## Abstract

Our current view of T cell differentiation and population dynamics is assembled from pieces of data obtained from separate experimental systems and is thus patchy. We reassessed homeostasis and dynamics of T cells 1) by generating a mathematical model describing the spatiotemporal features of T cell differentiation, and 2) by fitting this model to experimental data generated by disturbing T cell differentiation through transient depletion of dividing T cells in mice. This specific depletion was obtained by administration of ganciclovir to mice expressing the conditional thymidine kinase suicide gene in T cells. With this experimental approach, we could derive quantitative parameters describing the cell fluxes, residence times, and rates of import, export, proliferation, and death across cell compartments for thymocytes and recent thymic emigrants (RTEs). Among other parameters, we show that 93% of thymocytes produced before single-positive stages are eliminated through the selection process. Then, a postselection peripheral expansion of naive T cells contributes three times more to naive T cell production than the thymus, with half of the naive T cells consisting of dividing RTEs. Altogether, this work provides a quantitative population dynamical framework of thymocyte development, RTEs, and naive T cells.

The high turnover, migration, and recirculation of lymphoid cell populations are key features of a resilient immune system regulated by homeostasis (1). The immune system efficacy is guaranteed both by the diverse repertoire of the naive cells (2) and by the rapid memory response of previously selected and expanded Ag-experienced T cells (3). Because immune responses are associated with major expansions of specific T cells followed by T cell death, homeostatic regulation is essential to maintain the equilibrium between cell production and cell death, and between naive and effector/memory T cells, as well as to preserve repertoire diversity.

Elements of the dynamics of peripheral T lymphocyte populations and their precursors in the thymus have been studied using various experimental approaches in rodents. These studies were based on cellular DNA labeling (4, 5, 6, 7, 8), targeted expression of MHC molecules (9, 10), or the tracking of migration of labeled recent thymic emigrants (RTE)^{6} (11, 12, 13). They also involved the artificial increase (14) or decrease (15) of thymic output, transfer of isolated cell populations (16, 17), or in vitro systems of T cell development (18). Collectively, these studies have provided estimates for proliferation rates, cell lifespans, daily export, or cell death and renewal rates in various cellular compartments. Different mechanisms for homeostasis were proposed (19, 20, 21). However, these data gathered with various experimental methods are difficult to combine into a single model accounting for the quantitative aspects of cell dynamics. In addition, the temporal parameters of the population dynamics of T cells in response to a short lymphopenia have only partially been established (22, 23, 24, 25, 26). This leaves us with a puzzling, complex, and incomplete view of thymocyte and peripheral T cell dynamics.

In the present study, we used a single experimental system of transient perturbation of T lymphocyte homeostasis, in combination with a mathematical model, to quantify the T cell differentiation dynamics in mice. We induced a temporary cell depletion of the immature TCR^{−}CD4^{−}CD8^{−} thymocytes through to the mature peripheral CD4 and CD8 T cells in young adult mice with a wild-type T cell repertoire. This was achieved using mice transgenic for the conditional *HSV1-TK* (*TK*) suicide gene (27, 28, 29). Expression of the *TK* gene allows eukaryotic cells to metabolize ganciclovir (GCV) into a triphosphate active form, which can be incorporated in the DNA of dividing cells, blocking DNA elongation and inducing cell death. Cell killing in our system is thus strictly conditioned by three simultaneous constraints: *TK* expression, GCV administration, and cell division. In fact, our experimental system corresponds to “soft” thymectomy (Tx), as this procedure temporarily blocks thymic output without the side effects associated with surgery.

We monitored cell numbers in the different T cell compartments, from CD4^{−}CD8^{−} double negative (DN) thymocytes to mature T splenocytes, at different time points during and after GCV treatment. Given the complexity of the cell dynamics in the various compartments, we developed a mathematical model based on a “conveyor-belt” type of differentiation (30). We fitted this model to our experimental results, allowing us to quantify the dynamics of thymocyte and peripheral T cell populations. Our system thus provides a more comprehensive view of T cell differentiation and population dynamics in mice at steady state, from DN thymocytes to naive T cells in the spleen. Moreover, our model improves our understanding of T cell homeostasis following transient lymphopenia.

## Materials and Methods

### HSV1-TK-transgenic mice

The previously described *EpCD4 HSV1-TK*-transgenic L40 mice (FVB background) (28, 31) were bred in our colonies (Nouvelle Animalerie Commune, Centre d’Exploration Fonctionnelle Pitié-Salpêtrière) under specific pathogen-free conditions. Littermates were used as TK^{−} mice. Mice were manipulated according to European Council Directive 86/609/EEC and with the approval of the ethic committee.

