Screening of alternative models for transitional B cell maturation

Gitit Shahaf1, David Allman2, Michael P. Cancro2 and Ramit Mehr1

1 Faculty of Life Sciences, Bar-Ilan University, Ramat-Gan 52900, Israel
2 Pathology and Laboratory Medicine, University of Pennsylvania School of Medicine, 231 John Morgan Building, 36th and Hamilton Walk, Philadelphia, PA 19104-6082, USA

Correspondence to: R. Mehr; E-mail: mehrra{at}mail.biu.ac.il


    Abstract
 Top
 Abstract
 Introduction
 Methods
 Results
 Conclusions
 References
 
Several functional and phenotypic B cell populations have been described in the spleen. These include the ‘transitional’ subsets, which are thought to be late differentiation intermediates of marrow-derived, mature follicular B cells. The exact progenitor–successor relationships of these transitional subsets, as well as whether a proliferative step is requisite for follicular B cell maturation, remain controversial. Moreover, whether late B cell differentiation might involve branched or asynchronous maturation pathways, thus allowing some cells to ‘skip’ one or more of these stages, has not been investigated. Herein we have used mathematical modeling to interrogate these possibilities. Using mathematical models that numerically simulate splenic B cell population dynamics, we have determined which alternative models of differentiation best fit existing in vivo labeling data. Our results indicate that follicular differentiation does not involve a proliferating splenic intermediate. Our results further suggest that some developing cells move directly from the immature marrow pool to more advanced semi-mature peripheral subsets without passing through the least mature subset in the spleen.

Keywords: mathematical model, population dynamics, transitional B lymphocyte subsets


    Introduction
 Top
 Abstract
 Introduction
 Methods
 Results
 Conclusions
 References
 
B cells are derived from bone marrow precursors that transit a series of differentiate stages associated with immunoglobulin (Ig) gene rearrangement and the assembly of a functional B cell receptor (110). These subsets and their associated surface markers are shown in Table 1. Thus, pro-B cells contain heavy but not light chain gene rearrangements, and subsequent expression of an Ig heavy chain associated with surrogate light chain yields rapid proliferation and entry into the large pre-B cell compartment. This is followed by progression to the small pre-B cell compartment, light chain gene rearrangement, and assembly of heavy and light chains. The expression of surface immunoglobulin marks the generation of immature B cells, which are exported to the periphery, complete maturation, and join the mature peripheral (that is, non-bone marrow) B cell pool (1116). After completing maturation, the majority of these cells enter the follicular subset, which constitutes the bulk of the peripheral B cell pool and populates the B cell follicles in secondary lymphoid organs. Alternatively, some of these newly emerging cells may be recruited to the far smaller MZ or B1 subsets, but the exact progenitor relationships and dynamics of these latter pools remain obscure (17). Among the follicular B lineage differentiation stages, all but the mature peripheral B cells display rapid turnover (12,1823). This reflects high production rates coupled with either rapid transit to successive differentiated stages or death. In fact, <5% of newly formed immature marrow B cells survive to maturity, due to combined losses from homeostatic and selective processes (12).


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Table 1. Bone marrow B cell developmental subsets

 
Following the discovery that B cells complete maturation in the periphery (11,12) much activity has focused on the peripheral subsets that bridge initial antigen receptor expression in the bone marrow with entry into the long-lived follicular pool. Mounting evidence indicates that the maturation stages immediately following marrow egress are equally critical points in B cell differentiation, where both negative and positive selection processes act on newly emerging B cell clones (13,14,18,2428). Subsequent to their initial description based on HSA (CD24) intensity (11,12), cells within this peripheral subset have been termed ‘transitional’ cells (13) and have been further subdivided based on two partially overlapping strategies (Table 2). In the first, Loder et al. (15) forwarded the notion that transitional differentiation proceeds in a stepwise fashion, using CD23, CD21 and sIgD expression to define two transitional categories; termed T1 and T2. The other strategy resolves three transitional subpopulations, termed T1, T2 and T3, on the basis of AA4.1 (14), CD23, and sIgM expression.


