A mathematical model for germinal centre kinetics and affinity maturation

We present a mathematical model which reproduces experimental data on the germinal centre (GC) kinetics of the primed primary immune response and on affinity maturation observed during the reaction. We show that antigen masking by antibodies which are produced by emerging plasma cells can drive affinity maturation and provide a feedback mechanism by which the reaction is stable against variations in the initial antigen amount over several orders of magnitude. This provides a possible answer to the long-standing question of the role of antigen reduction in driving affinity maturation. By comparing model predictions with experimental results, we propose that the selection probability of centrocytes and the recycling probability of selected centrocytes are not constant but vary during the GC reaction with respect to time. It is shown that the efficiency of affinity maturation is highest if clones with an affinity for the antigen well above the average affinity in the GC leave the GC for either the memory or plasma cell pool. It is further shown that termination of somatic hypermutation several days before the end of the germinal centre reaction is beneficial for affinity maturation. The impact on affinity maturation of simultaneous initiation of memory cell formation and somatic hypermutation vs. delayed initiation of memory cell formation is discussed.     

Antigen-driven selection in germinal centers as reflected by the shape characteristics of immunoglobulin gene lineage trees: a large-scale simulation study

During the immune response, the generation of memory B lymphocytes in germinal centers involves affinity maturation of the cells’ antigen receptors, based on somatic hypermutation of receptor genes and antigen-driven selection of the resulting mutants. Affinity maturation is vital for immune protection, and is the basis of humoral immune learning and memory. Lineage trees of somatically hypermutated immunoglobulin genes often serve to qualitatively illustrate claims concerning the dynamics of affinity maturation in germinal centers. Here, we derive the quantitative relationships between parameters characterizing affinity maturation dynamics (proliferation, differentiation and mutation rates, initial affinity of the Ig to the antigen, and selection thresholds) and the mathematical properties of lineage trees, using a computer simulation which combines mathematical models for all mature B cell populations, stochastic models of hypermutation and selection, lineage tree generation and measurement of graphical tree characteristics. We identified seven key lineage tree properties, and found correlations of these with initial clone affinity and with the selection threshold. These two parameters were found to be the main factors affecting lineage tree shapes in both primary and secondary response trees. The results also confirm that recycling from centrocytes back to centroblasts is highly likely.     

Does recycling in germinal centres exist?

A general criterion is formulated to decide if the recycling of B cells exists in germinal centre (GC) reactions. The criterion is independent of the selection and affinity maturation process and based solely on total centroblast population arguments. An experimental test is proposed to verify whether the criterion is fulfilled.     

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.     

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.     

Maturation of the immune response in germinal centers.

Germinal centers develop in peripheral lymphatic tissue during the primary immune response and may play a crucial role in affinity maturation. We have compared the diversification of the antigen-specific repertoire of B cells, both from within and from outside the germinal centers, during the murine response to 2-phenyloxazolone (phOx). By sequencing V kappa Ox1 L-chains characteristic of phOx-specific antibodies, we show that somatic mutations accumulate in germinal center B cells and that a mutation conferring high affinity binding is found with increasing frequency. An analysis of V/D/J rearrangements suggests that this mutation occurred independently in many B cells, which were then preferentially expanded. We conclude that, although the hypermutation mechanism may be activated before germinal centers develop, affinity maturation by hypermutation and selection takes place in the germinal centers.     

Follicular regulatory T cells expressing Foxp3 and Bcl-6 suppress germinal center reactions

Foxp3(+) regulatory T (T(reg)) cells suppress different types of immune responses to help maintain homeostasis in the body. How T(reg) cells regulate humoral immunity, including germinal center reactions, is unclear. Here we identify a subset of T(reg) cells expressing CXCR5 and Bcl-6 that localize to the germinal centers in mice and humans. The expression of CXCR5 on T(reg) cells depends on Bcl-6. These CXCR5(+)Bcl-6(+) T(reg) cells are absent in the thymus but can be generated de novo from CXCR5(-)Foxp3(+) natural T(reg) precursors. A lack of CXCR5(+) T(reg) cells leads to greater germinal center reactions including germinal center B cells, affinity maturation of antibodies and the differentiation of plasma cells. These results unveil a Bcl-6-CXCR5 axis in T(reg) cells that drives the development of follicular regulatory T (T(FR)) cells that function to inhibit the germinal center reactions.     

