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.     

Recycling probability and dynamical properties of germinal center reactions

We introduce a new model for the dynamics of centroblasts and centrocytes in a germinal center. The model reduces the germinal center reaction to the elements considered as essential and embeds proliferation of centroblasts, point mutations of the corresponding antibody types represented in a shape space, differentiation to centrocytes, selection with respect to initial antigens, differentiation of positively selected centrocytes to plasma or memory cells and recycling of centrocytes to centroblasts. We use exclusively parameters with a direct biological interpretation such that, once determined by experimental data, the model gains predictive power. Based on the experiment of Han et al. (1995b) we predict that a high rate of recycling of centrocytes to centroblasts is necessary for the germinal center reaction to work reliably. Furthermore, we find a delayed start of the production of plasma and memory cells with respect to the start of point mutations, which turns out to be necessary for the optimization process during the germinal center reaction. The dependence of the germinal center reaction on the recycling probability is analysed.     

Modelling two possible mechanisms for the regulation of the germinal center dynamics

Research on the germinal center has tried to unravel the mechanisms that control its dynamics. In this study we focus on the termination of the germinal center reaction, which is still an open problem. We propose two hypothetical biological mechanisms that may be responsible for the control of germinal center dynamics and analyze them through mathematical models. The first one is based on the differentiation of follicular dendritic cells and/or T cells. Interaction of these cells in the differentiated state with germinal center B cells would promote B cell differentiation into memory B cells and Ab-forming cells, ending the germinal center reaction. The second mechanism applies only to a scenario without recycling and consists of the decay of a hypothetical proliferation signal for centroblasts that limits the number of cell divisions. Each of the models makes predictions that can be experimentally tested.     

A Model for Migratory B Cell Oscillations from Receptor Down-Regulation Induced by External Chemokine Fields

A long-standing paradigm in B cell immunology is that effective somatic hypermutation and affinity maturation require cycling between the dark zone and light zone of the germinal center. The cyclic re-entry hypothesis was first proposed based on considerations of the efficiency of affinity maturation using an ordinary differential equations model for B cell population dynamics. More recently, two-photon microscopy studies of B cell motility within lymph nodes in situ have revealed the complex migration patterns of B lymphocytes both in the preactivation follicle and post-activation germinal center. There is strong evidence that chemokines secreted by stromal cells and the regulation of cognate G-protein coupled receptors by these chemokines are necessary for the observed spatial cell distributions. For example, the distribution of B cells within the light and dark zones of the germinal center appears to be determined by the reciprocal interaction between the level of the CXCR4 and CXCR5 receptors and the spatial distribution of their respective chemokines CXCL12 and CXCL13. Computer simulations of individual-based models have been used to study the complex biophysical and mechanistic processes at the individual cell level, but such simulations can be challenging to parameterize and analyze. In contrast, ordinary differential equations are more tractable, but traditional compartment model formalizations ignore the spatial chemokine distribution that drives B cell redistribution. Motivated by the desire to understand the motility patterns observed in an individual-based simulation of B cell migration in the lymph node, we propose and analyze the dynamics of an ordinary differential equation model incorporating explicit chemokine spatial distributions. While there is experimental evidence that B cell migration patterns in the germinal center are driven by extrinsically regulated differentiation programs, the model shows, perhaps surprisingly, that feedback from receptor down-regulation induced by external chemokine fields can give rise to spontaneous interzonal and intrazonal oscillations in the absence of any extrinsic regulation. While the extent to which such simple feedback mechanisms contributes to B cell migration patterns in the germinal center is unknown, the model provides an alternative hypothesis for how complex B cell migration patterns might arise from very simple mechanisms.     