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Table 2. Post BCR-expression developmental subsets

 
While the T1 populations defined by these two approaches are congruent, a critical disparity exists in the later stages: in the two-stage scheme, the T2 population is cycling; whereas in the AA4.1 based scheme none of the subsets are cycling. Thus, the relationship of cycling T2 cells to the non-dividing, AA4.1+, T2/3 pools remains unclear. Each phenotypically defined subset may represent an intermediate stage in the final maturation of recent marrow émigrés. In this case, the cycling T2 pool must be a short-lived intermediate spanning the AA4.1+ (T3?) pool with AA4.1 mature follicular B cells. Alternatively, some of these subsets could represent previously unappreciated compartments that lie outside the pathway of follicular B cell generation from marrow progenitors.

We have previously shown that mathematical modeling of population kinetics established from in vivo bromodeoxyuridine (BrdU) labeling studies provides a powerful tool with which to assess alternative models of B cell differentiation in the bone marrow (29). Because the same kind of labeling data exists for transitional and follicular peripheral subsets, and the presence of a cycling population should yield distinctly different kinetics in downstream populations, mathematical modeling provides a critical test of the placement of the cycling T2 pool in the pathway for follicular differentiation from recent marrow émigrés. Mathematical modeling enables us to write equations for various alternative models, and using simulations based on these equations and fitting simulation results to experimental data, evaluate the plausibility of alternative models. In the present study, we have used a computer program that numerically simulates B cell population dynamics in the spleen to fit alternative models to the experimental data, in order to find out which model fits the data best.


    Methods
 Top
 Abstract
 Introduction
 Methods
 Results
 Conclusions
 References
 
Data for model fitting
In order to understand the behavior of the transitional subpopulations of B cells that will become mature naive B cells in the spleen, we used published experimental data on these subpopulations in mice [(14), and see below)]. The current dogma is that the process of B cell development is the same in mice and humans and hence the theories based upon these data are valid for humans as well. The data include measurements on two immature marrow subpopulations distinguished by differential IgM surface expression and four peripheral populations including mature follicular B cells and three subsets of transitional B cell defined previously by differential surface expression of IgM and CD23.

The data we previously used to fit our mathematical model of B cell populations in the bone marrow (29), distinguished the immature B cell from the pro and pre B cell subsets by the expression of the receptor (IgM) on the cell surface. However, in (29), all IgM+ cells were taken as one population, rather than being divided into IgMhi and IgMlo subsets as in (14). Since the mathematical model should be fit to the pro and pre B cell data from (29), and to the immature B cell data from (14), we needed to translate the data in (14) to the labeling kinetics on the whole IgM+ subsets. This was done by taking the labeling kinetics averages of the two subpopulations of immature B cells from the Allman data (14).

Kinetic analyses using BrdU labeling
Kinetic analyses used in these modeling comparisons were those previously published from the Allman and Cancro laboratories (14,29). Detailed methodological descriptions are thus available in these publications. Briefly, mice were treated with i.p. injections of 0.5 mg BrdU (Sigma) twice daily. Bone marrow cells or splenocytes were analyzed at successive intervals thereafter by fluorescent staining for surface markers and incorporated BrdU. For each mouse, the percentage of BrdU-labeled cells in each subset was measured cytometrically and multiplied by the total cell number in the subset to give the number of labeled cells. The values were plotted as a function of time, and a regression analysis was done on the linear portion of the plot to obtain the BrdU+ cell accumulation rate.

Mathematical models
We expanded our model of B cell development by adding transitional B cells in the spleen. The number of cells in the early and late pro-B, early and late pre-B, and immature B cell population are represented by Bor, Boc, Bec, Ber and Bi, respectively; and the numbers of cells in the T1, T2, T3 and mature B cells are represented by the variables T1, T2, T3 and Bm, respectively. Bone marrow cell populations are described, as in our previous work (29), by the following equations:





In these equations, the input of stem cells into pro-B compartment is denoted by s (for ‘source’), the parameters µ denote death rates, the parameters {delta} denote differentiation rates and the parameters {gamma} denote proliferation rates. More information about this part of the model is found in (29).

