Figure 1.
A cartoon of the model with explicit kinetic heterogeneity.
In the model, the population of cells consists of sub-populations with different rates of turnover. In the
sub-population, there is a source of new cells that enter the cell population at rate
cells per day, cells divide at rate
per day, and die at rate
per day. To maintain the size of all sub-populations constant,
for every sub-population
, where
is the number of cells in the
sub-population. In this model we assume that the source produces only labeled cells during the labeling phase, and delabeled cells during the unlabeling phase [11].
Figure 2.
Model predictions for exponentially distributed turnover rates.
We have plotted the changes in the fraction of labeled DNA according to the explicit kinetic heterogeneity model with exponentially distributed turnover rates (eqn. (5), mean day). Predicted changes are shown for a short labeling period (
day, solid line) and a long labeling period (
days, dashed line) on a linear (panel A) and a logarithmic (panel B) scale. The initial uplabeling rate is independent of the length of the labeling period and is given by
. The initial rate of delabeling, in contrast, depends on the length of the labeling period and is approximately twice as fast in the case of short-term labeling as compared to long-term labeling.
Figure 3.
Model predictions for gamma-distributed turnover rates.
We have plotted the changes in the fraction of labeled DNA according to the kinetic heterogeneity model with gamma-distributed turnover rates (eqn. (4)) with average turnover rate /day on a linear (panel A) or logarithmic (panel B) scale. Predicted changes are shown for different values of the shape parameter
. Larger values of
correspond to a more symmetric distribution (Panel C). For low values of the shape parameter
, the loss of labeled DNA after label cessation is biphasic, which is most clearly visible on a logarithmic scale for
(panel B). This characteristic of the kinetic heterogeneity model differs from the Asymptote models which have a constant per capita rate at which labeled DNA is lost. Note that for shape parameters
, the distribution of turnover rates
becomes extremely skewed with most cells undergoing hardly any division and relatively few cells undergoing extremely many rounds of division (panel C). Panel D gives the cumulative contribution of sub-populations with a particular turnover rate
to the average rate of turnover of the population
. The vertical line shows the value of the average proliferation rate
. For high values of the shape parameter (
), the cell sub-populations with turnover rates that are somewhat lower or higher than
give the main contribution to the average turnover rate. In contrast, for low values of
(
), the major contribution to the average turnover rate comes from sub-populations with extremely rapid turnover rates (
); about 50% of the average turnover is due to a few sub-populations with turnover rates that exceed 10 per day, which is biologically unrealistic.
Figure 4.
Fits of artificial (simulated) data with various models.
We have fitted artificial data (black dots) with the Asymptote model (eqn. (1), solid black lines), the Exponential model, in which a fraction of the cells have exponentially distributed turnover rates (eqn. (8), small red dashed lines), and the Gamma model with gamma distributed turnover rates (eqn. (4), large green dashed lines). Data were generated using the Gamma model (panel A&B), the Exponential model (panel C&D) and the Two-populations model (panel E&F, Eqn. (2)), respectively. Thin blue lines show the exact curves of the models that were used to generate the data. The different models were fitted to 11 datapoints taken from these predicted curves after having added noise to these data points. Noise was added by a relative change of the predicted value with a normally distributed error (with standard deviation of the distribution
). The models were fitted to data from artificial labeling experiments in which the label was administered for 7 (left panels) or 15 (right panels) days. Parameter estimates providing the best fit are shown in Table S1, and the corresponding estimates of the average rates of cell turnover
are shown in Figure 5. Parameters used to generate the data are also given in Table S1.
Figure 5.
Average turnover rate estimated obtained by fitting mathematical models to simulated data.
By fitting the artificial data described in the text, we estimated the average turnover rate using three models: the Asymptote model (eqn. (1)), the Exponential model (in which a fraction of cells have exponentially distributed turnover rates, eqn. (8)), and the Gamma model (with gamma-distributed turnover rates, eqn. (4)). Estimated mean values and 95% confidence intervals obtained by bootstrapping the residuals with 1000 simulations are shown. Data were generated using the Gamma model (empty bars), the Exponential model (gray bars) and the Two-populations model (black bars). Labeling periods were 7 (panel A) and 15 (panel B) days. Horizontal dashed lines denote the actual average rate of lymphocyte turnover in all data,
/day. Note that in this example, the Asymptote model always underestimated the average rate of cell turnover, and that there is a systematic 2-fold underestimation of the average turnover by all models when the data from the Two-populations model were fitted. This is because all three models fail to describe the relatively rapid accumulation of the label at early time points (see Figure 4E–F).
Figure 6.
Fits of the deuterium labeling data with mathematica models.
Data on labeling of CD4+ (top rows) and CD8+ (bottom rows) T cells in four healthy humans were fitted by three models: the Asymptote model (panels A and D), the Exponential model, in which a fraction of cells have exponentially-distributed turnover rates (eqn. (8), panels B and E), and the Gamma model, with gamma-distributed turnover rates (eqn. (4), panels C and F). Experimental data obtained from Mohri et al. [2] are shown as symbols and the curves are the best model fits. The sum of squared residuals of the model fits to the data on the dynamics of CD4+ T cells are
for the Asymptote model, the Exponential model and the Gamma model, respectively. The sum of squared residuals of the model fits to the data on the dynamics of CD8+ T cells are
for the Asymptote model, the Exponential model and the Gamma model, respectively. Note that the two explicit kinetic heterogeneity models describe these data with similar (Exponential model) or even better (Gamma model) quality compared to the Asymptote model.
Figure 7.
Estimates of the average turnover rates of human T cells.
Data on deuterium labeling of CD4+ (panel A) and CD8+ (panel B) T cells in four healthy humans were fitted with three different models: the Asymptote model (empty bars), the Exponential model (gray bars), in which a fraction of the cells have exponentially-distributed turnover rates, and the Gamma model (black bars) with gamma-distributed turnover rates. Best fits of the data are shown in Figure 6, and estimates of all parameters of the models are shown in Table S2 and S3. Confidence intervals were obtained by bootstrapping the residuals with 1000 simulations. Note that all models deliver very similar estimates for the average turnover rate
, with the exception of the estimated CD8+ T-cell turnover rates in individual c1 which are highly model-dependent.
Table 1.
Summary of the major findings of the paper.