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. NOWAKInstitute 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 normally 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 stochastic framework to analyse the somatic evolution of cancer initiation. Within this model, westudy how apoptosis and DNA repair can maintain the transient stability of somatic cells anddelay 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 "ndthat the loss of DNA repair or the loss of apoptosis both hasten tumorigenesis, but incharacteristically di!erent ways.
 2002 Elsevier Science Ltd

1. Introduction
Tumorigenesis is de"ned as the onset of unregulated cell proliferation. In humans and manyother mammals, the process towards accelerated
cellular growth is marked by the loss of important regulation genes which usually control cellcycle functioning. These regulation mechanisms
may be broadly characterized (Kinzler & Vogelstein, 1997; Vogelstein et al., 2000) as DNA repair
genes, which repair mutations and DNA damage
before further cell division, and tumor suppressor
genes, which signal for cell-cycle arrest and induce apoptosis if substantial genomic damage is
detected (Gottlieb & Oren, 1998).
The loss of DNA mismatch or DNA excision
repair genes increases the e!ective mutation rate
per cell division (Orr-Weaver & Weinberg, 1998).

* Author to whom correspondence should be addressed.
E-mail: plotkin@ias.edu0022}5193/02/030453#15 $35.00/0

This phenomenon is often called the mutator
phenotype. The loss of tumor suppressor genes,on the other hand, diminishes a cell's ability torecognize damage and induce apoptosis. Bothincreased mutation rates and de"cient apoptotic"delity accelerate tumorigenesis. But the relative
importance of these two carcinogenic mechanisms is, a priori, unclear. The extent to which themutator phenotype determines the timing of tumorigenesis is hotly contested in the literature.
Some scientists (Loeb, 1991) have argued thatan increased pre-malignant mutation rate is required for tumorigenesis to occur whatsoever.
Many other scientists agree that the mutator
phenotype plays an important, if not absolutely
necessary, role in cancer development (Lengauer
et al., 1998; Orr-Weaver & Weinberg, 1998;
Murdoch & VanKirk, 1997). But others (Tomlinson et al., 1996; Tomlinson & Bodmer, 1999)
contend that Darwinian selection on cellular proliferation rates dominates the process towards
 2002 Elsevier Science Ltd

454

J. B. PLOTKIN AND M. A. NOWAK

carcinogenesis, superseding any e!ects of themutator phenotype. Still others stress the importance of apoptosis in preventing tumorigenesis(Tomlinson & Bodmer, 1995; Hong et al., 2000;
Harnois et al., 1997; Chang et al., 1998). Others
yet contend that the immune system plays a central role in controlling tumors (Darnell, 1999).
The extent to which each of these factors*
decreased apoptosis, increased mutation, increased proliferation, etc.*contribute towards
cancer depends, no doubt, on the cancer type.
Given the diversity of cancer types, it would be
misguiding to debate the crucial carcinogenic
processes. Nevertheless, a rigorous, qualitative
understanding of the di!erences between thesecarcinogenic processes can inform the debate
about their relative importance in various cancer
settings.
In this paper, we develop a mathematicalframework for investigating the e!ects of cellcycle regulation genes. In particular, we use
stochastic multistage models to investigate as tohow DNA error repair and apoptosis stave o!
tumorigenesis. Our generic framework can beused to investigate qualitative patterns in the
progression towards carcinogenesis. We do notinitially specify a particular cell type or a particular cancer type. Some of our assumptions,however, constrain the applicability of our model
to particular classes of cancer types. We do notaddress the progression of a malignant tumor
through its various cancerous stages, angiogenesis, and eventually metastasis. Instead, weinvestigate the accumulation of mutations in
a pre-malignant cell lineage.
Stochastic modelling of tumorigenesis wasintroduced in the 1950s (Armitage & Doll, 1954)
for comparison with adult age-speci"c cancer
incidence rates. Given the di!erences between
spontaneous and inherited cancers, however,authors soon began to develop multiple stage
models of tumorigenesis, starting with two-stage
models (Knudson, 1971). Truly rigorous andcomplete analyses of the two-stage stochastic
models soon followed (Moolgavkar & Venzon,
1979; Moolgavkar & Knudson, 1981), treating
childhood and adult tumors separately. Multistage models have since been expanded (e.g. Mao
et al., 1998). In all such models, mutation rates
were generally assumed to be constant, andapoptosis and DNA repair were largely ignored.
Outside of the general multistage framework,
other authors have investigated the e!ects ofapoptosis (Tomlinson & Bodmer, 1995) andmutation rates (Tomlinson et al., 1996). In addition to research into the genetic events that cause
tumorigenesis, there is a large and detailed literature which models the physiological processes
involved in tumor invasion (Chaplain, 1995;
Perumpanani et al., 1996), growth (Byrne &Chaplain, 1996a, b), encapsulation (Sherratt,2000), macrophage dynamics (Owen & Sherratt,1999), and angiogenesis (Chaplain, 2000).
This paper is divided into nine sections. In
Section 2, we discuss parameter values for mutation rates and approximations appropriate tomodelling mutation. Section 3 presents our basicmodel of tumorigenesis. In Section 4, we analyse
the simple case when apoptosis and increased
mutation rates are both neglected. In Section 5,
we compute the average waiting time before
tumorigenesis in terms of the apoptotic rates andincreased mutation rates. In Section 6, we compute the distribution of waiting times before
tumorigenesis. Section 7 compares the e!ect ofincreased mutation with that of de"cient apoptosis. Section 8 addresses intrinsic costs associated with elevated mutation rates. Concluding
remarks are given in Section 9. The main text
refers to Appendices A}C for mathematicaldetails.
2. Target Genes and Mutation
Throughout our analysis, we assume that thereare ¸ genes involved, in some way, in regulating
normal cell-cycle functioning. We call these¸ sites target genes because their removal canincrease the chance of tumor initiation. For humans, ¸ is rather large. As a rough approximation, given that there are over 150 genes involved
in apoptosis alone (Aravind et al., 2001) and over
130 involved in DNA repair (Wood et al., 2001),
we assume that ¸+500. We imagine that thetarget genes*DNA repair genes and tumor
suppressor genes*form a large network whose
redundancy bu!ers the cell against tumorigenesis. We will assume that if any n of these target
genes become defective in a cell, then the cell will
start to proliferate causing the onset of cancer.

