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Biological tools such as genetic lineage tracing, three dimensional confocalmicroscopy and next generation DNA sequencing are providing new ways toquantify the distribution of clones of normal and mutated cells.Population-wide clone size distributions in vivo are complicated by multiplecell types, and overlapping birth and death processes. This has led to theincreased need for mathematically informed models to understand theirbiological significance. Standard approaches usually require knowledge ofclonal age. We show that modelling on clone size independent of time is analternative method that offers certain analytical advantages; it can helpparameterize these models, and obtain distributions for counts of mutated orproliferating cells, for example. When applied to a general birth-death processcommon in epithelial progenitors this takes the form of a gamblers ruinproblem, the solution of which relates to counting Motzkin lattice paths.Applying this approach to mutational processes, an alternative, exact,formulation of the classic Luria Delbruck problem emerges. This approach can beextended beyond neutral models of mutant clonal evolution, and also describesome distributions relating to sub-clones within a tumour. The approaches aboveare generally applicable to any Markovian branching process where the dynamicsof different "coloured" daughter branches are of interest.

An Exact, Time-Independent Approach to Clone Size Distributions in Normal and Mutated Cells