## Damage and genetic instability

Cancer is initiated and progresses via the accumulation of multiple mutations. The last chapters presented mathematical analyses of how cells proceed down this pathway to cancer in the most efficient way. In particular, the question was addressed whether genetic instability is observed because it allows cells to acquire oncogenic mutations at a faster rate. The mathematical approaches concentrated entirely on the cells which develop cancer, and did not take into account environmental factors. Environmental factors can greatly influence whether cells can become cancerous and grow successfully. They may also provide conditions under which genetically unstable cells have a selective advantage. Identifying such conditions is important, since they might contribute to explaining why so many cancers are characterized by genomic instability.

A major environmental factor in the development of cancer is the amount of DNA damage which cells experience. DNA damage can come from a variety of sources. Carcinogens contained in food or in the air we breath can damage DNA. UV radiation can break DNA. Chemotherapeutic agents can lead to various forms of DNA damage. Most importantly perhaps, aging leads to an increased amount of DNA damage. This is because metabolic activities produce reactive oxygen species which are toxic for our genome [Campisi (2001)].

How does DNA damage influence the process of carcinogenesis? On the most elementary level, it might increase the basic mutation rate. More damage results in a higher probability that mutations are produced. More profoundly, however, it might also influence whether genetically unstable cells are more advantageous than stable cells, or whether stable cells can grow better than unstable ones. High amounts of DNA damage have the following consequences for stable cells. On the one hand, the cells main tain stable genomes. On the other hand, each time a cell gets hit it repairs the damage. This takes time and is usually manifested in cell cycle arrest or in stalling of the replication process. Repair is therefore costly because it slows down the overall growth of the cell population. Unstable cells are influenced by high levels of DNA damage in the following way. They avoid repair and therefore do not enter cell cycle arrest. On the other hand, they pay an alternative cost. Many mutants are created, and a large proportion of the mutants are likely to be non-viable.

This chapter will present a mathematical model to investigate whether and how DNA damage can influence the growth processes of stable and unstable cells. This is done by examining the competition dynamics between stable and unstable cells. Which cell type wins? Can an increase in the level of DNA damage reverse the outcome of competition?

### 7.1 Competition dynamics

We start by exploring the competition dynamics between a stable and a mutator cell population [Komarova and Wodarz (2003)]. They differ in the probability with which they repair genetic damage. Stable cells repair damage with a probability es, and mutator cells repair damage with a probability em, where es > em. We further assume that these cell populations differ in their intrinsic rate of replication. The stable cells replicate at a rate rs, and the mutator cells replicate at a rate rTO. Let us denote the abundance of stable and mutator cells as S and M, respectively. The competition dynamics are given by the following pair of differential equations which describe the development of the cell populations over time,

S = rsS{ 1 -u + (3esu) + aursS{ 1 - es) - (f)S, (7.1)

M = rmM(l -u + ¡3emu) + aurmM{ 1 - em) - <j>M. (7.2)

The model is explained graphically in Figure 7.1. The cells replicate at a rate rs or rm. These parameters reflect how often cells reproduce and die; we will call this the intrinsic replication rate of the cells. The two cell populations compete for a shared resource. Competition is captured in the expressions 4>S and <pM, where 4> is defined as follows:

During replication a genetic alteration can occur with a probability u.

Healthy cell

Repairing cell

Repairing cell

Healthy cell

Damaged cell

Mutated cell Dead cell

1 mutation 2 mutations 3 mutations n mutations

Fig. 7.1 Schematic Diagram of the model, (a) Process of cell reproduction, DNA damage, repair, cell cycle arrest, mutation, and death, (b) When DNA damage is not repaired, the cells can accumulate mutations. In the model cancer progression corresponds to the successive accumulation of mutations, also referred to as the mutation cascade.

We call this the DNA hit rate. DNA damage can occur both spontaneously (most likely at low levels), or it can be induced by DNA damaging agents which corresponds to a high value of u. If a genetic alteration has occurred, it gets repaired with a probability es or em. During repair, there is cell cycle arrest, and this is captured in the parameter (5. The value of /? can lie between zero and one and thus reduces the rate of cell division (given by fir). If (3 = 0, the repairing cells never replicate and this is the maximal cost. If = 1, there is no cell cycle arrest and no cost associated with repair. With a probability (1 — es or 1 — em) the genetic alteration does not get repaired. If the alteration is not repaired, a mutant is generated. A mutation is therefore the result of the occurrence of DNA damage combined with the absence of repair. The mutant is viable (and neutral) with a probability a, while it is non-viable with a probability 1 — a. Therefore, the model captures both the costs and benefits of repair: Efficient repair avoids deleterious mutations but is associated with cell cycle arrest. Absence of efficient repair can result in the generation of deleterious mutants, but avoids cell cycle arrest.

