Overcoming the Dichotomy of Quantity and Quality in Antibody Responses

Germinal centers (GCs) are important structures during infections and are essential to guarantee the success of vaccination strategies. GCs are located in secondary lymphoid tissues and develop in primary follicles in response to a cognate interaction between T cells and B cells at the border of the B cell follicle and the T zone (1). The activated seeder B cells enter the follicle and start a process of monoclonal expansion giving rise to 10,000 B cells with BCRs diversified by frequent somatic hypermutation in the CDR region (2). B cells upregulate their BCR, stop dividing, and activate apoptosis, but they can be rescued from apoptosis if they are able to collect Ag presented on FcRs by follicular dendritic cells (FDCs) (3). This selection process is also associated with the development of the two characteristic zones of the GC (4): the dark zone and the light zone. B cells that managed to collect Ag internalize and process it, and present it as peptide on MHC (pMHC). Because the amount of presented pMHC depends on the amount of previously collected Ag, pMHC density is a measure for the affinity of the BCR for the Ag. In this way, T follicular helper cells (TFHs) can sense the affinity of BCRs. TFHs polarize to the B cell with highest density of pMHC (5) and provide rescue signals to only this particular B cell. Evidence for the mathematical prediction that help by TFH is the limiting factor of B cell selection and leads to strong B cell competition (6, 7) was provided in multiphoton experiments (8, 9).

The kind of functional signals that TFHs provide to selected B cells was investigated (10, 11) but is not yet fully resolved. The interaction between TFHs and B cells is believed to be involved in Ig class switching (12), as well as in fate decision, that is, either recycling to another round of division and mutation (13, 14) or final differentiation to either memory B cells or Ab-forming plasma cells (15, 16). Alternatively, final differentiation versus recycling might be also decided by asymmetric division of the collected Ag onto the daughters of dividing B cells (17, 18). According to this hypothesis, Ag-carrying B cells would preferentially differentiate to the output phenotype, whereas Ag-free B cells would stay in the GC for another round of selection.

Because TFH can sense the affinity of the BCR for the Ag via the density of presented pMHC, one can imagine that the subsequent number of divisions of selected B cells may depend on the strength of the TFH signal (19). A mathematical analysis of photoactivation experiments (9) provided further evidence for this mechanism and suggested that the induced number of divisions is in the range between one and six divisions (17). After the positive validation of this prediction in recent experiments (20), we now asked how this and other mechanisms, like affinity-dependent mutation probability, impact onto the overall success of the GC-derived Ab response.

The novel modeling framework for GC reactions presented in this article can be considered as a combination of the LEDA model (17) and a dynamic description of generated Abs including Ab feedback (AFB) onto the GC dynamics derived from chemical binding kinetics (21). The combined model was further extended to include dynamic and affinity-dependent number of B cell divisions (DND), and dynamic affinity-dependent somatic hypermutation probability (DSH).

The model was constructed as an agent-based model for cells and realized as a stochastic event generator in a discretized three-dimensional space combined with a reaction-diffusion system for soluble chemokines CXCL12 and CXCL13. CXCL13 is generated by the FDCs randomly positioned in the light zone, whereas CXCL12 is produced by stromal cells at the boundary of the GC toward the T zone. GC B cells express the respective chemokine receptors CXCR4 and CXCR5, which are dynamically upregulated and downregulated depending on whether they acquire the dark or the light zone B cell phenotype. Expression of these receptors determines the direction of B cell migration, whereby a chemokine concentration-dependent desensitization was used (22). Desensitized B cells do random walk but also resensitize when a below-threshold concentration of chemokines is locally detected.

Mutations of the CDR region of the B cell are represented in the shape space model (23). The dimension of the shape space is set to four, and the Hamming distance is used for evaluation of the distance to the optimal clone (24). A mutation is represented as a step to the next von Neumann neighbor. Every cell is associated with a position on the shape space. Seeder cells are positioned at a Hamming distance of five mutations.

