HOME HELP FEEDBACK SUBSCRIPTIONS ARCHIVE SEARCH TABLE OF CONTENTS		QUICK SEARCH:   [advanced] Author: Keyword(s): Year:  Vol:  Page:

This Article Abstract Full Text (PDF) Alert me when this article is cited Alert me if a correction is posted Citation Map Services Similar articles in this journal Similar articles in PubMed Alert me to new issues of the journal Download to citation manager Request Permissions Citing Articles Citing Articles via HighWire Citing Articles via Google Scholar Google Scholar Articles by Ganusov, V. V. Articles by De Boer, R. J. Search for Related Content PubMed PubMed Citation Articles by Ganusov, V. V. Articles by De Boer, R. J.

IL-2 Regulates Expansion of CD4⁺ T Cell Populations by Affecting Cell Death: Insights from Modeling CFSE Data¹

Vitaly V. Ganusov², Dejan Milutinovi and Rob J. De Boer

Theoretical Biology, Utrecht University, Utrecht, The Netherlands

	Abstract

Top Abstract Introduction Materials and Methods Results Discussion Disclosures References

 
It is generally accepted that IL-2 influences the dynamics of populations of T cells in vitro and in vivo. However, which parameters for cell division and/or death are affected by IL-2 is not well understood. To get better insights into the potential ways of how IL-2 may influence the population dynamics of T cells, we analyze data on the dynamics of CFSE-labeled polyclonal CD4⁺ T lymphocytes in vitro after anti-CD3 stimulation at different concentrations of exogenous IL-2. Inferring cell division and death rates from CFSE-delabeling experiments is not straightforward and requires the use of mathematical models. We find that to adequately describe the dynamics of T cells at low concentrations of exogenous IL-2, the death rate of divided cells has to increase with the number of divisions cells have undergone. IL-2 hardly affects the average interdivision time. At low IL-2 concentrations 1) fewer cells are recruited into the response and successfully complete their first division; 2) the stochasticity of cell division is increased; and 3) the rate, at which the death rate increases with the division number, increases. Summarizing, our mathematical reinterpretation suggests that the main effect of IL-2 on the in vitro dynamics of naive CD4⁺ T cells occurs by affecting the rate of cell death and not by changing the rate of cell division.

	Introduction

Top Abstract Introduction Materials and Methods Results Discussion Disclosures References

 
Generally T cells require several stimuli for rapid expansion, among which triggering the specific TCR, costimulation, and signal 3 cytokine (IL-12 or type I IFNs for CD8⁺ T cells) are the most important (1, 2). Several additional factors may also influence the magnitude and duration of T cell responses; it has been known for long time that IL-2 is generally required for expansion of populations of T lymphocytes in vitro (3). In many cases IL-2 appears to be dispensable for proliferation of Ag-specific CD8⁺ T cells in vivo (4, 5, 6, 7). How IL-2 regulates expansion and magnitude of the CD8⁺ and CD4⁺ T responses is not well understood, however. For example, administration of IL-2 in mice infected with lymphocytic choriomeningitis virus may enhance survival, or increase apoptosis, of both CD8⁺ and CD4⁺ T cells depending on the time of IL-2 administration (8). Whether such changes occur via affecting the rate of cell proliferation or cell death is generally unknown. In most cases, discriminating between the effects of IL-2 on cell division and cell death is difficult because changes in the total population size only delivers an estimate of the net difference between the rates of proliferation and death and not their individual values.

CFSE labeling (or delabeling) is one of the most informative tools used in immunology to detect cell division in various settings in vitro and in vivo (9). Generally, CFSE allows one to determine the number of divisions a cell has undergone, and up to 6 to 10 divisions can be monitored with CFSE with good accuracy (9, 10). By multiplying the fraction of cells that have undergone different numbers of division by the total number of cells, one finds the absolute number of cells that have undergone a given number of divisions at different time instances. This we call CFSE data. In principle, CFSE data allow one to separate the contribution of proliferation and death processes to the net change of the population size (10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21).

A previous study addressed the role of IL-2 in proliferation of polyclonal CD4⁺ T lymphocytes in vitro (22). In this study, CFSE-labeled naive CD4⁺ T cells were stimulated in vitro with anti-CD3 Abs at different concentrations of exogenous IL-2. A simple mathematical model was developed to fit these CFSE data. The model assumed a lognormal distribution of division times for undivided cells and a deterministic division with a fixed interdivision time for divided cells (20, 22). The authors found that IL-2 mainly affects the fraction of cells recruited into the response, the interdivision time, and the probability of cell death during the division for divided cells. Here we extend this previous study and reanalyze the dynamics of CFSE-labeled CD4⁺ T cells in vitro using several mathematical models fitted to the same data (22). The reanalysis was prompted by two important observations we have made in the data.

First, there is change in the symmetry of the time course of cell cohorts having completed the same number of divisions. If one looks at the time course of the number of cells in a cohort of undivided cells (Fig. 1), or cells having undergone one division (e.g., in Fig. 3A), this time course can be described by an asymmetric, lognormal distribution. This time course becomes more symmetric (i.e., can be described by a normal distribution) for cohorts of cells having undergone more divisions (e.g., 3 divisions, see Fig. 3A). This observation indicates that division of cells cannot be strictly deterministic, because in a deterministically dividing cell population, the time course of cell cohorts is simply shifted in time and scaled in amplitude relative to each other (20).

View larger version (19K): [in this window] [in a new window]   FIGURE 1. Recruitment of naive CD4+ T cells into the S phase of the cell cycle following anti-CD3 stimulation in the presence of demecolcine. At various times after stimulation, cell cultures were pulsed with [3H]TdR (thymidine) for 2 h before harvesting. The resulting radioactivity is plotted as the function of time of harvesting. Experiments were done with four different IL-2 concentrations (concentrations of IL-2 in U/ml are marked). A log normal distribution with a delay was fitted to these data. The data are presented by the symbols and the fits of the distribution as lines. Estimates of the parameters for the log normal distribution are given in Table I.

