Computational Model Reveals a Stochastic Mechanism behind Germinal Center Clonal Bursts
Abstract
:1. Introduction
2. Methods: A Probabilistic Model of GC Dynamics
2.1. Stochastic Model of the GC
2.2. Cell Fate after Somatic Hypermutation
2.3. B Cell Receptor Affinity to Epitote
2.4. T${}_{FH}$ Dynamics
2.5. T${}_{FH}$ Survival Signals
2.6. Regulation of the Cell Cycle
2.7. Plasma Cell and Memory B Cell Differentiation
 Antigendriven differentiation model (model 1): In this first scenario, we consider that affinity determines B cell fate in a deterministic manner. Namely, there is a threshold pMHC${}_{\mathrm{thresh}}$ above which a CC becomes a PC. Below that threshold, a cell becomes a MBC.
 Dynamic antigendriven differentiation model (model 2): A more sophisticated hypothesis is that T cells progressively become less sensitive to the pMHC complex and, as the average affinity of B cells increases in the GC, a larger amount of pMHC is needed to induce the same amount of IRF4. In the second scenario, the choice between PC and MBC is also deterministic, although according to a dynamic threshold that increases linearly with time, i.e., pMHC${}_{\mathrm{thresh}}$ = affinity${}_{GC}(t{t}_{0})$ + a${}_{0}$.
 Stochastic differentiation model (model 3): Finally, we also investigate a probabilistic scenario where, once a B cell has been selected for differentiation, it becomes a PC or MBC according to some fixed probabilities that depend on the amount of expressed pMHC as follows: ${p}_{\mathrm{MBC}}={e}^{k(\mathrm{pMHC}{p}_{0})}$ and ${p}_{\mathrm{PC}}=1{p}_{\mathrm{MBC}}$.
2.8. Phylogenetic Trees Representation
2.9. Clonal Diversity
2.10. Parameter Optimization
 Number of GC B cells: A timeresolved study of 3457 GC sections from mice provided information about the GC area ${A}_{\mathrm{GC}}$ throughout the GC reaction [18]. Assuming that GC are spherical, then ${A}_{\mathrm{GC}}=\pi \phantom{\rule{0.166667em}{0ex}}{r}_{\mathrm{GC}}^{2}$, from where the radius of the GC, ${r}_{\mathrm{GC}}$, can be estimated. If we further assume that B cells are the predominant cellular type in the GC, we can estimate the number of B cells in the GC as:$${N}_{\mathrm{B}}\phantom{\rule{4.pt}{0ex}}\mathrm{cell}\approx \frac{{V}_{\mathrm{GC}}}{{V}_{\mathrm{B}}\phantom{\rule{4.pt}{0ex}}\mathrm{cell}}={\left(\frac{{r}_{\mathrm{GC}}}{{r}_{\mathrm{B}}\phantom{\rule{4.pt}{0ex}}\mathrm{cell}}\right)}^{3}\phantom{\rule{0.166667em}{0ex}},$$
 DZ/LZ ratio: Despite high variability in size, a typical cellular composition with relatively stable cell ratios of resident T${}_{FHs}$, macrophages, proliferating cells, and apoptotic nuclei seems to be maintained during the established phase of the response [18]. Timeresolved data about the relative size of DZ to LZ is also available from the same study. While the variability in the relative volumes of DZ and LZ is considerably high at all sampled time points (Figure 4B), it stabilizes after the phase of monoclonal expansion to an average of $\sim 1$ during the entire GC reaction [18]. The high variability is consistent with some other studies that found a ratio of 2 [9]. In our simulation, the DZ/LZ ratio is defined as the number of CBs over the number of CCs; however, we only estimate this ratio after day 9 after immunization, hence avoiding the early days of the GC establishment, where the DZ and LZ are still not spatially separated.
 Cell death: Apoptosis is prevalent in the GC, with up to half of all GC B cells dying every 6 h [55]. We use a relatively constant cell death rate of 8%/h in both the DZ and LZ through the entire GC response.
 GC cellular output: Finally, we use timeresolve data of the GC cellular output [50] to constrain our model. Because the experiment only measured relative abundances rather than direct cell counts, we scale both the model predictions and experimental measures by the maximum value.
