Why are there so few key mutant clones? The influence of stochastic selection and blocking on affinity maturation in the germinal center
Steven H. Kleinstein1 and
Jaswinder Pal Singh1
1 Department of Computer Science, Princeton University, Princeton, NJ 08544, USA
Correspondence to: S. Kleinstein, Physiome Sciences, 150 College Road West, Princeton, NJ 08540, USA. E-mail: stevenk{at}cs.princeton.edu
Transmitting editor: M. C. Nussenzweig
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Abstract
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A small number of key somatic mutations lead to high-affinity binding in the anti-hapten immune responses to 2-phenyl-5-oxazolone (phOx) and (4-hydroxy-3-nitrophenyl)acetyl (NP). Affinity maturation models of the germinal center hold that B cells carrying these key mutations are preferentially selected for expansion within the germinal centers. However, additional factors are required to account for some quantitative aspects of affinity maturation in vivo. Radmacher et al. have shown that key mutants are observed in vivo significantly less frequently than expected by these models. To account for this finding, they propose that selection is a stochastic process where key mutants may be overlooked by positive selection or recruited out of the germinal center. While acknowledging that a minimal amount of stochastic selection is probably unavoidable in the germinal center, we instead propose a structural explanation for this key mutant discrepancy. This model is based on the existence of a large number of blocking mutations whose presence can prevent the ability of key mutations to confer high-affinity binding. Using mathematical modeling and computer simulation, we show that in addition to reconciling the key mutant discrepancy, the blocking model accounts for other aspects of experimental data that are not predicted by the stochastic selection model. In particular, the blocking model is consistent with the observation that key mutants generally exhibit a higher number of mutations per sequence in the phOx response, but a lower number in the NP response.
Keywords: 2-phenyl-5-oxazolone, (4-hydroxy-3-nitrophenyl)acetyl, mathematical modeling, somatic hypermutation
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Introduction
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Within the germinal centers, activated and proliferating B cells undergo a process of somatic hypermutation. This diversification of the repertoire is believed to lead to the affinity maturation of the B cell population through a combination of positive and negative selection (1,2). Cells with affinity-increasing mutations have a higher effective division rate and are preferentially selected to expand. Other cells with decreased affinity or altered specificity may leave the germinal center or die by apoptosis. Affinity maturation has been studied in great detail during the anti-hapten immune responses to 2-phenyl-5-oxazolone (phOx) and (4-hydroxy-3-nitrophenyl)acetyl (NP). During the first week post- immunization, germinal centers in these responses become dominated by low-affinity B cells carrying canonical rearrangements (3,4). Single key mutations in these canonical antibody receptors lead to an
10-fold increase in affinity. Cells carrying such mutations appear to be rapidly selected and expanded. These key mutants constitute up to half the total pool of splenic germinal center B cells by 2 weeks post-immunization (5,6). In the phOx response, key mutations lead to the exchange of histidine for either asparagine (H
N) or glutamine (H
Q) at codon 34 of the canonical light chain (7). Key mutations in the NP response cause an amino acid exchange from tryptophan to leucine (W
L) at codon 33 of the canonical heavy chain (8). Table 1 contains a full list of these key DNA point mutations.
Although the probability of accumulating one of the key mutations is small for any individual B cell (of the order of 104 to 103 division1), the large size (
2300 cells) and fast proliferation rate (up to 3.43 divisions cell1 day1) of the centroblast population within each germinal center should yield at least one key mutant every day after the onset of mutation (9). According to this simple view, the number of key mutant clones within individual germinal centers will follow a Poisson distribution. If an average of one key mutant is created each day per germinal center, then after just 2 days 86% of germinal centers are expected to have key mutants present, and almost 60% of germinal centers are predicted to contain at least two independent key mutant clones. With each passing day, this simple model predicts that more and more germinal centers will contain key mutants, and that the number of independent key mutant clones, and consequently the number of founder cells, within each germinal center will also increase. However, microdissection studies of individual germinal centers have produced results that conflict with these estimates. They have found that even 2 weeks post-immunization most germinal centers in fact do not have any key mutants present (Tables 2 and 3). In addition, through the construction of clonal trees, these studies have demonstrated that the entire key mutant population within an individual germinal center always descends from a single ancestor or founder cell. In fact, these two experimental observations are consistent with each other. According to the Poisson distribution, if half of all germinal centers have no key mutants, <3% are expected to posses multiple key mutant founders (a value potentially small enough to remain undetected given the limited experimental data currently available).
This inconsistency between the numbers of key mutants predicted by simple theoretical calculations and that observed by experiments was first pointed out by Radmacher et al. for the NP response (9). In addition, they provided a way to estimate the magnitude of this discrepancy. Specifically, a methodology was developed to calculate the expected number of key mutants that are lost in vivo (i.e. the average number of key mutants that are predicted to be produced according to theoretical calculations, but which are not observed in vivo).
The simple view of affinity maturation must be extended to account for this key mutant discrepancy. Negative selection of lethal mutations alone cannot account for the observed difference. To resolve the discrepancy, Radmacher et al. have suggested that selection is a stochastic (i.e. probabilistic) process in which high-affinity key mutants may be overlooked by positive selection or recruited out of the germinal center (9). The failure of some key mutant lineages to be positively selected may be due to difficulty in finding the T cell help probably required for continued proliferation (10). The possibility that T cell help is a limiting factor is supported by the observation that T cells constitute a small fraction of germinal center cells (10) and that carrier priming, which increases the availability of T cell help, produces a significantly larger germinal center response (11). Alternatively, high-affinity key mutants may be preferentially selected for emigration out of the germinal center. Tarlinton and Smith have given one possible biological basis for this mechanism. In their view, extensive cross-linking initiates differentiation to antibody-forming cells (12). This level of stimulation is most likely to occur in high-affinity key mutants, especially early in the response, before they must compete with large amounts of serum antibody.