The states of the in silico B cells acquired during the GC reaction are:

• B cell division and somatic hypermutation are active in the dark zone phenotype.

• Upon activation of apoptosis, B cells upregulate sensitivity to CXCL13 and have to bind Ag on FDCs within a limited time for survival.

• B cells that collected Ag stop apoptosis and search for TFH (which are also located in the light zone) and interact with TFH in a competitive way. TFH always polarize to the B cell with the highest amount of collected Ag, and B cells have to get an above-threshold amount of survival signals within a limited time (6).

• B cells surviving this interaction with TFH all acquire the dark zone phenotype and return to the dark zone for division where they divide the previously collected Ag asymmetrically onto the daughters (17, 18).

• B cells that kept the Ag leave the GC reaction as GC output, whereas all other B cells re-enter another round of division, mutation, and selection.

For a discussion of the associated parameter values, refer to Meyer-Hermann et al. (17). In this article, the features of the new modeling framework are described with a special focus on the three investigated mechanisms: DND, DSH, and AFB.

In previous models, TFH-selected B cells always were attributed the same number of divisions. The number of divisions per round of selection should be as low as possible to ensure a good affinity maturation process that was considered optimal when each mutation is followed by selection (13). However, the dark zone/light zone ratio of 2 (9) determined that two divisions per round of selection are required (17). Only then was the dark zone/light zone ratio in agreement with experiment (Fig. 1B).

The number of divisions attributed to selected B cells was assumed dependent on the amount of previously collected Ag. This quantity is correlated with the amount of pMHC presented to TFH and, indirectly, reflects the affinity of the BCR for the Ag. The dependence of the number of divisions P(A) on the amount of presented pMHC (A) is derived from a log-sigmoidal function:

The more Ag was collected by the B cell, the more divisions were induced. We set the minimum number of division to one (P_min = 1) to avoid recycling events without further division. It is limited by six divisions in the best case, which is motivated by DEC205-OVA experiments in which DEC205^+/+ BCs received abundant Ag, which increased pMHC presentation to a maximum (9). The population dynamics in vivo and in silico only matched when the number of divisions was increased to six in the simulation (17), suggesting that the strongest possible pMHC presentation to TFH induces six divisions (P_max = 6). The Hill coefficient n_P = 2 was set to guarantee larger numbers of divisions in the course of the GC reactions (Fig. 1D).

The half-value K_P remained to be determined, which denotes the amount of Ag collected by B cells at which the number of divisions becomes half maximal. The number of collected Ag portions varies between zero and a maximum determined by the duration of the Ag collection phase, the duration of each B cell interaction with FDCs, and the migration time between two Ag-presenting sites. The numbers of successful B cell–FDC encounters as observed in the simulations served as estimate of A_max. Low-affinity B cells had zero or one Ag uptake event, whereas high-affinity cells took up between 5 and 10 portions. A good starting point would be to induce the number of divisions that is used in the model with a constant number of divisions, that is, two divisions (P₀ = 2), in an intermediate range of A₀ = 4, which leads to the condition:

K_P was fine-tuned to generate comparable simulation with different assumed mechanisms (see Table I and Fig. 1A).

The probability of somatic hypermutation with effect on the BCR affinity to the Ag was dynamically regulated. It starts off at m_max = 0.5 (2) and is reduced down to m_min = 0 in an affinity-dependent way using

where Φ is the position of the BCR in the shape space, Φ* is the position of the optimal clone in the shape space, $||.||1$ is the Hamming distance of both clones, and Γ = 2.8 (24) is the width of the affinity-function in the shape space (23).

These dynamics were optionally switched off, which means that m(Φ) = m_max/2 holds throughout the GC reaction. m(Φ) = m_max was also tested but led to less output and lower mean affinity of GC output, such that these results were not further considered.

Soluble Abs generated by GC-derived plasma cells may re-enter the running GC reaction and compete with BCRs for Ag binding on FDCs. Recently, this idea got further support by the proof that injected Abs penetrate GCs and displace endogenous Abs. Depending on the affinity of the injected Ab, these were then displaced again by higher affinity Abs generated in the course of the GC reaction (21).

