Towards a unified model of naive T cell dynamics across the lifespan
Abstract
Naive CD4 and CD8 T cells are cornerstones of adaptive immunity, but the dynamics of their establishment early in life and how their kinetics change as they mature following release from the thymus are poorly understood. Further, due to the diverse signals implicated in naive T cell survival, it has been a longheld and conceptually attractive view that they are sustained by active homeostatic control as thymic activity wanes. Here we use multiple modelling and experimental approaches to identify a unified model of naive CD4 and CD8 T cell population dynamics in mice, across their lifespan. We infer that both subsets divide rarely, and progressively increase their survival capacity with cell age. Strikingly, this simple model is able to describe naive CD4 T cell dynamics throughout life. In contrast, we find that newly generated naive CD8 T cells are lost more rapidly during the first 3–4 weeks of life, likely due to increased recruitment into memory. We find no evidence for elevated division rates in neonates, or for feedback regulation of naive T cell numbers at any age. We show how confronting mathematical models with diverse datasets can reveal a quantitative and remarkably simple picture of naive T cell dynamics in mice from birth into old age.
Editor's evaluation
This paper challenges the widely held view that the number of naive T cells in our body is regulated through homeostatic feedback mechanisms, meaning that cells divide more frequently – or live longer – when cell numbers are low. The arguments in favor of this homeostatic regulation rely on crosssectional data, which fail to distinguish between the effects of host age, cell age, and cell numbers, each of which separately may influence the dynamics of cells. Using a set of mathematical models and experimental data sets, this paper manages to tear these factors apart and reports that in mice there is no feedback regulation of naive T cell numbers.
https://doi.org/10.7554/eLife.78168.sa0Introduction
Lifelong and comprehensive adaptive immunity depends upon generating naive CD4 and CD8 T cell populations with diverse repertoires of T cell receptors (TCRs). These must be established rapidly from birth and then maintained throughout life. In mice, the number of circulating naive T cells grows from tens of thousands at birth to tens of millions in several weeks, peaking at around 2 months of age (Scollay et al., 1980; den Braber et al., 2012) and waning thereafter. Quantifying the relative contributions of thymic influx, and of loss and selfrenewal across the lifespan, processes that either boost or preserve diversity, will therefore help us understand at a mechanistic level how the TCR repertoire is generated and evolves as an individual ages.
The consensus view is that, in adult mice, naive T cells have a mean lifespan of several weeks but a mean interdivision time of several years (den Braber et al., 2012; Hogan et al., 2015). This difference in timescales leads to the conclusion that, in mice, most naive T cells never divide and that their numbers are sustained largely by thymic export, which in adult mice contributes 1–2% of the peripheral pool size per day (Egerton et al., 1990; Graziano et al., 1998; Scollay et al., 1980; den Braber et al., 2012; Hogan et al., 2015). However, the dynamics of the naive pool may be radically different early in life, and it is unclear whether the rules that govern naive T cell dynamics in adults are the same in neonates. Indeed, there is considerable evidence that this is not the case. First, studies suggest that neonatal mice are lymphopenic, a state which, when artificially induced, supports the rapid expansion of newly introduced T cells through a mechanism referred to as lymphopeniainduced proliferation (LIP) (Rocha et al., 1989; Almeida et al., 2001; Yates et al., 2008; Houston et al., 2011; Hogan et al., 2013), and T cells transferred to healthy neonatal mice undergo cell divisions not observed in adult recipients (Min et al., 2003; Le Campion et al., 2002). However, the early establishment of naive compartments is still heavily reliant upon thymic output, since depletion of thymocytes in 2weekold mice drives a rapid and transient 50–70% reduction of peripheral CD4 and CD8 T cell numbers (Dzierzak et al., 1993).
Second, memory T cell compartments are rapidly established in neonatal mice, which derive from the activation of naive T cells. For instance, we have shown that the rate of generation of memory CD4 T cells is elevated early in life, at levels influenced by the antigenic content of the environment (Hogan et al., 2019). This result suggests that high per capita rates of activation upon first exposure to environmental stimuli may increase the apparent rate of loss of naive CD4 T cells in neonatal mice. One might expect a similar process to occur with naive CD8 T cells, with substantial numbers of socalled ‘virtual’ memory CD8 T cells generated from naive T cells in the periphery soon after birth (Akue et al., 2012, Smith et al., 2018). Together, these observations suggest that the average residence times of naive T cells differ in neonates and adults.
Third, the postthymic age of cells in neonates is inevitably more restricted than in adults. Following the dynamic period of their establishment, there is evidence that naive T cells do not die or selfrenew at constant rates but continue to respond or adapt to the host environment (Houston et al., 2008). Recent thymic emigrants (RTE) are functionally distinct from mature T cells (Adkins, 1999; Wang et al., 2016), may be lost at a higher rate than mature naive T cells under healthy conditions (Berzins et al., 1998; Berzins et al., 1999; Houston et al., 2011; van Hoeven et al., 2017), and respond differently to mature naive cells under lymphopenia (Houston et al., 2011). In the early weeks of life, all naive T cells are effectively RTE. Phenotypic markers of RTE are poorly defined, however, and so without a strict definition of ‘recent’ it is difficult to reach a consensus description of their kinetics. It may be more appropriate to view maturation as a continuum of states, and indeed the net loss rates (the balance of loss and selfrenewal) of both naive CD4 (Rane et al., 2018) and CD8 (Rane et al., 2018; Reynaldi et al., 2019) T cells in mice appear to fall smoothly with a cell’s postthymic age, a process we have referred to as adaptation (Rane et al., 2018). Such behaviour will lead to increasing heterogeneity in the kinetics of naive T cells over time, as the population’s agedistribution broadens, and may also contribute to skewing of the TCR repertoire, through a ‘firstin, lastout’ dynamic in which older naive T cells become progressively fitter than newer immigrants (Hogan et al., 2015). A conceptually similar model, in which naive T cells accrue fitness with their age through a sequence of stochastic mutation events, has been used to explain the reduced diversity of naive CD4 T cells in elderly humans (Johnson et al., 2012).
Taken together, these results indicate that cell numbers, host age and cell age may all influence naive T cell dynamics to varying degrees. When dealing with crosssectional observations of cell populations, these effects may be difficult to distinguish. For example, the progressive decrease in the populationaverage loss rate of naive T cells observed in thymectomised mice (den Braber et al., 2012) may not derive from reduced competition, as was suggested, but may also be explained by adaptation or selective effects (Rane et al., 2018). It is also possible that elevated loss rates of naive T cells early in life may not be an effect of the neonatal environment per se, but just a consequence of the nascent naive T cell pool being comprised almost entirely of RTE with intrinsically shorter residence times than mature cells. These uncertainties invite the use of mathematical models to distinguish different descriptions of naive T cell population dynamics from birth into old age.
Here, we combine model selection tools with data from multiple distinct experimental systems to investigate the rules governing naive T cell maintenance across the full lifespan of the mouse. We used an established bone marrow chimera system to specifically measure and model production, division and turnover of naive T cells in adult mice. We then used an outofsample prediction approach to test and refine these models in the settings of the establishment of the naive T cell compartments in neonates, and – using a unique Rag/Ki67 reporter mouse model – characterising the dynamics of RTE and mature naive T cells. We find that naive CD4 T cells appear to follow consistent rules of behaviour throughout the mouse lifespan, dividing very rarely and with a progressive increase in survival capacity with cell age, with no evidence for altered behaviour in neonates. Naive CD8 T cells behave similarly, but with an additional, increased rate of loss during the first few weeks of life that may reflect high levels of recruitment into early memory populations. These models are able to explain diverse observations and present a remarkably simple picture in which naive T cells appear to be passively maintained throughout life, with gradually extending lifespans that compensate in part from the decline in thymic output, and with no evidence for feedback regulation of cell numbers.
Results
Naive CD4 and CD8 T cells divide very rarely in adult mice and expected lifespans increase with cell age
Reports from our group and others (Hogan et al., 2015; Rane et al., 2018; Reynaldi et al., 2019; Mold et al., 2019) show that the dynamics of naive CD4 and CD8 T cells in adult mice and humans depend on cell age, defined to be time since they (or their ancestor, if they have divided) were released from the thymus. All these studies found that the net loss rate, which is the balance of their rate of loss through death or differentiation, and selfrenewal through division, decreases gradually with cell age for both subsets. It is unknown whether these adaptations modulate the processes that regulate their survival, or their ability to selfrenew.
To address this question, we used a wellestablished system that we have developed to quantify lymphocyte dynamics at steady state in healthy mice (Hogan et al., 2017), with the addition of detailed measurements of cell proliferation activity throughout. Briefly, hematopoietic stem cells (HSCs) in the bone marrow (BM) are partially and specifically depleted by optimised doses of the transplant conditioning drug busulfan, and reconstituted with T and Bcelldepleted BM from congenic donor mice. Chimerism rapidly stabilises among progenitors in the bone marrow (Verheijen et al., 2020) and thymus (Hogan et al., 2015) and is maintained for the lifetime of the mouse (Figure 1A). The host’s peripheral lymphocyte populations are unperturbed by treatment, and as donor T cells develop they progressively replace host T cells in the periphery through natural turnover. This system allows us to estimate the rates of influx into different lymphocyte populations and the net loss rates of cells within them; identify subpopulations with different rates of turnover; and infer whether and how these dynamics vary with host and/or cell age (Hogan et al., 2015; Gossel et al., 2017; Verheijen et al., 2020). Here, we generated a cohort of busulfan chimeric mice who underwent bone marrow transplant (BMT) between 7 and 25 weeks of age. At different times postBMT, we enumerated host and donorderived thymocyte subsets and peripheral naive T cells from spleen and lymph nodes (see Figure 1—figure supplement 1 for the flow cytometric gating strategy). We began by normalising the chimerism (fraction donor) within naive CD4 and CD8 T cells to that of DP1 thymocytes to remove the effect of variation across mice in the stable level of bonemarrow chimerism. This normalised donor fraction (f_{d}) will approach 1 within a population if it turns over completely – that is, if its donor:host composition equilibrates to that of its precursor. Saturation at ${f}_{d}<1$ implies incomplete replacement (Figure 1A), which can occur either through waning influx from the precursor population, or if older (host) cells persist longer than new (donor) cells, on average, implying cellage effects on turnover or selfrenewal. Previously, we observed incomplete replacement of both naive CD4 and CD8 T cells in adult busulfan chimeric mice (Hogan et al., 2015), and excluded the possibility that this shortfall derived from the natural involution of the thymus, leading us to infer that the net loss rates of both subsets increase with cell age (Hogan et al., 2015; Rane et al., 2018). For the present study, we also used concurrent measurements of Ki67, a nuclear protein that is expressed following entry into cell cycle and is detectable for approximately 3–4 days afterwards (Gossel et al., 2017; Miller et al., 2018), and stratified by host and donor cells. We reasoned that this new information would enable us to determine whether cellage effects are manifest through survival or selfrenewal.
