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. (Drake et al., 1998). When apoptosis and DNA repair do not occur, the stochastic process of accumulating cellular mutations is relatively simple. Consider the case 3. A Model of Mutation, Apoptosis, of tumorigenesis de"ned by any n mutations. and DNA Repair Each cell in a tissue of constant size is, on aver- We now formulate a stochastic, Markov-chain age, replicating and being replaced by its daugh- model of the mutational and apoptotic process. ter cell at each generation. A cell is characterized We imagine a large tissue of cells which, at all by the number of mutant genes k,0)k)¸, times preceding the onset of cancer, is almost it currently harbors. We say that a cell lineage completely mutation free. We keep track of the becomes cancerous (i.e. has lost its regulation current number of mutations k harbored in a cell ability), if k*n. In the most simple case, the cell lineage. In our Markov model, any particular cell has a probability p of increasing its number of is replaced by its daughter cell in each generation. mutations by one, and probability q"1!p of At each generation, the healthy cell (k"0) either maintaining the same number of mutations at remains healthy, with probability q, or accumu- each replication/replacement event. We call this lates a single mutated gene, with probability " ! situation &&neutral'' because the cell behaves in the p 1 q. same manner regardless of its current mutational A cell harboring k*1 mutations, however, is status. subject to tumor-suppressor-induced apoptosis. Starting from a tissue containing N healthy In particular, at each generation, a cell harboring a cells, we will compute the expected time before k mutated genes has probability I of destroying 456 J. B. PLOTKIN AND M. A. NOWAK itself and being replaced by another cell in the morigenesis (k"n) is a slippery slope; once a few tissue, which we assume to be healthy. With suppressor genes have been damaged, it becomes b " !a probability I 1 I, on the other hand, the increasingly hard for a cell to recognize muta- mutated cell replicates*maintaining the same tions, and thereby to prevent further progression number of mutations with probability qI, and towards cancer (Kinzler & Vogelstein, 1997). On acquiring an additional mutation with probabil- the other hand, we can also consider cases when a ity pI. According to this formulation, a cell lin- the apoptotic rates I increase. This models the eage is described by a Markov chain with the situation in which, say, progressively more on- following (n#1);(n#1) transition matrix: cogenes become mutated and signal for increased apoptosis. P" We are primarily interested in the cases when a I decreases, corresponding to the loss of tumor- 22 q p 0 0 suppressors and apoptotic "delity. Similarly, we a b q b p 0 2 0 generally consider cases in which the e!ective      mutation rates p are increasing. This assumption a 0 b q b p 2 0 I      . mirrors the progressive loss of DNA repair genes $$ \\ $ and the corresponding increase in e!ective muta- a 2 b b tion rate and genomic instability (Vogelstein A L\ 0 0 L\qL\ L\pL\B et al., 2000). 0 222 01 We are nominally modelling the behavior of (2) a single-cell lineage. But whenever a cell under- a goes apoptosis (with probability I), it is replaced The state k"n, which we call tumorigenesis, is by a cell from the surrounding tissue which we absorbing. Once the cell reaches this state, we say assume to be healthy. Thus, we are, in fact, mod- that the cell has escaped from proper cell regula- elling the behavior of a cell in a large tissue tion and begins malignant growth. We do not composed predominantly of healthy cells. There- concern ourselves with the progression of cancer fore, our cell &&lineage'' is not a strict lineage, but after this stage. Although active tumors certainly rather includes replacement by healthy cells upon undergo mutation and apoptosis, the evolution apoptosis. Before the onset of cancer, we assume of a tumor is a very di!erent phase of cancer that almost all cells in the tissue are mutation progression which we do not model here. free. This assumption, used here to simplify popu- a When apoptosis is ignored (i.e. all I are zero), lation dynamics, may be invalid for some cancer then