The question of the origins of cancer is among the most important in our understanding of the disease. There is no universal answer to this question, as different cancers are initiated by different mechanisms. There are however certain patterns that can be recognized. Among the most important ones is cancer initiation via the inactivation of a tumor suppressor gene. The concept has evolved during the last 30 years. A defining landmark was the discovery of the Rb gene.

Retinoblastoma is a rare and deadly cancer of the eye that afflicts children. It comes in two versions. One affects newborn infants and is characterized by multiple tumors. The other hits children when they are older and is usually characterized by only a single tumor. In 1971, Alfred Knud-son proposed an explanation, which became known as the famous Knud-son's "two-hit hypothesis" [Knudson (1971)]. According to his theory, in the early-onset version of retinoblastoma, children inherit a defective gene from one parent. These children are halfway to getting the disease the moment they are born. Then, an error in DNA replication in a single eye cell, causing a defect in the normal gene that was inherited from the other parent, would send that cell on its way to becoming a tumor. In contrast, children who develop retinoblastoma later in childhood are probably born with two good copies of the gene but acquire two hits in both copies of the gene in a cell. This would take longer, causing the cancer to show up at a later age.

Knudson proposed the tumor suppressor gene hypothesis of oncogenesis after detecting a partial deletion of chromosome 13 in a child with retinoblastoma. This was a revolutionary concept, that is, cancer was not caused by the presence of an oncogene, but rather the absence of an "anti-oncogene." He concluded that the retinoblastoma tumor suppressor gene would be found at band 13ql4. It wasn't until the late 1980s when scientists eventually cloned the gene Rb which mapped exactly to the location predicted by Knudson.

Other genes with similar properties were discovered, including p53, WT1, BRCA1, BRCA2 and APC. The generic definition of a tumor suppressor gene comprises the idea of a loss of function. Only when both alleles of the gene are inactivated, does the cell acquire a phenotypic change. Many of tumor suppressor genes are involved in familial cancers. The mechanism is similar to the one described by Knudson in retinoblastoma. If a defective allele is present in the germline, the affected individuals will have a higher chance of developing a cancer as only one remaining allele must be inactivated to initiate an early stage lesion.

In collaboration with Knudson, Suresh Moolgavkar went on to develop mathematical models for this hypothesis, which were the first to coalesce clinical-epidemiological observations with putative mutation rates and molecular genetics [Moolgavkar and Knudson (1981)]. In subsequent publications, Moolgavkar and colleagues have created a rigorous methodology of studying multistage carcinogenesis [Moolgavkar (1978); Moolgavkar et al. (1980); Moolgavkar et al. (1988)]. In this chapter we will review some of the main ideas of the two-hit models, and develop them further to provide tools for this book. In particular, we will derive simple expressions for the probability of generating double-mutants. We consider small, intermediate and large populations, in the case of disadvantageous, neutral or advantageous intermediate mutants. We start from a one-hit model and then go on to describe a more involved process with two hits.

3.1 A one-hit model

We will use this section to review several important mathematical tools describing stochastic population dynamics.

3.1.1 Mutation-selection diagrams and the formulation of a stochastic process

Let us first assume that there are two types of cells in a population, which we will call type "A" and type "B". Cells can reproduce, mutate and die. The probability that a cell of type "A" reproduces faithfully is 1 - u; with probability u it will mutate to type "B". Cells of type "B" always reproduce faithfully. We will assume that the total number of cells is constant and equal to N. Let the cells of type "A" have reproductive rate 1 and the cells of type "B" - reproductive rate r.

We will use the following convenient short-hand representation of these processes:

Here the reproductive rate of each type is given in brackets and the mutation rate is marked above the arrow. We will refer to such diagrams as mutation-selection networks.

The one-hit model can be relevant for the description of an oncogene activation, or cancer initiation in patients with familial disorders, where the first allele is mutated in the germ line, and the inactivation of the second allele leads to a fitness advantage of the cell. In these cases, we can assume r > 1. In the more general case, we can view the one-hit model as the process of any one genetic alteration, resulting in a advantageous (r > 1), disadvantageous (r < 1) or a neutral (r = 1) mutant. The Moran process. One can envisage the following birth-death process (called the Moran process). At each time step, one cell reproduces, and one cell dies. We set the length of each time step to be 1/N, so that during a unit time interval, N cells are chosen for reproduction and N cells die. We assume that all cells have an equal chance to die (this is equal to 1/N). On the other hand, reproduction happens differentially depending on the type, and the relative probability of being chosen for reproduction is given by 1 and r for the cells of types "A" and "B" respectively. Obviously, in this setting the total number of cells is preserved.

Let us denote the number of cells of type "A" as a, and the number of cells of type "B" as b, so that a + b — N. The probability that a cell of type "A" reproduces is proportional to its frequency and the reproductive rate, and is given by a/(a + rb). Similarly, the probability that a cell of type "B" reproduces is rb/(a + rb). Thus the probability that the new cell is of type "A" or type "B" is given respectively by

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