A mathematical model for germinal centre kinetics and affinity maturation

A branching stochastic evolutionary model of the B-cell repertoire

We propose a stochastic framework to describe the evolution of the B-cell repertoire during germinal center (GC) reactions. Our model is formulated as a multitype age-dependent branching process with time-varying immigration. The immigration process captures the mechanism by which founder B cells initiate clones by gradually seeding GC over time, while the branching process describes the temporal evolution of the composition of these clones. The model assigns a type to each cell to represent attributes of interest. Examples of attributes include the binding affinity class of the B cells, their clonal family, or the nucleotide sequence of the heavy and light chains of their receptors. The process is generally non-Markovian. We present its properties, including as t → ∞ when the process is supercritical, the most relevant case to study expansion of GC B cells. We introduce temporal alpha and beta diversity indices for multitype branching processes. We focus on the dynamics of clonal dominance, highlighting its non-stationarity, and the accumulation of somatic hypermutations in the context of sequential immunization. We evaluate the impact of the ongoing seeding of GC by founder B cells on the dynamics of the B-cell repertoire, and quantify the effect of precursor frequency and antigen availability on the timing of GC entry. An application of the model illustrates how it may help with interpretation of BCR sequencing data.

Investigating the Mechanism of Germinal Center Shutdown

Germinal centers (GCs) are transient structures where affinity maturation of B cells gives rise to high affinity plasma and memory cells. The mechanism of GC shutdown is unclear, despite being an important phenomenon maintaining immune homeostasis. In this study, we used a mathematical model to identify mechanisms that can independently promote contraction of GCs leading to shutdown. We show that GC shutdown can be promoted by antigen consumption by B cells, antigen masking by soluble antibodies, alterations in follicular dendritic cell (FDC) network area, modulation of immune complex cycling rate constants, alterations in T follicular helper signaling, increased terminal differentiation and reduced B cell division capacity. Proposed mechanisms promoted GC contraction by ultimately decreasing the number of B cell divisions and recycling cells. Based on the in-silico predictions, we suggest a combination of experiments that can be potentially employed by future studies to unravel the mechanistic basis of GC shutdown such as measurements of the density of pMHC presentation of B cells, FDC network size per B cell, fraction of cells expressing differentiation markers. We also show that the identified mechanisms differentially affect the efficiency of GC reaction estimated based on the quantity and quality of resulting antibodies.

System-Level Scenarios for the Elucidation of T Cell-Mediated Germinal Center B Cell Differentiation

Germinal center (GC) reactions are vital to the correct functioning of the adaptive immune system, through formation of high affinity, class switched antibodies. GCs are transient anatomical structures in secondary lymphoid organs where specific B cells, after recognition of antigen and with T cell help, undergo class switching. Subsequently, B cells cycle between zones of proliferation and somatic hypermutation and zones where renewed antigen acquisition and T cell help allows for selection of high affinity B cells (affinity maturation). Eventually GC B cells first differentiate into long-lived memory B cells (MBC) and finally into plasma cells (PC) that partially migrate to the bone marrow to encapsulate into long-lived survival niches. The regulation of GC reactions is a highly dynamically coordinated process that occurs between various cells and molecules that change in their signals. Here, we present a system-level perspective of T cell-mediated GC B cell differentiation, presenting and discussing the experimental and computational efforts on the regulation of the GCs. We aim to integrate Systems Biology with B cell biology, to advance elucidation of the regulation of high-affinity, class switched antibody formation, thus to shed light on the delicate functioning of the adaptive immune system. Specifically, we: i) review experimental findings of internal and external factors driving various GC dynamics, such as GC initiation, maturation and GCBC fate determination; ii) draw comparisons between experimental observations and mathematical modeling investigations; and iii) discuss and reflect on current strategies of modeling efforts, to elucidate B cell behavior during the GC tract. Finally, perspectives are specifically given on to the areas where a Systems Biology approach may be useful to predict novel GCBC-T cell interaction dynamics.