Immature B (Bi) cells migrate from the bone marrow to the periphery with a constant rate of {delta}i (29) (Fig. 1). Out of the {delta}iBi cells that exit the bone marrow daily, f1 is the fraction of {delta}{iota} cells that differentiate to the T1 subset and f2 is the fraction of cells that differentiate directly to T2. There may also be a fraction f3 of cells that differentiate to T3, and a fraction fm of cells that differentiate to Bm. T1 cells differentiate to T2, and may differentiate to T3 or to mature B cells (Bm). Hence we denote by {delta}12 the differentiation rate of T1 to T2, by {delta}13 the differentiation rate of T1 to T3, and by {delta}1m the differentiation rate of T1 to Bm. T2 cells can differentiate to T3 or to Bm, and the rates of differentiation to those subsets are {delta}23 and {delta}2m, correspondingly. If T2 are cycling, the proliferation rate of this subset may be assumed to be limited by the finite space and resources, such as growth factors for B lineage cells in the spleen. Hence, the term for proliferation ({gamma}) is multiplied by a logistic growth-limiting factor: (1 – T2/KT). Finally, {delta}3m is the differentiation rate from T3 into Bm. The exit rate from each compartment is the sum of the death rates (µ) and the differentiation rates ({delta}) and is denoted by {varepsilon}1, {varepsilon}2, {varepsilon}3 and {varepsilon}m for T1, T2, T3 and Bm, respectively.



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Fig. 1. Alternative models of developing B cell populations in the spleen. (A) The model of Loder et al. (15,16), with only two transitional cell subsets. (B) The Allman et al. model (14), with three non-cycling transitional B cell subsets. (C) The Allman et al. model (14), with all possible transitions between subpopulations explicitly shown. Parameters governing population dynamics in our mathematical model are also shown (see Methods for details).

 
The equations describing the dynamics of transitional and mature B cells are as follows (the terms enclosed in square brackets are optional and represent alternative models):




The models for the pro B and pre B populations were not changed; we used the equations and parameters from our previous study (29). This model was used to find out which transitions actually do occur in transitional B cell populations, and at what rates. All rates were in terms of fraction of the cell population per simulation time step; the simulation time step (i.e. the time interval represented by each iteration of the differential equations) was taken to be 6 h, which is the minimal possible cell division time for B cells.

These equations were integrated using a simple C program. In the simulations, we first ran the model without labeling for 100 time units (600 h, which is long enough for cell numbers to arrive at their steady states), then ran them with labeling for 28 units (168 h, as in the experiments).

Choosing the best models
A priori, counting all combinations of alternative transitions and processes, we see that the T1 subset can be described by only one possible model, the T2 subset has six possible models describing its dynamics, the T3 subset has seven possible models, and mature B cells have 15 possible models. Therefore we have a total of 630 possible models for the dynamics of B cell populations in the spleen. 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 this data depends on the model used, they were useful in suggesting the appropriate ranges for some of the parameters.

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 and transitional B cells. Any parameter set which did not conform to this criterion was rejected.

The time of arrival to the steady state should be biologically reasonable. These conditions significantly constrain the choice of parameter ranges used in our simulations, such that the parameter range which gives results obeying all constraints is rather narrow. In fact, only 8 out of the 630 possible models obeyed these constraints for some parameter sets (see below).

Since our goal here is only rough scanning for the best model, fitting simulations to the published data (averages of labeling fractions in each population at each time point) was sufficient. Among all simulations that obeyed the above criteria, we looked for the best fit to the data on the subpopulations, defined as the minimum value of the sum of squared deviations from experimental data points (a least-squares fit), described by:

Ykt refers to the set of experimental measurements, fkt refers to the set of simulation results, and these are compared for all subpopulations, indexed by k, at each time point t for which there is an experimental result.