In previous work, we found that the well-known OpreaPerelson model of germinal center dynamics also predicted the production of more key mutants than observed in vivo for the phOx and NP responses (13). Significantly, this model already includes a number of processes that can be viewed as stochastic selection. For example, centroblasts have a natural death rate and even high-affinity key mutant centrocytes may succumb to apoptosis before they can bind to a follicular dendritic cell (FDC). Although the key mutant discrepancy in the OpreaPerelson model can be resolved (13), an important side-effect of the resolution is that germinal centers without key mutants rapidly shrink in size after the onset of mutation. This mode of operation predicts large size differences between germinal centers with and without key mutants, a conjecture that has not been supported yet by experimental observations. For this reason, it is important to investigate germinal center dynamics under conditions where germinal center size remains approximately constant after reaching its peak around the end of the first week post-immunization, regardless of whether the germinal center contains key mutants or not. This mode of operation is adhered to by the models presented in this paper.
We first improve the methodology of Radmacher et al. which provides a worst-case estimate of the magnitude of the key mutant discrepancy. This analysis is also extended to cover the phOx response in addition to NP. Next, we examine mechanisms that have the potential to reconcile theory and experiment. As an additional mechanism that can work independently or complement stochastic selection, we propose a model in which the ability of key mutations to confer high affinity can be blocked by other mutations. Key mutant clones carrying these blocking mutations have no proliferative advantage. This blocking model is based on the ability of somatic mutations to subtly alter antibody structure and affect the affinity for antigen. Although both models can provide a quantitative explanation for the key mutant discrepancy, comparing additional predictions with available experimental data exposes the superiority of the blocking model. In this way, we can begin to understand how the forces of positive and negative selection work in conjunction with specific features of the affinity landscape to influence the process of affinity maturation and the observed patterns of somatic mutation.
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The model
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While the estimates for the magnitude of the key mutant discrepancy follow from the methodology set out in Radmacher et al. (9), the analysis of the stochastic selection and blocking models is based on a dynamic computer simulation of clonal expansion in the germinal center. A high-level view of this model is depicted in Fig. 1. The simulation follows the expansion of a B cell clone beginning with a single seeding cell carrying the canonical germline sequence at day 2 post-immunization. This seeding cell is presumed to have previously undergone an initial antigen-dependent selection event leading to its entering the follicle and initiating clonal expansion, so its initial division rate is not necessarily the same as a naive B cell. The potential effects of additional clones in the same germinal center will be considered later. Each step of the simulation corresponds to 1 h of real time and data are collected every 6 h. Cells are followed individually so that the number and type of mutations can be tracked. Division occurs deterministically according to the cell cycle time TC. High-affinity key mutants are assumed to have a shorter cycle time compared with other cells (i.e. they are positively selected). Key mutants with one or more blocking mutations are considered to have germline affinity and will divide at the slower rate.

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Fig. 1. High-level model of clonal expansion in the germinal center depicting potential mechanisms underlying the key mutant discrepancy. During each division, cells may accumulate lethal mutations and/or blocking mutations with probabilities governed by the mutation decision tree in Fig. 2 with parameters given in Table 4. Cells may be lost after each division with probability (1 r), where r is the probability of recycling.
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Stochastic selection is modeled by removing cells with probability (1 r) after each division, where r is the probability of recycling. In order to avoid frequent clonal collapse, stochastic selection operates only after the onset of mutation (when the clone size is larger). The germinal center is assumed to be composed of a fixed number of niches where cells can survive (14). When the dividing population goes over this maximum capacity (2300 cells by default), cells are removed in order to maintain a constant population size. Lower-affinity cells are always chosen first for removal (i.e. they are subject to negative selection), so that once the germinal center has reached its maximum size high-affinity cells will come to dominate as quickly as they can divide. This is the strongest possible selection pressure that can occur with a constant germinal center size.
In this simulation, a mutation decision tree is used to implement the affinity landscape (15). After the onset of mutation, a Poisson distributed number of mutations are accumulated during each division. The average number of mutations per division is µM, where µ is the average mutation rate per base pair per division and M accounts for the number of bases in the canonical sequence along with their intrinsic mutability (see Table 4 for values). The decision tree presented in Fig. 2 with parameters described in Table 4 determines the type and effect of each mutation. These parameters account for the presence of micro-sequence specificity (16) and transition/transversion bias (17). Note that there are only two parameters in the decision tree that cannot be set directly by analyzing the canonical germline sequences. One of these, the fraction of framework region (FWR) replacement mutations that are lethal (pLethal), has previously been estimated to be
0.5 (18). The other is the total frequency of blocking mutations (pBlock). The total frequency of blocking mutations, which will be estimated from experimental data, is split between the heavy and light chains according to the simulation parameter HBias. This parameter gives the fraction of total blocking that occurs in the heavy chain. Thus, pBlock for the heavy chain is actually pBlock x HBias x (µMHeavy + µMLight)/µMHeavy and similarly for the light chain.