View larger version (18K): [in this window] [in a new window]   FIGURE 3. Best fit of the SM model with division-dependent death rate to the CFSE data of CD4+ T cells stimulated in vitro with anti-CD3 Abs at four concentrations of exogenous IL-2: IL-2 = 50 (A), 5 (B), 2.5 (C), and 1.25 (D) U/ml. We plot the number of cells having undergone the same number of divisions as a function of time: cohort having completed one division (solid line with circles), two divisions (short-dashed line with triangles), three (long-dashed line with squares), four (dotted line with diamonds), five (dash-dotted line with stars), to six or more (double dash-dotted line with open triangles). A lognormal distribution with the mean and shape parameters estimated from the TLE data, were used to describe the recruitment function R(t). Note that the graphs are plotted on different scales on the y-axes for different IL-2 concentrations; this is done to demonstrate the quality of the mode fits to data. The sums of squared residuals for the fits are: 7.28 x 106 (IL-2 = 50 U/ml), 1.03 x 106 (IL-2 = 5 U/ml), 3.06 x 106 (IL-2 = 2.5 U/ml), 5.23 x 105 (IL-2 = 1.25 U/ml). Parameter values providing the best fits are given in Table I.

In our study, we improve the analysis of these CFSE data in several directions. First, we explicitly account for the data on the recruitment on undivided CD4⁺ T cells into the S phase of the cell cycle after initial stimulation (see Materials and Methods). We show that neglecting these data can lead to a possibly incorrect interpretation of the data (20). Second, we use a more general model for cell division, the Smith-Martin (SM)³ model, that includes both a deterministic and a stochastic component, because the latter is required for the changes in symmetry of the time course of cell cohorts in the CFSE data (see above). Finally, we investigate several alternative models explaining these data (division-dependent death rate model and IL-2 consumption model, see Materials and Methods) and suggest experimental tests allowing discrimination between different models. Our study demonstrates 1) the wealth of information that can be obtained from CFSE-delabeling experiments, and 2) potential challenges one faces with the analysis of these data. The main goal of this study is to investigate which parameters for cell division and death are affected by IL-2 during expansion of populations of naive CD4⁺ T cells in vitro.

We find that IL-2 has the most profound effect on the death rate of dividing CD4⁺ T cells, and specifically, on the rate of increase of the death rate with the division number. IL-2 also affects the fraction of cells recruited into the response and surviving their first division. In contrast, IL-2 does not affect the average interdivision time of dividing cells. Summarizing, our results suggest that magnitude of expansion of populations of naive CD4⁺ T cells in vitro is largely controlled by regulating the rates of cell death, and hardly the rates of cell division.

	Materials and Methods

Top Abstract Introduction Materials and Methods Results Discussion Disclosures References

 
Data

The data used in this study have been provided by Dr. P. Hodgkin and have been published previously (22); the data are available at http://theory.bio.uu.nl/vitaly/data/Ganusov07_ji_il2_data.zip. In brief, mouse lymph nodes naive CD4⁺ T lymphocytes have been labeled with CFSE and then stimulated in vitro with anti-CD3 Abs in the presence of different concentrations of exogenous human (h)IL-2 (IL-2 = 1.25, 2.5, 5, 50 U/ml). Endogenously produced mouse (m)IL-2 was effectively blocked using anti-mIL-2 Abs because in cultures where no hIL-2 was added, no cell proliferation was observed (E. Deenick, unpublished observations). The initial number of cells in cultures at all IL-2 concentrations was 3 x 10⁴. In a separate experiment, cell cultures were pulsed at day 4 with 100 µg/ml BrdU for 4 h before harvesting. BrdU incorporation was detected using an anti-BrdU Ab. In the thymidine-labeling experiments (TLE), cells were stimulated in the presence of 5 ng/ml demecolcine. After various times, cultures were pulsed with ³H[TdR] for 2 h before harvesting.

Dynamics of undivided cells

Previous studies have suggested that in some situations division and death of undivided and divided cells needs to be described separately (15, 22). In particular, it may take 24–48 h for a cell to divide for the first time whereas subsequent divisions typically occur much faster (15, 23, 24, 25). Some cells may die in vitro soon after their recovery from hosts due to different reasons including cytokine deprivation (15), and not all cells are stimulated to divide. To overcome these problems with describing the dynamics of undivided cells, we focus on the dynamics of divided cells only and introduce a recruitment function R(t) which describes the number of cells successfully completing their first division (and entering their second division) per unit of time (Fig. 2). In independent experiments, recruitment of cells into the first S phase has been estimated by pulsing cell cultures with radioactive thymidine at various times and preventing mitosis with demecolcine (22). We refitted these data using a log normal distribution with a delay and estimated the mean of the distribution µ', shape parameter ', and the delay ₀' (Fig. 1 and Table I). The form of the recruitment function used then in the fits of the CFSE data was simply a scaled and shifted log normal distribution with estimated parameters µ' and ':

(1)

where C is the total number of cells entered their second division, and ₀ is the total delay in the recruitment function resulted after adding an extra delay to the estimated parameter ₀' (_{0 0}'). Because demecolcine may induce apoptosis in cells (26, 27), only the mean µ' and shape parameter ' of the log normal distribution in the recruitment function R(t) were used in fitting CFSE data and not the total number of cells entering their first S phase after stimulation (proportional to the area under the curve in Fig. 1).