To describe these data we explored variants of a structured population model in which either the rate of division or rate of loss of naive T cells varies exponentially with their postthymic age. These models are three dimensional linear partial differential equations (PDEs) that extend those we described previously (Hogan et al., 2015; Rane et al., 2018), allowing us to track the joint distribution of cell age and Ki67 expression within the population. A simpler variant of the agestructured model is one that explicitly distinguishes RTE from mature naive T cells, with a constant rate of maturation between two, and allows each to have their own rates of division and loss (van Hoeven et al., 2017; Rane et al., 2018). We also considered models of homogeneous cell dynamics; the simplest ‘neutral’ model with uniform and constant rates of division and loss, and densitydependent models that allowed these rates to vary with population size. All models are illustrated schematically in Figure 1B and their formulations are detailed in Appendix 1.
Each model was fitted simultaneously to the measured timecourses of total naive CD4 or CD8 T cell numbers, the normalised donor fraction, and the proportions of donor and host cells expressing Ki67. To model influx from the thymus we used empirical functions fitted to the numbers and Ki67 expression levels of late stage singlepositive CD4 and CD8 thymocytes (Appendix 2). Assuming that the rate of export of cells from the thymus is proportional to the number of singlepositive thymocytes (Berzins et al., 1998), we used these functions to represent the rates of production of Ki67^{+} and Ki67^{} RTE with mouse age, up to a multiplicative constant which we estimated. The fitting procedure is outlined in Appendix 3, and detailed in Verheijen et al., 2020.
Our analysis confirmed support for the models of cellagedependent kinetics (Figure 2), with all other candidates, including the RTE model, receiving substantially lower statistical support (Table 1; fits shown in Figure 2—figure supplement 1). For naive CD4 T cells, we found strongest support for the agedependent loss model (relative weight = 86%; Figure 2A) which revealed that their rate of loss declines as they age, halving roughly every 3 months (Table 2). For naive CD8 T cells the agedependent division model was favoured statistically (relative weight = 85%; Figure 2B, dashed lines). However, it yielded extremely low division rates, with recently exported cells having an estimated mean interdivision time of 18 months (95% CI: 14–25), and the division rate increasing only very slowly with cell age (doubling every 10 months). This model was therefore very similar to a neutral, homogeneous model and predicted that the normalised donor fraction approaches 1 in aged mice. This conclusion contradicts findings from our own and others’ studies that demonstrated that models assuming homogeneity in naive CD8 T cells failed to capture their dynamics in adult and aged mice (2–20 months old) (Hogan et al., 2015; Rane et al., 2018; Reynaldi et al., 2019).
Any signal of improvement in fitness with cell age, either in loss or division rates, is manifest primarily in an asymptotic value of the normalised donor fraction lower than one. For naive CD8 T cells, the normalised donor fractions at late times postBMT exhibit considerable scatter (Figure 2B, middle row), and so this asymptote is relatively poorly defined. This uncertainty reduces our ability to discriminate between the two agedependent models based solely on information criteria. For the next phase of analysis, we therefore retained the agedependent loss model, which had the next highest level of support and was similar by visual inspection (Figure 2B, solid lines), as a candidate description of naive CD8 T cell dynamics.
Agedependent loss models can describe RTE and mature naive CD4 and CD8 T cell kinetics in cotransfer experiments
To challenge these models further, we confronted them with data from a study that compared the ability of RTE and mature naive (MN) CD4 and CD8 T cells to persist following cotransfer to an adult congenic recipient (Houston et al., 2011). This study used a reporter mouse strain in which green fluorescent protein (GFP) expression is driven by Rag2 gene expression elements, and is thus expressed throughout thymic development and for several days following export into the periphery. This is a longestablished mouse model in which GFP expression is used as a surrogate marker of RTE status (Boursalian et al., 2004). After transferring RTE (GFP^{+}) and MN (GFP^{}) cells in equal numbers, the RTE:MN ratio within both CD4 and CD8 populations decreased progressively, falling by approximately 50% at 6 weeks (Figure 3), indicating that MN T cells persist significantly longer than RTE. We simulated this cotransfer using the models fitted to the data from the busulfan chimeric mice, and found that the agedependent loss model predicted the trends in the CD4 and CD8 RTE:MN ratios (Figure 3, blue lines) while the fitted agedependent division model, which exhibited very weak age effects, predicted that the ratio would remain close to 1 (Figure 3, orange lines). Details of this simulation procedure are given in Appendix 4. These data confirm the presence of strong cellage effects in naive T cell persistence, and substantially reduce our confidence in the bestfitting model for CD8 T cells, which predicted only a very weak dependence of cell division rates on cell age.
Models parameterised using data from adult mice accurately predict the dynamics of naive CD4 T cells in neonates, but not of CD8 T cells
Next, we wanted to characterise the dynamics of naive CD4 and CD8 T cells during the first few weeks of life, and connect the two regimes to build unified models of the dynamics of these populations from birth into old age. Because it takes at least 4 weeks for peripheral donorderived T cells to be detectable in busulfan chimeras, this system is not suitable for studying cell dynamics in young mice. Instead, we asked whether the models parameterised using data from adult mice could explain dynamics in young mice, and determine what (if any) modifications of the model were needed. We drew on two new data sets. One comprised the numbers and Ki67 expression of naive T cells derived from wildtype mice aged between 5 and 300 days. The other was derived from a cohort of Rag^{GFP} reporter mice, in which information about cell age can be gleaned from GFP expression levels. In this strain, intracellular staining for Ki67 is not possible without severely compromising GFP fluorescence. Therefore, we also introduced a Ki67^{RFP} reporter construct (Basak et al., 2014) to the strain to generate Rag^{GFP}Ki67^{RFP} dual reporter mice. Tracking GFP and RFP expression simultaneously allows us to study the kinetics and division rates of RTE, which are enriched for GFP^{+} cells, and of mature naive T cells, which are expected to have largely lost GFP. We could then directly confront the models derived from adult mice with these new data.
Figure 4A and B show the numbers of naive CD4 and CD8 T cells and their Ki67 expression frequencies in three cohorts of mice – Rag^{GFP}Ki67^{RFP} dual reporter mice aged between 10 and 120 days, wildtype mice, and adult busulfan chimeras in which host and donor cells were pooled. The red curves show the predictions of the cellagedependent loss models, which were fitted to the busulfan chimera data (red points) and extrapolated back to 1 day after birth. The dual reporter mice also yielded measurements of the coexpression of GFP and Ki67. To predict the kinetics of GFP^{+}Ki67^{–} and GFP^{+}Ki67^{+} proportions (Figure 4C and D), we needed to estimate only one additional parameter – the average duration of GFP expression. We assume that RTE become GFPnegative with first order kinetics at a rate defined both by the intrinsic rate of decay of GFP and the threshold of expression used to define GFP^{+} cells by flow cytometry. Our estimates of the mean duration of GFP expression within CD4 and CD8 RTE were similar (11 and 8 days, respectively). Details of how we connected GFP measurements to the agestructured models are provided in Appendix 5, and a description of the process of predicting neonatal T cell dynamics is given in Appendix 6.
Strikingly, the model of naive CD4 T cell dynamics in adult chimeric mice captured the total numbers and Ki67 expression of these cells in neonates remarkably well (Figure 4A), as well as the dynamics of Ki67^{} and Ki67^{+} RTE as defined by GFP expression (Figure 4C). This agreement indicates that the high level of Ki67 expression in naive CD4 T cells early in life does not reflect increased rates of division or LIP, but is rather inherited from precursors within the neonatal thymus, a large fraction of which undergo cell division (Appendix 2).
For naive CD8 T cells the cellagedependent loss model accurately predicted cell dynamics in both the reporter and wildtype mice back to approximately 3 weeks of age, but underestimated Ki67^{+} frequencies in neonatal mice (Figure 4B, right panel), suggesting that naive CD8 T cells exhibit distinct dynamics very early in life. Intuitively, this mismatch can be explained in two ways: either CD8 RTE are lost at a higher rate in neonates than in adults or they divide more rapidly. In the former, a greater proportion of GFP^{+} Ki67^{+} RTE will be lost before they become Ki67^{} and so the predicted proportion of cells that are GFP^{+} Ki67^{} will be lower (Figure 4D, centre panel). In the latter, the GFP^{+} Ki67^{+} proportion will increase (Figure 4D, right panel). Therefore, to explain naive CD8 T cell dynamics in neonates the basic model of cellagedependent loss in adults can be extended in two ways, modulating either the division or loss rate early in life.