The transitional 1 B cell population (T1) is the first splenic subpopulation to differentiate directly from immature B cells in the bone marrow. Since there is only one possible model for the dynamics of T1, we conducted an automated search for the values of the parameters governing the behavior of pre-B cells, immature B cells and T1 B cells, searching 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 the models that best fitted the data, we conducted similar searches over all biologically reasonable parameter ranges for the more differentiated subpopulations, the transitional 2 B cells (T2), the transitional 3 B cells (T3) and the mature B cells, and calculated the fitness formula of those three subpopulations together.

Possible limitations of the models
The models we used were all linear in the numbers of cells in each population, other than possible logistic limitations of the terms describing cell proliferation. Since our goal was to screen the various alternative models in terms of the existence or absence of certain transitions and proliferation processes—and there is at present no additional data which could help us identify the precise kinetics of such transitions or proliferation processes—there was no justification for using more complicated, non-linear terms to describe each of these processes. The only exception is the logistic term limitations imposed on cell proliferation, which can be justified by the assumption that space and resources in B cell follicles cannot possibly be unlimited, and hence proliferation should depend on the density of the existing number of cells. Furthermore, the intrinsic variability of the data itself does not allow us to pin down the parameter values with great precision in most cases. As shown here, however, even these data enable us to narrow down the number of alternative models from hundreds of possible combinations to just a few possibilities, and suggest directions for future study.


    Results
 Top
 Abstract
 Introduction
 Methods
 Results
 Conclusions
 References
 
The T1 model
For T1 B cells there is only one model, in which immature B cells (Bm) migrate from the bone marrow to the periphery with a constant rate {delta}i (29). Out of these emigrants, f1 is the fraction of immature B cells that differentiate into the T1 subset. Cells in T1 can either die or differentiate into the T2 subset. When fitting only the T1 data, we lump death and differentiation of T1 cells into one term of exit at rate {varepsilon}1. The equation for the T1 subset is:

We ran the bone marrow and T1 model simulation with the parameter values of the pre-B and immature B cell subpopulations as given in Table 3, which give the best fit of the pre-B and immature B cell subpopulations to the experimental data (Fig. 2A). We chose a biologically reasonable parameter range for f1 and {varepsilon}1. According to the criteria above, we rejected parameter sets which resulted in total subpopulation numbers and fractions of labeled T1 B cell outside the experimental range (Fig. 3A). The best fit was obtained with the parameters given in Table 4.


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Table 3. Rate parameters for bone marrow populationsa

 


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Fig. 2. (A) BrdU labeling kinetics obtained by a simulation of the model which gave the best fit to the data (29) on pro and pre B cells and to the data (14) on immature B cells. The parameters used are given in Table 3. Simulation results (lines) are presented along with the experimental results (symbols with error bars). (B) BrdU labeling kinetics obtained by a simulation of the spleen population model which gave the best fit to the data (14). The parameters used are given in Tables 6–8GoGo. Simulation results (dashed lines) are presented along with the experimental results (symbols with error bars).

 


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Fig. 3. (A) Cell numbers obtained by a simulation of the spleen population model which gave the best fit to the data. The parameters used are the ones given in Tables 6–8GoGo. (B) Extending the simulation to 120 h after the start of labeling.

 

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Table 4. Parameters for T1 B cellsa

 

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Table 6. Parameters for T2 B cells

 

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Table 7. Parameters for T3 B cells

 

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Table 8. Parameters for mature B cells

 
Models for T2
For the T2 B cell subset there are two alternative models, one with and one without proliferation. In both models, T2 cells can come from T1 cells or directly from a fraction of immature B cells that transit from the bone marrow to the spleen. The exit rate represents the sum of rates of death and differentiation to the next subpopulations. The models are represented in Fig. 4(A). The equations describing these models are:


When {gamma}2 = 0 this becomes a model without proliferation. We ran simulations of the model with and without T2 cell proliferation.



