Exact solution of a two-type branching process: Clone size distribution in cell division kinetics

We study a two-type branching process which provides excellent description of experimental data on cell dynamics in skin tissue (Clayton et al., 2007). The model involves only a single type of progenitor cell, and does not require support from a self-renewed population of stem cells. The progenitor cells divide and may differentiate into post-mitotic cells. We derive an exact solution of this model in terms of generating functions for the total number of cells, and for the number of cells of different types. We also deduce large time asymptotic behaviors drawing on our exact results, and on an independent diffusion approximation.

💡 Research Summary

The paper presents a mathematically rigorous analysis of a two‑type branching process that accurately captures the dynamics of epidermal cell clones observed in mouse skin experiments (Clayton et al., 2007). The model assumes a single population of progenitor (P) cells that can undergo three stochastic division outcomes: symmetric self‑renewal (P → P + P) with probability p, asymmetric division (P → P + D) with probability q, and symmetric differentiation (P → D + D) with probability r = 1 − p − q. The differentiated (D) cells are post‑mitotic and do not divide further. No cell death is considered, and the system is spatially homogeneous.

To obtain exact results, the authors introduce the joint probability generating function
(G(x,y,t)=\sum_{m,n}P_{m,n}(t)x^{m}y^{n}),
where (P_{m,n}(t)) is the probability of having m progenitor and n differentiated cells at time t. From the master equation they derive a nonlinear partial differential equation for G, which, after the change of variables (u=1-x), (v=1-y) and a Laplace transform in time, becomes tractable. By employing the method of characteristics and suitable integrating factors they solve the equation in closed form. The solution can be expressed as a combination of exponential and beta‑function terms, allowing explicit formulas for the marginal distributions of total cell number (N=m+n) and progenitor count m.

The exact solution reveals several key dynamical features. The mean progenitor number grows linearly with time, (\langle m\rangle\approx(p-r)t), while the variance scales as (\sigma^{2}\sim t), indicating diffusive fluctuations. In the long‑time limit the total clone size distribution acquires a heavy tail of the form
(P(N,t)\propto N^{-3/2}\exp