Naive CD8 T cells are lost at a higher rate in neonates than in adults
To distinguish between these possibilities we turned to a study by Reynaldi et al., 2019, who used an elegant tamoxifendriven CD4Cre^{ERT2}RFP reporter mouse model to track cohorts of CD8 T cells released from the thymus into the peripheral circulation of animals of varying ages. In this model, a pulse of tamoxifen permanently induces RFP in cells expressing CD4, including CD4^{+}CD8^{+} doublepositive thymocytes. The cohort of naive CD8 T cells deriving from these precursors continues to express RFP in the periphery and timecourses of their numbers in individual mice were estimated with serial sampling of blood. These timecourses showed that the net loss rate of naive CD8 T cells appears to slow with their postthymic age, and the rate of loss of cells immediately following release from the thymus appears to be greater in neonates than in adults (Figure 5A). Without measures of proliferation, these survival curves reflect only the net effect of survival and selfrenewal. Nevertheless, we reasoned that confronting our models with these additional data, and triangulating with inferences from other datasets, would allow us to identify a ‘universal’ model of naive CD8 T cell loss and division across the mouse lifespan.
We reanalysed the data from Reynaldi et al. using a Bayesian hierarchical approach (Appendix 7) to explain the variation in the kinetics of loss of these cohorts of cells across animals and age groups. Since there was no readout of cell division in this system, we simplified the cellagedependent loss model by combining division and loss into a net loss rate $\lambda (a)$. We then fitted this model to the timecourses of labelled naive CD8 T cells across the different treatment groups. We tested four possibilities in which either the initial numbers of labelled cells ($N}_{0$) and/or the net loss rate of cells of age $0$ (${\lambda}_{0}$) varied across groups or animals as normallydistributed hyperparameters. The model in which $N}_{0$ was specific to each mouse and ${\lambda}_{0}$ was specific to each age group gained 100% relative support (Appendix 7—table 1; fits in Figure 5A). This model confirmed that CD8 RTE are indeed lost at a significantly higher rate in the younger groups of mice (Figure 5B). We then described this decline in ${\lambda}_{0}$ with mouse age empirically with a sigmoid (Hill) function, ${\lambda}_{0}(t)$ (Figure 5B, solid line) and used it to replace the discrete grouplevel variation in ${\lambda}_{0}$ within the hierarchical agestructured model (Appendix 7). This ‘universal’ model, in which the loss rate of naive CD8 T cells declines with cell age but begins at higher baseline levels early in life, explained the data from Reynaldi et al. equally well, visually and statistically (difference in the expected log pointwise predictive density, elpd_{loo}=3.4; differences <4 typically indicate that two models have similar predictive performance (Sivula et al., 2020). See Appendix 3 for details of the calculation of elpd_{loo} values.)
This analysis shows that the baseline net loss rate of CD8 RTE declines from the age of ∼3 weeks and stabilises at a level approximately 50% lower by age 9 weeks (Figure 5B). Therefore, newly exported naive CD8 T cells are lost at a higher rate in neonates than in adults, or they divide more slowly. Only the former is consistent with our inference from the Rag/Ki67 dual reporter mice. Indeed, we confirmed that simulating the agedependent loss model from birth with a lower baseline division rate in neonates than in adults failed to improve the description of the early trajectories of the frequencies of GFP^{+} Ki67^{–} and GFP^{+} Ki67^{+} naive CD8 T cells (Figure 5C, blue dashed line). In contrast, increasing the baseline loss rate in neonates according to the function we derived from the data in Reynaldi et al. (Appendix 7) captured these dynamics well (Figure 5C, green dashed line).
In summary, we find that naive CD8 T cells rarely divide, increase their capacity to survive with cell age, and those generated within the first few weeks of life are lost at a higher baseline rate than those in adults.
Ki67 expression within naive CD4 and CD8 T cells in adult mice is almost entirely a residual signal of intrathymic proliferation
Our analyses are consistent with earlier reports that naive T cells in mice divide very rarely (Modigliani et al., 1994; den Braber et al., 2012; Hogan et al., 2015). By explicitly modeling the kinetics of quiescent and recently divided cells, we can also explain the apparently contradictory observation that more than 60% of naive CD4 and CD8 T cells express Ki67 early in life, declining to 2–3% by 3 months of age (Figure 4). We argue that this pattern, rather than being an indication of lymphopeniainduced proliferation early in life fading to lowlevel but appreciable selfrenewal in adults, is instead just a shadow of intrathymic division; Ki67 among peripheral naive T cells is almost entirely derived from cells that divided in the thymus and were exported within the previous few days. This conclusion emerged from the modelling of the busulfan chimera data but is also directly evident from the Rag^{GFP} Ki67^{RFP} reporter mice, in which Ki67RFP expression among naive T cells was exclusively found on GFP^{high} peripheral RTE, and was a continuum of the expression by mature singlepositive (SP) thymocytes (Figure 6A). This inheritance of expression from the thymus is also reflected in the high degree of correlation between the frequencies of Ki67^{+} cells among SP thymocytes and peripheral naive T cells throughout life, observed in wildtype mice (Figure 6B).
This result also gives an intuitive explanation of the trajectories of Ki67 expression within donor and host cells in the busulfan chimeric mice, which are distinct soon after BMT but converge after 6–12 months (Figure 2). This behaviour does not derive from any intrinsic differences between host and donor T cells, but rather from the distinct age profiles of the two populations. Following BMT, the rate of production of host naive T cells declines substantially, as the procedure typically results in 80%–90% replacement of host HSC with donor HSC. Since Ki67 is seen almost exclusively within very recent thymic emigrants, the frequency of Ki67expressing host naive T cells then declines rapidly. Conversely, new donorderived naive T cells are initially highly enriched for Ki67^{+} cells. The frequencies of Ki67^{+} cells within the two populations then gradually converge to pretransplant levels as aged Ki67^{} donor cells accumulate, and hostderived naive T cells equilibrate at lower numbers.
Discussion
Our previous analyses suggested naive T cells operate autonomously and compensate for the gradual decline in thymic output with age by increasing their ability to persist with time since they leave the thymus in both adult mice (Rane et al., 2018) and in humans (Mold et al., 2019). Here, we show through the modelling of a range of datasets that naive T cell adaptation in mice manifests primarily through a progressive decrease in their loss rate, and that they divide very rarely if at all, with mean interdivision times of at least 14 months. This means that throughout the mouse lifetime, newly made CD4 and CD8 RTE are lost at faster rates than their mature counterparts, predicting the preferential retention and accumulation of clones exported early in life. The lack of peripheral expansion combined with high levels of thymic export throughout life implies that, in mice, the majority of the naive T cell repertoire is made up of small and longlived TCR clones. This interpretation is consistent with studies showing enormous diversity within naive TCR repertoires in mice (Gonçalves et al., 2017) and supports the idea that any hierarchy within it is shaped by the generational frequencies of individual clones in the thymus, rather than by peripheral expansions (Quigley et al., 2010). However, we observed a remarkably high degree of intrathymic proliferation in young mice, with close to 100% of latestage CD62L^{hi} SP thymocytes expressing Ki67 in neonates, declining to approximately 20% over the first 3 months of life (Figure 6B and Appendix 2). Ki67 expression within these mature SP populations derives exclusively from cell division after TCR rearrangement and positive selection are complete. This observation implies that naive T clones generated in neonatal mice, which will ultimately be overrepresented in older mice, may be substantially larger on average than those exported from adult thymi.
It is wellestablished that proliferative selfrenewal plays a much more important role in naive T cell dynamics in humans than in mice (den Braber et al., 2012), which may compensate in part for the quite severe atrophy of the thymus that progresses from young adulthood onwards (Steinmann et al., 1985). However, we and others have shown that, similar to mice, the net loss rates of naive T cells in humans appear to fall with cell age (Johnson et al., 2012; Mold et al., 2019). Thus, progressive, cellintrinsic increases in the homeostatic fitness of naive T cells may be a common mechanism. Whether this adaptation in humans occurs through changes in the capacity to survive or to selfrenew, or both, is unclear. In any case the implication is that, as in mice, naive T cell age distributions in humans become disproportionally weighted toward older cells, or clones, over time. The combination of cell proliferation and extended lifespans, which give more time for cell fitness disparities to widen, may underlie the even broader distributions of naive T cell clone sizes observed in humans (Qi et al., 2014; Mora and Walczak, 2019; de Greef et al., 2020).
Our fitted agedependent loss models showed agreement with the trends demonstrated by Houston et al., 2011, in which mature naive (MN) CD4 and CD8 T cell persist longer than RTE, but underestimated the extent of enrichment of MN cells (Figure 3A). This mismatch may derive in part from uncertainties in the agedistributions of the transferred T cells in their experiments, and from our need to specify a cutoff in cell age associated with the definition of RTE as GFPpositive. It is also possible that the cell manipulations involved in adoptive transfer had a differential impact on RTE and MN cell survival. The essential point here, however, is that the clear disparity of kinetics of the two transferred populations weighs against models exhibiting weak cellage effects.
Another modelling study also demonstrated that CD4 RTE are lost more rapidly than MN CD4 T cells (van Hoeven et al., 2017). They estimated that the loss rate of CD4 RTE is 0.063 day^{1}, translating to a residence time of 15 days (95% CI: 9–26), and is roughly four times shorter than the 66 day (52–90) residence time of MN CD4 T cells. Our results agree closely. We estimate that CD4 RTE (cells of age 0) have an expected residence time of 22 (18–28) days, doubling approximately every 3 months, such that in a 12weekold mouse, the mean residence time of MN CD4 T cells aged 21 days or greater is roughly 60 days. In contrast, van Hoeven et al. concluded that naive CD8 T cells are a kinetically homogeneous population with a mean residence time of 76 (42–83) days. With our favoured agedependent loss model, we estimate that CD8 RTE initially have an expected residence time of 40 (18–28) days, doubling every ∼5 months. However, our predicted average residence time of MN CD8 T cells (aged >21 days) in a 12 week old mouse was approximately 76 days, which agrees with their estimate. We included a similar RTE/MN model in our analysis (Figure 1B) and found that for CD8 T cells it received statistical support comparable to a neutral model of constant division and loss, in line with their analysis. Therefore, our different conclusions may stem in part from the specification of our models. It would be instructive to analyse the data from their thymic transplantation and heavy water labelling studies with the agestructured models we consider here. Another puzzle is that our result and those of van Hoeven et al., 2017, Houston et al., 2011, and Tsukamoto et al., 2009 are all at odds with the study of Dong et al., 2013 who observed that CD4 GFP^{+} RTE from RagGFP reporter mice persisted better than bulk naive CD4 T cells from agematched donor mice, one week after cotransfer. We are unable to explain this observation, although we speculate that differential survival may have been influenced by the manipulation step of labelling the bulk naive T cell cohort, but not the RTE, with a fluorescent dye (CFSE).
