ü > u


point-mutation rate, which means that û > u.

We are interested in the probability to find the crypt in the state X2, Y2 and Z2 as a function of t. In other words, we want to know the probability for the dysplastic crypt to have CIN (Y2), MSI (Z2) or no genetic instability {X2). The mutation-selection network of Figure 4.3 is more complicated than the one-dimensional network of Figure 4.2, but the solutions for X2, Y2 and Z2 can still be written down.

The diagram of Figure 4.3 corresponds to a system of 11 linear ODE's describing the time-evolution of the probabilities to find the system in any of the 12 possible homogeneous states. An exact solution can be written down but it is a very cumbersome expression, so we will make some approximations. Let us use the fact that the quantities ut/r and N(u+po)t/r are very small compared to 1 for t ~ 70 years, and the quantity Npt/r 1. This tells us that the steps in the diagram characterized by the rates u and Pq are slow (rate limiting) compared to the steps with the rate p. Taking the Taylor expansion of the solution in terms of ut/r and N(u + po)t/r, we obtain the following result:

X2{t) = Nu{u + p0){t/r)2, y2(t) = 4 ncu2{t/r)2. (4.2)

The rate u is neither fast nor slow, so the solution for Z2 is more complicated. We have

where a = 2, b = N(u +p0)/u and Ex = e~xU/T. Note that if the it-steps are fast (i.e. if ut/r » 1), the limit of this expression is given by Z(t) = nmu(u +i>o)(i/T)2- In the opposite limit where ut/r -C 1, we have Z(t) = nmNu{u + p0)u(u + po)(t/r)4/6.

The key idea of this analysis is to identify how many slow (rate-limiting) steps separate the initial state (Xo) from the state of interest. The slow steps in our model are the ones whose rates scale with u or po. The step from Y\ to is fast, because it is proportional to the rate of LOH in CIN cells, p, which is much larger than u and p0. The steps with the rate u are neither fast nor slow. For all possible pathways from the initial state to the final state of interest, we have to multiply the slow rates together times the appropriate power of i/r, and divide by the factorial of the number of slow steps. Summing over all possible paths we will obtain the probability to find the crypt in the state in question.

Applying this rule, we can see that X2(t) and Y2(t) are both quadratic in time, because it takes two rate-limiting steps to go from X0 to Xi and from X0 to >2- The state is separated from X0 by two rate-limiting steps and two 'intermediate' steps (whose rate is proportional to u), so the quantity ¿^(i) grows as the forth power of time for wi/r -c 1 and as the second power of time in the opposite limit.

The probability that a crypt is dysplastic at time t is given by P(t) = X2{t) + Y2(t) + Z2(t). Therefore, the expected number of dysplastic crypts in a person of age t is MP(t). Of these dysplastic crypts, MY2(t) have CIN and MZ2(t) have MSI. This suggests that the fraction of CIN cancers is at least Y2(t)/P(t) and the fraction of MSI cancers is at least Z2(t)/P(t). The actual values may be higher because in our model, only the very first stage of cancer development is considered. At later stages of progression from a dysplastic crypt to cancer, there are more chances for cells to acquire a CIN or an MSI mutation.

Some numerical examples are given in Table 4.4, where the relative fractions of dysplastic crypts with CIN, MSI and without genetic instability are presented for different values of nc, the number of CIN genes. Larger values of nc lead to increased percentage of dysplastic crypts with CIN. According to observations, 13% of all sporadic colorectal cancers have MSI and 87% have CIN [Lengauer et al. (1998)]. In terms of our model this means that we should have Z2(t)/P(t) < 0.13 and Y2fP{t) < 0.87. Prom Table 4.4 we can see that for values of nc of the order of 100, the fraction of CIN crypts is higher than expected.

Table 4.4 Sporadic colorectal cancer: the expected number of dysplastic crypts and fractions of crypts with different instabilities, at 70 years of age, in the model with CIN and MSI. M = 107, N = 5, r = 20 days, u = 10~7, umei = 10"6, u = 10~4, p0 = 10~7, rim — 3 and t = 70 years.

MSI gene inactivation


Total number of dyspl. crypts

% of CIN

% of MSI


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