Abstract
We consider a model of fixed sizeN= 2lin which there arelgenerations of daughter cells and a stem cell. In each generationithere are 2i−1daughter cells. At each integral time unit the cells split so that the stem cell splits into a stem cell and generation 1 daughter cell and the generationidaughter cells become two cells of generationi+1. The last generation is removed from the population. A stem cell acquires first and second mutations at ratesu1andu2, and a daughter cell acquires first and second mutations at ratesv1andv2. We find the distribution for the time it takes to acquire two mutations asNgoes to ∞ and the mutation rates go to 0. The mutation rates may tend to 0 at different speeds. We also find the distribution for the locations of the mutations. In particular, we determine whether or not the mutations occur on a stem cell and if not, at what generation in the daughter cells they occur. Several outcomes are possible, depending on how fast the rates go to 0. The model considered has been proposed by Komarova (2007) as a model for colon cancer.
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