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## Toward Quantitative Models of Germinal Center Dynamics

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**Toward Quantitative Modelsof Germinal Center Dynamics**Steven H. Kleinstein Princeton University Advisor: Jaswinder Pal Singh Martin Weigert**So, what are germinal centers?**Spleen Immunity 1996 4: 241–250 Cellular structures that dynamically form in lymph nodes and spleen during immune response http://www.chemistry.ucsc.edu/**B**B B B Ag B B B Germinal Center Germinal Centers are the Site of Affinity Maturation (selection for cells with affinity-increasing mutations) Why do we care about germinal centers? Ag Foreign Pathogen (Antigen)**Sounds good, what’s the problem?**Not understood how mechanisms fit together Use computer simulation Shortcomings of Common Models • Prototypical Response • Qualitative validation • Average-case dynamics Our Goals • Specific Responses • Quantitative validation • Average & Distribution**Talk Outline**• Starting Point: The Recycling Model • Creating Response-Specific Models • Model Validation • Average-Case Dynamics • Behavior of Individual Germinal Centers • Current Research Focus**Death**Affinity-Dependent Selection Proliferate & Mutate Memory Light-Zone Dark-Zone The Recycling Model Differential Equations by Oprea and Perelson (Liu et al. Immunity 1996 4: 241) Oprea, M., and A. Perelson. 1997. J. Immunol. 158:5155.**Seed**Grow Affinity Maturation Qualitative Affinity Maturation Days post-immunization**Similar**Affinity Ox Splenic Germinal Centers NP Splenic Germinal Centers 1 …CACTTGATG… 1 Higher Affinity Initial Sequence …TACTGGATG… 10 …TACTTGATG… Key Mutation Quantitative Affinity Maturation Consider two well-studied antigens: Ox and NP Small number of Key Mutations increase affinity 10-fold and are efficiently selected in germinal centers**Compare with Data**• Average-Case Half-life Migration Rates Physical Capacity B Cell Affinity Effect of mutation Formulas Ox NP Response-specific values different from‘prototypical’ response Modeling Specific Responses Model + Parameters Dynamics General Response Specific**Simulating Average Case Dynamics**Ox NP Ox works with optimistic assumptions for general parameters NP works when key mutation less cold After optimization of unknown parameters and other assumptions, model can predict average case dynamics**• Individual**Individual behavior not obvious from average-case Beyond Average-Case Dynamics Compare with Data • Average-Case Model + Parameters Dynamics General Response Specific Formulas Ox NP**Data from Individual Germinal Centers**Experiments show many germinal centers have no key mutants, but those with key mutants are dominated by them Ox Day 14-15 NP Day 14-16 All-or-None Distribution**Key Mutant Founder**b c a Initial Sequence d Key Mutant Founder More Data from Individual Germinal Centers A B C D**More Data from Individual Germinal Centers**Key Mutant Founder d A D b c B a Initial Sequence C Multiple founders never observed in experiments (yet) Single founder cell for key mutant population**Discrete**Stochastic New model implementation necessary to predict individual dynamics Simulating Individual Germinal Centers Compare with Data • Average-Case • Individual Model + Parameters Dynamics Differential Equations General Response Specific Formulas Ox NP**The Discrete/Stochastic Implementation**• Fixed-increment time advance framework • Assume Poisson processes • Use 1-e-t to calculate event probability • Random numbers determine occurrence Use PC cluster to run this embarrassingly parallel program New implementation allows simulation of individual germinal centers**Individual Germinal Center Dynamics**• Model does not predict all-or-none response Ox NP • Predicts multiple founders per germinal center • 7 for Ox, 2 for NP Recycling model fails to predict individual behavior**Too many key mutants**Selection too weak Also, basis for multiple founders Basis for Failure of Recycling Model Ox**Fixing the Model**Propose Hypotheses Compare with Data • Average-Case • Individual Model + Parameters Dynamics Differential Equations General Response Specific Formulas Discrete Stochastic Ox NP How can we bring model and experiment into agreement?