Implicit source (IS) model

Any population is maintained due to gain of new cells (through a source from a precursor or through proliferation) and loss of existing cells (through differentiation into another population, migration or death). Consider a population, , which is at equilibrium (denoted by ), has a source of cells/day, a per capita proliferation rate, /day and a per capita loss rate, /day (equation 1a). The kinetics of the number of cells, , the number of labelled DNA fragments, , and fraction of labelled DNA, , within this population can be described as:

(1a)

(1b)

(1c)

where

(1d)

(1e)

This model suggests that the fraction of labelled DNA can be accurately tracked if just the loss rate, , of the cells in the population is known (equation 1c). The turnover rate, , is typically identifiable, as the number of parameters to be estimated reduces from 3 to only 1, when dealing with the labelled fraction of DNA in a cell population at steady state (equation 1a vs equation 1c).

Importantly, in the equations above, the precursor population (source) is assumed to label and de-label instantaneously (equation 1b), i.e., the DNA of the cells arriving from the precursors () is assumed to follow the fraction of label in the body.

In the model, is a function denoting the probability that deuterium is incorporated into newly synthesized deoxyribose molecules ( is the level of deuterated water or glucose in plasma (or urine), and is the intracellular amplification or dilution factor [7]). For simplicity, we treat as a square pulse where the maximum is scaled to 1 so that the labelled fraction approaches 1 () for very long labelling experiments (). This simplifying assumption does not change the conclusions drawn in this study because the turnover of water and glucose is typically faster than that of the cells in the POI [4,8]. We also scale time such that the labelling period (typically denoted as days) is 1 time unit. Scaling time scales the rates of the population. For example, if the memory T-cell pool in mice (i.e., the POI) is labelled for 28 days and has an estimated turnover rate of 0.02/day, the labelling period becomes 1 (scaled unit of time, reflecting about a month) and the turnover rate of the POI becomes 0.02 x 28 = 0.56 per scaled unit of time (i.e., approximately per month). This scaling simplifies the presentation of the results considerably.

Explicit source (ES) model

While the previous model assumed that the precursor population labels and de-labels as fast as the body deuterium concentration, in fact, the dynamics of the source population itself can have a major impact. This was illustrated by a study on the dynamics of neutrophils. The lifespan of blood neutrophils was estimated to be ~ 5 days when the precursors of neutrophils (in the bone marrow) were assumed to have a fast turnover, and ~0.5 day when the precursors were assumed to be slow, underlining the importance of explicitly considering the dynamics of the source [4,9].

To quantify the effect of the source’s dynamics on the rate estimates of the POI, consider an experiment designed to find out the rates of a POI, , that has an unobserved precursor population, . The cells of the precursors, , have a source of cells/scaled time unit (stu), from their immediate precursors, they divide at a rate of /stu and are lost at a rate of /stu. A fraction of the cells leaving the precursor population mature into cells of the POI (i.e., the remaining fraction die or move elsewhere). Thus, the precursors obey the standard IS model (equation 1a). If the cells of the POI divide at a rate of /stu, and are lost at a rate of /stu, the kinetics of the precursors and the POI are written as:

(2a)

(2b)

with,

(2c)

(2d)

where and, and are the steady state values of the precursors and the POI, respectively. If the differentiation of the precursors into the POI is accompanied by one cell division, the parameter , otherwise . See the companion article [10] for a discussion on larger values of . The POI is primarily maintained by the source whenever in equation 2b, which simplifies to the quite natural condition . For simplicity, we no longer mention the units (stu) of the parameters as they are all scaled with respect to the labelling period.

The number of labelled DNA strands in the precursors, , and in the POI, , in the ES model obey:

(2e)

(2f)

Thus, if differentiation is accompanied by cell division, the POI gains label due to the inflow of labelled DNA strands from the precursors as well as due to the generation of de-novo DNA strands on replication of the labelled and unlabelled DNA strands. The kinetics of the fractions of labelled DNA strands in the precursors, , and the POI, , then follow:

(2g)

(2h)

(2i)

Transforming the system from cell numbers into fractions reduces the system to a 3-parameter model: , and (equation 2i). Note that after a long labelling period (or in rapidly turning over populations), , and that regardless of the value of ,

In practice, the POI is often measured in blood and tends to be at the end of a series of cell populations. For example, neutrophils in the blood are preceded by several cell populations in the bone marrow, e.g., mature neutrophils, banded neutrophils, metamyelocytes, myelocytes, promyelocytes, myeloblasts, and all the way up until haematopoietic stem cells. Fortunately, the ES model shows that simply measuring the fraction of labelled precursors (and not even knowing their true rates) is enough to estimate the true rates of the POI (equation 2h). Thus, one does not need to know the labelling dynamics of all the sub-populations in the entire chain from the precursor populations to the POI to find an accurate estimate of the POI’s per capita rates.

