Probability of bottleneck survival for the full population 

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Qualitative observations: the influence of Ag dosage and injection delay

For each tested mouse our experimental apparatus measures the full affinity distributions of Ab-SCs (full distributions of pooled data are displayed in later sections in fig. 21).
In fig. 11 we report the average binding energy of these distributions for single mice (orange crosses) and for pooled data from mice tested under the same condition (orange empty dots). These are reported as a function of the varied quantity, either Ag dosage D or injection time delay T, depending on the scheme considered. Notice that the number of mice employed per condition can vary (the number for each condition is reported in
fig. 21). As a first qualitative observation one notices that in scheme 1 the average affinity seems to be maximal (i.e. minimum binding energy) at intermediate values of the Ag dosage D.
The same behavior can be observed for scheme 2, even though in this case data are more noisy. From the results on scheme 3 instead it appears that delaying the second injection of some weeks is beneficial compared to administering shortly after the first. To interpret these behaviors we introduce in the next section a model for affinity maturation.

stochastic model of am to predict affinity distributions

Here we introduce a stochastic model for Affinity Maturation, which takes inspiration from [155, 154], with the aim of quantitatively explaining the experimental data. The model describes the evolution of a population of B-cells inside a Germinal Center. These cells are subject to cycles of duplication, mutation, selection and differentiation, repro- ducing the biological process described in section 1.2. In our model each cell is characterized solely by the binding affinity of its B-cell Receptor (BCR) for the antigen. The aim of the model is to follow the evolution of the distribution of affinities of cells in the population during the maturation process. This is important for comparison with experimental data, which themselves consists in affinity distributions.
For the sake of readability we limit here the description to the model definition, and move the discussion about the detailed numerical implementation and choice of parameters value from the literature in appendix A.1. All parameters values are also reported in table 1.

Antigen dynamics

The first model ingredient we describe is the dynamics of Ag concentration C inside of the simulated GC. In the model the amount of Ag available controls the strength of selection, and evolves during the GCR under the action of different mechanisms, which are schematized in fig. 12A.
At the time of injection an amount Cinj of Ag is added to the Ag reservoir. This amount is related to the injected dosage D by a proportionality constant D = Cinj. We express as a mass, which makes concentration dimensionless. The reservoir is constituted by Ag trapped in the adjuvant matrix, from which it is quickly released at a rate k+ and becomes available for cells to bind. Due to recycling of Ag from surface of Follicular Dendritic Cells (FDCs) to endosomal compartments [52, 90] available Ag decays at a slow rate k- ?, and is consumed by B-cells at a faster rate, k- B NB, proportional to the number NB of B-cells in the GC. As the amount of Ag is depleted, selection of B-cell is more and more stringent, and the GC eventually dies out. The evolution of the reservoir Ag concentration Cres and the available Ag concentration Cav are regulated by the following pair of equations: d dt Cres(t) = -k+Cres(t) (4) d dt Cav(t) = k+Cres(t) – (k- ? + k- BNBt ) Cav(t).

Three model variants

In our analysis we will compare three different variants of the model, which differ only by the way selection is performed.
• In variant (A) B-cells are selected according to the two-step process consisting of Ag-binding selection and competitive selection for T-cell help, i.e. eqs. (6) and (7).
• In variant (B) we consider the same two-step selection but without permissiveness, i.e. we set a = b = 0.
• In variant (C) we neglect Ag-binding selection and consider only competition for T-cell help, i.e. eq. (7), but allowing for permissiveness.
Notice that these variants contain a different number of model parameters, which must be taken into account when comparing their respective likelihood. We chose to compare these variants to investigate how important permissiveness and Ag-binding selection (excluded respectively in model variants B and C) are to quantitatively fit the data.

Values of model parameters

The values of all but nine model parameters were extracted from existing literature, see description in appendix A.1 and table 1. The remaining nine parameters, which were either not precisely known or strongly dependent on our experimental protocol, were fitted from the experimental data through a Maximum-Likelihood inference procedure described in section 2.6. This procedure was executed on each of the three model variants. The nine inferred parameters are:
• the conversion factor , which allows for conversion between experimental administered Ag dosage D, measured in micrograms, and the dimensionless administered Ag concentration of our model, C = D=.
• the Ag consumption rate per B-cell k- B, which controls the GC lifetime and also the extent of the affinity maturation.
• the mean naive and variance 2 naive of the Gaussian binding energy distribution for the GC seeder clones, elicited directly from the naive population.
• the binding energy threshold Ag for a B-cell to be able to bind Ag with sufficient affinity to internalize it (cfr eq. (6)). This parameter does not appear in model variant C, where only competitive selection for T cell help is implemented.
• The T-cell selection characteristic coefficients, a and b, encoding respectively the baseline probabilities to survive or not survive selection, see eq. (7) and fig. 14D.
• The weight parameters grecall, gimm, representing the MC fraction in the measured population of IgG-SCs for the two protocols, respectively for schemes 2 and 3 with measurement one day after boost, and scheme 1 with measurement 4 days after second injection.

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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

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