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Table 4. Parameters for the mutation decision tree (Fig. 2) governing somatic hypermutation in the heavy and light chains of the canonical germline sequences of the phOx and NP responses [see (13) for sequences and feature information]
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Fig. 2. The mutation decision tree. The number of mutations per division is distributed as a Poisson random variable with average rate given by µM. The decision tree determines the effect of each mutation. The sum of all branches emanating from each node is 1. Parameter values for the canonical heavy and light chains of the phOx and NP responses are given in Table 4. The parameter pBlock is calculated differently each chain according to the simulation parameter HBias as described in The model.
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Results
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The key mutant discrepancy during the phOx and NP responses
As proposed in (9), the expected number of key mutants that are lost in vivo, E, is estimated by comparing a maximum likelihood estimate of the actual waiting time before the founder key mutant is observed in vivo (
) with the expected waiting time for the first key mutant to be produced based on theory (
). Specifically, we have: E =
>/
1. The basis for this formula is that during the
> days it takes for the founder key mutant to become established in vivo, it is expected that new key mutants are produced every
days. Only one of these is the actual founder key mutant.
The waiting time for the founder key mutant in vivo (
>) cannot be observed directly by experiments. It is determined by a maximum likelihood analysis of sequence data from individual germinal centers (Tables 2 and 3) (9). This analysis, based on a simple statistical model that requires a small number of parameters (described in the caption of Table 5), also provides an estimate for the speed of clonal dominance (k).
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Table 5. Maximum likelihood estimates for , the waiting time for the founder key mutant (measured from the onset of mutation), and k, the speed of clonal dominance, based on experimental observations of individual germinal centers (Tables 2 and 3)
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An analytical equation gives the expected waiting time (in days) for the first key mutant:
= [2300 x 2 x (24/TC) x µK]1, where 2300 is the estimated stable size of the centroblast population, TC is the cell cycle time (in h) and µ is the mutation rate per base pair per division (9). The factor K accounts for the precise set of key mutations as well as detailed features of hypermutation such as micro-sequence specificity (16) and transition/transversion bias (17). For example, K = 1.14 x 0.17 = 0.19 in the NP response based on the assumptions given in Table 1.
Revised estimates for the division and mutation rates
The theoretically determined waiting time
depends on a number of variables that are estimated from experimental data. In this section, we improve some of the estimates proposed by Radmacher et al. in their original study. For example, they use a mutation rate of 103 bp1 division1. At the same time, they assume that the division rate for all cells is 3.43 day1. These values are inconsistent since the quoted mutation rate is in fact based on a much slower division rate (19). Furthermore, the division rate estimate is based on an experiment that did not differentiate between germline and higher-affinity cells (20).
To improve on the estimates of Radmacher et al., we take advantage of the fact that only the product of the division and mutation rates appears in the formula for the waiting time
shown above. This product is easily estimated by using linear regression to fit a straight line through experimental data counting the number of mutations per sequence (Fig. 3). We consider separately cells with and without key mutations since these populations may have different division and/or mutation rates. Under simple assumptions, the slope of these regression lines (m) obeys the following formula: m = 24/TC x µM(1 L), where TC is the cell cycle time (in h), µ is the average mutation rate per base pair per division, M is a factor accounting for the number of base pairs sequenced along with their intrinsic mutability (see Table 4 for values) and L is the fraction of mutations that are lethal. Under the assumption that 50% of FWR replacement mutations are lethal, L
0.3 for both the NP and phOx responses (18).

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Fig. 3. The average number of mutations per mutated sequence from individual germinal centers for the in vivo (A) phOx and (B) NP responses. Each point represents microdissection data from an individual germinal center (Tables 2 and 3). Sequences in the phOx response are from the canonical light chain, while the NP response data is from the canonical heavy chain. Sequences with (open circle markers, offset slightly to right to avoid overlap) and without (square markers) key mutations are counted separately, even if found in the same germinal center. Also shown is the best-fit line, as determined by linear regression, for data without key mutations. Experimental data is shown for all cases where sequence data was available. Note that very few of these germinal centers contained germline sequences so that the plot of mutations per sequence is virtually identical.
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The above formula can be used directly to estimate the product of the mutation and division rates. However, it is often more intuitive to think about one or the other of these rates. This can be done by simply assuming that the value used by Radmacher et al. for the mutation rate is correct and then choosing the division rate so that the product is right. Even though this method may produce incorrect estimates for both individual rates, their product, which is the important quantity when considering the key mutant discrepancy, will be correct. By applying this method and assuming a mutation rate of 103 bp1 division1, cells without key mutations in the phOx response are estimated to have an average cycle time of 11.4 h, while similar cells in the NP response exhibit a slightly shorter cycle time of 11.0 h. [Since the simulation time step is 1 h, the cycle time (TC) of cells without key mutations is assumed to be 12 h in the phOx response and 11 h in the NP response.] The cycle time may be longer if other mutations besides the key mutation are positively selected or if we have overestimated the fraction of lethal mutations. It is not reasonable to determine the cycle time of the key mutant population using this method since there are too few experimental observations other than at day 14. For lack of any better data at this time, high-affinity key mutants are assumed to divide every 7 h, similar to the fastest time observed in vivo (11,20).
