Fig 3.

Transition rates and stochastic tunnels of the probabilistic process describing the dynamics of early steps in colon cancer. The states X₀, X₁, and X₂ refer to homogeneous crypts of non-CIN cells with 0, 1, and 2 inactivated copies of APC, respectively. The states Y₀, Y₁, and Y₂ 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^N−1(1 − 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 = 2n_cu, 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 p₀, respectively. Let γ = (1 − r)²r^N−2 if r < 1 and γ = (r − 1)/{rN 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_cpr/(1 − r), and (iic) if r > 1 and ≫ 1/N and r − 1 ≫ and p ≫ γ, then R = N²u_cplog[N(r − 1)/r]. Network iii occurs if r < 1, p ≫ γ, and ≪ 1/N ≪ 1 − r; we have R = Nu_cpr/(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.