The pioneering studies by Berzins et al. showed substantial and proportional increases in T cell numbers in mice transplanted with 2, 6 and 9 thymic lobes Berzins et al., 1998; Berzins et al., 1999. They concluded that this increase corresponds to the accumulation of RTE exported in the previous 3 weeks. In absence of any homeostatic regulation, the increase in the sizes of the naive CD4 and CD8 T cell pools under hyperthymic conditions is determined by the change in thymic output and by RTE lifespans. Our estimates of these lifespans (roughly 22 and 40 days for CD4 and CD8, respectively) are in line with their estimate of 3 weeks (Berzins et al., 1999). Indeed, simulating the transplantation of 6 thymic lobes using the agedependent loss models and parameters derived from busulfan chimeric mice recapitulates their observations (Appendix 7).
Reynaldi et al. used a novel fatemapping system to demonstrate that the net loss rate of naive CD8 T cells (loss minus selfrenewal) declines with their postthymic age and is higher for CD8 RTE in neonates than in adults Reynaldi et al., 2019. Our addition to this narrative was to reanalyse their data with a more mechanistic modeling approach to isolate the effects of division and loss, and to calculate a functional form for the dependence of the CD8 RTE loss rate on mouse age. In conjunction with our analysis of data from Rag/Ki67 dual reporter mice we inferred that the baseline loss rate of naive CD8 T cells immediately following release from the thymus is higher in neonates than in adults, while the rate of division is close to zero throughout the mouse lifespan, and independent of host and cell age. The higher loss rate of CD8 RTE in neonates may derive from high rates of differentiation into memory phenotype cells rather than impaired survival. This idea is consistent with the rapid accumulation of virtual memory CD8 T cells in the periphery during the postnatal period (Akue et al., 2012, Smith et al., 2018). However, we found no evidence for a similar process among naive CD4 T cells. We recently showed that increasing the exposure to environmental antigens boosts the generation of memory CD4 T cell subsets early in life, but not in adulthood (Hogan et al., 2019). It may be that in the young specific pathogenfree mice we studied here, any such elevated flux out of the naive CD4 T cell pool due to activation, which occurs before clonal expansion, was too low for our analysis to detect. We speculate that any dependence of naive T cell residence times on host age may be even more pronounced in truly wild mice and in humans, given their extensive exposure to environmental and commensal antigens immediately following birth.
We do not explicitly model the mechanisms underlying adaptation in cell persistence. Modulation of sensitivity to IL7 and signaling via Bcl2 associated molecules has been implicated in increasing naive T cell longevity (Tsukamoto et al., 2010; Houston et al., 2011) and is consistent with the outcome of cotransfer experiments. It is also possible that increased persistence derives additionally from a progressive or selective decrease in naive T cells’ ability to be triggered into effector or memory subsets. Studying the dynamics of naive T cells in busulfan chimeras generated using bone marrow from TCRtransgenic donor mice may help us untangle the contributions of survival and differentiation to the increase in their residence time with their age.
An alternative to the adaptation model is one of selection, in which each cell’s survival capacity (loss rate) is determined during thymic development, drawn from a distribution, and subsequently fixed for its lifespan (Dowling et al., 2005; Rane et al., 2018). As an individual ages, naive T cells with intrinsically longer expected residence times will then be selected for. Indirect support for such a mechanism comes from the observation that lowlevel TCR signalling is essential for naive T cell survival (Seddon and Zamoyska, 2002; Martin et al., 2006), suggesting that the ability to gain trophic signals from selfpeptide MHC ligands may vary from clone to clone. We have also shown that different TCRtransgenic naive T cell clones have different capacities for proliferation in lymphopenic hosts (Hogan et al., 2013). However, one prediction of a selective model based on heterogeneity in TCR affinity alone is that TCR transgenic T cells cotransferred from young and old hosts would be lost at identical rates. One such experiment still saw that older cells exhibited a fitness advantage over younger ones (Tsukamoto et al., 2009). Therefore, while we cannot rule out a preprogrammed (and possibly TCRspecific) element to each naive T cell’s life expectancy, it is clear that they undergo progressive changes in their fitness, expressed in the adaptation models we have considered here.
Naive T cells proliferate under severely lymphopenic conditions in mice (Almeida et al., 2001; Yates et al., 2008; Hogan et al., 2013), a phenomenon that has contributed to the idea that quorumsensing (through resource competition, for example) may act to regulate naive T cell numbers. However, lymphopeniainduced proliferation is associated with the acquisition of a memorylike phenotype (Cho et al., 2000; Min et al., 2003; Min and Paul, 2005; Hogan et al., 2013). The observation that this process occurs in healthy neonatal mice was taken to indicate that they are lymphopenic to some degree (Min et al., 2003), but it has been shown since that there are constitutive flows from the naive CD4 and CD8 to memoryphenotype T cell pools throughout life under replete conditions (van Hoeven et al., 2017; Gossel et al., 2017; Hogan et al., 2019), and that this transition can occur soon after release from the thymus (van Hoeven et al., 2017). It is therefore not clear that young mice are functionally lymphopenic, nor that any compensatory processes support the production or maintenance of truly naive T cells early in life. In line with this, our analyses of neonatal mice revealed no evidence of increased rates of selfrenewal, nor any reduction of cell loss rates, that would act to boost or preserve naive T cell numbers in the early weeks of life; neither did we need to invoke feedback regulation of their kinetics in adulthood. We showed previously that apparent densitydependent effects on naive T cell survival following thymectomy can also be explained by adaptive or selective processes (den Braber et al., 2012; Rane et al., 2018). Similarly, in humans, any regulation of a natural setpoint appears to be incomplete at best; naive T cell numbers in HIVinfected adults typically do not normalise following antiretroviral therapy (Hazenberg et al., 2000), and recovery from autologous haematopoietic stem cell transplant results in persistent perturbations of T cell dynamics (BaliuPiqué et al., 2021). Dutilh and de Boer, 2003 showed that explaining the kinetics of decline in TREC frequencies in human naive T cells requires an increase in either cell division or survival with age, as naive T cell numbers decline. They ascribed this to a densitydependent, homeostatic mechanism, but again cellintrinsic adaptation or selection could underlie the phenomenon. Therefore, the idea of naive T cell homeostasis over the life course, in the sense of compensatory or quorum sensing behaviour, may well be largely a theoretical concept. Selection pressures that shaped the evolution of lymphocyte development are most likely to have been exerted on the establishment of T cell compartments and immunity that would support host survival to reproductive age, and would have little traction upon T cell behaviour into old age. Perhaps a better model, in both mice and humans, is the traditional understanding in which the thymus drives the generation of the bulk of the naive T cell pool in the early life, and thereafter naive T cell repertoires coast out into old age in a cellautonomous manner.
Materials and methods
Generating busulfan chimeric mice
Request a detailed protocolSJL.C57Bl/6J (CD45.1.B6) mice were treated with optimised low doses of busulfan to deplete HSC but leave peripheral T cell subsets intact. HSC were reconstituted with congenicallydistinct, T cell depleted bone marrow from C57Bl/6J donors to generate stable chimeras. Details of the protocols are given in Hogan et al., 2017.
Mice
Mki67tm1.1Cle/J (Ki67RFP) mice were generously provided by the laboratory of Prof. Hans Clevers (Hubrecht Institute, KNAW and University Medical Centre Utrecht, Utrecht, The Netherlands; Basak et al., 2014). FVBTg(Rag2EGFP) 1Mnz/J mice were from Jax Laboratories (strain 005688). Ki67RFP x Rag2EGFP F1 mice were subsequently backcrossed to a C57Bl/6J background for seven generations. Busulfan chimeric mice and wildtype control mice were housed in conventional animal facilities at the UCL Royal Free Campus, London, UK (UCL). Mice were housed in individually ventilated cages and drank irradiated water. All the animals were handled according to UK home office regulations (licence PPL PP2330953) and institutional animal care and use committee (IACUC) protocols at University College London.
Flow cytometry
Request a detailed protocolSinglecell suspensions were prepared from the thymus, spleen and lymph nodes of busulfan chimeric mice, wildtype control mice, or germfree mice. Cells were stained with the following monoclonal antibodies and cell dyes: CD45.1 FITC, CD45.2 FITC, CD45.2 AlexaFluor700, TCR$\beta $ APC, CD4^{+} PerCPeFluor710, CD44 APCeFluor780, CD25 PE, CD25 eFluor450, CD25 PECy7, CD62L eFluor450, NK1.1 PECy7 (all eBioscience), CD45.1 BV650, CD45.2 PEDazzle, TCR$\beta $ PerCPCy5.5 CD4^{+} BV711, CD44 BV785, CD25 BV650 (all Biolegend), CD62L BUV737 (BD Biosciences), LIVE/DEAD nearIR and LIVE/DEAD blue viability dyes. For Ki67 staining, cells were fixed using the eBioscience Foxp3 /Transcription Factor Staining Buffer Set and stained with either antimouse Ki67 FITC or PE (both eBioscience). Cells were acquired on a BD LSRFortessa flow cytometer and analysed with Flowjo software (Treestar). See Figure 1—figure supplement 1 for the gating strategy used to identify mature single positive thymocytes and peripheral naive subsets, and gates to measure Ki67 frequencies.