We can also estimate the time that somatic hypermutation starts by looking at the x-intercept of the regression line for the non-key mutant population (Fig. 3). Consistent with other studies (4,21), the intercept predicts a period of mutation-free growth for both responses during the initial expansion phase of the germinal center. In the phOx response, mutation is predicted to start around day 7. This is slightly later than the previously assumed time of day 6.5 by Radmacher et al. and is a minor help in explaining the key mutant discrepancy. In contrast, mutation appears to begin at day 5 in the NP response. [Technically, we should include germline cells in the plot when determining the x-intercept (i.e. we should not use mutations per mutated sequence). However, since few germline cells are present in the data their inclusion has no significant impact on any of the results.]
Estimating the magnitude of the key mutant discrepancy
Compared with the original parameter estimates used by Radmacher et al., our modified estimates predict that a greater number of key mutants are lost (Table 6). There are two reasons for this. First, even though the division rate used here is slower, the expected waiting time for the first key mutant based on theory is shortened since more current experimental data finds a lower mutability for the sequence leading to the key mutation than originally used by Radmacher et al. (see Table 1). Second, the predicted waiting time for the founder key mutant in vivo is lengthened due to an earlier start to mutation combined with additional data from individual germinal centers (shown in Tables 2 and 3). In the phOx response, the number of key mutants that are lost is even greater than in the NP response, even though mutation starts later and the founder key mutant in vivo is expected to appear earlier (Tables 5 and 6). This difference is a consequence of the fact that there are multiple key mutations in the phOx response, all of which lie in relative hot-spots of mutation (Table 1). The overall probability of accumulating a key mutation in the phOx response is 8.2 x 104 division1 (1.9 x 104 division1 if we exclude the H
Q exchange as will be proposed below), while the probability in the NP response is much lower, 0.7 x 104 division1. The effect of this difference on the expected waiting time for the first key mutant overwhelms any other differences, leading to the prediction that 80 key mutants are lost in the phOx response compared with 10 in the NP response.
H
Q exchange in phOx requires second mutation to be selectable
The large number of key mutants predicted to be lost during the phOx response can be partly explained by the following observation. Although there are thought to be two different amino acid exchanges that can lead to higher affinity in the phOx response, only the H
N exchange is frequently observed in vivo (22). For example, it is exclusively present in seven of eight germinal centers with key mutants, P < 0.003 (Table 3). This conflicts with the estimate that the H
Q exchange is
23 times more likely to occur at the genetic level (see Table 1). We propose that the observed frequency of this exchange is so far from the expected value because it is only efficiently selected for when it occurs in combination with a second mutation (Y
F at codon 36). In support of this idea, we found only a single sequence (one out of 21) where the H
Q exchange occurs in the absence of this second exchange (5,7,2224). In contrast, almost half (54 out of 119) of all sequences with the H
N exchange are found to occur without this second exchange. Modeling this hypothesis precisely requires a number of changes to the model. However, the probability of accumulating two specific, independent mutations is low and, in fact, this combination is rarely observed in vivo. Thus, as a first approximation we can exclude mutations leading to the H
Q exchange from the set of potential key mutations in the phOx response in the rest of this paper. This will turn out to be an optimistic assumption since it helps account for the key mutant discrepancy, and thus leads to a better fit between model and experiment. Even in this case, however, the probability of accumulating a key mutation is almost 3-fold higher and almost twice as many key mutants are lost in the phOx response compared with NP (Table 6).
Accounting for the key mutant discrepancy
The above calculations suggest that many key mutants are overlooked within the germinal centers. What processes account for this loss? One possibility is the occurrence of lethal mutations. It has been estimated that 50% of FWR replacement mutations are lethal (18). While some key mutant lineages may be lost by this mechanism, such mutations are not frequent enough to fully explain the key mutant discrepancy (9). Even so, the effects of lethal mutations are included in our model and in all of the analysis that follows. It is interesting to note that although the presence of lethal mutations does little to resolve the key mutant discrepancy, the hypothetical absence of such mutations can go a long way towards making the discrepancy disappear. This is due to the fact that, in the absence of lethal mutations, the estimated cycle time for cells without key mutations is increased to 16 h in both the phOx and NP responses. In this case, the number of key mutants that are predicted to be lost falls to 13, with a 95% confidence interval (95% CI) of [632], for the phOx response and 4 [118] for the NP response. In the following sections, we will investigate other mechanisms for resolving the key mutant discrepancy.
The stochastic selection model
The stochastic selection model explains the key mutant discrepancy by proposing that selection is a probabilistic process (9). More specifically, key mutants may fail to obtain access to necessary stimulatory signals such as FDC or T cell help. Alternatively, they may preferentially emigrate out of the germinal center. Although these processes, depicted in Fig. 1, may have some impact on all cells, their effect is especially important on the establishment of newly formed key mutant clones. In these cases, the induced fluctuations in the small population size of a newly formed clone can cause many clones to collapse and thereby become lost from the germinal center.
To estimate the magnitude of stochastic selection necessary to account for the key mutant discrepancy, we have developed a computer simulation of clonal expansion in the germinal center. In this simulation, all processes leading to failure of selection are lumped together and assumed to occur with probability (1 r), where r is the probability of recycling per division. The simulation also includes other relevant stochastic processes such as the occurrence of lethal mutations. A full description of the simulation can be found in The model. To estimate the probability of recycling, the simulation was run many times at each value of r (increments of 0.025) and the likelihood of observing the experimental data given in Tables 2 and 3 was determined. The likelihood function concerns the fraction of cells within individual germinal centers that carry key mutations and accounts for the limited number of sequences collected. It is equivalent to that of (9).
