## Abstract

Recognition of peptide-MHC by the TCR induces T lymphocytes to undergo cell division. Although recognition of foreign peptide induces a program of cellular division and differentiation by responding T cells, stimulation by self-peptide MHC complexes in lymphopenic conditions induces a slower burst of divisions that may or may not be accompanied by effector differentiation. Although both responses are triggered by signals from the TCR, it is not known whether they represent distinct programs of cell cycle control. In this study, we use a mathematical modeling approach to analyze the proliferative response of TCR transgenic F5 T cells to lymphopenia. We tested two fundamentally different models of cell division: one in which T cells are triggered into an “autopilot” deterministic burst of divisions, a model successfully used elsewhere to describe T cell responses to cognate Ag, and a second contrasting model in which cells undergo independent single stochastic divisions. Whereas the autopilot model provided a very poor description of the F5 T cell responses to lymphopenia, the model of single stochastic divisions fitted the experimental data remarkably closely. Furthermore, this model proved robust because specific predictions of cellular behavior made by this model concerning the onset, rate, and nature of division were successfully validated experimentally. Our results suggest cell division induced by lymphopenia involves a process of single stochastic divisions, which is best suited to a homeostatic rather than differentiation role.

The peripheral T cell compartment is maintained in a remarkably stable state of homeostasis despite its heterogenous composition and the constant influx of cells into the different pools (1). This stability is achieved by close regulation of cell survival and proliferation of the constituent subsets. For naive CD4 and CD8 T cells, these homeostatic survival and proliferative mechanisms are regulated by cytokines and TCR signals. In contrast, memory cell homeostasis appears to depend only on cytokines including IL-7 and IL-15. During Ag responses, specific T cells are released from normal homeostatic control and proliferation and survival is governed by the presence of specific Ag. Upon resolution of a response and clearance of Ag, specific T cells return to normal homeostatic control.

Recognition of self-peptide MHC complexes by naive T cells delivers signals that are important for T cell survival (2, 3, 4, 5, 6, 7, 8), and in conditions of lymphopenia, these same signals can induce T cells to undergo so-called homeostatic or lymphopenia-induced proliferation (LIP)^{3} (4, 9, 10, 11). Although induction of proliferation in lymphopenia is critically dependent on TCR signals, and therefore could rely on similar proliferative mechanisms that govern responses to Ag, the precise relation between these two responses remains unclear. Studies using different CD8^{+} TCR transgenic models have revealed a complex picture in which a broad range of proliferative responses to lymphopenia can be observed. Some TCR transgenic T cells proliferate slowly (12) or not at all (6), whereas others undergo extensive cell division accompanied by development of effector function (13, 14). LIP may also have a strong stochastic (random) component because even within an apparently homogeneous monoclonal population, some cells divide extensively, whereas in the same environment and time frame others will not divide at all. In the instances where lymphopenia induces the most rapid division, the similarity with true Ag-driven proliferation is quite considerable both in terms of dependency on CD4 T cell help (15) and at a genetic level where they share similar profiles of gene expression (16). However, in other regards, such as the lower rate of division and its dependence on IL-7 (17), lymphopenia-induced proliferation is distinct from that induced by Ag.

Despite the extensive molecular and cellular characterization, it remains unclear whether proliferation induced by lymphopenia follows a pattern similar to that induced by Ag, differing from it only quantitatively, or whether induction and/or maintenance of cell division in these two types of proliferation are regulated in fundamentally distinct ways. To gain a better understanding of how LIP is regulated, we took a different approach, constructing mathematical models to describe the proliferative process quantitatively and thus provide new insights into the control required to achieve the complex responses observed. Previous studies have successfully used mathematical modeling to better understand T cell proliferation to Ag (18, 19, 20, 21), in which cells divide in a “programmed” manner, largely independent of the requirement for continued antigenic stimulation (22, 23). This program can be represented mathematically as a rapid and deterministic or autopilot series of divisions that follow a relatively slow and stochastic (probabilistic) entry into the first division, the mean time to which can be affected by costimulation and cytokines. This model can accurately describe the processes involved in T cell activation and provide quantitative insights into how costimulation shapes the proliferative response.

