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The calculations presented in Table 4.4 were performed under the assumption that chromosomal instability does not have a cost. In other words, the CIN phenotype is neutral with respect to the wild type. Perhaps a more realistic model should include a possibility that CIN phenotype is disadvantageous compared to the wild-type. Indeed, genetically unstable cells can have a higher apoptosis rate because of a high frequency of mutations in essential genes. Therefore, we can assume that genetic instability leads to a change of reproductive rate thus giving the mutant cells a selective disadvantage, see Chapter 6 (or a selective advantage, if the environment is right, see Chapter 7).

For both disadvantageous and advantageous CIN, we have the computational machinery developed in Chapter 3 which can be used. Let us suppose that the relative reproductive rate of CIN cells is rc. Then, if no tunneling occurs, the transition rates from X0 to Y0 and from Xi to Yj is not 2ncu but 2ncNr^~1(l — rc)/{ 1 — r^). This quantity is larger than 2ncu in the case when CIN is advantageous {rc > 1) and smaller if it is disadvantageous (rc < 1). In the case where CIN is disadvantageous, the transition rate from X(i to Y0 and from Xi to Yx becomes lower. For example, if the relative disadvantage of a CIN cell is 10%, then the fraction of CIN dysplastic crypts in Table 4.4 will be reduced by 20%. In the case where stochastic tunneling occurs, the computation will be different. It follows in a straightforward manner from the results of Chapter 3, and we do not present it here. An interested reader can refer to the paper by Komarova et al [Komarova et al. (2003)] for details.

In our model, we assume that CIN is generated by means of a mutation in any of nc dominant-negative CIN genes. In other words, a genetic hit in either of the two copies of a CIN gene will lead to the acquisition of the CIN phenotype. Alternatively, it could happen that the CIN phenotype requires the inactivation of both copies [Rajagopalan et al. (2004)], like MSI genes or tumor suppressor genes. In terms of the diagram in Figure 4.3, this would mean that we have two steps separating the wild type (Xo) from the CIN phenotype (Zq). If we assume that the CIN phenotype is neutral, the fraction of CIN dysplastic crypts in Table 4.4 would be negligible. This means that in this case, the CIN phenotype must be very advantageous in order to show up early in carcinogenesis. We will develop these ideas further in Chapters 6 and 7.

The fraction of MSI crypts as predicted by this model is quite low (for nc = 10 we get only 0.1 % of dysplastic crypts with MSI). This could mean that MSI develops at later stages of cancer. However, there is indirect evidence that the replication error phenotype precedes, and is responsible for, APC mutations in MSI cancers [Huang et al. (1996)]. Our model is consistent with this data if we assume higher rates of MSI induction in a cell. This could be caused by higher mutation rates in MSI genes, a larger number of MSI genes or the possibility of epigenetic mechanisms of gene silencing. DNA methylation of the hMLHl gene is found at a high frequency in sporadic MSI tumors [Ahuja et al. (1997); Cunningham et al. (1998); Kane et al. (1997)]. In the diagram of Figure 4.3 this means that the rates from XQ to the MSI type (vertical arrows), 2nmu and u + p0, should be replaced by 2nmumet and umet +po, respectively, where umet is the rate of methylation per gene per cell division. In terms of our equations, we need to replace u by umet in the expression for Z2(t), equation (4.3). If we assume that umet is larger than the basic mutation rate, u (say umet = 10-6), then the expected fraction of MSI crypts predicted by our model becomes larger. Note however that at this stage there is no accurate estimation of methylation rates compared to mutation rates. Our model suggests that if epigenetic mechanisms significantly increase the APC inactivation rate, then the predicted fraction of MSI crypts is consistent with the observed frequency of MSI cancers, see Table 4.4.

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