3. Results
3.1. Germinal Center Dynamics
3.2. Comparison of Simulated and Experimentally Determined BCR Sequences
3.3. Visualizing Affinity Maturation
3.4. Visualizing Clonal Competition and Clonal Burst
4. Discussion
Supplementary Materials
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Code Availibility: The experimental data and python code used to fit our model, are provided freely on Github under MIT license at https://github.ibm.com/SysBio/GerminalCenter. The GC kinetic data of [18] were derived from the online database at http://sysimmtools.eu, and GC B cell sequences used to construct trees were made publicly available by [15]. The normalized dominance score [15] and MBC/PC output with time [50] were kindly provided by the corresponding authors of the respective studies.

Reaction  Description  Parameter Value 

$\mathrm{NB}\stackrel{{r}_{\mathrm{activation}}}{\to}\mathrm{CB}$  B cell activation  ${r}_{\mathrm{activation}}=3.94\phantom{\rule{4pt}{0ex}}/\mathrm{h}$ 
$\mathrm{CB}\stackrel{{r}_{\mathrm{division}}}{\to}2\mathrm{CB}$ where each daughter CB undergoes apoptosis ($\mathrm{CB}\stackrel{}{\to}\varnothing $) with probability $\delta $ per acquired mutation.  Centroblast division  ${r}_{\mathrm{division}}=0.134\phantom{\rule{4pt}{0ex}}/\mathrm{h}$ $\delta =0.52$ 
$\mathrm{CB}\stackrel{{r}_{\mathrm{migration}}}{\to}\mathrm{CC}$  Centroblast migration to LZ  ${r}_{\mathrm{migration}}=3.75\phantom{\rule{4pt}{0ex}}/\mathrm{h}$ 
$\mathrm{CC}\stackrel{{r}_{\mathrm{apoptosis}}}{\to}\varnothing $  Centrocyte apoptosis  ${r}_{\mathrm{apoptosis}}=0.084\phantom{\rule{4pt}{0ex}}/\mathrm{h}$ 
$\mathrm{CC}+\mathrm{FDC}\stackrel{{r}_{\mathrm{FDC}\text{:}\mathrm{CC}}}{\to}\mathrm{CC}\left(\mathrm{pMHC}\right)+\mathrm{FDC}$  Centrocyte antigen uptake  ${r}_{\mathrm{FDC}\text{:}\mathrm{CC}}={\displaystyle \frac{40}{{N}_{\mathrm{CC}}}}\phantom{\rule{4pt}{0ex}}/\mathrm{h}$ 
$\mathrm{CC}+\mathrm{T}{}_{FH}\stackrel{{r}_{\mathrm{TC}\text{:}\mathrm{CC}}}{\to}\left[\mathrm{CCTC}\right]$  Centrocyte binding to T${}_{FH}$  ${r}_{\mathrm{TC}\text{:}\mathrm{CC}}={\displaystyle \frac{145}{{N}_{\mathrm{CC}}}}\phantom{\rule{4pt}{0ex}}/\mathrm{h}$ 
$\left[{\mathrm{CC}}_{1}\mathrm{TC}\right]+{\mathrm{CC}}_{2}\stackrel{{r}_{\mathrm{TC}\text{:}\mathrm{CC}}}{\to}{\mathrm{CC}}_{1}+\left[{\mathrm{CC}}_{2}\mathrm{TC}\right]$  Centrocyte T${}_{FH}$ switch  
$\left[\mathrm{CCTC}\right]\stackrel{{r}_{\mathrm{unbinding}}}{\to}{\mathrm{CC}}_{\mathrm{sel}}+\mathrm{T}{}_{FH}$  Centrocyte spontaneous unbinding  ${r}_{\mathrm{division}}=2\phantom{\rule{4pt}{0ex}}/\mathrm{h}$ 
${\mathrm{CC}}_{\mathrm{sel}}\stackrel{{r}_{\mathrm{recirculation}}}{\to}\mathrm{CB}$  Centrocyte recirculation to DZ  ${r}_{\mathrm{recirculate}}=3.75\phantom{\rule{4pt}{0ex}}/\mathrm{h}$ 
${\mathrm{CC}}_{\mathrm{sel}}\stackrel{{r}_{\mathrm{exit}}}{\to}\mathrm{MBC}\phantom{\rule{4.pt}{0ex}}\mathrm{or}\phantom{\rule{4.pt}{0ex}}\mathrm{PC}$ with probability ${p}_{\mathrm{MBC}}:\mathrm{CC}\stackrel{}{\to}\mathrm{MBC}$, with probability ${p}_{\mathrm{PC}}:\mathrm{CC}\stackrel{}{\to}\mathrm{PC}$.  Centrocyte exit  ${r}_{\mathrm{exit}}=1.6\phantom{\rule{4pt}{0ex}}/\mathrm{h}$ see Section 2.7 for descriptions of ${p}_{\mathrm{MBC}}$ and ${p}_{\mathrm{PC}}$ 
