This is the first of a number of chapters which investigates the relationship between carcinogenesis and genetic instability. Here, we will examine the most basic scenario: the generation of the first malignant cell. Does the presence of genetic instability result in a faster generation of the first malignant cell? The mathematics in this chapter are applications of the formalisms developed in Chapter 3. Here we present a simple example of how stochastic models developed for two-hit processes can be applied to biological reality.

We will concentrate on cancers which are initiated via the inactivation of a tumor suppressor gene. That is, both the maternal and the paternal copy of the gene have to lose function. A particular example which will be discussed in this context is colorectal cancer. Colorectal cancer is a major cause of mortality in the Western world. Approximately 5% of the population develop the disease, and about 40% of those diagnosed with it die within 5 years. Considerable progress has been made in identifying genetic events leading to colorectal cancer. Somatic inactivation of the adenomatous polyposis coli (APC) gene is believed to be one of the earliest steps occurring in sporadic colorectal cancer. It has been observed that the frequency of APC mutations is as high in small lesions as it is in cancers. Evidence that the APC gene plays a crucial role in colorectal cancer also comes from the study of individuals with familial adenomatous polyposis coli (FAP). FAP patients inherit a mutation in one of the copies of the APC gene; by their teens, they harbor hundreds to thousands of adenomatous polyps.

The APC gene is a tumor suppressor gene which controls cell birth and cell death processes. Inactivation of only one copy of the APC gene does not seem to lead to any phenotypic changes. Inactivation of both copies of this gene appears to result in an increased cell birth to death ratio in the corresponding cell and leads to clonal expansion and the formation of a dysplastic crypt. Here, we define a dysplastic crypt as a crypt that consists of cells with both copies of the APC gene inactivated. Dysplastic crypts are at risk of developing further somatic mutations which will eventually lead to cancer. The typical estimate is that an average 70 year old has about 1-10 dysplastic crypts, but precise counts have never been published.

How can the tumor suppressor gene be inactivated? A point mutation can induce a loss of function in one copy of the gene. Both copies of the gene can be inactivated by two subsequent point mutations in the same cell: one in the maternal, and the other in the paternal allele. Each mutation would occur with the physiological mutation rate of 10-7 per gene per cell division. Now consider genetic instability. As explained in Chapter 1, there are two major types of instabilities [Lengauer et al. (1998); Sen (2000)]: (i) small scale subtle sequence changes, such as microsatellite instability (MSI). The MSI phenotype is generated if specific MSI genes are inactivated. Both copies of an MSI gene need to be mutated. MSI basically results in an elevated point mutation rate in the context of repeat sequences called microsatellites. (ii) Gross chromosomal alterations can occur, and this is known as chromosomal instability (CIN). The genetic basis of CIN is uncertain, and specific scenarios will be discussed below. If one copy of the tumor suppressor gene has been inactivated by a point mutation, the other copy can be inactivated very quickly in CIN cells due to the loss of the healthy allele. This can occur through a variety of mechanisms. They include loss of the remaining chromosome and loss of part of the remaining chromosome. These processes are also called loss of heterozygocity, or LOH.

While genetic instability might speed up the loss of tumor suppressor function, the MSI or CIN phenotypes need to be generated first (for example by basic point mutations). This chapter discusses a mathematical analysis of how MSI and CIN influence the rate of tumor suppressor gene inactivation. We will apply this analysis to various scenarios which include the sporadic (spontaneous) development of colon cancer, and familial colon cancers. We start with some more detailed biological facts about CIN and MSI in colon cancer and then present the mathematical analysis.

4.1 Some biological facts about genetic instability in colon cancer

Here we will study the role that CIN and MSI may play in the inactivation of the APC gene. About 13 % of all colorectal cancers have MSI and most of the rest are characterized by CIN [Lengauer et al. (1998)]. MSI occurs in virtually all hereditary non-polyposis colorectal cancers (HNPCC), which account for about 3% of all colorectal cancers. The MSI phenotype results from defective mismatch repair. Several genes have been identified whose inactivation leads to an increased rate of subtle genetic alterations. The main ones are hMSH2 and hMLHl. Both copies of an MSI gene must be inactivated in order for any phenotypic changes to occur. HNPCC patients inherit a mutation in one of the copies of an MSI gene and normally develop colorectal tumors in their forties. Unlike FAP patients, they do not have a vastly increased number of polyps, but the rate of progression from polyp to cancer is faster.

Molecular mechanisms leading to CIN in human cancers remain to be understood. It has been proposed that CIN might be caused by mutations in genes involved in centrosome/microtubule dynamics, or checkpoint genes that monitor the progression of the cell cycle, e.g. the spindle checkpoint or the DNA-damage checkpoint [Kolodner et al. (2002)]. For example, heterozygous mutations in the mitotic spindle checkpoint gene hBUBl have been detected in a small fraction of colorectal cancers with the CIN phenotype [Cahill et al. (1998); Gemma et al. (2000); Imai et al. (1999); Ohshima et al. (2000)]. Also, the MAD2 gene seems to be transcriptionally repressed in various solid tumors [Li and Benezra (1996); Michel et al. (2001); Ro and Rannala (2001); Wang et al. (2000)]. Some CIN genes might act in a dominant-negative fashion: an alteration in one allele leads to CIN.

4.2 A model for the initiation of sporadic colorectal cancers

The colonic epithelium is organized in crypts covered with a self-renewing layer of cells (Figure 4.1). The total number of crypts is of the order of M = 107 in a human. Each crypt contains of the order of 103 cells. A crypt is renewed by a small number of stem cells (perhaps 1 — 10) [Ro and Rannala (2001); Yatabe et al. (2001)]. The life cycle of stem cells is of the order of 1 - 20 days [Bach et al. (2000); Potten et al. (1992)]. Stem cells

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