Quantitative modeling of the effect of antigen dosage on B-cell affinity distributions in maturating germinal centers

  1. Marco Molari
  2. Klaus Eyer
  3. Jean Baudry
  4. Simona Cocco
  5. Rémi Monasson  Is a corresponding author
  1. Laboratoire de Physique de l’École Normale Supérieure, ENS, PSL University, CNRS UMR8023, Sorbonne Université, Université Paris-Diderot, Sorbonne Paris Cité, France
  2. Laboratory for Functional Immune Repertoire Analysis, Institute of Pharmaceutical Sciences, ETH Zurich, Switzerland
  3. Laboratoire Colloides et Materiaux Divises (LCMD), Chemistry, Biology and Innovation (CBI), ESPCI, PSL Research and CNRS, France
17 figures, 2 tables and 2 additional files

Figures

Sketch of the germinal center reaction (inner part) and effects of the main reaction steps on the distribution of the binding energies (ϵ, equivalent to the logarithm of the dissociation constant logKd) of the B-cell population (histograms on the outer part).

A red-to-green color-scale is used to depict the affinity of both B-cell receptors in the inner part of the scheme and in the outer binding-energy histograms. Upon Ag administration GCs start to …

Effect of different antigen dosages on model evolution.

(A) Schematic representation of the Antigen (Ag) dynamics. Upon injection Ag is added to the reservoir. From there it is gradually released at a rate k+ and becomes available for B-cells to bind. …

Simulated GCs present different levels of homogenization.

(A) Example of homogenizing selection in GC evolution. Population size as a function of time for each clonal family in stochastic simulations of a single GC. The GC were initiated with an injected …

Asymptotic evolution at constant Ag concentration.

(A) Analysis of the asymptotic deterministic evolution for the large-size limit of the model, at constant available concentration C=30. Top left: size of the population vs. number of maturation …

Comparison between model-predicted and experimentally measured affinity distributions of antibody-secreting cells (Ab-SCs) for different immunization protocols.

A schematic representation of the protocol used is reported on top of each column. Scheme 1 (left column) consists of two injections at the same Ag dosage D, separated by a 4 weeks delay. Cells are …

Comparison between data and model prediction for the average binding energy (top) and high affinity fraction (bottom) of the Ab-secreting cell population under the three different immunization schemes (scheme 1 - left, scheme 2 - center, scheme 3 - right).

The high-affinity fraction corresponds to the fraction of measured cells having binding affinity Kd<50nM, or equivalently binding energy ϵ<-16.8kBT. On the x axis we report the variable quantity in the scheme, …

Appendix 1—figure 1
Effect of affinity-affecting mutations, differentiation time-switch and survival probability.

(A) Plot of the affinity-affecting kernel Kaa. Only around 5% of the affinity-affecting mutations increase affinity (green part of the curve for Δϵ<0). (B) Probability of MC μMC and PC μPC

Appendix 1—figure 2
Deterministic and stochastic evoliution comparison.

(A to D) Comparison between average evolution of the system and prediction of our deterministic model. The average is performed over 1000 independent GCR simulations at injected dosage of Ag D=1μg. The …

Appendix 1—figure 3
Influence of number and affinity of founder clones on evolution stochasticity.

(A to D) Effect of increasing the number of founder clones in the population. We compare 1000 stochastic simulations of the standard version of the model (blue) with a modified version in which …

Appendix 1—figure 4
Asymptotic evolution matches the solution of the eigenvalue equation.

We check that upon repetition of the evolution operator the system converges at the eigenvalue equation solution. For a given constant Ag concentration (C=10 in our case) we solve the eigenvalue …

Appendix 1—figure 5
Parallel tempering helps exiting local minima in stochastic likelihood maximization.

A: intuitive representation of the advantage of Parallel Tempering. When a Monte Carlo simulation is run at low temperature (T1) the system reaches a low-energy state but can get stuck in local …

Appendix 1—figure 6
Convergence of the stochastic likelihood maximization procedure for variant C of the model (see main text).

In this variant 8 of the model parameters are inferred (μnaive, σnaive, kB-, α, a, b, grecall, gimm). (A) Values of the log-likelihood log and the temperature T for the parameter set that reached peak …

Appendix 1—figure 7
Affinity distributions, comparison between experiments and model prediction for all tested conditions.