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Fig. 4. (A) The transitional 2 (T2) B cell subset and its rate parameters. T2 cells come from T1 or from a fraction of immature B cells that transit from the bone marrow to the spleen. T2 cells may or may not proliferate: {gamma}2 > 0 or {gamma}2 = 0. (B) The transitional 3 (T3) B cell subset and its rate parameters. (C) The mature B cell Bm subset and its rate. (D) A model in which both the differentiation from T1 to Bm and the differentiation from T2 to Bm do not exist ({delta}1m = 0, {delta}2m = 0). This was the best-fit model.

 
Models for T3
Cells in T3 can come from T2, from T1, or from a fraction of immature B cells that transit from the bone marrow to the spleen. The alternative models for T3 are represented in Fig. 4(B). The equations for these models are:



Models for mature B cells
The mature B cell subpopulation contains fully developed naive mature B cells, which express productive immunoglobulin on their surface, have passed the selection against binding to self-antigen, but have not yet been exposed to antigen. The equations for the models that include mature B cells are:




Simulation results
We ran all the alternative models of the three splenic B cell subsets together. Whenever the existence or absence of a certain transition was examined, the range of its rate parameter included the possibility of rate = zero. This applied to f2, f3, fm, {delta}12, {delta}13, {delta}1m, {delta}23, {delta}2m and {delta}2m. We required that a model fit not only the labeling kinetics of all three splenic B cell subsets, but also the total cell numbers. This reduced the number of acceptable models.

When we ran the T2 model with {gamma}2 > 0 together with the models for T3 and mature B cell subpopulations, we failed to get any results that fitted the experimental data. Even with a high value of KT, the models with T2 proliferation did not fit the data. Hence we may narrow down the possibilities by rejecting the model in which T2 cells [as defined by the scheme of Allman et al. (14)] proliferate.

In all the simulations, only 8 out of 630 possible models were found to obey our criteria and fit the experimental data. These models are represented in Fig. 4(C and D). The best fit was obtained in the model (Fig. 4D) in which the differentiation from T1 to mature B cells and the differentiation from T2 to mature B cells does not exist ({delta}1m, {delta}2m = 0). The least-squares fit values of the best fit for each model are presented in Table 5.


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Table 5. Models that fit the data, and the best least-squares fit value obtained for each model

 
The equations of the best fitting model are:




The ranges of parameter values of the best model of the four spleen subpopulations, which gives results within the experimental range for total cell numbers in each population as well as the best fit for fractions of labeled cells, are given in Table 4 for the T1 subset, in Table 6 for the T2 subset, in Table 7 for the T3 subset and in Table 8 for the mature B cell subset. Note that the ranges are given here for each parameter separately, hence not all values in the range given for one parameter necessarily give a good fit when used with all values in the ranges given for other parameters. The best simulation of the spleen population is represented in Fig. 2(B).

Calculation of subpopulation death rates from the best fit exit and differentiation rates
Once we found the differentiation rates for all B cell populations, we can estimate their death rates as the difference between exit from each population and entry into the next population, as follows:




The simulations were extended to up to 120 h after the start of labeling, to show the predicted course of labeling decay, which is represented in Fig. 3(B). In this period of time, mature B cells have negligible death rate. This supports a previous study claiming that the half-life of transitional B cells is 2–4 days, compared to 15–20 weeks for mature B cells (28), which recirculate in the blood and lymph system for several weeks.


    Conclusions
 Top
 Abstract
 Introduction
 Methods
 Results
 Conclusions
 References
 
Relationship of cycling peripheral pool to follicular differentiation from marrow precursors
We have employed mathematical modeling to evaluate alternative models for transitional B cell population dynamics. Our results predict that the development of mature follicular B cells from marrow-derived transitional B cells does not involve significant proliferation, suggesting that selection events within transitional B cell subsets are not accompanied by cell cycle entry. We have also used our mathematical model to test a model which does not have a T3 population, and in which T2 are cycling. We fitted this model to the Allman data (14), taking Allman's T2 and T3 as one cycling population; however, this version of the mathematical model did not fit the experimental data.

