Asynchronous differentiation models explain bone marrow labeling

International Immunology, Vol. 15, No. 3, pp. 301±312 doi:10.1093/intimm/dxg025, available online at www.intimm.oupjournals.org

ã 2003 The Japanese Society for Immunology

Asynchronous differentiation models explain bone marrow labeling kinetics and predict re¯ux between the pre- and immature B cell pools Ramit Mehr1, Gitit Shahaf1, Alex Sah2 and Michael Cancro2 1Faculty

of Life Sciences, Bar-Ilan University, Ramat-Gan 52900, Israel and Laboratory Medicine, University of Pennsylvania School of Medicine, 36th and Hamilton Walk, Philadelphia, PA 19104-6082, USA 2Pathology

Keywords: B lymphocyte subsets, bromo-deoxyuridine, mathematical model Abstract B lymphopoiesis has historically been depicted as a unidirectional process, in which cohorts of developing cells transit through successive differentiative stages in an irreversible, synchronous manner. Here, we examine this view by combining kinetic analysis of developing B cell subsets in the bone marrow with mathematical modeling. Our bromo-deoxyuridine (BrdU) labeling data are incompatible with B cell development being a synchronous process, because labeling curves are non-linear. Moreover, we show that B cell development may not be completely unidirectional, because our results support the possibility of a phenotypic `re¯ux' among the immature to the pre-B cell subsets.

Introduction Bone marrow stem cells that have committed to the B lineage proceed through several surface marker-de®ned stages, which can be further broken into substages according to the proliferative status of the cell, expression of recombination activating genes (RAG-I and RAG-II), etc. [reviewed in (1±4)]. In the pro-B stage [B220+CD43+IgM±, as de®ned by the Osmond scheme (1)], cells rearrange their heavy chain genes. This stage can be further divided to three fractions according to the Hardy scheme (5,6): fraction A (HSA±BP-1±) in which all Ig genes are still in the germline con®guration, although sterile m transcripts and RAG expression can be detected late in this stage, and fractions B and C (HSA+BP-1± and HSA+BP-1+, respectively) that contain increasing numbers of cells which have already rearranged D±J (but not V±DJ) on one or both of their heavy chain alleles (7). Cells that have succeeded in rearranging a heavy chain express it on their surface in conjunction with the surrogate light chain and possibly a stromal ligand for the pre-BCR (8). These cells perform several rounds of division, along with differentiating to the pre-B stage (9,10). It has been shown that de novo synthesis of the preBCR not only allows differentiation, but also directly induces or at least enhances proliferation (11); the surrogate light chain

constant part equivalent l5 is required for this function (12) and so is the variable part equivalent Vpre-B, although neither component is necessary for the induction of heavy chain allelic exclusion (13,14). The fraction of dividing cells, which are CD43low and also TdT±, has been termed fraction C¢ (5,15). Differentiation and proliferation require signaling through the pre-BCR, which is transduced (as are BCR signals later on) by Iga and Igb. Iga and Igb have partially overlapping, but not completely redundant, roles in transducing the signals required for developing B cell survival, proliferation and differentiation [reviewed in (16)]. In the pre-B stage (mostly B220dullCD43±IgM±, Hardy's fraction D), cell divisions are terminated and the cells proceed to rearrange their light chain genes. In the immature B cell stage (B220dullCD43±IgM+, Hardy's fraction E), the cells express IgM BCR on their surface and are subject to strong negative selection as ligand binding yields either cell death or rescue via receptor editing (17±24). The threshold for negative selection, in terms of receptor avidity, is much lower than that of mature B cell activation (25). Immature B cells are exported from the bone marrow to the periphery as `transitional' cells (26±29), which are subject to further selection (28±30),

Correspondence to: R. Mehr; E-mail: [email protected] Transmitting editor: I. Pecht

