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The Different Effects of Apoptosis and DNA Repair on Tumorigenesis

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The Different Effects of Apoptosis and DNA Repair on Tumorigenesis J. theor. Biol. (2002) 214, 453}467 doi:10.1006/jtbi.2001.2471, available online at http://www.idealibrary.com on The Di4erent E4ects of Apoptosis and DNA Repair on Tumorigenesis JOSHUA B. PLOTKIN* AND MARTIN A. NOWAK Institute for Advanced Study, Princeton, NJ 08540, ;.S.A. (Received on 14 May 2001, Accepted in revised form on 5 October 2001) Complex multicellular organisms have evolved mechanisms to ensure that individual cells follow their proper developmental and somatic programs. Tumorigenesis, or uncontrolled cellular proliferation, is caused by somatic mutations to those genetic constraints that nor- mally operate within a tissue. Genes involved in DNA repair and apoptosis are particularly instrumental in safeguarding cells against tumorigenesis. In this paper, we introduce a stochas- tic framework to analyse the somatic evolution of cancer initiation. Within this model, we study how apoptosis and DNA repair can maintain the transient stability of somatic cells and delay the onset of cancer. Focusing on individual cell lineages, we calculate the waiting time before tumorigenesis in the presence of varying degrees of apoptosis and DNA repair. We "nd that the loss of DNA repair or the loss of apoptosis both hasten tumorigenesis, but in characteristically di!erent ways. ( 2002 Elsevier Science Ltd 1. Introduction This phenomenon is often called the mutator Tumorigenesis is de"ned as the onset of unregu- phenotype. The loss of tumor suppressor genes, lated cell proliferation. In humans and many on the other hand, diminishes a cell's ability to other mammals, the process towards accelerated recognize damage and induce apoptosis. Both cellular growth is marked by the loss of impor- increased mutation rates and de"cient apoptotic tant regulation genes which usually control cell- "delity accelerate tumorigenesis. But the relative cycle functioning. These regulation mechanisms importance of these two carcinogenic mecha- may be broadly characterized (Kinzler & Vogel- nisms is, a priori, unclear. The extent to which the stein, 1997; Vogelstein et al., 2000) as DNA repair mutator phenotype determines the timing of tu- genes, which repair mutations and DNA damage morigenesis is hotly contested in the literature. before further cell division, and tumor suppressor Some scientists (Loeb, 1991) have argued that genes, which signal for cell-cycle arrest and in- an increased pre-malignant mutation rate is re- duce apoptosis if substantial genomic damage is quired for tumorigenesis to occur whatsoever. detected (Gottlieb & Oren, 1998). Many other scientists agree that the mutator The loss of DNA mismatch or DNA excision phenotype plays an important, if not absolutely repair genes increases the e!ective mutation rate necessary, role in cancer development (Lengauer per cell division (Orr-Weaver & Weinberg, 1998). et al., 1998; Orr-Weaver & Weinberg, 1998; Murdoch & VanKirk, 1997). But others (Tomlin- son et al., 1996; Tomlinson & Bodmer, 1999) * Author to whom correspondence should be addressed. contend that Darwinian selection on cellular pro- E-mail: [email protected] liferation rates dominates the process towards 0022}5193/02/030453#15 $35.00/0 ( 2002 Elsevier Science Ltd 454 J. B. PLOTKIN AND M. A. NOWAK carcinogenesis, superseding any e!ects of the apoptosis and DNA repair were largely ignored. mutator phenotype. Still others stress the import- Outside of the general multistage framework, ance of apoptosis in preventing tumorigenesis other authors have investigated the e!ects of (Tomlinson & Bodmer, 1995; Hong et al., 2000; apoptosis (Tomlinson & Bodmer, 1995) and Harnois et al., 1997; Chang et al., 1998). Others mutation rates (Tomlinson et al., 1996). In addi- yet contend that the immune system plays a cen- tion to research into the genetic events that cause tral role in controlling tumors (Darnell, 1999). tumorigenesis, there is a large and detailed litera- The extent to which each of these factors* ture which models the physiological processes decreased apoptosis, increased mutation, in- involved in tumor invasion (Chaplain, 1995; creased proliferation, etc.*contribute towards Perumpanani et al., 1996), growth (Byrne & cancer depends, no doubt, on the cancer type. Chaplain, 1996a, b), encapsulation (Sherratt, Given the diversity of cancer types, it would be 2000), macrophage dynamics (Owen & Sherratt, misguiding to debate the crucial carcinogenic 1999), and angiogenesis (Chaplain, 2000). processes. Nevertheless, a rigorous, qualitative This paper is divided into nine sections. In understanding of the di!erences between these Section 2, we discuss parameter values for muta- carcinogenic processes can inform the debate tion rates and approximations appropriate to about their relative importance in various cancer modelling mutation. Section 3 presents our basic settings. model of tumorigenesis. In Section 4, we analyse In this paper, we develop a mathematical the simple case when apoptosis and