Introduction
4.2 Microscopic model for B-T cell interaction
4.2.1 Microscopic mechanism of B-T cell interaction
4.2.2 Maturation in the independent case
4.2.3 Maturation in the linear case
4.2.4 Maturation in the mixed case
4.3 Conclusion and perspectives
5 Perspectives – breadth acquisition in single ag immunization 
5.1 Introduction
5.2 Breadth acquisition in single-Ag vaccination
5.2.1 Model extension to multiple antigens
5.2.2 Effect of one maturation round on breadth
5.2.3 Effect of Ag concentration and Ag mutability on breadth acquisition
5.3 Conclusion and perspectives
a Appendix – chapter 2 
a.1 Numerical model implementation and parameters choice
a.2 Small variations of the standard model
a.3 Initial conditions and stochasticity
a.4 Role of a, b parameters
a.4.1 Number of accumulated mutations
b Appendix – chapter 3 
b.1 Parameters choice
b.2 Critical tree size and extinction time in the absence of mutations
b.3 Stochastic evolution of the competitive selection pressure and finite-size correction for evolutionary trajectories
c Appendix – chapter 4 
c.1 Pertubative analysis of binding probability master equation