455

APOPTOSIS AND DNA REPAIR(In an alternative model, we can de"ne tumor
initiation as the mutation of a particular set ofn genes from the ¸ regulation genes.) The threshold n is usually small compared to the total
number of target genes. Evidence suggests thatas few as n"2 defective genes can cause unregulated cell proliferation for certain cancer
types, such as retinoblastoma (Knudson, 1971),
while n"6 or more are required for other cancer
types (Loeb, 1991).
In reality, even when a single cell acquires
n mutations, the immune system may yet prevent
tumorigenesis by targeting the deviant cell (Darnell, 1999). The frequency and importance of thisphenomenon, however, are hotly contested. We
will therefore assume throughout that tumorigenesis is simply de"ned by n mutations, delaying
our discussion of immune action until Section 9.
We assume that at every generation of cell
division, each of the ¸ regulation genes may
undergo a debilitating mutation with probabilityk. Once a gene is mutated, we can safely assume
that it will never again back-mutate into a functional gene. Although the per-base mutation rate
is well known for many organisms (Drake et al.,1998), the per-gene mutation rate k in non-germline cells is more di$cult to measure in practice.
Several authors have suggested that k+10\
per cell division (Orr-Weaver & Weinberg, 1998),
although the precision of this value is unclear.
When apoptosis and DNA repair do not occur,
the stochastic process of accumulating cellularmutations is relatively simple. Consider the caseof tumorigenesis de"ned by any n mutations.
Each cell in a tissue of constant size is, on average, replicating and being replaced by its daughter cell at each generation. A cell is characterized
by the number of mutant genes k, 0)k)¸,it currently harbors. We say that a cell lineage
becomes cancerous (i.e. has lost its regulation
ability), if k*n. In the most simple case, the cell
has a probability p of increasing its number ofmutations by one, and probability q"1!p ofmaintaining the same number of mutations at
each replication/replacement event. We call thissituation &&neutral'' because the cell behaves in thesame manner regardless of its current mutational
status.
Starting from a tissue containing N healthy
cells, we will compute the expected time before

at least one of the cells acquires any n mutations.
We will also compute this time in the presence of
apoptosis and repair genes. Before we analyse
the waiting times before tumorigenesis, we makeseveral preliminary remarks about modelling
mutation. We are assuming an extremely simple
mutational process. Regardless of the current
number of mutations, k, we assume that in eachsomatic generation k either increases by one,with probability p, or remains constant, withprobability q"1!p.
Even when we choose to ignore back mutation,
this formulation is only approximately correct.
The exact formulation, according to an independent forward mutation rate k and backward
mutation rate zero per gene, is given by/(kPk)

 
0for k(k,

(1)
" ¸!k
kIY\I(1!k)*\IY for k*k.
k!k
To be exact, then, /(kPk)"(1!k)*\I. We
approximate this exact equation by de"ningq"1!k¸+/(kPk) and p"1!q. This approximation is valid provided that n;¸ and¸k;1, both of which are true for humans(Drake et al., 1998).

3. A Model of Mutation, Apoptosis,
and DNA Repair
We now formulate a stochastic, Markov-chain
model of the mutational and apoptotic process.
We imagine a large tissue of cells which, at all
times preceding the onset of cancer, is almost
completely mutation free. We keep track of thecurrent number of mutations k harbored in a celllineage. In our Markov model, any particular cellis replaced by its daughter cell in each generation.
At each generation, the healthy cell (k"0) either
remains healthy, with probability q , or accumu
lates a single mutated gene, with probability
p "1!q .
A cell harboring k*1 mutations, however, issubject to tumor-suppressor-induced apoptosis.
In particular, at each generation, a cell harboring
k mutated genes has probability a of destroyingI456

J. B. PLOTKIN AND M. A. NOWAK

itself and being replaced by another cell in thetissue, which we assume to be healthy. With
probability b "1!a , on the other hand, theIImutated cell replicates*maintaining the same
number of mutations with probability q , andI
acquiring an additional mutation with probability p . According to this formulation, a cell linI
eage is described by a Markov chain with thefollowing (n#1);(n#1) transition matrix:
P"



q

a

a

$

a
L\
0

p
0
2

b q b p0
   0 b q b p   
$
\
0

2

0

2

2

2

b

2

0

2

0

2

0

\

$



.

q
b p
L\ L\ L\ L\
0
1

(2)

