Conceived and designed the experiments: GS MPC. Performed the experiments: GS. Analyzed the data: GS. Contributed reagents/materials/analysis tools: MPC RM. Wrote the paper: GS MPC RM.
The authors have declared that no competing interests exist.
B lymphocytes are subject to elimination following strong BCR ligation in the absence of appropriate second signals, and this mechanism mediates substantial cell losses during late differentiation steps in the bone marrow and periphery. Mature B cells may also be eliminated through this mechanism as well as through normal turnover, but the population containing mature cells destined for elimination has not been identified. Herein, we asked whether the transitional 3 (T3) subset, which contains most newly formed cells undergoing anergic death, could also include mature B cells destined for elimination.
To interrogate this hypothesis and its implications, we applied mathematical models to previously generated in vivo labeling data. Our analyses reveal that the death rate of T3 B cells is far higher than the death rates of all other splenic B cell subpopulations. Further, the model, in which the T3 pool includes both newly formed and mature primary B cells destined for apoptotic death, shows that this cell loss may account for nearly all mature B cell turnover.
This finding has implications for the mechanism of normal mature B cell turnover.
Following immunoglobulin (Ig) gene rearrangement and the expression of a functional B cell receptor (BCR) (reviewed in
The notion that anergic cells reside briefly in the TR compartment before dying, as well as the belief that mature cells are also subject to tolerogenic elimination if their BCR is engaged without costimulation, prompts several questions. First, whether particular TR phenotypes correspond to cells undergoing apoptotic death versus those that will complete maturation is unclear. Second, if particular phenotypes correspond to dying cells, the proportional contributions of newly formed versus mature cells to these pools require definition. Since mature B cells are non-dividing, the relatively rapid turnover of TR pools suggests that most losses in these subsets reflect the death of recent marrow émigrés. Nonetheless, recent studies in transgenic systems have suggested that FO cells dying from lack of costimulation re-acquire the T3 phenotype
We have previously shown that mathematical modeling of population kinetics established from in vivo bromodeoxyuridine (BrdU) labeling studies is a powerful tool with which to assess alternative models of B cell differentiation and fate
In the present study, we ask whether T3 B cell compartment contains most peripheral B cell slated for elimination, and whether a model based on this hypothesis (The inset in
The main figure shows the one found as the best model in our previous study
In order to understand the behavior of the transitional B cell subpopulations that will become mature naive B cells in the spleen, we used published experimental data on these subpopulations in mice
Our model starts with three bone marrow populations: pro-B [B220+CD43+IgM−], pre-B [B220+CD43−IgM−] and immature B cells [B220+HSA+IgM^{hi}IgD^{lo}], with cell numbers in these subsets represented by the variables Bo, Be and Bi, respectively. However, previous experimental observations distinguish between small, non-cycling cells and large, cycling cells in both the pro-B and pre-B compartments, where the transition from pro-B to pre-B occurs while the cells are cycling. Hence, we break the pro-B and pre-B subsets into two subsets each: Bor for small resting pro-B cells (Hardy's fractions A through C) and Boc for large cycling pro-B cells (part of Hardy's fraction C); similarly, Bec for large cycling pre-B cells (the remainder of fraction C) and Ber for small resting pre-B cells (fraction D). Immature B (Bi) cells migrate from the bone marrow to the periphery as transitional B cell. The four subpopulations in the periphery are the three transitional B cell subsets and the mature B cell subset in the spleen as defined by Allman et al,
Anatomic Site | Status | Subset | Cycling? | Surface Phenotype |
Bone marrow |
Immature | E | no | IgM^{hi}IgD^{lo}CD23^{+/−}B220^{+}AA4.1^{+} |
Periphery | Transitional |
T1 | no | IgM^{hi}CD23^{−}B220^{+}AA4.1^{+} |
T2 | no | IgM^{hi}CD23^{+}B220^{+}AA4.1^{+} | ||
T3 | no | IgM^{lo}CD23^{+}B220^{+}AA4.1^{+} | ||
Transitional |
T1 | no | IgM^{hi}CD23-B220^{+}IgD^{−}CD21^{−}HSA^{hi} | |
T2 | yes | IgM^{hi}CD23^{+}B220^{+}CD21^{hi}IgD^{lo}HSA^{hi} | ||
Mature | FO/(B2) | no | IgM^{lo}CD23^{+}B220^{hi}AA4.1^{−} | |
Marginal Zone(MZ) | CD9^{+}IgM^{hi}IgD^{lo}CD23^{−}CD21^{+} | |||
B1 | IgM^{hi}CD43^{+}IgD^{lo/−}CD23^{lo/−} |
The numbers of cells in the T_{1} and T_{2} combined subsets, T_{3} and mature B cells were represented in our mathematical models by the variables
In these equations, the input of stem cells into pro-B compartment is denoted by
Immature B (
Finally, in the new model, no differentiation from
The above-described hypothesis is described by the following equations.