Germinal centers and the B-cell system

Summary Germinal centers of the rabbit appendix were studied for the presence of complement receptors, immunoglobulin and alkaline phosphatase. In popliteal lymph nodes, de novo-developing germinal centers were studied with respect to these markers up to 21 days after sheep red blood cell (SRBC)-stimulation. In addition, the possible presence of antigen (SRBC) receptor-bearing cells in these germinal centers was investigated.The results may be summarized as follows: 1) Germinal centers in the appendix as well as those in popliteal lymph nodes were rich in complement receptor-bearing cells. Complement-receptor density did not significantly change during a germinal-center reaction. 2) Immunoglobulins were present only at very low densities on the surface of lymphoid cells in the densely populated area of germinal centers. In germinal centers of popliteal lymph nodes lymphoid cells in the thinly populated area again showed higher densities. Immunoglobulins possibly complexed with antigen on the surface of follicular dendritic cells were not observed in the initial phase of a germinal center reaction. In contrast, in germinal centers of the appendix, immunoglobulin was present in excessive amounts throughout the thinly populated area, possibly complexed with antigen, which is also abundantly present. 3) Reticular staining patterns of alkaline phosphatase were present in the densely populated area, but absent in the thinly populated area of germinal centers in both appendix and popliteal lymph nodes. Primary follicles and young germinal centers were alkaline phosphatase-negative. 4) Antigen receptor-bearing cells were detected in germinal centers of popliteal lymph nodes as early as 5 days after SRBC-stimulation, reaching a maximum at day 10.In conclusion, with the present experimental approach, microenvironmental differences were shown between the densely populated area and the thinly populated area of germinal centers. However, no indication was obtained for a postulated maturation event of the lymphocytes within germinal centers, or for functional differences that may exist between germinal centers in the appendix and popliteal lymph nodes.     

Some state space models of HIV pathogenesis under treatment by anti-viral drugs in HIV-infected individuals

In this paper we have extended the model of HIV pathogenesis under treatment by anti-viral drugs given by Perelson et al. [A.S. Perelson et al., Science 271 (1999) 1582] to a stochastic model. By using this stochastic model as the stochastic system model, we have developed a state space model for the HIV pathogenesis under treatment by anti-viral drugs. In this state space model, the observation model is a statistical model based on the observed numbers of RNA virus copies over different times. For this model we have developed procedures for estimating and predicting the numbers of infectious free HIV and non-infectious free HIV as well as the numbers of different types of T cells through extended Kalman filter method. As an illustration, we have applied the method of this paper to the data of patient Nos. 104, 105 and 107 given by Perelson et al. [A.S. Perelson et al., Science 271 (1999) 1582] under treatment by Ritonavir. For these individuals, it is shown that within two weeks since treatment, most of the free HIV are non-infectious, indicating the usefulness of the treatment. Furthermore, the Kalman filter method revealed a much stronger effect of the treatment within the first 10 to 20 h than that predicted by the deterministic model.     

Germinal centers and the B-cell system

Summary The migration pattern of germinal center cells of the rabbit appendix was studied and compared with that of appendix dome cells, spleen cells, thymus cells and thoracic duct lymphocytes. To discriminate T-and B-cell migration pathways, normal or T-cell-depleted rabbits were used as donors. Cell suspensions were labeled in vitro with ³H-leucine followed by intravenous transfer. The migration of labeled cells in lymphoid organs was studied using autoradiography, particular attention being paid to the spleen of the recipient. B-cells from the appendix dome, spleen and thoracic-duct lymph migrate to primary follicles or the corona of secondary follicles via thymus-dependent areas of peripheral lymphoid organs. In contrast, a B-cell subpopulation from the germinal centers of the appendix migrates to the center of splenic primary follicles and into germinal centers. The migration of germinal center cells to splenic follicle centers is not enhanced by specific antigens. The migration properties of B-cells, possibly changing during differentiation, may be instrumental in the two types of immune reactions, i.e., plasma-cell reaction and germinal-center reaction.     

B cell activation state-governed formation of germinal centers following viral infection

Germinal centers are structures that promote humoral memory cell formation and affinity maturation, but the triggers for their development are not entirely clear. Activated extrafollicular B cells can form IgM-producing plasmablasts or enter a germinal center reaction and differentiate into memory or plasma cells, mostly of the IgG isotype. Vesicular stomatitis virus (VSV) induces both types of response, allowing events that promote each of these pathways to be studied. In this work, extrafollicular vs germinal center responses were examined at a cellular level, analyzing VSV-specific B cells in infected mice. We show that VSV-specific germinal centers are transiently formed when insufficient proportions of specific T cell help is available and that strong B cell activation in cells expressing high levels of the VSV-specific BCR promoted their differentiation into early blasts, whereas moderate stimulation of B cells or interaction with Th cells restricted extrafollicular responses and promoted germinal center formation.     