Mathematical modelling and statistical analysis
Request a detailed protocolWe fitted a set of candidate mathematical models (described in Appendix 1) to the data from adult busulfan chimeric mice, using empirical descriptions of the pool sizes and Ki67^{+} fraction within SP thymocytes to define thymic influx (Appendix 2). Specifically, we fitted simultaneously to the time courses of total cell counts, normalised donor fraction and the fraction of cells that were Ki67^{+} within donor and host subsets of naive CD4 and CD8 T cells. We used a Bayesian estimation approach using R and Stan. Code and data used to perform model fitting, and details of the prior distributions for parameters, are available at this linked Github repository. Models were ranked based on information criteria estimated using the LeaveOneOut (LOO) cross validation method (Vehtari et al., 2015; Vehtari et al., 2016), described in Appendix 3. Appendix 4 describes how we simulated the cotransfer experiment performed by Houston et al., 2011, using the agestructured PDE model (Appendix 1) with parameters estimated from fits to the busulfan chimeric mouse data. To predict the dynamics of naive T cells in neonatal mice, we constructed a mapping between cell age and GFP expression to predict the kinetics of GFP^{+} and Ki67^{+} cells in Rag^{GFP} Ki67^{RFP} reporter mice aged between 11 days and 4 months (Appendix 5), and the models fitted to data from adult mice were extrapolated back to near birth (Appendix 6). To reanalyze longitudinal data from Reynaldi et al., 2019, tracking the survival of cohorts of naive CD8 T cells within different age groups of mice, we used a hierarchical Bayesian modelling approach (Appendix 7).
Appendix 1
Models of naive T cell maintenance
Neutral model
We assume that the naive T cell pool is a kinetically homogeneous population that selfrenews through a constant rate of division $\rho $, and is lost by death and differentiation at a constant rate $\delta $. The inverse of $\rho $ is the mean interdivision time, and the inverse of $\delta $ is the mean residence time of a cell. Thymic influx is the product of the per capita rate of influx $\alpha $ and the timecourse of the size of SP population $S(t)$, which is described empirically (see Appendix 2). We model the dynamics of Ki67^{+} (N^{+}) and Ki67^{} (N^{—}) cells using the following ODE model;
Here, $\beta $ is the rate of loss of Ki67 expression after mitosis, and $\u03f5$ is the Ki67^{+} fraction among cells immediately after export from the thymus. We assumed Equation 2 hold identically for host and donor cells and solved them to derive the solutions to the following:
using the empirical descriptions of the size ($S(t)$) and Ki67^{+ }fraction ($\u03f5(t)$) of SP thymocytes, their direct precursors.
Densitydependent models
In these extensions of the neutral model, the rate of cell division $\rho $, or the rate of loss $\delta $ vary with the size of the naive T cell population. We explored models exhibiting densitydependence in either $\rho $ or $\delta $. Both models assume that all cells in the population follow the same rules of selfrenewal and turnover, at any given time. We defined the densitydependence using Hill functions;
where $\overline{C}$ and $a}_{0$ or ${\delta}_{0}$ were estimated from the model fits to the data.
RTE model
Here, we treated RTE and mature naive (MN) T cells separately, allowing them to have distinct rates of division (${\rho}_{R}$ and ${\rho}_{N}$) and of loss (${\delta}_{R}$ and ${\delta}_{N}$). We assume a constant rate of maturation of RTE ($\mu $). In this model, the expected residence time of cells in the RTE compartment is $1/({\delta}_{R}+\mu )$, and a proportion $\mu /({\delta}_{R}+\mu )$ of RTE survive to maturity. We solve the following equations for Ki67^{+} and Ki67^{} cells with the RTE and MN compartments, which are identical for host and donorderived cells;
Agedependent division and loss models
We aimed to model the population density of naive T cells $u(a,k,t)$, where $t$ is mouse age, $a$ is a cell’s age (defined as the time since it or its ancestor left the thymus), and $k$ is a cell’s level of Ki67 expression. We assume Ki67 expression reaches a maximal level of $k=1$ immediately after cell division and decays exponentially at rate $\beta $. We assume that the per capita rates at which cells divide ($\rho $) and are lost ($\delta $) can be functions of mouse age $t$ and/or of cell age $a$. To model the evolution of $u(a,k,t)$, we extended an agestructured population PDE model described previously (Hogan et al., 2015; Rane et al., 2018) to include Ki67 expression.
Initial conditions. We assume that at some host age $t}_{0$, the population has size $N}_{0$ and has a cellage distribution $\gamma (a)$, where $0\le a\le {t}_{0}$ and ${\int}_{0}^{{t}_{0}}\gamma (a)\phantom{\rule{thinmathspace}{0ex}}da=1$ (we assume all cells are of age $a=0$ when they leave the thymus); and these cells have a distribution of levels of Ki67 expression $\psi (k)$, where ${\int}_{0}^{1}\psi (k)dk=1$ and $\psi (k)=0$ for $k\notin (0,1)$. Here for simplicity we are assuming no relation between Ki67 expression and cell age within the cells present at $t={t}_{0}$, but one could easily extend this framework with a more general initial joint distribution $P(a,k)$. At times $t\ge {t}_{0}$, we assume that cells of age zero enter the naive pool from the thymus at rate $\theta (t)$ and with Ki67 distribution $\varphi (k,t)$, where ${\int}_{0}^{1}\varphi (k,t)dk=1$ for all $t$.
Breaking the solution into cohorts of cells. Our approach is to track separately the fates of cells that were present at mouse age $t}_{0$, who will all have age $a>t{t}_{0}$ at some later time $t$; and the fates of those that were exported from the thymus at time $t}_{0$ or later, which will all have age $a<t{t}_{0}$. We then add these to get the full population density $u(a,k,t)$. The master PDE for both populations combined is
with boundary conditions
The first condition above derives from cell division; at any host age $t$, cells of age a at time $t$ divide at rate $\rho (a,t)$ generating two cells of age $a$ with $k=1$.
Initial cohort
Nondivided cells. First consider those cells present at $t={t}_{0}$ that have yet to divide; this population decreases in size with a per capita rate $\delta (a,t)+\rho (a,t)$, and follows Equation 8 with the single boundary condition $u(a,k,t={t}_{0})={N}_{0}\gamma (a)\psi (k)$. We solve this using the method of characteristics by identifying a variable $s$ such that
where for brevity we define $\mathrm{{\rm Y}}(a,t)=\rho (a,t)+\delta (a,t)$. Equating terms with Equation 8 gives
Along the characteristic that starts at $({a}_{0},{k}_{0},{t}_{0})$, illustrated in Appendix 1—figure 1A below, the population density evolves as
We know from Equation 13 that $k}_{0}=k{e}^{\beta (t{t}_{0})$, so
The population density $u(.)$ must then be transformed with a Jacobian to express it as a density over $k$ rather than over $k{e}^{\beta (t{t}_{0})}$. If $y=k{e}^{\beta (t{t}_{0})}$, then
giving
where the subscript $(i,n)$ denotes ‘initial and nondivided’. Equation 19 holds for $k\le {e}^{\beta (t{t}_{0})}$ because none of these cells have divided since time $t}_{0$ and their Ki67 expression is decaying $k=1$.
Divided cells. Appendix 1—figure 1B above illustrates the evolution of cells from the initial cohort who subsequently divide, each time resetting their Ki67 expression to $k=1$. To follow these cohorts we solve Equation 8 with the boundary condition describing the division of cells of age $a$ at host age $t$;
Characteristic curves are again of the general form $k={k}_{0}{e}^{\beta t}$, but we parameterise them differently to those described above, since these originate at $k=1$ and not $t={t}_{0}$. Cells with Ki67 expression $k$ must have divided a time $\tau =(1/\beta )\mathrm{ln}(k)$ in the past. A convenient parameterisation is therefore
where the cells currently of age $a$ divided at time $T=t\tau $ when they were aged ${a}_{0}=a\tau $. Along these curves,
where the exponential term represents the proportion of cells on this characteristic curve that divide again or die. It integrates a cell’s experience from age $a}_{0$ at host age $T$, to age $a$ at host age $t=T+\tau $. Here, $u}_{i$ denotes the entire initial cohort, divided or undivided, integrated over all levels of Ki67 expression, whose cells of age $a}_{0$ fed the population at time $T$.
To convert this to a density ${u}_{i,\phantom{\rule{thinmathspace}{0ex}}d}(a,\phantom{\rule{thinmathspace}{0ex}}k,\phantom{\rule{thinmathspace}{0ex}}t)$, we use the Jacobian $d\tau /dk=1/(\beta k)$. This gives
where ${U}_{i}(a,t)$ is the population density of the initial cohort (divided or undivided) at age $a$ and time $t$, integrated over all $k$, which we will return to below.
Cells that enter after $t={t}_{0}$
A similar set of calculations applies for cells that subsequently enter the pool, at which point we define them to be of age zero. Again we partition these cells into those that will not divide and those that will. For the former, consider those that are of age $a\le t{t}_{0}$ at time $t$; that is, cells that entered later than $t}_{0$. We denote this population ${u}_{\theta ,n}(a,k,t)$ where $k\le {e}^{\beta a}$. These cells are what remains of the cohort exported at time $ta$, at age $a=0$, of size $\theta (ta)$ and with Ki67 distribution $\varphi (k{e}^{\beta a},ta)$. These are then lost to death or division. By analogy with Equation 19, the Jacobian is ${e}^{\beta a}$ and these nondivided cells evolve as
Here, cells are born with age zero and so the characteristics are $k={k}_{0}{e}^{\beta a}$. Similarly for divided cells; by analogy with Equation 23,
Here, ${U}_{\theta}(a,t)$ is the population density of cells of of age $a$ at time $t$, who entered the pool a time $\tau =ta$ ago and may have divided or not.