Evolution in miniature: selection, survival and distribution of antigen reactive cells in the germinal centre

Productive immune responses to T-cell-dependent antigens ultimately generate two long-lived compartments: memory B cells and bone marrow-resident plasma cells, which both arise from within germinal centres. The ability of a B-cell clone to populate these effector compartments requires its descendents to outcompete those of other clones participating in the response. Selection occurs at several stages of the response and the criteria differ at these different stages. While affinity predominates as the key, underlying driving force of selection, there is a distinction made at the point at which germinal centre cells initiate entry into the plasma cell and memory B-cell compartments. Becoming a plasma cell requires high affinity and cannot be subverted by blocking cell death, while becoming a memory B cell is dependent on survival alone. While such survival is typically mediated by affinity within the GC, the distinction has important mechanistic implications.     

Key mutations stabilize antigen‐binding conformation during affinity maturation of a broadly neutralizing influenza antibody lineage

Affinity maturation, the process in which somatic hypermutation and positive selection generate antibodies with increasing affinity for an antigen, is pivotal in acquired humoral immunity. We have studied the mechanism of affinity gain in a human B‐cell lineage in which two main maturation pathways, diverging from a common ancestor, lead to three mature antibodies that neutralize a broad range of H1 influenza viruses. Previous work showed that increased affinity in the mature antibodies derives primarily from stabilization of the CDR H3 loop in the antigen‐binding conformation. We have now used molecular dynamics simulations and existing crystal structures to identify potentially key maturation mutations, and we have characterized their effects on the CDR H3 loop and on antigen binding using further simulations and experimental affinity measurements, respectively. In the two maturation pathways, different contacts between light and heavy chains stabilize the CDR H3 loop. As few as two single‐site mutations in each pathway can confer substantial loop stability, but none of them confers experimentally detectable stability on its own. Our results support models of the germinal center reaction in which two or more mutations can occur without concomitant selection and show how divergent pathways have yielded functionally equivalent antibodies. Proteins 2014; 83:771–780. © 2014 The Authors. Proteins: Structure, Function, and Bioinformatics Published by Wiley Periodicals, Inc.     

In vivo imaging studies shed light on germinal-centre development

Affinity maturation of antibodies during the course of an adaptive immune response requires germinal centre (GC) formation within B-cell follicles. Much of the current understanding of GC function has been derived from histology, but these static views have left unresolved many questions about cell movement in GCs. In this Progress article, we describe how several recent studies using time-resolved multiphoton microscopy to track GC B-cell movement within lymph nodes have shed light on the processes that influence GC B-cell dynamics.     

What limits affinity maturation of antibodies in Xenopus--the rate of somatic mutation or the ability to select mutants?

Although the Xenopus immunoglobulin heavy chain locus is structurally and functionally similar to mammalian IgH loci, Xenopus antibodies are limited in heterogeneity, and they mature only slightly in affinity during immune responses. During the antibody response of isogenic frogs to DNP-KLH, mu and upsilon cDNA sequences using elements of the VH1 family were cloned, sequenced and compared with germline counterparts. There were zero to four mutations per sequence, mostly single base substitutions, in the framework and CDRs 1 and 2 of VH. No mutations were found in JH. Since the point mutation rate was only 4- to 7-fold lower than that calculated for mice, affinity maturation does not seem to be limited by mutant availability. Because of a relatively low ratio of replacement to silent mutations in the CDRs and a very high ratio of GC to AT base pairs altered by mutation, it is suggested that the problem results from the absence of an effective mechanism for selecting mutants, which in turn might be related to the absence of germinal centers in Xenopus.     