In the present study, we show that F5 TCR transgenic T cells undergo a distinct and defined program of cell division in response to lymphopenia. Proliferation is slower than compared with Ag-induced proliferation and cells fail to up-regulate activation markers or develop effector function, a response profile found to be in common with a large proportion of the CD8 T cell repertoire in polyclonal mice. Significantly, we show that homeostatic proliferation of F5 T cells is best described by a remarkably simple model of infrequent, stochastic single divisions and does not fit a model of an autopilot burst of proliferation. Our data suggest that Ag- and lymphopenia-induced proliferation may represent fundamentally distinct programs of cell division.

## Materials and Methods

### Mice

C57Bl6/J wild-type (WT), Ly5.1 C57Bl/6J, RAG-1-deficient (*Rag1*^{−/−}) mice, *Il7*^{−/−} (The Jackson Laboratory), F5 *Rag1*^{−/−} mice, and breeding combinations thereof were bred in a conventional colony free of pathogens at the National Institute for Medical Research (London, U.K.). All lines used were of H-2^{b} haplotype. Animal experiments were done according to institutional guidelines and Home Office regulations.

### Flow cytometry

Flow cytometry was conducted using 2–5 × 10^{6} lymph node or spleen cells. Cell concentrations were determined using a Scharf Instruments Casy Counter. Cells were incubated with saturating concentrations of Abs in 100 μl of PBS-BSA (0.1%)-azide (1 mM) for 1 h at 4°C followed by three washes in PBS-BSA-azide. Monoclonal Ab used in this study were as follows: allophycocyanin-TCR (H57–597) (BD Pharmingen), allophycocyanin-CD44 (Leinco Technologies), bio-CD44 (BD Pharmingen), and PerCP- and PE-CD8 (BD Pharmingen). Biotinylated mAb staining was detected using PerCP steptavidin (BD Pharmingen) and PE-TR steptavidin (Caltag Laboratories). Four- and five-color cytometric staining was analyzed on a FACSCalibur and LSR Instruments (BD Biosciences), respectively, and data analysis was performed using FlowJo V8.1 software (TreeStar).

### Labeling and adoptive transfer of T cells

Lymphocytes were teased from lymph nodes and spleen of donor mice and single cell suspensions were prepared. Cells were labeled with 2 μM CFSE (Molecular Probes) in Dulbecco’s PBS (Invitrogen Life Technologies) for 10 min at 37°C and washed twice. Cells were transferred into various recipient mice via tail vein injections. Mice further challenged with influenza virus (A/NT/60-68) were injected i.v. with 10^{7} hemagglutinating units of virus.

Cellular expansion of CFSE-labeled cells was calculated by determining the frequency (*F*_{i}) of recovered donor cells that had undergone *i* divisions, Σ*F*_{i} = 1. Adjusted frequencies, or precursor frequencies, *f*_{i} were calculated by dividing *F*_{i} by 2^{i} to remove the effects of expansion (Σ*f*_{i} ≤ 1). The fold expansion was then given by the quantity 1/Σ*f*_{i}. Mean divisions were calculated as Σ_{i} *f*_{i}/Σ*f*_{i}.

### In vivo killing assay

Splenocytes from Ly5.1 C57BL/6J were incubated with either 10^{−6} M, 10^{−7} M, 10^{−8} M, 10^{−9} M, or no Ag as control for 2 h at 37°C, then labeled with 4 μM, 1 μM, 250 nM, 62 nM, and 16 nM CFSE, respectively, for 10 min at 37°C and washed twice. Four × 10^{6} of each population of cells were mixed and injected into *Rag1*^{−/−} recipients. *Rag1*^{−/−} and F5 *Rag1*^{−/−} recipients received targets alone as controls. Twenty-four hours later, recipients were killed and splenocytes isolated. Target cells were stained with Ly5.1 and analyzed by FACS. Percent killing was determined by comparison of target populations remaining in experimental hosts with control hosts that received targets alone and normalized to the unpulsed control population.

### Mathematical modeling of homeostatic proliferation

#### Gett-Hodgkin (GH) (programmed divisions) model.

In this model, used successfully to describe programmed T cell responses to Ag stimulation (18), the first division for each cell takes a time *t* drawn from a distribution with probability density φ(*t*), and subsequent divisions for each cell are deterministic, each division taking a time Δ. We assumed no cell death over the course of the experiments. We attempted to fit the data using three left-bounded unimodal distributions φ(*t*)-gamma, lognormal, and Weibull. Whatever the specific form of φ(*t*), if cells are exposed to a stimulus at a time *T* after transfer, this model predicts that the precursor frequency of cells that have divided *i* times (that is, the frequency of cells in generation *i* after adjustment for their expansion) at time *t* after transfer is (18):