The ES model can perfectly describe the dynamics of a two sub-population kinetically heterogeneous population

We generalized the IS model above by simply introducing a source population in the ES model which allows us to model either a slow or a fast source population. The POI in the ES model, however, is still modelled as a homogeneous population. Yet, the ES model can describe a two sub-population kinetically heterogeneous population perfectly because the two sub-population kinetic heterogeneity model [6,11] turns out to be a special case of the ES model. To show this, we compare the solutions of the two sub-population kinetically heterogeneous model (Equation 3) and the ES model (Equation 4).

The solution of the two sub-population kinetically heterogeneous model [11] is:

(3)

where and are the turnover rates of the first (slow) and the second (fast) sub-populations, respectively, because we define for the fraction of the population that is made up of the slow sub-population (without loss of generality).

During the labelling phase, the solution of the ES model is:

(4a)

(4b)

(4c)

where . We, here, consider the case of the ES model because the case is the more general model and includes the case (see the sub-section division-linked differentiation below). The case is studied in the companion paper [10].

The solutions of the POI in the ES model (equation 4c) and the two sub-population kinetically heterogeneous model (Equation 3) are identical. As has a wider range than , the ES model can perfectly describe any two sub-population kinetically heterogeneous population but the converse is not always true. Further, if and , the ES model can also represent a population of stem cells, , dividing asymmetrically into themselves () and into the following population () (see equation A in S1 Text). Thus, the ES model is the more general model and can provide alternate explanations of many other mechanistic models.

Although this equality makes it difficult to recognize the true mechanism underlying a labelling curve, the ranges of and help us identify scenarios where the mechanism is unambiguous. To find scenarios where the dynamics of the ES model is identical to that of the two sub-population kinetic heterogeneity model, we compare the relationship between the parameters of the ES and the two sub-population kinetic heterogeneity models. As , we only consider , meaning that , i.e., that the POI is faster than its source population. Comparing Equations 3 and 4c, we find that , and

(5)

when and . That is, in the ES model gives the fraction of the two sub-population kinetically heterogeneous population that is made up of the slow sub-population, and the proliferation rate of the POI, , in the ES model is the same as the average turnover rate, , estimated for the two sub-population kinetically heterogeneous population. For example, deuterium labelling of CD8⁺ memory T cells [12] can be described equally well with either the two sub-population kinetic heterogeneity model (as was shown by Westera et al. [12]) or the ES model. Additional information can sometimes help to determine the choice of the model. For example, as memory T cells specific to LCMV-Armstrong are maintained by homoeostatic divisions [13], one can confidently choose a two sub-population kinetic heterogeneity model without a source over the ES model when evaluating their labelling data.

As the labelling curve of POI in both models is defined as the sum of two exponentials, the turnover rates of the precursor and the POI in the ES model have to be the same as those of the slow and fast sub-populations in the two sub-population kinetic heterogeneity model, respectively. Therefore, based on its deuterium labelling dynamics, a kinetically homogeneous population with a source cannot be distinguished from a two sub-population kinetically heterogeneous population whenever the POI is faster than its source population (i.e., when , meaning ). In cases where , one can, therefore, be certain that the population is not kinetically heterogeneous.

The labelling curve of the POI reflects its own or its precursor’s turnover rates in special cases

In the ES model, the POI can have a source (of potentially recently divided cells), can divide, and has a loss rate. Trivially, the labelling curve of the ES model is defined by the turnover rate of the POI, , if the source into the POI is negligible, i.e., if (in equation 2i). Therefore, in this section, we discuss the label gain and loss rates in two special cases of the ES model where the POI does have a significant source from the precursors: 1) a rapidly turning over precursor, and 2) a rapidly turning over, non-dividing POI. Throughout this article, we use label gain rate (and label loss rate) to refer to the rate at which a population gains (and loses) labelled DNA.

If the precursors turn over rapidly, e.g., , the unlabelled cells in the precursor population would be rapidly replaced by labelled cells. The precursors can, in such a case, be approximated by the deuterium availability in plasma, i.e., (equation 2g). The labelling in the POI (equation 2i) will then boil down to:

(6)

which is the familiar IS model, showing that the enrichment of the POI is determined by just the turnover rate of the POI, . Thus, the gain and loss rates estimated from labelling experiments where the precursors turn over rapidly (relative to the labelling period) indeed reflect the turnover rate of the POI [6]. For example, as rapidly dividing thymocytes are the precursors of slowly dividing naive T cells, the estimated lifespan should correctly reflect the true lifespan of naive T cells [14]. See Box 1 for scenarios when the estimated rates are accurate.