Solving the agestructured PDE only
To complete these solutions we need the population densities ${U}_{i}(a,t)$ and ${U}_{\theta}(a,t)$, ignoring Ki67 expression. These are straightforward;
where for brevity $\lambda (a,t)$ is the net loss rate of cells of age $a$ at time $t$, which is $\delta (a,t)\rho (a,t)$. The integral in Equation 26 follows a cell whose age runs from $a(t{t}_{0})$ to $a$, during which host age runs from $ta$ to $t$. The integral in Equation 27 follows a cell whose age runs from $0$ to $a$, between host ages of $ta$ to $t$. The two solutions join at $a=t{t}_{0}$; the influx at time $t}_{0$ must be the density of cells of age zero in the initial cohort; $\theta ({t}_{0})={N}_{0}\gamma (0)$.
Therefore, to obtain the total solution $u(a,k,t)={u}_{i,n}+{u}_{i,d}+{u}_{\theta ,n}+{u}_{\theta ,d}$, we add Equations 19, 23, 24 and 25, using the solutions for the agestructured model given in Equations 26 and 27.
We can connect this solution to gated flow cytometry data by partitioning the population into high and low Ki67 expression. We define Ki67^{+} cells to be those which divided no more than a time $1/\beta $ ago, which corresponds to a cutoff of $k=1/e$. Therefore, the numbers of Ki67 positive and negative cells at time $t$ are
Appendix 2
Constructing empirical descriptions of the dynamics of mature SP thymocytes over the mouse lifespan
We assumed that the rate of export of new naive T cells from the thymus is proportional to the numbers of single positive (SP4 and SP8) thymocytes (Berzins et al., 1998). The total numbers of SP thymocytes increase rapidly up to 6–7 weeks of age and then drop gradually over time. We used the following empirical descriptor function to capture the dynamics of thymic SP cells varying with mouse age;
We estimated the parameters $S(0)$, $A$, $n$ and $q$ by fitting Equation 29 to the logtransformed data from wildtype mice bred in the same facility as the busulfan chimeras (Appendix 2—figure 1A). The rate of thymic export is then $\theta (t)=\alpha S(t)$, with the constant $\alpha $ estimated when fitting models to the busulfan chimera data.
We modelled the Ki67^{+} fraction within SP4 and SP8 thymocytes using the form
and estimated ${\u03f5}_{0}$, ${\u03f5}_{f}$ and $C$ by fitting this function to the logittransformed proportions of cells that were Ki67^{+} (Appendix 2—figure 1B).
When modelling data from the busulfan chimeras, we assumed that the total output from the thymus at any time is identical to that in agematched wild type mice, but is split between donor and host cells according to the chimerism $\chi $ at the DP1 stage of thymic development:
Appendix 3
Model fitting and selection criteria
Each of the models described in Appendix 1 was fitted simultaneously to four sets of observations – cell counts, normalised chimerism, and the proportions of cells expressing Ki67 within donor and host naive T cells. To normalise residuals, cell counts were log transformed; and normalised chimerism and Ki67^{+} fractions were logit and arcsine square roottransformed, respectively. We used a Bayesian inference approach to estimate the model parameters and the errors associated with the measurements in each dataset. Model definitions, the prior distribution of parameters and the likelihood definitions were encoded in the Stan language and are available at this Github repository, and models were fitted using the Hamiltonian Monte Carlo algorithm (Team Stan Development, 2022).
We compared the support for models using the leaveoneout (LOO) cross validation method. The expected log pointwise predictive density (elpd) of the model ${M}_{k}$, which is a measure of its outofsample prediction accuracy (Vehtari et al., 2016), can be estimated using LOO as
where $P({y}_{i}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{y}_{i},{M}_{k})$ is the likelihood of observing $y}_{i$ given model ${M}_{k}$ fitted on the data with observation $i$ excluded. The elpd estimate is the sum of $n$ independent components, and so its standard error is
We calculated the elpd estimate using the loo2.0 package in the Rstan library, which employs Pareto smoothed importance sampling (PSIS) (Vehtari et al., 2015) to approximate LOO cross validation. The input for this process is an array of joint likelihoods evaluated at draws from the posterior parameter distributions. The estimates of elpd and its standard error were used to rank models using the PseudoBayesian model averaging (BMA) method (Yao et al., 2018), which is analogous to using Akaike’s Information Criterion (AIC) to calculate model weights Akaike, 1978; Burnham and Anderson, 2002; Wagenmakers and Farrell, 2004, given by
An intuitive interpretation of this formulation is that models are given a greater weight for both high total outofsample prediction accuracy and for evenness of this accuracy across the set of observations. These PseudoBMA model weights were calculated using the loo2.0 package and are reported in Table 1 in the text.
Appendix 4
Predicting the kinetics of loss of adoptively cotransferred RTE and mature naive T cells
To compare the dynamics of recent thymic emigrants (RTE) and mature naive (MN) T cells, we simulated the cotransfer experiment described in Houston et al., 2011, using agedependent loss and division models with parameters derived from fitting to the busulfan chimera data.
Houston et al. isolated RTE from 5 to 9weekold RAG2GFP reporter mice, and MN T cells from a mixture of thymectomised WT and RAG2GFP mice aged greater than 12 weeks. These cells were cotransferred to WT recipient mice.
To predict the subsequent kinetics of the transferred populations with our models, we needed to define their agedistributions at the time of transfer. This involved an approximation, given the uncertainty in the ages of the donor mice. We assumed that the RTE and MN populations were sampled from mice of age 5 weeks and 20 weeks. We then solved the agestructured PDE only, without tracking Ki67 expression (Appendix 1, Equations 26 and 27), until age 5 weeks or 20 weeks and then simulated the process of transfer by setting thymic influx to zero and observing the decay of the population. We then enumerated RTE and MN cells integrating the appropriate regions of the cell age distribution at the specified times posttransfer, defining RTE as cells with a postthymic age of 10 days or less, and MN cells being older than 28 days.
Appendix 5
Predicting RTE dynamics in Rag/Ki67 dual reporter mice
In Rag^{GFP} Ki67^{RFP} reporter mice, RTE are identified based on the transient expression of GFP, which we assume decays with first order kinetics. Our models estimated very low rates of division among naive CD4 and CD8 T cells, such that any dilution of GFP through division is minimal. There is therefore a simple and direct correlation between GFP expression $f$ and cell age $a$ within naive T cells, which we used to predict the fractions of GFP^{+} cells ($F$) within the naive compartment.
Using the favoured agestructured models described in Appendix 1, the age distribution of the naive T cell pool at mouse age $t$ is given by Equations 26 and 27 as $U(a,t)={U}_{i}(a,t)+{U}_{\theta}(a,t)$. The fraction of cells that are GFP^{+} is then
where $\overline{a}$ is the (unknown) time required for cells to transition from GFP^{+} to GFP^{}. Implicit in this calculation is the assumption that GFP levels are similar in Ki67^{} and Ki67^{+} RTE; mature SP cells are indeed very bright for GFP with minimal differences when stratified by Ki67 expression (data not shown). We then used the parameters derived from busulfan chimera data to generate $U(a,t)$, and estimated $\overline{a}$ by fitting Equation 35 to the timecourse of the GFP^{+} fraction within naive T cells observed in Rag/Ki67 dual reporter mice. The fits are shown in Figure 4C and D, red lines in the leftmost panels. We then generated the predicted timecourses of the GFP^{+} Ki67^{+} and GFP^{+} Ki67^{} fractions by multiplying $F$ with the predicted Ki67^{+} and Ki67^{} fractions that we derived by running the agedependent loss model from the age of the youngest GFP/Ki67 reporter mouse (11 days).
Appendix 6
Extending models back to nearbirth to predict the dynamics of naive T cells in neonates
In order to model the busulfan chimera data, in which mice underwent BMT at different ages, we needed to define the healthy dynamics of naive T cells. We took the approach of evolving the naive T cell pool from 1 day of age assuming that kinetic parameters were constant across the lifespan. In this way, every mouse began from the same ‘baseline’ state just after birth, and varied only in its age at BMT and in the level of stable chimerism within the bone marrow and thymus. The free parameters for this aspect of the model were then the (unknown) numbers of Ki67^{+} and Ki67^{} naive CD4 or CD8 T cells in a 1dayold mouse, ${N}_{0}^{+}$ and ${N}_{0}^{}$ respectively. For each model, we then calculated the predicted numbers of hostderived Ki67^{+} and Ki67^{} cells at the age of BMT (${t}_{\text{BMT}}$) by simulating forward from this initial condition, allowing for the continued influx of Ki67^{+} and Ki67^{} cells from the thymus as described in Appendix 2.
In cellage dependent models, we also defined the Ki67 distribution within the preexisting naive T cells (initial cohort) and among cells subsequently exported from the thymus emigrants. We also assumed a uniform distribution of cell ages ($\gamma (a)=1$) in 1dayold mice.
Initial cohort: Naive T cells present at mouse age ${t}_{0}=1$ day are enriched with Ki67^{+} cells (Figure 4A and B), and we defined the distribution of their normalised Ki67 expression $k$ over (0,1] to be $\psi (k)={e}^{k}/({e}^{k}1)$. We assume Ki67 is lost exponentially at a constant rate $\beta $, such that at a time $s$ after division $k(s)={e}^{\beta s}$. Our results were not sensitive to the form of $\psi (k)$ since it ‘washes out’ on a timescale of $1/\beta $ = 3.5 days.
We define the cutoff that separates Ki67^{+} from Ki67^{} cells to be $\overline{k}=1/e$, such that cells spend a time $1/\beta $ as Ki67^{+}. The fraction of Ki67^{+} cells in the initial cohort is then obtained by integrating the initial Ki67 distribution $\psi (k)$ from $k=\overline{k}$ to $k=1$, yielding
where $N}_{0$ is the total number of naive CD4 or CD8 T cells present in a 1dayold mouse, and was a free parameter in the models.