In contrast with the first *TK*^{+}-transgenic mice (23), here the *TK* transgene is under control of the EPCD4 regulatory sequences (32). TK mRNA expression was assessed by RT-PCR in highly FACS purified (>99% purity) thymic or peripheral cell populations. Total RNA was extracted using RNable (Eurobio) and cDNA was synthesized from 2 μg of each RNA sample using SuperScript II RNase H reverse transcriptase and oligo dT as primer (Invitrogen Life Technologies); 100 ng was used in each quantitative real-time PCR. The same amount of RNA and cDNA was used for each sample. Primer sequences for TK were 5′-CGAGCCGATGACTTACTGGC (forward) and 5′-CCCCGGCCGATATCTCAC (reverse) and the probe sequence was 5′-FAM-TACACCACACAACACCGCCTCGACC-TAMRA-3′ (Applied Biosystems). Primers and probe for 18S were purchased as reagent kits from Applied Biosystems. The real-time PCR was performed on an ABI Prism 7700 using TaqMan Universal PCR master mix (Applied Biosystems) in triplicates. The average threshold cycles (Ct) of the triplicates were used to calculate the fold change between lymph nodes (LN) TK^{+}CD4^{+} and others samples. Ct for 18S was used to normalize the samples. Relative quantification was calculated using the comparative Ct method (Applied Biosystems).

The relative mRNA TK expression is of 1.43 in immature CD3^{−}DN thymocytes, 6.31 in CD3^{−} CD4^{+}CD8^{+} double-positive (DP) thymocytes and, respectively, 1 and 0.67 in CD4 and CD8 LN peripheral T cells from TK^{+} mice; the background value in TK^{−} CD4 and CD8 LN T cells from TK^{−} was 0. This HSV1-TK mRNA expression from immature thymocyte stages to mature peripheral T cells is in accordance with the observed cell depletion due to the TK enzymatic activity upon GCV treatment. This contrasts with the expression of the human CD4 reporter gene under the control of EPCD4 regulatory sequences, detected only in mature CD4 and CD8 T cells, but not immature DN or DP (32).

### Thymectomy

Mice were thymectomized at 6 wk of age under ketamine (150 mg/kg)/xylazine (10 mg/kg) anesthesia after a preanesthesia with robinul (1.5 μg s.c.) and dopram (Vétoquinol Lure France) (0.5 mg i.p.). Under a stereomicroscope, the upper part of the thorax was opened, and a small opening made in the pleura. The two thymic lobes were carefully removed with forceps, and the skin was closed with a silk suture. A 4- to 5-wk recovery period was observed before GCV treatment.

### GCV treatment and cell preparation

To deliver GCV continuously and avoid chronic stress by repeated mouse manipulation—which can induce thymic atrophy and severely affect T cell homeostasis (33), an osmotic pump was implanted in *TK* mice on day 0, by s.c. incision under ketamine/xylazine anesthesia. GCV (Roche) was administered at a dose of 50 mg/kg/day, for 7 or 14 days by Alzet 2001 and 2002 osmotic pumps, respectively (Charles River Laboratories). Treatment was initiated between 11 and 20 wk of age; the mice were then sacrificed at various times and spleen and thymus recovered. Cell suspensions were made by mechanical dissociation, washed in PBS 3% calf serum, and living mononucleated cells were counted by trypan blue exclusion.

### Immunostaining and flow cytometry

Cell surface staining was performed as previously described (28) by incubation of cell suspensions with directly labeled mAbs after blocking of FcR. Quadruple staining was done with specific anti-CD4, CD8, CD45RB, CD44, or TCRαβ, purchased from BD Pharmingen (BD Biosciences). Labeled cells were analyzed on a FACSCalibur (BD Biosciences). Lymphocyte cell counts were calculated as the proportion of lymphocytes, as determined by forward scatter/side scatter among the numbers of mononucleated cells per organ.

### Cell cycle analysis

Quantification of cell DNA content was done by flow cytometry. Thymic or splenic cell suspensions were stained with fluorescent-labeled Abs. Cells populations were sorted on ARIA (BD Biosciences), fixed in 70% ethanol for 2 h, incubated with 7-aminoactinomycin D (7AAD; 15 μg/ml; Sigma-Aldrich) in presence of RNase (100 μg/ml) for 1 h at 37°C. Cell cycle analysis was performed on LSR2 (BD Biosciences) with 488 and 633 laser wave excitation. Dead cells and doublets were gated out and percentages of cells in S plus G_{2}/M phases were estimated on the basis of 7AAD expression against forward scatter.

### Depletion rates under GCV therapy

The exponential depletion rate *d* for each cell population was obtained by fitting an exponential decay curve (*N*(*t*) = *N*_{0}*e*^{−dt}, where *d* corresponds to the probability that a cell dies under GCV), to the data obtained under 7-day GCV treatment. To compare the depletion rates between two cell populations, we computed within each mouse the difference between cell numbers of the two populations. We then tested whether the depletion rates were different, by drawing a regression line through this difference. If the slope of the regression was significantly different from zero (*t* test on slope of the regression), the depletion rates were considered significantly different. We applied a Bonferroni correction for multiple comparisons.