If the only source of label into the POI is due to the flow of labelled DNA from the precursors (i.e., and ), the labelling dynamics in the POI (equation 2i) obeys:

(7)

If the cells of the POI turn over rapidly (e.g., ), the enrichment of the POI mimics that of the precursors (Equation 7). Therefore, measuring only the POI will, in fact, reveal the turnover rate of the precursors. The neutrophils are a prime example of this case, as it is unlikely that the slow labelling rate of mature neutrophils in blood reflects their own rate. Their rate of label accrual probably reflects the slow turnover of their precursors in the bone marrow [4,9]. Note that when , most precursors become labelled upon arriving in the POI (see the companion paper by Yan et al. [10]).

The special cases can also be derived from the function describing the fraction of labelled DNA strands in the POI (equation 4b). When either the precursors or the POI are extremely fast, the fraction of labelled DNA strands in the POI boils down to or , which is only dependent on either the turnover of the POI or that of the precursors, respectively. Here, is the fraction of the POI that is replaced by cell division. These special cases showcase that the labelling curve of the POI sometimes reflects the turnover rate of the POI and sometimes that of its precursors.

The label gain and loss rates of the POI do not equal its turnover rate if the POI and the precursors have comparable turnover rates

Above we estimated the label gain and loss rates in a few extreme cases where either the source into the POI is negligible, or the precursors or the POI are very short-lived (i.e., the precursors or the POI become labelled very rapidly). In many labelling experiments, the duration of the labelling period is comparable to the POI’s lifespan, to avoid scenarios where the POI either hardly labels or becomes labelled very rapidly during the labelling period. This is important as an accurate estimation of the POI’s rates requires the collection of enough informative data points during the labelling period. As estimating the label gain and loss rates in these cases is not straight-forward, we derive first-order approximations for the POI’s label gain and loss rates.

Denoting the label gain and loss rates as and , respectively, the labelling curve of a POI can be written as (i.e., generalizing equation 1c using time-varying parameters):

(8)

The label gain rate reflects the dynamics of the entire population [5], as de-novo labelled DNA molecules are made if a cell (with either labelled or unlabelled DNA molecules) divides during the labelling phase. The label loss rate, however, reflects the loss rate of the labelled sub-population only [5], as the POI loses label only when a cell with labelled DNA is replaced by a cell with unlabelled DNA.

In the ES model introduced above, the fraction labelled DNA in the precursors is determined only by its loss rate, , whereas the fraction labelled DNA in the POI is dependent not just on its own rates but also on the rates of its precursor population (equation 2h). The rate of label gain in the POI, , during the labelling phase (, which means ) is (from equations 2h and 4b):

(9a)

(9b)

The rate of label gain depends on both the precursor and the POI, as also depends on . The initial label gain rate of the POI (i.e., when ) is:

(9c)

The initial label gain rate, , can be much slower than the true turnover rate of the POI, , if the precursors are slow (see the previous section on special cases). Therefore, it might be beneficial for our analysis to express the labelling in the POI as a function of the label gain rate, , instead of the true turnover rate of the POI. To calculate the rate of label gain later during the labelling phase, the expression of label gain rate (equation 9a) needs to be further simplified. If the label gain can be approximated well by a linear increase (i.e., if the turnover rates of the precursors and the POI are small relative to the labelling period, which is the case for well-designed experiments), the expression for the gain rate (equation 9a) can be simplified by noticing that , and . During the labelling phase (), the rate of label gain in the POI can then be approximated by:

(9d)

The approximate rate of label gain at the end of the labelling phase (when ) of the POI, then, becomes

(9e)

If we are in a regime where the gain of label in the POI can be approximated well by a straight line, these expressions become similar, i.e., (see Figs 1 and 2, and Table 1 below).

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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).

https://doi.org/10.1371/journal.pcbi.1013052.g001

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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)).

https://doi.org/10.1371/journal.pcbi.1013052.g002

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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.

https://doi.org/10.1371/journal.pcbi.1013052.t001

Additionally, the above expressions can be used to determine the behaviour of the precursors relative to the POI. If the labelling curves of both the precursors and the POI can be approximated reasonably by a straight line, the population with the faster labelling dynamics can be identified just by comparing the initial gain rate of these populations. Hence, the gain in label in the POI (during the labelling phase) is faster than that in its precursor if

(10)

As opposed to the rate of label gain, , which is defined on the total population, the label loss rate, , is defined on the labelled fraction (see Equation 8). So, the rate of label loss in the POI, , in the de-labelling phase (i.e., when in equation 2h) is:

(11a)

(11b)

Again, as and , the above expression simplifies to

(11c)

It is important to keep in mind that the approximated downslope is accurate for only a short period of time in the de-labelling phase as the approximations of and are not accurate for .

