Mechanism of murine epidermal maintenance: Cell division and the voter model

The dynamics of a genetically labeled cell population may be used to infer the laws of cell division in mammalian tissue. Recently, we showed that in mouse tail skin, where proliferating cells are confined to a two-dimensional layer, cells proliferate and differentiate according to a simple stochastic model of cell division involving just one type of proliferating cell that may divide both symmetrically and asymmetrically. Curiously, these simple rules provide excellent predictions of the cell population dynamics without having to address the cells’ spatial distribution. Yet, if the spatial behavior of cells is addressed by allowing cells to diffuse at random, one deduces that density fluctuations destroy tissue confluence, implying some hidden degree of spatial regulation of cell division. To infer the mechanism of spatial regulation, we consider a two-dimensional model of cell fate that preserves the overall population dynamics. By identifying the resulting behavior with a three-species variation of the voter model, we predict that proliferating cells in the basal layer should cluster. Analysis of empirical correlations of cells stained for proliferation activity confirms that the expected clustering behavior is indeed seen in nature. As well as explaining how cells maintain a uniform two-dimensional density, these findings present an interesting experimental example of voter-model statistics in biology.

Figure 1

(Color online) (a) Schematic cross section of murine interfollicular epidermis. 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 online) (iii), eventually becoming shed at the surface. (b) Two examples of typical clones 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 $20\mu \text{m}$.

Figure 2

(Color online) Schematic motivating the proposed lattice model defined in Eq. (2). Top: A cartoon of cells within the basal layer, showing the exit of a cell (light gray) through upward migration into the suprabasal layers, concurrent with the division of a nearby progenitor cell [dark gray (red)]. To ensure continuity of the basal layer, it is postulated that cells rearrange to maintain a uniform cell density (wide arrow). Bottom: In a simple lattice model that captures the essence of the steady-state dynamics, the exit of a cell from the basal layer gives rise to a vacant lattice site (light gray), which rapidly diffuses by exchanging position with adjacent cells (white). Upon coming into contact with a proliferating cell, the latter may divide and replace the vacancy with a daughter cell (striped).

Figure 3

(Color online) Comparison of the exact basal layer lattice model (2) (red curves) with the simplified model (4) (black). (a) The order parameter $1/c_{AB}$ is plotted against $\text{ln}t$. The curves correspond to the simplified model with $r=1/2$ (dashed black) and $r=1/4$ (dotted), and to the exact model with $r=1/2$ (dashed red) and $r=0.08$ (solid curve). For the simplified model, the analytical expression for $c_{AB}$ was used. For the exact model, the results were obtained from numerical simulation on hexagonal lattices of size $N=1024\times 1024$, with $\rho =0.22$ and setting $\lambda =1$. Inset: The correlation function plotted against logarithm of the distance, $\text{ln}x$, evaluated from numerical simulation and Eq. (17). The time-invariant ratio $\left[\rho -C\left(x,t\right)\right]/c_{AB}$ is plotted at $\lambda t=50$, 100, 1000 for the classical voter model (dashed), and at $\lambda t=350$, 3500, 7000 for the exact model with $r=1/2$ (solid curves). Consistent with the expected behavior [see Eq. (15)], the curves overlap and are convex near $x=1$, becoming linear at $x>1$. (b) The long-time asymptotic roughness constant $\Omega$ plotted against $r$ for both models. Data points correspond to the values calculated using Eq. (17) from numerical simulations such as shown in Fig. 4, using the algorithm described in Sec. 3A. Error bars result from fluctuations due to finite-size effects, as evaluated by considering the random variation in $\Omega$ over the time of the simulation. The black curve gives the theoretical value of $\Omega$ for the simplified (monomer) model on a hexagonal lattice, while the red curve gives the best fit to the numerical results for the exact model.

