Table 1. Non-universal dependencies of the cluster size distribution.

Analytical expressions for the cluster size distribution (top row in each cell) and average cluster size (bottom row). Shown are expressions in situations, where labelling density is clonal, labelling density is almost clonal but clones are subject to fragmentation, and where both merging and fragmentation of clones occur (left to right). As merging and fragmentation both result from tissue rearrangements merging should always imply fragmentation. Time is measured in units of the cell cycle time. Expressions are valid after convergence to the scaling regime, when the typical cluster size is much larger than the size of single cells, and in the mean-field limit, which is a good approximation for two and three dimensional tissues. In addition, it is assumed that the full spectrum of cluster sizes can be experimentally resolved. If clones fragment but not merge fragmentation and growth ultimately compensate to lead to a stationary distribution. In case of clonal merging and fragmentation expressions give empirical approximations, where depends on the details of the merging and fragmentation processes (see Supplemental Theory).

Growth mode	Clonal	Fragmentation	Merging & fragmentation
Exponential	〈x〉−1 exp (−x/〈x〉) 〈x〉 = exp(t)	φ exp (−φ−1x) 〈x〉 = φ−1	$\{\begin{array}{ll}{(x/\langle x\rangle )}^{\alpha }\hfill & x\ll \langle x\rangle \hfill \\ \text{exp}(-x/\langle x\rangle )\hfill & x\gg \langle x\rangle \hfill \end{array}$ 〈x〉 = exp(t)
Linear	$\frac{1}{\sqrt{2\pi \langle x\rangle }}\text{exp}\left[-\frac{{(x-\langle x\rangle )}^2}{2\langle x\rangle }\right]$ 〈x〉 = t	$\phi x\left(2+\sqrt{\phi }x\right)\text{exp}\left(-\sqrt{\phi }x-\frac{\phi }{2}{x}^2\right)$ 〈x〉 = φ−1/2	$\{\begin{array}{ll}{(x/\langle x\rangle )}^{\alpha }\hfill & x\ll \langle x\rangle \hfill \\ \text{exp}(-x/\langle x\rangle )\hfill & x\gg \langle x\rangle \hfill \end{array}$ 〈x〉 = t
Homeostasis	〈x〉−1 exp [−x / 〈x〉] (see Ref. 6) 〈x〉 ∝ t	J(x) (see Ref. 24) 〈x〉 = const.	$\{\begin{array}{ll}{(x/\langle x\rangle )}^{\alpha }\hfill & x\ll \langle x\rangle \hfill \\ \text{exp}(-x/\langle x\rangle )\hfill & x\gg \langle x\rangle \hfill \end{array}$ 〈x〉 = const.