The numerical simulations of the mathematical models were performed in a program written in the C programming language, which runs on the entire parameter space in small intervals, searching for the best-fit parameter set for each model.
The mathematical models were simulated and fitted to data using a C language program. The program receives as input the experimental data, and the ranges of parameter values within which the model should be run. The program divides each range to very small intervals, thus providing a thorough coverage of the parameter space. This creates a set of 1.5×10^{6} parameter combinations to be checked by the program. For each parameter value set, the program integrates the model equations as follows. The initial conditions are zero cells in all populations; labeling starts after the populations have reached a steady state. After integration, the program first checks whether the total cell number and the fractions of cells in each population are within the experimentally measured ranges. Runs in which this is not the case are discarded. For all other runs, the program records the value of the fit of the model to the data (defined below), and outputs the parameter set(s) that have yielded the best fit. This process was performed for each of the models, and the fit values were compared using the AIC method (see below).
In choosing alternative models and parameter values for the simulations of our model, we adhered to the following guidelines.
The parameters should be in the experimentally observed orders of magnitude, if published information is available. While these estimates (where available) are usually not given in units of population rates, so that interpretation of most of these data depends on the model used, these estimates were useful in suggesting the appropriate value ranges for some of the parameters. For example, cell proliferation rates can not be higher than the equivalent of 3–4 divisions per day.
The steady state values obtained using these parameters should be in agreement with our experimental observations on both the total numbers and the composition of BM and transitional B cells. Any parameter set which did not conform to these criteria was automatically rejected.
The time of arrival to the steady state should be biologically reasonable. That is, since a mouse completes its growth within less than 2 months, parameter sets that resulted in longer times of arrival to the steady state in each subpopulation were also rejected.
These conditions significantly constrain the choice of parameter ranges used in our simulations, such that the parameter subspace which gives results obeying all constraints is rather narrow.
Our goal here was to check whether the new hypothesis of T3 behavior accounts for B cell dynamics in the spleen), and estimate the parameter ranges characterizing B cell dynamics in the spleen, by fitting simulations to the published data described above. Among all simulations that obeyed the above criteria, we looked for the best fit to the experimental data, defined as the minimum value of the sum of squared deviations of simulated points from experimental data points (a least-squares fit), described by:
Thus, we searched for parameter values that minimize the deviation of results from experimental data, based on the least-squares criterion defined above. Each automated search involved varying all the relevant parameters simultaneously in very small steps (0.01, or smaller if higher resolution was found to be necessary), recording the fit of each run, and the parameter ranges which gave results within the experimental errors. In order to find whether the model fits the data, we conducted similar searches over all biologically reasonable parameter ranges
We used “Akaike's Information Criterion” (AIC) to find if our model is more likely to give a good explanation of B cell development in the spleen. We used an adaptation of the AIC method. In this method, we associate an AIC score to the parameter set that minimize the deviation of results from experimental data. We denote M to be the number of parameters fit by the regression, and N to be the number of data points. The AIC score (corrected for small numbers of data points) is thus defined by equation 10.
Suppose AIC_{c(A)} is the score of one with the minimal the deviation of results from experimental data (SS_{A}), and AIC_{c(B)} is the score of another set of parameters with a minimal sum SS_{B}. In this case, the difference between the AIC_{c} scores is given by equation 11, and is has a negative value, since
The probability that we have chosen the correct model (out of those that were considered) is then computed from equation 12. Since we have used the sum of squared deviations as an approximation for the MLE assumed by the AIC criterion, this probability is an approximation.