View larger version (8K): [in this window] [in a new window]   FIGURE 2. A cartoon of the modified SM model for cell dynamics. We focus on the dynamics of divided cells (i.e., cells having undergone at least one division). We define the recruitment function R(t) which is equal to the number of cells entering their second division per unit of time. Cell division is described using the SM model (28 ). In this model, the cell cycle of cells consists of two parts: a stochastic A-state in which the waiting time is exponentially distributed (with parameter ), and a deterministic B phase of fixed duration (equal to ). The total number of cells in the nth division is Xn = An + Bn. The rate of cell death during the A-state and the B phase is d.

We have extended the SM model for cell division for the analysis of CFSE data (14, 28). In the SM model, the cell cycle of proliferating cells can be divided in two main parts: the variable A-state and the deterministic B phase. The waiting time in the A-state is exponentially distributed (with parameter ), and different cells will take different times to leave the A-state. In contrast, the B phase has a fixed duration , and cells completing the B phase, divide and deliver two daughter cells into the A-state. In this model, the average interdivision time is T = + 1/ (Fig. 2). Biologically, the A-state and the B phase correspond approximately to the G₁ phase and S + G₂ + M phases of the cell cycle, respectively. In our model thus the progression through the cell cycle is determined by two parameters, and . In more complex models, however, this progression can be also affected by other factors, for example, by cell size (29, 30, 31). Cells in the A-state and the B phase die at a constant rate d. The model can be formulated as a system of ordinary and partial differential equations (14, 18).

As discussed in the Introduction, there are quantitative changes in the dynamics of cell cohorts that imply changes in the parameters for cell division/death with the division number and/or time. Changes in the parameters with the division number are intuitive and come from the general observation that changes in cell phenotype are often (but not always) associated with cell division (32, 33, 34, 35, 36). The SM model with division-dependent parameters can be reformulated as a set of delayed differentialequations (for derivation, see Supplementary Material available online at http://theory.bio.uu.nl/vitaly/publ/Ganusov07_ji_appendix.pdf):

(2)

(3)

(4)

where A_n(t) and B_n(t) are the numbers of cells in the A-state and the B phase, respectively, having undergone n divisions by time t; R(t) is the recruitment function denoting the number of cells that successfully complete their first division per unit of time, _n, _n, and d_n are the rate of recruitment into the B phase, the duration of the deterministic B phase, and the death rate, respectively, dependent on the division number n. In the SM model, we calculate the probability of cell death per division as _n = 1 – _n/(_n + d_n) x e^–d_nn, where _n/_n + d_n) is the probability that a cell survives during the A-state, and e^–d_nn is the probability that a cell survives during the B phase.

Additional experiments undertaken by Deenick et al. (22) involving short-pulse BrdU labeling of cultures of CD4⁺ T cells stimulated in vitro at different concentrations of IL-2 suggested that parameters for cell division are unlikely to change with the division number. This comes from the observation that there is only a minimal change in the fraction of BrdU⁺ cells with the cell division number (for n > 2) across all IL-2 concentrations (22). Therefore, in our "best guess" model we assumed constant parameters for cell proliferation ( and ), and let only the death rate change linearly with the division number n, d_n = d₁ + a(n – 1). The model was fitted to each individual CFSE dataset for a given IL-2 concentration.

Because in experiments of Deenick et al. (22), cell cultures were started with different initial concentrations of IL-2, it seems natural to consider models where the IL-2 concentration changes over time and that affects the rates of proliferation and/or death of T cells. Indeed, it has been shown that proliferating T lymphocytes reduce the amount of IL-2 present in the culture medium (3, 37, 38), and it seems likely that changes we observe in the dynamics of CD4⁺ T cells at low IL-2 concentrations are due to "consumption" of IL-2 by T cells. Therefore, in an alternative, IL-2 "consumption" model, we let CD4⁺ T cells reduce the medium concentration of IL-2. Changes in the IL-2 concentration affect parameters for cell division and death (see Supplementary Material for more details). Shortly, the dynamics of the IL-2 concentration in the medium, I(t), are described by

(5)

where c₁ and c₂ are parameters describing consumption of IL-2 by divided cells X(t) = _n=1 X_n (t), and by undivided cells, respectively; h_I is a half-saturation constant defining that at high IL-2 concentrations, cells have a maximum rate of IL-2 consumption. Dependencies of the parameters for divided cells on the medium concentration of IL-2 were chosen to be simple saturating functions

(6)

with different half-saturation constants h, h, and h_d (see Fig. 7). The dynamics of T cells having undergone different numbers of divisions is given by the modified SM model reformulated as a system of delayed differential equations (shown in Supplementary Material). The model was fitted to the CFSE data at all IL-2 concentrations simultaneously, with different initial IL-2 concentrations for different datasets.

View larger version (8K): [in this window] [in a new window]   FIGURE 7. Estimated parameters for cell division and death in the IL-2 "consumption" model as the function of the IL-2 concentration. Changes in the rate of commitment to division (A), the duration of the B phase (B), and the death rate d (C) with the IL-2 concentration given in Equation 6 are plotted. Estimates of coefficients determining changes in these parameters with the IL-2 concentration are listed in the legend of Fig. 6.

Numerical procedures

We use numerical solutions of the mathematical models for fitting the CFSE data. The fitting is done by minimizing the difference between the model predictions on the total cell number X_n(t) = A_n(t) + B_n(t) and the data using least squares nonlinear regression (assuming normally distributed errors with zero mean and constant variance). Because the CFSE concentration is diluted 2-fold following cell division, it is often difficult to estimate the number of cells having undergone many divisions due to autofluorescence of CFSE-unlabeled cells. To reduce errors associated with the estimation of the number of cells having undergone >6 divisions, we have created a new division class of cells, called "6+", by which we denote the number of cells undergone 6 or more divisions. Confidence intervals have been calculated by bootstrapping the residuals with 500 simulations (39). Statistical comparison of nested models was done using the F-test (40, 41). Simulations were done in Mathematica 4.0 using FindMinimum routine and external functions coded in C. Solutions of delay differential equations have been obtained using routine retard.f described by Hairer et al. (42) and ported to C. Mathematica notebooks and relevant C codes are available from the authors upon request. Additional results and mathematical derivations are given as Supplementary Material (available online at http://theory.bio.uu.nl/vitaly/publ/Ganusov07_ji_appendix.pdf).

