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(b) Apoptosis impaired

Low DNA hit rate

High DNA hit rate

Time (arbitrary units)

Fig. 7.5 DNA damage and the selection of genetic instability, (a) Cells have intact apoptotic responses. At low DNA hit rates stability wins. At high DNA hit rates instability wins, (b) Cells have impaired apoptotic responses. At low DNA hit rates, instability wins. At high DNA hit rates stability wins. Parameter values were chosen as follows: e3 = 0.99; em = 0.1; 0 = 0.2. For (a) a = 0.61 a = 0.5. For (b) a = 0.1; a = 0.2. Low DNA hit rate corresponds to u = 0.07, and high DNA hit rate corresponds to u = 0.7. Fitness landscapes for successive mutants are given in Figure 7.4.

These results can be obtained by a very simple analysis of the relative values of the relevant eigenvectors, see (7.27-7.28), and by finding conditions under which reversal can occur. In mathematical terms, strong apoptosis corresponds to the situation where

From definition (7.33), 0 < es < 1. Also, we will use the fact that A*e grows with es, so that A*e > 0 for es > es and A*e < 0 for es < es. We can distinguish the following two cases:

• If (3 > a (which is the same as Carr < Cdei), we have an > a'n, which means that Mn never corresponds to the largest eigenvalue. This means that the stable cells always win and the competition reversal does not happen. (Technically speaking, the reversal happens between ao and an rather than ao and a!n.)

• If f3 < a (which is the same as Carr > Cdei), then competition reversal will happen if the following condition is satisfied: Ae > A*e (this is because the function uc decays with Ae). We also observe that in this case, A*e is a growing function of es, which reaches zero at es = e, with 0 < ë < 1. We have two subcases:

(a) For es < ës, we have A*e < 0, and the reversal happens for any difference between es and em.

(b) For es > es, A*e > 0, and we need a finite gap between es and em, Ae > A*e. We also have to make sure that A*e < es, which gives the condition aRn e° f3Ro'

The biological interpretation of these conditions was given in the beginning of this section.

These results have important practical implications. The model tells us that in the presence of intact apoptotic mechanisms, a high DNA hit rate selects in favor of genetic instability, while the tissue remains stable and unaltered if the DNA hit rate is low. A high DNA hit rate can be brought about both by the presence of carcinogens, or by chemotherapy. Therefore, if healthy tissue is exposed to carcinogens, we expect genetic instability to rapidly emerge and this can give rise to cancer progression. In the same way, chemotherapy can select for genetic instability in otherwise healthy tissue and thus induce new tumors as a side effect.

### 7.2.3 Weak apoptosis

Now we assume that the apoptotic mechanisms in cells are impaired. That is, r0 < rjv(l — a), Figure 7.4b. This means that accumulation of mutations will eventually result in the generation of variants which have a faster intrinsic growth rate compared to unaltered cells. Thus, in principle, both populations are expected to eventually evolve toward the accumulation of mutations and progression to cancer. Hence, both stable and unstable cancers can be observed. However, as we noted before, we assume that these processes occur over different time scales for the two populations of cells, condition (7.32).

If the stable and unstable populations compete, the unstable population will have a higher intrinsic growth rate than the stable population (because the induction of apoptosis in response to mutation is inefficient). Therefore, at low DNA hit rates, u, the mutator phenotype, M, wins the competition (Figure 7.5). If the DNA hit rate is increased, the competition can be reversed in favor of the stable cell population, S. This requires that the cost of generating deleterious mutants be greater than the cost of cell cycle arrest (i.e. Cdei > Carr). Furthermore, a sufficient difference in the repair rate of stable and unstable cells is required to reverse the outcome of competition.

Here is the reasoning behind these conclusions. For weak apoptosis, we have

Ro < Ruin this case, if (Rn — Ro)/Rn < a/(3, then es > 1. If on the other hand (.R„ - Ro)/Rn > a/(), then es < 0. We have the following two cases:

• If (3 > a, then the function uc decays with Ae, so for reversal to occur we need to have Ae > A*e.

(a) For (Rn — R0)/Rn < a/0, the function A*e decays with es and crosses zero at es = e > 1. This means that for all es, A*e > 0. We need to require that A*e < es, which gives the condition aRn £s f3Ro'

If this condition holds then the reversal occurs, as long as Ae > A*e; that is, the difference in repair rates must be larger than the critical value.

(b) For (Rn - Re)/R n > ot/ft, the function A*e grows with es and crosses zero at es = e < 0. This means that for all es, A*e > 0.

We need to require that A*e < es, which gives again the condition aRn

If this condition holds then the reversal occurs as long as Ae > A*e.

• If ¡3 < a, then the function uc grows with Ae, so we need to have Ae < A*e.

(a) Condition (Rn — R0)/Rn > a/¡3 is impossible to satisfy, so reversal does not happen in this case.

(b) If (Rn-R0)/Rn < a/¡3 then A*e is a growing function of e8 which crosses zero at es = e > 1. This means that for all es, A*e < 0, and reversal is again impossible.

Our results have practical implications. If cells develop a mutation resulting in impaired apoptotic responses, then genetic instability has a selective advantage if the DNA hit rate is low. Therefore, even if there is no exposure to carcinogens, a chance loss of apoptosis can result in the outgrowth of genetic instability and thus progression of cancer. On the other hand, if there is a growing cancer with impaired apoptotic responses, our results suggest that an elevation of the DNA hit rate by chemotherapeutic agents can reverse the relative fitness in favor of stable cells, and this can result in cancer reduction or slower progression.

A note of clarification: in the above arguments we assumed for simplicity that apoptosis is inefficient in both the unstable and the stable cells. The arguments about chemotherapy, however, remain robust even if we assume that only the mutator phenotypes have impaired apoptosis, while the stable and healthy population of cells has intact apoptotic responses. The reason is that over the time frame considered, the population of stable cells remains genetically unaltered (i.e. at stage So). Since the cells are unaltered, the presence or absence of apoptosis does not change the dynamics.

### 7.3 Summary of mathematical results

The equations have examined the competition dynamics between genetically stable and unstable populations of cells. They identified under which circumstances genetic instability is selected for or against in the context of cancer progression. In particular, they examined the role of the rate at which DNA is damaged.

Table 7.2 Summary of the results gained from the model which takes into account evolution and mutation cascades. If apoptosis is intact, mutators (M) have a lower intrinsic growth rate than stable cells (S). Hence, a high DNA hit rate can select for M. If apoptosis is impaired, M have a higher overall intrinsic growth rate than S. Thus, a high DNA hit rate can select in favor of S.

Table 7.2 Summary of the results gained from the model which takes into account evolution and mutation cascades. If apoptosis is intact, mutators (M) have a lower intrinsic growth rate than stable cells (S). Hence, a high DNA hit rate can select for M. If apoptosis is impaired, M have a higher overall intrinsic growth rate than S. Thus, a high DNA hit rate can select in favor of S.

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