The state k"n, which we call tumorigenesis, isabsorbing. Once the cell reaches this state, we saythat the cell has escaped from proper cell regulation and begins malignant growth. We do notconcern ourselves with the progression of cancer
after this stage. Although active tumors certainly
undergo mutation and apoptosis, the evolutionof a tumor is a very di!erent phase of cancer
progression which we do not model here.
When apoptosis is ignored (i.e. all a are zero),Ithen the formulation given by eqn (2) is equivalent (in discrete time) to the classical models oftumorigenesis formulated by Armitage & Doll
(1954) and later developed by Moolgavkar(1978). Here we extend these models to include
the action of apoptosis.
We will examine tumorigenesis for arbitrary
apoptotic rates a . The simplest case occurs whenI
a is constant. If we focus our attention on targetI
genes which are related to apoptosis, then theapoptotic rates a should naturally be decreasing
Iwith k. This corresponds to the situation inwhich as tumor-suppressor genes become mutated, their e$ciency at scanning for damage andinducing apoptosis is reduced. Hence, mutatedcells gain a selective advantage by avoiding
programmed cell death (Tomlinson & Bodmer,
1995). In this formulation, the road towards tu-

morigenesis (k"n) is a slippery slope; once a few
suppressor genes have been damaged, it becomesincreasingly hard for a cell to recognize mutations, and thereby to prevent further progression
towards cancer (Kinzler & Vogelstein, 1997). On
the other hand, we can also consider cases when
the apoptotic rates a increase. This models theIsituation in which, say, progressively more oncogenes become mutated and signal for increased
apoptosis.
We are primarily interested in the cases whena decreases, corresponding to the loss of tumorI
suppressors and apoptotic "delity. Similarly, we
generally consider cases in which the e!ective
mutation rates p are increasing. This assumptionI
mirrors the progressive loss of DNA repair genes
and the corresponding increase in e!ective mutation rate and genomic instability (Vogelstein
et al., 2000).
We are nominally modelling the behavior ofa single-cell lineage. But whenever a cell undergoes apoptosis (with probability a ), it is replacedIby a cell from the surrounding tissue which we
assume to be healthy. Thus, we are, in fact, modelling the behavior of a cell in a large tissue
composed predominantly of healthy cells. Therefore, our cell &&lineage'' is not a strict lineage, butrather includes replacement by healthy cells upon
apoptosis. Before the onset of cancer, we assume
that almost all cells in the tissue are mutation
free. This assumption, used here to simplify population dynamics, may be invalid for some cancer
types. (Elsewhere we relax this assumption, andallow for apoptosis followed by replacement of
nearby unhealthy cells, Nowak & Plotkin, in
preparation) Colorectal cancer, for example, isoften preceded by the invasion of numerous benign polyps which exhibit one or two mutatedgenes, often APC, throughout the tissue (Kinzler& Vogelstein, 1996). Other cancers*such as
retinoblastoma*are not generally preceded bywidespread polyps containing mutations, andare therefore more appropriately modelled by
our formulation.
Our Markov model does not address di!erences in proliferative rates between cell lineages.
Elsewhere (Nowak & Plotkin, in preparation),
we develop deterministic models which treat
inter-lineage population dynamics. Again, theone-dimensional Markov model is therefore less

457

APOPTOSIS AND DNA REPAIR

appropriate for colorectal type cancers*which
exhibit clonal expansion and competition
through increased proliferation*than it is forretinoblastoma-type cancers (Cairns, 1998).
The Markov model keeps track of the current
number of defective regulation genes, k. At eachstage we do not know which k of the ¸ target
genes are mutated. In this sense, the reduced
apoptotic "delities a and the increased e!ectiveImutation rates p represent averages over all
Ipossible combinations of k mutated genes. In
other words, the loss of some tumor suppressors,e.g. p53, can in reality be more deleterious thanthe loss of other ones; we average over these
possibilities.

4. The Neutral Case
Before we analyse the full-blown Markov
model, we consider the simple, neutral case inwhich apoptosis and DNA repair are both
ignored. In other words, we assume for now that
all a are zero and all p are equal. In Appendix A,II
we derive the expected number of cell divisions,¹, required before at least one cell in a tissueof N cells, originally all healthy, acquires n
mutations:
1C(n, a), da.
¹"
log(1/q)(n!1)!,
(3)

Here C(), )) denotes the incomplete Gamma
function. Equation (3) demonstrates that in theneutral case the waiting time before tumorigenesis depends inversely on the logarithm of one
replication "delity (q). Figure 1 demonstrates the
expected neutral waiting time before tumorigenesis for various mutation rates and tissue sizes. As
is clear from the "gure, increasing the tissue size
N geometrically causes the waiting time ¹ todecrease geometrically.

5. The Average Waiting Time before
Tumorigenesis
Given arbitrary mutation rates p and apopI
totic rates a , what is the mean time before carI
cinogenesis (state k"n)? In addition, what is the
expected total time spent by a cell lineage in eachof the various mutated classes? For example, do
we expect to "nd cells with k"2 mutations half
as often as k"1? From this section onwards, weanswer these questions insofar as a single-celllineage is concerned*an approach originally espoused by Armitage & Doll (1954) and later
Moolgavkar (1978).
De"ne ¹ as the expected number of generaG
tions before absorption into the cancerous state,assuming that the cell begins in state i at time
zero. We are eventually interested in ¹ , themean time until tumorigenesis starting froma healthy cell. We might also be interested in ¹ ,
if an individual inherits a defective gene. De"ne= as the mean number of visits to state j prior toGHabsorption, assuming that the cell starts in statei. As is clear from the Markovian property ofour process, we have the following recursive
relationships:
L\= "d #
P =for i"0,2 , n!1, (4)
GH
GH
GK KH
K
In Appendix B, we use this recursion to compute the mean time spent in each mutation classprior to tumorigenesis, starting from a healthy cell:

FIG. 1. The expected number of cell divisions before one
cell in a tissue of size N acquires n"5 mutations, ignoring
apoptosis and DNA repair. The mutation rate p varies fromp"500;10\ to 0.01 on the x-axis. Each curve represents
a di!erent tissue size ranging geometrically from N"10,
highest curve, to N"10, lowest curve.