When AFB was switched on, the GC reaction ended earlier than before; thus, the strength of the GC reaction Ω(T) (Eq. 4) was reduced. This could be compensated by a larger average number of divisions attributed to B cells. However, this impacted onto the dark zone/light zone ratio of B cells, which was shown to strictly depend on the duration of the processes in both zones (17). Reducing the cell-cycle time would be unphysiological. Therefore, the dark zone/light zone ratio was restored by a prolonged Ag collection phase (Table I).

This also implied that the working regimen of dynamic division rates K_P had to be adjusted as well because the total amount of Ag collected in a prolonged time span was increased. The resulting parameters of this procedure are collected in Table I.

The total strength Ω of the GC reaction is defined as

where the indices CB and CC denote the dark zone and light zone B cell phenotype, respectively, and N_CB,CC(t) is the number of B cells of the respective phenotype at time point t. T = 21 d is the total duration of the GC reaction simulation. Ω(T) is a suitable measure for comparing GC simulations because the number of generated GC output strongly depends on the total number of B cells that went through a GC reaction. The requirement of similar strength of the GC reaction was used to determine unknown parameter values specific for the different models (Table I).

Three different affinity-dependent mechanisms with impact on B cell selection and differentiation were compared: DND, affinity-dependent dynamic increase of the number of B cell divisions; DSH, affinity-dependent dynamic downregulation of somatic hypermutation (17); and AFB, Ab feedback onto the GC reaction (21).

Although DND and DSH both represent mechanisms on the level of individual interactions between B and T cells, which depend on the previously collected, processed, and presented amount of Ag, AFB represents the idea that Abs generated by the GC output increase B cell competition for Ag by coverage of the Ag presented on FDCs. It is important to realize that when DSH is not active, somatic hypermutation occurs at every division event with a constant mutation probability.

The in silico model developed for transzone migration events as observed in photoactivation experiments (17) was merged with a model for the dynamics of soluble Abs in the GCs (21). The combined model was extended to include the three affinity-dependent mechanisms DND, DSH, and AFB. The resulting novel simulation framework was analyzed for affinity maturation (quality) and number of generated output cells (quantity). These two quantities normally are competitive in the sense that either affinity maturation or the number of output cells is increased in dependence of GC parameters, but never both.

The total number of B cells participating in a GC reaction changed the number of generated memory and plasma cells (shortly output cells). Each of the three mechanisms had an impact onto GC population dynamics, affinity maturation, and the ratio of B cells in the dark and the light zones. By fine-tuning of parameters in each model (Table I), comparable simulations were generated (Fig. 1A). This procedure enabled an investigation of the mechanisms in terms of quantity and quality of generated output cells purged of global GC differences. Thereby, quality of output cells is measured as the mean affinity of the BCRs to the Ag in all output cells.

The number of divisions induced in selected B cells depends on which mechanisms are being activated in the model. The attribution of numbers of divisions was implemented on an individual cell–cell interaction level, and the simulation generated frequencies of numbers of divisions in the GC reaction as readout (Fig. 1C, 1D). The average number of divisions remained in the range of 2 (Fig. 1C, 1D) in agreement with results from photoactivation transzone migration experiments (9, 17). Without DND (Fig. 1C, 1D, orange, blue, magenta) the number of divisions attributed to selected B cells was fixed by definition. However, in the case of active AFB, the mean number had to be set slightly above 2 (Table I), such that a small fraction of B cells divided three times (Fig. 1D, magenta). Importantly, all models with activated DND exhibited a distribution of attributed numbers of divisions between 1 and 5 (Fig. 1D, green, red, black), which was a result derived from the actual density of pMHC presentation to TFH of the individual in silico B cells. Thus, DND as implemented led to a functional mechanism and could be used for further analysis.