	Results

Top Abstract Introduction Materials and Methods Results Discussion Disclosures References

 
Fitting CFSE data

We fitted the CFSE data with the modified version of the SM model (see Materials and Methods, Equations 2–4) with division-dependent death rate (Fig. 3). These data have been fitted previously by several models (20, 22), but the extended SM model for the first time accurately describes the dynamics of cells having undergone a few divisions at low IL-2 concentrations. In particular, at IL-2 = 2.5 U/ml, the maximum density of cells in each cohort is correctly predicted to be reached by the cohort of cells having completed two divisions (Fig. 3C).

The total number of cells successfully completing their first division, which is the total area under the recruitment function R(t), is similar for the three highest IL-2 concentrations (C 9 x 10³, see Figs. 4A and 5A and Table I). At these conditions, 15% of the initial cell number (9 x 10³/(2 x 3 x 10⁴) 15%) is completing their first division. At the lowest IL-2 concentration (IL-2 = 1.25 U/ml), only half of that number is completing the first division (C 4.0 x 10³). The mean time cells require to successfully complete their first division, T₁ = µ' + ₀, is hardly affected by the IL-2 concentration (Fig. 5B).

View larger version (8K): [in this window] [in a new window]   FIGURE 4. Estimated recruitment function R(t) (A) and the death rate of divided cells (B) for the four IL-2 concentrations (IL-2 = 50 U/ml: continuous lines, 5 U/ml: small dashing lines, 2.5 U/ml: large dashing lines, and 1.25 U/ml: dot-dashed lines). The death rate of dividing cells is linearly changing with the division number n, dn = d1 + a(n – 1).

View larger version (9K): [in this window] [in a new window]   FIGURE 5. Parameters estimated by fitting the SM model with division-dependent death rate dn plotted as a function of the IL-2 concentration. The total number of cells successfully completed first division C (A), the average time required to complete first division by surviving cells µ' + 0 (B), the average cell cycle time of dividing cells T = + 1/ (C), the degree of stochasticity of the cell cycle (D), the slope of the death rate (E), and death rate d1 of cells having completed one division (F). Error bars, 95% confidence limits as obtained by bootstrapping the residuals with 500 simulations.

The latter result is surprising because intuitively one expects a longer B phase when IL-2 concentration is low. Because T cells at low IL-2 concentration have a longer A-state (see Table I), such cells could be better "prepared" for division and could traverse through the B phase faster. For example, such cells could have a higher RNA content at the end of the G₁ phase (i.e., A-state), which is known to affect the duration of the S phase of the cell cycle (43, 44). In contrast, this result may be an artifact of the SM model because an increase in stochasticity of cell division at the fixed average interdivision time can only be accounted for with a shorter B phase. Indeed, our estimate for the length of the B phase at IL-2 = 1.25 U/ml appears to be somewhat shorter than previous estimates of S + G₂ + M phases for lymphocytes (43). More complex models for cell division incorporating extra parameters for stochasticity of cell division will have to be used to investigate this result further. Several variants of such models are currently being developed (20, 45). Whatever the precise mechanism, our results do suggest that at high concentrations of IL-2, CD4⁺ T cell division becomes fairly deterministic, as was proposed earlier (12).

In BrdU-labeling experiments, Deenick et al. (22) have observed that IL-2 has strong influence on the fraction of BrdU⁺ cells in the culture. At the lowest IL-2 concentration, 40% of cells were BrdU⁺ in each division while at IL-2 = 50 U/ml, around 80% of cells were BrdU⁺. This observation is consistent with the parameter estimates of our model. In the SM model, recruitment of cells into the S phase of the cell cycle is determined by the rate parameter . This parameter is strongly affected by the IL-2 concentration such that at a low IL-2 concentration a longer time is required for a cell to be recruited into the S phase (Table I).

Confirming our previous results (20), we find that IL-2 has the most profound effect on the death rate of dividing CD4⁺ T cells and, specifically, on the increase of the death rate with the division number. The rate , at which the death rate increases with the division number, increases with the decreasing IL-2 concentration (Figs. 4B and 5E and Table I). At IL-2 = 50 U/ml, the death rate hardly changes with the division number, and 20–40% of cells die per cell cycle. At the lowest IL-2 concentration the change in the death rate with the division number is significantly different from zero resulting in 20% and 90% of cells dying in the first and the sixth divisions, respectively. Note that decreasing the IL-2 concentration from IL-2 = 50 U/ml to 1.25 U/ml increases the slope >10-fold, whereas the change in the relative number of cells recruited into the response given by the parameter C over the same range of IL-2 concentrations is more modest (little over 2-fold). Although the number of cells recruited into the response does depend on the IL-2 concentration (Ref. 22 ; see Fig. 1), the magnitude of the CD4⁺ T cell response is mainly determined by the changes in the death rate with the division number.

The death rate of cells that have divided once, d₁, is approximately independent of IL-2 concentration (Fig. 5F and Table I); the low estimate for d₁ at IL-2 = 5 U/ml is probably due to an experimental error in estimation of total cell numbers as discussed previously (20). Given that there are only small changes in the average interdivision time with the IL-2 concentration, these results suggest that magnitude of expansion of populations of naive CD4⁺ T cells in vitro is largely controlled by regulating the rates of cell death, and hardly by the rates of cell division.

