Exact, time-independent estimation of clone size distributions in normal and mutated cells

Biological tools such as genetic lineage tracing, three-dimensional confocal microscopy and next-generation DNA sequencing are providing new ways to quantify the distribution of clones of normal and mutated cells. Understanding population-wide clone size distributions in vivo is complicated by multiple cell types within observed tissues, and overlapping birth and death processes. This has led to the increased need for mathematically informed models to understand their biological significance. Standard approaches usually require knowledge of clonal age. We show that modelling on clone size independent of time is an alternative method that offers certain analytical advantages; it can help parametrize these models, and obtain distributions for counts of mutated or proliferating cells, for example. When applied to a general birth-death process common in epithelial progenitors, this takes the form of a gambler's ruin problem, the solution of which relates to counting Motzkin lattice paths. Applying this approach to mutational processes, alternative, exact, formulations of classic Luria-Delbrück-type problems emerge. This approach can be extended beyond neutral models of mutant clonal evolution. Applications of these approaches are twofold. First, we resolve the probability of progenitor cells generating proliferating or differentiating progeny in clonal lineage tracing experiments in vivo or cell culture assays where clone age is not known. Second, we model mutation frequency distributions that deep sequencing of subclonal samples produce.