#### Stochastic single divisions model.

To describe cells undergoing single divisions randomly, we used a version of the Smith-Martin (SM) model of the cell cycle (24) in which cells spend exponentially distributed times in a resting “A phase” with rate constant λ (such that the average waiting time for a cell in A phase is 1/λ). In this model, all cells have an equal probability of committing to division in any time interval. After committing to division, cells transit deterministically through a “B phase” of duration Δ before undergoing mitosis and returning to A phase. Let the expected numbers of cells in generation *i* (that is, cells that have divided *i* times since transfer) in the A and B phases at time *t* after transfer be *x*_{i}^{A}(*t*) and *x*_{i}^{B}(*t*), respectively. Assuming *N* cells survive the transfer and are capable of dividing in the recipient, these quantities can be calculated using the following system of equations: where *x*_{i}^{B}(*s*, *t*) is the population density of cells at time *t* that have already divided *i* times and have been in B phase for a time *s*, 0 ≤ *s* ≤ Δ. The solution is subject to the boundary conditions *x*_{i}^{B}(*t*, 0) = λ*x*_{i}^{A}(*t*), *i* = 0… *n*, *x*_{0}^{A}(*t* = *T*) = *N*, *x*_{i}^{A}(*T*) = 0, *i* = 1… *n*, and *x*_{i}^{B}(*s*, 0), *i* + 1… *n*, for all *s*. The subscripts *i* = 0… *n* refer to division or generation number, *n* is the maximum number of divisions that can be resolved (typically 8–9 with CFSE labeling), and we assumed no division was possible during an initial lag period of duration *T* after transfer. With CFSE information alone only the sum of the populations in the A and B phases are observable; that is, the number of cells in CFSE peak corresponding to cells divided *i* times is These quantities are proportional to the initial undivided cell number, *N*, or the number of transferred cells that survive the transfer and participate in the homeostatic expansion. This number is unknown and may vary from animal to animal. The assumption of no cell death allowed us to remove this dependence on *N* and work instead with the precursor frequencies of cells in each generation predicted at time *t*, We also fitted a modified SM model with an additional parameter μ that allowed a progressive reduction in the division rate with time to mimic competition for homeostatic resources as the population expands. We assumed an exponential decline, such that λ(*t*) = λ_{0} exp(−μ*t*).

#### Estimation of parameters.

Best-fit values of the parameters (that is, for the GH model, the parameters describing φ(*t*) together with the lag time *T* and the division time Δ; for the simple SM model, λ, Δ, and *T*; and for the modified SM model; λ_{0}, μ, Δ, and *T)* were found using a custom-written *Mathematica* script. The model equations were solved numerically to generate the predicted precursor frequencies *p*_{i}(*t*); independent components of the frequency data *p*_{i}(*t*) and their observed counterparts *f*_{i}(*t*) were arcsin-square root transformed to normalize their distributions across animals at each division at each time point; the weighted sum of squared residuals between the *p*_{i}(*t*) and the *f*_{i}(*t*) was then minimized with respect to the parameters using a Nelder-Mead optimization algorithm. By fitting the model to 500 datasets generated by resampling with replacement, we obtained empirical (bootstrap) distributions of the parameter estimates and 95% confidence intervals were obtained using the percentile method (25). The *Mathematica* script is available on request from the authors.

#### Comparing models.

The improvement in fit obtained by adding a parameter to the SM model was assessed with an F-test. If SSR denotes the sum of the squares of the weighted residuals, each term in which is assumed be the square of a normally distributed variable with unit variance, then the quantity (SSR1 − SSR2)/(SSR2/(*m* − *q*)) is F-distributed on (1, *m* − *q*) degrees of freedom, where SSR1 is the original model, SSR2 is the extended model with extra parameter μ (SSR2 < SSR1), *m* is the number of independent frequency measurements used in the fits (*m* = 186), and *q* = 4 is the number of parameters in the larger model. Measures of relative support for the GH and SM models were obtained with the Akaike information criterion (AIC), using the definition AIC = *m* log (SSR/*m*) + 2(*K* + 1) where *K* is the number of parameters in the model (26).