Comparison to other models

The SM model with a death rate linearly changing with the division number gives an excellent description of the dynamics of the CFSE-labeled CD4⁺ T lymphocytes. We have also tested whether simpler models with a smaller number of parameters can describe these CFSE data. Two simpler models have been tested: a model with random division in which interdivision times are exponentially distributed (obtained by letting 0), and a model with deterministic division in which division takes a fixed time period (by letting ). Both models gave rather poor fits at all IL-2 concentrations (p < 0.001 for all models, F-test). The poor fits of the random division model are due to a very rapid cell division of some cells since the model lacks a minimum interdivision time (19). Poor fits of the deterministic division model are due to the inability of the model to "convert" an asymmetric recruitment function R(t) into the later symmetric time course for cohorts of cells having undergone three and more divisions (see Introduction and Ref. 20).

Allowing the parameters for cell division such as and to change linearly with the division number in the SM model with a linearly changing death rate d_n did not lead to an improvement of the quality of the fits (p > 0.05, F-test). Surprisingly, allowing only the commitment rate or the duration of the B phase , to change linearly with the division number n led to fits of a similar quality as those with division-dependent death rate (results not shown). These fits predicted a relatively fast increase in the average interdivision time T_n = _n + 1/_n with the division number n (either by increasing _n or decreasing _n with n, see Supplementary Material). This is not in agreement with the BrdU-labeling experiments suggesting that there are limited, if any, changes in the cell division parameters with the IL-2 concentration (22).

Changes in the dynamics of lymphocytes may have also occurred due to reduction in the amount of IL-2 in the medium due to "consumption" of IL-2 by dividing T cells (37, 38). How IL-2 concentration would affect the rates of cell division and death, and how cells "consume" IL-2 is not well understood. A simple SM model in which lymphocytes could reduce the medium concentration of IL-2 was fitted to data (see Materials and Methods for model description). This model describes the dynamics of CFSE-labeled CD4⁺ T cells very well (Fig. 6), although with somewhat lower quality than the SM model with division-dependent death rate (based on sum of squared residuals). The fits predict that the rate of recruitment into the B phase is affected the most by changes in the IL-2 concentration (Fig. 7). This is in disagreement with the prediction of the SM model with division-dependent death rate suggesting that the main effects of IL-2 is on the rate of cell death.

View larger version (18K): [in this window] [in a new window]   FIGURE 6. Best fit of the IL-2 consumption model to the CFSE data of CD4+ T cells stimulated in vitro with anti-CD3 Abs at four initial concentrations of exogenous IL-2: IL-2 = 50 (A), 5 (B), 2.5 (C), and 1.25 (D) U/ml. Notations for symbols and lines are the same as in Fig. 3. Parameters providing the best fit are: C = (4.2, 9.0, 12.1, 10.5) x 103 cells and 0 = (20.13, 30.53, 53.25.49, 21.25) hours for IL-2 = (1.25, 2.5, 5, 50) U/ml, respectively; 1 = 0.046 1/h, 2 = 0.24 1/h, 1 = 7.24 h, 2 = 12.71 h, d1 = 0.04 1/h, d2 = 0.029 1/h, h = 0.97 U/ml, h = 66.1 U/ml, hd = 4.5 x 10–3 U/ml, hI = 1.6 x 10–2 U/ml, c1 = 2.8 x 10–4 U/(ml · cell · h), and c2 = 1.8 x 10–2 U/(ml · cell). The sum of squared residuals is 1.36 x 107.

View larger version (12K): [in this window] [in a new window]   FIGURE 8. Predicted medium IL-2 concentration as the function of time since stimulation in the IL-2 "consumption" model. The plotted IL-2 concentration is normalized to its initial value. Predictions are for the initial concentration of IL-2 = 50 U/ml (continuous lines), 5 U/ml (small dashing lines), 2.5 U/ml (large dashing lines), and 1.25 U/ml (dot-dashed lines). Estimates of the model parameters are listed in the legend of Fig. 6.

	Discussion

Top Abstract Introduction Materials and Methods Results Discussion Disclosures References

 
There is a rapidly growing body of literature quantifying the dynamics of T cell responses in mice (46); nevertheless, in most cases we do not have clear understanding of relative contribution of proliferation and death processes during this dynamic (47). In this work we have demonstrated that one can retrieve both division and death rate from experimental data obtained using CFSE. This was done by building a set of mathematical models with different levels of complexity and fitting these models to the data. Statistical comparison of the fits produced by simple and more complex models is important for selection of a simplest model that can explain the data while making a minimal number of assumptions (48).

By using a rigorous theoretical framework, we have obtained a number of new results that complement and reevaluate the previous analysis of Deenick et al. (22), and emphasize the importance of testing several alternative models in how well they describe the data. Our main new result is that expansion of the population of CD4⁺ T cells can be regulated by IL-2 largely by changing the rate of cell death. IL-2 has the most dramatic influence on the rate at which the death rate increases with the division number. IL-2 only moderately affects the cell interdivision time, but reducing the IL-2 concentration increases the degree of stochasticity of the cell division. Interestingly, a recent study has also found dependence of the rate of cell death and proliferation on the division number for lymphocytes stimulated with PMA in vitro (21).

IL-2 also has an influence on the dynamics of undivided cells by reducing the fraction of cells recruited into the response and successfully completing their first division. Furthermore, using statistical methods, we also have shown that the SM model is a minimal model required for the description of the CFSE data. Simpler models such as widely used random division model (20) or deterministic division model (12, 22), fail to adequately describe these data. In contrast, to discriminate between alternative SM models, additional experimental data from BrdU-labeling experiments, were required. This argues in favor of using the SM model and not more complex models (29, 30, 31) for the analysis of these CFSE data.