the formulation given by eqn (2) is equiva- types. (Elsewhere we relax this assumption, and lent (in discrete time) to the classical models of allow for apoptosis followed by replacement of tumorigenesis formulated by Armitage & Doll nearby unhealthy cells, Nowak & Plotkin, in (1954) and later developed by Moolgavkar preparation) Colorectal cancer, for example, is (1978). Here we extend these models to include often preceded by the invasion of numerous be- the action of apoptosis. nign polyps which exhibit one or two mutated We will examine tumorigenesis for arbitrary genes, often APC, throughout the tissue (Kinzler a apoptotic rates I. The simplest case occurs when & Vogelstein, 1996). Other cancers*such as a I is constant. If we focus our attention on target retinoblastoma*are not generally preceded by genes which are related to apoptosis, then the widespread polyps containing mutations, and a apoptotic rates I should naturally be decreasing are therefore more appropriately modelled by with k. This corresponds to the situation in our formulation. which as tumor-suppressor genes become muta- Our Markov model does not address di!er- ted, their e$ciency at scanning for damage and ences in proliferative rates between cell lineages. inducing apoptosis is reduced. Hence, mutated Elsewhere (Nowak & Plotkin, in preparation), cells gain a selective advantage by avoiding we develop deterministic models which treat programmed cell death (Tomlinson & Bodmer, inter-lineage population dynamics. Again, the 1995). In this formulation, the road towards tu- one-dimensional Markov model is therefore less APOPTOSIS AND DNA REPAIR 457 appropriate for colorectal type cancers*which Here C(), )) denotes the incomplete Gamma exhibit clonal expansion and competition function. Equation (3) demonstrates that in the through increased proliferation*than it is for neutral case the waiting time before tumorigen- retinoblastoma-type cancers (Cairns, 1998). esis depends inversely on the logarithm of one The Markov model keeps track of the current replication "delity (q). Figure 1 demonstrates the number of defective regulation genes, k. At each expected neutral waiting time before tumorigen- stage we do not know which k of the ¸ target esis for various mutation rates and tissue sizes. As genes are mutated. In this sense, the reduced is clear from the "gure, increasing the tissue size a ¹ apoptotic "delities I and the increased e!ective N geometrically causes the waiting time to mutation rates pI represent averages over all decrease geometrically. possible combinations of k mutated genes. In other words, the loss of some tumor suppressors, e.g. p53, can in reality be more deleterious than 5. The Average Waiting Time before the loss of other ones; we average over these Tumorigenesis possibilities. Given arbitrary mutation rates pI and apop- a totic rates I, what is the mean time before car- " 4. The Neutral Case cinogenesis (state k n)? In addition, what is the expected total time spent by a cell lineage in each Before we analyse the full-blown Markov of the various mutated classes? For example, do model, we consider the simple, neutral case in we expect to "nd cells with k"2 mutations half which apoptosis and DNA repair are both as often as k"1? From this section onwards, we ignored. In other words, we assume for now that answer these questions insofar as a single-cell a all I are zero and all pI are equal. In Appendix A, lineage is concerned*an approach originally es- we derive the expected number of cell divisions, poused by Armitage & Doll (1954) and later ¹, required before at least one cell in a tissue Moolgavkar (1978). of N cells, originally all healthy, acquires n ¹ De"ne G as the expected number of genera- mutations: tions before absorption into the cancerous state, assuming that the cell begins in state i at time 1  ¹ ¹" C , zero. We are eventually interested in , the ! ,  (n, a) da. (3) log(1/q)(n 1)!  mean time until tumorigenesis starting from ¹ a healthy cell. We might also be interested in , if an individual inherits a defective gene. De"ne =GH as the mean number of visits to state j prior to absorption, assuming that the cell starts in state i. As is clear from the Markovian property of our process, we have the following recursive relationships:

L\ "d # + " ! =GH GH PGK=KH for i 0,2, n 1, (4) K In Appendix B, we use this recursion to com- pute the mean time spent in each mutation class prior to tumorigenesis, starting from a healthy cell:

FIG. 1. The expected number of cell divisions before one " 1 L\ p b #a cell in a tissue of size N acquires n 5 mutations, ignoring = " “ V V V apoptosis and DNA repair. The mutation rate p varies from  I p b p b p"500;10\ to 0.01 on the x-axis. Each curve represents L\ L\ VI> V\ V\ adi!erent tissue size ranging geometrically from N"10, highest curve, to N"10, lowest curve. for 0)k)n!1, (5) 458 J. B. PLOTKIN AND M. A. NOWAK

Assuming that apoptotic rates are constant, Fig. 2 shows as to how the timing of tumorigen- esis depends upon the damaged cell's mutation rate. The loss of DNA repair genes in damaged cells can increase the e!ective mutation rate by a factor ranging from 10 to 10 (Tomlinson et al., 1996). Note that the bene"cial e!ect of apoptosis is minimized as the damaged cell's mutation rate increases. In other words, if there is a strong mutator phenotype, then the time before tumori- genesis is less signi"cantly a!ected by the pre- sence of functioning apoptotic machinery. FIG. 2. A graph of the expected number of cell divisions before tumorigenesis (n"4) in a cell lineage as a function of the damaged cell's mutation rate p. We assume a constant apoptotic rate [Eqn (9)]. The healthy cell's mutation rate is " ; \ p 500 10 . The damaged cell's mutation rate ranges 6. The Distribution of Waiting Times before " 100-fold on the x-axis from p p to 0.05. Each of the six Tumorigenesis curves represents a di!erent apoptotic rate, ranging linearly from a"0.001 (lowest curve) to a"0.01 (highest curve). Equation (6) describes the expected time before Note that the time until tumorigenesis is less sensitive to the tumorigenesis in a cell lineage, given the apop- apoptotic "delity whenever the damaged cell's mutation rate is high. totic and mutation rates. We are further interest- ed in the complete distribution of waiting times L\ until tumorigenesis. For instance, does the muta- ¹ " +  = I, (6) tor phenotype increase the variance of the wait- I ¹ ing time? In this section, we let  denote the a " random time at which the Markov chain reaches where we have de"ned  0 for convenience. Note that a cell spends (p b #a )/(p b )as the cancerous state, n. We will "nd expressions I I I I> I> $¹ much time with k mutations as it does with k#1 for the expected value  and for the distribu- ¹ mutations, before tumorigenesis. In addition, the tion function of , in terms of the eigenvalues of mean time a cell spends in mutational class k de- P. As we will discuss below, the distribution of ¹ pends only upon the e!ective mutation rates and  can be used to predict cancer age}incidence apoptotic rates of the classes k*k. In other curves comparable to epidemiological census words, the selective pressure on k-th-order check- data. # point genes does not depend upon the behavior Consider any (n 1)-state Markov chain < < of cells with fewer than k mutations. M acting on the left. (In our case, take M to be In the case when all the apoptotic rates are the transpose of P.) Label the states 0 to n, and constant, a "a, and when all the damaged assume that state n is the only absorbing state. I ; mutation rates are constant (p "p "2" Consider the n n submatrix M which excludes   " O p "p), eqns (5) and (6) simplify to the following: the absorbing state. Let < (1, 0,2,0) denote L\ the initial condition: in state zero. Note that i-th coordinate of MR< , denoted [MR< ] , is the 1 pb#a L\   G = " probability that the chain is in state i at time t.   A b B , (7) p p We index the coordinates from 0 to n!1. Hence the following expression: 1 pb#a L\I\ = " ) ) !  I b A b B for 1 k n 1, L\ p p + R "# R # [M <]G M < , (8) G yields the probability that the cell is not yet in the b a# ¹ " p ( p) !1 absorbing state at time t. In other words, this sum  a a # !a #a L a . (9) p( q p)(1 /(p q)) is the complementary cumulative distribution APOPTOSIS AND DNA REPAIR 459

¹ # ) # function of . We use to denote the sum of provides an alternative analytic formula for the the components of a vector. density function (t). Moreover, the generating We can write M"SDS\ where D is the dia- function yields analytic expressions for all the j j ¹ gonal matrix of eigenvalues, ,2, L\, pro- moments of . vided that they are all distinct. The columns of There is great value in knowing the distri- S are the corresponding eigenvectors. In this case, bution, (t), of the waiting time before tumori- R " R \ M < SD S <. Hence, the cumulative distri- genesis in a single-cell lineage. As originally ¹ U " !# R \ # bution of  is given by (t) 1 SD S < . elucidated by Moolgavkar (1978), knowledge of This in turn allows us to compute the expected (t) allows us to compute the hazard function of time before tumorigenesis: tumorigenesis in a large tissue of N cells. The probability that at least one tumor occurs in the   tissue by time t is simply F(t)"1!(1!U(t)),. $¹ " + !U " + R \ "# #  1 (t) SD S < Q , Hence, the hazard function*that is to say, the R R instantaneous rate of cancer incidence*is given (10) by F(t)/(1!F(t))"N (t)/(1!U(t)). The hazard function of tumorigenesis is directly comparable where to cancer age-incidence data collected from a population (Armitage & Doll, 1954). In most 1 cases, U(t) will be small within a human lifespan. !j 00 1  Therefore, the rate of a cancer incidence will be approximated very well simply by N (t). Hence, " \ \< Q S 0 0 S . (11) by understanding the probabilistic behavior of single-cell lineage we can predict the cancer A 1 B age-incidence rates within a human population 00!j 1 L\ (Armitage & Doll, 1954).