L\ p b #a
1
V V
V

= "
I p bp b
L\ L\ VI> V\ V\
for 0)k)n!1,

(5)

458

J. B. PLOTKIN AND M. A. NOWAK

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 isp "500;10\. The damaged cell's mutation rate ranges
100-fold on the x-axis from p"p to 0.05. Each of the sixcurves represents a di!erent apoptotic rate, ranging linearly
from a"0.001 (lowest curve) to a"0.01 (highest curve).
Note that the time until tumorigenesis is less sensitive to theapoptotic "delity whenever the damaged cell's mutation rate
is high.

L\
¹ "
= ,

I
I

(6)

where we have de"ned a "0 for convenience.
Note that a cell spends (p b #a )/(p b ) as
I I
I I> I>much time with k mutations as it does with k#1
mutations, before tumorigenesis. In addition, themean time a cell spends in mutational class k depends only upon the e!ective mutation rates andapoptotic rates of the classes k*k. In otherwords, the selective pressure on k-th-order checkpoint genes does not depend upon the behaviorof cells with fewer than k mutations.
In the case when all the apoptotic rates areconstant, a "a, and when all the damaged
I
mutation rates are constant (p "p "2"p "p), eqns (5) and (6) simplify to the following:
L\





1 pb#a L\
,
= "
 p
pb




(7)



1 pb#a L\I\
for 1)k)n!1,= " I pbpb(8)pb(a#p )
1

¹ "
! .
 ap (aq#p)(1!a/(p#aq))L a(9)Assuming that apoptotic rates are constant,Fig. 2 shows as to how the timing of tumorigenesis depends upon the damaged cell's mutation
rate. The loss of DNA repair genes in damaged
cells can increase the e!ective mutation rate bya 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 rateincreases. In other words, if there is a strong
mutator phenotype, then the time before tumorigenesis is less signi"cantly a!ected by the presence of functioning apoptotic machinery.

6. The Distribution of Waiting Times before
Tumorigenesis
Equation (6) describes the expected time before
tumorigenesis in a cell lineage, given the apoptotic and mutation rates. We are further interested in the complete distribution of waiting times
until tumorigenesis. For instance, does the mutator phenotype increase the variance of the waiting time? In this section, we let ¹ denote therandom time at which the Markov chain reaches
the cancerous state, n. We will "nd expressions
for the expected value $¹ and for the distribu
tion function of ¹ , in terms of the eigenvalues of
P. As we will discuss below, the distribution of¹ can be used to predict cancer age}incidencecurves comparable to epidemiological census
data.
Consider any (n#1)-state Markov chain
M< acting on the left. (In our case, take M< to be
the transpose of P.) Label the states 0 to n, andassume that state n is the only absorbing state.
Consider the n;n submatrix M which excludes
the absorbing state. Let < "(1, 0,2 , 0)O denotethe initial condition: in state zero. Note that i-thcoordinate of MR< , denoted [MR< ] , is the Gprobability that the chain is in state i at time t.
We index the coordinates from 0 to n!1. Hencethe following expression:
L\G

[MR< ] "#MR< #, Gyields the probability that the cell is not yet in theabsorbing state at time t. In other words, this sumis the complementary cumulative distribution

APOPTOSIS AND DNA REPAIR

function of ¹ . We use # ) # to denote the sum ofthe components of a vector.
We can write M"SDS\ where D is the diagonal matrix of eigenvalues, j ,2 , j , proL\vided that they are all distinct. The columns of
S are the corresponding eigenvectors. In this case,MR< "SDRS\< . Hence, the cumulative distribution of ¹ is given by U(t)"1!#SDRS\< #.
This in turn allows us to compute the expected
time before tumorigenesis:
SDRS\< "#Q#,$¹ "1!U(t)"


R
R
(10)
where

Q"S



1
1!j


0

0

0

\

0

0

0

1
1!j
L\



S\< . (11)

Similarly, the i-th component of Q gives the expected value of = , the amount of time spent inGclass i before absorption. For any given parameter values, we can numerically compute theeigenvectors of P and use eqn (11) to "nd = .
GAmazingly, even when analytic expressions forthe eigenvectors of P are di$cult, we know thateqn (10) always simpli"es to the exact solution
given in eqn (6).
Finally, from our expression for the cumulative
distribution U, we see that the probability density
function (t)"U(t)!U(t!1) of ¹ is given by

(t)



jR\!jR 0


0
\
" S
0

0

0
0
jR\ !jR
L\
L\



S\< .
(12)

In Appendix C, we compute the generating function of ¹ in the case of constant apoptosis. This

459

provides an alternative analytic formula for thedensity function (t). Moreover, the generating
function yields analytic expressions for all the
moments of ¹ .

