Quantitative Immunology for Physicists Gr´egoireAltan-Bonnet,1, ∗ Thierry Mora,2, ∗ and Aleksandra M. Walczak2, ∗ 1Immunodynamics Section, Cancer & Inflammation Program, National Cancer Institute, Bethesda MD 20892, USA 2Laboratoire de physique de l'Ecole´ normale sup´erieure (PSL University), CNRS, Sorbonne Universit´e,Universit´ede Paris, 75005 Paris, France The adaptive immune system is a dynamical, self-organized multiscale system that protects ver- tebrates from both pathogens and internal irregularities, such as tumours. For these reason it fascinates physicists, yet the multitude of different cells, molecules and sub-systems is often also petrifying. Despite this complexity, as experiments on different scales of the adaptive immune system become more quantitative, many physicists have made both theoretical and experimental contributions that help predict the behaviour of ensembles of cells and molecules that participate in an immune response. Here we review some recent contributions with an emphasis on quantitative questions and methodologies. We also provide a more general methods section that presents some of the wide array of theoretical tools used in the field. CONTENTS A. Cytokine signaling and the JAK-STAT pathway 14 I. Introduction 2 1. Cytokine binding and signaling at equilibrium 15 II. Physical chemistry of ligand-receptor 2. Tunability of cytokine responses. 15 interaction: specificity, sensitivity, kinetics. 4 3. Regulation by cytokine consumption 17 A. Diffusion-limited reaction rates 4 4. Other regulations 17 B. Extrapolating collision rates in solution to B. Communication across space and time, and association rates in the physiological cytokine niches 18 context 5 1. Cytokine dynamics as C. Rates and numbers in the physiological diffusion-degradation 18 context 5 2. Screening by cytokine-consuming cells 18 1. Association rates 5 3. Probability of autocrine capture 19 2. Dissociation rates 5 4. Size of cytokine niches 20 3. Numbers of receptors per cell 5 V. Cell fate 20 4. Typical binding rates and some biology 6 A. Gene regulation and cell differentiation 20 D. Receptor-antigen specificity 7 1. Th1/Th2 differentiation 21 1. Cross-reactivity 7 2. Other differentiation switches 22 2. Models of receptor-antigen binding 7 3. Experimental test of bistability in cell 3. Data-driven receptor-antigen binding differentiation 22 models 8 B. Hematopoiesis 24 4. Modeling immunogeneticity 8 1. Timescales 24 C. Inferring the hematopoiesis differentiation III. Antigen discrimination 9 tree 25 A. T cells 9 D. Cell fate during the immune response 26 1. Kinetic proofreading for ligand 1. Choice and timing of cell fate under discrimination 9 arXiv:1907.03891v2 [q-bio.QM] 28 Jul 2019 stimulation 26 2. Adaptive kinetic proofreading 10 2. An aside on cell types 27 3. Coupling mechanics and biochemistry: 3. Inferring cell fate timelines during the the significance of forces for ligand immune response 27 discrimination 11 4. Quorum sensing 29 B. B cells 13 C. Coarse-graining of molecular details and VI. Repertoires 29 model reduction 14 A. Size of immune repertoires 29 B. Inference of the stochastic repertoire IV. Cell-to-cell communication through cytokines 14 generation process 31 C. Thymic selection and central tolerance 32 D. Diversity and the clone size distribution 34 E. Repertoire sharing 35 ∗ Authors are listed alphabetically. F. Optimal receptor distribution 36 2 G. Repertoire response to an immune 12. Population growth rates 62 challenge 37 D. Ecological models 62 1. Generalized Lotka{Volterra models 62 VII. Lymphocyte population dynamics 37 2. Susceptible-Infected-Recovered (SIR) A. Neutral dynamics 37 Models 63 B. A note about \neutral processes" 38 3. Solution of stochastic population C. Population dynamics model with external dynamics with a source 64 signals 39 4. Solution to foward jump process with D. In host HIV dynamics. 41 opposing drift and source 65 5. The Yule process 66 VIII. Affinity maturation 41 E. Inference 66 A. Modeling the hypermutation process 42 1. Probabilistic inference, maximum B. Cycles of selection 42 likelihood and Bayesian statistics 66 C. Evolution of broadly neutralizing 2. Model selection 67 antibodies 43 3. Expectation-Maximization 68 D. Population genetics approaches to affinity 4. Hidden Markov models 68 maturation 44 5. Information theory 69 1. Evolutionary analysis of repertoire 6. Maximum entropy models 69 dynamics 44 7. Machine learning and Neural networks 69 2. Models of co-evolution of phenotypic F. High-throughput repertoire sequencing 69 traits 45 IX. Population dynamics of pathogens and hosts 46 XIII. Acknowledgments 70 A. Viral fitness models 47 References 70 B. Co-evolution between host and pathogen populations 47 X. Discussion 49 I. INTRODUCTION XI. Glossary 49 The role