Germinal Centre Shutdown

Germinal Centres (GCs) are transient structures in secondary lymphoid organs, where affinity maturation of B cells takes place following an infection. While GCs are responsible for protective antibody responses, dysregulated GC reactions are associated with autoimmune disease and B cell lymphoma. Typically, ‘normal’ GCs persist for a limited period of time and eventually undergo shutdown. In this review, we focus on an important but unanswered question – what causes the natural termination of the GC reaction? In murine experiments, lack of antigen, absence or constitutive T cell help leads to premature termination of the GC reaction. Consequently, our present understanding is limited to the idea that GCs are terminated due to a decrease in antigen access or changes in the nature of T cell help. However, there is no direct evidence on which biological signals are primarily responsible for natural termination of GCs and a mechanistic understanding is clearly lacking. We discuss the present understanding of the GC shutdown, from factors impacting GC dynamics to changes in cellular interactions/dynamics during the GC lifetime. We also address potential missing links and remaining questions in GC biology, to facilitate further studies to promote a better understanding of GC shutdown in infection and immune dysregulation.

A Perspective on the Role of Computational Models in Immunology

This is an exciting time for immunology because the future promises to be replete with exciting new discoveries that can be translated to improve health and treat disease in novel ways. Immunologists are attempting to answer increasingly complex questions concerning phenomena that range from the genetic, molecular, and cellular scales to that of organs, whole animals or humans, and populations of humans and pathogens. An important goal is to understand how the many different components involved interact with each other within and across these scales for immune responses to emerge, and how aberrant regulation of these processes causes disease. To aid this quest, large amounts of data can be collected using high-throughput instrumentation. The nonlinear, cooperative, and stochastic character of the interactions between components of the immune system as well as the overwhelming amounts of data can make it difficult to intuit patterns in the data or a mechanistic understanding of the phenomena being studied. Computational models are increasingly important in confronting and overcoming these challenges. I first describe an iterative paradigm of research that integrates laboratory experiments, clinical data, computational inference, and mechanistic computational models. I then illustrate this paradigm with a few examples from the recent literature that make vivid the power of bringing together diverse types of computational models with experimental and clinical studies to fruitfully interrogate the immune system.

Trade-offs in antibody repertoires to complex antigens

Pathogens vary in their antigenic complexity. While some pathogens such as measles present a few relatively invariant targets to the immune system, others such as malaria display considerable antigenic diversity. How the immune response copes in the presence of multiple antigens, and whether a trade-off exists between the breadth and efficacy of antibody (Ab)-mediated immune responses, are unsolved problems. We present a theoretical model of affinity maturation of B-cell receptors (BCRs) during a primary infection and examine how variation in the number of accessible antigenic sites alters the Ab repertoire. Naive B cells with randomly generated receptor sequences initiate the germinal centre (GC) reaction. The binding affinity of a BCR to an antigen is quantified via a genotype-phenotype map, based on a random energy landscape, that combines local and distant interactions between residues. In the presence of numerous antigens or epitopes, B-cell clones with different specificities compete for stimulation during rounds of mutation within GCs. We find that the availability of many epitopes reduces the affinity and relative breadth of the Ab repertoire. Despite the stochasticity of somatic hypermutation, patterns of immunodominance are strongly shaped by chance selection of naive B cells with specificities for particular epitopes. Our model provides a mechanistic basis for the diversity of Ab repertoires and the evolutionary advantage of antigenically complex pathogens.

Overcoming the dichotomy of quantity and quality in antibody responses

Germinal centers (GCs) are specialized environments in which B cells mutate their BCR to identify new Abs with high affinity to a challenging Ag. B cells are selected in an evolutionary process of multiple rounds of mutation and selection. In the past decade, mechanisms of B cell migration, division, mutation, selection, and final differentiation have been extensively studied. Thereby, modulations of these mechanisms either optimize the quality, in terms of affinity, or the quantity of generated Abs, but never both, leading to an unclear effect on the overall efficiency of the Ab response. In this article, we predict with mathematical models that an affinity-dependent number of GC B cell divisions overcomes the dichotomy of quality and quantity, and has to be considered as a good target for immune interventions, in particular, in the elderly population with poor GC responses.

Modeling of celiac disease immune response and the therapeutic effect of potential drugs

Background

Celiac disease (CD) is an autoimmune disorder that occurs in genetically predisposed people and is caused by a reaction to the gluten protein found in wheat, which leads to intestinal villous atrophy. Currently there is no drug for treatment of CD. The only known treatment is lifelong gluten-free diet. The main aim of this work is to develop a mathematical model of the immune response in CD patients and to predict the efficacy of a transglutaminase-2 (TG-2) inhibitor as a potential drug for treatment of CD.