There is great value in knowing the distribution, (t), of the waiting time before tumorigenesis in a single-cell lineage. As originallyelucidated by Moolgavkar (1978), knowledge of(t) allows us to compute the hazard function oftumorigenesis in a large tissue of N cells. The
probability that at least one tumor occurs in thetissue by time t is simply F(t)"1!(1!U(t)),.
Hence, the hazard function*that is to say, theinstantaneous rate of cancer incidence*is given
by F(t)/(1!F(t))"N (t)/(1!U(t)). The hazard
function of tumorigenesis is directly comparable
to cancer age-incidence data collected froma population (Armitage & Doll, 1954). In mostcases, U(t) will be small within a human lifespan.
Therefore, the rate of a cancer incidence will beapproximated very well simply by N (t). Hence,by understanding the probabilistic behavior of
single-cell lineage we can predict the cancer
age-incidence rates within a human population(Armitage & Doll, 1954).
7. The Loss of Repair vs. the Loss of Apoptosis
Equations (5), (6) and (12) describe the precise
in#uence of apoptosis and DNA repair as deterrents against cancer. For example, in Fig. 3 weplot the distribution of the waiting time before
tumorigenesis in the simple two-hit case, n"2.
As expected, when compared with the neutral
case, increased mutation rates (the loss of DNA
repair genes) cause a decrease in mean time
before cancer, while tumor-suppressor-induced
apoptosis causes an increase. Interestingly, we
also observe that apoptosis greatly increases thevariance in the time before tumorigenesis*anobservation that is con"rmed, analytically, byeqn (23).
In Fig. 4, we compare the progressive loss of
apoptotic activity to the progressive loss of DNA
repair activity, in the case n"4. If the apoptotic
rates a and mutation rates p are constant, thenII
the distribution of waiting times has a very large
variance. We compare the model's behavior withconstant a's and p's against two alternative scenarios. 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 apoptosis when n"2 mutations are required for tumorigenesis.
The "gure shows the probability density function of the time
before tumorigenesis, ¹ , given by eqn (12). In the neutralcase (-----), all states have the same mutation rate,
p "p "500;10\, and there is no apoptosis, a "0. The
four curves drawn in (} } } }) show the e!ect of varying
degrees of increased mutation, ranging linearly fromp "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 ofapoptosis, ranging from a "0.01 (lowermost curve) to
a "0. Apoptosis increases the mean waiting time beforetumorigenesis. The mutator phenotype decreases the variance in the waiting time, while apoptosis greatly increases
the variance.

constant and we let the apoptotic rates a deI
crease linearly with k. This models the loss of
suppressor-induced apoptosis, resulting in ashorter waiting time before tumorigenesis. In thesecond alternative, we keep apoptosis constant
and we let the mutation rates p increase with k.
I
This models the mutator phenotype caused by
the loss of DNA repair genes. In this case, tumorigenesis occurs much earlier on average andhas a much smaller variance. In other words, theloss of repair has a stronger and more dramatic
e!ect on tumorigenesis than the loss of proper
apoptotic functioning, in the sense that an x-foldloss of repair decreases the time before tumor
initiation far more than an x-fold loss of apoptotic "delity (Fig. 4).
Figure 5 further elucidates the e!ect of
apoptotic malfunction compared to increasing
mutation rates. Again, we plot the density of the
waiting time before tumorigenesis in various alternative scenarios. In the most simple scenario,
all the apoptotic rates and mutation rates areconstant: a "2"a ; p "2"p . In

L\ 
L\
one set of alternatives, the mutation rates p inI
crease with k. The ratio p /p measures the
L\ 

FIG. 4. The distribution of the waiting time before tumorigenesis [n"4, using eqn (12)] shown on linear axes(top), log-linear axes (middle), and log}log axes (bottom).
The (-----) curve corresponds to the case when a "a "
a "0.001 and p "p "p "p "500;10\. The





(} } } }) curve corresponds to linearly increasing mutation
rates, p /p "100. The (**) curve corresponds to linearly decreasing apoptotic rates, a /a "100. The mutator 
phenotype causes a dramatic decrease in the mean andvariance of time before tumorigenesis. The loss of proper
apoptotic function has a more modest e!ect on the waiting
time.

strength of the mutator phenotype. As thestrength of the mutator phenotype increases,
the mean and variance of the waiting time before
tumorigenesis both decrease. In this model, thee!ect of the mutator phenotype does not saturate: the more the mutation rates increase, theearlier tumorigenesis occurs. In the other set ofalternatives, the apoptotic rates a decrease withI
k. Although the loss of apoptosis also results ina shorter time before tumorigenesis, this e!ect
eventually saturates.

461

APOPTOSIS AND DNA REPAIR

FIG. 5. The distribution of waiting times before tumorigenesis [n"4, using eqn (12)] for various amounts ofincreased mutation and decreased apoptosis. The (-----)
curve corresponds to the simple case of constant apoptotic
rates and mutation rates, as in Fig. 4. The (} } } }) curves
correspond to cases of increasing mutation rates, with p /p ranging from 1 to 10 (uppermost curve). The (**) curves
correspond to decreasing apoptotic "delity with a /a 
ranging from 1 to 20. Note that the e!ect of reduced apoptotic "delity asymptotes, whereas the e!ect of the mutator
phenotype does not saturate. This result remains true
whether p and a vary geometrically or arithmeticallyI
Iwith k.