Our results suggest that IL-2 affects the magnitude of CD4⁺ T cell expansion by affecting the rate of cell death. This does not contradict the notion that IL-2 is also necessary for cell division since we observed a tendency for longer cell interdivision times, and fewer cells were recruited into the response at low IL-2 concentrations.

In our analysis we have assumed that the rates of cell proliferation and death are identical for all CD4⁺ T cells. Alternatively, there might be inherent heterogeneity in propensity to proliferate and/or die for some particular T cells. For example, distribution times of first division of T cells stimulated in vitro may dependent on the distribution of IL-2 receptors (49). More work is clearly required to investigate whether such inherent heterogeneity extends beyond the first cell division. We have assumed that the death rate of divided cells is exponentially distributed. Restricting death events to occur only in the A-state, or only in the B phase generally led to fits of similar quality as presented above (results not shown). Previous work suggested that changing the distribution of cell death over the cell cycle may affect the estimates for the rates of cell division and death, especially for declining cell populations (14, 18). CFSE data typically is not sufficient for estimating the distribution of death rates over the cell cycle (18). Future work on additional measurements (such as the fraction of cells in the S/G₂/M phases of the cell cycle), and combining CFSE and BrdU-labeling experiments may allow for a more precise estimation of the proliferation and death rates of proliferating T cells.

Our preference of the SM model with division-dependent death rate as the best description of the CFSE data was in part due to using additional, independent experimental data on labeling T lymphocytes with BrdU (see above). This clearly demonstrates that even for such large CFSE datasets (there are 6 time points in the CFSE data, much more than the majority of CFSE data in the field), several alternative models can describe the data relatively well, and additional information is required to reduce the number of likely models. However, it is not convincing that only one experimental observation is used to discriminate between distinct models. Indeed, once we have a set of candidate models that reasonably describe the data, additional measurements/experiments (in addition to BrdU labeling) may help to support/reject alternative models. Discriminating between the model with division-dependent death rate and the model with IL-2 consumption is crucial since they predict an opposite effect of IL-2 on parameters for cell division and death. The SM model with division-dependent death predicts that IL-2 affects mainly the cell death rate, while the IL-2 consumption model predicts that IL-2 largely affects the rate of cell proliferation (or the average interdivision time).

There are several ways in which predictions of different models analyzed in this paper can be tested. First, we have tried to test the prediction of our model that the death rate of cells changes with the division number, but were plagued by difficulties in measuring rates of cell death. Measuring the fraction of cells in early or late stages of apoptosis (by caspase-3, Annexin V, or PI staining) at one time instance does not deliver an estimate of the rate at cell death (50, 51). We have extended the current SM model to include the dynamics of cells in an proapoptotic state, for example, caspase-3+ cells, making it possible to test the prediction of the model if the relevant data become available. However, this requires additional kinetic assumptions on how cells become proapoptotic and how proapoptotic cells actually die (results not shown).

Alternatively, one could try to directly measure the dynamics of dead cells having undergone different numbers of divisions. A simple extension of the SM model to include the dynamics of dead cells suggests that having such data allows one to discriminate between the SM model with division-dependent death rate d_n and division-dependent commitment rate _n (see Supplementary Material). At the present, however, it is too ambiguous to determine the number of dead cells per division class because 1) CFSE peaks cannot be clearly identified; 2) the variance in the CFSE distribution of dead cells is quite different from that of divided cells; and 3) it is not known how much CFSE cells lose when they die. Measuring the medium IL-2 concentration, or adding IL-2 at late time points to see whether such interventions affect cell dynamics is a direct test of the IL-2 "consumption" model. Addressing these experimental issues, as well as measuring the rate of accumulation of dying cells by markers of apoptosis, will allow for a major advance in understanding kinetics of cell proliferation, and could be used to examine the predictions of our and other mathematical models.

	Acknowledgments

 
We thank Phil Hodgkin and members of his laboratory for sharing the published and unpublished data with us and for useful discussions on the use of CFSE.

	Disclosures

Top Abstract Introduction Materials and Methods Results Discussion Disclosures References

 
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 Human Frontier Science Program Grant RGP0010/2004, Netherlands Organization for Scientific Research Grant 016.048.603, and Marie Curie Incoming International Fellowship (FP6).

² Address correspondence and reprint requests to Dr. Vitaly V. Ganusov, Utrecht University, Padualaan 8, Utrecht, Netherlands. E-mail address: v.v.ganusov{at}uu.nl

³ Abbreviations used in this paper: SM, Smith-Martin; TLE, thymidine-labeling experiment.

Received for publication September 21, 2006. Accepted for publication April 30, 2007.

	References

Top Abstract Introduction Materials and Methods Results Discussion Disclosures References

 