Results

A thorough analysis of the developed model provided the following results:

1. TG-2 inhibitor treatment leads to insignificant decrease in antibody levels, and hence remains higher than in healthy individuals.

2. TG-2 inhibitor treatment does not lead to any significant increase in villous area.

3. The model predicts that the most effective treatment of CD would be the use of gluten peptide analogs that antagonize the binding of immunogenic gluten peptides to APC. The model predicts that the treatment of CD by such gluten peptide analogs can lead to a decrease in antibody levels to those of normal healthy people, and to a significant increase in villous area.

Conclusions

The developed mathematical model of immune response in CD allows prediction of the efficacy of TG-2 inhibitors and other possible drugs for the treatment of CD: their influence on the intestinal villous area and on the antibody levels. The model also allows to understand what processes in the immune response have the strongest influence on the efficacy of different drugs. This model could be applied in the pharmaceutical R&D arena for the design of drugs against autoimmune small intestine disorders and on the design of their corresponding clinical trials.

A theory of germinal center B cell selection, division, and exit

High-affinity antibodies are generated in germinal centers in a process involving mutation and selection of B cells. Information processing in germinal center reactions has been investigated in a number of recent experiments. These have revealed cell migration patterns, asymmetric cell divisions, and cell-cell interaction characteristics, used here to develop a theory of germinal center B cell selection, division, and exit (the LEDA model). According to this model, B cells selected by T follicular helper cells on the basis of successful antigen processing always return to the dark zone for asymmetric division, and acquired antigen is inherited by one daughter cell only. Antigen-retaining B cells differentiate to plasma cells and leave the germinal center through the dark zone. This theory has implications for the functioning of germinal centers because compared to previous models, high-affinity antibodies appear one day earlier and the amount of derived plasma cells is considerably larger.

Optimality of mutation and selection in germinal centers

The population dynamics theory of B cells in a typical germinal center could play an important role in revealing how affinity maturation is achieved. However, the existing models encountered some conflicts with experiments. To resolve these conflicts, we present a coarse-grained model to calculate the B cell population development in affinity maturation, which allows a comprehensive analysis of its parameter space to look for optimal values of mutation rate, selection strength, and initial antibody-antigen binding level that maximize the affinity improvement. With these optimized parameters, the model is compatible with the experimental observations such as the ∼100-fold affinity improvements, the number of mutations, the hypermutation rate, and the “all or none” phenomenon. Moreover, we study the reasons behind the optimal parameters. The optimal mutation rate, in agreement with the hypermutation rate in vivo, results from a tradeoff between accumulating enough beneficial mutations and avoiding too many deleterious or lethal mutations. The optimal selection strength evolves as a balance between the need for affinity improvement and the requirement to pass the population bottleneck. These findings point to the conclusion that germinal centers have been optimized by evolution to generate strong affinity antibodies effectively and rapidly. In addition, we study the enhancement of affinity improvement due to B cell migration between germinal centers. These results could enhance our understanding of the functions of germinal centers.

The antibodies in our immune system could efficiently improve their abilities in recognizing new antigens. This is done with the help of proliferation, mutation and selection of B cells which carry antibodies, but we have difficulties in developing a quantitative description of this adaptation process which is consistent with the various aspects of experimental observations. Based on the knowledge from experiments, here we present a theoretical model to calculate the numbers of B cells with different antigen recognizing abilities all the time, and look for the best possible design that improves the antigen recognizing ability most efficiently. We find that the best possible design is consistent with the experimental observations, pointing to the conclusion that the immune system has been optimized in evolution. We then study the trade-offs leading to the optimization of the design. The results will not only improve our understanding of the functions in immune system, but also reveal the design principles behind the details. In addition, the study enhances our understanding of the population dynamics in evolution.