### Mathematical model of T cell differentiation, from early thymocytes to naive splenocytes (see supplemental material: mathematical model and formulas, section 1 and 2)^{7}

We formulated a mathematical model having maximally 18 parameters to describe the differentiation from DN thymocytes to CD44^{low}CD45RB^{high} naive cells as a linear DN→DP→SP→RTE→naive T splenocyte conveyor belt lacking density dependence. The dynamics were described with differential equations for the number of cells at each division stage within each T cell compartment. Fig. 1⇓ illustrates the generic form of the model equations. For full mathematical details on the model, see the supplemental material, section 2.

The model can verbally be described as follows. Thymic precursors enter the DN cell compartment, and complete a number of divisions determined by an exit function that increases with the number of divisions. During the last division, DN cells acquire the DP phenotype. We modeled the expansion of early DP cells as a series of divisions—again dictated by an exit function—during which cells have an increasing rate of differentiation to the last stage DP. Thus, as DP cells divide, they have an increasing probability of maturing to the last stage DP where they undergo negative or positive selection before becoming single positive (SP). In the last stage, DP do not divide, and are subject to selection. There is a dramatic loss of cells between the last stage DP and early SP stage (70% of thymocytes are last stage DP cells, while 12% of thymocytes are SP cells (30)), Thus, only a fraction of last stage DP cells differentiate into SP due to the selection process. Then, SP cells complete a certain number of divisions, with late SP similarly having a greater probability than early SP of exiting the thymus. Finally, we assumed that RTE complete a number of divisions, during which they have an increasing probability of exiting the compartment: they may then settle in the long-lived naive cell pool, die, or possibly migrate to other lymphoid organs. To model the naive cell dynamics, we imposed a condition on the parameters, namely that 42% of cells exported from the thymus go to the spleen (11) (see supplemental material, section 3). This value was incorporated in a mathematical formula yielding the parameter *f*_{8}. We also assumed that initially, each cell population has a constant size, i.e., is at steady state.

The model was fitted to our in vivo experimental data, testing five different scenarios, each making different assumptions on the proliferation and death rates (see supplemental material, section 7). The best fit was selected according to the lowest mean squares criterion.

### Mathematical simulation of Tx

Starting values for our Tx simulations were the number of RTE in each division stage estimated with our model (scenario 5) plus the number of long-lived naive cells. The decay process is described with the differential equations for *R4*_{i} and *R8*_{i}, with division cycle *i* ranging from 0 to 2, and the input from the thymus being set to 0 at the moment of Tx.

## Results

### Kinetics of thymocytes and mature peripheral T cells, during and after 7-day depletion of dividing T cells

In our *TK*^{+}-transgenic mice, *TK* mRNA is expressed in thymocytes and mature peripheral T cells. In accordance with this *TK* expression pattern, GCV treatment resulted in transient depletion of thymocytes and T splenocytes in *TK*^{+}, but not in *TK*^{−} mice (Fig. 2⇓*A*). In the thymus of *TK*^{+} mice, the percentage of CD4^{+}CD8^{+} (DP) cells decreased from 75 to 55%, and the nadir—the time point when a given cell population reaches its lowest size—was reached on day 7 of GCV treatment. Prolongation of treatment to 14 days did not result in further depletion but prolonged it. Whatever the treatment length, the recovery of DP cells started immediately after stopping GCV treatment. This indicates that GCV, with an intracellular half-life of 20 h (34), had no long-lasting effect on thymocytes and T cells.

In the spleen of TK^{+} mice, the percentage of CD4^{+} T cells dropped from 27 to 12% after a 7-day GCV treatment, with a nadir on day 14 (Fig. 2⇑*A*). The continued decrease of T cell numbers between days 7 and 14, in the absence of GCV, likely reflects the time required for thymocyte differentiation and migration to the spleen. The nadir after a 14-day GCV treatment is less distinct, suggesting the involvement of more complex processes such as cell tissue redistribution or lymphopenia-induced proliferation. Whatever the treatment length, T cell percentages were back to almost normal levels by day 70 in all organs.

To avoid additional perturbations induced by prolonged T cell depletion, we focused our analyses on TK^{+} mice treated with GCV for 7 days. We determined the absolute cell numbers in the main thymocyte subpopulations and in mature T splenocytes at various time points during and after GCV treatment (Fig. 2⇑*B*). In the thymus, the depletion of dividing cells induced a rapid shrinkage of the DN, DP, SP CD4^{+}CD8^{−} (SP4) and CD4^{−}CD8^{+} (SP8) populations, the total thymocyte count falling by ∼88% in 7 days (from 56 to 6.5 million).