We find the maximum likelihood estimate for the probability of recycling is 0.800 for the phOx response (Fig. 4). At this level, the key mutant discrepancy is easily explained. First, even 14 days post-immunization many germinal centers have no or only a few key mutants. In fact, key mutants are predicted to constitute <10% of cells in >40% of all germinal centers at this time (data not shown). In addition, although the simulation predicts multiple founder cells per germinal center, the progeny of a single founder dominates the key mutant population within each individual germinal center. For example, among germinal centers with key mutants, a single founder cell accounts for an average of 66% of the key mutant population at day 14 post-immunization.

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Fig. 4. Finding the maximum likelihood estimate for (A) the probability of recycling and (B) the frequency of blocking mutations. For each value of r (increments of 0.25), the probability of recycling, and pBlock (increments of 0.05), the frequency of blocking mutations, the simulation was run 10,000 times and the likelihood of observing the experimental data (Tables 2 and 3) was determined. The NP response data excludes germinal center L1AB02 from day 10. The solid portion of the line shows the 95% CI based on the 2 distribution of the likelihood function. The horizontal bars show the values of r and pBlock that fit experimental data on the fraction of total splenic germinal center B cells carrying key mutations according to the 95% CI (13).
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In the NP response, the maximum likelihood estimate for the probability of recycling is 0.950. However, the key mutant discrepancy is not accounted for in this case. With such a high probability of recycling, few key mutants are lost due to stochastic selection and almost every germinal center has a significant number of key mutants. The reason for this high estimate is the need to account for a single experimentally observed germinal center, L1AB02 at day 10. This is the earliest germinal center to contain key mutants. However, inspection of the experimentally obtained sequence data suggests that the founder key mutant was not formed within this germinal center. Nine out of the 10 sequences from this germinal center are identical, carrying two mutations including the key. Only a single sequence carries one additional mutation. Thus, it is likely that this germinal center was seeded with a pre-existing key mutant. If we exclude germinal center L1AB02 from our analysis (which we do from now on), then the probability of recycling is estimated to be 0.875 (Fig. 4). In this case, the key mutant discrepancy is easily explained. At day 14 post-immunization key mutants constitute <10% of cells in 55% of all germinal centers and a single founder cell accounts for 84% of the key mutant population on average (data not shown).
Failure to predict mutation dynamics in the NP response
With a reasonable probability of recycling, stochastic selection can resolve the key mutant discrepancy. To further validate the stochastic selection model, we now compare the predicted and observed dynamics of mutation. Simulation and experiment are in excellent agreement with respect to the rate at which mutations are accumulated within the non-key mutant population. This should come as no surprise since the mutation and division rates used in the simulation were chosen based on this data. However, it is important to ask whether this agreement also holds for the key mutant population. Stochastic selection has no effect on the rate at which mutations are accumulated in any given population. This is because cells are lost randomly, without regard for how many or the type of mutations they carry. Consequently, in both responses the model predicts that mutations accumulate faster among key mutants compared with other clones due to their faster division rate. Comparing these dynamics with experimental data shows that this prediction is accurate for the phOx response, but not for the NP response (Fig. 5). Cells with key mutations in fact accumulate fewer mutations, on average, than other clones during the NP response in vivo (Fig. 3). This does not imply that stochastic selection is not operating during the NP response. At the very least, however, different mechanisms must be introduced in order to account for these observations. In the next section, we propose a different, single mechanism for reconciling the key mutant discrepancy, which has the potential to explain these observations as well.

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Fig. 5. The average number of mutations per mutated sequence within individual germinal centers under the stochastic selection model for the (A) phOx and (B) NP responses. For each response, the simulation was run 50 times with the probability of recycling set to its maximum likelihood value (0.800 for the phOx response and 0.875 for the NP response). Every 6 h a random sample of 23 or nine cells was chosen from each germinal center in the phOx and NP responses respectively. This limited sampling is consistent with currently available experimental data. Also following the experimental data, values for the phOx response consider the light chain only, while those for the NP response consider the heavy chain only. In both cases, the thick lines shows the average number of mutations per mutated sequence among cells without key mutations over all germinal centers. Each gray point shows the average value among cells with key mutations for an individual germinal center. If a sample from a single germinal center produced a mix of sequences with and without key mutations, these were counted separately. These dynamics can be compared with the dark circles, which show the average number of mutations from sequences with key mutations in vivo. Experimental data for cells without key mutants (see Fig. 3) is not shown here for comparison with the predictions given by the thick line since mutation and division rates were chosen in such a way as to ensure consistency in this case.
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The blocking model
The blocking model proposes an explanation for the key mutant discrepancy based on the effect of somatic mutations on antibody structure. It asserts that many key mutations occur in the context of blocking mutations, which prevent their ability to confer high affinity. The existence of blocking mutations is not a new idea. For instance, in previous work we assumed that mutations at contact residues could lower the affinity of even those cells with key mutations (13). What is new here is the idea that these mutations are prevalent enough to resolve the key mutant discrepancy. In fact, blocking mutations can be very effective at preventing the establishment of key mutants. Over time they build up in a population of germline affinity cells, thus preventing the formation of many high-affinity key mutants. In addition, high-affinity key mutant clones that do get established will often revert to germline affinity. Some blocking mutations may even be positively selected as the initial steps of an alternate path to high-affinity (25).