Similarly, the i-th component of Q gives the ex- pected value of =G, the amount of time spent in 7. The Loss of Repair vs. the Loss of Apoptosis class i before absorption. For any given para- Equations (5), (6) and (12) describe the precise meter values, we can numerically compute the in#uence of apoptosis and DNA repair as deter- eigenvectors of P and use eqn (11) to "nd =G. rents against cancer. For example, in Fig. 3 we Amazingly, even when analytic expressions for plot the distribution of the waiting time before the eigenvectors of P are di$cult, we know that tumorigenesis in the simple two-hit case, n"2. eqn (10) always simpli"es to the exact solution As expected, when compared with the neutral given in eqn (6). case, increased mutation rates (the loss of DNA Finally, from our expression for the cumulative repair genes) cause a decrease in mean time U distribution , we see that the probability density before cancer, while tumor-suppressor-induced "U !U ! ¹ function (t) (t) (t 1) of  is given by apoptosis causes an increase. Interestingly, we also observe that apoptosis greatly increases the (t) variance in the time before tumorigenesis*an observation that is con"rmed, analytically, by jR\!jR eqn (23).   00 In Fig. 4, we compare the progressive loss of " S 0 \ 0 S\< .  apoptotic activity to the progressive loss of DNA A jR\!jR B 00L\ L\ repair activity, in the case n"4. If the apoptotic a rates I and mutation rates pI are constant, then (12) the distribution of waiting times has a very large variance. We compare the model's behavior with In Appendix C, we compute the generating func- constant a's and p's against two alternative scen- ¹ tion of  in the case of constant apoptosis. This arios. In the "rst alternative, we keep mutation 460 J. B. PLOTKIN AND M. A. NOWAK

FIG. 3. The e!ect of increased mutation rates and apop- tosis when n"2 mutations are required for tumorigenesis. The "gure shows the probability density function of the time ¹ before tumorigenesis, , given by eqn (12). In the neutral case (-----), all states have the same mutation rate, " " ; \ a " p p 500 10 , and there is no apoptosis,  0. The four curves drawn in (}}}}) show the e!ect of varying degrees of increased mutation, ranging linearly from " " p 0.01 (uppermost curve) to p p. The mutator phenotype decreases the mean time before tumorigenesis. The (**) curves show the e!ect of various amounts of a " apoptosis, ranging from  0.01 (lowermost curve) to a "  0. Apoptosis increases the mean waiting time before tumorigenesis. The mutator phenotype decreases the vari- ance in the waiting time, while apoptosis greatly increases the variance.

a constant and we let the apoptotic rates I de- crease linearly with k. This models the loss of suppressor-induced apoptosis, resulting in a FIG. 4. The distribution of the waiting time before tu- shorter waiting time before tumorigenesis. In the morigenesis [n"4, using eqn (12)] shown on linear axes second alternative, we keep apoptosis constant (top), log-linear axes (middle), and log}log axes (bottom). a "a " The (-----) curve corresponds to the case when   and we let the mutation rates pI increase with k. a " " " " " ; \  0.001 and p p p p 500 10 . The This models the mutator phenotype caused by (}}}}) curve corresponds to linearly increasing mutation " the loss of DNA repair genes. In this case, tu- rates, p/p 100. The (**) curve corresponds to linearly a a " morigenesis occurs much earlier on average and decreasing apoptotic rates, /  100. The mutator phenotype causes a dramatic decrease in the mean and has a much smaller variance. In other words, the variance of time before tumorigenesis. The loss of proper loss of repair has a stronger and more dramatic apoptotic function has a more modest e!ect on the waiting e!ect on tumorigenesis than the loss of proper time. apoptotic functioning, in the sense that an x-fold loss of repair decreases the time before tumor initiation far more than an x-fold loss of apop- strength of the mutator phenotype. As the totic "delity (Fig. 4). strength of the mutator phenotype increases, Figure 5 further elucidates the e!ect of the mean and variance of the waiting time before apoptotic malfunction compared to increasing tumorigenesis both decrease. In this model, the mutation rates. Again, we plot the density of the e!ect of the mutator phenotype does not satu- waiting time before tumorigenesis in various al- rate: the more the mutation rates increase, the ternative scenarios. In the most simple scenario, earlier tumorigenesis occurs. In the other set of a all the apoptotic rates and mutation rates are alternatives, the apoptotic rates I decrease with a "2"a "2" constant:  L\; p pL\.In k. Although the loss of apoptosis also results in one set of alternatives, the mutation rates pI in- a shorter time before tumorigenesis, this e!ect crease with k. The ratio pL\/p measures the eventually saturates. APOPTOSIS AND DNA REPAIR 461

