Department of Computer Science, Princeton University

1. Towards Quantitative Validation of Immune-System ModelsGerminal Center Dynamics in the 2-phenyl-5-oxazolone Response J.P. Singhjps@cs.princeton.edu Steven Kleinsteinstevenk@cs.princeton.edu Department of Computer Science, Princeton University

2. 3 2 1 0 -1 -2 The Oprea and Perelson Model A set of differential equations modeling cell population dynamics in the germinal center, emphasizing the selection of high-affinity mutations • Model of B cell affinity and somatic mutation is critical 1000 + Affinity class framework groups B cells with similar on-rates 100 10 Germline 1 1/10 - 1/100

3. The Oprea and Perelson Model • Lots of cell types • Lots of processes • Equations are indexed by affinity class

4. The Simulation Parameters Parameters General Response Specific Affinity Class: Germline kon & koff Transition Probabilities Affinity Factor Half-life Migration Rates Physical Capacity Oprea and Perelson suggest a list of default parameter values

5. Simulated Germinal Center Dynamics

6. Qualitative Quantitative Achieves qualitative correspondence with experiment, but... Is the timing of selection same as experimental data? Can we achieve the same efficiency of selection? Quantitative validation increases credibility

7. Obstacles to Quantitative Validation • Not meant to predict particular experiments • Parameters reflect ‘general’ response • Hard to estimate parameters for real response • Large diversity of activated cells • Many genetic sequences in each class • Specific +/- mutations unknown • No good metric to compare with experiment • Currently use total/average affinity

8. Overall Validation Methodology • Want to model complex, real response • Many unknowns  Validation difficult • Start with simple response Validation with simple response increases confidence when model is applied to more complex response Primary response to hapten oxazolone (phOx)

9. Validation Step #1 Parameters General System Specific Affinity Class: Germline kon & koff Transition Probabilities Affinity Factor phOx

10. 1 0 -1 -2 Affinity Class Parameters for phOx Transition probabilities based on DNA sequence, antibody/antigen structure and, of course, a few assumptions (Details are discussed in paper) Single high-affinity class contains cells with canonical high-affinity mutations at codon 34

11. Ag B Choosing kon & koff for phOx In the model, these refer to rates of cell activation... Not sure how (or if) these relate to thermodynamic rates... kf 5 x 106 M-1 s-1 kr 5 s-1 (Solution  look at a range of values)

12. Model  Experiment Model Predicts < 0.15 Experimental Result  0.50 [Berek, Berger and Apel, 1991]

13. Validation Step #2 Parameters General Response Specific Based on literature and expert advice, we updated: Centroblast Proliferation Centrocyte Migration Physical Capacity (Not phOx specific) Updated phOx

14. Model = Experiment (possibly)

15. Comparison Over Time

16. Experimentally Testable Predictions If we believe the model... • Mapping thermodynamic  cell-level kon and koff • Key selection pressure is rescue from apoptosis • Selection occurs only for limited range of kon How we choose parameters values is critical (but there is more to the story…)

17. Beyond Simple Averages Limited experimental data suggests... Fraction high-affinity follows bimodal distribution Single high-affinity founder cell within each GC [Ziegner, Steinhauser and Berek, 1994] Statistical model for similar response (not experimental data)

18. Validation Step #3 Differential equations implicitly model average-case dynamics and have no notion of individual cells Create new discrete/stochastic simulation of the Oprea and Perelson model

19. Making a discrete/stochastic simulation • Fixed-increment time advance framework • Assume Poisson processes • Use 1-e-t to calculate event probability • Random numbers determine occurrence Run simulation 500 times to simulate a spleen’s worth of germinal centers

20. Thus, the following scenario is possible: Can new simulation predict bimodality? • Appearance of founder is stochastic event... • cell death • competition for antigen • exit germinal center 1. Median appearance is day 14 2. Take-over is extremely rapid

21. …but model distribution is not bimodal Model also predicts > 20 founding cells

22. Theoretical Work Model is bimodal  Push further Model captures single mode  New mechanisms Model is completely wrong? Experimental Work Refine observations Test model predictions Conclusions No agreement yet, but...

23. Affinity Class Parameters for phOx

24. The Updated Parameters