Our previously described mathematical models of B cell development in the BM
Our model starts with three bone marrow populations: pro-B [B220^{+}CD43^{+}IgM^{−}], pre-B [B220^{+}CD43^{−}IgM^{−}] and immature B cells [B220+HSA+IgM^{hi}IgD^{lo}], with cell numbers in these subsets represented by the variables B_{o}, B_{e} and B_{i}, respectively. However, previous experimental observations distinguish between small, non-cycling cells and large, cycling cells in both the pro-B and pre-B compartments, where the transition from pro-B to pre-B occurs while the cells are cycling. Hence, we break the pro-B and pre-B subsets into two subsets each: B_{or} for small resting pro-B cells (Hardy's fractions A through C) and B_{oc} for large cycling pro-B cells (part of Hardy's fraction C); similarly, B_{ec} for large cycling pre-B cells (the remainder of fraction C) and B_{er} for small resting pre-B cells (fraction D). Immature B (
In our previous study, we showed that out of 630 possible alternative models, only 8 can explain the population dynamics of transitional B cell differentiation in the spleen
We conducted simulations of this model, using the best set of parameter values previously obtained for the BM equations, and varying the parameters of the spleen populations, in order to obtain the best fit to published experimental data on these subpopulations in mice (section on data for model fitting and ref.
The parameter value ranges for the new hypothesis are presented in
These numbers were obtained by a simulation with the parameters set that gave the best fit to the data. Parameter values are given in
These kinetics were obtained by a simulation of the spleen population model
Parameter symbol and description |
Value range in acceptable models |
Value in the best-fit model |
0.15–0.19 | ||
0–0.1 | ||
0–0.005 | ||
0.07–0.11 | ||
0.13–0.19 | ||
0–0.009 | ||
0.035–0.06 | ||
0.004 | 0.004 | |
0–0.001 |
Rates are per 6 hours.
Models that obey our parameter choice criteria and fit the experimental data.
Using Akaike's Information Criterion as described in the methods shows that the probability that we have chosen the correct model is 86%, hence the new hypothesis is more likely to be correct.
We also obtained the same results with separate
An implication of this proposition is that most of the loss in peripheral B cell maturation may be due to a high rate of loss in the
Moreover, in the best-fit simulations, mature B cells differentiate mostly from
We next proceeded to estimate the fraction of mature B cells that differentiate to the T3 pool. The range of values of the differentiation rate from mature B cells to
Here we plotted the mature B cell differentiation term (
In the previous study, several alternative models were found to have a good fit to the data; all of them were models that included a differentiation from T3 to mature B cells and not vice versa, but they differed in the presence or absence of other transitions. For example, we found that the T1 or the T2 stage, or both, may be skipped by a small fraction of the differentiating cells. To find whether all these possibilities are also valid in the new model, we simulated all the alternative models of splenic B cell subsets that include the new hypothesis. Whenever the existence or absence of a certain transition was examined, the range of its rate parameter included the possibility that this rate equals zero. This was applied to
As shown in
Thus, as shown in
In this model, mature B cells do not have a death rate, that is μ_{m} = 0. The differentiation from B_{m} to T_{3} occurs a rate of δ_{m3} = 0.003, thus the T_{3} compartment accounts for all the mature B cell turnover. The other parameter values are given in
This study models the kinetics of splenic B cell compartments, using a combination of in vivo BrdU labeling data
We find that the in vivo labeling data are consistent with a model in which the death rate among T_{3} B cells is higher than in any other subpopulations, suggesting this pool represents the principal death niche for peripheral B cells. The model predicts that a majority of T3 B cells are derived from recent marrow émigrés, confirming prior assumptions that the T3 subset contains newly formed B cells that fail to meet the selective criteria imposed during transitional differentiation
Our analyses also reveal that up to 40% of the T3 pool, or about 10^{5} T3 B cells, may be derived from the mature B cell compartment. Because the mature pool is numerically large compared to the T3 pool, this indicates a low overall frequency with which mature B cells meet this fate. Nonetheless, it suggests that nearly all mature B cell losses could proceed via this phenotypic intermediate, because the mature B cell turnover of ∼2% per day would generate a steady state value in the 10^{5} range. Thus, B cell losses in the T_{3} compartment can not only account for all losses at the TR to mature B cell checkpoint, but can accommodate the bulk of mature B cell turnover as well. This is consistent with the view that T3 cells represent peripheral B cells destined for death regardless of origin, in accord with recent suggestions from Merrell et al
Together, these findings suggest that B cells fated for imminent elimination from pre-immune subsets comprise the T3 compartment, where they reside briefly. It is tempting to speculate that this reflects a common death pathway, especially since all of these cells rely on continuous signaling via the BCR and BLyS receptor 3 (BR3, also termed BAFFR) to survive. Accordingly, failure to fall within appropriate ranges for signaling via these two systems–regardless of the basis - may lead to acquisition of the T3 phenotype and subsequent death.
The authors are grateful to Ms. Hanna Edelman for help in manuscript preparation.