## Results

### Lymphopenia and Ag induce distinct programs of cell division and differentiation in F5 T cells

A mouse model extensively used in our laboratory for studying the processes that govern homeostatic survival and proliferative responses is the F5 TCR transgenic mouse strain that expresses a class I restricted TCR specific for a peptide of influenza nucleoprotein (27). F5 T cells make relatively weak proliferative responses under conditions of T cell deficiency (12) as compared with other TCR transgenic strains (14). However, to directly compare the proliferative response of F5 T cells in lymphopenia with that induced by Ag, we transferred F5 T cells labeled with the cell dye CFSE into *Rag1*^{−/−} T cell-deficient hosts either in the presence or absence of influenza (Flu) virus. At various days post transfer, cells were recovered from host mice and analyzed by FACS. In the absence of Flu virus, T cells underwent a sustained but slow process of cell division. In hosts additionally challenged with flu virus, F5 T cells underwent a rapid and robust proliferation with most cells losing CFSE dye by day 7 (Fig. 1⇓*A*). In fact, in lymphopenic hosts it took approximately 2 wk for F5 T cells to proliferate to a similar degree as observed in only 3 days in the presence of Ag (Fig. 1⇓*B*). Furthermore, F5 T cells failed to up-regulate activation markers such as CD44 compared with the same cells challenged with Flu virus (Fig. 1⇓*B*) and failed to elicit any effector function when challenged with Ag pulse targets in vivo (Fig. 1⇓*C*).

Because of the wide variation in lymphopenic proliferative responses reported in different TCR transgenic systems, we more closely examined homeostatic proliferation of WT CD8^{+} T cells to collectively quantify the variety of responses made within a polyclonal repertoire. Purified CD8^{+} T cells from WT mice were labeled with CFSE and transferred into *Rag1*^{−/−} hosts. At 6 and 13 days post transfer, cells were recovered and the lymphopenia-induced proliferation assessed by FACS. By day 13, it appeared the majority of cells recovered were CFSE negative and CD44^{high} (Fig. 1⇑*D*) as described previously (9, 13, 28). However, a significant proportion of T cells were still labeled with varying levels of CFSE and remained CD44^{low}. Examination of responses at day 6 allowed more detailed quantitative analysis because all cells were still CFSE positive. More than 40% of T cells had up-regulated CD44 at this stage and had undergone more divisions than the CD44^{low} population. However, using the CFSE profile to calculate the precursor frequency (see *Materials and Methods*) revealed that the CD44^{high} population arose from only 17% of the precursor population. The remaining 83% of the precursor pool remained CD44^{low} and divided more slowly or not at all. In fact, 37% of the injected T cells were still undivided by day 6. Therefore, it appears that the abundance of the CD44^{high} population at day 13 reflects the extensive proliferation of a small proportion of the starting population rather than a majority response by the entire initial cohort. These data suggest that the ability of T cells to undergo extensive proliferation and phenotypic conversion is a feature of a relatively small proportion of the normal WT repertoire. Rather, the majority of CD8 T cells divided slowly, or not at all, without up-regulating activation markers. Therefore, the slow division rate of F5 TCR transgenic T cells in *Rag1*^{−/−} hosts closely resembles the slow dividing population of WT CD8 T cells, further highlighting the suitability of F5 T cells for the present analysis.

### Quantifying the F5 T cell response to lymphopenia

F5 T cells proliferate in response to lymphopenia in a manner quite distinct from that induced by cognate Ag (Fig. 1⇑*A*). Previous studies have demonstrated that T cell stimulation with cognate Ag results in the induction of a predetermined program of cell division with distinct characteristics, such as time to first division and subsequent interdivisional durations. To gain an insight into the cell cycle regulation controlling homeostatic cell division and how it compares to that described following T cell activation, we aimed to use models to describe the cell division processes quantitatively. To do this, we required a precise experimental description of lymphopenia-induced proliferation by F5 T cells to underpin any modeling. Therefore, we first attempted to measure the F5 T cell response to lymphopenia in detail.

F5 T cells were labeled with CFSE and transferred into T cell-deficient *Rag1*^{−/−} hosts. At various time points, cohorts of mice were taken and the proliferative and expansive behavior of the F5 T cells determined by cellular enumeration in lymph nodes and spleen and analysis of CFSE labeling by FACS. Over the duration of the experiments, F5 T cells were seen to undergo steady progressive divisions (Fig. 2⇓*A*). Calculating the mean number of divisions undergone revealed a brief lag followed by a near linear increase in mean division with time corresponding to roughly exponential growth (Fig. 2⇓*B*), confirmed by measurement of absolute cell numbers (Fig. 2⇓*C*). The F5 T cell precursor number, calculated by removing the effects of expansion predicted by the cells’ CFSE profile from the cell recovery, did not decline during the course of the experiment, suggesting that there was negligible cell death of F5 T cells in the lymphopenic hosts (Fig. 2⇓*C*).