that the mutator phenotype pI and the apoptotic a rates I should be correlated to some extent. In this section we will explore an extended model in which increased mutation is accompanied by increased cellular death. We have assumed that there are ¸ &&target'' genes involved in regulating cell-cycle function- ing. Assume now that there are an additional M &&critical'' genes (M+10 000 in humans) in- volved in cellular functioning which do not cause cancer, but whose mutation causes cell death (even in a cancerous cell). For any mutation rate FIG. 5. The distribution of waiting times before tu- k morigenesis [n"4, using eqn (12)] for various amounts of ,wede"ne increased mutation and decreased apoptosis. The (-----) curve corresponds to the simple case of constant apoptotic q "(1!k)+>*, rates and mutation rates, as in Fig. 4. The (}}}}) curves I correspond to cases of increasing mutation rates, with p/p ranging from 1 to 10 (uppermost curve). The (**) curves p "[1!(1!k)*](1!k)+, a a I correspond to decreasing apoptotic "delity with /  ranging from 1 to 20. Note that the e!ect of reduced apop- " ! ! " ! !k + totic "delity asymptotes, whereas the e!ect of the mutator sI 1 qI pI [1 (1 ) ]. phenotype does not saturate. This result remains true a whether pI and I vary geometrically or arithmetically Here q represents the probability per cell divi- with k. I sion that no genes whatsoever are mutated; pI represents the probability that at least one target gene is mutated, but none of the critical Generally speaking, the distribution of the M genes is mutated; and sI denotes the waiting time for cancer is nearly identical when- probability that at least one of the critical genes a a ) ever /  exceeds 10, for n 6. The exact e!ect becomes mutated*which will cause immediate of progressive loss of apoptotic "delity depends cell death. upon the cancer threshold n and the base-line Consider the following Markov chain model of a apoptotic "delity . But for any parameter tumorigenesis, which includes an intrinsic cost of a ¹ values n and , the distribution of  asymp- higher mutation rates: a a totes as / L\ increases, whereas the distribu- tion of ¹ does not asymptote with increasing  q #s p 0 22 0 p /p . This behavior is apparent in Fig. 5 and IŠ IŠ IŠ L\  a#b b b 2 can also be seen, in expectation, from eqn (5) sI qI pI 0 0 directly. This result highlights the importance of a#bs 0 bq bp 2 0 P" I I I . the mutator phenotype as a potential driving $$ \\ $ force behind early tumorigenesis. Conversely, a#b 2 b b A sI 0 0 qI pIB this result indicates that the loss of apoptotic functioning hastens tumorigenesis, but that this 0 222 01 e!ect has an intrinsic limitation. In this formulation, a denotes the apoptotic rate, k which we assume to be constant. Here  denotes 8. Costs Associated with the Mutator Phenotype the intrinsic mutation rate per gene per replica- So far, we have ignored any intrinsic costs tion event of a healthy cell and k denotes the associated with increased mutation rates. In real- elevated mutation rate of a cell which lacks one ity, however, high mutation rates can lead either or more regulation genes. Note that either apop- (i) to the loss of crucial genes needed for cellular tosis or the mutation of any &&critical'' gene causes function and/or (ii) to the triggering of pro- a cell to die and be replaced by a surrounding grammed cell death. This phenomenon indicates healthy cell. Given that increased mutation now 462 J. B. PLOTKIN AND M. A. NOWAK carries a cost*namely the possibility of mutating a gene critical for cellular survival*we expect to "nd some optimum increased mutation rate k which minimized the time to tumorigen- esis. As before, let = I denote the mean time spent in class k before absorption into class n, starting from a healthy cell. Using the methods described in Appendix B we deduce that

1 bp I>\L = " I  I !a!b # A !b B 1 (qI s) 1 qI

for 1)k(n!1, (13)

1!bq bp \L = " I I .   p (1!a!b(q #s)) A1!bq B IŠ I I (14)

