  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.
Population-wide clone size distributions in vivo are complicated by multiple
cell types, 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
parameterize 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 gamblers ruin
problem, 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 be
extended beyond neutral models of mutant clonal evolution, and also describe
some distributions relating to sub-clones within a tumour. The approaches above
are generally applicable to any Markovian branching process where the dynamics
of different "coloured" daughter branches are of interest.