Immune network behavior—II. From oscillations to chaos and stationary states

Two types of behavior have been previously reported in models of immune networks. The typical behavior of simple models, which involve B cells only, is stationary behavior involving several steady states. Finite amplitude perturbations may cause the model to switch between different equilibria. The typical behavior of more realistic models, which involve both B cells and antibody, consists of autonomous oscillations and/or chaos. While stationary behavior leads to easy interpretations in terms of idiotypic memory, oscillatory behavior seems to be in better agreement with experimental data obtained in unimmunized animals. Here we study a series of models of the idiotypic interaction between two B cell clones. The models differ with respect to the incorporation of antibodies, B cell maturation and compartmentalization. The most complicated model in the series has two realistic parameter regimes in which the behavior is respectively stationary and chaotic. The stability of the equilibrium states and the structure and interactions of the stable and unstable manifolds of the saddle-type equilibria turn out to be factors influencing the model's behavior. Whether or not the model is able to attain any form of sustained oscillatory behavior, i.e. limit cycles or chaos, seems to be determined by (global) bifurcations involving the stable and unstable manifolds of the equilibrium states. We attempt to determine whether such behavior should be expected to be attained from reasonable initial conditions by incorporating an immune response to an antigen in the model. A comparison of the behavior of the model with experimental data from the literature provides suggestions for the parameter regime in which the immune system is operating.     

Toward quantitative simulation of germinal center dynamics: biological and modeling insights from experimental validation.

As models of immune system dynamics are developed, it is important to validate them with specific experimental data in order to understand their shortcomings and guide them toward becoming predictive. In this paper, we examine whether a particular mathematical model of germinal center dynamics, proposed by Oprea and Perelson, can reproduce experimental data from two specific primary responses, namely those directed against the haptens 2-phenyl-5-oxazolone and (4-hydroxy-3-nitrophenyl)acetyl. We develop formulas for estimating response-specific model parameters, as well as constraints for validating the model. In addition, we outline a general methodology for translating a continuous/deterministic model, expressed as a set of ordinary differential equations, into a discrete/stochastic framework. This methodology is used to create a new implementation of the Oprea and Perelson model that enables comparison with data on individual germinal centers. We conclude that while the model can reproduce the average dynamics of splenic germinal centers, it is at best incomplete and does not reproduce the distribution of individual germinal center behaviors. In addition to suggesting possible extensions to the model which can reconcile the dynamics with some aspects of the experimental data, we make a number of specific predictions that can be tested by in vivo experiments to obtain further insights and validation.     

Heterogeneity and function of human B lymphocytes

B lymphocytes represent an important arm of the immune system. Besides their main function of providing antibodies protecting against pathogens, they also exert some regulatory functions, in particular for secondary lymphoid tissue differentiation. Human B cells can be divided in various subsets representing different maturation stages and different pathways of humoral immune responses. Na?ve IgMlow IgDhigh CD27- B cells can participate in T-cell dependent immune responses leading to germinal center formation in follicles of secondary lymphoid organs. Interactions with follicular helper T cells, a recently identified CD185+ T cell population providing help to follicular B cell, involve costimulatory molecules including CD40, CD27, CD278 and SAP-recruiting receptors. B cell interaction with follicular helper T cells represents a critical step controlling the generation of plama cells that ultimately produce high affinity, somatically mutated, class-switched antibodies or of their memory B cell counterpart (identified as CD27+ Ig switched or IgMonly B cells). IgMhigh IgDlow CD27+ B cells are a puzzling population apparently specialized in T-independent responses to bacterial capsular polysaccharides. The extra-follicular, probably antigen-independent, differentiation pathway of these cells, allowing pre-immune repertoire diversification by somatic hypermutation, is not yet characterized. However, circulating IgMhigh IgDlow CD27+ B cells are similar to splenic marginal zone B cells. In addition to these subsets, minor populations can also be identified in peripheral blood, such as transitional B cells and plasma blasts. All together, deciphering human B cell heterogeneity provides tools for investigations of humoral immunodeficiencies and auto-immune diseases, that will in return shed more light on B cell biology.     