Therefore, the expected time to tumorigenesis is given by FIG. 6. (a) The expected time until tumorigenesis as k'k " \ L\ a function of the increased mutation rate  10 ,in ¹ " + = " the presence of mutation-associated costs [eqn (15)]. There   I are ¸"500 target genes, and the apoptotic rate is a"0.001. I Six curves are shown, depending upon the number of critical genes, M. If there are more critical genes, then the time to p #(p b)\L(1#q (a!1))L\[b(p #q )!p !1] tumorigenesis is increased. For each set of parameters, there IŠ I I I I IŠ . is an optimum mutation rate k"k* at which the time to p (b(p #q )!1) k IŠ I I tumorigenesis is minimized. (b) The dependence of * on the (15) number of critical genes. If there are more critical genes, then the optimum mutation rate is decreased, and the e!ect of a strong mutator phenotype can be detrimental (from the For any parameter values ¸, M, n, and k , cancer's viewpoint). The optimum mutation rate does not  a eqn (15) is always minimized at some value depend upon the apoptotic rate or the cancer threshold n. k"k*, which is the optimal increased mutation rate (from the cancer's point of view). Figure 6(a) shows the expected time until tumorigenesis for 9. Conclusions varying degrees of the mutator phenotype, with We have introduced a stochastic framework associated costs. Figure 6(b) shows the optimum for investigating the relations between apoptosis, increased mutation rate k* as a function of the the mutator phenotype, and carcinogenesis. As number of critical genes. The optimal mutation we have seen, the loss of apoptosis and the loss of rate k* is always less than the mutation rate DNA repair both hasten the onset of cancer, but which simply maximizes the speed of forward their speci"ce!ects on the pre-cancer waiting mutation, pI.We"nd that the waiting time be- time are dramatically di!erent. The mutator fore tumorigenesis is strongly a!ected by the phenotype can cause a dramatic decrease in the number of mutations required for tumorigenesis, mean and variance of the time before cancer, n, but interestingly the value of the optimum while the loss of apoptosis reduces the mean mutation rate does not depend upon the tumor but not the variance. In addition, stronger threshold n or the apoptotic rate a. mutator phenotypes lead to increasingly rapid APOPTOSIS AND DNA REPAIR 463 tumorigenesis, while the e!ect of apoptotic mal- the spread of cell lineages which have undergone function asymptotes. somatic mutations. Similarly, some mutations We have seen that the e!ect of mutator pheno- may increase the rate of cell division, which we type can also asymptote, however, if we include have assumed to be constant. Nevertheless, the costs associated with increased mutation rates. If simplifying assumption of direct daughter-cell a certain set of genes is critical for cellular replica- replacement has long been considered to be a tion*without which even cancer cells cannot reasonable model in the cancer literature (Armit- survive*then there exists an optimal value of the age & Doll, 1954; Moolgavkar, 1978). Prelimi- increased mutation rate (from the cancer's view- nary work (Nowak & Plotkin, in preparation) point) which minimizes time before tumorigen- indicates that the timing of tumorigenesis is esis. This optimal mutation rate does not depend largely unchanged even when this simplifying upon the amount of apoptosis or the number of assumption is relaxed. hits required for tumorigenesis. Our model also ignores the e!ect of the This paper extends multistage models for immune system and tumor immunity. We have cancer initiation (Armitage & Doll, 1954; Mool- treated each cell lineage as an independent, gavkar, 1978) to include the e!ects of apoptosis autonomous process. In reality, the immune sys- and mutator phenotypes. In general, this ap- tem certainly plays a role in some cancer types proach allows us to determine the waiting time (Burnet, 1970), while the extent of this role is before tumorigenesis and the cancer age}inci- hotly contested. Recent developments (Albert dence curve of a population in terms of the pro- et al., 1998a, b) indicate that apoptosis of tumor babilistic behavior of a cell lineage. Because we cells may be followed by uptake into dendritic have not considered the progression of a tumor cells thereby training ¹ cells to recognize and through its various cancerous stages, the thera- target the tumor. This process suggests that apo- peutic signi"cance of our results is limited to ptosis may play a much larger albeit indirect role cancer prevention. Our results would indicate *by sensitizing the immune system*in tumor that gene therapies preserving e$cient DNA control. Our model investigates the patterns of repair may be more important than those tumorigenesis in the absence of immune activity preserving the apoptotic "delity of cells. *which could, by failure to match empirical We hasten to discuss some important simpli"- patterns, provide evidence for