Received 12 June 2002, accepted 6 November 2002

302 Re¯ux between the pre- and immature B cell pools [reviewed in (31)]. Those transitional cells that survive complete their maturation and join the mature naive peripheral B cell pool (B220brightCD43±IgM+IgD+, Hardy's fraction F). It has been suggested that the extension of selection to the periphery allows those autoreactive B cells, which express receptors for self-antigens not encountered in the bone marrow, to be deleted from the repertoire before they fully mature and become potentially dangerous. The present paper, however, deals only with the population dynamics of developing B cells in the bone marrow. For the sake of our discussion, the term `immature' B cells refers to cells that are already expressing their IgM antigen receptors on their surface and are undergoing selection or getting ready to emigrate to the periphery. One factor that delayed the formation of a complete quantitative model of B cell development was the inconsistency in designation of the different developmental stages, which has been recently resolved (1). Additionally, the results of different attempts to quantify the populations of developing B cells in mouse bone marrow vary greatly [see, e.g. (32±38)], due to differences between mouse strains, the use of different markers to identify developing B cell subpopulations, variability in experimental conditions or differences in the methods used to calculate the ®nal numbers. For example, many studies use the percentages of cells in different subsets in bone marrow (obtained by ¯ow cytometry) from one or two femura, multiplied by the total cell count (which is usually much less precise than determination of subset composition), then multiplied by the estimated ratio of the number of cells in total bone marrow to the number in one femur. For the latter ratio there is a single estimate of 15.8 (36), which is used in all calculations, even though this ratio probably varies between individual mice and between different mouse strains. Estimating kinetic parameters has proven equally dif®cult. The rates of differentiation from one population to the next can only be estimated based on an a priori model of population dynamics, using data from labeling studies and making assumptions about how much of the labeling in each compartment re¯ects proliferation versus entry of cells from a previous compartment (35). The various estimates of population growth rates are based on different methods of enumerating cycling cells, which do not always give consistent results (33,39). Additionally, population growth rates are usually estimated as the slope of the `linear' part of the labeling curve (40), which is wrong (41), because the labeling rate is, at all time points, a non-linear function of all rate parameters for each population and no part of it is truly linear. The estimation of cell death rates is the most dif®cult, because dead cells are rapidly cleared from the bone marrow (42±44). Partially blocking programmed cell death can give an estimate of the numbers of cells that would be otherwise lost (45), but it is dif®cult to deduce the overall population death rates from these numbers. An assumption inherent in the models used so far (reviewed above) to obtain kinetic parameter estimates is that developing B cell population dynamics are virtually synchronous and unidirectional. Cells are depicted as progressing in a straightforward manner from one developmental stage to the next, as if on a conveyor belt. However, there are several reasons to doubt that B cells develop along a synchronous, irreversible program. (i) Gene rearrangement success or failure and

selection for functional heavy chains introduce variability in the time cells take to traverse the early pro-B stage and in the number of cell divisions performed by the cells which have succeeded in rearranging and expressing a functional heavy chain (46). (ii) The number of cell divisions may vary depending on the level of expression of the heavy chain and the components of the surrogate light chain, the af®nity between BCR components, the strength of the signal transduced and the availability of microenvironmental support (47). (iii) Variability is also introduced during the rearrangement of light chain genes in the late pre-B stage. (iv) Marked variation in residence times in the pre-B and immature B cell subsets is introduced by the existence of secondary rearrangements. An example of the inter-cell variability is the recent discovery that a single event of receptor editing causes a delay of at least 2 h in developing B cell progression and that the extent of editing is quite high, at least 25% of developing cells (48). Repeated receptor editing events may thus lead to considerable delays. One may ask, then, how many attempts of rearrangement is a cell allowed to make, and how long does it take to go through the process of rearrangement attempt, failure, and initiation of the next rearrangement and testing cycle. The ®rst question is dealt with in other studies [(49) and our work in preparation]; the second is more complex and will be discussed below. The ®ndings concerning secondary rearrangements raise the following question: is the progression from pre-B to immature B cells a unilateral, irreversible process? All published calculations have based the estimates of differentiation and death rates in these compartments on the assumption that B cell progression from the pre-B to the immature B cell stage is unidirectional and irreversible. However, the identi®cation of cells as pre-B or immature B cells is largely based on surface IgM expression, which may be reduced or even completely abrogated as a cell undergoes receptor editing. Some downregulation of BCR expression in immature B cells following signaling through their BCR has indeed been observed (50). The above ®ndings point at the possibility that some of the cells identi®ed as pre-B cells may be re-entrants from the immature B cell compartment, i.e. cells in the process of performing secondary rearrangements. This possibility has not, so far, been included in the analyses used to estimate kinetic parameters of developing B cell populations. It has also been noted that the number of immature B cells exported to the periphery per day accounts for only about a quarter of the calculated production from the immature B compartment (35). Such a discrepancy may be attributed to cell loss; however it can also be partially due to an inaccurate estimate of cell production in the pre-B cell compartment, which ignores the possibility of cells re¯uxing from the immature into the pre-B compartment. Below, we present the ®rst quantitative model for the population dynamics of developing B cells. Using this model we attempt to reconcile experimental estimates of the composition and kinetic parameters of B cell subpopulations and examine the assumption that B cell development is a synchronous and unidirectional process. We show that intercell variability and phenotypic re¯uxÐimmature B cells returning to the pre-B stage for secondary rearrangementsÐ provide a better explanation of experimental observations.

Re¯ux between the pre- and immature B cell pools 303

Fig. 2. Model of developing B cell populations. Cell subsets and parameters represented in our model are shown (see Methods for details). The thick dashed arrow designates re¯ux of cells from the immature B cell to the pre-B cell compartment, due to secondary rearrangements, with rate dr. Fig. 1. BrdU labeling of developing B cells in murine bone marrow. The total cell numbers were (4.08 6 0.5) 3 107 cells, out of which 8.6 6 3.0% were pro-B (IgM±CD43+), 68.6625.2% were pre-B (IgM± CD43±) and 22.8 6 8.3% were immature B cells (IgM+CD43±). Each point is the average of measurements of at least four mice in two independent experiments.