The phase of monoclonal expansion was identical in all models because all mechanisms relied on interactions of GC B cells with TFH during the selection process (Fig. 2A, until day 3). However, despite the similar GC reaction strength (Fig. 1A), the GC B cells population kinetics differed between the models (Fig. 2A). Models including AFB developed larger populations in the beginning of the GC phase of selection, whereas they showed reduced populations after 3 wk. AFB can, thus, be considered as a suitable mechanisms for shutdown of GC reactions.

Affinity of GC B cells developed best when DND and DSH were switched on both (Fig. 2B, red line). Switching on AFB in addition led to the same affinity maturation success (Fig. 2B, red and black line on top of each other), such that affinity of GC B cells was not further improved by AFB.

GC output cells are derived from GC B cells and, therefore, had the same mean affinity at any time point of the reaction (data not shown). The sum of all generated output cells in the course of the whole reaction is an independent measure of affinity maturation and depends not only on the affinity of GC B cells but also on their amount. The result found for GC B cells was even more pronounced for the integrated GC generated output. When DND and DSH were on, switching on AFB in addition reduced the mean affinity of all output cells generated over the whole GC reaction (Fig. 2C, red versus black line). This is a result of the population kinetics in the GC: When AFB is activated, the GC B cell population is larger in the beginning, that is, in a phase when affinity maturation is just starting. Many output cells of comparably low affinity are generated in this phase. Later, when affinity maturation is accomplished, GCs with activated AFB were smaller. The contribution of the few high-affinity output cells generated in this late phase did not compensate for the low-affinity output generated in the early phase.

Next, the number of generated output cells summed over the whole GC reaction was investigated (Fig. 2D). Strikingly, the models with the highest number of generated output were based on DND only (Fig. 2D, green line). Switching on DSH or AFB both reduced the number of output cells.

To understand the relative effect of each mechanism on affinity maturation and number of generated output cells, the model with all mechanisms switched off was used as a reference model (Fig. 2, orange lines). When AFB was activated, the mean affinity of output cells was reduced, whereas the number of output cells was transiently increased (Fig. 2C, 2D, magenta lines). However, in the long run, the number of output cells also stayed below the reference simulation. When DSH was activated, the mean affinity of output cells was substantially increased, but the number of output cells stayed below the reference simulation (Fig. 2C, 2D, blue lines). When DND was activated, both the mean affinity and the number of generated output cells were increased (Fig. 2C, 2D, green lines). Note that during simulations with DND only, DSH was switched off, that is, the mutation probability in each division event was constant throughout the GC reaction (see Table I). DND is the first identified mechanism that increases both quantity and quality of Ab-forming cells generated in GC reactions.

A successful immune response requires both high-affinity Abs and these in large amounts. We used the product of mean output affinity and total number of generated output as a measure for the strength of the induced immune response and called this quantity GC efficiency (Fig. 2E). Simulations without DND and DSH had the weakest GC efficiency (Fig. 2E, orange and magenta line).

For easier comparison of GC efficiencies, these were normalized with the simulation with all mechanisms switched off (Fig. 2F). Despite the finding that DND is the only mechanism that improves quantity and quality of GC output, the GC efficiency was larger when combined with DSH (Fig. 2F, compare black and red lines with green line). A dynamic affinity-dependent downregulation of somatic hypermutation helps in maintaining the level of already achieved affinity maturation. In general, the probability of reducing affinity to the Ag by mutations is larger than increasing it. Therefore, once high-affinity clones are found, too-high mutation frequencies would bear the risk for losing affinity again. This explains why the affinity is further improved by activation of DSH, whereas the number of generated output is reduced. Because the improvement in affinity due to DSH is stronger than the loss of output cells, the GC efficiency is higher in simulations with DND and DSH combined.