As in some cases, for example when and , we use the approximation for defining . Thus, the initial rate at which label is lost in the beginning of the de-labelling phase is

(11d)

where is small.

The approximations of the gain and loss rates of label in the POI show that these rates are not dictated only by the turnover rate of the POI but also by the division rate of the POI and the turnover rate of the precursors. Next, we use the above simplified expressions for the gain and loss rate of label in the POI to find qualitative relationships between the estimated and true label gain or loss rates.

The rate of label uptake in the POI is generally lower than its true turnover rate

Models that consider an implicit source (like the IS model above in Equation 1 or the implicit kinetic heterogeneity model [5]) predict that, in a population that is at steady state, the rate of label gain in the POI, , represents the average turnover rate of the POI ( in the IS model and in the implicit kinetic heterogeneity model [5]). To test the validity of this prediction, we graphically compare the label gain and loss rates of the POI found from the analytical predictions (equations 9c–e and 11c) and from numerical estimations, using the implicit kinetic heterogeneity model (Equation 8 with and instead of and , respectively) with actual labelling curves of the ES model (Figs 1, 2 and Fig A in S1 Text and Table 1) in different scenarios. Similar conclusions can be drawn when is not modelled as a square pulse, but instead reflects the slower gain and loss of deuterium enrichment in body water that is typically observed during deuterated water labelling experiments (Figs B-D in S1 Text).

Case II: The turnover rates of the precursors and the POI are not comparable.

If the POI is much faster than the precursors, the label gain rate approaches the turnover rate of the precursors, which was highlighted above as a well-known special case [4,9]. Note that the approximated slopes describe the labelling curve for a short period of time, as linear approximations do not give good descriptions of and when (see Fig A and Table A in S1 Text). Similarly, if the precursors are faster than the POI, the label gain rate in the POI approaches its own turnover rate as the label gain rate in the POI cannot be higher than its turnover rate, which was also reported above as a special case. Therefore, the approximated rates give a good description of the gain and loss of label in a population.

The label gain rate can maximally be the turnover rate of the POI. Intuitively, the label gain rate has to be lower than the true turnover rate of the POI as unlabelled DNA strands cannot be replaced faster than the rate at which the cells of the POI are replaced. The examples above confirmed this intuition. Here we prove that the rate at which the POI gains label is invariably lower than the POI’s turnover rate..

The label gain rate, , (equation 9b) can be re-written as:

(12a)

where, , and .

If denotes the timepoint where is at its maximum (see equation B in S1 Text for details), then the upper bound of the label gain rate, , is:

(12b)(12c)

as when , which is always true. Thus, the rate at which the POI gets labelled is always lower than its turnover rate, . Interpreting the label gain rate as the true turnover rate would, therefore, overestimate the POI’s lifespan whenever the precursors are not very fast and/or play a significant role in the maintenance of the POI.

The initial label loss in the POI is accurately predicted

We have shown that the calculated label gain rates predict the enrichment in the precursors and the POI accurately in most cases. As the relative enrichment of populations at the end of the labelling period influences the rate at which label is lost (Equation 11b), the initial rate at which label is lost, , can be approximated well by using the estimated gain rates (equation 11c).

If the lifespans of the POI and the precursors are comparable to the labelling period, the estimated gain and loss rates give a good description of the labelling curves (see the red and green arrows in Figs 1 and 2). The estimated loss rates predict whether the POI start to lose or continue to accrue label at the end of the labelling phase (Table 1). If the precursors are more enriched than the POI, the labelling of the POI continues after the labelling period in most cases (4 out of 5 cases, see Figs 1a, 1b, 1d and 2b). In these cases, the gain of label in the POI is primarily due to the inflow of highly labelled precursor cells (also see the subsection Timing of the peak). In the other four cases (see Figs 1c, 2a, 2c and 2d), the label gain in the POI was driven by proliferation.

Note that even if the cells in the POI are very short-lived, the estimates of the initial loss rate accurately informed on whether the POI gained or lost label after the labelling period (Table A in S1 Text). So, the estimated rates do provide a faithful description of the de-labelling curve, but only for a very short period of time (Fig A in S1 Text). Thus, the predicted label gain rate, , can be used to find a good approximation of the initial loss rate (equation 11c).