Parameter  Description  Lower Bound  Upper Bound  Fitted  Sensitivity  Mainly Affects  

Intercellular  ${r}_{\mathrm{activation}}$ [${\mathrm{h}}^{1}$]  B cell activation rate  1  10  3.94  high  Value of GC peak 
${r}_{\mathrm{division}}$ [${\mathrm{h}}^{1}$]  CB division rate  0.08  0.16  0.134  moderate  DZ/LZ ratio  
${r}_{\mathrm{migration}}$ [${\mathrm{h}}^{1}$]  CB migration rate  0.15  3.75  3.75  low  DZ/LZ ratio  
${r}_{\mathrm{apoptosis}}$ [${\mathrm{h}}^{1}$]  CC apoptosis rate  0.06  0.16  0.084  high  GC decay  
${r}_{\mathrm{exit}}$ [${\mathrm{h}}^{1}$]  CC${}_{\mathrm{sel}}$ exit rate  0.41  3.75  1.6  high  GC decay  
${r}_{\mathrm{recirculate}}$ [${\mathrm{h}}^{1}$]  CC${}_{\mathrm{sel}}$ recirculation rate  fixed to 3.75  high  GC decay  
${r}_{\mathrm{FDC}\text{:}\mathrm{CC}}$ [${\mathrm{h}}^{1}$]  FDC encounter rate  fixed to $40/{N}_{\mathrm{CC}}$  low  Clonal competition  
${r}_{\mathrm{TC}\text{:}\mathrm{CC}}$ [${\mathrm{h}}^{1}$]  T${}_{FH}$ encounter rate  fixed to $145/{N}_{\mathrm{CC}}$  moderate  Clonal competition  
${r}_{\mathrm{unbinding}}$ [${\mathrm{h}}^{1}$]  T${}_{FH}$ unbinding rate  fixed to 2  high  GC decay  
${\alpha}_{\mathrm{TC}}={N}_{\mathrm{TC}}/{N}_{\mathrm{CC}}$  T${}_{FH}$ to CC ratio  1/100  1/7  1/46  high  GC decay  
${N}_{\mathrm{FDC}}$  Number of FDCs  fixed to 250  low  Clonal competition  
Intracellular  $\delta $  Lethal mutation probability  0.1  0.9  0.52  high  GC decay 
$\sigma $  Silent mutation probability  fixed to 0.28  low  Affinity maturation  
${p}_{\mathrm{SHM}}$  Mutation rate  fixed to $1\times {10}^{3}$  high  GC decay  
${N}_{\mathrm{CDR}}$  CDR length (nucleotides)  fixed to 25  low  Affinity maturation  
PC/MBC  ${\mathrm{pMHC}}_{\mathrm{threshold}}$  Model 1 parameter  0  1  0.46  moderate  PC/MBC production 
${t}_{0}$  Model 2 parameter  0  ∞  3 days  moderate  PC/MBC production  
${a}_{0}$  Model 2 parameter  0  1  0.05  moderate  PC/MBC production  
k  Model 3 parameter  0  ∞  14  moderate  PC/MBC production  
${p}_{0}$  Model 3 parameter  0  1  0.4  moderate  PC/MBC production 
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Pélissier, A.; Akrout, Y.; Jahn , K.; Kuipers , J.; Klein , U.; Beerenwinkel, N.; Rodríguez Martínez , M. Computational Model Reveals a Stochastic Mechanism behind Germinal Center Clonal Bursts. Cells 2020, 9, 1448. https://doi.org/10.3390/cells9061448
Pélissier A, Akrout Y, Jahn K, Kuipers J, Klein U, Beerenwinkel N, Rodríguez Martínez M. Computational Model Reveals a Stochastic Mechanism behind Germinal Center Clonal Bursts. Cells. 2020; 9(6):1448. https://doi.org/10.3390/cells9061448
Chicago/Turabian StylePélissier, Aurélien, Youcef Akrout, Katharina Jahn , Jack Kuipers , Ulf Klein , Niko Beerenwinkel, and María Rodríguez Martínez . 2020. "Computational Model Reveals a Stochastic Mechanism behind Germinal Center Clonal Bursts" Cells 9, no. 6: 1448. https://doi.org/10.3390/cells9061448