Kinetics of cell division in epidermal maintenance

The rules governing cell division and differentiation are central to understanding the mechanisms of development, aging, and cancer. By utilizing inducible genetic labeling, recent studies have shown that the clonal population in transgenic mouse epidermis can be tracked in vivo. Drawing on these results, we explain how clonal fate data may be used to infer the rules of cell division and differentiation underlying the maintenance of adult murine tail-skin. We show that the rates of cell division and differentiation may be evaluated by considering the long-time and short-time clone fate data, and that the data is consistent with cells dividing independently rather than synchronously. Motivated by these findings, we consider a mechanism for cancer onset based closely on the model for normal adult skin. By analyzing the expected changes to clonal fate in cancer emerging from a simple two-stage mutation, we propose that clonal fate data may provide a novel method for studying the earliest stages of the disease.

Figure 1

(Color online) (a) Schematic cross section of murine interfollicular epidermis (IFE) showing the organization of cells within different layers and indicating the architecture of typical labeled clones. Proliferating cells (gray) are confined to the basal layer (labeled i); differentiated cells migrate through the superbasal layers (ii), where they flatten into cornified cells, losing their nuclei and assembling a cornified envelope (green) (iii), eventually becoming shed at the surface. The shaded regions (yellow) indicate two distinct clones, the progeny of single basal layer cells labeled at induction. While the clone on the right retains at least one labeled cell in the basal layer, the clone on the left hand side has detached from the basal layer indicating that all of the cells have stopped proliferating. The former are designated as “persisting clones” and contribute to the clone size distributions, while the latter, being difficult to resolve reliably, are excluded from experimental consideration. (b) Typical example of a clone acquired at a late time point, viewed from the basal layer surface. Cell nuclei are labeled blue; the hereditary clone marker (EYFP) appears yellow. Scale bar: $20\mu \mathrm{m}$.

Figure 2

(Color online) Top: Theoretical lineage for the first $12\text{weeks}$ post-labeling of (a) a detached clone in which all cells have undergone a transition to terminal differentiation by week 12, and (b) a persisting clone in which some of the cells maintain a proliferative capacity, according to model (2). Circles indicate progenitor cells (P), differentiated cells (D), and suprabasal cells (SB). Note that, because the birth-death process (2) is Markovian, the lifetime of cells is drawn from a Poisson distribution with no strict minimum or maximum lifetime. The statistics of such lineage trees do not change significantly when we account for a latency period between divisions that is much shorter than the mean cell lifetime (see discussion in Sec. 2C). Bottom: The total number of proliferating, differentiated, and suprabasal cells for the two clones as a function of time.

Figure 3

(Color online) Mean number of basal layer cells in persisting clones. The experimental data (circles) show an inexorable increase in the size of persisting clones over the entire time course of the experiment. The behavior at short times (from $2–6\text{weeks}$) and at long times (beyond $13\text{weeks}$) follows the two simple analytical approximations described in the main text (lower and upper dashed curves). For times earlier than two weeks (referring to Sec. 2D), clones remain approximately one cell in size. The experimental data are consistent with the behavior predicted by process (2) (black line) when it is assumed that only A-type cells are labeled at induction. In contrast, assuming that A- and B-type cells label in proportion to their steady-state population leads to an underestimate of average clone size between two and six weeks (lower curve, red online), as does the assumption that type-B cells label with better efficiency (not shown). Inset: The underlying distribution of basal cells per clone at $2\text{weeks}$ and $26\text{weeks}$ post-labeling. The data is binned by cell count in increasing powers of 2.

Figure 4

(Color online) Time dependence of the grouped size distribution of persisting clones, ${\mathcal{P}}_k^{\mathrm{pers.}}\left(t\right)$, plotted as a function of the rescaled time coordinate $t∕2^k\mapsto t$. The data points show measurements [extracted from data such as shown in Fig. 3(inset), given fully in Ref. 15], while the solid curves show the probability distributions associated with the nonequilibrium process (2) for the basal-layer clone population as obtained by a numerical solution of the master equation (3). (Error bars refer to standard error of the mean.) At long times, the data converge onto a universal curve (dashed line), which one may identify with the form given in Eq. (7). The rescaling compresses the time axis for larger clones, so that the large-clone distributions appear to converge much earlier onto the universal curve.

Figure 5

(Color online) (a) Fit of Eq. (9) to the short-time clone size distributions. The data are optimally fitted by Eq. (9) using the value $\lambda =1.1∕\text{week}$ and the empirical value $\rho =0.22$ (solid lines show fit). To ensure integrity of the analysis, data for times earlier than week 2 have been excluded (see Sec. 2D). (b) Linearization of the long-time asymptotic data using the “inverse” scaling function $f^{-1}(2^k∕t)$ for $t⩾13\text{weeks}$ and $k⩾3$ (see main text). (c) Likelihood of the overall division rate $\lambda$ (red) and the symmetric division rate $r\lambda$ (black), as assessed from a ${\chi }^2$ test of the numerical solution to Eq. (3) [29]. A fit to the basal-layer clone size distribution alone (solid curve) is less discriminatory than a simultaneous fit to both the basal-layer and total clone size distributions (dashed). The likelihood of $r\lambda$ is shown for the optimal value of $\lambda$, and vice versa. Inset: Referring to Sec. 2C, the likelihood is plotted against the duration of a latency period $\left({\tau }_{\mathrm{min.}}\right)$ immediately following cell division, and assuming that division events are otherwise independent (see main text).

Figure 6

(Color online). (a) Examples of progenitor cell cycle-time distributions with the same average cycle time $1∕\lambda$ $(\lambda =1.1∕\text{week})$, and with a latency period of ${\tau }_{\mathrm{min.}}=12\mathrm{h}$ introduced between consecutive cell divisions (hashed region). The case $\kappa =1$ corresponds to a model of independent cell division such as that assumed in Sec. 2A, but now accounting for an initial latency period. The case $\kappa \to \infty$ (black dashed) corresponds to all cells having an exact cell cycle time of $1∕\lambda$. Note that small values of $\kappa$ allow for both very short and very long cycle times. (b) Using Monte Carlo simulations of process (2), the clone size distributions predicted by each of the different cycle-time distributions in (a) are compared with the empirical data. Data points show the size distribution of persisting clones including suprabasal layer cells over the first $6\text{weeks}$ post-labeling (extracted from data given fully in Ref. 15; for legend see Fig. 4), and the theoretical curves correspond to the same legend as in (a). All of the models give an optimal fit with the same value of $\lambda =1.1∕\text{week}$, $r=0.08$.

Figure 7

(Color online) (a) The total number of basal layer cells per labeled clone during the onset of cancer according to process (10). The figure was plotted by numerically integrating Eq. (11) using the empirical value $r\lambda =0.088∕\text{week}$ found for normal skin, together with hypothetical values of the cancer growth parameters $\nu =0.1r\lambda$, $\mu =10r\lambda$, and $\Delta =0.5$. To compare with normal skin, the predicted clone size distributions are replotted against the rescaled time coordinate $t∕2^k\mapsto t$ in (a) inset. In contrast with Fig. 4, here the curves fail to converge. In (b), the same curves are shown converge onto the universal form given in Eq. (11) (dashed) when they are plotted against a new rescaled time $t\mapsto t_k^{\prime }=t+k\ln 2∕\Delta \mu$. Note that the large-clone distributions converge rapidly, whereas the distributions for smaller clones are affected by the non-negligible contribution of noncancerous (A) cells to the small-clone size distribution.