Impaired affinity maturation in Cr2-/- mice is rescued by adjuvants without improvement in germinal center development

Cr2-/- mice have an impairment in humoral immunity, as shown by the decrease in the Ab titers against T cell-dependent Ags and abnormalities in germinal center formation. Germinal centers are present, but they are decreased in size and number, indicating problems in their development. In this study, we investigated whether this abnormality in germinal center development is associated with problems in the establishment of optimal affinity maturation and the generation of memory B cells, processes closely related to the germinal center reaction. We immunized the Cr2-/- animals with different Ags with or without adjuvants. We showed that, when immunized without adjuvants, complement receptors are absolutely required for optimal affinity maturation. Although limited affinity maturation is elicited in the Cr2-/- Ab response, it is decreased as compared with normal animals. Memory B cell generation is also impaired. In the presence of adjuvants, germinal center development in the Cr2-/- mice is still abnormal, as demonstrated by their decreased size and number. Surprisingly, adjuvants establish optimal affinity maturation and partially restore the amount of Ab produced during the primary response and memory B cell generation. However, adjuvants cannot improve the ability of follicular dendritic cells to retain Ags in the form of immune complexes. These observations indicate that immunization with inflammatory Ags offset some of the immunological abnormalities found in the Cr2-/- mice and show that optimal affinity maturation in the Cr2-/- mice can be achieved in the absence of normal germinal centers.     

B cells in rheumatoid arthritis

Normally the immune response is restricted to the peripheral secondary lymphoid organs. However, additional ectopic lymphoid tissue may develop at chronic sites of inflammation. In the synovium of rheumatoid arthritis patients the local production of proinflammatory cytokines seems to support the formation of a precisely structured microenvironment, which allows an antigen dependent immune response to take place. The analysis of the V-gene repertoire expressed in synovial B cells demonstrated that in the inflamed synovium a germinal centre reaction takes place. Antigen presented by a network of follicular dendritic cells may activate synovial B cells and support their differentiation into plasma cells secreting high affinity antibodies. The specificity of these antibodies remains to be determined.     

Demonstration of an in vivo generated sub-picomolar affinity fully human monoclonal antibody to interleukin-8

The high specificity and affinity of monoclonal antibodies make them attractive as therapeutic agents. In general, the affinities of antibodies reported to be high affinity are in the high picomolar to low nanomolar range and have been affinity matured in vitro. It has been proposed that there is an in vivo affinity ceiling at 100 pM and that B cells producing antibodies with affinities for antigen above the estimated ceiling would have no selective advantage in antigen-induced affinity maturation during normal immune responses. Using a transgenic mouse producing fully human antibodies, we have routinely generated antibodies with sub-nanomolar affinities, have frequently rescued antibodies with less than 10 pM affinity, and now describe the existence of an in vivo generated anti-hIL-8 antibody with a sub-picomolar equilibrium dissociation constant. This confirms the prediction that antibodies with affinities beyond the proposed affinity ceiling can be generated in vivo. We also describe the technical challenges of determining such high affinities. To further understand the importance of affinity for therapy, we have constructed a mathematical model to predict the relationship between the affinity of an antibody and its in vivo potency using IL-8 as a model antigen.     

A mathematical model for the germinal center morphology and affinity maturation

During germinal center reactions, the appearance of two specific zones are observed: the dark and the light zone. Up to now, the origin and function of these zones are poorly understood. In the framework of a stochastic and discrete model, several possible pathways of zone development during germinal center reactions are investigated. The importance of the zones in the germinal center for affinity maturation, i.e. the process of antibody optimization is discussed.