Mathematical Models of B Cell Subpopulations in the Spleen

We fit the BrdU data from control and B cell-depleted mice to our mathematical models of B cell development in the BM and maturation in the spleen. The model (Figure ​(Figure1A)1A) is fully described by Eqs 1–5 below and has been constructed as follows. First, for each population, we assigned a variable denoting the number of cells in each population. The current data did not include pro- and pre-B subpopulations; hence, we decided to include in the model a population that combined both subsets, B_oe, to represent the cells in the pro- and pre-B cell subsets that eventually differentiate to the immature subset (henceforth called pro-/pre-B cells). Transitional B cells are modeled here as one subset, ignoring their division into several maturation stages, because the focus of this study is on B cell development in the BM. The current data include the mature recirculating B cells subset in the BM, so we added this subset to our mathematical model (Figure ​(Figure1A;1A; Eq. 3). The numbers of cells in the immature, mature recirculating, transitional, and splenic mature subsets are represented by the variables B_i, B_Mrec, B_t, and B_Mspl, respectively (Figure ​(Figure2;2; Eqs 2–5).

Model of B cell population dynamics in the bone marrow and spleen. (A) The model of maturing B-cell population dynamics in the bone marrow and spleen is shown. All population processes – differentiation, proliferation, and death – are described by arrows. The rate of each process is given near the corresponding arrow. S is the source of B lineage precursors (cells/6 h), parameters δ denote differentiation rates, μ mortality rates, γ proliferation rates, and ϕ flow rates. (B) The model for labeling dynamics of developing B cells in the bone marrow assumes that cells in the unlabeled pro-/pre-B cell compartment move to the corresponding labeled compartment upon dividing. The model also assumes that each labeled cell remains labeled for the duration of the experiment.

Selective depletion of BM B cell subsets in an hCD20Tg mouse. (A) BM cells from hCD20Tg mouse were stained for B220, IgM, AA4.1, and hCD20. For analysis, viable lymphocyte were defined by forward and side light scatter, and gates were set to analyze B220⁺/AA4.1⁺ early developing B cells (pro-/pre-B and immature B cells) and to exclude circulated B220⁺/AA4.1⁻ and non-B lineage cells. Gated cells were analyzed for expression of IgM and hCD20. (B) hCD20Tg mice were injected with anti-hCD20 antibodies. Three days after injection, BM cells were stained for B220 and IgM and analyzed for the following B cell subsets: pre-/pro- (B220⁺/IgM⁻), immature (B220⁺/IgM⁺), and circulated mature (B220hi/IgM⁺) B cells, relative to control hCD20Tg mice that were not treated. The results shown are representative of three to four mice in each group.

Second, for each process in the model (that is, an arrow in Figure ​Figure1),1), a rate parameter was assigned; parameters were denoted by Greek letters to distinguish them from variables. For example, B_oe cells differentiate into the immature subset with a constant rate denoted by δ_oe (per 6 h, which is our simulation time unit). Third, the change in each population in each time unit (approximated by the derivative of the corresponding variable representing the number of the cells in this population) is given by a differential equation. The general form of each differential equation is of the form:

where each of the four right-hand terms is either a constant or a function of the variables and parameters. For example, the B_oe subpopulation is renewed from previous subsets at the constant rate S. This subpopulation proliferates after passing heavy or light chain rearrangements with a maximum rate γ (Figure ​(Figure1A;1A; Eq. 1). The proliferation term in Eq. 1 is the product of the number of cells that are able to proliferate (B_oe), the maximum proliferation rate (γ), and a so-called “logistic” limit on the proliferation, which is a function that decreases with the available space for this population and is constructed as follows. The mature recirculating subpopulation includes mature recirculating B cells that compete with pro-/pre-B cells for survival niches (38), as do plasma cells. Therefore, the carrying capacity parameter, K, that limits the proliferation rate of the pro-/pre-B cell subpopulation includes the total number of cells in both the pro-/pre-B and the mature recirculating compartments (Eqs 1, 6, and 11), and the logistic term is thus 1−(Boe+BMrec)K.

The mature recirculating B cell subpopulation in this model represents all the mature B cell subsets that re-enter into the BM from the periphery. Other feedback onto the pro-/pre-B cell division by any other cells, such as plasma cells, was not explicitly modeled, since those cells were not measured; however, their effect is implicitly modeled in part via the carrying capacity K, such that a decrease in K would be interpreted as competition by other cells for survival niches. Additionally, since the labeling data did not include pro- and pre-B cell subpopulations, a regulation of the source of pro-/pre-B cells by peripheral B cells or by other cells in response to the depletion was not explicitly examined (see Discussion).

Immature B cells either differentiate to BM mature cells at rate δ_{i_re}, or emigrate from the BM to the spleen and differentiate to transitional B cells at rate δ_{i_t} (Eqs 2–4). Transitional B cells differentiate to splenic mature B cells at rate δ_t (Eqs 4 and 5). After their maturation, splenic mature B cells can go back to the PB and then to the mature recirculating population in the BM. The flow of mature B cells from the spleen to the mature (recirculating) population in the BM is represented by the parameter ϕ_S. The flow in the opposite direction is represented by the parameter ϕ_BM (Eqs 3 and 5).

The death rates are denoted by μ_i, μ_t, and μ_rec for B_i, B_t, and B_Mrec, respectively. The exit from the splenic mature B cell, including death or transition to other organs, is denoted by ε_spl. Based on previous studies, it is assumed that the transitional subpopulation is not cycling (29, 39). We did not include the possibility that peripheral B cell division is also a source of the reconstitution, since a preliminary Ki67 staining showed no evidence for cell division among mature splenic B cells in either the depleted or control mice (data not shown). Thus, the differences in the BrdU labeling kinetics between control and depleted mice (Figure ​(Figure3B)3B) must be ascribed to differences in the entry-labeled cells from previous developmental stages.

Control and hCD20-depleted mice (34 days after depletion) were injected with BrdU and analyzed for BrdU labeling as detailed in Section “Materials and Methods.” (A) BM and spleen cells from the control and the depleted mice were stained for IgM, AA4.1 and BrdU labeling. For analysis, viable lymphocytes were defined by forward and side light scatter, and the relative BrdU labeling in BM immature (IgM⁺/AA4.1⁺) and mature-circulated (IgM⁺/AA4.1⁻), and in spleen transitional (IgM⁺/AA4.1⁺) and mature (IgM⁺/AA4.1⁻) B cells was determined using gates set as shown (representative example). (B) BrdU labeling kinetics in B cell-depleted (black, open circles for each single mouse and band for the mean value) and control (gray, filled triangles for each single mouse and band for the mean value) mice. For total cell numbers, see Figure ​Figure44.

Thus, the model is completely described by the following equations.