immune activity. cations used in the modelling framework. The In fact, the observed limitations of apoptotic mechanisms and connections between apoptosis malfunction as a direct cause of tumorigenesis in and repair are, no doubt, much more complic- our model, together with the empirical observa- ated in reality than in our model. Although our tion that tumor suppressors like p53 are usually stochastic framework allows for coordinated be- mutated in many tumors, may suggest that apop- a havior, we have largely treated apoptosis ( I) and tosis does indeed have an important indirect DNA repair (pI) as independent processes. In e!ect in controlling cancer. some sense, we have treated tumor suppressor genes and DNA repair genes separately. In J. B. P. gratefully acknowledges support from reality, however, some tumor suppressors such the National Science Foundation and the Burroughs as p53 (Vogelstein et al., 2000) often induce Wellcome Fund. J. B. P. and M. A. N. also thank cell-cycle arrest (Laronga et al., 2000) and thereby the Alfred P. Sloan Foundation, the Ambrose Monell allow DNA repair genes to remove DNA damage Foundation, the Florence Gould Foundation, and the (Tanaka et al., 2000). Hence, reduced apoptotic J. Seward Johnson Trust. Both authors also appre- a ciate helpful discussions with J. Dusho!, R. 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Mutation and cancer: statistical APPENDIX A study of retinoblastoma. Proc. Natl Acad. Sci. ;.S.A. 68, 820}823. The Neutral Case LARONGA, C., YANG, H. Y., NEAL,C.&LEE, M. H. (2000). Association of the cyclin-dependent kinases and 14-3-3 In this appendix, we analyse the neutral case sigma negatively regulates cell cycle progression. J. Biol. when apoptosis and repair are ignored. We will Chem. 275, 23106}23112. calculate the expected time until one of the LENGAUER, C., KINZLER,K.W.&VOGELSTEIN, B. (1998). Genetic instabilities in human cancers. Nature 396, N cells has at least n mutations, in terms of the 643}649. genic mutation rate k. Let us "rst consider APOPTOSIS AND DNA REPAIR 465 a single cell. We de"ne the stopping time ¹ as ing one mutation within one time unit. The distri- the smallest number of divisions required before bution G(t) for the time of i-th mutation satis"es a cell has n mutations, given that it starts with the recursion k"0 mutations. Clearly, then, the time to tumo- rigenesis ¹ equals or exceeds n. The probability R " ! that the cell lineage becomes cancerous at G(t)  G\(s) (t s)ds, generation n#x is given by Q whence we "nd that the waiting time before n#x!1 /(¹"n#x)"pLqV , (A.1) tumorigenesis has the Gamma distribution A x B " ! jL L\ \HR L(t) 1/(n 1)! t e . This in turn yields the complementary CDF of ¹: /(¹*a)" " !k * where, as discussed earlier, q (1 ) , and  (t)dt"1/(n!1)!C(n, ja), where C is the " ! ? L p 1 q. Note that, for a single cell, the incomplete Gamma function. We can now com- ¹ + # expected value of is simply V(n x) pute the mean of >: L V L>V\ " p q ( V ) n/p. We are interested, however, in the expected   C(n, ja) , time at which any one of the N cells comprising $>" / ¹* , "  ( a) da  A ! B da the tissue becomes cancerous. In other words, we   (n 1)! " +¹ ¹ want the expected value of > min , , , ¹ , where the ¹ are independent random  2 , G " 1 C , variables each distributed as ¹. We know that for j ! ,  (n, a) da. (A.3) (n 1)!  a*n Equation (A.3), which is straightforward to  ! / ¹* " + L G\L i 1 compute numerically, provides an accurate ( a) p q A ! B G? i n approximation of eqn (A.2). L\ a!1 "q?\ + (p/q)G A B . APPENDIX B G i The Mean Time before Tumorigenesis The complementary cumulative distribution of In this appendix, we calculate the mean time > / >* "/ ¹* , is given by ( a) ( a) . This, in turn, before tumorigenesis in the Markov model with > allows us to compute the expected value of : arbitrary apoptotic and mutation rates. We start by calculating = directly. Then we compute   L\ $>" + /(>*a) the ratios = I/= I>. We eventually combine ? these results into expressions for = I, yielding ¹ "+L\=  I  I.  L\ a!1 , "n# + Cq?\ + (p/q)G A BD . (A.2) In order to compute = L\, we imagine a ?L> G i reduced Markov chain in which states zero through n!2 are collapsed into a single class. The exact, discrete-time solution [eqn (A.2)] is This chain has the following transition matrix: unwieldy, but it can be approximated very well using a continuous-time, discrete-state model e 1!e 0 (Poisson process). In this approximation, each M" a b q b p , cell experiences an exponentially distributed A L\ L\ L\ L\ L\B waiting time before accumulating another muta- 00 1 tion. The time until a healthy cell acquires one mutation is distributed according to the density where e3(0, 1) is some value determined by the "j \HR j"! function (t) e . We choose log apoptotic rates, mutation rates, and n. We need (1!p) so that the cell has probability p of acquir- not actually calculate the value of e. We denote 466 J. B. PLOTKIN AND M. A. NOWAK