In the spleen, we analyzed separately the phenotypically defined CD45RB^{high}CD44^{low}“naive” and CD44^{high}“Ag-experienced” T cells (Fig. 2⇑*C*). We observed a strong depletion of both CD4^{+} and CD8^{+} naive T cells, which at steady state (“day 0” in Fig. 2⇑*C*) represent the major splenic T cell population (around 70%) in young adult mice. In contrast, there was only a modest depletion of Ag-experienced T cells during GCV treatment (Fig. 2⇑*B*), and an increase in their percentage as the naive cell pool shrinks (Fig. 2⇑*C*).

### Estimation of depletion rates

A simple approach to analyze T cell dynamics during the depletion phase is to estimate the cell depletion rate during GCV treatment in the various thymocyte and splenocyte subpopulations using a model of exponential decay (Table I⇓). The rate of depletion under GCV reflects 1) death of dividing cells resulting from GCV treatment, 2) “natural” cell death, and 3) decreased import from the previous cell compartment. Because the depletion rate was estimated over a short period of time (7 days), we assumed that it reflects the physiological behavior of T cells and their precursors before compensatory mechanisms may have started to counter cell loss. In the thymus, the DP cells have the highest turnover (a depletion rate of 37% per day, Table I⇓), followed by the SP4 thymocytes (24%), and DN cells (21%). With a depletion rate of 14% per day, SP8 thymocytes have a significantly lower turnover than SP4 thymocytes (*t* test on slope of regression on difference between SP4 and SP8 numbers, *p* < 0.05).

In the spleen, the depletion rates are lower: CD4^{+} and CD8^{+} naive T cells have a daily depletion rate of 9 and 7%, respectively (NS: *t* test, *p* > 0.05). The SP4 thymocyte depletion rate differs significantly from the naive CD4^{+} depletion rate (*t* test, *p* < 0.05), which was not the case for SP8 and naive CD8^{+} splenocytes. Of note, GCV treatment results in a significant depletion of the CD8^{+}CD44^{high} cells in the spleen (5% per day, *p* < 0.05, Table I⇑), but not of CD4^{+}CD44^{high} cells (2% per day, *p* > 0.05, Table I⇑). These results suggest that the depletions observed in the naive and Ag-experienced T cell populations are unrelated.

### Nadir and recovery

For the most immature DN and DP thymocytes, the nadir occurs at days 7 and 8, which corresponds to the end of the GCV treatment, and occurs later for mature T cells in the thymus and for naive splenocytes (14 days or later, Table I⇑ and Fig. 2⇑*B*). Actually, the delay in cell recovery after GCV treatment becomes longer as the cell populations are more differentiated, in accordance with a DN→DP→SP→naive T splenocytes conveyor-belt type of thymocyte differentiation (30).

In contrast, for Ag-experienced T splenocytes the nadir was observed at day 7 (Table I⇑): the compartment starts to refill just after the end of GCV treatment, confirming the difference between the dynamics of Ag-experienced and naive T cells.

After cessation of the 7-day GCV treatment, most thymocyte populations returned to levels similar to pretreatment values in ∼4 wk (Fig. 2⇑*B*, Mann-Whitney *U* test, *p* > 0.05 when comparing DN and SP cell numbers at days 0 and 28); only DP numbers were still slightly lower at day 28 than at day 0 (*p* < 0.05). In the spleen, CD4^{+} and CD8^{+} naive cells were still significantly depleted on day 28 (*p* < 0.05), but on day 70 their numbers had approached pretreatment values (*p* > 0.05). Thus, in young euthymic mice, thymocytes and spleen T lymphocytes can return to values close to steady state levels within 2 mo after short-term lymphopenia.

### Estimation of the size of the long-lived naive cell population

Several observations led us to consider the CD44^{low}CD45RB^{high} splenic T cells as a heterogeneous population, composed of a first subpopulation of rapidly proliferating RTE, differentiating into a second subpopulation of “long-lived”, non- or slowly dividing naive cells, “resistant” to GCV and persisting after Tx (22).