increased framework for investigating the e!ects of cell- mutation rates are both neglected. In Section 5, cycle regulation genes. In particular, we use we compute the average waiting time before stochastic multistage models to investigate as to tumorigenesis in terms of the apoptotic rates and how DNA error repair and apoptosis stave o! increased mutation rates. In Section 6, we com- tumorigenesis. Our generic framework can be pute the distribution of waiting times before used to investigate qualitative patterns in the tumorigenesis. Section 7 compares the e!ect of progression towards carcinogenesis. We do not increased mutation with that of de"cient apop- initially specify a particular cell type or a parti- tosis. Section 8 addresses intrinsic costs asso- cular cancer type. Some of our assumptions, ciated with elevated mutation rates. Concluding however, constrain the applicability of our model remarks are given in Section 9. The main text to particular classes of cancer types. We do not refers to Appendices A}C for mathematical address the progression of a malignant tumor details. through its various cancerous stages, angio- genesis, and eventually metastasis. Instead, we investigate the accumulation of mutations in 2. Target Genes and Mutation a pre-malignant cell lineage. Throughout our analysis, we assume that there Stochastic modelling of tumorigenesis was are ¸ genes involved, in some way, in regulating introduced in the 1950s (Armitage & Doll, 1954) normal cell-cycle functioning. We call these for comparison with adult age-speci"c cancer ¸ sites target genes because their removal can incidence rates. Given the di!erences between increase the chance of tumor initiation. For hu- spontaneous and inherited cancers, however, mans, ¸ is rather large. As a rough approxima- authors soon began to develop multiple stage tion, given that there are over 150 genes involved models of tumorigenesis, starting with two-stage in apoptosis alone (Aravind et al., 2001) and over models (Knudson, 1971). Truly rigorous and 130 involved in DNA repair (Wood et al., 2001), complete analyses of the two-stage stochastic we assume that ¸+500. We imagine that the models soon followed (Moolgavkar & Venzon, target genes*DNA repair genes and tumor 1979; Moolgavkar & Knudson, 1981), treating suppressor genes*form a large network whose childhood and adult tumors separately. Multi- redundancy bu!ers the cell against tumorigen- stage models have since been expanded (e.g. Mao esis. We will assume that if any n of these target et al., 1998). In all such models, mutation rates genes become defective in a cell, then the cell will were generally assumed to be constant, and start to proliferate causing the onset of cancer. APOPTOSIS AND DNA REPAIR 455 (In an alternative model, we can de"ne tumor at least one of the cells acquires any n mutations. initiation as the mutation of a particular set of We will also compute this time in the presence of n genes from the ¸ regulation genes.) The thre- apoptosis and repair genes. Before we analyse shold n is usually small compared to the total the waiting times before tumorigenesis, we make number of target genes. Evidence suggests that several preliminary remarks about modelling as few as n"2 defective genes can cause un- mutation. We are assuming an extremely simple regulated cell proliferation for certain cancer mutational process. Regardless of the current types, such as retinoblastoma (Knudson, 1971), number of mutations, k, we assume that in each while n"6 or more are required for other cancer somatic generation k either increases by one, types (Loeb, 1991). with probability p, or remains constant, with In reality, even when a single cell acquires probability q"1!p. n mutations, the immune system may yet prevent Even when we choose to ignore back mutation, tumorigenesis by targeting the deviant cell (Dar- this formulation is only approximately correct. nell, 1999). The frequency and importance of this The exact formulation, according to an indepen- phenomenon, however, are hotly contested. We dent forward mutation rate k and backward will therefore assume throughout that tumori- mutation rate zero per gene, is given by genesis is simply de"ned by n mutations, delaying our discussion of immune action until Section 9. /(kPk) We assume that at every generation of cell division, each of the ¸ regulation genes may 0 for k(k, " ¸! (1) undergo a debilitating mutation with probability k kIY\I !k *\IY * k G (1 ) for k k. Once a gene is mutated, we can safely assume Ak!kB that it will never again back-mutate into a func- tional gene. Although the per-base mutation rate / P " !k *\I is well known for many organisms (Drake et al., To be exact, then, (k k) (1 ) .We 1998), the per-gene mutation rate k in non-germ- approximate this exact equation by de"ning " !k¸+/ P " ! line cells is more di$cult to measure in practice. q 1 (k k) and p 1 q. This ap- ;¸ Several authors have suggested that k+10\ proximation is valid provided that n and ¸k; per cell division (Orr-Weaver & Weinberg, 1998), 1, both of which are true for humans although the precision of this value is unclear.
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