A Hierarchical Probability Model of Colon Cancer

We consider a model of fixed size $N = 2^l$ in which there are $l$ generations of daughter cells and a stem cell. In each generation $i$ there are $2^{i-1}$ daughter 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 generation $i$ daughter cells become two cells of generation $i+1$. The last generation is removed from the population. The stem cell gets first and second mutations at rates $u_1$ and $u_2$ and the daughter cells get first and second mutations at rates $v_1$ and $v_2$. We find the distribution for the time it takes to get two mutations as $N$ goes to infinity and the mutation rates go to 0. We also find the distribution for the location of the mutations. 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.