We next assessed the sensitivity of the homeostatic proliferative response to host environment. First, we compared the proliferative response when different numbers of F5 T cells were transferred to *Rag1*^{−/−} recipients. Hosts were injected with cell doses between 5 × 10^{5} and 1.5 × 10^{7} CFSE-labeled F5 T cells and cell division assessed 14 days post transfer. F5 T cell proliferation as measured by mean division number was inversely proportional to inoculum size and the response was indeed exquisitely sensitive to initial cell dose, with reproducible differences observed between as little as 2-fold variations in initial cell dose (Fig. 2⇑*D*). Since IL-7 is critically required for homeostatic cell division, we asked whether varying IL-7 availability would have a quantitative effect on F5 T cell proliferation. We tested this by transferring CFSE-labeled F5 T cells into control *Rag1*^{−/−} hosts or into *Rag1*^{−/−} *Il7*^{+/−} hosts that have only one functional *Il7* allele. F5 T cells in *Il7*^{+/−} hosts proliferated more slowly than in controls as measured by mean division number versus time (Fig. 2⇑*E*), further illustrating that homeostatic proliferation was very sensitive to the availability of resources in the host.

### Modeling the homeostatic proliferation of F5 T cells

To test different hypotheses for the regulation of LIP, we fitted a series of mathematical models of cell division to these time series data. For each model, we searched for values of the model parameters that gave predicted patterns of cell division over time that best fitted our experimental time course data. In all cases we assumed that cell death among the cohort of cells that survived the transfer was negligible. This seemed reasonable on biological grounds as the lymphopenic host is likely to present an abundance of T cell survival signals or resources, but this assumption was also validated by the observation that the precursor number calculated from cells recovered from spleen and lymph nodes was approximately constant during the first 14 days after transfer (Fig. 2⇑*C*).

We attempted to describe the data with two classes of stochastic model. The formulation of the models and the fitting procedure are described in detail in *Materials and Methods*. The first class of model assumed that the mode of proliferation observed was analogous to the programmed proliferation of Ag-stimulated T cells observed in vitro, a model first studied quantitatively by Gett and Hodgkin (18) and extended in further studies (19, 21). We refer to it here as the GH model. To apply this model in the context of lymphopenia-induced proliferation in vivo we assume cells receive a proliferative stimulus from their environment soon after transfer to the lymphopenic host. For each cell, the time between receipt of this stimulus and first mitosis is assumed to be a random variable drawn from a left-bounded unimodal distribution. We used log-normal, gamma, and Weibull distributions as candidates. Cells that have divided once then embark on a program of divisions each of duration Δ before returning to a quiescent state (Fig. 3⇓*A*).

The free parameters in this model were the mean and variance specifying the probability distribution of times to first division, the division time Δ, and a delay before any cell could divide at all (*T*). The Weibull distribution provided the best fit of the three candidates (ΔAIC values; log-normal-Weibull = 75; gamma-Weibull = 35). The best fit parameters were as follows: distribution shape parameter = 2.26, scale parameter = 8.27 (mean time to first mitosis = 7.33 days, SD = 3.43 days), division time Δ = 2.88 days, lag time *T* = 0. On comparing the time course of cell division predicted by this GH model to the response observed experimentally, the model appeared to provide a very poor description of the data (Fig. 4⇓*A*), both in terms of the kinetics of mean division for the whole population and the predicted profiles of cell divisions at the various time points.

The second class of model assumed that cells circulate and compete for homeostatic division signals such that a given cell has a constant probability of entering the cell cycle in any short time interval and dividing once. We used a version of the canonical SM model of the cell cycle (24), illustrated schematically in Fig. 3⇑*B*, in which cells spend exponentially distributed times in a quiescent A phase (corresponding approximately to the G_{0}/G_{1} states of the cell cycle) and then progress through a B phase (S, G_{2}, and M) before dividing and returning to the A phase. The B phase was taken to be of fixed duration. The rate of transition from A to B phase, i.e., the rate at which T cells are triggered into cell division, is represented by the term λ, and duration of the B phase by Δ. The rate λ is defined such that the probability of any given cell entering B phase in a short time interval δ*t* is simply λδ*t*. We also estimated a lag time *T* after transfer during which no entry into B phase is possible.