Note that in this first model, we assume that mutants that are created are either non-viable (and thus do not participate in the competition dynamics) or neutral (and thus have the same intrinsic reproductive rate as the wild type). We will include the possibility of advantageous and disadvantageous (but viable) mutants later.

Le us explore how the competition dynamics depends on the rate at which cells acquire genetic alterations (DNA hit rate, u). In general, if two cell populations compete, the cells with the higher fitness wins. The fitness of the cells is given by rS)m[l - u[I — a + es¡m(a - /?)]]. Note that the quantity 1 — a has the meaning of the cost of production of deleterious mutants; we will refer to it as

Similarly, the quantity 1 - /3 is the cost of cell cycle arrest,

In these notations, we can rewrite the expression for the fitness in a more intuitive way,

If the DNA hit rate is low (low value of u), the fitness of the cells is approximately given by their intrinsic rate of replication (rs and rm). Thus, the cell population with the higher intrinsic replication rate has a higher fitness than the cell population with the lower intrinsic replication rate. On the other hand, when the DNA hit rate, u, is increased, the fitness

(a) Small difference in repair rate (b) Large difference in repair rate

DNA hit rate DNA hit rate

Fig. 7.2 Effect of the DNA hit rate, u, on the fitness of two cell populations. At low DNA hit rates, the population with the higher intrinsic replication rate wins. An increase in the DNA hit rate decreases the fitness of both cell populations. However, the degree of fitness reduction of the population characterized by the higher intrinsic replication rate is stronger than that of the slower population of cells. If there is a sufficient difference in the repair rates (degrees of genetic stability) between the two cell populations (a), an increase in the DNA hit rate can result in a reversal of the relative fitnesses, and thus in a reversal of the outcome of competition. If the difference in repair rates between the two cell populations is not sufficient, (6), we do not observe such a reversal. Parameter values were chosen as follows: rs = 1; rm = 1.3; a = 0.05; P = 0.3; es = 0.99. For (a) era = 0.1. For (b) em = 0.9.

depends more strongly on other parameters. In particular, the fitness of both populations can depend on the DNA hit rate, u. Notably, an increase in the value of u may result in a stronger decline in fitness of the cell population with the faster intrinsic rate of replication relative to the slower cell population (Figure 7.2). Therefore, if the DNA hit rate crosses a critical threshold, u > uc, the outcome of competition can be reversed. The value of uc is given by

We are interested to find out, under what circumstances reversal of competition can occur. One condition required for the reversal of competition is that the stable and mutator cells are characterized by a sufficient difference in the repair rate (Figure 7.2) which is defined as

Ae> \rs - Tm|EC3- - Cdei)(l - ea) + 6,(1 - Carr)}

Further, we need to distinguish between two scenarios.

(1) In the first case we assume that the stable cells have a faster intrinsic rate of replication than the mutator cells (i.e. rs > rm). Therefore, at low DNA hit rates, the stable cells win. An increased DNA hit rate, u, can shift the competition dynamics in favor of the unstable cells. In other words, unstable cells gain a selective advantage as the DNA hit rate becomes large. This is because the population of stable cells frequently enters cell cycle arrest when repairing genetic damage and this slows down the overall growth rate. For this outcome to be possible, the following condition has to be fulfilled: The cost of cell cycle arrest, Carr, must be greater than the cost of producing non-viable mutants, Cdei• If this condition is not fulfilled, reversal of competition at high DNA hit rates is not observed.

(2) In the second case we assume that the stable cells have a slower intrinsic replication rate than the mutator cells (i.e. rs < rm). Therefore, at low DNA hit rates, the unstable cells win. An increased DNA hit rate, u, can shift the competition dynamics in favor of the stable cells. In other words, a high DNA hit rate selects against genetic instability. This is because the unstable cells produce more non-viable mutants and this reduces the effective growth rate significantly. In contrast to the previous scenario, this requires that the cost of producing non-viable mutants, Cdeh must be higher than the cost of cell cycle arrest, Carr-If this condition is not fulfilled, reversal of competition at high DNA hit rates is not observed.

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