## X

(pi + R)(pi + R- Np2)' where A = Np2(e~^+R^ - 1) - (Pl + R)(e~Np2t - 1), and xss = Xsd/N^ ^g)

Relative importance of the three scenarios. It is clear that the probability of the ss scenario is small compared to the sd scenario. As time increases, we have lim xdd = —^-r.

This means that if the condition R > pi is satisfied, then the majority of double mutants will acquire both mutations in a DC. Using expression (5.1), we obtain the following inequality, p2\\ogp2\ > jj. (5.10)

If this condition is satisfied, then the probability to obtain a double mutant by two hits in the DC compartment is larger than the probability to first get a mutation in an SC. It is interesting that this condition only depends on the second mutation rate, p2, and is independent on the first mutation rate, p\.

A technical note: very little elaboration is required to obtain the relationship between xss and xsd, equation (5.7). This simple relationship provides important insight that "stem cells are not the entire story, and the second hit is more likely to occur in a DC". However, we need to do more work to answer the following question: how likely is it that both hits fall outside the stem cell compartment? This is where the above model becomes necessary.

Simulations. To check our analytical results, we can use numerical simulations to study the crypt dynamics. Let us trace the offspring of one stem cell; the process goes on until the first double mutant is created. We record whether each of the hits in the double mutant was the result of an SC or a DC mutation, and then we stop the simulation for the crypt. This process is repeated many times, and then the probability of different pathways is estimated. We ran the simulation with 103 and 107 realizations. The results are almost the same which suggests convergence.

Figure 5.4 plots the probability of having a double mutant via different pathways by time n. The curves represent analytical prediction by formulas (5.5), (5.8) and (5.9). The points are results of numerical simulation and they agree with the calculations very well. Notice that for pi= P2 = 10~4 (Figure 5.4a) the probability of having a double mutant initiated in an SC (pathway sd) is greater than the probability of having a double mutant initiated in a DC (pathway dd). On the other hand, when increases to P2 — 10~3, the situation reverses, see Figure 5.4b. The reversal of the two scenarios is predicted correctly by formula (5.10): if N = 29 = 512, this condition holds for p2 = 10~3 and it does not hold for p2 = 10~4.

Figure 5.4c shows the situation corresponding to colon cancer initiation in the presence of chromosomal instability. The first mutation rate, p\ = 10~7, corresponds to the basic point mutation rate. The second mutation rate corresponds to a highly elevated rate of LOH in unstable cancers, p2 = 10~2. We can see that the dd scenario prevails in this case.

Simulating crypt dynamics for lower values of mutation rates, e.g., Pi = P2 ^ 10-7, which corresponds to both hits occurring with the basic point-mutation rate, is very time-consuming. More sophisticated numerical methods could give a faster performance, but here we would like to emphasizes the value of analytical results. Our formulas are valid for arbitrarily small values of mutation rates, and in fact, the precision of the method grows as pi ,2 decrease!

Finally, we have derived an exact analytical formula for the total prob-

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