The assumption of a constant mean rate of entry into division (λ) after the lag time *T* gave a reasonable fit to the data (Fig. 4⇑*B*). This model predicts a linear increase in the mean and variance of division number with time after a brief transient. The best-fit parameter estimates with 95% confidence intervals were λ = 0.210 day^{−1} (0.157, 0.266), Δ = 6.65 h (3.29, 18.8), and *T* = 2.03 days (0.172, 2.72). Therefore we estimate the mean duration of the A phase, λ^{−1}, to be around 4.8 days, and the average time between divisions for each cell (λ^{−1} + Δ) as just over 5 days. The trend in the mean division number averaged over all animals at each time point (Fig. 4⇑) suggested a progressive slowing of the division rate with time. To investigate this possibility we extended the model to allow a time-dependent change in division rate, defined by an additional parameter μ such that λ = λ_{0} exp(−μ*t*). This improved the fit significantly (partial F-test, *p* < 10^{−4}; ΔAIC = 14; Fig. 4⇑*C*). Best-fit parameter estimates were λ_{0} = 0.455 day^{−1} (0.228, 3.58), μ = 0.115 day^{−1} (0.020, 0.683), Δ = 6.19 h (2.26, 15.1), and *T* = 2.44 days (1.99, 2.90). The slowing of the division rate means that at day 3 the mean interdivision time was 3.4 days, increasing to 11.3 days at day 14.

Both the simple and modified SM model provided a significantly better fit to the data than the best fitting “programmed divisions” GH model (ΔAIC values; GH − basic SM model = 214; GH − modified SM model = 227). Despite the poor fit of the GH model, however, we clearly cannot rule out this model with other distributions for the times to first division. In particular the assumption that all cells receive their first division stimulus at the same time, as is likely in in vitro Ag-driven proliferation assays, may be questioned in vivo. Nevertheless we note that the gradient of the curve of mean division number with time is the inverse of the average time between divisions, which is roughly 5 days for the linear fit predicted by the simple SM model (the quantity λ^{−1} + Δ) and between 3 and 11 days for the extended SM model with slowing division. Irrespective of the precise form of the distribution of times to first division, the GH model predicts that a plot of mean division number versus time tends to a straight line with slope 1/Δ after all cells have entered the proliferative program and are dividing asynchronously but uniformly with constant division time Δ. Crucially, the slope we observe indicates an interdivision >10-fold slower than that observed in Ag-driven T cell responses (18) and such slow proliferation is unprecedented in programmed responses. In summary, the SM model is a good description of the data, with even the more parsimonious three-parameter version describing proliferation better than the GH autopilot model.

### Testing the biological predictions of the mathematical model in vivo

Although the SM-based model of stochastic cell divisions appears to provide the best description of the observed response by F5 T cells to lymphopenia, this observation neither proves the model is correct nor indeed excludes the possibility that other models might equally well or better describe the data. The model did, however, make several specific predictions about the nature of the F5 T cell response to lymphopenia. Therefore, we sought to further validate the model by directly testing its predictions experimentally.

A key biological prediction of the model was that TCR stimulation by self-peptide MHC should induce independent, single stochastic cell divisions. That being the case, ongoing homeostatic cell division should rely on constant signals from the lymphopenic environment to maintain continued cell division. Reversal of lymphopenia should, therefore, bring about a cessation of cell division, which would not be the case if the divisions were programmed or on autopilot. To test this directly, we injected groups of *Rag1*^{−/−} mice with CFSE-labeled F5 T cells. Three days later, when cell division of F5 T cells had already started and was at its highest rate, we injected cohorts of mice with very high doses of WT T cells (10^{8}/mouse) to quickly reverse the lymphopenia. Strikingly, such treatment abruptly halted the observed cell division of F5 T cells in these hosts, whereas the same cells in untreated *Rag1*^{−/−} hosts continued to divide (Fig. 5⇓*A*). Although F5 T cells had divided when examined 2 days later in treated hosts, they had undergone fewer divisions than F5 T cells from untreated control hosts and there was no further cell division detected in the treated hosts up to 17 days later. Only 27% of cells in treated hosts underwent any cell division, compared with 83% in controls. Were a deterministic program of division at work, then cells triggered into division before treatment would be expected to complete their program of divisions. Rather, in treated mice, the vast majority of the dividing population (80%) underwent only a single division.