Methods Mice BALB/c mice were used in all experiments and were obtained from the Jackson Laboratory (Bar Harbor, ME). Lymphocyte suspensions Brie¯y, bone marrow cells were obtained from the two hind limbs of donor animals, and prepared by ¯ushing the femurs and tibias with cold medium and aspirating the cell suspension using sequentially smaller bore needles. Erythrocytes are depleted by incubation in either Gey's solution or ammonium chloride±Tris. Antibodies to cell-surface antigen and immuno¯uorescent analyses The following reagents were purchased from PharMingen (San Diego, CA): phycoerythrin (PE)- and FITC-labeled anti-CD24 (heat stable antigen) (M1/69); PE-labeled anti-CD43 (leukosialin) (S7); allophycocyanin- and PE-labeled anti-CD45R (B220) (RA3-6B2). Biotin-labeled goat anti-mouse IgM, PElabeled anti-IgD (SBA-1) and streptavidin±alkaline phosphatase were purchased from Southern Biotechnology Associates (Birmingham, AL). FITC-labeled anti-bromo-deoxyuridine (BrdU) (B44) was purchased from Becton Dickinson (San Jose, CA). Continuous BrdU labeling and cyto¯uorimetric analysis Mice were injected i.p. with 0.6 mg BrdU (Sigma, St Louis, MO) in 0.2 ml PBS at 12-h intervals for the duration of each experiment. Cells from BrdU-treated mice were then stained for appropriate surface markers (PE, allophycocyanin and biotin-conjugated antibodies followed by Red 670±streptavidin) and washed with cold PBS. BrdU incorporation was analyzed according to previously published procedures. Brie¯y, the cells are permeabilized by dropwise addition of ice-cold 95% ethanol, washed, and then ®xed in PBS with 1% paraformaldehyde and 0.05% Tween 20. The cells were then incubated in buffer plus 100 U/ml DNase to partially degrade

and denature their chromatin. Finally, the cells were stained with FITC-labeled mAb to BrdU. Cytometric analyses were performed on a FACSCalibur ¯ow cytometer that was calibrated using the manufacturer's beads (Becton Dickinson). Data were collected by gating on all nucleated cells. Data were analyzed using CellQuest (Becton Dickinson) and FlowJo (Tree Star, San Carlos, CA) software. Statistical analysis The ®tting criterion for the case of two interdependent populations, based on the multivariate normality assumption, is minimization of the expression: ( ) 2 X T T 2 X T X X  2 1X 0 nt Y kt ÿ fkt ÿ  nt f1t f2t ÿ nt Y k t fkt …1† 2 kˆ1 tˆ1 tˆ1 kˆ1 tˆ1

where Ykt refers to the set of experimental measurements, fkt refers to the set of simulation results and these are compared for the two subpopulations, indexed by k (with k ¢ = 1 if k = 2 and vice versa), at each time point t for which there is an experimental result (t = 0,...,5 in units of 12 h from the beginning of labeling). The number of data points at time t is given by nt and r is the correlation between the data points for the two populations. The likelihood ratio test statistic for this case, used in obtaining the P value for comparison between the two hypotheses, is:

" 2 X T X

 nt Y kt

kˆ1 tˆ1

L ˆ ÿ2 …1 ÿ 2 †ÿ1 # 2 X T 2 X  2 ÿ fkt … 0 † ÿ nt Y kt ÿ fkt … † "

‡2ÿ2 …1 ÿ 2 †ÿ1 ( ÿ

kˆ1 tˆ1

T X

à 2t … † à ÿ nt f1t … †f

tˆ1 T X

nt f1t … Ã0 †f2t … Ã0 † ÿ

tˆ1

where s is the variance of Ykt.

2 X T X

à nt Y p t fkt … † 0

kˆ1 tˆ1 2 X T X

nt Y k t fkt … Ã0 0

kˆ1 tˆ1

)# …2†

304 Re¯ux between the pre- and immature B cell pools

Fig. 3. The modi®cation of our model to include BrdU labeling. All dividing cells are assumed to be labeled.