of the immune system is to detect po- tential pathogens, confirm they really are undesirable XII. Methods 50 pathogens, and destroy them. The goal is in principle A. Physical kinetics 50 well defined. However recognizing molecular friends from 1. Diffusion-limited reaction rate 50 foes is not easy, and organisms have evolved many com- 2. The rates of dissociation between two plementary ways of dealing with this problem. Immu- biomolecules 51 nologists separate the molecularly non-specific response 3. The formation of ligand-receptor pairs: of the \innate" immune system, which includes every- equilibrium and kinetics 51 thing from scratching to the recognition of protein mo- B. Gene regulation 52 tifs characteristic of bacteria, and the molecularly specific 1. Basic model 52 \adaptive" response, by which specialized cells recognize 2. Auto-amplification with a single evolving features of never encountered before pathogens. transcription factor 52 From another angle we can consider different ways of de- 3. Auto-amplification with multiple stroying a pathogen: either swallowing pathogens whole, transcription factors 52 which is done by cells of the innate immune system called C. Population dynamics, genetics 53 neutrophils and macrophages; or killing our own cells 1. Deterministic mutation-selection that have been infected by a pathogen or are tumourous balance 53 | as performed by representatives of the adaptive im- 2. Genetic drift 54 mune system called killer T-cells. Alternatively, the 3. Wright-Fisher model 54 adaptive immune system produces specialized molecules 4. Probability of extinction 54 called antibodies that smother the invader: they attach 5. Moran model, continuous limit, and to pathogens to prevent them from entering cells and time varying selection 55 multiplying; they bind to bacterial toxins, thereby dis- 6. Branching processes 56 arming them; or they bind directly to bacterial cells, 7. Coalescence process 57 flagging them for consumption by macrophages. 8. Site frequency spectra and tree That short overview gives us an idea of the many balancing 57 strategies that both pathogens and the host organism 9. Clonal interference 58 have at their disposal (Fig. 1). Pathogenic cells (a bac- 10. Quantitative traits 59 teria, virus, or a tumour cell) are programmed to divide. 11. Lineage reconstruction 60 The immune system is there to prevent this. To a large 3 extend, its main challenge is to recognize the unknown. spatial distribution Pathogens are constantly evolving to escape recognition antigen presentation recognition by the immune system. Although the host organism does affinity maturation have a certain list of \warning sign features," most of repertoire diversity which are taken care of by the innate immune system, it collective sensing is up against a large set of constantly moving targets | as illustrated in our everyday experience by the evolving influenza virus, which requires a new vaccination every year. For this reason, the strategies developed by the immune system are mostly statistical, and require multi- ple interactions between different types of cells, a lot of signaling checks and balances that leads to a multiplicity of time and length scales (Fig. 2). Despite this complexity, the immune system works remarkably reliably. How do these P pathogen - immune system interactions on many scales dynamically come together co-evolution in a self-organized way to build a complex sensory system against a high dimensional moving target? This review gene regulation breaks this high level question into smaller problems and presents some results and concepts contributed by physi- molecular cell organism population cists. It also presents a summary of the current meth- ods, experimental and theoretical, used in the field. We FIG. 1. The many scales of the immune system. The do not shy away from the biology of the immune sys- immune system works at many scales from the molecular of tem, but to help the physicist navigate the complexity receptor-protein interactions, gene regulation, activation of of immunology, biological details will be introduced as biochemical pathways, to the cellular of cell-to-cell communi- we go along. For an introduction of the immune system cation directly and through signalling molecules, and organ- for the non-specialist, we refer the reader to the short ismal level responses using cells of the innate and adaptive immune systems, to the population level where global viral but excellent book by Lauren Sompayrac [1]. This re- evolution drives the co-evolution of immune systems of differ- view does not aim to be exhaustive, but rather focuses ent individuals. on the important physical concepts of immune function, reducing biological complexity to a minimum whenever appropriate. has one type