## J

The integral yields the following law of growth:

where we assumed that for relevant times, aßt > 1.

Let us ask the following question: how long does it take, on average, for a TSP~/~ clone to reach a certain size? The answer will depend on all the parameters of the system, and in particular, on the rate of chromosome loss, p. For the reasons explained above, the waiting time will be very large both for p = 0 and for very high values of p. Indeed, for very small p the mutations that lead to a TSP inactivation will take too long, and for very large values of p, the clonal expansion will be too slow because of the amount of non-viable or non-reproductive cells produced. The waiting time will have a minimum for an intermediate value of p = p*, which we call the optimal (for cancer) value of the rate of chromosome loss. With this value of p, a cancer will appear and grow at the fastest rate. This approach is equivalent to the "minimum-time-to-target" method in optimization theory.

To find an optimal value of p that maximizes the growth, we solve Zy(t) = M for t, which gives,

~aßM 1 - di and then we minimize this as a function of p. This can be done easily if we assume that p ^C 1/(2k) (it will turn out that the result for p* satisfies this assumption). Expanding the expression dt(M)/dp in terms of p, we obtain the equation for p,

1 aPM

p up where we formally have k = 1 for scenario (a), and k = 23 for scenario (b).

Large initial number of cells. In the above model, the number of wildtype cells in the compartment is small (N <C 1 /u). In order to handle the scenario where a large number of cells are competing in a compartment, which may correspond to later stages of carcinogenesis, we numerically simulated a set of quasispecies-type equations. The estimate obtained for the optimal value of p is very similar to the ones given for the stochastic model above.

Parameter dependence of the result. The result for p* turns out to be amazingly robust, see Table 6.1. We can see that p* depends logarithmically (that is, weakly), on the combination k = a/3M/u. As we vary these parameters over many decades, so that k changes from 105 to 1020, the result for the optimal value of p varies only slightly. Interestingly, it also does not significantly depend on the overall fitness cost for the cell brought about by chromosome loss. The results for scenarios (a) and (b) are presented in Table 6.1. The optimal value of p is lower in scenario (b), which is not surprising because this case assumes a higher penalty for chromosome loss events. The remarkable fact is that the values of p* for the two scenarios are so close to each other, and that they depend so little on the assumptions of the model.

Scenario |
re = 102U |
re = 105 |

(a) Optimistic |
2 x 10"2 |
8 x 10"'2 |

(b) Pessimistic |
5 x icr4 |
2 x 10"3 |

What is even more encouraging is that we can compare these results with the value of the rate of chromosomal loss obtained by Lengauer et al. [Lengauer et al. (1997)] in vitro for several CIN colon cancer cell lines. In their paper, Lengauer et al. allowed cell colonies to grow from a single cell for 25 generations, after which FISH analysis was performed on a subset of the progeny. This allowed to count the number of individual chromosomes in cells. The average number of chromosomal copies was calculated for each cell line, for each chromosome, and this was compared with the mode number, equivalent to the number of chromosome copies in the original cell. This was the first (and only) experiment which allowed to calculate the rate of chromosome loss and gain, as opposed to the estimates of the frequency of various chromosomal aberrations in a given lesion/cell colony. Two types of cancer cells have been used: some known to possess mismatch repair instability, and some characterized by CIN. In the cell lines with microsatellite instability, the rate of chromosome loss was the same as control (and indistinguishable from the background). In the chromosomally unstable cell lines, the rate of chromosome copy change was highly elevated. The value that emerges from experiments of Lengauer et al. is p = 10-2 per chromosome per cell division, which is almost exactly in the middle of the range that we obtained theoretically.

6.3 Why does CIN emerge?

Next, we will discuss the ways by which CIN could come about in carcinogenesis. Let us compare two cell lines, one with CIN, such that its rate of chromosome loss is optimal, p = p*, and another without CIN, such that From our argument above, and from the definition of the optimal rate, p*, it is clear that the unstable cell line will grow faster. Now let us reformulate the question slightly. Suppose that we start from a non-CIN wild-type cell. In order to use the "advantages" of CIN, a cell must at some points acquire the CIN phenotype.

Comparison of stable and unstable pathways. Let us include the step of initiation of CIN. All pathways can be expressed by the following diagram,

lose 2k chromosomes lose 2k — 1 chromosomes

Here Xi are the probabilities for the stem cell to be stable and have i inactivated copies of the TSP gene, and yi are the probabilities for the cell to be CIN and have i inactivated copies of the TSP gene. uc is the rate at which a cell acquires CIN. The Kolmogorov forward equations are:

1 - do)2uyQ + (1 - u)ucxi + [(1 - di)(l - u - p) - 1] yu (6.10) 1 -d1)(u+p)y1, (6.11)

with the initial condition xo(0) = 1. It is easy to show that for the optimistic (for cancer) scenario (a), the two CIN pathways (xq —.> y0 —> yi —!> y2 and xo —> xi —> ?/i —> y2) contribute equally to y2- For the pessimistic (for cancer) scenario (b), and p u, the second of these pathways gives a much larger contribution. The reason for this is that losing a "wrong" chromosome will destroy the cell line in this extreme scenario. Therefore, it is much more likely to reach the state y2 if CIN appears as late as possible. In what follows we will ignore the first pathway entirely because it either does not contribute anything or gives a factor of 2. This simplifies the calculation because now, the first step for both stable and CIN cancer is Xo —> Xi, and if we only want to compare the CIN and non-CIN pathways with each other, this step can be ignored. This is equivalent to starting from xi(0) = 1 rather than £o(0) = 1-

The probability distribution of creating a TSP-/- mutant as a function of time, is given by x2 for the stable pathway and by y2, for the unstable pathway. We have, x2(t) = ue-^, and y2 is given by formula (6.4), with u replaced by uc.

The clonal expansion law for unstable cells is given by equation (6.5), and for stable cells we simply have Zx(t) = eat. Taking a convolution of the rates for the mutation and expansion stages, we arrive at the following laws of growth:

It turns out that unless uc is several orders of magnitude bigger than u, Zy grows slower than Zx, that is, genetically activated CIN cannot be advantageous.

Can CIN be the first event? Here is what the calculation above shows. Let us assume that

(1) CIN is a genetic event which happens at a rate comparable to the basic mutation rate, u, or even a couple of orders of magnitude larger, and

(2) unstable cells do not have any additional fitness advantages compared to wild type cells.

sialic

2u stable

stable unstable

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