## I

In the limit of small P2, we have the following expression, P ^ nPiP2(\iogP2l - log2 - 7)

It is interesting to compare this with the results of Chapter 3, obtained for the rate at which a double mutant is produced in a two-hit model. In Chapter 3, the structure of compartments is not taken into account, but the phenomenon of "stochastic tunneling" tunneling that we introduced there is very similar to pathway dd studied here. Indeed, tunneling occurs when the second hit occurs before the first hit has had a chance to reach fixation. For such models, the rate R, roughly speaking, is given by Npi^/pz. We can see that taking account of the structure of the colon changes these results. In particular, the new rate given by (5.1) is lower because P2\ logy>2| < \/p2-

Equations containing all scenarios. We have the following equations for the mutation processes:

The probability to obtain a double mutant, x2, is given by equation (5.4). It can be rewritten as a sum of three contributions, x2 = xdd + xsd + xss.

Here, the probability to obtain a double mutant by two DC mutations is given by xdd = Rx 0, which yields

The probability to obtain a double mutant by first mutating the stem cell is xsd + xss =Np2Xl, (5.6)

where xss refers to the pathway where both mutations occur in the SC, and xsd implies the second mutation in a DC. We have the following intuitive relation, xss/xsd « l/N. (5.7)

This means that the vast majority of double mutants will acquire the second mutation in the differentiated stage. We can solve equations (5.2), (5.3) and (5.6) and use equation (5.7) to eliminate xss to obtain the following expressions:

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