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Successive mutants

### Successive mutants

Fig. 7.4 Fitness landscape as a result of the successive accumulation of mutations by cells. We distinguish two scenarios, (a) If apoptosis is intact, accumulation of mutations results in a lower fitness compared to unaltered cells. Even if the mutations result in an increased rate of cell division, the induction of apoptosis in mutated cells prevents them from attaining a higher fitness than the unaltered cells. (b) If apoptosis is impaired, the accumulation of successive mutations will eventually result in a higher fitness compared to unaltered cells. The exact shapes of the curves are not essential. What is important is whether the mutants will eventually have a lower (a) or higher (b) intrinsic reproductive rate.

Since we are interested in cancer progression, we assume that the intrinsic rate of cell division of the consecutive mutants, increases (ri+1 > r.,). Such mutations could correspond to alterations in oncogenes or tumor suppressor genes. Because an accumulation of mutations cannot result in an infinite increase in the division rate of cells, we assume that the division rate plateaus. Once the cells have accumulated n mutations, we assume that further viable mutants are neutral because the division rate cannot be increased further. (This end stage of the mutational cascade is thus mathematically identical to the simple model discussed in the last section.) While we assume that the consecutive mutants can divide faster, they can also carry a disadvantage: the mutations can be recognized by the appropriate checkpoints which induce apoptosis. With this in mind, we will consider two basic types of fitness landscapes. If r<j > rn(l — a), the intrinsic growth rate of the mutated cells, Si and Mi, will be less than that of the unaltered cells, So and M0 (Figure 7.4). While the mutations allow the cells to escape growth control, the mutated cells are killed at a fast rate by apoptosis upon cell division. This scenario corresponds to the presence of efficient apoptotic mechanisms in cells. On the other hand, if ro < rn( 1 — a), the accumulation of mutations will eventually result in an intrinsic growth rate which is larger than that of unaltered cells (Figure 7.4). While mutated cells can still undergo apoptosis upon cell division, apoptosis is not strong enough to prevent an increase in the intrinsic growth rate. Hence, this scenario corresponds to impaired apoptosis in cells. In the following sections, we study the competition dynamics between stable and mutator cells in an evolutionary setting, assuming the presence of relatively strong and weak apoptotic responses.

Time scale separation. In what follows we will assume that the dynamics of the two cell populations happen on two different time scales. In other words, we require that the stable population is still in the state So while the unstable population has already produced all mutants and reached a quasistationary state.

The typical time, if, of change for the type So is found from equation (7.22). Similarly, we can find the time, trr"', it takes to reach the state Mn. It is given by the same equation (7.22) except the coefficients in the nth equation must be replaced by the corresponding coefficients with primes. The mapping to the biological parameters is found from (7.26-7.30). Note that using formula (7.22) has its restrictions, and in the case where it is not applicable, one can directly calculate t™ by estimating the time it takes for zn to reach its maximum (see formula (7.21)). The inequality n « c (7.32)

guarantees that by the time the unstable population has traveled down the mutation cascade to approach its quasistationary distribution, the stable population of cells is still dominated by So.

The conditions for the reversal of competition. In the multidimensional competition problem, equations (7.31), the outcome is determined by the largest eigenvalue of the fitness matrices, f3 and fm. As time goes by, the unstable cell population will approach its stationary distribution (defined by the eigenvector corresponding to the principal eigenvalue), and its fitness is given by the eigenvalue. Because of the time-scale separation, we will assume that during this time, the stable population remains largely at the state Sq. Thus the "winner" of the competition is defined by comparing the two eigenvalues, ao and a'n, see equations (7.27-7.28).

Let us define the value of u, uc, so that for u = uc, we have ao = a'n. As the hit rate passes through uc, the result of the competition reverses. We have

In order to determine whether competition reversal takes place for each scenario (see below), we need to make sure that the following condition is satisfied:

In the next sections we will examine different parameter regimes and conclude that competition reversal may or may not take place; we derive the exact conditions for this. In what follows, we will use several definitions. Let us set

Ae = es - em, and denote by es the following threshold value of es,

This quantity is defined from setting uc = 1 and Ae — 0. Finally, we define the critical gap, A*e, between the two values of e, by setting uc = 1:

Now, let us go back to the two types of fitness landscape, Figure 7.4, and examine the scenarios of strong and weak apoptosis separately.

### 7.2.2 Strong apoptosis

Here we assume that the apoptotic mechanisms in cells are strong. That is, T() > Tn (1 - a) (Figure 7.4a). This means that although the successive mutations will allow the cell to divide faster, the induction of apoptosis ensures that the intrinsic growth rate of the mutants is lower compared to unaltered cells. Note that it is not necessary to assume that oncogenic mutations allow cells to divide faster. Indeed, some cancer cells may progress more slowly through the cell cycle than healthy cells. The important assumption is that accumulation of mutations lowers the intrinsic growth rate of the cells.

In this scenario, the intrinsic growth rate of the stable cells, S, is higher than that of the unstable cells, M. The reason is as follows. The population of stable cells, S, has efficient repair mechanisms. Thus, most cells will remain at the unaltered stage, So- Because population M is unstable, a higher fraction of this cell population will contain mutations. Since these mutations impair reproduction (e.g. because of induction of apoptosis), the intrinsic growth rate of the unstable cells, M, is lower than that of the stable population, S.

At low DNA hit rates, the cells with the faster intrinsic growth rate win the competition. Thus, at low DNA hit rates (low value of u), the stable phenotype, S, wins (Figure 7.5a). On the other hand, at higher DNA hit rates (high value of u), the outcome of competition can be reversed because frequent cell cycle arrest significantly reduces growth. That is, the genetically unstable cells, M, may win and take over the population. As in the simple model discussed above, it requires that the cost of cell cycle arrest is higher than the cost associated with the generation of deleterious mutants (i.e. Carr > Cdei)• Furthermore, reversal of competition may require that the repair rate of stable cells (es) lies below a threshold, and that there is a sufficient difference in the repair rate between stable and unstable cells.

As the population of unstable cells wins, they accumulate mutations. Even if the sequential mutants are disadvantageous because of the induction of apoptosis, the high mutation rate pushes the population down the mutational cascade. While all variants, Mi, persist, the distribution of the variants becomes skewed toward Mn as the DNA hit rate is increased.

(a) Apoptosis intact

Low DNA hit rate

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