1. Baxter, A. G., P. D. Hodgkin. 2002. Activation rules: the two-signal theories of immune activation. Nat. Rev. Immunol. 2: 439-446. [Medline]
2. Curtsinger, J. M., C. M. Johnson, M. F. Mescher. 2003. CD8 T cell clonal expansion and development of effector function require prolonged exposure to antigen, costimulation, and signal 3 cytokine. J. Immunol. 171: 5165-5171. [Abstract/Free Full Text]
3. Smith, K. A.. 1988. Interleukin-2: inception, impact, and implications. Science 240: 1169-1176. [Abstract/Free Full Text]
4. Kundig, T. M., H. Schorle, M. F. Bachmann, H. Hengartner, R. M. Zinkernagel, I. Horak. 1993. Immune responses in interleukin-2-deficient mice. Science 262: 1059-1061. [Abstract/Free Full Text]
5. Cousens, L. P., J. S. Orange, C. A. Biron. 1995. Endogenous IL-2 contributes to T cell expansion and IFN- production during lymphocytic choriomeningitis virus infection. J. Immunol. 155: 5690-5699. [Abstract]
6. Wong, P., M. Lara-Tejero, A. Ploss, I. Leiner, E. G. Pamer. 2004. Rapid development of T cell memory. J. Immunol. 172: 7239-7245. [Abstract/Free Full Text]
7. Williams, M. A., A. J. Tyznik, M. J. Bevan. 2006. Interleukin-2 signals during priming are required for secondary expansion of CD8⁺ memory T cells. Nature 441: 890-893. [Medline]
8. Blattman, J. N., J. M. Grayson, E. J. Wherry, S. M. Kaech, K. A. Smith, R. Ahmed. 2003. Therapeutic use of IL-2 to enhance antiviral T-cell responses in vivo. Nat. Med. 9: 540-547. [Medline]
9. Lyons, A. B.. 2000. Analysing cell division in vivo and in vitro using flow cytometric measurement of CFSE dye dilution. J. Immunol. Methods 243: 147-154. [Medline]
10. Hasbold, J., A. V. Gett, J. S. Rush, E. Deenick, D. Avery, J. Jun, P. D. Hodgkin. 1999. Quantitative analysis of lymphocyte differentiation and proliferation in vitro using carboxyfluorescein diacetate succinimidyl ester. Immunol. Cell Biol. 77: 516-522. [Medline]
11. Nordon, R. E., M. Nakamura, C. Ramirez, R. Odell. 1999. Analysis of growth kinetics by division tracking. Immunol. Cell Biol. 77: 523-524. [Medline]
12. Gett, A.V., P. D. Hodgkin. 2000. A cellular calculus for signal integration by T cells. Nat. Immunol. 1: 239-234. [Medline]
13. Revy, P., M. Sospedra, B. Barbour, A. Trautmann. 2001. Functional antigen-independent synapses formed between T cells and dendritic cells. Nat. Immunol. 2: 925-931. [Medline]
14. Pilyugin, S. S., V. V. Ganusov, K. Murali-Krishna, R. Ahmed, R. Antia. 2003. The rescaling method for quantifying the turnover of cell populations. J. Theor. Biol. 225: 275-283. [Medline]
15. Bernard, S., L. Pujo-Menjouet, M. C. Mackey. 2003. Analysis of cell kinetics using a cell division marker: mathematical modeling of experimental data. Biophys. J. 84: 3414-3424. [Medline]
16. Leon, K., J. Faro, J. Carneiro. 2004. A general mathematical framework to model generation structure in a population of asynchronously dividing cells. J. Theor. Biol. 229: 455-476. [Medline]
17. Thompson, B. S., T. C. Mitchell. 2004. Measurement of daughter cell accumulation during lymphocyte proliferation in vivo. J. Immunol. Methods 295: 79-87. [Medline]
18. Ganusov, V. V., S. S. Pilyugin, R. J. de Boer, K. Murali-Krishna, R. Ahmed, R. Antia. 2005. Quantifying cell turnover using CFSE data. J. Immunol. Methods 298: 183-200. [Medline]
19. De Boer, R. J., A. S. Perelson. 2005. Estimating division and death rates from CFSE data. J. Comp. Appl. Math. 184: 140-144.
20. De Boer, R. J., V. V. Ganusov, D. Milutinovic, P. D. Hodgkin, A. S. Perelson. 2006. Estimating lymphocyte division and death rates from CFSE data. Bull. Math. Biol. 68: 1011-1031. [Medline]
21. Luzyanina, T., S. Mrusek, J. T. Edwards, D. Roose, S. Ehl, G. Bocharov. 2007. Computational analysis of CFSE proliferation assay. J. Math. Biol. 54: 57-89. [Medline]
22. Deenick, E. K., A. V. Gett, P. D. Hodgkin. 2003. Stochastic model of T cell proliferation: a calculus revealing IL-2 regulation of precursor frequencies, cell cycle time, and survival. J. Immunol. 170: 4963-4972. [Abstract/Free Full Text]
23. van Stipdonk, M. J., E. E. Lemmens, S. P. Schoenberger. 2001. Naive CTLs require a single brief period of antigenic stimulation for clonal expansion and differentiation. Nat. Immunol. 2: 423-429. [Medline]
24. Mempel, T. R., S. E. Henrickson, U. H. Von Andrian. 2004. T-cell priming by dendritic cells in lymph nodes occurs in three distinct phases. Nature 427: 154-159. [Medline]
25. Miller, M. J., O. Safrina, I. Parker, M. D. Cahalan. 2004. Imaging the single cell dynamics of CD4⁺ T cell activation by dendritic cells in lymph nodes. J. Exp. Med. 200: 847-856. [Abstract/Free Full Text]
26. Sherwood, S. W., J. P. Sheridan, R. T. Schimke. 1994. Induction of apoptosis by the anti-tubulin drug colcemid: relationship of mitotic checkpoint control to the induction of apoptosis in HeLa S3 cells. Exp. Cell Res. 215: 373-379. [Medline]
27. Puntoni, F., E. Villa-Moruzzi. 1999. Protein phosphatase-1 activation and association with the retinoblastoma protein in colcemid-induced apoptosis. Biochem. Biophys. Res. Commun. 266: 279-283. [Medline]