The first observation is the match (Mann-Whitney *U* test, *p* < 0.05) between the sizes of the naive T cell populations in the spleen in euthymic mice after GCV therapy, and in Tx mice 4–5 wk after Tx (Fig. 2⇑*B*, at 28 days after GCV, CD4^{+}CD44^{low}: 14.8 ± 4.6 million cells and CD8^{+}CD44^{low}: 4.6 ± 1.3 million cells; compare with Fig. 3⇓*A*, Tx mice, day 0, 14.2 ± 1.7 million CD4^{+}CD44^{low} cells and 4.8 ± 0.7 million CD8^{+}CD44^{low} cells). These numbers represented ∼50% of those observed in euthymic mice (Fig. 2⇑*B*: day 0, 29 ± 3.1 millions CD4^{+}CD44^{low} and 8.9 ± 0.6 millions CD8^{+}CD44^{low}). The second observation is that the number of naive CD44^{low} T cells in Tx mice remained stable over the course of 14-day GCV treatment (ANOVA least significant difference method Fisher, *p* > 0.05 comparing day 0, 14, or 70 cell numbers; Fig. 3⇓*A*), in contrast with euthymic mice (Mann-Whitney *U* test *p* < 0.01 comparing days 14 and 70; Fig. 3⇓*A*). This suggests the existence of a population of resting long-lived naive cells, persisting after Tx and “resistant” to GCV.

The third observation is that the mice have approximately three and eight times more naive CD4 and CD8 T cells, respectively, in the spleen, than SP4 and SP8 cells in the thymus (Table I⇑), and that half of them are strongly affected by GCV treatment, which is not compatible with the idea that nondividing RTE slowly accumulate from a small thymic source.

Altogether, our results indicate that RTE, which are depleted following Tx as well as GCV treatment, form a population of cells with a high turnover representing roughly 50% of the naive T cell population in young mice.

### Best fit model and turnover parameters

Using the mathematical model described in *Materials and Methods*, we fitted five different scenarios to the data (comparative results given in supplemental material, section 7). We selected the scenario that gave the best fit (Fig. 4⇓; scenario 5), that is 1) all cells—except last-stage DP cells—have negligible “natural” death rates on the time scale of the experiment; 2) RTE and thymocytes, except early DP cells, have similar proliferation rates.

We estimated residence and division times (Table II⇓), cell numbers at each division stage, and differentiation rates in each population (Table III⇓), cell turnover, production, and death rates (Table IV⇓). The model and its main results are represented in Fig. 6⇓. The best-fit model indicates that precursors enter the differentiation process as DN cells at a rate of 2 × 10^{4}/day (σ_{N}), i.e., 0.04% of all thymocytes (Table II⇓). All DN divide four times (Fig. 5⇓ and Table III⇓) during a period of ∼18 days, with an average time between divisions (1/*p*_{N}, with proliferation rate *p*_{N} = 0.23/day) of ∼4 days (Table II⇓), before becoming DP. Early DP then actively proliferate during ∼1 day; they divide on average every 5–6 h (proliferation rate *p*_{p} = 4.5/day, Table II⇓), 59% of early DP completing five divisions and the rest six divisions (Fig. 5⇓ and Table III⇓), the maximum number of divisions we allowed in the model. After this period of intense proliferation, DP cells mature into nondividing DP cells (last-stage DP), a stage lasting on average 3 days (Table II⇓).

Of the last-stage DP cells, 0.9 × 10^{6} or 2.3% (μ_{LP}α_{4}) and 0.2 × 10^{6} or 0.4% (μ_{LP}α_{8}) become SP4 and SP8 each day, respectively. This means that 35% (μ_{LP}(1-α_{4}-α_{8})) of last-stage DP cells die per day during selection (i.e., 14 × 10^{6} DP cells, Fig. 6⇓ and Table IV⇑). Altogether, 93% of all cells produced before the SP stage die per day at the last DP stage, corresponding to 83% of all cells produced in the thymus (Table IV⇑). Most SP cells then divide once (69%, Fig. 5⇑) after on average 4 days (Table II⇑).

According to the model, 5% of thymocytes are exported to the periphery per day, which corresponds to a thymic output of 2.9 × 10^{6} cells/day (roughly 145 times the input of 2 × 10^{4}/day; Fig. 6⇑ and Table IV⇑). We hypothesized that 42% of these RTE reach the spleen (11), where most undergo two divisions (96%, Fig. 5⇑) over a period averaging 9 days, while keeping the naive CD44^{low}CD45RB^{high} phenotype (Table II⇑ and III⇑). Then, they settle in the pool of long-lived naive cells, die, or migrate to another lymphoid compartment.

We estimated that 16.8 × 10^{6} thymocytes are produced daily by intrathymic proliferation (Table IV⇑). Besides, we estimated that 3.5 × 10^{6} CD44^{low}CD45RB^{high} T cells per day are produced by proliferation in the spleen, which is approximately three times more than the 1.2 × 10^{6} cells/day contributed by the thymus to the spleen (42% of thymic export; Table IV⇑).

SP and DP cells have the highest turnover with, respectively, 37 and 35% of cells being replaced per day, while DN cells have the lowest turnover (24% per day) (Table IV⇑). Consequently, the time necessary to replace 50% of the population is similar between DP, SP, and RTE (around 1.5 days), and is slightly higher for DN (2.1 days).