This result questions the nature of the cycling B cell subset described by Loder et al. (15,16). Since the vast majority of peripheral B cells are quiescent (14,3031), yet positive selection of developing B and T cells is apparently not associated with entry into the cell cycle [our conclusions (32)], we conclude that alternative interpretations of the data should be considered. In this regard, it is noteworthy that many of the putative late transitional B cells described by Loder et al. were defined by the cell surface phenotype CD21/35high CD23+. Since marginal zone B cells are often defined as CD21/35high CD23, we speculate that these cells may be more directly related to the maintenance and/or differentiation of the marginal zone B cell compartment. This interpretation is consistent with recent analyses of various mutant mouse strains characterized by normal numbers of follicular B cells without detectable CD21/35high B cells (33,34) (S. H. Smith and M. P. Cancro, unpublished data). It remains unclear as to why such cells would be cycling, but recent data indicate that maintenance of the marginal zone compartment does not require constant influx from marrow B-lineage progenitors (35), raising the possibility that CD21/35high CD23+ splenic B cells are enriched for proliferative marginal zone B cell precursors that are highly sensitive to homeostatic pressures designed to maintain sufficient numbers of marginal zone B cells. Such an interpretation is also consistent with recent data from Woodland and co-workers (36), who showed that resting splenic B cells are induced to proliferate when transferred into B cell deficient hosts. Alternatively, the conclusion that these cells are enriched for proliferating cells may be erroneous.

Asynchronous, alternative routes of follicular B cell differentiation
Our results further suggest that immature bone marrow B cells may proceed via one of several alternative routes to the mature follicular pool. For example, our results are compatible with the possibility that peripheral transitional B cells in the AA4.1-defined T2 population may derive from both splenic T1 cells and an unidentified immature B cell subset in the bone marrow. Moreover, additional parallel or branched pathways may also be employed, and it remains conceivable that an undetermined fraction of immature cells in the bone marrow proceed directly to the T2/T3 or mature follicular compartments. Our observation that the best fit is achieved by having some cells skip certain developmental stages and pass ‘directly’ into the later stages of follicular differentiation may best be interpreted as differing rates of transit to mature follicular through the T2/3 pools, rather than indicating discrete pathways per se. That is, once selected for recruitment into the follicular pool, some cells may pass quickly through these late intermediates, whereas other, yet unselected cells, may move more slowly.

Such a model predicts previously unappreciated heterogeneity in the marrow immature compartment. Indeed, recent data indicate that the marrow fraction E population is heterogeneous for surface expression of CD23, as well as binding capacity for BLyS (BAFF), a B lineage-specific anti-apoptotic cytokine associated with transitional B cell maturation. Since increasing surface expression of CD23 and the BLyS receptor BR3 are both associated with maturation of sIgM+ B cells (37,38), it is tempting to speculate that these cells may be more advanced B cells characterized by the capacity to transit directly from the marrow into late transitional or follicular pools and thus bypass the CD23 splenic T1 compartment. Moreover, given the relatively high death rates within the T1 population predicted by our modeling studies, we speculate that relatively few cells within the splenic T1 subset contribute to downstream B cell compartments. Further experimentation, perhaps coupled with more directed mathematical modeling, will be required to address this possibility.

Overall, our studies extend the notion that B lineage differentiation is an asynchronous, branched process, whereby developing cells may not only transit various progenitor compartments at different rates, but in which selective events are not necessarily restricted to a particular phenotype or anatomic compartment.


    Acknowledgements
 
The authors are grateful to Ms Hanna Edelman for help with manuscript preparation. The work was supported in part by Israel Science Foundation grant number 759/01-1, The Yigal Alon Fellowship, and a Bar-Ilan University internal grant (to R.M.).


    Abbreviations
 
BCR   B cell receptor
BrdU   bromodeoxyuridine

    Notes
 
Transmitting editor: I. Pecht

Received 27 June 2003, accepted 6 May 2004.


    References
 Top
 Abstract
 Introduction
 Methods
 Results
 Conclusions
 References
 

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