! "2" " the three states of this process by s, n 1, and n have the same mutation rates p pL\ ' a "2"a "a (s stands for starting). We can apply eqn (4) to this p p and apoptotic rates  L\ . "/ ¹ " reduced process in order to calculate =Q L\, Let uI G ( I i) denote the probability of which equals = L\ in the full Markov chain. absorption at time i, starting from state k.Ac- The resulting two-dimensional linear system has cording to the Markov chain in eqn (2), we have " b the unique solution = L\ 1/pL\ L\, inde- the following recursive relations: pendent of e. = = " b Next, we show that  I/  I\ pI\ I\/ u "bqu #bpu #au for 1)k(n, b #a " ! I G> I G I> G  G (pI I I), for k 2, 3,2, n 1. We consider another reduced Markov chain. This time, we (C.1) collapse states zero through k!2 into a single state, and states k#1ton!1 into a single state. " # u G> qu G pu G (C.2) This yields the following transition matrix with "ve states, called s, k!1, k, f, and n: "d subject to the boundary condition uL G L .We ¹ de"ne the generating function of I as the formal M" sum

 e 1!e 000 " + G ;I(s) uI G s . a b b G I\ I\qI\ I\pI\ 00 a 0 b q b p 0 , I I I I\ I On multiplying eqns (C.1) and (C.2) by sG> and A c 00d 1!c!dB summing over i,we"nd that 00 0 0 1 " b # b # a ;I(s) s q;I(s) s p;I>(s) s ; where, again, e, c, and d are constants which we need not compute. Note that c#d(1, because for 1)k(n, (C.3) the only absorbing state is n. Although we cannot use this reduced system to compute = or " #  I ;(s) sq;(s) sp;(s) (C.4) = I\ directly, we can compute the ratio of these quantities. We apply eqn (4) twice, once subject to the boundary condition ; (s)"1. In with j"k and once with j"k!1. For j"k,we L order to solve these recursions, we may treat the obtain a four-dimensional system with unique formal variable s as a constant. In general, the solution = "(1!d)/(p b (1!c!d)). When Q I I I recursion j"k!1, eqn (4) gives another four-dimensional " !d # a system with solution =Q I\ (1 )(pI qI I)/ b b !c!d (x)"a (x)#b (x#1)#c (1) (pIpI\ I I\(1 )). On dividing, we "nd " " b b that =Q I/=Q I\ = I/= I\ pI\ I\/(pI I #a has the solution I), as desired. A similar argument for the " " boundary value k 1 implies that =/= b #a " p/(p  ). By combining these results, we (x) obtain the exact expressions given by eqns (5) Db\V(b(1!a)V#bVc(a!1)#b(a#c!1)(1!a)V) and (6). , (b#c)(a#b!1)

APPENDIX C where D is a constant determined by a boundary condition. In our case, we "rst solve eqn (C.4) for The Generating Function of T0 ;(s) in terms of ;(s). Treating s as a constant In this appendix, we compute the generating and substituting into eqn (C.3) we "nd that a re- ¹ function of  in the case when all damaged cells cursion of the appropriate form whose solution, APOPTOSIS AND DNA REPAIR 467

" along with boundary condition ;L(s) 1, is

ap s(bqs!1)#(bp(s!1)(1#sa#sp !s))(1!1/p#1/(psb))G ; "   G(s) a b ! # b ! # a# ! ! # b L , ps( qs 1) ( p(s 1)(1 s sp s))(1 1/p 1/(ps )) valid for 1)i)n. Using eqn (C.4) again, we "nd that

p (1#(a!1) s)(bqs!1) ; "  (s) a b ! # b ! # a# ! ! # b L , ps( qs 1) ( p(s 1)(1 s sp s))(1 1/p 1/(ps ))

¹ which gives the generating function of . With b a# p ( p) 1 ;(s) in hand, we can easily calculate the pro- " ! , ¹ ¹ ap (aq#p)(1!a/(p#aq))L a bability distribution of , or of any I:  ; / ¹ " "1 d (s) (  i) G . in agreement with eqn (9). Similarly, i! ds Q We also obtain analytic formulae for all the Var ¹ "$(¹)!($¹ ) (C.5) ¹    moments of . For instance, ; s d; (s) $¹ "d ( ) "  #$¹ !($¹ ) (C.6)  ds   ds Q Q

a# #a a ! # ! # # #= "X #pZ(pXZ( p) p ( (p 1) 2p(n 1) p(p 2) )) a C1 a# b   D , (C.7) ( p ) p

where X"a!1, Z"(1#a/(pb))L, and =" a ! # (pq p 2n).