### Model validation: analysis of the percentages of cells in division

We determined the percentage of cells in division in the thymic and splenic subpopulations at steady state, by quantifying ex vivo the cell DNA contents. This is highest in DP (around 6% of all DP are in S plus G_{2}/M) and lowest (≤1%) in SP4 and CD44^{low} splenic T cells (Fig. 7⇓, *A* and *B*).

To see whether our model estimates would match these results, we computed the percentage of cells entering division per day for each subpopulation, based on our fitting results (see formulas in supplemental material, section 5). We estimated that 34% of DP cells, 23% of DN, SP, and RTE cells enter division daily (Fig. 7⇑*C*). If 43 and 49% of naive CD4^{+} and CD8^{+} T cells, respectively, consist of RTE (Table III⇑), 9.9% (0.43 × 23%) of CD4^{+}CD44^{low} cells and 11.3% (0.49 × 23%) of CD8^{+}CD44^{low} cells would enter division daily (Fig. 7⇑*C*). Thus, similar patterns were obtained with both methods.

### Model validation: simulated Tx

To validate our model results, we assessed whether they would reproduce the effect of Tx in mice (Fig. 3⇑*B*): we simulated the naive T cell loss observed in the first 6 wk after Tx. The simulation predicts that the numbers of naive CD4^{+} and CD8^{+} T cells decrease in the spleen to, respectively, reach ∼14.5 and 4.5 million 4–5 wk after Tx (Fig. 3⇑*B*). These estimates are in accordance with the numbers of long-lived naive CD4^{+} and CD8^{+} T cells found in the spleen of mice 4–5 wk after Tx (14.2 and 4.8 million cells, respectively).

## Discussion

Our experimental results demonstrate that the transient depletion of dividing thymocytes and T cells induces a fully reversible lymphopenia in young euthymic mice. Compared with other methods of T cell depletion, 1) the specific *TK* expression in the T cell compartment eliminates the potential side effects of chemo- or radioablation that affect other cell types; 2) the depletion is transient, controlled by the duration of GCV administration, in contrast with Tx; and 3) the depletion affects only dividing cells within the cell populations expressing *TK*. The animals regained almost normal numbers of thymocytes and mature T cells within ∼1 and 2 mo after depletion, respectively, without alteration of the T cell repertoire (data not shown). This shows that the thymus of a young adult mouse, even after major depletion, remains fully functional and contributes to the production of RTE, as recently underlined in humans (35) and mice (26). The progressive delay in nadir of depletion observed in the thymocyte populations as they become more differentiated is consistent with a conveyor-belt model of division and differentiation of thymocytes: a cell enters the next, more mature, compartment after a time lag of differentiation and maturation that involves one or more division cycles. These observations are at the basis of our mathematical model. The present study reveals that traditional analysis of experimental results concerning the dynamics of complex lymphoid cell population can be improved markedly by mathematical modeling.

### Considerations on the mathematical model

We formulated an 18-parameter mathematical model describing the dynamics of thymocytes and naive splenic T cells. Based on this model, we investigated the fit to our in vivo experimental data, of a number of scenarios with different sets of free parameters. We then selected the scenario which gave the best fit, using the lowest mean squares of residuals as a criterion. Although we could select a best-fit scenario, the fits were very similar in quality. Some scenarios gave quite divergent estimates for proliferation rates and residence times, especially for DP (supplemental material, section 8): residence times vary from 4 to 16 days, and proliferation rates from 0.3/day to 4.5/day. We therefore provide some additional support for the best-fit scenario. When *p*_{P} = 4.5/day, ∼34% of DP cells enter division daily, and roughly 92% of DP cells are in the last, nondividing stage (last table supplementary material, scenarios 1, 2, and 5). When *p*_{P} = 0.3/day, the proportion DP entering division is halved (17%), while the percentage of cells in last stage DP is reduced to 40% (supplemental material, section 8, scenarios 3 and 4). Under GCV treatment, the observed daily rate of cell loss—which reflects the extent of division—is highest in the DP population (37% per day, Table I⇑). Also, the majority (∼85%) of DP are mature, nondividing cells (30). These two independent observations support the notion that early DP proliferate faster than other cell types, as assumed in scenarios 1, 2, and 5. Among these three scenarios yielding very similar parameter estimates, we took the best-fit scenario (scenario 5) to estimate the population dynamic parameters.

Earlier mathematical models describing thymocyte dynamics assumed that thymocytes proliferated in a density-dependent manner (36). It has since been shown that homeostatic regulation in the thymus only occurs when DN precursors are below 5% of their original counts (15). Similar observations were made in peripheral organs, where lymphopenia-induced proliferation only occurs when depletion exceeds 90% (26). The T cells in our *TK*-transgenic mice were not depleted to such an extent, so we did not allow for density-dependent homeostasis in our model.

