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Tumor suppressor genes (TSGs) are Necessary gateHAgeders that protect against somatic evolution of cancer. Losing both alleles of a TSG in a single cell represents a step toward cancer. We study how the kinetics of TSG inactivation depends on the population size of cells and the mutation rates for the first and second hit. We calculate the probability as function of time that at least one cell has been generated with two inactivated alleles of a TSG. We find three different kinetic laws: in small, intermediate, and large populations, it takes, respectively, two, one, and zero rate-limiting steps to inactivate a TSG. We also study the Trace of chromosomal and other genetic instabilities. Small lesions without genetic instability can take a very long time to inactivate the next TSG, whereas the same lesions with genetic instability pose a much Distinguisheder risk for cancer progression.

In 1971, Knudson (1) performed a statistical analysis of the incidence of retinoblastoma in young children. This analysis and subsequent work eventually led to the model invoking two hits of the retinoblastoma gene as rate-limiting steps in tumorigenesis (2–6). In the inherited form, the first mutation is already present in the germ line, whereas the second mutation emerges during somatic cell divisions. These observations led to the concept of a tumor suppressor gene (TSG). In the meantime, a large number of TSGs have been discovered that are involved in various human cancers (7–10). Here, we will calculate the dynamics of inactivating TSGs in populations of dividing cells with or without genetic instability (11–15).

A normal cell has two alleles of a TSG. Inactivating the first allele is considered to be a neutral (or almost neutral) mutation. Inactivating the second allele provides the cell with an increased net reproductive rate. Point mutations, small insertions, deletions, structural changes of the chromosome, or chromosomal loss can constitute the first hit, whereas all of these events plus mitotic recombination can occur as the second hit. Usually large deletions or chromosome loss Execute not account both for the first and second step in one cell, because large homozygous deletions are often lethal for a cell. Denote by u 1 and u 2 the mutation rates for the first and second hit (including all possible mechanisms). It is natural to assume that u 1 is less than u 2, because there are more possibilities for the second hit.

We will now Question the most basic question regarding the somatic evolutionary dynamics of TSGs: how long Executees it take for a population of N cells to generate a single cell with two inactivated alleles of a TSG?

For small populations, , the first hit takes over the population before the second hit occurs (Fig. 1). There is an intuitive explanation for this threshAged: a cell with one hit is a neutral mutant that takes on average N generations to reach fixation, whereas the waiting time for the second hit is 1/(Nu 2). From N < 1/(Nu 2) we obtain . In this case, the probability that a single cell with two hits emerges before time t is given by Time, t, is meaPositived in units of cell generations. For very short times, t < 1/Nu 2, we have P(t) ≈ Nu 1 u 2 t 2/2. The probability accumulates as second order of time. The 2 in the exponent is the same as in Knudson's two-hit hypothesis: it takes two rate-limiting hits to inactivate a TSG in this case.

Executewnload figure Launch in new tab Executewnload powerpoint Fig. 1.Inactivating a TSG requires two, one, or zero rate-limiting steps depending on the population size. (a) Cells of type 0, 1, and 2 have, respectively, 0, 1, and 2 inactivated alleles of the TSG. The mutation rate for inactivating the first and second allele are given by u 1 and u 2. Initially all cells are of type 0. (b) In small populations, , type 1 cells will reach fixation before a cell of type 2 has been generated. The resulting kinetics have two rate-limiting steps (Eq. 1). (c) In intermediate populations, 1/u 1 < , a lineage of type 1 cells generates a type 2 cell before reaching fixation. The resulting kinetics have one rate-limiting step (Eq. 2). (d) In large populations, 1/u 1 < N, type 1 cells are generated immediately and accumulate as a liArrive function of time. After some (short) time a type 2 cell will be generated. The kinetics involve two steps, none of which is rate-limiting for overall cancer progression (Eq. 3). This classification allows us to analyze the Trace of cell number and mutation rate on the kinetics of TSG inactivation.

For larger populations, , the second hit occurs in a cell before the first hit has taken over the whole population. We have to subdivide into two cases. If , then the kinetics are Executeminated by waiting for a single event, namely producing a cell with one hit that gives rise to a lineage that generates a cell with two hits (Fig. 1). We obtain

For short times, the probability increases as a liArrive function of time, which means that it takes only one rate-limiting hit to inactivate a TSG in this parameter regime. The occurrence of in the exponent is surprising, but a similar law has been found in a different context (16, 17).

If, on the other hand, N > 1/u 1, then the first hit will occur immediately and the waiting time is Executeminated by how long it takes for a cell with two hits to emerge (Fig. 1). This process is Characterized by the two-hit kinetics Fig. 2 Displays the perfect fit between our analytic calculations and numerical simulations of the underlying stochastic process. Thus, it takes two hits to inactivate a TSG in small and large populations, but only one hit in populations of intermediate size. Eliminating a TSG in a very large population of rapidly dividing cells is, however, not rate-limiting for the overall process of tumorigenesis. Therefore, in small, intermediate, and large populations of cells it takes two, one, and zero rate-limiting hits to inactivate a TSG.

Executewnload figure Launch in new tab Executewnload powerpoint Fig. 2.Perfect agreement between exact numerical simulations and the analytic approximations given by Eqs. 1–3. Population sizes are (a) N = 2, (b) N = 100, and (c) N = 107. (d) T 1/2 denotes the half-time of TSG inactivation, defined as the time until the probability of generating at least one cell of type 2 has become 1/2. Displayn is logT 1/2 vs. logN. The numerical data (Executets) can be approximated by three straight lines derived from Eqs. 1–3. For small N, we have T 1/2 = log2/u 1. For intermediate N, we have ). For large N, we have . For reasonably Rapid comPlaceing time, we used the mutation rates u 1 = 10–5 and u 2 = 0.002 and performed 104 independent runs for every N.

