Exact solution of a two-type branching process: models of tumor progression

In the limit of simultaneous small mutation rate and large time scaling, the distribution of the mutant cells develops some atypical properties, including a power law tail and diverging average.

Abstract

An explicit solution for a general two-type birth-death branching process\nwith one way mutation is presented. This continuous time process mimics the\nevolution of resistance to treatment, or the onset of an extra driver mutation\nduring tumor progression. We obtain the exact generating function of the\nprocess at arbitrary times, and derive various large time scaling limits. In\nthe simultaneous small mutation rate and large time scaling limit, the\ndistribution of the mutant cells develops some atypical properties, including a\npower law tail and diverging average.\n