### Estimation of division rates and lifespans of thymic cell populations

The frequency of dividing cells estimated by DNA content and the model-based estimates of daily percentage of cells entering division show the same pattern (Fig. 7⇑): DP have highest division rates, DN and SP have intermediate division rates, and naive cells in the spleen have the lowest.

Previously estimated times required for DN to differentiate into DP range from 5 to 6 days (37) to 15–16 days (17, 38). Our fitting procedure yielded an average residence time in the DN compartment of 18 days, with cells undergoing on average four divisions during that period. Previous reports showed that the total residence time in the DP compartment is 3 days (5), 3–4 days (30), or 3.5 days (6), and that cycling DP differentiate into SP in a minimum of 2 days (39). We found a residence time of 4 days for DP cells. We could split this period into two. According to our best-fit scenario, early DP undergo five to six divisions during their first day, in agreement with a previous report showing that all cycling DP disappeared 1 day after demecolcine treatment (37). In their second 3-day period, DP do not divide and are subject to selection. This would correspond to the period where cells have lower motility and prolonged contact with the thymic stroma, while positively-selected cells rapidly migrate from the cortex to the medulla (40).

Previous data showed a residence time of 6–7 days for the SP in the medulla (9). Our model indicates that positively selected SP cells undergo a postselection expansion (41, 42) during ∼6 days before reaching the periphery, completing one to two divisions (whether TCR driven or not (8)), with an interdivision time averaging 4 days. Thus, an average of 27 days is required in these 3-mo-old adult mice to obtain a fully mature SP cell from a hematopoietic precursor cell.

### Selection in the thymus and export rate to periphery

In our model, “natural” cell death in the thymus is only significant in the last, nondividing DP stage, likely reflecting the combined effect of death by neglect and negative selection. We estimated that 35% of the last stage nondividing DP die per day. To test the robustness of this finding, we looked at the scenario in which death may occur at every stage of thymocyte development and the proliferation rates differ across all cell populations (scenario 1, supplemental material, section 7). We still found that 97.5% of cells dying in the thymus do so at the last DP stage (the rest, 2.5%, dying at the SP stage; results not shown). Thus, our results are consistent with the notion that thymocyte selection occurs mainly at the DP stage (43, 44).

Given the diversity of experimental approaches and means of quantifying thymic selection, these estimates from the literature are difficult to interpret and compare with our findings. Positive selection per se would concern 5% (45) to 20% of DP cells (10, 46), while estimates of subsequent negative selection are quite variable (10, 45, 47, 48). Our estimation that 83% of all thymocytes produced daily, or 93% of those produced before the SP stage (Table IV⇑), die in the thymus, is in accordance with previous results (11) and the estimation that 90% of “developing thymocytes” (if taken as the daily production of thymocytes) die by neglect and 5% by negative selection (10). Our results reveal that after the combined process of positive and negative selection, only 2.5% of the last stage DP cells daily reach the early SP stage.

Our fitting procedure suggests that in 3-mo-old mice, ∼3 million SP cells are exported from the thymus per day, i.e., 5.4% of the thymocytes. This is a rather high percentage compared with the previous estimates of 1% in 4.5-wk-old mice and 0.3% in 3.5-mo-old mice (11). In absolute cell numbers, previous estimates are also quite variable, ranging from 1 to 2–3 million per day (11, 12, 14) or even 5 million per day (22). These differences could stem from differences in age and strain of the mice used; also, lower estimates of cell export in the literature could be associated with stress induced by some experimental procedures.

Recently, it was shown that T cell differentiation continues postthymically with a preferential CD8 expansion (also observed in Fig. 7⇑) and a lowered Bcl-2 expression in CD4, contributing to the decrease of the CD4:CD8 ratio at the periphery (13). We therefore investigated models of thymocyte and naive T cell dynamics, in which either the number of divisions, or the proliferation rates, differed between CD4 and CD8 cells in thymus and RTE in the spleen (data not shown). In the latter case, we imposed equal selection fractions (α_{4} = α_{8}) and equal export fractions (*f*_{4} = *f*_{8}). We still found the best fit for our original model, where the CD4:CD8 ratios in the thymus and spleen are set, respectively, by the difference in thymic selection coefficient (α_{4} and α_{8}), and in fraction exported CD4 and CD8 thymocytes migrating to the spleen (*f*_{4} and *f*_{8}). This means that five times more CD4 than CD8 thymocytes are generated from DP cells (α_{4}/α_{8} = 5). This result is in accordance with earlier work (49), and with the recent hypothesis that CD4 or CD8 lineage commitment is determined by competition for MHC during epithelial cell interaction and cell dissociation rate, which is lower for CD8 than CD4 thymic T cells (50). Our data also suggest that twice as many CD8 as CD4 cells are exported to the spleen (*f*_{8}/*f*_{4} = 2).

