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## Results: 3

1.

Fig 2. From: The role of chromosomal instability in tumor initiation.

The early steps of colon cancer occur in small crypts that contain a few thousand cells. The whole crypt is replenished from a small number of stem cells. The effective population size of the crypt,

*N*, with respect to the somatic evolution of cancer might be of the order of 10 cells. As long as*N*^{2}≪ 1/*u*(where*u*≈ 10^{−7}–10^{−6}), then there is a high probability that at any one time crypts contain cells of only one type. Hence, we can investigate a stochastic process describing transitions among six different states,*X*_{0},*X*_{1},*X*_{2},*Y*_{0},*Y*_{1}, and*Y*_{2}, referring to homogeneous crypts of cell type*x*_{0},*x*_{1},*x*_{2},*y*_{0},*y*_{1}, and*y*_{2}, respectively. The transitions reflect the mutational network of Fig. 1. In addition, there are three stochastic tunnels. For certain parameter values, the system can tunnel from*X*_{0}to*X*_{2}without reaching*X*_{1}. Similarly, there are tunnels from*Y*_{0}to*Y*_{2}and from*X*_{1}to*Y*_{2}. Tunnels occur if the second step in a consecutive transition is much faster than the first one and if the final cell has a strong selective advantage.2.

Fig 1. From: The role of chromosomal instability in tumor initiation.

Mutational network of cancer initiation describing inactivation of a TSP gene and activation of CIN. Normal cells,

*x*_{0}, have two functioning copies of the gene and no CIN. Cells with one inactivated copy,*x*_{1}, arise from*x*_{0}cells at a rate of 2*u*; each copy can mutate with probability*u*per cell division. Cells with two inactivated copies of the TSP gene,*x*_{2}, arise from*x*_{1}cells at a rate of*u*+*p*_{0}. The parameter*p*_{0}describes the probability that the second copy of the TSP gene is lost during a cell division because of LOH. CIN cells,*y*_{i}, arise from non-CIN cells,*x*_{i}, at a rate of*u*_{c}= 2*n*_{c}*u*, where*n*_{c}denotes the number of genes that cause CIN if a single copy of them is mutated or inactivated. CIN cells with two functioning copies of the TSP gene,*y*_{0}, mutate into*y*_{1}cells at a rate of 2*u*. CIN cells of type*y*_{1}mutate to*y*_{2}cells at a rate of*u*+*p*, where*p*is the rate of LOH in CIN cells. We expect*p*to be much greater than*u*and*p*_{0}. For simplicity, we neglect the possibility that*APC*might be inactivated by an LOH event followed by a point mutation. If CIN (or LOH in 5q) has a cost, then this pathway will contribute very little; otherwise it could enhance the relative success of CIN by a factor of ≈2.3.

Fig 3. From: The role of chromosomal instability in tumor initiation.

Transition rates and stochastic tunnels of the probabilistic process describing the dynamics of early steps in colon cancer. The states

*X*_{0},*X*_{1}, and*X*_{2}refer to homogeneous crypts of non-CIN cells with 0, 1, and 2 inactivated copies of*APC*, respectively. The states*Y*_{0},*Y*_{1}, and*Y*_{2}refer to homogeneous crypts of CIN cells with 0, 1, and 2 inactivated copies of*APC*, respectively. The probability that a CIN cell with reproductive rate*r*reaches fixation in a crypt of*N*cells is given by ρ =*r*−1(1 −^{N}*r*)/(1 −*r*^{N}). The mutation rate per gene per cell division is given by*u*. The mutation rate from non-CIN cells to CIN cells is given by*u*_{c}= 2*n*_{c}*u*, where*n*_{c}is the number of genes that cause CIN if one copy of them is mutated. The rate of LOH in CIN and non-CIN cells is given by*p*and*p*_{0}, respectively. Let γ = (1 −*r*)^{2}*r*−2 if^{N}*r*< 1 and γ = (*r*− 1)/{r*N*log[*N*(r − 1)/*r*]} if*r*> 1. Network*i*occurs in two cases: (*ia*) ≪ 1/*N*and |1 −*r*| ≪ 1/*N*and (*ib*) ≪ 1/*N*and |1 −*r*| ≫ 1/*N*and*p*≪ γ. Network*ii*occurs in three cases: (*iia*) if ≫ 1/*N*and |1 −*r*| ≪ , then*R*=*Nu*_{c}, (*iib*) if*r*< 1 and ≫ 1/*N*and 1 −*r*≫ , then*R*=*Nu*_{c}*pr*/(1 −*r*), and (*iic*) if*r*> 1 and ≫ 1/*N*and*r*− 1 ≫ and*p*≫ γ, then*R*=*N*^{2}*u*_{c}*p*log[*N*(r − 1)/*r*]. Network*iii*occurs if*r*< 1,*p*≫ γ, and ≪ 1/*N*≪ 1 −*r*; we have*R*=*Nu*_{c}*pr*/(1 −*r*). Network*iv*occurs if*r*> 1,*p*≪ γ, and*r*− 1 ≫ ≫ 1/*N*. In addition, networks*i*–*iv*require that ≪ 1/*N*. Network*v*occurs if ≫ 1/*N*. This is a complete classification of all generic cases.