Generally speaking, the distribution of the
waiting time for cancer is nearly identical whenever a /a exceeds 10, for n)6. The exact e!ect 
of progressive loss of apoptotic "delity depends
upon the cancer threshold n and the base-line
apoptotic "delity a . But for any parameter
values n and a , the distribution of ¹ asymp

totes as a /aincreases, whereas the distribu L\tion of ¹ does not asymptote with increasingp /p . This behavior is apparent in Fig. 5 andL\ 
can also be seen, in expectation, from eqn (5)directly. This result highlights the importance ofthe mutator phenotype as a potential driving
force behind early tumorigenesis. Conversely,
this result indicates that the loss of apoptotic
functioning hastens tumorigenesis, but that this
e!ect has an intrinsic limitation.
8. Costs Associated with the Mutator Phenotype
So far, we have ignored any intrinsic costs
associated with increased mutation rates. In reality, however, high mutation rates can lead either(i) to the loss of crucial genes needed for cellularfunction and/or (ii) to the triggering of programmed cell death. This phenomenon indicatesthat the mutator phenotype p and the apoptotic
I
rates a should be correlated to some extent. In
I
this section we will explore an extended modelin which increased mutation is accompanied byincreased cellular death.
We have assumed that there are ¸ &&target''genes involved in regulating cell-cycle functioning. Assume now that there are an additional
M &&critical'' genes (M+10 000 in humans) involved in cellular functioning which do not cause
cancer, but whose mutation causes cell death(even in a cancerous cell). For any mutation ratek, we de"neq "(1!k)+>*,
Ip "[1!(1!k)*](1!k)+,Is "1!q !p "[1!(1!k)+].
I
I
I
Here q represents the probability per cell diviIsion that no genes whatsoever are mutated;
p represents the probability that at least one
Itarget gene is mutated, but none of the critical
M genes is mutated; and s denotes theIprobability that at least one of the critical genes
becomes mutated*which will cause immediatecell death.
Consider the following Markov chain model oftumorigenesis, which includes an intrinsic cost of
higher mutation rates:



q #s
I
I
a#bs
I
a#bs
I
P"
$
a#bs
0

I

p



0
2
I
bq bp
0
I
I
0 bq bp
I
I
$
\

2

0

2

0

2

0

\

$

0

2

0

bq

2

2

2

0

bp
I
1

I

.

In this formulation, a denotes the apoptotic rate,
which we assume to be constant. Here k denotes
the intrinsic mutation rate per gene per replication event of a healthy cell and k denotes theelevated mutation rate of a cell which lacks one
or more regulation genes. Note that either apoptosis or the mutation of any &&critical'' gene causesa cell to die and be replaced by a surrounding
healthy cell. Given that increased mutation now462

J. B. PLOTKIN AND M. A. NOWAK

carries a cost*namely the possibility ofmutating a gene critical for cellular survival*we
expect to "nd some optimum increased mutation
rate k which minimized the time to tumorigenesis.
As before, let = denote the mean time spentIin class k before absorption into class n, startingfrom a healthy cell. Using the methods describedin Appendix B we deduce that





bp
I>\L
1
I
= "
 I 1!a!b(q #s) 1!bq
II
for 1)k(n!1,

(13)





bp
\L
1!bq
I
I
.
= "
  p (1!a!b(q #s)) 1!bq
I
I
I
(14)
Therefore, the expected time to tumorigenesis isgiven by
L\
¹ "
= "

I
I
p #(p b)\L(1#q (a!1))L\[b(p #q )!p !1]
I
I
I
I
I
I
.
p (b(p #q )!1)
I
I
I

(15)

For any parameter values ¸, M, n, and k ,eqn (15) is always minimized at some valuek"k*, which is the optimal increased mutation
rate (from the cancer's point of view). Figure 6(a)shows the expected time until tumorigenesis forvarying degrees of the mutator phenotype, withassociated costs. Figure 6(b) shows the optimum
increased mutation rate k* as a function of thenumber of critical genes. The optimal mutation
rate k* is always less than the mutation rate
which simply maximizes the speed of forward
mutation, p . We "nd that the waiting time beI
fore tumorigenesis is strongly a!ected by the
number of mutations required for tumorigenesis,n, but interestingly the value of the optimum
mutation rate does not depend upon the tumor
threshold n or the apoptotic rate a.

FIG. 6. (a) The expected time until tumorigenesis as
a function of the increased mutation rate k'k "10\, in
the presence of mutation-associated costs [eqn (15)]. Thereare ¸"500 target genes, and the apoptotic rate is a"0.001.
Six curves are shown, depending upon the number of criticalgenes, M. If there are more critical genes, then the time totumorigenesis is increased. For each set of parameters, there
is an optimum mutation rate k"k* at which the time totumorigenesis is minimized. (b) The dependence of k* on thenumber of critical genes. If there are more critical genes, thenthe optimum mutation rate is decreased, and the e!ect ofa strong mutator phenotype can be detrimental (from thecancer's viewpoint). The optimum mutation rate does notdepend upon the apoptotic rate a or the cancer threshold n.

9. Conclusions
We have introduced a stochastic framework
for investigating the relations between apoptosis,
the mutator phenotype, and carcinogenesis. As
we have seen, the loss of apoptosis and the loss of
DNA repair both hasten the onset of cancer, but
their speci"c e!ects on the pre-cancer waitingtime are dramatically di!erent. The mutator
phenotype can cause a dramatic decrease in themean and variance of the time before cancer,
while the loss of apoptosis reduces the mean
but not the variance. In addition, stronger
mutator phenotypes lead to increasingly rapid