Obvious candidates for blocking mutations include all complementarity-determining region (CDR) replacement mutations. However, the 50% of FWR replacement mutations that are not categorized as lethal also have the potential to block the key mutation. Although it is possible that cells with lethal mutations might remain in the germinal center for long periods of time (and thus serve to block the key mutation), it is assumed here that these cells are subject to strong negative selection and quickly removed from the germinal center population. Out of this potential pool, the total fraction of mutations with the ability to block the key mutation is given by the simulation parameter pBlock. The precise set of blocking mutations is likely to depend on a number of structural features of the antibody including the location of the antigen-binding site. As a first approximation in the absence of more detailed data, it is assumed that blocking mutations are distributed between the heavy and light chains in proportion to the number of contact residues present in each. This is modeled with the simulation parameter HBias, which gives the fraction of total blocking that occurs in the heavy chain (as described in The model). For the phOx response, blocking mutations are equally distributed since half of the 14 contact residues are in each chain (i.e. HBias = 0.5) (26). However, in the NP response most blocking mutations are predicted to occur in the heavy chain since only three out of 14 contact residues are in the light chain (i.e. HBias = 0.8) (27,28). This difference between the phOx and NP responses will turn out to be important in explaining experimental observations.
The frequency of blocking mutations most consistent with experimental data in Tables 2 and 3 can be estimated using the simulation in the same way that the probability of recycling was predicted. In this case, we can also assume various amounts of stochastic selection to operate along with blocking. Initially, it is assumed that blocking is operating alone (i.e. there is no stochastic selection, r = 1.0) and the likelihood at various values of pBlock (increments of 0.05) is calculated. Using this method, the maximum likelihood frequency of blocking mutations (pBlock) is 0.75, with a 95% CI of [0.650.85] in the phOx response and 0.35 [0.200.55] in the NP response (Fig. 4). The higher frequency of blocking mutations estimated for the phOx response is consistent with the prediction of a larger key mutant discrepancy in Table 6. As with the stochastic selection model, these values easily account for the key mutant discrepancy. There is a single dominant founder cell, and many germinal centers have almost no key mutants at day 14 post-immunization (data not shown).
The maximum likelihood frequency of blocking mutations can also be estimated while assuming various degrees of stochastic selection (i.e. different values for r). It is significant to note that this likelihood is always maximal when no stochastic selection is assumed (i.e. r = 1.0). In other words, experimental data in both responses is best explained when blocking acts alone. Although this finding does not rule out the existence of stochastic selection in vivo, in the effort to resolve the key mutant discrepancy there is no compelling reason yet to include stochastic selection in the model.
In addition to accounting for the key mutant discrepancy, the blocking model has the potential to explain why key mutants are more mutated compared with other clones in the phOx response, but less mutated in the NP response (Fig. 3). Under the blocking model, key mutant clones generally accumulate mutations at a slower rate than otherwise expected since cells with blocking mutations are negatively selected in the presence of high-affinity key mutants. In the phOx response, this effect is fully counter-balanced by the faster division rate of high-affinity key mutants. At the maximum likelihood estimate for the frequency of blocking mutations, the end result is that key mutants generally carry more mutations than other cells (Fig. 6). For example, 63% of key mutant clones are predicted to carry
3.4 mutations per mutated sequence at day 14. This level is the average number of mutations predicted to be carried by clones without key mutations at this time. The higher number of mutations observed in vivo might also reflect the positive selection of additional mutations such as Y
F at codon 36 of the canonical light chain. Similar effects are not a concern for the NP response where it has been demonstrated that the key mutation alone is sufficient to reach a local affinity optimum (29).

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Fig. 6. The average number of mutations per mutated sequence within individual germinal centers under the blocking model for the (A) phOx and (B) NP responses. The simulation was run 50 times for each response with the frequency of blocking mutations (pBlock) set to its maximum likelihood value (0.75 for the phOx response and 0.35 for the NP response). There is no stochastic selection in these simulations (i.e. r = 1.0), other settings are equivalent to Fig. 5.
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Although the maximum likelihood frequency of blocking mutations in the NP response is much less than that estimated for the phOx response, this frequency has a disproportionately large effect on the observed number of mutations in the key mutant population (Fig. 6). This is because the microdissection experiments in the NP response only sequenced the heavy chain and blocking mutations are predicted to be concentrated in this region. The result is that an overall frequency of blocking mutations of 0.35 will appear as a frequency of 0.51 in the heavy chain. Even so, according to the simulation most key mutant clones are still more mutated than other clones at day 14 post-immunization (Fig. 6). However, increasing the frequency of blocking mutations, while remaining well within the bounds prescribed by the 95% CI shown in Fig. 4, allows us to explain the mutation dynamics in the NP response. For instance, if the overall frequency of blocking mutations is raised to 0.50 (equivalent to a frequency of 0.73 in the heavy chain), the model predicts that key mutants will generally be less mutated than other cells (Fig. 7). Specifically, 64% of key mutant clones have <5.5 mutations per mutated sequence, the average value predicted for clones without key mutations at day 14. In fact, the best case for explaining the mutation data occurs when the frequency of blocking mutations is at the upper bound of the 95% CI. Currently, it is unclear how high the frequency of blocking mutations actually needs to be since there are too few experimental observations of germinal centers with key mutants. Although it is possible to including stochastic selection in the model along with blocking, this only makes it more difficult to explain the lower number of mutations among key mutant clones in the NP response. Stochastic selection decreases the required level of blocking necessary to reconcile the key mutant discrepancy by providing an alternate mechanism to account for many of the lost key mutants.