The POI can have four different labelling behaviours

Unlike the IS model where the labelling of the POI is determined only by the death rate of the POI (Equation 1c), in the ES model the POI can have four qualitatively different slopes at which the POI gains and loses label (summarized in Table 2). The labelling of the POI is determined by:

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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.

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1. 1). its own turnover rate, , if the precursors turn over rapidly. In this case, the labelling in the precursors approximates the label availability in the plasma. The ES model approximates the IS model, and the rates of labelling and de-labelling are primarily driven by the POI’s loss rate. An example of this case are naive T cells, that have the rapidly dividing thymocytes as their precursors [14]. Note that if , the ES model approximates the IS model (see the companion paper by Yan et al. [10]).
2. 2). the turnover rate of the precursors, , if the POI is fast. An example would be mature neutrophils in blood, which are thought to mimic the labelling curve of their precursors in the bone marrow [4,9]. The POI, here mature neutrophils, are probably much faster than their precursors in the bone marrow. They attain the enrichment of the precursors as they are replaced (by the cells flowing in from the precursors, post-mitotic pool) much faster than the change in the enrichment of the precursors.
3. 3). its division rate, , if differentiation is not accompanied by division (). This is because the increase in enrichment due to the division in the POI would initially be the major contributor compared to the label flowing in from the precursors. Note that the precursors and the POI should have lifespans that are comparable to the length of the labelling period. A likely example for such a case would be central memory T cells undergoing homeostatic differentiation into effector memory T cells [16,17].
4. 4). the average of its division and death rates, , if the precursors go through division-linked differentiation (). Here, along with the label gain due to the division in the POI, the source from the precursors also has a significant contribution of labelled cells as the precursor cells also divide and pick up label when they differentiate into the POI. Thus, if a naive T cell were to undergo division while differentiation into memory T cells [18], the gain of label by memory T cells would reflect both its division and death rates.

These four different labelling behaviours can be distinguished from each other if the turnover rates of the precursors is known. Therefore, it is essential to have information on the precursors’ dynamics. Since the labelling in the POI depends on the enrichment of its immediate precursors only (equation 2h), it is sufficient to have knowledge or data on the immediate precursors (and not the precursors’ precursor) to distinguish the rates of the POI from that of its precursors. So, even a phenomenological description of the precursors’ labelling curve, , would suffice to correctly estimate the turnover rate of the POI. Therefore, it is essential and sufficient to measure and model the immediate precursor population of the POI to reliably interpret the estimated rates.

Numerical tests reveal the range of variations around the identified parameter relationships that is introduced by fitting simple models to data

Up until now, we have identified relationships between the parameters of the ES model (for example, and ) and the parameters of the phenomenological model ( and ), by analytically comparing equations. Next, we aimed to take the measurement noise in the labelling data and uncertainties introduced by fitting simplified models to that data into account.

We systematically varied the values of the three parameters of the ES model (, and ) to generate a 1000 different parameter sets. Each parameter could have 10 different values: , where . These 1000 parameter sets were then used to make artificial data using the ES model (having three parameters). This dense and perfect data was subsequently fitted with the implicit kinetic heterogeneity model (Equation 8, having two parameters, and ). As we are interested in the average turnover rate, the estimated value of was plotted against the true values of the three parameters of the ES model (Figs 3 and 4).

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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.

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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.

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To test the analytical prediction of the label gain rate (equation 9c), we also plotted the estimated values against , which denotes the initial label gain rate of a population. Note that in the companion paper [10] is referred to as the quantity ‘production by division’, which combines potential division-linked differentiation of the precursors with the proliferation within the POI. Production by division is defined as (i.e., the coefficients of in equation 2h), which is equal to .

If differentiation is not accompanied by cell division (i.e., ). The relationships observed in the numerical simulations (Fig 3) tend to be in line with our analytical results. The estimated label gain rate, , is always lower than the true turnover rate of the population of interest, (Fig 3b), and is approximately equal to the true proliferation rate of the population of interest, (Fig 3c). The initial label gain rate, , is equal to when , and hence Fig 3c and 3d are identical. The numerical results do not always match the analytical ones, as the relationship between and has a wide range for low values of , and differs by more than 2-fold for many parameter sets (the dashed line indicates the relationships and in Fig 3c and 3d, respectively). This shows that fitting a simplified model to the labelling data can introduce error in the value of the estimated parameters.

In Fig 3, the implicit kinetic heterogeneity model was fitted to a dense, artificial dataset that did not have any experimental noise. However, true deuterium labelling data are never free of noise and are not as dense as our artificial data. Reassuringly, when we used a dense dataset (Fig G in S1 Text, top row), or a sparse dataset (Fig G in S1 Text, bottom row), or when we added noise to the data (Figs H and I in S1 Text), the relationships between the parameters did not change.