Results BrdU labeling kinetics of developing B cells If B cell differentiation is irreversible and synchronous, i.e. if all B cells had been spending a ®xed amount of time (corresponding to a ®xed number of cell cycles) in each compartment, then the labeling of pro-B cells would immediately (after one cell cycle) reach its maximum value, which corresponds to the fraction of late pro-B cells (those which belong to fraction C¢) out of all pro-B cells. Since all early pre-B cells would also be labeled in the ®rst cycle, the labeling of pre-B cells would reach within one cycle the fraction of early pre-B cells in all pre-B cells and will gradually increase in subsequent cycles to the maximum value of 100%. Labeling of all subsequent compartments would lag after the labeling of the previous compartment. In contrast, if these widely held assumptions are incorrect, then bi- or multiphasic labeling rates should be obtained. Most published labeling data have not sampled at frequent enough intervals to make this distinction. Accordingly, we undertook short-term (3 days) continuous in vivo BrdU labeling of bone marrow subsets and sampled every 12 h. The results (Fig. 1) suggest that some cells transit through the various stages faster than others. The labeling kinetics in the pro-B compartment are non-linear and slower, indicating a variability in the timing and length of cell cycle, which would be best visualized as a pool of cells with a wide distribution of times in each compartment. The same is true for the pre-B compartment. The labeling kinetics in the immature B cell compartment are more or less linear, implying that cells entering this compartment do not divide and exit the compartment after some characteristic length of time. On the other hand, the labeling kinetics in the pro- and pre-B compartments are clearly non-linear, exhibiting gradual saturation to a maximal value. A mathematical model of B cell development In order to extract more information from our experimental data, we formulated a mathematical model of the population dynamics of B cell development, used here to examine the concept of `re¯ux' from the immature back to the pre-B cell

compartment. The model is based on differential equations representing developing B lymphocyte subsets. This addresses the issue of intercellular variability, by representing each compartment as a `pool' of cells, where we ®x only the probability (per unit time) that a cell would leave this pool by using a population rate of transition. Cell division and death are similarly treated as probabilistic events by using population rates. A schematic representation of the model is presented in Fig. 2. Our model deals with three populations: pro-B, pre-B and immature B cells, with cell numbers in these subsets represented by Bo, Be and Bi, respectively. However, previous experimental observations distinguish between small, noncycling cells and large, cycling cells in both the pro-B and preB 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). For the sake of comparison to our experimental results, however, in the graphs presented in this paper, all pro-B cells were grouped together and similarly all pre-B cells were grouped together into one compartment. It should be noted that a version of the model which lumps together all of fraction C¢ cells, i.e. all dividing cells, into one subset was tried ®rst, but using this version we could not obtain a good ®t to the experimental results (data not shown). Hence we developed the ®vepopulation version described here. The input of stem cells into the pro-B compartment is denoted by s (for `source') in units of cells per time unit. For the sake of convenience we chose the time unit for which parameter values were de®ned to be 6 h, which is a minimum estimate for the cell cycle time of bone marrow cells (51). All other parameters are rate parameters, de®ned in units of fractions of cells per time unit. The rates of differentiation between subsets are represented by dX, with X representing the compartment the cells are differentiating from. Proliferation (population growth) is assumed to occur only in the late pro-B compartment (with the rate denoted by go) and the early pre-B compartment (with rate ge). Proliferation of developing B cells is known to be limited by the ®nite space and resources (e.g. contact with the stroma, growth factors, nutrients) in the bone marrow (10,32). Hence the term for proliferation in each compartment is multiplied by a logistic growth-limiting factor: (1 ± Bo/Ko) for pro-B cells and (1 ± Be/Ke) for pre-B cells, with Bo = Bor + Boc and Be = Bec + Ber. Ko and Ke denote the carrying capacities of the pro- and pre-B compartments, respectively, i.e. the population sizes for which the corresponding population growth rates become zero. While our model is the ®rst mathematical model of the population dynamics of developing B lymphocytes, much work has been done on modeling the population dynamics of developing T lymphocytes, which are similar in many aspects to developing B lymphocytes (52±57). For developing T lymphocytes, the logistic model has been found to be the most appropriate and we had no reason to assume that the growth limitations are different for B lymphocytes. Nevertheless, we have checked other models as well, de®ning the

Re¯ux between the pre- and immature B cell pools 305 Table 1. Published data on the rates of cycling, differentiation and apoptosis of bone marrow B cell populationsa Mice (references) Proliferation BrdU+ (%) (32) divisions (35) cycling (%) (5) cycling (%) (35) (%) (42) cycling BALB/cAn (% in metaphase) (38) DBA/2 (%) in metaphase) (35) Differentiation differentiation (3106) (34,35) Cell death (apoptosis) apoptosis (%) (35) apoptosis (%/h) (42)

Pro-B

Pre-B

Small

Large

Large

1±2b

10±15 20±30

5±6c >30

70±80

Small 54.061.2% 0). Implementing all of the features discussed above, we arrive at a set of differential equations, given in the Appendix, which describe the temporal dynamics of B cell subsets (Fig. 2). The solutions of these equations describe subset cell numbers that grow from zero until they gradually reach a steady state, i.e. a dynamic equilibrium in which, for each subset, the sum of cell input and production equals the sum of cell loss and exit. These equations have only one all-positive (and hence biologically meaningful) steady state, which is stable, i.e. small perturbations to the steady state will merely lead to transient changes in the total cell numbers, which will quickly return to the steady state. Steady-state cell numbers are given in the Appendix as functions of the model's parameters.