Although the simple SM model described the data well (Fig. 4⇑*B*), introducing a time-dependent reduction in division rate resulted in a significantly better fit (Fig. 4⇑*C*). The slowing in division rate indicated by this analysis was also supported by our observation that F5 T cell homeostatic division is very sensitive to cell numbers in the host environment (Fig. 2⇑*D*). It seems reasonable that the rate of cell division should slow as the T cells expand in number, either due to the effects of competition or adaptation to the host. This slowing was not always evident in longer time courses because at later time points many cells had become CFSE negative, and mean cell divisions could not be accurately determined. To overcome this limitation and examine the cell kinetics of expansion over longer time periods, we measured the division rate by injecting hosts first with unlabeled F5 T cells and then transferring CFSE-labeled F5 sensor cells 2 wk later and measuring their cell division over time compared with the same cells in untreated *Rag1*^{−/−} hosts. As the data show, F5 T cells injected into the pretreated hosts proliferated at a significantly reduced rate compared with cells transferred to empty hosts (Fig. 5⇑*B*). The division rate also further reduced with time in both hosts.

Finally, another prediction of the model was that there is a lag phase of ∼2 days before cells started to respond to lymphopenia. To investigate whether there was evidence of such a delay in responsiveness of F5 T cells to the lymphopenic environment, we assessed cell size and CD5 expression by F5 T cells following transfer to lymphopenic hosts. Previous studies have shown that environmental signals in lymphopenia have trophic affects on T cells (29), resulting in increases in cell size, while the level of CD5 expression by peripheral T cells has been shown to be positively regulated by contact with self-peptide-MHC complexes (5, 30). Upon transfer to *Rag1*^{−/−} hosts, F5 T cells did indeed increase their cell size and up-regulated CD5 expression compared with the same cells transferred to C57Bl6 controls, indicative of enhanced contact with self-peptide-MHC that is required to induce LIP (Fig. 5⇑*C*). However, these changes in F5 T cell phenotype were not evident until 2 days post transfer and not maximal until day 3. By 24 h post transfer, the phenotype of F5 T cells in lymphopenic *Rag1*^{−/−} hosts was almost indistinguishable from the same cells in replete C57BL/6 controls.

## Discussion

Cell division induced in T cells by cognate Ag or by lymphopenia both depend on TCR signaling but result in distinct patterns of proliferation and, for many cells, differences in development of effector function. Previous studies have successfully used mathematical modeling to describe T responses to cognate Ag and greatly facilitated our understanding of the proliferative control involved. Here, we used a mathematical modeling approach to better understand the program of cell division that is induced in lymphopenia. In contrast to the autopilot program of deterministic cell divisions induced following stimulation by cognate Ag, we found that a model in which T cells underwent independent single stochastic divisions better described the F5 T cell response to lymphopenia.

Variants of the SM stochastic single divisions model have been used by other authors to describe turnover of both T (31, 32) and B (33) cells and implicitly held it to be a good description of their data but did not formally compare alternative proliferation mechanisms. Our successful application of the SM-based model of single stochastic divisions (Fig. 3⇑), however, still does not prove it is correct or exclude other possible models. What did provide strong support for the SM model was the predictions it made that were successfully validated experimentally and that some of these predictions were incompatible with an autopilot model. Reversing the lymphopenia at the peak of the proliferative response had a profound inhibitory affect on the response, consistent with the SM model in which cells underwent independent single stochastic divisions and that additional divisions required further stimulus from the lymphopenia. Were cells stimulated into a deterministic program of divisions, they would have already received their proliferative stimulus and would have continued upon the deterministic program of division that could not be reversed by later removal of the inducing stimulus, i.e., by reversing lymphopenia. In contrast, this is certainly the case for activation of CD8^{+} T cells by cognate Ag that requires as little as 4 h stimulation to evoke a full proliferative response (34).

Another prediction made by the SM model that we tested was that the division rate slowed with time. Both models were compatible with a strictly linear increase in mean division that was close to what was observed experimentally. However, the fit of the SM model was improved by introducing a time dependent reduction in rate of entry of cells into division. Closer experimental scrutiny revealed that lymphopenia-induced proliferation of F5 T cells did indeed slow with time (Fig. 5⇑). The magnitude of the F5 T cell response to lymphopenia was exquisitely sensitive to the lymphopenic environment, such that altered levels of IL-7 or even increasing T cell competition by varying T cell numbers was sufficient to have appreciable impact on proliferation. Therefore, it seems that the strength of the environmental signals that induce T cell proliferation in lymphopenia are subject to dynamic change as a consequence of T cell expansion, resulting in increasing cellular competition for key resources such as IL-7 and self-peptide MHC. Requiring T cells to re-audition after single divisions for further lymphopenia-induced divisions, as described in the model, also explains how the proliferation can be so finely tuned to relatively subtle changes in the competitive environment. The SM model also predicted that F5 T cells required an initial 24–48 h period of exposure to lymphopenic environment before undergoing LIP responses, a prediction also validated experimentally by the kinetics of phenotypic changes observed in F5 T cells. Such a delay may allow for transient, localized fluctuations in microenvironment that might otherwise stimulate LIP but not in fact necessitate a homeostatic response.