Exploration of factors affecting the onset and maturation course of follicular lymphoma through simulations of the germinal center

Genetic mutations frequently observed in human follicular lymphoma (FL) B-cells result in aberrant expression of the anti-apoptotic protein bcl-2 and surface immunoglobulins (Igs) which display one or more novel variable (V) region N-glycosylation motifs. In the present study, we develop a simulation model of the germinal center (GC) to explore how these mutations might influence the emergence and clonal expansion of key mutants which provoke FL development. The simulations employ a stochastic method for calculating the cellular dynamics, which incorporates actual IgV region sequences and a simplified hypermutation scheme. We first bring our simulations into agreement with experimental data for well-characterized normal and bcl-2(+) anti-hapten GC responses in mice to provide a model for understanding how bcl-2 expression leads to permissive selection and memory cell differentiation of weakly competitive B-cells. However, as bcl-2 expression in the GC alone is thought to be insufficient for FL development, we next monitor simulated IgV region mutations to determine the emergence times of key mutants displaying aberrant N-glycosylation motifs recurrently observed in human FL IgV regions. Simulations of 26 germline V(H) gene segments indicate that particular IgV regions have a dynamical selective advantage by virtue of the speed with which one or more of their key sites can generate N-glycosylation motifs upon hypermutation. Separate calculations attribute the high occurrence frequency of such IgV regions in FL to an ability to produce key mutants at a fast enough rate to overcome stochastic processes in the GC that hinder clonal expansion. Altogether, these simulations characterize three pathways for FL maturation through positively selected N-glycosylations, namely, via one of two key sites within germline V(H) region gene segments, or via a site in the third heavy chain complementarity-determining region (CDR-H3) that is generated from VDJ recombination.

A virtual look at Epstein-Barr virus infection: simulation mechanism

Epstein-Barr virus (EBV) is an important human pathogen that establishes a life-long persistent infection and for which no precise animal model exists. In this paper, we describe in detail an agent-based model and computer simulation of EBV infection. Agents representing EBV and sets of B and T lymphocytes move and interact on a three-dimensional grid approximating Waldeyer's ring, together with abstract compartments for lymph and blood. The simulation allows us to explore the development and resolution of virtual infections in a manner not possible in actual human experiments. Specifically, we identify parameters capable of inducing clearance, persistent infection, or death.

Recirculation of germinal center B cells: a multilevel selection strategy for antibody maturation

Optimization of antibody affinity is a hallmark of the humoral immune response. It takes place in hundreds of transient microstructures called germinal centers (GCs). Their function and time-dependent behavior are subjects of active investigation. According to a generally accepted notion, their individual kinetics follows the average kinetics of all GCs present in the observed lymphatic tissue. In this review, we challenge this view and point out, with the help of mathematical simulations, that inferring the kinetics of individual GCs from cross-sectional evaluation of GC kinetics is virtually impossible. Thus, the time course of individual GCs is open to conjecture. For instance, one possible interpretation is that GCs exist for a time span considerably shorter than that of the observed average kinetics. We explore the implications of different temporal organizations of GCs in the light of the hypothesis that GC B-cell emigrants recolonize GC niches. This assumption leads to a view where GCs work in parallel but are linked by recirculation of B-cell emigrants. In this view, interleaved global and local competition provide for an implementation of multiple levels of B-cell selection in affinity maturation. The concepts of iteration, all-or-none behavior, and phasic mutation schedule are discussed in the light of this hypothesis.

The evolution of mathematical immunology

The types of mathematical models used in immunology and their scope have changed drastically in the past 10 years. Classical models were based on ordinary differential equations (ODEs), difference equations, and cellular automata. These models focused on the 'simple' dynamics obtained between a small number of reagent types (e.g. one type of receptor and one type of antigen or two T-cell populations). With the advent of high-throughput methods, genomic data, and unlimited computing power, immunological modeling shifted toward the informatics side. Many current applications of mathematical models in immunology are now focused around the concepts of high-throughput measurements and system immunology (immunomics), as well as the bioinformatics analysis of molecular immunology. The types of models have shifted from mainly ODEs of simple systems to the extensive use of Monte Carlo simulations. The transition to a more molecular and more computer-based attitude is similar to the one occurring over all the fields of complex systems analysis. An interesting additional aspect in theoretical immunology is the transition from an extreme focus on the adaptive immune system (that was considered more interesting from a theoretical point of view) to a more balanced focus taking into account the innate immune system also. We here review the origin and evolution of mathematical modeling in immunology and the contribution of such models to many important immunological concepts.