28. Smith, J. A., L. Martin. 1973. Do cells cycle?. Proc. Natl. Acad. Sci. USA 70: 1263-1267. [Abstract/Free Full Text]
29. Tyson, J. J., K. B. Hannsgen. 1985. The distributions of cell size and generation time in a model of the cell cycle incorporating size control and random transitions. J. Theor. Biol. 113: 29-62. [Medline]
30. Tyson, J. J., O. Diekmann. 1986. Sloppy size control of the cell division cycle. J. Theor. Biol. 118: 405-404. [Medline]
31. Tyson, J. J., K. B. Hannsgen. 1986. Cell growth and division: a deterministic/probabilistic model of the cell cycle. J. Math. Biol. 23: 231-246. [Medline]
32. Opferman, J. T., B. T. Ober, P. G. Ashton-Rickardt. 1999. Linear differentiation of cytotoxic effectors into memory T lymphocytes. Science 283: 1745-1748. [Abstract/Free Full Text]
33. Gett, A. V., F. Sallusto, A. Lanzavecchia, J. Geginat. 2003. T cell fitness determined by signal strength. Nat. Immunol. 4: 355-360. [Medline]
34. Ma, C. S., P. D. Hodgkin, S. G. Tangye. 2004. Automatic generation of lymphocyte heterogeneity: division-dependent changes in the expression of CD27, CCR7 and CD45 by activated human naive CD4⁺ T cells are independently regulated. Immunol. Cell Biol. 82: 67-74. [Medline]
35. Ben-Sasson, S. Z., R. Gerstel, J. Hu-Li, W. E. Paul. 2001. Cell division is not a "clock" measuring acquisition of competence to produce IFN- or IL-4. J. Immunol. 166: 112-120. [Abstract/Free Full Text]
36. Chiu, C., A. G. Heaps, V. Cerundolo, A. J. McMichael, C. R. Bangham, M. F. Callan. 2007. Early acquisition of cytolytic function and transcriptional changes in a primary CD8⁺ T-cell response in vivo. Blood 109: 1086-1094. [Abstract/Free Full Text]
37. Gillis, S., M. M. Ferm, W. Ou, K. A. Smith. 1978. T cell growth factor: parameters of production and a quantitative microassay for activity. J. Immunol. 120: 2027-2032. [Abstract/Free Full Text]
38. Claret, E., J. C. Renversez, X. Zheng, T. Bonnefoix, J. J. Sotto. 1992. Valid estimation of IL2 secretion by PHA-stimulated T-cell clones absolutely requires the use of anti-CD25 monoclonal antibody to prevent IL2 consumption. Immunol. Lett. 33: 179-185. [Medline]
39. Efron, B., R. Tibshirani. 1993. An Introduction to the Bootstrap Chapman and Hall, New York.
40. Bates, D. M., D. G. Watts. 1988. Nonlinear Regression Analysis and Its Applications John Wiley and Sons, New York.
41. Armitage, P., G. Berry. 2002. Statistical Methods in Medical Research Blackwell, Oxford.
42. Hairer, E., S. P. Norsett, G. Wanner. 1993. Solving Ordinary Differential Equations in Nonstiff Problems Springer Series in Computational Mathematics, Springer-Verlag, Berlin.
43. Darzynkiewicz, Z., D. Evenson, L. Staiano-Coico, T. Sharpless, M. R. Melamed. 1979. Relationship between RNA content and progression of lymphocytes through S phase of cell cycle. Proc. Natl. Acad. Sci. USA 76: 358-362. [Abstract/Free Full Text]
44. Darzynkiewicz, Z., D. P. Evenson, L. Staiano-Coico, T. K. Sharpless, M. L. Melamed. Correlation between cell cycle duration and RNA content. J. Cell. Physiol. 100: 1979b425-438. [Medline]
45. Hawkins, E. D., M. L. Turner, M. R. Dowling, C. van Gend, P. D. Hodgkin. 2007. A model of immune regulation as a consequence of randomized lymphocyte division and death times. Proc. Natl. Acad. Sci. USA 104: 5032-5037. [Abstract/Free Full Text]
46. Doherty, P. C., J. P. Christensen. 2000. Accessing complexity: the dynamics of virus-specific T cell responses. Annu. Rev. Immunol. 18: 561-592. [Medline]
47. Rocha, B., C. Tanchot. 2006. The tower of Babel of CD8⁺ T-cell memory: known facts, deserted roads, muddy waters, and possible dead ends. Immunol. Rev. 211: 182-196. [Medline]
48. Burnham, K. P., D. R. Anderson. 2002. Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach Springer-Verlag, New York.
49. Cantrell, D. A., K. A. Smith. 1984. The interleukin-2 T-cell system: a new cell growth model. Science 224: 1312-1316. [Abstract/Free Full Text]
50. Renno, T., A. Attinger, S. Locatelli, T. Bakker, S. Vacheron, H. R. MacDonald. 1999. Cutting edge: apoptosis of superantigen-activated T cells occurs preferentially after a discrete number of cell divisions in vivo. J. Immunol. 162: 6312-6315. [Abstract/Free Full Text]
51. Boissonnas, A., B. Combadiere. 2004. Interplay between cell division and cell death during TCR triggering. Eur. J. Immunol. 34: 2430-2348. [Medline]

This article has been cited by other articles:

				V. V. Ganusov Discriminating between Different Pathways of Memory CD8+ T Cell Differentiation J. Immunol., October 15, 2007; 179(8): 5006 - 5013. [Abstract] [Full Text] [PDF]

This Article Abstract Full Text (PDF) Alert me when this article is cited Alert me if a correction is posted Citation Map Services Similar articles in this journal Similar articles in PubMed Alert me to new issues of the journal Download to citation manager Request Permissions Citing Articles Citing Articles via HighWire Citing Articles via Google Scholar Google Scholar Articles by Ganusov, V. V. Articles by De Boer, R. J. Search for Related Content PubMed PubMed Citation Articles by Ganusov, V. V. Articles by De Boer, R. J.