**All-or-None**Single Founder Ox Additional hypotheses can bring model closer to data Extending the Recycling Model • Need to add new biological assumptions • E.g., selected cells immediately begin division (no strict spatial separation)**Making Verifiable Predictions**Model can operate in two modes: Antigen is limiting Antigen is not limiting Ox Model predicts size correlated with affinity maturation**Propose Hypotheses**Compare with Data • Average-Case Differential Equations General Response Specific Make Predictions Formulas Discrete Stochastic Ox NP Differential equations are not adequate Modeling specific responses is important Summary and Conclusions • Individual Model + Parameters Dynamics**Additional Observations Requiring Explanation**Key mutants divide faster so more mutations expected, but… NP Ox —— Key Mutants ------ Other Cells —— Key Mutants ------ Other Cells Number of Mutations Number of Mutations Days post-immunization Days post-immunization In NP response, key mutants are less mutated then other cells**Key Mutation,**but Initial Affinity 1 10 …TACTTGATG… …TACTTGGTG… Blocking Key Mutation Helps explain why few key mutant founders are observed The Blocking Model We propose that some mutations can block ability of key mutations to confer high affinity 1 Initial Sequence …TACTGGATG…**Ox**NP Effect of Blocking on Mutation Dynamics Key mutants with blocking mutations are negatively selected Blocking —— Key Mutants ------ Other Cells Number of Mutations Days post-immunization Blocking also impacts many other aspects of the model**The End**For more information: stevenk@cs.princeton.edu www.cs.princeton.edu/~stevenk**Future Research Directions**• How to account for fast clonal dominance • What is the selective force • Integrate spatial effects into models • Design experiments that can differentiate • Derive and apply additional constraints • Migration patterns, clonal tree shapes • Develop landscapes for other responses**Blocking can occur in both chains, but**only one chain observed in experiments Paradox in Frequency of Blocking Mutations More key mutants are lost in Ox compared with NP In NP response, fewer mutations in key mutants • Overall frequency of blocking is higher in Ox • Blocking is biased toward observed sequence in NP Solution: blocking mutations follow distribution of contact residues**Papers & Presentations**Papers Steven H. Kleinstein and Jaswinder Pal Singh. "Toward quantitative simulation of germinal center dynamics: Biological and modeling insights from experimental validation". The Journal of Theoretical Biology. 2001. 211:253-275 Steven H. Kleinstein and Philip E. Seiden. "Simulating the Immune System". Computing in Science and Engineering, 69-77. July/August 2000. Steven H. Kleinstein. “Why are there so few key mutant clones? The influence of stochastic selection and blocking on affinity maturation in the germinal center”. In preparation. Conference Talks Erich Schmidt and Steven H. Kleinstein. “Optimal Decision Making of Individual Cells: A Case Study of Affinity Maturation”. International Conference on Mathematical and Theoretical Biology. July 2001. Steven H. Kleinstein. "Toward quantitative validation of immune-system models: germinal center dynamics in the 2-phenyl-5-oxazolone response". In Silico Biology: The Future of Target Triage. June 2000. Steven H. Kleinstein and Jaswinder Pal Singh. "Toward quantitative simulation of germinal center dynamics: Biological and modeling insights from experimental validation". Duke University's Conference on Mathematical Immunology. April 2000. Steven H. Kleinstein. "Cell dynamics of the germinal center". Theory in Immunology. October 1999. Steven H. Kleinstein, Jaswinder Pal Singh and Martin Weigert. "Putting limits on selection and mutation in a simulation of the humoral immune response". International Conference on Complex Systems. October 1998.**Biological Discovery Inspired by Modeling**Although HQ is over 2x more likely to occur at the DNA level, it is observed significantly less frequently than HN. Germline …TACATGCACTGGTACCAGCAG… Low Affinity Key Mutant (HQ) …TACATGCAATGGTACCAGCAG… Low Affinity (1) Key Mutant (HQ) …TACATGCAATGGTTCCAGCAG… High Affinity (20) Key Mutant (HN) …TACATGAACTGGTACCAGCAG… High Affinity (54) Key Mutant (HN) …TACATGAACTGGTTCCAGCAG… High Affinity (65) Key Mutations Helper Mutation H Q requires second mutation YF for high affinity**Stochastic Effects**Initially producing 4 key mutants per day Ox Only important later on in response**What if germinal center size is constant?**Key Mutant Discrepancy: expected observed • Expectation based on population genetics • Key mutant founders produced every 1.5 days • Appearance time from experimental observations • Key mutant founder appears day 11 (on average) • Many key mutant founders are ‘lost’ • 7 lost during NP response, 18 lost during Ox response Problem could be expectation, observation or in between Radmacher et al. 1998. Immunol Cell Bio. 76:373.**Recycling Model**• Works for average, but fails for individual • Outlined problems and proposed solutions • Verifiable Predictions for further insights & validation • Blocking Model • Explains key mutant discrepancy and mutation dynamics • Stochastic Selection Model ANALYSIS Summary of Contributions • Methods/Formulas to create specific affinity landscapes • Allows prediction of particular experiments (e.g., Ox, NP) • Library for Discrete/Stochastic simulations • Allows prediction of individual germinal center behavior • Set of constraints for quantitative validation FRAMEWORK**The End**For more information: stevenk@cs.princeton.edu www.cs.princeton.edu/~stevenk