Since the data points in these figures above are coloured by the quantile (based on the SSR of the best fits) that those fits lay in, we can see that the deviations from the predictions are largely due to the fits in the bottom 50% quantile (see Figs G and I in S1 Text). However, as a quantile is a relative measure, this does not necessarily mean that the fits in the bottom 50% quantile poorly describe the data (see Fig J in S1 Text). Thus, this numerical exercise not only confirms the analytical results, but also identifies parameter values (for example, of ) where the estimated quantity () is expected to have a large error.

If differentiation is accompanied by cell division (i.e., ). The numerical results also tend to be in line with the analytical results when the precursors go through division-linked differentiation into the POI. The estimated label gain rate, , always underestimates the true turnover rate of the POI, , as in the case (Fig 4b). However, unlike the case, always overestimates the true proliferation rate of the POI, (Fig 4c). This was bound to happen as the estimate of takes both the division-linked differentiation of the precursors and the proliferation within the POI into account. The initial label gain rate, , is approximately equal to the estimated (Fig 4d). See Fig J in S1 Text for representative fits to dense or sparse datasets.

Qualitatively, Fig 4a–b look similar to Fig 3a–b, whereas Fig 4c–d seem to be a subset of Fig 3c–d. Thus, it appears that the case is a subset of the case. We will analytically confirm this below. Importantly, in both cases, the estimated is consistently well-captured by the parameter , meaning that we can always estimate from deuterium labelling data using the simple implicit kinetic heterogeneity model. However, the true turnover rate of a POI, , cannot be inferred from as the value of depends on both and , which weakens the significance of estimating this identifiable parameter.

Danger zone of deuterium labelling interpretation

The incorporation of deuterium in the POI is dependent on various factors: the rate at which the precursors are replaced, whether precursors undergo cell division when differentiating into the POI, the rate of replacement of cells in the POI, the rate at which cells in the POI divide, and the length of the labelling period. The most important comparison among these factors that define the interpretation of the deuterium labelling data are the replacement rates (also referred to as turnover rates, and defined as the reciprocal of the expected lifespan) of the POI and the precursors. To better understand this, let us consider the following cases (see Fig 5 below):

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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.

https://doi.org/10.1371/journal.pcbi.1013052.g005

1. Short-lived precursors: If the precursors are so fast that the labelling levels of the precursors reflect the body deuterium levels, the rate at which the POI gains label reflects the true turnover of the POI. As an example, the lifespan estimates of naive T cells in the literature are likely to be accurate as it is well-known that short-lived thymocytes are the precursors of naive T cells. The point (, ) is annotated with an ‘N’ as the estimated lifespan of naive T cells in humans is ~ 80 months (6.5 years) and that of thymocytes is in the order of days based on a 2-months long deuterium labelling experiment [14].
2. Short-lived POI: If the cells in the POI are replaced more quickly than their precursors, the estimated turnover rate of the POI reflects the turnover rate of its precursors (provided that cells in the POI are not dividing). The case of neutrophils is an ideal example here (annotated at (, ) as ‘G’ for granulocytes as the most widely accepted estimated lifespan of blood neutrophils in humans is ~ 12 hours and that of neutrophil precursors is ~ 0.5 week (84 hours) based on a 6-hour deuterium labelling experiment [4]). The rate at which deuterium is lost from blood neutrophils may either reflect the rate at which blood neutrophils are replaced or the rate at which their precursors in the bone marrow turn over [4,9]. The current dogma dictates that the precursors in the bone marrow have a slower turnover rate than blood neutrophils. However, in the absence of any information on the dynamics of the precursors in bone marrow, it remains impossible to estimate the lifespan of blood neutrophils.
3. All other situations: If none of the populations (i.e., the precursor or the POI) is much short-lived than the other, the estimated turnover rate reflects a combination of the POI’s and precursor’s rates (see Table 2 in the manuscript for the approximations). It is likely that most cell populations fall in this category. For example, deuterium labelling curves of memory T cells (annotated at (, ) as ‘M’ as the estimated lifespan of memory T cells in humans is ~ 6 months, with naive T cells as their most likely precursors, based on a 2-months long deuterium labelling experiment [12]) typically show that a considerable fraction of cells remains unlabelled even after four weeks of labelling. Furthermore, there is still no consensus on the precursors of memory T cells in different sub-populations, making it difficult to interpret the turnover rates estimated from the deuterium labelling curves of memory T cells. The (current) best known precursors of memory T cells are naive T cells, which replace a small portion of the memory T-cell pool every month, both in mice [18] and in humans (in preparation)

The colours (in Fig 5) represent how wrong the estimated turnover rate could be if the turnover rate of the precursors would not be taken into account. Unfortunately, the danger zone is much larger than the safe zone and most studied cell populations lie in the danger zone. Our results show that misinterpreting the rate at which the POI gains label as its turnover rate could lead to a drastic (i.e., up to 7-fold) overestimate of the POI’s lifespan (see Fig 6). It is, therefore, important to carefully re-examine existing lifespan estimates of different cell populations by taking the dynamics of their precursors into account.

