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M slower than S

M faster than S

Low DNA hit rate

S win

M win

High DNA hit rate

M win if Carr > Cdel

S win if Carr < Cdel

To summarize, this analysis gives rise to the following results (Table 7.1). A high DNA hit rate, u, can reverse the outcome of competition in favor of the cell population characterized by a slower intrinsic growth rate if the competing populations are characterized by a sufficient difference in their repair rates. The higher the difference in the intrinsic replication rate of the two cell populations, the higher the difference in repair rates required to reverse the outcome of competition. If the intrinsic replication rate of the genetically unstable cell is slower, a high DNA hit rate can select in favor of genetic instability. On the other hand, if the intrinsic growth rate of the genetically unstable cell is faster, a high DNA hit rate can select against genetic instability.

7.2 Competition dynamics and cancer evolution 7.2.1 A quasispecies model

In the previous section, we considered the competition dynamics between stable and unstable populations of cells, assuming that they are characterized by different and fixed rates of replication. We further assumed that mutations are either non-viable or neutral. However, mutations are unlikely to be neutral, and will change the replication rate of the cells. In other words, cells may evolve to grow either at a faster or a slower rate, depending on the mutations generated. Here, we extend the above model to take into account such evolutionary dynamics.

The competition problem. As before, we consider two competing cell populations: a genetically stable population, S, and a mutator population, M, see Figure 7.1b. We start with unaltered cells which have not accumulated any mutations. They are denoted by So and M0, respectively. Both are assumed to replicate at the same rate, ro- When the cells become damaged and this damage is not repaired, mutants are generated. If the mutants are viable, they can continue to replicate. During these replication events, further mutations can be accumulated if genetic alterations are not repaired. We call the process of accumulation of mutations the mutational cascade. Cells which have accumulated i mutations are denoted by 5» and Mi, respectively, where i — 1,... ,n. They are assumed to replicate at a rate rt. Stable and unstable cells differ in the rate at which they proceed down the mutational cascade. In addition to the basic dynamics of cell replication described in the previous section, we assume that during cell division, mutated cells can undergo apoptosis, since oncogenic mutations can induce apoptotic checkpoints [Seoane et al. (2002); Vogelstein et al.

(2000b)]. Thus, the intrinsic replication rate of mutated cells is given by ri(l — a), where a denotes the probability to undergo apoptosis upon cell division. These processes can be summarized in the following equations:

51 = auRi^Si-^1 - es) + RiSi(l - us) - (f>Si, l<i<n-l, (7.6) Sn - auRn-iSn-i(l - es) + RnSn[l - us + au(l - es)] - <f>Sn, (7.7) M0 = RoM0{\ - um) - 4>Mo, (7.8) Mi = auRi-1Mi-1(l - em) + RiMi(l - um) - <f>Mi, 2<«<n-l, (7.9) Mn = auRn-iMn-i(l - em) + RnMn{ 1 -um + au{ 1 - em)} - <j>Sn, (7.10)

where we introduced the following short hand notations: Ri is the effective intrinsic reproductive rate, Ri = r¿(l — a) for 1 < i < n and Rq ~ tq, and us<m are the two effective mutation rates, us¡m = u( 1 —/3es,m). The variable w denotes the non-viable mutants produced by the cells. The equations are coupled through the function <¡), the average fitness, which is given by n n

Solving quasispecies equations. Equations (7.5-7.11) are an example of a quasispecies-type system, which is a well-known population dynamical model in evolutionary biology. Quasispecies equations were first derived for molecular evolution by M. Eigen and P. Schuster [Eigen and Schuster (1979)], and since then have found applications in many areas of research, including biochemistry, evolution, and game theory.

In order to analyze system (7.5-7.11), we would like to review some of the techniques for solving quasispecies equations. Let the variable x = (xo, x\,..., xn+i) satisfy the system

11 = biXi-i + aiXi — (¡>Xi, 1 < i < n, (7-13)

where n

fe=i

We have = 1- System (7.12-7.14) is nonlinear. However, the nonlinearity can be removed by the following trick. Let us consider the variable z = (zq, z\,..., zn+1) which satisfies the following system:

If we set

then the variable x satisfies system (7.12-7.14). The general solution of system (7.15-7.17) is given by n z (t) = + an+iv(™+1), (7.19)

where aj are constants determined from the initial condition, and v^ are eigenvectors of the appropriate triangular matrix corresponding to the eigenvalues a,j. The eigenvector v(n+1) = (0,0,..., 0,1)T corresponds to the zero eigenvalue, and for the rest of the eigenvectors we have,

0 0

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