The expressions for the steady state, combined with some more biological knowledge, lead to several constraints on our parameter values, which will help us choose parameter ranges for the simulations. First, as mentioned above, for all populations to be positive, all parameters must be non-negative; differentiation rates (the ds) must be strictly positive. Second, for the steady state to be stable, the following additional condition (derived in the Appendix) must be met by the parameters: (go ± doc)(ge ± dec) > 0

(3)

Third, pro-B cells need a continuous contact with the bone marrow stroma in order to survive and develop further, while pre-B cells do not need that much contact with stroma and can thrive as long as the soluble growth factor IL-7 is supplied by stromal cells (3,58). Hence, Ke > Ko, which allows the pre-B compartment to grow to larger cell numbers. Modeling labeling kinetics of developing B cells To model the BrdU labeling experiments, we divided all B cell subsets (other than Bor) to two subsets, unlabeled (Boc,Bec,Ber,Bi) and labeled (LBoc,LBec,LBer,LBi). We assume cells in the ®rst subset, Bor, do not divide and hence are not labeled. Fractions A±C are non-dividing and hence cannot be labeled (7). As uncommitted stem cells may have been labeled to a minor extent, however, we have tested this point by adding to our simulations the assumption that up to 20% of the input cells may be labeled. This did not result in any signi®cant change to the results presented here (data not shown), because in the early pro-B subset there is much cell death and most of the expansion occurs in later subsets. Hence we neglect this effect henceforth. Cells in the Boc subset move to the labeled LBoc subset upon dividing and a similar move occurs in the Bec subset (Fig.3). We assume that once a cell is labeled, it remains labeled throughout the experiment. The new equations,

306 Re¯ux between the pre- and immature B cell pools Table 2. Parameter values in the simulation which gave the best ®t to experimental resultsa Subset

Pro±B

Pre±B

Immature B

Resting

Cycling

Cycling

Resting

Proliferation rate Ranges

0

0.7 0.05±0.95

0.5 0.2±0.95

0

0

Death rate Ranges (dr=0) (dr=0.1) (dr=0.2) (dr=0.3±0.5) (dr=0.6)

0.4

0

0

0.4

0

0±0.1 0±0.2 0±0.5 0±0.5 0±0.4

0±0.4 0±0.4 0±0.35 0±0.25 0±0.15

Output rate Ranges (dr=0) (dr=0.1) (dr=0.2) (dr=0.3) (dr=0.4) (dr=0.5) (dr=0.6)

0±1.8 0±1.8 0±1.8 0±1.8 0±1.8 1.5

0.1

0.15

0.95

0.05

0.1±1.9 0.1±1.9 0.1±1.9 0.1±1.9 0.1±1.9 0.1±1.9 0.1±1.9

0.1 0.1 0.1 0.1 0.1 0.1 0.1

0.15±0.7 0.15±0.4 0.15±0.4 0.15±0.2 0.15±0.2 0.15±0.2 0.15±0.2

0.1±0.3 0.05±0.6 0.1±0.95 0.2±0.95 0.25±0.95 0.35±0.95 0.45±0.95

0.05±0.45 0.05±0.4 0.05±0.3 0.05±0.3 0.05±0.2 0.05±0.3 0.05±0.2

aAlso shown are ranges of parameters that gave simulations with total cell numbers and population composition within the experimental range and score > 5. All rates are per a time unit representing 6 h. The rate of cell entry into the pro-B compartment, s, was 105, with a range of 5 to 25 3 104. The `carrying capacities' limiting population growth in the pro-B and pre-B compartments were, respectively, Ko = 5 3 106 and Ke = 3 3 107, with ranges of 5 3 106 < Ko < 5 3 107 and 3 3 107 < Ke < 108 respectively. Ranges of the rate parameters are given for simulations with dr = 0, 0.1, 0.2, 0.3, 0.4, 0.5 and 0.6 which have obeyed all the constraints; in case the ranges were different for different values of dr, separate ranges are given for separate values of dr