463

APOPTOSIS AND DNA REPAIR

tumorigenesis, while the e!ect of apoptotic malfunction asymptotes.
We have seen that the e!ect of mutator phenotype can also asymptote, however, if we include
costs associated with increased mutation rates. If
a certain set of genes is critical for cellular replication*without which even cancer cells cannot
survive*then there exists an optimal value of theincreased mutation rate (from the cancer's viewpoint) which minimizes time before tumorigenesis. This optimal mutation rate does not depend
upon the amount of apoptosis or the number ofhits required for tumorigenesis.
This paper extends multistage models forcancer initiation (Armitage & Doll, 1954; Moolgavkar, 1978) to include the e!ects of apoptosis
and mutator phenotypes. In general, this approach allows us to determine the waiting time
before tumorigenesis and the cancer age}incidence curve of a population in terms of the probabilistic behavior of a cell lineage. Because wehave not considered the progression of a tumor
through its various cancerous stages, the therapeutic signi"cance of our results is limited tocancer prevention. Our results would indicatethat gene therapies preserving e$cient DNA
repair may be more important than thosepreserving the apoptotic "delity of cells.
We hasten to discuss some important simpli"cations used in the modelling framework. The
mechanisms and connections between apoptosis
and repair are, no doubt, much more complicated in reality than in our model. Although ourstochastic framework allows for coordinated behavior, we have largely treated apoptosis (a ) andI
DNA repair (p ) as independent processes. In
I
some sense, we have treated tumor suppressor
genes and DNA repair genes separately. In
reality, however, some tumor suppressors suchas p53 (Vogelstein et al., 2000) often induce
cell-cycle arrest (Laronga et al., 2000) and thereby
allow DNA repair genes to remove DNA damage(Tanaka et al., 2000). Hence, reduced apoptotic
rates a may be partially correlated with inI
creased mutation rates p .
I
Similarly, we have assumed that, at each time
step, a cell divides and is replaced by a daughter
cell. Although this process occurs on average,
sometimes in reality an individual cell may divide
and produce two viable cells, possibly hastening

the spread of cell lineages which have undergone
somatic mutations. Similarly, some mutations
may increase the rate of cell division, which wehave assumed to be constant. Nevertheless, thesimplifying assumption of direct daughter-cell
replacement has long been considered to be areasonable model in the cancer literature (Armitage & Doll, 1954; Moolgavkar, 1978). Preliminary work (Nowak & Plotkin, in preparation)
indicates that the timing of tumorigenesis islargely unchanged even when this simplifying
assumption is relaxed.
Our model also ignores the e!ect of theimmune system and tumor immunity. We havetreated each cell lineage as an independent,
autonomous process. In reality, the immune system certainly plays a role in some cancer types(Burnet, 1970), while the extent of this role ishotly contested. Recent developments (Albert
et al., 1998a, b) indicate that apoptosis of tumor
cells may be followed by uptake into dendritic
cells thereby training ¹ cells to recognize andtarget the tumor. This process suggests that apoptosis may play a much larger albeit indirect role*by sensitizing the immune system*in tumor
control. Our model investigates the patterns oftumorigenesis in the absence of immune activity*which could, by failure to match empiricalpatterns, provide evidence for immune activity.
In fact, the observed limitations of apoptotic
malfunction as a direct cause of tumorigenesis in
our model, together with the empirical observation that tumor suppressors like p53 are usuallymutated in many tumors, may suggest that apoptosis does indeed have an important indirect
e!ect in controlling cancer.
J. B. P. gratefully acknowledges support fromthe National Science Foundation and the Burroughs
Wellcome Fund. J. B. P. and M. A. N. also thankthe Alfred P. Sloan Foundation, the Ambrose Monell
Foundation, the Florence Gould Foundation, and theJ. Seward Johnson Trust. Both authors also appreciate helpful discussions with J. Dusho!, R. Darnell,and M. Albert.
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APPENDIX A
The Neutral CaseIn this appendix, we analyse the neutral casewhen apoptosis and repair are ignored. We willcalculate the expected time until one of the
N cells has at least n mutations, in terms of the
genic mutation rate k. Let us "rst consider

465

APOPTOSIS AND DNA REPAIR

a single cell. We de"ne the stopping time ¹ as
the smallest number of divisions required before
a cell has n mutations, given that it starts with
k"0 mutations. Clearly, then, the time to tumorigenesis ¹ equals or exceeds n. The probability
that the cell lineage becomes cancerous at
generation n#x is given by/(¹"n#x)"pLqV





n#x!1
,
x

(A.1)

where, as discussed earlier, q"(1!k)*, andp"1!q. Note that, for a single cell, theexpected value of ¹ is simply  (n#x)
V
pLqV(L>V\)"n/p.
V
We are interested, however, in the expected
time at which any one of the N cells comprising
the tissue becomes cancerous. In other words, wewant the expected value of >"min+¹ , ¹ , 2, ¹,, where the ¹G are independent randomvariables each distributed as ¹. We know that fora*n

/(¹*a)"
G?

 
 

i!1
pLqG\L
i!n

"q?\

L\
G

(p/q)G$>" /(>*a)
?



G



(t)"

 

,

.

(A.2)

The exact, discrete-time solution [eqn (A.2)] is
unwieldy, but it can be approximated very wellusing a continuous-time, discrete-state model(Poisson process). In this approximation, eachcell experiences an exponentially distributed
waiting time before accumulating another mutation. The time until a healthy cell acquires one
mutation is distributed according to the density
function(t)"je\HR. We choose j"!log(1!p) so that the cell has probability p of acquir-RQ(s) (t!s) ds,
G\

whence we "nd that the waiting time before
tumorigenesis has the Gamma distribution(t)"1/(n!1)!jLtL\e\HR. This in turn yields
Lthe complementary CDF of ¹: /(¹*a)" (t) dt"1/(n!1)!C(n, ja), where C is the? L
incomplete Gamma function. We can now compute the mean of >:



$>"







/(¹*a), da" C(n, ja) ,da(n!1)!


