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Time

Fig. 5.4 The probability to acquire a double mutant as a function of time, with I = 9. Numerical simulations are compared with analytical results, "o" denotes the simulation for sd pathway, "+" denotes the simulation for dd pathway, and "0" denotes the simulation for ss pathway. In (a), pathway sd is more likely. In (b), where p2 is an order of magnitude larger, pathway dd prevails. In (c), dd pathway prevails. Pathway ss is always the least likely scenario.

ability of creating a double mutant, see Appendix 12.6. This was used to check the simulations by comparing the numerical value of xss + xsd + xdd with the exact formula, expression (A.l) in Appendix 12.6. The formula gave a perfect agreement with simulation results (data not shown). Formula (A.l) works very well for large values of pii2. For instance, it can give the probability of producing a double mutant, where the approximation of this section breaks down, that is, in the regime where pi ~ l/N. However, the exact calculation of Appendix 12.6 has the same disadvantage as the numerical simulations: for low values of P12 it becomes difficult to implement. The approximate formulas (5.5), (5.8) and (5.9) can be used for all realistic values of mutation rates.

### 5.5 Implications and data

We considered the renewal dynamics of a colon crypt, repopulated by asymmetric divisions of stem cells. Dysplasia occurs as a result of inactivation of both copies of the APC gene (or other genes in the Wnt pathway). Thus, it takes two hits before phenotypic changes occur. There are three pathways for the two hits: ss (both hits occur in an SC), sd (the first hit occurs in an SC, and the next hit is acquired by one of the DC in its clone), and dd (the first hit is acquired in a DC, and the second hit occurs in one of its progeny). The results obtained from the mathematical models can be summarized as follows:

(1) The probability of the ss pathway is negligible. That is, it is unlikely that both hits occur in the SC.

(2) As a consequence, at least one of the hits will occur in the migrating, proliferative compartment. This is consistent with the observation that below the dysplastic layer, cells in the crypt do not exhibit the APC~/~ phenotype. The relative importance of sd and dd pathways depends on the parameters of the system.

(3) In particular, ifp2| logp2| > l/A", the (¿¿pathways become more important. It means that the first double mutant will appear outside of the SC compartment. This contradicts previous thinking that SCs are crucially important for colon cancer initiation, being the first mutational "targets".

(4) If the reverse condition holds, that is, if P2\\ogp2\ < V-^i then the most likely scenario is the sd pathway, that is, first an SC acquires a mutation, then all of its offspring contain an inactivated APC copy, and then one of these daughter cells acquires a second hit. According to the sd scenario, the crypt below the dysplastic cells should contain cells with one inactivated and one functional copy of the APC gene.

One remarkable property of the model is a very small number of parameters that it contains. For instance, the condition which identifies whether an SC mutation is important (that is, which of the pathways sd and dd is more likely), only depends on two parameters, p2 and N. Table 5.1 identifies the most important pathway, depending on the parameter p2 and the number of SCs per crypt. The parameter N can be found as follows. If there are n SCs per crypt, then N is the size of the crypt divided by n, that is, the "share" of each of the SCs in the crypt. We suppose that there are 2,000 cells in a crypt, and the number of SCs per crypt is varied between 1 and 32 [Potten and Loeffler (1990)].

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