Naive T cells that subsequently enter the periphery:
The Ki67^{+} fraction within SP thymocytes ($\u03f5(t)$; Equation 30) varies with time, which implies that the distribution of Ki67 expression within naive T cells of age zero (i.e., just exported from the thymus) also change with time. We define this distribution to be $\varphi (k,t)$, such that the Ki67^{+} fraction among new RTE at mouse age $t$ is
We then defined an empirical step function $\varphi (k,t)$ to be consistent with this relationship;
To generate the predicted numbers and Ki67^{+} fractions of naive CD4 and CD8 T cells from age 5 days onwards (Figure 4), it was then straightforward to plot the model fits derived from the adult busulfan chimeric mice; note that none of the data derived from the younger, wildtype mice were used in the fitting.
Appendix 7
Hierarchical modelling of naive CD8 T cell timestamping data
The data from Reynaldi et al., 2019, shown in Figure 5, comprised longitudinal samples drawn from animals in five different age groups who were each treated with pulses of tamoxifen to label cohorts of CD8 T cells leaving the thymus. To use these data to estimate how cell loss rates vary as a function of both cell and host age, we took a hierarchical modelling approach. We allowed for animal and/or grouplevel variation in the initial numbers of RFP labelled cells in each animal ($N}_{0$) and in their initial loss rate (that is, the instantaneous net loss rate of cells of age zero, just exported from the thymus). We began by modeling the kinetics of labelled cells with the assumption that their net loss rate $\lambda $ varies with their postthymic age $a$ as $\lambda (a)={\lambda}_{0}{e}^{\gamma a}$. The population density of cells of age $a$ at mouse age $t$, $N(a,t)$, then obeys
with the boundary condition $N(T,a)={N}_{0}\delta (a)$, where $T$ is the mouse age at the time of treatment and $\delta (.)$ is the Dirac delta function. For mouse $i$, in age group $j$, at timepoint $k$, the observed cell numbers are ${y}_{ijk}$, where
We considered models in which the initial cell numbers $N}_{0$ and initial loss rate ${\lambda}_{0}$ for each mouse were either drawn from a single parent distribution or from distributions with means and variances specific to each age group. We defined priors for $\mu}_{N$, $\sigma}_{N$, $\mu}_{\lambda$, $\sigma}_{\lambda$ and $\gamma $ and fitted permutations of the hierarchical agestructured model to the time courses of labelled CD8 T cell numbers (Appendix 7—table 1). The bestfitting model exhibited groupspecific values of the initial RTE loss rate ${\lambda}_{0}$, and variation in the initial numbers of labelled cells ($N}_{0$) across mice, likely deriving from variations in the efficiency of tamoxifendriven labelling.
We then generated an explicit, empirical description of the variation in ${\lambda}_{0}$ with mouse age $t$, using the groupspecific estimates of ${\lambda}_{0}$ from the bestfitting hierarchical model. We used the following model of the net loss rate of cells of age $a$ at time $t$, with estimated parameters ${\lambda}_{h},Q,q$ and $\gamma $;
Appendix 8
Data availability
All code and data used in this study are available at https://github.com/sanketrane/T_cell_dynamics_birthdeath (copy archived at swh:1:rev:6e17ad8936bb34e966b5b920a943b6355981124e).
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Decision letter

Gabrielle T BelzReviewing Editor; The University of Queensland, Australia

Tadatsugu TaniguchiSenior Editor; Institute of Industrial Science, The University of Tokyo, Japan

Jose BorghansReviewer; UMCU, Netherlands
Our editorial process produces two outputs: (i) public reviews designed to be posted alongside the preprint for the benefit of readers; (ii) feedback on the manuscript for the authors, including requests for revisions, shown below. We also include an acceptance summary that explains what the editors found interesting or important about the work.
Decision letter after peer review:
Thank you for submitting your article "Towards a unified model of naive T cell dynamics across the lifespan" for consideration by eLife. Your article has been reviewed by 2 peer reviewers, and the evaluation has been overseen by a Reviewing Editor and Tadatsugu Taniguchi as the Senior Editor. The following individual involved in the review of your submission has agreed to reveal their identity: Jose Borghans (Reviewer #1).
The reviewers have discussed their reviews with one another, and the Reviewing Editor has drafted this to help you prepare a revised submission.
Essential revisions:
The reviewers have noted several elements of the work that require clarification:
1) Expansion of the explanation regarding several points of the model.
2) Attention to revisions of the figures and figure legends and provision of additional qualifications of the models (eg. quantitative validation such as correlation coefficients).
3) Consider points raised regarding additional approaches in modelling/testing the data.
Reviewer #1 (Recommendations for the authors):
– Some sentences/claims in the abstract are very general; please specify that this is the situation in (clean) mice.
– I found Figure S2 very helpful. I would advise incorporating it as the first main figure of the paper.
– If I understood correctly, in Figure 1B, the authors have plotted the combined data of mice who underwent BMT at different ages (between 7 and 25 weeks of age). I'm wondering whether the age at BMT may have an influence on the measurements. It may be interesting to use different labels to specify the age at BMT.
– Figure 2: Please plot the fits for all the models that were tested (even if only in supplemental information). That would help the reader judge visually how bad the fits of the alternative models were and would show in which ranges those models fit poorly. The lower relative weights of the alternative models suggest that the fits of the alternative models were really bad, but I would like to see that also visually.
– In Figure 2, I noticed that in the Ki67 panels, the early data points describing the donor cells are consistently missed by the best fits. Do the authors know why this is the case? Please explain. In the last Ki67 panel (1125 weeks), on the other hand, the donor data seem to be almost overfitted.
– On page 4, the authors state that "A prediction of any progressive increase in cell division rates with age is also inconsistent with observations of the frequencies of T cell receptor excision circles (TRECs) in adult mice." The authors also report, however, that the best fit of the naive CD8 T cell data is obtained for an agedependent division model in which the T cell division rate increases only very slightly with age. I'm not convinced that the situation would be inconsistent with the TREC data.
– On the same page, it is stated "This model was very similar to a neutral, homogeneous model and predicted that the normalised donor fraction approaches 1 in aged mice. This conclusion contradicts findings from our own and others' studies that demonstrated that models assuming homogeneity in naive CD8 T cells failed to capture their dynamics in adult and aged mice (220 months old)." The authors explain this as follows: "The statistical support for the agedependent division model here may derive from the relatively sparse observations in aged mice in this dataset, which define the asymptotic replacement fraction." I'm afraid I miss the point; I fail to understand this explanation.
– I am struggling to understand intuitively why the agedependent division model performs so poorly on the data of Figure 3B, while the model performs really well on the data of Figure 2. If the proliferation rate were to be fitted again on the basis of Figure 3, would the fit to the data of Figure 2 become very bad? What happens if both data sets would be fitted simultaneously?
– Discussion: It would be very much appreciated if the authors could discuss their view on the translation of these findings to the human and the dirtier mouse situation.
– In the Discussion it is mentioned that "Naive T cells proliferate under severely lymphopenic conditions in mice but acquire a memorylike phenotype. There is some evidence that this process occurs in healthy neonatal mice, suggesting that they are lymphopenic to some degree." I wanted to note that in Hoeven et al. (ref 22), we observed that this increased transition of RTE to the memory pool even occurs in the lymphoreplete situation, suggesting that it's related to the RTE nature of the cells rather than to the cell numbers in the host. The authors may want to consider adding this to their discussion.
Reviewer #2 (Recommendations for the authors):
I think it would have been helpful to have seen quantitative descriptions of model fits (e.g. R^{2} values) in the main body or captions, which I think would help substantiate some of the remarks.
For the qualitative fit in Figure 3, perhaps some speculative statement (if you have one) might be worth making to explain the quantitative discrepancy?
I'd suggest that all of the figures and captions need to be reread carefully for errors. It might be excessive, but in all captions (or recorded in SI if it is too unattractive in the main text) it would have been useful to know the number of samples/mice in each panel. These are also absent from the main text.
Figure 1 is missing caption entries for panels C and D.
Figure 2. Reporting goodness of fit statistics in the caption might be informative.
Figure 3. Perhaps the top two panels should be A and the bottom two B, with appropriate classification in the caption? As this is an important figure, I think it would be worth expanding on the simulation, even with some brief text, as there appears to be none presently.
In Figure 5B, what are the units for the yaxis?
In Figure 6B, if these are paired measurements, it might be appropriate to report the Spearman correlation coefficient to quantitatively support the (obvious anyway) relationship between SP and Naive.
Check that BMT, bone marrow transplant, and SP, for single positive, are defined the first time they are used.
Did you consider fitting your models to a bootstrapped version of your data to get estimates on the influence of experimental variability? That might, e.g., have given an indication as to the strength of belief in the CD8^{+} agedependent division model in the first section.
https://doi.org/10.7554/eLife.78168.sa1Author response
Essential revisions:
The reviewers have noted several elements of the work that require clarification:
1) Expansion of the explanation regarding several points of the model.
2) Attention to revisions of the figures and figure legends and provision of additional qualifications of the models (eg. quantitative validation such as correlation coefficients).
3) Consider points raised regarding additional approaches in modelling/testing the data.
We’re really grateful for the care both reviewers took in understanding and critiquing our manuscript. We feel that addressing their comments has helped us to improve it substantially.
Reviewer #1 (Recommendations for the authors):
– Some sentences/claims in the abstract are very general; please specify that this is the situation in (clean) mice.
We now clarify in several places that we are dealing with mice.
– I found Figure S2 very helpful. I would advise incorporating it as the first main figure of the paper.
This is a good idea – we’ve moved the model schematics to Figure 1 – see response below, too.
– If I understood correctly, in Figure 1B, the authors have plotted the combined data of mice who underwent BMT at different ages (between 7 and 25 weeks of age). I'm wondering whether the age at BMT may have an influence on the measurements. It may be interesting to use different labels to specify the age at BMT.
Yes, Figure 1 didn’t present the data in the clearest way. We were indeed careful to take both mouse age and age at BMT explicitly into account in the modelling. Specifically, we used the empirical description of singlepositive thymocyte numbers with mouse age as a surrogate for thymus output, and used the appropriate region of this function when modeling host and donor cell dynamics in each mouse, with its particular age at BMT.