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Fig. 7. The average number of mutations per mutated sequence within individual germinal centers during the NP response under the blocking model. The simulation was run 50 times with the frequency of blocking mutations set to 0.50. There is no stochastic selection in these simulations (i.e. r = 1.0), other settings are equivalent to Fig. 5.
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Additional predictions and validation
According to the blocking model, the number of sites that permit mutations is greatly restricted due to the high frequency of blocking mutations predicted for both responses. This is consistent with the observation that 80% of mutations occur at 7.5% of positions in the canonical light chain of the phOx response (7). An even greater degree of restriction is predicted for the canonical heavy chain of the NP response. In order to test the stochastic selection and blocking models further, and potentially to find ways to discriminate between the models in addition to the mutation dynamics, the simulation was used to predict other aspects of germinal center dynamics.
Frequency of key mutants in the whole spleen
Thus far, we have been using experimental data from microdissection studies of individual germinal centers. Other experiments have measured the fraction of total splenic germinal center B cells that carry key mutations at various times post-immunization (5,6). For example, these studies show that key mutants constitute 54 and 32% of splenic germinal center B cells at day 14 post-immunization in the phOx and NP responses respectively. For the NP response, both the stochastic selection and blocking models are in good agreement with this average-case data (Fig. 8). However, neither model is consistent with observations during the phOx response. It is easy to understand the basis for this discrepancy by considering the situation at day 14. At this point, 11 of 17 individual germinal centers in vivo are observed to have no key mutants (Table 3). Theoretically, even if all six of the remaining germinal centers were completely dominated by key mutants, the assumption that all germinal centers are the same size implies that at most 35% of splenic germinal center B cells could carry key mutations. This level is below the 95% CI of the experimental observations (13). Consequently, in order to bring the models into agreement simultaneously with both the microdissection and whole-spleen experiments in the phOx response, it appears necessary to allow for at least some size differences between germinal centers with and without key mutants.

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Fig. 8. Affinity maturation among splenic germinal center B cells in the (A) phOx and (B) NP responses under the stochastic selection (dotted lines) and blocking (solid lines) models. The simulation was used to predict the total fraction of cells that carry key mutations (a sum over 10,000 simulations). These values can be compared with experimental data measuring the average fraction of splenic germinal center B cells carrying key mutations (closed circles). The error bars give the 95% CI based on the small number of sequences collected (13). For the stochastic selection model, the probability of recycling is set to its maximum likelihood value, 0.800 in the phOx response and 0.875 in the NP response. For the blocking model, the frequency of blocking mutations is set to its maximum likelihood value, 0.75 in the phOx response and 0.35 in the NP response (thick lines). Results for the blocking model are also shown for values at the edge of the 95% CI: 0.65 for the phOx response and 0.50 for the NP response (thin lines).
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The CDR replacement-to-silent (R:S) ratio
The stochastic selection model is consistent with the view that high R:S ratios in the CDR indicate positive selection (30). In contrast, the blocking model predicts that positively selected key mutant clones should exhibit lower CDR R:S ratios than expected. The reason for this difference is that positive selection for high affinity goes hand-in-hand with negative selection against blocking mutations, which are estimated to constitute a significant fraction of replacement mutations. The simulation shows, however, that CDR R:S ratios exhibit a very high variability if only a small number of sequences are sampled from each germinal center, as is the case for available experimental data (Fig. 9). An average of only 23 sequences are collected from each germinal center in the phOx response and only nine are available from the average germinal center in the NP response. The variability predicted by the simulations is clearly reflected in the experimental data collected from both responses (Fig. 9), and prevents us from making a detailed comparison with the simulation. At this point, we can only say that neither the stochastic selection nor the blocking models can be ruled out based on the CDR R:S ratios observed in experiments. All of the in vivo values fall within the ranges predicted by either the stochastic selection or blocking models.


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Fig. 9. The average CDR R:S ratio for individual germinal centers under the stochastic selection model for (A) phOx and (C) NP, and the blocking model for (B) phOx and (D) NP. In each case, the probability of recycling or the frequency of blocking mutations was set to its maximum likelihood value as determined in Fig. 4. Each thin line shows the average CDR R:S ratio among all sampled B cells within an individual germinal center (50 different simulated germinal centers are shown with parameter settings equivalent to those in Fig. 5). The thick solid lines show the average value for clones without key mutants, while the thick dotted lines show the average value for clones with key mutants. To be consistent with experimental data, a random sample of 23 or nine cells was used to determine the CDR R:S ratio at each time point in the simulated phOx and NP responses respectively. Also following the experimental data, values for the phOx response consider the light chain only while those for the NP response consider the heavy chain only. For comparison, experimentally observed average CDR R:S ratios for sequences with (circles) and without (squares) key mutations are shown. If a sample from a single germinal center produced a mix of sequences with and without key mutations, these were counted separately. The error bars give the minimum and maximum values observed over all germinal centers. All R:S ratios count independent mutations only as determined by construction of clonal trees for all germinal centers where sequence data was available as specified in Tables 2 and 3. Experimental data for germinal centers without key mutants is offset slightly to the right in order to avoid overlap.