Modeling collective animal behavior with a cognitive perspective: a methodological framework

The last decades have seen an increasing interest in modeling collective animal behavior. Some studies try to reproduce as accurately as possible the collective dynamics and patterns observed in several animal groups with biologically plausible, individual behavioral rules. The objective is then essentially to demonstrate that the observed collective features may be the result of self-organizing processes involving quite simple individual behaviors. Other studies concentrate on the objective of establishing or enriching links between collective behavior researches and cognitive or physiological ones, which then requires that each individual rule be carefully validated. Here we discuss the methodological consequences of this additional requirement. Using the example of corpse clustering in ants, we first illustrate that it may be impossible to discriminate among alternative individual rules by considering only observational data collected at the group level. Six individual behavioral models are described: They are clearly distinct in terms of individual behaviors, they all reproduce satisfactorily the collective dynamics and distribution patterns observed in experiments, and we show theoretically that it is strictly impossible to discriminate two of these models even in the limit of an infinite amount of data whatever the accuracy level. A set of methodological steps are then listed and discussed as practical ways to partially overcome this problem. They involve complementary experimental protocols specifically designed to address the behavioral rules successively, conserving group-level data for the overall model validation. In this context, we highlight the importance of maintaining a sharp distinction between model enunciation, with explicit references to validated biological concepts, and formal translation of these concepts in terms of quantitative state variables and fittable functional dependences. Illustrative examples are provided of the benefits expected during the often long and difficult process of refining a behavioral model, designing adapted experimental protocols and inversing model parameters.     

CXCR3 is induced early on the pathway of CD4+ T cell differentiation and bridges central and peripheral functions

Chemokine receptors on T cells are frequently categorized as functioning either in immune system homeostasis within lymphoid organs, or in peripheral inflammation. CXCR3 is in the latter category and is reported to be expressed selectively on Th1 cells. We found that CXCR3 was expressed in vivo on newly activated tonsillar CD4(+) T cells. Using CD4(+) T cells from cord blood, we found that CXCR3 was induced by cellular activation in vitro independently of the cytokine milieu, although on resting cells, expression was maintained preferentially on those that had been activated in type 1 conditions. In inflamed tonsils, CXCR3(+)CD4(+) T cells were localized around and within germinal centers. The inference that CXCR3 has a role in germinal center reactions was supported by the finding that the CXCR3 ligand CXC chemokine ligand 9 was expressed in a pattern demarcating a subset of germinal centers both in tonsil and in lymph nodes from an HIV-infected individual. We next investigated the role of CXCR3 on peripheral effector/memory CD4(+) T cells by comparing its pattern of expression with that of CCR5, another Th1-cell associated chemokine receptor. Analysis of cells directly from peripheral blood and after activation in vitro suggested that CXCR3 expression preceded that of CCR5, supporting a model of sequential induction of chemokine receptors during CD4(+) T cell differentiation. Taken together, our data show that CXCR3 can be expressed at all stages of CD4(+) T cell activation and differentiation, bridging central function in lymphoid organs and effector function in peripheral tissues.     

Established T cell-driven germinal center B cell proliferation is independent of CD28 signaling but is tightly regulated through CTLA-4

CD4 T cell activation is positively (CD28) and negatively (CTLA-4) regulated by the costimulatory ligands CD80 and CD86. A central question is how the balance between these two opposing forces is controlled as T cells differentiate. We have previously shown that CD28 signaling is absolutely required to prime naive CD4 T cells to differentiate into effectors that provide help for germinal centers and class-switched Ab responses. In this study, we show that the requirement for CD28 signaling is transient and effector CD4 T cells do not require CD28 signals to sustain their function. The CD28 independence of effector T cells within germinal centers suggested that a key function for CD80/CD86 under these circumstances might be to provide negative regulatory signals via the CD28 homologue CTLA-4. By examining germinal center responses in mice where the ability to signal through T cell CTLA-4 was compromised, we provide data that supports a critical role for CTLA-4 in down-regulating T cell help for germinal center B cells.     

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.     

Textbook germinal centers?

Models for the development and function of germinal centers (GCs) have been so widely discussed in the original literature that they now appear in immunology textbooks. Unfortunately, many of the tenets of these models have not yet been subjected to adequate experimental scrutiny. Indeed, recent studies have called several of their principal assumptions into question. In addition, the term germinal center has been applied to a diverse assortment of focal processes of B cell proliferation and differentiation. This variability might be explained by alterations in the progression of a single textbook GC process. Alternatively, distinct developmental pathways may create unique classes of GCs with specialized functions.     

Individual-to-Resource Landscape Interaction Strength Can Explain Different Collective Feeding Behaviours

Taking in sufficient quantities of nutrients is vital for all living beings and in doing so, individuals interact with the local resource environment. Here, we focus explicitly on the interactions between feeding individuals and the resource landscape. In particular, we are interested in the emergent movement dynamics resulting from these interactions. We present an individual-based simulation model for the movement of populations in a resource landscape that allows us to vary the strength of the interactions mentioned above. The key assumption and novelty of our model is that individuals can cause the release of additional nutrients, as well as consuming them. Our model produces clear predictions. For example, we expect more tortuous individual movement paths and higher levels of aggregation in populations occupying homogeneous environments where individual movement makes more nutrients available. We also show how observed movement dynamics could change when local nutrient sources are depleted or when the population density increases. Our predictions are testable and qualitatively reproduce the different feeding behaviours observed in filter-feeding ducks, for example. We suggest that considering two-way interactions between feeding individuals and resource landscapes could help to explain fine-scale movement dynamics.     