Figure 4

Cellular automaton simulation of process (2), showing the distribution of progenitor cells (black hexagons) on a lattice of $N=200\times 200$, and using the experimental branching ratio $r=0.08$ and progenitor cell fraction $\rho =0.22$. The frame shown corresponds to an evolution time of $t=30/\lambda$, where $t=0$ corresponds to random initial conditions. White areas are fully occupied by postmitotic cells.

Figure 5

(Color online) Schematic demonstrating the variation in the effective division rate in process (2), accounting for the quantitative variation in behavior between processes (2, 4). The reactive interface between progenitor cells [light gray (red)] and postmitotic cells [dark gray (blue)] is indicated by the thick gray line, with a faster division rate shown in light gray and slower division rate shown in dark gray. On the left, a smooth interface results in faster cell division, as few progenitors on the boundary must compensate for the migration of many postmitotic cells in the bulk. On the right, the “rough” interface results in a variation between fast and slow division rates, due to restricted access of vacancies into the rough interfacial regions. Referring to the discussion at the end of Sec. 2B, this leads to increased stability of rough compared to smooth interfaces.

Figure 6

(Color online) (a) Comparison of the empirical clone size distribution (data points) to predictions of process (2) (solid curves), as obtained from Monte Carlo simulations of ${10}^4$ labeled clones. The distributions are plotted in terms of the probability ${\mathcal{P}}_k\left(t\right)$ for a clone to have between $2^{k-1}+1$ and $2^k$ basal layer cells at time $t$ after labeling, normalized to include only clones with two or more cells in the basal layer $\left(k\ge 1\right)$. The empirical data are reproduced from Ref. [5]. (b) Examples of the basal layer structure of several large late-stage clones evolving according to process (2), starting from a uniform random distribution of unlabeled cells, with one single type-$A$ cell labeled at $t=0$. Light and dark gray (red and blue online) hexagons indicate sites occupied by type $A$ and $B$ cells, respectively. White areas are populated by unlabeled cells. Each frame corresponds to the progeny of one initially labeled cell.

Figure 7

(Color online) Loss of cohesiveness with increasing relative hopping rate $\sigma /{\lambda }^{\prime }$ shown through examples of late-stage clone simulations. Light and dark gray (red and blue online) hexagons indicate sites occupied by type $A$ and $B$ cells, respectively, from the same clone. White areas are populated by unlabelled cells. The simulations used the same parameter set as described above for the clones in Fig. 6, but using the parameter values $\sigma =20/\text{week}$ and $200/\text{week}$ with ${\lambda }^{\prime }=10000/\text{week}$ for the top left and top right clones, respectively; $\sigma =200/\text{week}$ and $2000/\text{week}$ with ${\lambda }^{\prime }=200/\text{week}$ for the bottom left and bottom right clones. For the latter, which is clearly unphysical, the scale has been reduced twofold to demonstrate the wide dispersion of labeled cells. The examples demonstrate that clones are cohesive when $\sigma /{\lambda }^{\prime }\ll 1$, but dispersive otherwise.

Figure 8

(Color online) (a) Confocal micrograph of whole mounted mouse tail skin IFE, showing the two-dimensional basal layer immunostained for the nuclear marker DAPI [dark gray (blue)], and the proliferation marker Ki67 [light gray (red)]. The area surrounding the stained nuclei is occupied by unstained cell cytoplasm (black). (b) The empirical radial correlation function $C\left(x\right)$, as defined in Eq. (18). Data points show results obtained by analysis of the Ki67-stained epidermal whole mounts exemplified in (a), taken from a mouse aged 8 weeks; the dashed line shows the fit to the analytical form of the correlation function predicted by process (2), $C\left(x,t\right)=a\left(t\right)-{\xi }^{-1}\left(t\right)\text{ln}x$, with $x$ in units of the average cell diameter [see Eqs. (15, 16)], and where the single time point $t$ is fixed by the age of the mouse and by initial conditions (see discussion in Sec. 4). From the fitted slope one may extract the correlation length $\xi \left(t\right)$ defined in Eq. (16), giving $\xi =14.7$ cell diameters.