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*^{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} = 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*^{N}−2 if

*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.

## PubMed Commons