Genetic instability and clonal expansion - PubMed (original) (raw)
Genetic instability and clonal expansion
Martin A Nowak et al. J Theor Biol. 2006.
Abstract
Inactivation of tumor suppressor genes can lead to clonal expansion. We study the evolutionary dynamics of this process and calculate the probability that inactivation of a tumor suppressor gene is preceded by mutations in genes that confer genetic instability. Unstable cells might have a slower rate of clonal expansion than stable cells because of an increased probability of generating lethal mutations or inducing apoptosis. We show that the different growth rates of genetically stable and unstable cells during clonal expansion represent, in general, only a small disadvantage for genetic instability. The intuitive reason for this conclusion is that robust clonal expansion, where cellular birth rates are significantly greater than death rates, occurs on a much faster time scale than waiting for those mutations that allow clonal expansion. Moreover, in special cases where clonal expansion is very slow, genetically unstable cells have a higher probability to accumulate additional mutations during clonal expansion that confer a selective advantage. Clonal expansion represents a major disadvantage for genetic instability only when inactivation of the tumor suppressor gene leads to a very small increase of the cellular reproductive rate that is cancelled by the increased mortality of unstable cells.
Figures
Figure 1
The mechanism of cancer initiation that is analyzed in this paper assumes that inactivation of a tumor suppressor gene, A, leads to clonal expansion. The mutation rates for inactivating the first and second alleles are given by _u_1 and _u_2, respectively. There can also occur mutations, at rate uc, which lead to chromosomal instability (CIN). CIN causes very fast inactivation of the second allele of the tumor suppressor gene (TSG) by loss of heterozygosity (LOH); this step occurs on a much faster time scale than in stable cells and is usually not rate limiting. Thus, it requires two rate limiting hits to inactivate a TSG with or without CIN (eqs 1 and 2). We assume that the subsequent clonal expansion occurs at a faster rate in stable cells than in CIN cells, because the latter might suffer from a reduced rate of successful cell division and/or an increased death rate. We calculate the probability that a lesion consisting of M cells with or without CIN has arisen by a certain time (see eqs 4 and 5).
Figure 2
Perfect agreement between the analytical approximations given by eqs 4 and 5 (lines) and exact numerical simulation (points). We perform a stochastic simulation for the cancer initiation process described in the main text and illustrated in Figure 1. In each time step, a random event is chosen proportionally to its rate. Before clonal expansion, there is only one (stem) cell. Initially, this cell is of type _X_0. Mutation to _X_1 or _Y_1 occurs with probabilities _u_1 and uc, respectively. A _Y_0 cell can mutate to a _Y_1 cell with probability _u_1. An _X_1 cell can mutate to an _X_2 cell with probability _u_2 or to a _Y_1 cell with probability uc. A _Y_1 cell can mutate to a _Y_2 cell with probability _u_3. An _X_2 cell initiates clonal expansion; cells divide with rate a and die with rate b. Similarly, a _Y_2 cell initiates clonal expansion; cells divide with rate c and die with rate d. The simulation is stopped if the clone has died out or has reached M cells. The figure shows the probability that a single (stem) cell has given rise to a stable lesion or a CIN lesion. Parameter values are _u_1 = 2 × 10−7, _u_2 = 10−6, _u_3 = 10−2, uc = 5_u_1, a = 1, b = 0.1, c = 0.5, d = 0.1, and M = 106. The probability is evaluated over 107 runs.
Figure 3
Comparison between the analytical approximations given by eqs 4 and 5 (lines) and exact numerical simulation (points) for a different choice of parameter values. The same process is simulated as described in the legend of Figure 2. The figure shows the probability that lesions of sizes 10 to 108 cells have arisen after t = 30000 time units (days). For the chosen parameter values, the magnitude of the clonal expansion has no effect. This underlines our conclusion that robust clonal expansion has no effect for evaluating the cost of CIN. The time for clonal expansion is short compared to the waiting time of inactivating the tumor suppressor gene. Parameter values are _u_1 = 2 × 10−7, _u_2 = 10−6, _u_3 = 10−2, uc = 2_u_1, a = 1, b = 0.1, c = 0.5, d = 0.2, and t = 30000. The probability is evaluated over 107 runs.
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