### Splenic T cell turnover

Several lines of evidence suggested that the naive splenic T cell population in fact consists of dividing RTE and long-lived quiescent or slowly-dividing naive cells. Assuming that the naive T cell pool in Tx mice, or the cells remaining after GCV treatment, only consist of long-lived naive cells, we estimated that among naive T cells in the spleen ∼43% of CD4 (11 million CD4 RTE) and 49% of CD8 (4 million CD8 RTE) are actively dividing RTE (Table III⇑). Among all CD4 splenocytes, 31% are RTE and 41% long-lived naive cells, in accordance with percentages found in 3-mo-old mice (13). Our results suggest that splenic RTE divide twice, with an average interdivision time of 4 days, before settling in the long-lived naive cell pool, dying or migrating to other lymphoid organs. Our model results are compatible with the analysis of T cell depletion after hydroxyurea treatment, where Rocha et al. (22) estimated that 50% of peripheral T cells have a short survival time in vivo (48–72 h) and have recently divided.

Expansion of RTE and SP cells does not increase the repertoire diversity of naive T cells, as it is a postselection event; its main role could be to ensure a minimum size of the naive cell clone, for each cell having successfully passed the selection procedures, to guarantee Ag encounter and stimulation of an appropriate immune response.

We found an overall daily production of naive T cells of at least 6.4 × 10^{6} cells/day (2.9 × 10^{6} cells exported by the thymus and 3.5 × 10^{6} cells produced by division of RTE in the spleen). This is a minimum estimate because the production of naive cells by division in other tissues or organs was not estimated. Our model results are compatible with a 25% contribution by the thymus to naive T cell production in the spleen.

The fast recovery of “Ag-experienced” splenic T cells just after cessation of GCV treatment, contrasting with the delayed recovery of upstream cell compartments, suggests their rapid homeostatic expansion and/or conversion from resting naive T cells in response to lymphopenia (51, 52), and possible recirculation between lymphoid organs and nonlymphoid tissues.

### A systems biology approach to T cell differentiation and dynamics

Our work shows that understanding of a complex biological process, such as lymphocyte differentiation, can be fundamentally improved by mathematical modeling. Our model generates a comprehensive view of the T lymphocyte population quantifying the processes taking place during T cell differentiation. For example, it suggests that in young adult mice, 83% of thymocytes produced daily die at the transition from the DP to SP stage, making this the prime site of thymic selection. It also highlights the previously unrecognized major contribution of RTE expansion to the establishment of the peripheral naive T cell pool. We believe that our model, together with other systems biology approaches (50), can serve as a basis to further study T cell population dynamics, not only in physiological, but also in pathological settings.

## Acknowledgments

We thank C. Durieu for technical help, M. C. Burland for quantitative RT-PCR, and B. Gouritin for cell sorting; personnel from the Nouvelle Animalerie Commune, Centre d’Exploration Fonctionnelle Pitié-Salpêtrière for breeding mice; R. Geskus for valuable statistical advice; and our colleagues for discussions.

## Disclosures

The authors have no financial conflict of interest.

## Footnotes

The costs of publication of this article were defrayed in part by the payment of page charges. This article must therefore be hereby marked

*advertisement*in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.↵1 This work was supported by grants from Agence Nationale de la Recherche sur le SIDA, Centre National de la Recherche Scientifique, Paris 6 University, Action Thématique Concertée Vieillissement Institut National de la Santé et de la Recherche Médicale, Netherlands Organisation for Scientific Research, and CompuVac.

↵2 Address correspondence and reprint requests to Dr. Véronique Thomas-Vaslin, Unité Mixte de Recherche 7087, Université Pierre et Marie Curie, Centre National de la Recherche Scientifique, CERVI Groupe Hospitalo-Universitaire Pitié-Salpêtrière, 83 Boulevard de l’Hôpital 75013 Paris, France. E-mail address: thomas{at}chups.jussieu.fr

↵3 V.T.-V. and H.K.A. contributed equally.

↵4 Current address: Centre for Infectious Disease Epidemiology, National Institute for Public Health and the Environment, Antonie van Leeuwenhoeklaan 9, 3721 MA Bilthoven, The Netherlands.

↵5 R.J.d.B. and D.K. contributed equally.

↵6 Abbreviations used in this paper: RTE, recent thymic emigrant; GCV, ganciclovir; DN, CD4

^{−}CD8^{−}double negative; Ct, cycle threshold; LN, lymph node; DP, CD4^{+}CD8^{+}double positive; SP, single positive; TK, HSV-1 thymidine kinase; Tx, thymectomy; 7AAD, 7-aminoactinomycin D.↵7 The online version of this article contains supplemental material.

- Received July 24, 2007.
- Accepted December 4, 2007.

- Copyright © 2008 by The American Association of Immunologists