describing labeled and unlabeled populations, are given in the Appendix. These equations were integrated using a simple C program. In the simulations, we ®rst run the model without labeling for 100 time units (which is long enough for cell numbers to arrive at their steady-states), then run them with labeling for 10 units (60 h, as in the experiments) and then run them for the remaining time without labeling. The latter part of each run simulates the decay of labeling over time, under the assumption that this decay is only due to cell death or migration, neglecting any possible reduction of the amount of BrdU due to cell divisions. The decay of labeling was not measured in the experiments, hence our simulations predict what the dynamics of decay would be. This suggests an additional way to check our model, beyond ®tting to existing data. The total numbers of cells in both compartments and the fractions of cells in each compartment exhibit the same behavior as in the basic model, because we assume that labeling does not change the kinetics of B cell development. Model parameters In choosing parameter values for the simulations of our model, we adhered to the following guidelines. (i) The parameters should be in the experimentally observed orders of magnitude, if published information is available. A summary of the published information is given in Table 1. While these estimates (where available) suffer from the problems discussed above and, additionally, are usually not given in units of population rates, so that interpretation of most of this data depends on the model used, they were useful in suggesting the appropriate ranges for some of the parameters. (ii) 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 bone marrow B cells. Any parameter set which did not conform to this criterion was rejected. (iii) The time of arrival to the steady state should be reasonable (about 1 simulated month). (iv) The fraction of labeled cells in each of the three subpopulations should be within the error bars of the experimental observation for at least ®ve out of the six time points. We call the number of time points for which the simulated fraction of labeled cells for a given population the `score' for that simulation. The reason we do not require that the score will be 6 for all subpopulations is that there is a large variability at t = 0, where theoretically there should have been no labeled cells. All simulations which had a good ®t to the experimental data fell within this criterion (see below). These conditions signi®cantly constrain our choice of parameter ranges used in our simulations, such that the parameter range which gives results obeying all constraints is rather narrow. Since the parameters governing the behavior of pre-B and immature B cells have no effect on the behavior of pro-B cells, we ®rst conducted an automated search for the values of the six parameters governing the behavior of pro-B cells. These are s, mo, dor, go, Ko and doc. According to the criteria above, we rejected parameter sets which resulted in total pro-B cell numbers or fraction of labeled pro-B cells outside the experimental error (i.e. we used seven data points to determine six parameter values). Among the acceptable parameter sets, we looked for the best ®t to the data on pro-B cells, de®ned as the parameter set which gave the minimum value of the sum of squared deviations from experimental data points.

Re¯ux between the pre- and immature B cell pools 307

Fig. 4. The simulation of our model which gave the best ®t to the experimental results. The parameter values given in Table 2 were used. (A) Subset percentages. (B) Cell numbers.

The search was performed by repeating the simulations for 10±20 possible values for each parameter, i.e. ~106 simulations, and recording those parameter sets which gave acceptable results according to the criteria above. Since the parameters governing the behavior of immature B cells may (if dr>0) affect the behavior of pre-B cells, we could not perform independent parameter ®ttings for these two populations; instead, we performed a automated parameter search based on both populations, i.e. a search for the values of ge, Ke, dec, me and der, plus dr, mi and di. Thus, here we used 14 data points (total pre-B and immature B cell numbers and fraction of labeled pre-B and immature B cells at six time points) to determine the values of eight parameters. The ®tting criterion for this case of two interdependent populations and the likelihood ratio test statistic for this case, are given in Methods. Simulation results The parameter set which gave the best ®t to experimental data is given in Table 2. A simulation with this set of default parameters, run for 240 time units (60 days), arrives at the steady-state value of 4.53 3 107 cells. In terms of percentages, the pro-B cells consist of 9.0% of total cell numbers, the pre-B 60.3% and immature B cells 30.7%. These numbers are within the experimentally observed ranges. The kinetics of all B

Fig. 5. Labeling kinetics obtained by the simulation of the model which gave the best ®t to the experimental results. The parameters used are the ones given in Table 2. Simulation results (dashed lines) are presented along with the experimental results (symbols with error bars as in Fig. 1) and extended up to 120 h after the start of labeling, to show the predicted course of labeling decay. (Top) proB cells. (Middle) Pre-B cells. (Bottom) Immature B cells.

cell populations in this simulation are presented in Fig. 4. Simulated labeling kinetics obtained with our best-®t set of parameters, including the post-labeling decay of BrdUlabeled cell fractions, are given in Fig.5. As is evident from Table 2, the best ®t was obtained with dr = 0.4. When we ran simulations under the null hypothesis, i.e. with dr = 0 (with the same ranges for all other parameters), the best ®t was signi®cantly worse (P = 0.01366). Hence, our results strongly support the possibility of re¯ux between immature B cells and pre-B cells. The ranges of parameter values that give results within the experimental range (i.e. total cell numbers in each population are within the experimental range and fractions of labeled cells are within the experimental range for at least ®ve out of six time points in each population) are also given in Table 2. Note that the ranges are given here for each parameter separately, hence not all values in the range given for one parameter necessarily work with all values in the range given for other parameters. For example, within the ranges given for me and der, the pairs that give acceptable results vary depending on the value of dr used (Fig. 6A). The same is true for ge and dec (Fig. 6B). In this case it is clear that only pairs that obey ge > dec are acceptable, because these are the only sets that obey the stability constraint (go ± doc)(ge ± dec) > 0 (equation 11) with go > doc. The values of mi and di are also dependent on dr; pairs are acceptable as long as their sum remains suf®ciently small, so that immature B cell numbers do not become too low (Fig. 6C).