1

"C(n, a), da.
j(n!1)!,
(A.3)

Equation (A.3), which is straightforward tocompute numerically, provides an accurate
approximation of eqn (A.2).

a!1
.
i

The complementary cumulative distribution of> is given by /(>*a)"/(¹*a),. This, in turn,allows us to compute the expected value of >:

a!1

L\
"n#
q?\
(p/q)Gi?L>
G

ing one mutation within one time unit. The distribution (t) for the time of i-th mutation satis"esG
the recursion

APPENDIX B
The Mean Time before Tumorigenesis
In this appendix, we calculate the mean time
before tumorigenesis in the Markov model with
arbitrary apoptotic and mutation rates. We startby calculating =directly. Then we compute L\
the ratios = /=. We eventually combine I  I>these results into expressions for = , yieldingI¹ " L\= .

I

I
In order to compute =, we imagine a L\reduced Markov chain in which states zerothrough n!2 are collapsed into a single class.
This chain has the following transition matrix:



e

1!e

0



,
b q
b p
M" a
L\
L\ L\L\ L\001
where e3(0, 1) is some value determined by the
apoptotic rates, mutation rates, and n. We need
not actually calculate the value of e. We denote

466

J. B. PLOTKIN AND M. A. NOWAK

the three states of this process by s, n!1, and n(s stands for starting). We can apply eqn (4) to thisreduced process in order to calculate =,Q L\which equals =in the full Markov chain.
 L\The resulting two-dimensional linear system has
the unique solution ="1/p b , inde L\
L\ L\pendent of e.
Next, we show that = /="p b /
 I  I\I\ I\(p b #a ), for k"2, 3,2 , n!1. We consider
I I
Ianother reduced Markov chain. This time, we
collapse states zero through k!2 into a single
state, and states k#1 to n!1 into a single state.
This yields the following transition matrix with"ve states, called s, k!1, k, f, and n:
M"



e

1!e

0

0

0



a
b q
b p
0
0
I\ I\ I\ I\ I\
a
0
bq
b p
0
,
I
I I
I\ I
c
0
0
d
1!c!d
0

0

0

0

1

where, again, e, c, and d are constants which weneed not compute. Note that c#d(1, because
the only absorbing state is n. Although we cannot
use this reduced system to compute = orI=directly, we can compute the ratio of I\these quantities. We apply eqn (4) twice, oncewith j"k and once with j"k!1. For j"k, we
obtain a four-dimensional system with uniquesolution = "(1!d)/(p b (1!c!d)). When
QI
I I
j"k!1, eqn (4) gives another four-dimensional
system with solution ="(1!d)(p #q a )/
Q I\
I
I I
(p p b b (1!c!d)). On dividing, we "nd
I I\ I I\that = /="= /="p b /(p bQ I Q I\
 I  I\I\ I\ I I#a ), as desired. A similar argument for theIboundary value k"1 implies that = /= " p /(p b #a ). By combining these results, we  obtain the exact expressions given by eqns (5)and (6).
APPENDIX C
The Generating Function of T0
In this appendix, we compute the generating
function of ¹ in the case when all damaged cellshave the same mutation rates p "2"p "L\p'p and apoptotic rates a "2"a "a.

L\Let u "/(¹ "i) denote the probability ofIGI
absorption at time i, starting from state k. According to the Markov chain in eqn (2), we havethe following recursive relations:u"bqu #bpu#aufor 1)k(n,
I G>
IG
I> G
G
(C.1)
u
"q u #p u
 G>
 G
 G

(C.2)

subject to the boundary condition u "d . WeLGLde"ne the generating function of ¹ as the formalIsum
; (s)" u sG.
IIGGOn multiplying eqns (C.1) and (C.2) by sG> andsumming over i, we "nd that
; (s)"sbq; (s)#sbp; (s)#sa;
I
I
I>

for 1)k(n,; (s)"sq ; (s)#sp ; (s)

 
 

(C.3)
(C.4)

subject to the boundary condition ; (s)"1. In
L
order to solve these recursions, we may treat theformal variable s as a constant. In general, therecursion(x)"a (x)#b (x#1)#c (1)has the solution(x)"
Db\V(b(1!a)V#bVc(a!1)#b(a#c!1)(1!a)V),(b#c)(a#b!1)

where D is a constant determined by a boundary
condition. In our case, we "rst solve eqn (C.4) for; (s) in terms of ; (s). Treating s as a constantand substituting into eqn (C.3) we "nd that a recursion of the appropriate form whose solution,

467

APOPTOSIS AND DNA REPAIR

along with boundary condition ; (s)"1, isL
ap s(bqs!1)#(bp(s!1)(1#sa#sp !s))(1!1/p#1/(psb))G; (s)" ,G
ap s(bqs!1)#(bp(s!1)(1#sa#sp !s))(1!1/p#1/(psb))L
valid for 1)i)n. Using eqn (C.4) again, we "nd thatp (1#(a!1) s)(bqs!1)
; (s)",
ap s(bqs!1)#(bp(s!1)(1#sa#sp !s))(1!1/p#1/(psb))L
which gives the generating function of ¹ . With; (s) in hand, we can easily calculate the probability distribution of ¹ , or of any ¹ :I1 d; (s)/(¹ "i)"
i! dsG.
Q

We also obtain analytic formulae for all the
moments of ¹ . For instance,d; (s)$¹ " 

ds

Q



pb(a#p )1"! ,ap (aq#p) (1!a/(p#aq))L ain agreement with eqn (9). Similarly,
Var ¹ "$(¹)!($¹ )




(C.5)

d; (s)
" 
ds

(C.6)

#$¹ !($¹ )


Q



X
pZ(pXZ(a#p )#ap (a(p!1)#2p (n!1)#p(p #2)#=))




"
1#
,
a
(a#pb)p
(C.7)

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