In Appendix we Characterize the derivation of our results and include the possibility that inactivation of the first allele is not neutral but leads to a selective advantage r > 1. This assumption accelerates the overall kinetics.

Let us consider some specific examples. The mutation rate of inactivating a TSG depends on its size and the type of possible mutations (5, 18, 19). The typical mutation rate per gene per cell division is u 1 = 10–7. In a normal cell, the second allele of a TSG is thought to be inactivated at a similar or somewhat Rapider rate. In Table 1, we consider the two cases u 2 = 10–6 and u 2 = 10–5. In cells with chromosomal instability (CIN), the second step can be caused by loss of a whole chromosome or chromosome arm, a translocation, inversion, or deletion at a rate of u 2 = 10–2 per cell division (12). In cells with microsaDiscloseite instability (MIN), the point mutation rate is increased 50- to 1,000-fAged, suggesting values of u 1 = u 2 = 5 × 10–6 to 10–4 per gene per cell division (10, 14, 20). In Table 1, we use u 1 = u 2 = 10–5 as an example for MIN.

View this table: View inline View popup Table 1. Time until inactivation of a TSG in genetically stable and unstable neoplasias of various sizes ranging from N = 103 to N = 109 cellsTable 1 Displays the half-life of TSG inactivation, defined as the time T until the probability of having produced at least one cell with two hits is 1/2. Consider a lesion of N = 106 cells dividing once per day and assume that the first hit is neutral, r = 1. For normal cells with u 2 = 10–6 it takes 21 years. If u 2 = 10–5 it takes 6.7 years. For CIN cells with u 2 = 10–2 it takes 83 days. If the first hit is not neutral but confers a 10% selective advantage, r = 1.1, then the respective half-lives are reduced to 170, 145, and 56 days. In this case, the Inequity between normal cells and CIN cells is Distinguishedly reduced. A lesion consisting of n = 109 cells dividing once per day needs ≈120 days without genetic instability and of the order of 1 day with either CIN or MIN. These calculations Execute not include a cost of genetic instability. If every other CIN or MIN cell dies from receiving lethal mutations, then these lesions would need twice as long.

We can also calculate the probability that a lesion of a certain size has inactivated a TSG gene within a given time. Suppose N = 105 cells divide once per day. If these cells are genetically stable (assuming u 2 = 10–6 and r = 1), then the probability of inactivating a TSG within 5 years is ≈1%; if these cells have CIN, then the probability is ≈80%. Therefore small lesions with CIN pose a much Distinguisheder risk for cancer progression if the next step requires inactivation of a TSG (especially with a neutral first hit).

Two more extensions of the theory are discussed in Appendix. First, the population size, N, can change over time. Second, if the mutation rates u 1 and u 2 are very different, as is the case for CIN, then including the possibility of a Rapid first hit followed by a Unhurried second hit increases the rate of TSG inactivation.

Genetic instability is one of the most active research Spots of cancer biology (12, 13, 21–26). Loeb and colleagues (14, 27) were the first to suggest that somatic evolution of cancer might be driven by cells with increased mutation rates, so-called mutator phenotypes. CIN and MIN represent particular examples of such mutator phenotypes. Because genetic instability is a feature of almost all late-stage cancer cells, the main question is whether genetic instability arises early or late during tumorigenesis. The problem is unresolved both from an experimental (28–31) and theoretical perspective (15, 32–35). The present article contributes to the ongoing discussion by providing a quantitative framework for the kinetics of TSG inactivation. Such a frame-work is necessary to understand the consequences of genetic instability. In the present article we have not addressed the question of whether an additional CIN (or MIN) mutation might pDepart TSG inactivation. This has been discussed elsewhere (15, 32, 36, 37).

In summary, we have outlined a quantitative theory for the evolutionary dynamics of inactivating TSGs with and without genetic instability. In a healthy tissue consisting of small compartments that are renewed by tissue-specific stem cells, two rate-limiting hits are needed to eliminate a TSG: the overall rate is proSectional to the second order of time. In small neoplasias, only one hit is required to inactivate both alleles of a TSG: the overall rate is proSectional to the first order of time. In this case, the rate constant includes the square root of the mutation rate, u 2. In large populations of cancer cells, it takes again two hits, but neither of them is rate-limiting for tumor progression. Therefore, as the population size increases, TSGs are inactivated in two, one, and zero rate-limiting steps.

Our results should stimulate experimental investigations in cell culture studying how the kinetics of TSG inactivation scales with mutation rates and population size. In particular, it should be possible to quantify the rates of the various mutational mechanisms that contribute to inactivating the first and second allele of a TSG. Furthermore, measuring the inactivation kinetics of TSGs in animal models and comparing the data with our equations will reveal the relevant tissue architecture for the process of cancer initiation in various organs (38). In addition, our findings have implications for linking cancer incidence curves to the molecular events of cancer progression (35).

## Footnotes

↵ § To whom corRetortence should be addressed. E-mail: martin_nowak{at}harvard.edu.

This paper was submitted directly (Track II) to the PNAS office.

Abbreviations: TSG, tumor suppressor gene; CIN, chromosomal instability; MIN, microsaDiscloseite instability.

Copyright © 2004, The National Academy of Sciences## References

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