Comparison between experimental measurements (orange histograms), stochastic model (green histograms), and theoretical solution (blue curve) for affinity distributions of antibody-secreting cells …

Appendix 1—figure 8
Effect of the terms a, b on the model asymptotic behavior.

On the top row we plot the corresponding T-cell selection survival probability (setting for simplicity ϵ¯=0 and C=1) respectively in the case b=0.66 and a varying from 0 to 0.3 (A) and a=0.12, b varying from …

Appendix 1—figure 9
Result of the inference procedure on the five variations of the model and comparison with the standard version of the model.

The variations are described in appendix sect. 6 (Possible model variations). Top: final maximum value of the log-likelihood obtained with the inference procedure (black). Bottom: maximum-likelihood …

Appendix 1—figure 10
Distribution of beneficial and deleterious mutations over 1000 stochastic germinal center simulations at injected Ag dosage D=1μg.

Color represents the average number of cells that have developed the specified number of beneficial and deleterious mutations for any examined population, according to the color-scale on the right. …

Appendix 1—figure 11
Generation of realistic datasets, matching the statistics of the experimental one, for inference procedure validation.

(A) Average binding energy of responders populations in the 10 artificially generated datasets (gray) for the 15 experimentally tested conditions, vs the same quantity as predicted by simulations of …

Tables

Table 1
List of parameters in the model and of their values.

Binding energies are expressed in units of kBT, and times in days (d) or hours (h). The last nine parameters were inferred within selection variant (C), except ϵAg, whose reported value refers to …

Values of model parameters
SymbolValueMeaningSource
Tturn12 hDuration of an evolution turnWang et al., 2015
TGC6 dTime for GC formation after injectionDe Silva and Klein, 2015; Jacob et al., 1993; McHeyzer-Williams et al., 1993
Nmax2500GC max population sizeEisen, 2014; Tas et al., 2016
Ni2500Initial GC population sizeEisen, 2014; Tas et al., 2016
Nfound100Number of GC founder clonesTas et al., 2016; Mesin et al., 2016
pdiff10%Probability of differentiationWang et al., 2015; Meyer-Hermann et al., 2012; Oprea and Perelson, 1997
τdiff11 dSwitch time in MC/PC differentiationWeisel et al., 2016
Δτdiff2 dSwitching timescale in MC/PC differentiationWeisel et al., 2016
pmut14%Prob. of mutation per divisionWang et al., 2015; McKean et al., 1984; Kleinstein et al., 2003
ps,pl,paa50%, 30%, 20%Probability of a mutation to be silent/lethal/affinity-affectingZhang and Shakhnovich, 2010; Wang et al., 2015; Wang, 2017
Kaa(Δϵ)Equation 18Distribution of affinity-affecting mutationsOvchinnikov et al., 2018
k+0.98 /dAg release rateMacLean et al., 2001
k-1.22×102/dAg decay rateTew and Mandel, 1979
a0.12Baseline selection success probabilityMax-likelihood fit
b0.66Baseline selection failure probabilityMax-likelihood fit
μnaive-14.60Mean binding energy of seeder clones generated by naive precursorsMax-likelihood fit
σnaive1.66Standard deviation of the seeder clones binding energy distributionMax-likelihood fit
kB-2.07×10-5/dAg consumption rate per B-cellMax-likelihood fit
α2.3×102μgConcentration to dosage conversion factorMax-likelihood fit
grecall0.56MC fraction in Ab-SC population for measurement 1 day after boostMax-likelihood fit
gimm0MC fraction in Ab-SC population for measurement 4 days after second injectionMax-likelihood fit
ϵAg-13.59Threshold Ag binding energy (A)Max-likelihood fit
Appendix 1—table 1
Average results of the inference procedure on the 10 artificially generated datasets.

For each model parameter, we report the value used to generate the data (left), and the mean (middle) and standard deviation (right) over the 10 inferred values.

ParameterValue used to generate dataMean of inferred valuesStd of inferred values
μi−14.59−14.760.22
σi1.661.590.11
kB-2.07e-051.82e-050.71e-05
grecall0.560.550.11
gimm00.0090.022
α0.0230.0320.019
a0.1200.1250.094
b0.6610.6590.008

Additional files

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