Although the GH model describes Ag-induced proliferation well, our study required an entirely distinct model of proliferation to describe lymphopenia-induced proliferation and yet both responses are triggered through TCR stimulation. This raises the question of how it is possible for signals from the same receptor to elicit such distinct programs of cell division? One factor might by that the TCR signals required for induction of lymphopenia-induced proliferation depend on self-peptide MHC recognition which are likely to be low avidity interactions compared with cognate Ag. However, even low affinity TCR ligands can induce autopilot proliferation, just requiring a longer duration of stimulus (34). Another possibility may be the differential requirements for IL-7 and IL-2 for lymphopenia vs Ag-induced proliferation. T cell responses to cognate Ag have no apparent requirement for IL-7, either in vivo (17) or in vitro (B. Seddon, unpublished data). Indeed, IL-7R expression is largely lost following activation (17, 35). A key step in the Ag-induced program of proliferation is the induction of IL-2 and its receptor, following TCR stimulation. This autocrine axis plays an important role in promoting cell division and survival following T cell activation. In contrast, lymphopenia-induced proliferation is highly dependent on IL-7 (17, 36, 37). Like IL-2, IL-7 is also an important T cell survival factor (17, 38) and has also been described to affect cell cycle by its effects on expression of negative regulators such as p27kip (39) and FOXO proteins (40). Both IL-2 and IL-7 are members of the common γ-chain (γc) family of cytokines, and consequently, use similar signaling pathways to mediate their biological properties, such as Stat5 (41) and PI3K (42). Therefore, IL-2 and IL-7 may play parallel biochemical roles in supporting survival and proliferation of T cells during Ag and lymphopenia-induced proliferation, respectively, even if their administration is different. Also, it is interesting to note that activated T cells do not express IL-7R, while cells undergoing lymphopenia-induced proliferation do not express IL-2R.

Crucial differences in the biology of IL-2 and IL-7 may ultimately account for the fundamentally different programs of cell division they support. Following T cell activation, induction of IL-2 and IL-2R forms part of an autocrine loop that plays a key role in the burst of proliferation T cells undergo following antigenic stimulation, and it is therefore possible that it is the maintenance of just such a loop that underlies the deterministic program of divisions that is characteristic of Ag-induced responses. In contrast, enhanced self-peptide MHC recognition and increased bioavailability of IL-7 in lymphopenia are the key events that trigger some T cells into cell division. However, since IL-7 is made by stromal cells within the lymphoid compartment (43) and not by T cells and all T cells express IL-7R, dividing T cells are therefore in equal competition with nondividing T cells for continued stimulation from IL-7 to support continued division. Since T cells cannot make IL-7, there is no autocrine loop that might permit a deterministic burst of cell division. Demanding that T cells repeatedly audition for further lymphopenia-induced cell division permits the homeostatic responses to be highly reactive to the current size of the T cell compartment. In conclusion, the SM model of cell division not only provides a remarkably accurate description of lymphopenia-induced proliferation by F5 T cells but also has many of the features required for dynamic homeostatic regulation of T cell numbers.

## Acknowledgment

We thank Roslyn Kemp for critical reading of the manuscript.

## 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 the Medical Research Council (U.K.) and the National Institutes of Health.

↵2 Address correspondence and reprint requests to Dr. B. Seddon, Division of Immune Cell Biology, National Institute for Medical Research, The Ridgeway, London NW7 1AA, U.K. E-mail address: bseddon{at}nimr.mrc.ac.uk

↵3 Abbreviations used in this paper: LIP, lymphopenia-induced proliferation; flu, influenza; SM, Smith-Martin; GH, Gett-Hodgkin; WT, wild type.

- Received July 26, 2007.
- Accepted November 20, 2007.

- Copyright © 2008 by The American Association of Immunologists