However, intermediately, between days 6 and 14 after onset of monoclonal expansion, when the actual immune competence of the response is essential, the best GC efficiency was found for all mechanisms activated (Fig. 2F, black line). The additional Ab feedback had the main effect to shift the dominant output production phase to earlier times, whereas in the late GC phase, coverage of Ag by soluble high-affinity Abs stopped the reaction. Therefore, AFB is beneficial in two ways: GC efficiency rises earlier and chronic GCs are prevented. According to the simulations, increasing specific Ab titers may help to stop chronic GCs, in particular, in the elderly population. Aged individuals often suffer of impaired onset of GCs and chronic inflammatory states associated with reduced B cell repertoires (25).

This analysis of GC reactions demonstrates the dichotomy of quantity and quality of GC output for two of the three investigated mechanisms: affinity-dependent dynamic downregulation of the mutation probability (DSH) and feedback of GC-derived Abs on Ag presentation (AFB). Surprisingly, assuming a constant mutation probability and no AFB, we found that affinity-dependent dynamic upregulation of the number of divisions induced in GC B cells upon interaction with TFH (DND) increased both measures in silico; thus, this mechanism has to be considered critical for the success of the GC reaction. The GC efficiency further improved by combining DND with DSH and AFB.

Why is DND increasing both quality and quantity of GC output in silico? High-affinity cells get a stronger division signal than low-affinity cells. This induces a faster takeover of the high-affinity clones in the GC, that is, increases the GC B cell quality faster than without DND. The accelerated takeover of high-affinity clones increases the average affinity of the output cells, because low-affinity output cells are generated for a shorter period and high-affinity cells for a longer period. The total number of generated output cells is also increased by DND, despite the fact that the overall GC strength, as defined in Fig. 1A, is the same as without DND. Output cells are the result of a successful TFH–B cell interaction, and the success probability depends on the affinity of the BCR. DND simulations induced specific kinetics of the GC population (Fig. 2A): It is reduced in the beginning of the reaction (days 3–5) and increased at later times (days 5–12). In the beginning, low-affinity cells are still predominant and the per-cell probability of output generation is lower during this phase with less GC B cells. In contrast, the per-cell success probability is larger in a phase when the GC population is large as well. This explains the overall increased total number of output cells. Note that this explanation is a reduction of the complexity of the model, which incorporates many other factors with effect on the number of generated output like, for example, produced Abs that limit the availability of Ag in the late phase of the GC reaction or the number of TFHs, which limits available interaction partners of B cells.

In previous studies (21), a positive effect of injected Abs on affinity maturation was found. This effect was seen in response to injection of Abs and was also reproduced by corresponding simulations. In this study, the setup of the model is different and compares GCs, which assume different selection mechanisms but otherwise lead to the same GC phenomenology, as reflected in the same GC strength (Fig. 1A). In this setting, AFB had a negative effect on affinity maturation (not on GC efficiency). This shows that although injection of Abs can foster affinity maturation in GCs, AFB implemented as a GC mechanism can have the adverse effect in comparison with a GC without AFB. The present simulations support the view that AFB is more important for GC shutdown than for affinity maturation.

The simulations revealed that all three mechanisms increase GC efficiency, at least during the first 15 d of the reaction. The role of each mechanism was differentiated according to different phases of the immune response, suggesting an essential impact of DND and DSH at early times, whereas AFB was more important after the peak of the GC reaction. Of the three investigated mechanisms, AFB and DND already have experimental evidence (20, 21). DSH is not yet proved. Further experiments are needed to clarify the existence of an affinity-dependent downregulation of the mutation probability per division. Note, however, that a too-strong downregulation of mutations, despite being beneficial for the acute immune response, would also reduce the diversity of generated memory B cells, and by this the flexibility of a memory response to a mutated pathogen. Optimization of the tradeoff between an efficient immune response and a flexible memory response goes beyond the predictive power of the presented model. Independent of this, the presented results suggest to target DND in immune interventions with the aim of strengthening or inhibiting Ab responses because this mechanism has an unambiguous effect on the efficiency of the GC reaction. Further work is needed to understand DND on a mechanistic level.

I thank Sebastian Binder for revising the manuscript.

The authors have no financial conflicts of interest.