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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].

https://doi.org/10.1371/journal.pcbi.1013052.g006

The peaks of the labelling curves can help distinguish scenarios

Timing of the peak.

Interestingly, it is possible to infer some properties of the precursors and the POI just by comparing both the height and the time of their peaks. The POI reaches its peak enrichment either precisely at, or sometime after, the end of the labelling period. For a model with random cellular processes (i.e., an ODE-based model), the peak of the POI will be at the end of the labelling phase when the precursors flowing into the POI have lower enrichment than the cells in the POI, i.e., (from Equation 10). If precursors differentiate without dividing (i.e., when ),

(13)

i.e., the peak will be at the end of the labelling phase when the division rate of the cells in the POI is higher than the loss rate of the precursors. Instead, if the differentiation of the precursors is accompanied by division (i.e., when ),

(14)

meaning that the peak will be at the end of the labelling phase when the division rate of the POI is either higher than the loss rate of the precursors, , or is lower than the loss rate of the precursors, and the loss rate of the precursors is lower than half of the loss rate of the POI, . Summarizing, the peak of the POI is at the end of the labelling period only when the enrichment of the POI is largely due to division (within the POI or by differentiating precursors).

There are two corollaries of the above equations (Equations 13 and 14). One, a non-dividing POI always achieves its peak labelling after the stop of labelling. Two, the peak of the POI will be after the end of the labelling phase if the precursor cells flowing into the POI at and after the end of the labelling phase have a higher enrichment than the cells in the POI, i.e.,

(15)

Thus, the peak of the POI is after the end of the labelling phase only when non-dividing precursors differentiating into the POI are the major contributor of label in the POI. Therefore, the population architecture underlying a labelling curve can be partially realized by comparing the location of the peaks of the POI and its precursors (provided the dataset is sufficiently dense).

Division-linked differentiation.

To further pinpoint the properties of the precursors and the POI, it is important to be able to distinguish between labelling curves where differentiation is accompanied by division from those where it is not. Unfortunately, whether a population goes through division-linked differentiation is, in general, unidentifiable (i.e., curves with and are indistinguishable). To demonstrate this, we compare the labelling equations of the POI when and (equation 2h). Denoting the division rate of the POI, , as when , and as when , we find

(16a)(16b)(16c)

where . Therefore, the labelling curve of any population whose precursor goes through division-linked differentiation can also be explained by a scenario where the precursors differentiate without division (see Table 1 for examples). On the other hand, the labelling curve of a population whose precursors do not go through division-linked differentiation cannot be described with if more than half of the production of the POI is due to the source, i.e., if . Thus, one can conclude that the influx into the POI is not accompanied by division if in the estimates of the best fit. In a scenario where it is not known whether differentiation is linked with division, it is safe to conclude that the POI is primarily maintained by self-proliferation only if . Fortunately, the above transformation (Equation 16) does not affect the turnover rate of the POI, . Therefore, the turnover rate of the POI is identifiable if (and only if) the labelling in the precursors is known.

Intersecting labelling curves.

Consider a simple scenario where precursors differentiate without division into a non-dividing POI, i.e., and (see above). We have seen above that the peak enrichment in the POI will then be reached after the end of the labelling phase (where ). More importantly, the peak occurs when (Equation 4). Conversely, if when , equation 2i boils down to

(17)

As takes only integer values and , the only value that satisfies the identity in Equation 17 is , implying . This provides a cardinal property of a chain of populations:

(18)

i.e., if during the de-labelling phase, a population reaches its peak when its labelling curve intersects that of its precursors, then this is proof that the POI is not dividing, and that its precursors differentiate into the population without division. This immediately implies that such a population () has a lower peak than its precursor.

In certain cases, it is possible to infer the model architecture from the labelling curves. For example, it is safe to conclude that the cells in a population are not governed by random processes if the peak of the population is higher than its precursor and the peak is reached after the end of the labelling period (see Fig K in S1 Text, equations C-D and the associated text in S1 Text).