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Discussion
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In an analysis of the primary NP response, Radmacher et al. have shown that key mutants are observed in vivo significantly less frequently than expected according to predictions based on the population dynamics and genetics (9). By using mutation data from microdissection studies of individual germinal centers, we have first improved the Radmacher et al. estimate of the number of key mutants that are lost during the NP response. For example, the product of the mutation and division rates for cells without key mutations is found to be much lower than previously assumed. The net effect of these improved estimates, however, is to increase the magnitude of this key mutant discrepancy. Using equivalent methods, we also analyzed the phOx response where we find an even larger discrepancy compared with the NP response (i.e. many more key mutants are estimated to be lost) (Table 6). These analyses demonstrate how significantly the basic view of the germinal center disagrees with quantitative experimental observations.
The stochastic selection model, first suggested by Radmacher et al., explains the key mutant discrepancy by proposing that selection is a probabilistic process where key mutants may be overlooked by positive selection or recruited out of the germinal center (9). We have developed a computer simulation of B cell clonal expansion where all processes leading to failure of selection are lumped together and assumed to occur with probability (1 r), where r is the probability of recycling per division. This simulation is used to provide a maximum likelihood estimate for the probability of recycling directly from the experimental data (Tables 2 and 3). Significantly, this simulation is also used to compare other aspects of the predicted dynamics with experimental data. Although the stochastic selection model can indeed account for the dynamics of affinity maturation in both the phOx and NP responses, it fails to predict the slower accumulation of mutations among key mutants in the NP response (Fig. 3).
As an alternative to stochastic selection, we proposed a blocking model, which can address the above shortcoming for the NP response, in addition to resolving the key mutant discrepancy for both responses. The blocking model proposes a structural explanation for the key mutant discrepancy. This model is based on the existence of a large number of blocking mutations whose presence can prevent the ability of key mutations to confer high affinity. A maximum likelihood estimate for the frequency of blocking mutations in the phOx and NP responses was determined using the computer simulation of B cell clonal expansion. At these maximum likelihood levels, the blocking model easily accounts for the key mutant discrepancy. One potential issue, however, is that the best case for explaining the mutation dynamics in the NP response occurs when the frequency of blocking mutations it at the upper bound of its 95% CI. More experimental data and improved quantitative constraints are needed to determine if this is a real problem with the blocking model. Although the stochastic selection and blocking models are not mutually exclusive, so both may operate to some degree in vivo, the best case for reproducing experimental data always occurs when blocking is operating alone. To further differentiate between the two models, as well as to provide validation, additional predictions were made that can be compared with experimental data.
Our maximum likelihood estimates implicitly reflect a compromise between the need to explain the low frequency of key mutant founders and the high dominance of key mutant clones when they exist within individual germinal centers. This trade-off exists because a lower probability of recycling or a higher frequency of blocking mutations not only causes key mutant clones to become extinct with high probability, but also slows the growth rate of those that manage to become established. Although these effects are implicitly connected in the blocking model, it may be possible to at least partially disconnect them in the stochastic selection model. For example, DeBoer and Kesmir proposed a model in which key mutants are lost when created far from the site of an FDC. Once a key mutant establishes itself by gaining access to an FDC, growth may be less restricted (31).
It is important to continue to check the assumptions underlying the finding of a key mutant discrepancy. While we have demonstrated that this discrepancy can be reconciled through mechanisms such as stochastic selection or blocking, the key mutant discrepancy disappears if the effective germinal center centroblast population size is significantly smaller than we have assumed (data not shown). There are at least two reasons to suspect that this is the case. First, germinal centers may be populated with cells carrying non-canonical germline rearrangements while the model only follows canonical rearrangements (4,32). Second, sequences are generated from groups of individual cells that are microdissected from germinal centers in groups of 50100 cells with a total of at most a few hundred cells picked from each germinal center (4,32). If cells migrate sufficiently slowly, the effective population size could potentially be much smaller than the total size of the germinal center. Even with the disappearance of the key mutant discrepancy, we are still left with the need to account for the observation that key mutant clones are less mutated than others in the NP response. All of these issues are resolved by the blocking model.
In conclusion, affinity maturation models of the germinal center must be extended to account for quantitative in vivo observations. The stochastic selection and blocking models offer alternate explanations for the key mutant discrepancy. These models are not mutually exclusive, so that both may operate to some degree in vivo. Indeed, some degree of stochastic selection is probably unavoidable, since it is unlikely that affinity maturation is perfectly efficient. However, the best fit with the experimental data concerning the accumulation of key mutations always occurs when blocking is acting alone. In addition, only the blocking model can account for the dynamics of mutation accumulation in both responses. The blocking model is also consistent with other aspects of the experimental data such as CDR R:S ratios. Thus, blocking is likely to be the dominant force in vivo and an important factor in understanding affinity maturation.
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Acknowledgements
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We would like to thank Martin Weigert and Yoram Louzoun for useful discussions. We would also like to thank Michael Radmacher for providing the computer code used in Radmacher et al. (9), and Garnett Kelsoe for NP sequence data. This work was supported in part by a National Science Foundation IGERT grant (DGE-9972930) and a PECASE grant (CCR-9702115), and through DIMACS.
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Abbreviations
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CDRcomplementarity-determining region
CIconfidence interval
FDCfollicular dendritic cell
FWRframework region
NP(4-hydroxy-3-nitrophenyl)acetyl
phOx2-phenyl-5-oxazolone
R:Sreplacement-to-silent
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