Dynamical crossover in a stochastic model of cell fate decision

We study the asymptotic behaviors of stochastic cell fate decision between proliferation and differentiation. We propose a model of a self-replicating Langevin system, where cells choose their fate (i.e., proliferation or differentiation) depending on local cell density. Based on this model, we propose a scenario for multicellular organisms to maintain the density of cells (i.e., homeostasis) through finite-ranged cell-cell interactions. Furthermore, we numerically show that the distribution of the number of descendant cells changes over time, thus unifying the previously proposed two models regarding homeostasis: the critical birth death process and the voter model. Our results provide a general platform for the study of stochastic cell fate decision in terms of nonequilibrium statistical mechanics.

Figure 1

(a) Schematic showing a cellular label tracking experiment. Proliferating cells (pink) confined to the basal layer of the tissue undergo proliferation within the basal layer, as well as differentiation toward the upper layers. Cell fate is tracked by monitoring the labeled clones (yellow), which either expand or shrink by proliferation and differentiation, respectively. (b) Schematic of VM. (c) Schematic of CBD.

Figure 2

(a) Schematic of the self-replicating Langevin system. (b) An example of the density dependence of the birth and death rates. (c) The phase diagram via structure factor analysis, which is detailed in Appendix pp1. The gray line separates the parameter space into the linearly stable and unstable regions. Here, $\alpha$ represents the sensitivity of the growth rate around the steady-state density, and $\beta$ is the effective diffusion constant that depends on the strength of the cell-cell adhesion $K$. $\mu /\sigma$ is the relative noise strength. The definitions of $\alpha ,\beta ,\sigma$, and $\mu$ are given in Eq. (A13). Our clonal analysis is performed in the spatially homogeneous region.

Figure 3

Numerical results of the asymptotic clone size statistics in clonal analysis. We take $l_0=1$ and $L_{\mathrm{Sys}}=1000$. (a) The average clone size $l\left(t\right)$ for $L/l_0=10,20,30,40$, and 100. (b) The clone size distributions $C_n\left(t\right)$ at different time points for $L/l_0=100$ plotted against $n/l\left(t\right)$. (c), (d) $C_n\left(t\right)$ plotted against $n/l\left(t\right)$ for the large $L$ case ($L=L_{\mathrm{Sys}}$) and the small $L$ case ($L/l_0=1.25$), respectively. The insets show $l\left(t\right)-l_0$ against time $\lambda t$.

Figure 4

Scaling form of the average clone size $l\left(t\right)$. $l\left(t\right)-l_0$ is scaled by $L$ and plotted against $t/t_c\left(L\right)$ for $L/l_0=10,20,30,40$, and 100.

Figure 5

Schematic showing the coarse graining of cell population. After coarse graining, the clonal dynamics in the long time scale is similar to the case of small $L$ limit.

Figure 6

Two-body potential $u\left(X\right)$ for $K=1,l_0=1$, and $r_c=1.5$.

Figure 7

Structure factor $S_{\mathrm{ss}}\left(k\right)$ associated with (a) Gaussian and (b) top-hat kernel. A nontrivial peak appears after $L$ exceeds a certain threshold $L_c$. We set the parameters as $\alpha =1,\sigma =1,D=5,\mu =10,K=1000,l_0=1$, and $r_c=1.1$.