Fig 1.
The rate of label gain in the POI is lower than the POI’s turnover rate,
, if the precursors turn over faster than the POI. The filled circles mark the timepoints at which the slopes were calculated (note that unlike
and
,
is constant over time). A small offset was added to the slopes for better visualization. The slopes denoting initial,
(equation 9c), and final,
(equation 9e), label gain rates of the POI were given a small negative offset, while those denoting the true turnover rate,
, and the initial label loss rate,
(equation 11d), of the POI were given a small positive offset. The slopes were calculated as the label gain or loss rate multiplied by the total, i.e., one, or the fraction of labelled cells (i.e.,
), respectively, for example,
. The loss rates of the precursors,
, and of the POI,
, were set to 0.5 and 0.2, respectively. The division rate of the POI,
, was either set to 0.1 (in (a) and (c)) or to 0 (in (b) and (d)). Note that the dynamics in (a) and (d) are identical (see the sub-section Division-linked differentiation below for the explanation). In these simulations, the body deuterium concentration was described as a step function (equation 1e).
Fig 2.
The rate of label gain in the POI is also lower than the POI’s true turnover rate,
, if the precursors turn over more slowly than the POI. The filled circles mark the timepoints at which the slopes were calculated (note that unlike
and
,
is constant over time). A small offset was added to the slopes for better visualization. The slopes denoting the initial label gain rate,
(equation 9c), and the final label gain rate,
(equation 9e), of the POI were given a small negative offset, while those denoting the true turnover rate,
, and the initial label loss rate,
(equation 11d), of the POI were given a small positive offset. The slopes were calculated as the label gain or loss rate multiplied by the total, i.e., one, or the fraction of labelled cells (i.e.,
), respectively, for example,
. The loss rate of the precursors,
, and the POI,
, were set to 0.2 and 0.5, respectively. The division rate of the POI,
, was either set to 0.25 (in (a) and (c)) or to 0 (in (b) and (d)). Note that the dynamics in (a) and (d) are identical, which can also be seen in Table 1 (see the sub-section Division-linked differentiation below for details). In these simulations, the body deuterium concentration was described as a step function (equation 1e). Note the large difference in the labelling curves of a non-dividing POI depending on whether differentiation is accompanied by cell division (panels (b) and (d)).
Table 1.
The true proliferation and turnover rate (,
, and
), and the predicted label gain and loss rates (i.e.,
,
, and
using Equations 9c, 9e, sand 11d) corresponding to the simulations shown inFigs 1 and 2. The parameters (
and
) of the best fits (shown in Figs E and F in S1 Text) of the phenomenological model (Equation 8) to the labelling data of the POI. The estimated label gain rate initially (see
) reflects the proliferation rate of the POI (when
) and then changes to reflect the influence of the precursor’s rate (captured by
). The estimates of
and
are not reported for the cases in which the phenomenological model fails to describe the data (see Figs E and F in S1 Text). Note that the estimated loss rate,
, gives the rate at which the labelled population loses labelled DNA soon after the end of the labelling period (
is a very small positive number). The negative sign indicates that the population, in fact, gains labelled DNA for a short period after the end of labelling (see Equation 8). The rates (expressed as stu) are scaled with respect to the labelling period.
Table 2.
Summary of the approximations for the label gain and loss rates in the POI provided here for different scenarios. A POI with a precursor population has four qualitatively different gain and loss rate scenarios. These approximations underscore how crucial it is to know the precursor’s dynamics in identifying the true turnover rate of the POI. The special cases are derived in Equations 6 and 7, while the regular cases are derived in Equations 9 and 11.
Fig 3.
The estimated turnover rate of the POI underestimates its true turnover rate and reflects its true division rate when precursors differentiate without division into the POI.
The above plots show the distribution of the estimated gain rate, , as a function of the true rates of the precursor population and the POI when
. The solid and dashed lines represent slopes of 1 and 2, respectively. The lines with the slope of 1 and 2 show the line where the value of
is equal to and twice that of the parameters on the x-axis, respectively. The colours and shapes show four different groups of quantiles calculated based on the sum squared residual (SSR) of the 1000 best fits. For example, the dark red diamonds show the 0-25% quantile group, which are the top 250 best fits among the 1000 best fits.
Fig 4.
The estimated turnover rate of the POI underestimates its true turnover rate and overestimates its true division rate when precursors undergo division-linked differentiation.
The above plots show the distribution of the estimated gain rate, , as a function of the true rates of the precursor population and the POI when
. The solid and dashed lines represent slopes of 1 and 2, respectively. The lines with the slope of 1 and 2 show the line where the value of
is equal to and twice that of the parameters on the x-axis, respectively. The colours and shapes show four different groups of quantiles calculated based on the sum squared residual (SSR) of the 1000 best fits. For example, the dark red diamonds show the 0-25% quantile group, which are the top 250 best fits among the 1000 best fits.
Fig 5.
The danger and safe regions in the space defined by the turnover rates of the POI and the precursor populations, along with annotations (in white) of a few well-known cell populations (N: naive T cells, G: granulocytes/neutrophils, M: memory T cells).
This heatmap was generated using the approximation of the label gain rate at the end of the labelling period as derived in this manuscript (i.e., with
from equation 9b). The deuterium labelling curve of any cell population outside the dark blue region must be analyzed along with that of its precursors to accurately estimate the POI’s rates.
Fig 6.
Similar description of the deuterium labelled fraction of CD57
+CD4+ T cells based on different models despite radically different estimated rates. The above plots show the best fits of the two sub-population kinetic heterogeneity model (a), the two sub-population kinetic heterogeneity and the one population ES model (b), and the two sub-population kinetic heterogeneity and the two population ES model (c) to the deuterated-water labelling data of CD57-CD4+ and CD57+CD4+ memory T cells of individual DW02 from Ahmed et al. (2020) [15]. See S1 Text for the system of equations (equation E). The labelling data of the cell populations and the data of the body deuterium concentration were digitized from the original article for re-analysis. The parameters describing the deuterium concentration in the body water of individual DW02 are ,
/day,
, where
is the predicted asymptote of deuterium enrichment in body water,
is the estimated turnover rate of water and
is the estimated initial deuterium concentration in body water. See [15] for the equations. The above fits were based on the assumption that cells differentiate from one population into the other without division, i.e.,
. Note that the vertical axis gives the non-normalized deuterium enrichment (APE; atom percent excess) measured in the population, as was reported in the original work [15].
Table 3.
Parameter estimates of the best fits shown in Fig 6. The quantities ,
and
are the parameters of the two sub-population kinetic heterogeneity model (Equation 3). The average turnover rate is denoted by
and is calculated as
. The quantities
,
,
and
are the parameters of the ES model (equation 2h). The rates are reported in per day units. The AICs of the single population (a), two populations (b), and three populations (c) fits are -78.80, -261.47, and -257.47, respectively. The quantity of interest, i.e., the estimated turnover rate of CD57+CD4+ memory T cells, for the three scenarios is shown in bold.