Controlling tissue size by active fracture

TL;DR

The paper addresses how the size of cell clusters can emerge from the competition between growth and motility-driven fracture. It develops a one-dimensional active chain model of self-propelled cells connected by springs, deriving the rupture rate $k_b$ and mapping rupture dynamics to an effective temperature, yielding a direct link between motility, cell-cell mechanics, and cluster size via the ratio $\gamma=k_b/k_d$. Key contributions include exact steady-state cluster-size distributions $p_n$ that depend only on $\gamma$, a universal survival form $S(t)=e^{-k_d t}$ under all-cell growth, and supportive two-dimensional simulations that show qualitative agreement with the 1D theory in relevant regimes. The framework offers quantitative predictions for how changes in cell speed, persistence, and adhesion regulate organ or organism size, with relevance to germline cysts, cancer cell clusters, and confinement-driven tissues.

Abstract

Groups of cells, including clusters of cancerous cells, multicellular organisms, and developing organs, may both grow and break apart. What physical factors control these fractures? In these processes, what sets the eventual size of clusters? We first develop a one-dimensional framework for understanding cell clusters that can fragment due to cell motility using an active particle model. We compute analytically how the break rate of cell-cell junctions depends on cell speed, cell persistence, and cell-cell junction properties. Next, we find the cluster size distributions, which differ depending on whether all cells can divide or only the cells on the edge of the cluster divide. Cluster size distributions depend solely on the ratio of the break rate to the growth rate - allowing us to predict how cluster size and variability depend on cell motility and cell-cell mechanics. Our results suggest that organisms can achieve better size control when cell division is restricted to the cluster boundaries or when fracture can be localized to the cluster center. Additionally, we derive a universal survival probability for an intact cluster $S(t)=\mathrm{e}^{-k_d t}$ at steady state if all cells can divide, which is independent of the rupture kinetics and depends solely on the cell division rate $k_d$. Finally, we further corroborate the one-dimensional analytics with two-dimensional simulations, finding quantitative agreement with some - but not all - elements of the theory across a wide range of cell motility. Our results link the general physics problem of a collective active escape over a barrier to size control, providing a quantitative measure of how motility can regulate organ or organism size.

Figures (13)

• Figure 1: Schematic illustration of cell division and fracture. Red lines denote intercellular bridges that undergo fracture.
• Figure 2: (a) Illustration of the active chain model. Cells are connected by springs; each cell has active velocity $v_n$. (b) Variance of the spring stretch $\langle\delta\ell_n^2\rangle$ as a function of $\mu k\tau$. Gray dashed line is the thermal variance $\langle\delta\ell^2\rangle_\mathrm{th}=D/\mu k$. We change $\mu k\tau$ by varying $\tau$ while fixing $\mu k$ in the simulations. Red dashed line is the two-particle result $\langle\delta\ell^2\rangle_2$. (c) Mean escape time $\tau_\mathrm{esc}=1/k_b$. Empty circles are simulation results with $N=1000$ cells; solid lines are theory, Eq. \ref{['eq:break_rate']}.
• Figure 3: Growth and fracture of cell groups under different assumptions. (a) All cells can divide. After a rupture occurs, one daughter chain is selected at random. (b) Only the cell at one end of the chain can divide. (c) Only the cells at the two ends can divide. (d) Potential rupture points when a minimum cluster size $N_\textrm{min}$ is enforced. Rupture can only occur when the chain length $n\geqslant 2N_\textrm{min}$. Red cells have a non-zero division rate $k_d$, while blue cells have a division rate of zero; gray crosses indicate actual or potential rupture points.
• Figure 4: (a) Steady-state distribution of chain length $p_n$ for the three growth modes, with $\gamma=0.1$ for the all-cell growth and two-end growth models, and $\gamma$ replaced by $\gamma' =\gamma/2=0.05$ for the one-end growth model (doubled division rate). Crosses represent simulation results, while lines correspond to theoretical predictions. Panels (b)--(d) display the mean $\langle n\rangle$, variance $\langle n^2\rangle-\langle n\rangle^2$, and coefficient of variation $\mathrm{CV}=\sigma_n/\langle n\rangle$, where $\sigma_n=\sqrt{\langle n^2\rangle-\langle n\rangle^2}$. Purple dashed lines are the one-end result after doubling the division rate [$\gamma \to \gamma'=\gamma/2$ in Eq. \ref{['eq:ss_solution_one']}], while the purple dotted lines show Eq. \ref{['eq:ss_solution_one']}. (e) Steady-state distribution $p_n$ when minimum cluster size $N_{\min}=200$. Crosses are simulation results, lines are theoretical predictions by the matrix method. Panels (f)--(h) show how mean, variance, and $\mathrm{CV}$ vary with $N_\mathrm{min}$ in the all-cell growth model. Gray dashed lines show predictions assuming log-uniform distributions [Eq. \ref{['eq:loguniform']}].
• Figure 5: A modest change in cell motility can account for the observed decrease in fracture rates over embryonic time. The upper panel shows a logistic decrease in the typical cell speed, $v_\textrm{rms}=\sqrt{D/\tau}$ which we hypothesize underlies the fracture rate reduction. In the lower panel, symbols with error bars are experimental data from Fig. 4C of Ref. levy2024tug; the red dashed line shows the corresponding theoretical predictions for the break rate $k_b$, calculated from the upper panel's $v_\textrm{rms}$ using Eq. \ref{['eq:break_rate']}.