Pathogen responses to host immunity: the impact of time delays and memory on the evolution of virulence

Current analytical models of the mammalian immune system typically assume a specialist predator-prey relationship between invading pathogens and the active components of the immune response. However, in reality, the specific immune system is not immediately effective following invasion by a novel pathogen. First, there may be an explicit time delay between infection and immune initiation and, second, there may be a gradual build-up in immune efficacy (for instance, during the period of B-cell affinity maturation) during which the immune response develops, before reaching maximal specificity to the pathogen. Here, we use a novel theoretical approach to show that these processes, together with the presence of long-lived immune memory, decouple the immune response from current pathogen levels, greatly changing the dynamics of the pathogen-immune system interaction and the ability of the immune response to eliminate the pathogen. Furthermore, we use this model to show how distributed primary immune responses combine with immune memory to greatly affect the optimal virulence of the pathogen, potentially resulting in the evolution of highly virulent pathogens.

Are T cells at the origin of B cell lymphomas?

Lymphoma pathogenesis is at least in some cases related to transformed B cells (BCs) arising from germinal centre reactions (GCRs). In this article possible deregulations of GCRs are investigated using in silico simulations. It is found that the final differentiation of BCs as regulated by helper T cells (TCs) is the best candidate mechanism for such a deregulation. This shifts the paradigm of BC lymphoma pathogenesis from BC transformations to an emphasized role of TC-BC interactions.

Antibody Quality in Old Age

There is very little change in the quantity of antibodies people produce, of any isotype, with age. However, there is a change in the quality of the antibody response. Older people produce fewer antibodies that are specific for the activating pathogen or vaccine. At the same time, the number of nonspecific antibodies increases. Quite often these antibodies have self-reactivity (e.g., anti-dsDNA). The appearance of these antibodies is not associated with pathogenic autoimmune disease, although it is true that the incidence of some autoimmune diseases increases with age. The authors postulate that the process of antibody affinity maturation is compromised in old age. No evidence was found that the process of hypermutation is compromised with age. However, using graph theory to study the dynamics of a germinal center selection process, a decrease in the extent of selection occurring in the germinal centers of mucosal tissue was observed with age. This is a tissue-specific phenomenon because the decrease was not seen in the germinal centers of spleen. Because selection of highly specific cells in the germinal center depends on a number of factors (number and quality of founder cells, help from T cells, and follicular dendritic cells) these need to be investigated further to determine what is needed to improve the affinity mutation process.

Stochastic discrete event simulation of germinal center reactions

We introduce a generic reaction-diffusion model for germinal center reactions and perform numerical simulations within a stochastic discrete event approach. In contrast to the frequently used deterministic continuum approach, each single reaction event is monitored in space and time in order to simulate the correct time evolution of this complex biological system. Germinal centers play an important role in the immune system by performing a reaction that aims at improving the affinity between antibodies and antigens. Our model captures experimentally observed features of this reaction, such as the development of the remarkable germinal center morphology and the maturation of antibody-antigen affinity in the course of time. We model affinity maturation within a simple affinity class picture and study it as a function of the distance between the initial antibody-antigen affinity and the highest possible affinity. The model reveals that this mutation distance may be responsible for the experimentally observed all-or-none behavior of germinal centers; i.e., they generate either mainly output cells of high affinity or no high-affinity output cells at all. Furthermore, the exact simulation of the system dynamics allows us to study the hypothesis of cell recycling in germinal centers as a mechanism for affinity optimization. A comparison of three possible recycling pathways indicates that affinity maturation is optimized by a recycling pathway that has previously not been taken into account in deterministic continuum models.

Cutting edge: back to "one-way" germinal centers

The present status of germinal center (GC) research is revisited using in silico simulations based on recent lymphocyte motility data in mice. The generally adopted view of several rounds of somatic hypermutations and positive selection is analyzed with special emphasis on the spatial organization of the GC reaction. We claim that the development of dark zones is not necessary for successful GC reactions to develop. We find that a recirculation of positively selected centrocytes to the dark zone is rather unlikely. Instead we propose a scenario that combines a multiple-step mutation and selection concept with a "one-way" GC in the sense of cell migration.