308 Re¯ux between the pre- and immature B cell pools The most interesting parameter is dr, the rate of re¯ux, as the value of this parameter may shed light on the process of light chain rearrangement. In Fig. 7(A) we show the fraction of labeled immature B cells in several simulations, which differ

Fig. 6. (A) Pairs of me and der values which obeyed our acceptance constraints. (B) Pairs of ge and dec values which obeyed our acceptance constraints. (C) Pairs of mi and di values which obeyed our acceptance constraints. Other parameters are as in Fig. 5.

only in the value of dr; all other parameter values are as in the best ®t simulation (shown in Table 2). It is evident that the higher the re¯ux rate, the faster the increase in the fraction of labeled immature B cells, not only due to the more rapid exit of unlabeled cells out of this compartment, but also because previously labeled cells may re¯ux and then returnÐas labeled cellsÐto the immature B cell compartment. Simulations with dr = 0 gave a good reconstruction of the experimental results only with higher values of mi and/or di than in the simulations with dr > 0 (Fig. 6C). The meaning of this result is that the previous estimates of cell loss during the transition from immature bone marrow B cells to transitional B cells in the periphery may be over-estimates. The apparent discrepancy between immature B cell production and numbers of peripheral transitional B cells may be accounted for, at least in part, by phenotypic re¯ux back to the pre-B stage. As Fig. 7(A) shows, increasing dr while keeping mi constant increases the rate of labeling; the same is true for increasing mi while keeping dr constant (Fig. 7B).

Fig. 7. (A) The dependence of immature B cell labeling on dr. Other parameters are as in Fig. 5. With this set of parameters, only the simulations with dr = 0.4 and dr = 0.35 obeyed our acceptance constraints. (B) The dependence of immature B cell labeling on mi. Other parameters are as in Fig. 5.

Re¯ux between the pre- and immature B cell pools 309 Discussion This study presents an analysis of the kinetics of B cell development in the bone marrow, using a combination of BrdU labeling experiments and the ®rst mathematical model of the population dynamics of developing B cell subsets. Our results show that these kinetics are best accounted for by using nonlinear population growth terms and support the hypothesis of `re¯ux', stating that the pre-B compartment contains cells that have re-entered it from the immature compartment, most probably due to receptor editing. Using mice transgenic for autoreactive BCR, receptor editing has been identi®ed as one of the mechanisms of central tolerance (19±21). Computational models of lymphocyte antigen receptor gene rearrangement have shown that cells choose the gene segments to be rearranged in a semiordered manner (49), thus maximizing the usage of available gene segments and allowing several rearrangement attempts per cell. If a rearrangement is merely non-productive, the cell will either die or perform a secondary rearrangement. On the other hand, if a rearrangement is productive, the cell will express the resulting antigen receptor and await the judgement of negative selection. While expressing its antigen receptor, the cell will be counted among the immature B cells. However, if, due to negative selection signals, the cell is forced to edit its receptor, it is conceivable that during the time period between negative selection and the expression of the edited receptor, the cell will be IgM low to negative (50) and hence be counted again among the pre-B cells. Thus there is a small probability of phenotypic `re¯ux' after each rearrangement attempt. Further support for this possibility comes from studies which show that in the pre-B stage, cells can be found with multiple productive light chain gene rearrangements (59). The probability that a cell will perform such a `re¯ux' after a given rearrangement attempt can be calculated as follows. The probability that a rearrangement is productive is 1/3. The probability that the resulting light chain will pair with the cell's heavy chain, Ppair, has been estimated as ~0.8 [estimates of all probabilities mentioned here are reviewed in (49)]. Out of all expressed receptors, it has been estimated that about 2/3 will be negatively selected. Thus, a maximum of (Ppair/3)*(2/3) = 2Ppair/9 of all cells will revert to the pre-B phenotype and edit their receptors after each rearrangement. With Ppair = 0.8 (49), this corresponds to about 0.18 of the cells at most re¯uxing to the pre-B phenotype after each rearrangement. Obviously the actual number should be lower, because not all cells edit their receptors upon negative selection; some of them die in the immature stage. The recent ®nding that a single event of receptor editing causes a delay of at least 2 h in developing B cell progression (48) means that, at most, 0.18 of the cells in a given cohort re¯ux to the pre-B phenotype after 2 h, which corresponds to a maximum of 0.54 of cells re¯uxing after 6 h. In our simulations, the fraction of re¯uxing cells was found to be ~0.4 per 6 h, which is lower than the upper bound above, but is of the same order of magnitude. This unexpected agreement lends further support to our conclusions and also points at the possibility that the kinetics we observe result from multiple such delays.

