Info
Tim
Total number of MSI genes
2 — 5
Tic
Total number of CIN genes
?
1 copy of APC inactivated xn
normal cells
2 copies of APC inactivated
Fig. 4.2 Mutationselection network of sporadic colorectal cancer initiation. Initially, the crypt is at the state Xo, i.e. all cells are wildtype. With the rate 2u, cells with one copy of the APC gene mutated will take over the crypt (state Xi). This rate of change is calculated as N times the probability (per cell division) to produce a mutant of X\ (2u because either of the two alleles can be mutated) times the probability of one mutant of type Xi to get fixed (1/JV since there is no phenotypic change). From state Xi the system can go to state X2 (both copies of the APC gene inactivated) with the rate N(u + po) This rate is calculated as N times the probability per cell division to produce a mutant of X2 (u for an independent point mutation plus po for an LOH event) times the probability of the advantageous mutant of type X2 to take over (this is 1).
Let us denote by Xo, X\ and X2 the probability that the whole crypt consists of cells with 0, 1 and 2 copies of the APC gene inactivated, respectively. The simplest mutationselection network leading from Xo to Xi to X2 is shown in Figure 4.2. The rate of change is equal to the probability that one relevant mutation occurs times the probability that the mutant cell will take over the crypt.
In the beginning (see Figure 4.2), all cells are wild type. The fist copy of the APC gene can get inactivated by a mutation event. Because the mutation rate per gene per cell division, u ~ 10~7, is very small and the number of cells, N, is not large, it is safe to assume that once a mutation occurs, the population typically has enough time to become homogeneous again before the next mutation occurs. The condition is that the mutation rate, u, is much smaller than 1/N, as was derived in Chapter 3. This means that most of the time, the effective population of cells in a crypt can be considered as homogeneous with respect to APC mutations. Under this assumption we have Xo + X\ + Xi = 1.
Initially, all the N cells of a crypt have two copies of the APC gene. The first copy of the APC gene can be inactivated by means of a point mutation. The probability of mutation is given by N (a mutation can occur in any of the N cells) times the mutation rate per cell division, u, times 2, because any of the two copies of the APC gene can be mutated. Because inactivation of one copy of the APC does not lead to any phenotypic changes, the rate of fixation of the corresponding (neutral) mutant is equal to
,. 11/r see Chapter 3. "Fixation" means that the mutant cells take over the crypt. Therefore, the rate of change from Xo to X\ is 2uN x 1/N = 2u.
Once the first allele of the APC gene has been inactivated, the second allele can be inactivated either by another point mutation or by an LOH event. This process occurs with rate N{u + po), where po is the rate of LOH in normal (nonCIN) cells. We assume that mutants with both copies of the APC gene inactivated have a large selective advantage, so that once such a mutant is produced, the probability of its fixation is close to one. This assumption is made for simplicity. More generally, the relative fitness of type Xi is f, whereas the fitness of type Xo and X\ is 1. Then the second rate in Figure 4.1 should be taken to be Np(u + po), with p = (1 — l/f)/(l — l/fN). If the population size is not too large, and the relative fitness of type X2 is much greater than 1, we have p2 —*► 1, and we obtain the expression N(u + po).
The mutationselection network of Figure 4.2 is equivalent to a linear system of ordinary differential equations (ODE's), where the rates by the arrows refer to the coefficients and the direction of the arrows to the sign of the terms. One (nondimensional) time unit (t/r = 1) corresponds to a generation turnover. The calculations leading to the mutationselection network are performed for a Moran process where the population size is kept constant by removing one cell each time a cell reproduces, see Chapter 3. Our biological timeunit again corresponds to N "elementary events" of the Moran process, where an elementary event includes one birth and one death. We have:
X1=2uXoN(u+po)X1, with the constraint X0 + Xi + X2 = 1 and the initial condition
Here, we use the fact that the intermediate mutant is neutral and that the population size is small (N < Ntun, see Chapter 3) so that stochastic tunneling does not take place. Calculations for larger values of N can also be performed.
Using ut/r 1 and N(p0 + u)t/r 1, we can approximate the solution for X2 as
The quantity X2{t) stands for the probability that a crypt is dysplastic (i.e. consists of cells with both copies of the APC gene inactivated) at time t measured in days. This formula is a consequence of the fact that in the parameter regime we are considering, APC/^ cells are produced as a result of a genuine twohit process (see Chapter 3). There are two steps that separate the state Xq from the state X2, and thus the expected number of dysplastic crypts in a person of age t is proportional to the product of the two rates and the second power of time. This reminds us of the general ArmitageDoll model where the power dependence of the probability of cancer is equal to the number of mutations in the multistage process. In our case, the number of mutations needed to create a dysplastic crypt is two.
The probability to have i dysplastic crypts by the age t is given by a simple binomial, (™)X2(t)i{l  X2{t))M"i. The expected number of dysplastic crypts in a person of age t is then given by the following quantity,
Some estimates of the expected number of dysplastic crypts, based on equation (4.1), are given in Table 4.5.

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