The data were all shown more clearly in Figure 2, where mouse age and age at BMT are presented more logically. So we’ve removed the raw data from Figure 1, and inserted the model sketches here instead.
– Figure 2: Please plot the fits for all the models that were tested (even if only in supplemental information). That would help the reader judge visually how bad the fits of the alternative models were and would show in which ranges those models fit poorly. The lower relative weights of the alternative models suggest that the fits of the alternative models were really bad, but I would like to see that also visually.
Good idea – we’ve now added these to the SI. None of the alternative models look atrocious, even when viewed under the transformations they were fitted on (e.g. the logitscale for fractional measurements). When fitting to so many observations and different timecourses simultaneously, we find visual inspection doesn’t often square neatly with likelihood measures.
This problem (of strong statistical leanings vs. visually similar fits) highlights one reservation we have about relying on information criteria alone. The validity of any likelihoodbased measure depends on the assumptions underlying it – in particular, the error distributions (e.g. the assumption of lognormal noise in cell numbers). While we do our best to transform data to satisfy these error models, departures will occur. For this reason we used these model weights as a guide only, and focused instead on outofsample prediction.
– In Figure 2, I noticed that in the Ki67 panels, the early data points describing the donor cells are consistently missed by the best fits. Do the authors know why this is the case? Please explain. In the last Ki67 panel (1125 weeks), on the other hand, the donor data seem to be almost overfitted.
Thank you for picking this up! This was actually just an error in plotting the Ki67 predictions. They involve a subtraction step (age of mouse minus age at BMT) and this was done correctly for the fits to cell numbers and donor chimerism, but was incorrect for the Ki67 panels – so the fitted curves were slightly offset. We’ve corrected this.
– On page 4, the authors state that "A prediction of any progressive increase in cell division rates with age is also inconsistent with observations of the frequencies of T cell receptor excision circles (TRECs) in adult mice." The authors also report, however, that the best fit of the naive CD8 T cell data is obtained for an agedependent division model in which the T cell division rate increases only very slightly with age. I'm not convinced that the situation would be inconsistent with the TREC data.
Thanks, our logic was incorrect here – we’ve removed this argument.
– On the same page, it is stated "This model was very similar to a neutral, homogeneous model and predicted that the normalised donor fraction approaches 1 in aged mice. This conclusion contradicts findings from our own and others' studies that demonstrated that models assuming homogeneity in naive CD8 T cells failed to capture their dynamics in adult and aged mice (220 months old)." The authors explain this as follows: "The statistical support for the agedependent division model here may derive from the relatively sparse observations in aged mice in this dataset, which define the asymptotic replacement fraction." I'm afraid I miss the point; I fail to understand this explanation.
We’ve tried to clarify. Essentially it was about information. The cellagedependent kinetics models are constrained most tightly by the donor replacement curve, whose asymptote is rather noisy for CD8s. With these data, therefore, our intuition is that we are not able to reliably discriminate between agedependent loss and division for CD8s based on information criteria alone.
– I am struggling to understand intuitively why the agedependent division model performs so poorly on the data of Figure 3B, while the model performs really well on the data of Figure 2. If the proliferation rate were to be fitted again on the basis of Figure 3, would the fit to the data of Figure 2 become very bad? What happens if both data sets would be fitted simultaneously?
This comment was helpful – we now explain that Figure 3 just serves as confirmation of strong cellage effects for both CD4 and CD8 T cells; therefore, it excludes homogeneous models and lends weight to our decision to reject the fitted agedependent division model for CD8s.
The agedependent division models do poorly in Figure 3 because, for both CD4 and CD8 T cells, the predicted magnitudes of the division rate and its rate of increase with cell age are both very small when they are fitted to the data in Figure 2. This makes both the CD4 and CD8 agedependent division models similar to neutral models with constant rates of division and loss, with no difference in the loss rates of RTE vs mature naive T cells.
Direct fitting of the agedependent division model to the data in Figure 3 is difficult because the age of the mice in that experiment were not specified precisely, the MN cell donors were a mix of WT and thymectomised mice, and there is also ambiguity in the mapping of GFPpositivity to cell age, needed to define an RTE. We have extended our discussion of this point.
We went backandforth on the issue of fitting the models to everything at once vs fitting on data from one set of experiments and using outofsample prediction to challenge the models. Batch effects can be quite significant — for example, when comparing the dynamics of endogenous cells in Figure 2 to those of adoptively transferred cells in Figure 3; different housing facilities, reagents, and so on. So the correct way to fit simultaneously to observations from different labs and experimental systems would be to use hierarchical models. That would leave only information criteria with which to weigh support for different models, which as we describe above we find a little unsatisfactory. So in this study we tried to use outofsample prediction — even if just describing trends in new datasets — as a means of evaluating the models.
– Discussion: It would be very much appreciated if the authors could discuss their view on the translation of these findings to the human and the dirtier mouse situation.
Yes, this is important. We have added a section to the Discussion regarding selfrenewal clonality, and added comments on T cell behaviour in humans and ‘dirty’ mice at other points in the text.
– In the Discussion it is mentioned that "Naive T cells proliferate under severely lymphopenic conditions in mice but acquire a memorylike phenotype. There is some evidence that this process occurs in healthy neonatal mice, suggesting that they are lymphopenic to some degree." I wanted to note that in Hoeven et al. (ref 22), we observed that this increased transition of RTE to the memory pool even occurs in the lymphoreplete situation, suggesting that it's related to the RTE nature of the cells rather than to the cell numbers in the host. The authors may want to consider adding this to their discussion.
Thank you – we have added this point to our discussion and reworded that argument. (Your result also suggests to us that the continuous recruitment of new cells into memoryphenotype populations in the absence of infection likely derives substantially from RTE, rather than from random recruitment from the naive pool as a whole). Your study also reminded us to discuss the CD4 cotransfer experiment in Dong et al.. We also have difficulty reconciling their results with other studies, but we speculate that it may derive from the step of labelling one of the transferred populations (the bulk naive cells, which were lost fastest) with CFSE, but not the other.
Reviewer #2 (Recommendations for the authors):
I think it would have been helpful to have seen quantitative descriptions of model fits (e.g. R^{2} values) in the main body or captions, which I think would help substantiate some of the remarks.
We agree that the measures of model fit were not prominent enough. We moved the table showing model weights from the SI to the main text, and we have added a more complete description of how these weights are calculated (they reflect each models’ average outofsample prediction error).
R^{2} is just a measure of closeness of fit and tends to 1 for overfitted data – so we don’t feel it is so helpful here where our goal to trade off goodness of fit vs. model complexity.
For the qualitative fit in Figure 3, perhaps some speculative statement (if you have one) might be worth making to explain the quantitative discrepancy?
We did speculate in the discussion, but have expanded on it. We believe the disparity stems from a combination of the unknown ages of the mice used in their experiments (in these models, we need to know the full agedistribution in order to calculate the net loss rate); potential effects of cell manipulation (adoptive transfer vs dynamics of endogenous cells); and uncertainty in the age cutoff defining RTE.
I'd suggest that all of the figures and captions need to be reread carefully for errors. It might be excessive, but in all captions (or recorded in SI if it is too unattractive in the main text) it would have been useful to know the number of samples/mice in each panel. These are also absent from the main text.
Figure 1 is missing caption entries for panels C and D.
Thanks – we’ve gone through all of the captions, made corrections and added sample sizes.
Figure 2. Reporting goodness of fit statistics in the caption might be informative.
See above for our comment re: goodness of fit.
Figure 3. Perhaps the top two panels should be A and the bottom two B, with appropriate classification in the caption? As this is an important figure, I think it would be worth expanding on the simulation, even with some brief text, as there appears to be none presently.
We reframed the discussion of Figure 3 and have actually compressed the figure a little — we feel it’s clearer now. We have also added a new section in the SI that details how the simulations were performed.
In Figure 5B, what are the units for the yaxis?
Net loss rate (loss rate – division rate) – this is the inverse of the expected clonal lifespan. We’ve added this. Also for panel 5A.
In Figure 6B, if these are paired measurements, it might be appropriate to report the Spearman correlation coefficient to quantitatively support the (obvious anyway) relationship between SP and Naive.
This is a good idea – we now quote the Spearman rank correlation coefficients.
Check that BMT, bone marrow transplant, and SP, for single positive, are defined the first time they are used.
Thank you for picking these up.
Did you consider fitting your models to a bootstrapped version of your data to get estimates on the influence of experimental variability? That might, e.g., have given an indication as to the strength of belief in the CD8^{+} agedependent division model in the first section.
We are using a Bayesian fitting approach – the envelopes we show on the model predictions are essentially an analog of the envelopes you would get by bootstrapping the data and refitting (they represent the 2.5 and 97.5 percentiles of the model prediction taken across the posterior distributions of all the model parameters).
https://doi.org/10.7554/eLife.78168.sa2Article and author information
Author details
Funding
National Institutes of Health (R01AI093870)
 Andrew J Yates
University College London (Medical Research Council UK programme grant MR/P011225/1)
 Benedict Seddon
The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.
Ethics
All of the animals were handled according to UK home office regulations (licence PPL PP2330953) and institutional animal care and use committee (IACUC) protocols at University College London.
Senior Editor
 Tadatsugu Taniguchi, Institute of Industrial Science, The University of Tokyo, Japan
Reviewing Editor
 Gabrielle T Belz, The University of Queensland, Australia
Reviewer
 Jose Borghans, UMCU, Netherlands
Publication history
 Preprint posted: January 8, 2022 (view preprint)
 Received: February 25, 2022
 Accepted: June 8, 2022
 Accepted Manuscript published: June 9, 2022 (version 1)
 Accepted Manuscript updated: June 10, 2022 (version 2)
 Version of Record published: August 3, 2022 (version 3)
Copyright
© 2022, Rane, Hogan et al.
This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.
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