Two new, testable predictions, which stand in contrast to the intuition on which all the published work on estimates of population dynamic rates (cited above) is based, are thus made by our model. The ®rst is that some immature B cells undergoing receptor editing will down-regulate their surface IgM expression and hence will be phenotypically indistinguishable from pre-B cells. The second prediction is that the loss of B cells in the immature stage is lower than previously estimated. Historically, the difference between the rate of appearance of transitional cells in the periphery and the rate of production of immature B cells in the bone marrow has been assumed to be due to cell death at the immature stage. Our results imply that much of this difference may be due to re¯ux to the pre-B cell stage, where cells are again as likely to fail in the secondary rearrangement as in the primary rearrangement, so that most cell loss is due to failed rearrangements rather than direct BCR-mediated activation of apoptosis. Phenotypic re¯ux is only one possible explanation for the data presented here. Other explanations might also be suggested, e.g. there may be several pathways of pre-B to immature B cell differentiation, which proceed in parallel, but at different rates. The evaluation of such explanations would require the examination of yet unresolved phenotypic/functional subsets and further mathematical modeling. The combination of these methods will enable us in the future to examine alternative explanations of the observed labeling kinetics. The main point to bear in mind, however, is that estimates based on the unidirectional, synchronous model are likely to be inaccurate.

Acknowledgements We are grateful to M. Gor®ne and L. Freedman for the derivation of the ®tting criterion and likelihood ratio estimates for two interdependent populations. Supported in part by Israel Science Foundation grant 759/01-1, the Yigal Alon Fellowship and a Bar-Ilan University internal grant (to R. M.).

Abbreviation RAG

recombination-activating genes

Appendix: Model equations and analysis Implementing all of the features discussed above in `A mathematical model¼', we arrive at the following set of differential equations, which describe the temporal dynamics of B cell subsets (Fig. 2): dBor ˆ s ÿ …o ‡ or †Bor dt

…4†

dBoc Bo ˆ or Bor ‡ o Boc …1 ÿ † ÿ oc Boc dt Ko

…5†

310 Re¯ux between the pre- and immature B cell pools dBec Be ˆ oc Boc ‡ e Bec …1 ÿ † ÿ ec Bec dt Ke

…6†

ÿ…o ‡ or †… o ÿ oc †… e ÿ ec †‰…e ‡ er †…i ‡ i ‡ r † ÿ er r Š ˆ

dBer ˆ ec Bec ÿ …e ‡ er †Ber ‡ r Bi dt

…7†

dBi ˆ er Ber ÿ Bi …i ‡ i ‡ r † dt

…8†

Since all the parameters are non-negative and all differentiation rates (the ds) are positive, this condition is equivalent to (go ± doc)(ge ± dec) > 0 (equation 3). Third, as discussed in the model section, Ke>Ko, as the pre-B compartment is less constrained than the pro-B compartment and can grow to larger cell numbers. When we model labeled and unlabeled cell subsets separately, the equations become:

Equations (4)±(8) have only one all-positive (and hence biologically meaningful) steady state, which is stable. This steady state is given by: Bor ˆ

Boc

Bec ˆ

s o ‡ or

…9†

 q 1 1 ‡ 12 ‡ 4 2 2 ˆ 2

…10†

q

e ÿ ec ‡ … e ÿ ec †2 ‡ 4 e 5 oc Boc 2 e 5

Bi ˆ

…12†

er Ber i ‡ i ‡ r

…13†

where

1 ˆ

Ko s ‰ o …1 ÿ † ÿ oc Š; Ko …o ‡ or †

o

2 ˆ

Ko or s

o …o ‡ or †

and:

4 ˆ

r er ÿ e ÿ er ; …i ‡ i ‡ r †

5 ˆ

dBor ˆ s ÿ …o ‡ or †Bor dt

…14†

    dBoc Bo ‡ oc Boc ˆ or Bor ÿ o 1 ÿ dt Ko

…15†

    dBec Be ‡ ec Bec ˆ oc Boc ÿ e 1 ÿ dt Ke

…16†

dBer ˆ ec Bec ‡ r Bi ÿ …e ‡ er †Ber dt

…17†

dBi ˆ er Ber ÿ …i ‡ i ‡ r †Bi dt

…18†

…11†

ec B ec 4

Ber ˆ

ÿ…o ‡ or †… o ÿ oc †… e ÿ ec ‰er †…i ‡ i †‡ e …i ‡ i ‡ r †Š < 0

1 ‡ …ec =4 † Ke

These steady-state expressions, combined with biological knowledge, lead to several constraints on our parameter values, which helped us choose parameter ranges for the simulations. First, as mentioned in the text, for all populations to be positive, all parameters must be non-negative; differentiation rates (the ds) must be strictly positive. Second, for the steady state to be stable, the following additional condition must be met by the parameters:

  dLBoc Bo …2Boc ‡ LBoc † ÿ oc LBoc ˆ o 1 ÿ …19† dt Ko   dLBec Be …2Bec ‡ LBec † ÿ ec LBec …20† ˆ oc LBoc ‡ e 1 ÿ dt Ke

dLBer ˆ ec LBec ‡ r LBi ÿ …e ‡ er †LBer dt

…21†

dLBi …22† ˆ er LBer ÿ …i ‡ i ‡ r †LBi dt where Bo = Bor+Boc+LBoc and Be = Ber+Bec+LB ec+LBer denote the total numbers of pro-B and pre-B cells, respectively. The total steady-state cell numbers remain unchanged.

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