Inverse analysis of empirical matrices of idiotypic network interactions

The concept of shape space proposed by Perelson and Oster (1979,J. Theor. Biol. 81, 645–670) has been a useful tool for theoretical immunologists, who have invoked it to model idiotypic binding, which plays a significant role in mathematical models of immune networks. The actual construction of such a space from its definition requires specialized experimental information, which is not completely available. In this article, we discuss, with illustrative examples, how graphical representations similar to the idea of shape space can be derived by analyzing real affinity matrices, and the relative merits of such representations to approximations that might be obtained by the approach of Perelson and Oster. We also give directions for future research with a view toward applications.     

Mouse lymph node germinal centers contain a selected subset of T cells--the helper phenotype

Cells staining for Lyt-1 are more frequent than cells staining for Lyt-2 in both primary follicles and the cuffs of secondary follicles; there is an even more striking predominance of cells bearing only Lyt-1 in germinal centers. In addition, there is an increase in the total percentage of cells bearing T cell antigens in germinal centers compared to primary follicles. These differences in phenotype and distribution of T cell populations indicate the T cells in B cell areas, and especially in germinal centers, are not randomly selected, but rather represent a specific subpopulation of T cells enriched for the helper phenotype (Lyt-1+2-), perhaps involved in the development and/or function of germinal centers.     

Mechanism for Anaphase B: Evaluation of “Slide-and-Cluster” versus “Slide-and-Flux-or-Elongate” Models

Elongation of the mitotic spindle during anaphase B contributes to chromosome segregation in many cells. Here, we quantitatively test the ability of two models for spindle length control to describe the dynamics of anaphase B spindle elongation using experimental data from Drosophila embryos. In the slide-and-flux-or-elongate (SAFE) model, kinesin-5 motors persistently slide apart antiparallel interpolar microtubules (ipMTs). During pre-anaphase B, this outward sliding of ipMTs is balanced by depolymerization of their minus ends at the poles, producing poleward flux, while the spindle maintains a constant length. Following cyclin B degradation, ipMT depolymerization ceases so the sliding ipMTs can push the poles apart. The competing slide-and-cluster (SAC) model proposes that MTs nucleated at the equator are slid outward by the cooperative actions of the bipolar kinesin-5 and a minus-end-directed motor, which then pulls the sliding MTs inward and clusters them at the poles. In assessing both models, we assume that kinesin-5 preferentially cross-links and slides apart antiparallel MTs while the MT plus ends exhibit dynamic instability. However, in the SAC model, minus-end-directed motors bind the minus ends of MTs as cargo and transport them poleward along adjacent, parallel MT tracks, whereas in the SAFE model, all MT minus ends that reach the pole are depolymerized by kinesin-13. Remarkably, the results show that within a narrow range of MT dynamic instability parameters, both models can reproduce the steady-state length and dynamics of pre-anaphase B spindles and the rate of anaphase B spindle elongation. However, only the SAFE model reproduces the change in MT dynamics observed experimentally at anaphase B onset. Thus, although both models explain many features of anaphase B in this system, our quantitative evaluation of experimental data regarding several different aspects of spindle dynamics suggests that the SAFE model provides a better fit.     

Sample size determination for hierarchical longitudinal designs with differential attrition rates

Summary .   We consider the problem of sample size determination for three-level mixed-effects linear regression models for the analysis of clustered longitudinal data. Three-level designs are used in many areas, but in particular, multicenter randomized longitudinal clinical trials in medical or health-related research. In this case, level 1 represents measurement occasion, level 2 represents subject, and level 3 represents center. The model we consider involves random effects of the time trends at both the subject level and the center level. In the most common case, we have two random effects (constant and a single trend), at both subject and center levels. The approach presented here is general with respect to sampling proportions, number of groups, and attrition rates over time. In addition, we also develop a cost model, as an aid in selecting the most parsimonious of several possible competing models (i.e., different combinations of centers, subjects within centers, and measurement occasions). We derive sample size requirements (i.e., power characteristics) for a test of treatment-by-time interaction(s) for designs based on either subject-level or cluster-level randomization. The general methodology is illustrated using two characteristic examples.