Case study confirms that the underlying maintenance mechanisms can have enormous effect on the estimated turnover rate of the POI

The ES model can explain a wide variety of labelling curves, including the curves generated by the IS model. Both and labelling behaviour can be explained by the ES model (Table 2). This generality of the ES model is troubling, however. Most current estimates of cell lifespans are based on IS-like models that do not consider a precursor population, and the estimated rate at which label is gained is generally interpreted as the average turnover rate of the POI [5,6]. Since in the ES model, the same estimate could be reflecting the rate of the precursors (), of the POI (, ), or anything in between, this questions some of the current interpretations. We illustrate this problem with an example.

Labelling of CD57⁺ memory T cells by Ahmed at al. (2020)

A dataset that seems uniquely suited for our analysis was generated by Ahmed et al. (2020) who measured the label gain and loss in both the POI, i.e., CD57⁺CD4⁺ memory T cells, and their precursor population, i.e., CD57^-CD4⁺ memory T cells [15]. To find out the dynamics of the POI, they used a phenomenological chain of two populations describing both the POI and the precursors (see [15] for their model). Their analysis showed that both the precursors and the POI were replaced rapidly (on average every 2 days) and that the POI was maintained largely (~ 95%) by division rather than by the source from the precursors (~5%). Hence, the maintenance of the POI and its precursors were considered to be independent.

To re-analyse these data, we considered three different scenarios (Fig 6): modelling only the POI (Fig 6a); the POI and its precursors (Fig 6b); and the POI, its precursors and the precursor’s precursor (Fig 6c). The first population in each of these scenarios (i.e., CD57⁺CD4⁺ in Fig 6a, CD57^-CD4⁺ in Fig 6b and the precursors of CD57^-CD4⁺ in Fig 6c) is always described with the two sub-population kinetic heterogeneity model (Equation 3). The other populations (i.e., CD57⁺CD4⁺ in Fig 6b, and both CD57^-CD4⁺ and CD57⁺CD4⁺ cells in Fig 6c) were modelled with the ES model (equation 2h).

The labelling of CD57⁺CD4⁺ cells was well-described in all cases. However, the estimated rates were very different depending on the model used to describe the data and whether its precursors were explicitly modelled. The CD57⁺CD4⁺ population was estimated to have a 6.67-day lifespan when the population was described as a two sub-population kinetically heterogeneous population, which was ~ 3-fold longer than when the POI was described as a homogeneous model with a partially labelled precursor population (Fig 6a versus Fig 6b and 6c, Table 3). Similarly, the estimated lifespan of CD57^-CD4⁺ cells was 7-fold longer if the model was not given the freedom to opt for a partially labelled precursor population (Fig 6b versus 6c, Table 3). Note that the description and the estimates of the CD57⁺CD4⁺ cells were identical for the two cases in which the population was described with the ES model. This is because the description of the label in precursors (CD57^-CD4⁺ cells) was identical, even though the estimated rates for the precursors were different (Fig 6b and 6c). Thus, only knowing the labelling dynamics of the precursors (without knowing its true turnover rate) is enough to find a reliable estimate of the POI’s true turnover rate.

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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.

https://doi.org/10.1371/journal.pcbi.1013052.t003

One of the primary takeaways of the original paper was that most of the CD57⁺CD4⁺ population was replaced by division. In contrast, the estimates found from the fit of the ES model to the labelling data of CD57⁺CD4⁺ cells suggest that only 31% of the population is replaced due to division ( in Fig 6b and in Fig 6c), which is very different from the original estimate of 93% (for individual DW02), illustrating that mechanistically allowing for precursors can radically affect the biological interpretation of the data.

The precursors (and the precursor’s precursor) in the above fits were considered to differentiate without division into the POI (and into the precursors), i.e., . Since both the CD57^-CD4⁺ (~86%) and the CD57⁺CD4⁺ (~69%) memory T-cell populations were estimated to be largely maintained by the source (Table 3), our analytical derivation in the section Division-linked differentiation implies that this data cannot be described well with a model where differentiation of precursors is accompanied by cell division. A poor description of the data by the model with confirms our analytical results (Fig L in S1 Text). In contrast to the original article, which suggested a clonal burst during the differentiation of CD57^-CD4⁺ cells into CD57⁺CD4⁺ cells, the ES model suggests that precursors merely differentiate (i.e., without division) into the POI. Therefore, this case study also provides an example where only the deuterium labelling data is enough to deduce whether a population undergoes division-linked differentiation.

The above example nicely illustrates that the estimated parameters can strongly depend on the underlying model, and hence that one should always test whether the estimates change when a precursor with a realistic turnover rate is added. The estimates found from the ES model, in this example, could be realistic as they are in line with previous estimates stating that as much as a quarter of the CD4⁺ memory T-cell population could be maintained by a source from naive T cells [18].