# Selective pressures for and against genetic instability in cancer: a time-dependent problem.

**ABSTRACT** Genetic instability in cancer is a two-edge sword. It can both increase the rate of cancer progression (by increasing the probability of cancerous mutations) and decrease the rate of cancer growth (by imposing a large death toll on dividing cells). Two of the many selective pressures acting upon a tumour, the need for variability and the need to minimize deleterious mutations, affect the tumour's 'choice' of a stable or unstable 'strategy'. As cancer progresses, the balance of the two pressures will change. In this paper, we examine how the optimal strategy of cancerous cells is shaped by the changing selective pressures. We consider the two most common patterns in multistage carcinogenesis: the activation of an oncogene (a one-step process) and an inactivation of a tumour-suppressor gene (a two-step process). For these, we formulate an optimal control problem for the mutation rate in cancer cells. We then develop a method to find optimal time-dependent strategies. It turns out that for a wide range of parameters, the most successful strategy is to start with a high rate of mutations and then switch to stability. This agrees with the growing biological evidence that genetic instability, prevalent in early cancers, turns into stability later on in the progression. We also identify parameter regimes where it is advantageous to keep stable (or unstable) constantly throughout the growth.

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**ABSTRACT:**Intestinal crypts in mammals are comprised of long-lived stem cells and shorter-lived progenies. These two populations are maintained in specific proportions during adult life. Here, we investigate the design principles governing the dynamics of these proportions during crypt morphogenesis. Using optimal control theory, we show that a proliferation strategy known as a "bang-bang" control minimizes the time to obtain a mature crypt. This strategy consists of a surge of symmetric stem cell divisions, establishing the entire stem cell pool first, followed by a sharp transition to strictly asymmetric stem cell divisions, producing nonstem cells with a delay. We validate these predictions using lineage tracing and single-molecule fluorescence in situ hybridization of intestinal crypts in infant mice, uncovering small crypts that are entirely composed of Lgr5-labeled stem cells, which become a minority as crypts continue to grow. Our approach can be used to uncover similar design principles in other developmental systems.Cell 02/2012; 148(3):608-19. · 31.96 Impact Factor - SourceAvailable from: Katarzyna A Rejniak
##### Article: Hybrid models of tumor growth.

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**ABSTRACT:**Cancer is a complex, multiscale process in which genetic mutations occurring at a subcellular level manifest themselves as functional changes at the cellular and tissue scale. The multiscale nature of cancer requires mathematical modeling approaches that can handle multiple intracellular and extracellular factors acting on different time and space scales. Hybrid models provide a way to integrate both discrete and continuous variables that are used to represent individual cells and concentration or density fields, respectively. Each discrete cell can also be equipped with submodels that drive cell behavior in response to microenvironmental cues. Moreover, the individual cells can interact with one another to form and act as an integrated tissue. Hybrid models form part of a larger class of individual-based models that can naturally connect with tumor cell biology and allow for the integration of multiple interacting variables both intrinsically and extrinsically and are therefore perfectly suited to a systems biology approach to tumor growth.Wiley Interdisciplinary Reviews Systems Biology and Medicine 07/2010; 3(1):115-25. · 3.68 Impact Factor - SourceAvailable from: Charles Swanton[Show abstract] [Hide abstract]

**ABSTRACT:**The role of genetic instability in driving carcinogenesis remains controversial. Genetic instability should accelerate carcinogenesis by increasing the rate of advantageous driver mutations; however, genetic instability can also potentially retard tumour growth by increasing the rate of deleterious mutation. As such, it is unclear whether genetically unstable clones would tend to be more selectively advantageous than their genetically stable counterparts within a growing tumour. Here, we show the circumstances where genetic instability evolves during tumour progression towards cancer. We employ a Wright-Fisher type model that describes the evolution of tumour subclones. Clones can acquire both advantageous and deleterious mutations, and mutator mutations that increase a cell's intrinsic mutation rate. Within the model, cancers evolve with a mutator phenotype when driver mutations bestow only moderate increases in fitness: very strong or weak selection for driver mutations suppresses the evolution of a mutator phenotype. Genetic instability occurs secondarily to selectively advantageous driver mutations. Deleterious mutations have relatively little effect on the evolution of genetic instability unless selection for additional driver mutations is very weak or if deleterious mutations are very common. Our model provides a framework for studying the evolution of genetic instability in tumour progression. Our analysis highlights the central role of selection in shaping patterns of mutation in carcinogenesis.Evolutionary Applications 01/2013; 6(1):20-33. · 4.15 Impact Factor

Page 1

Selective pressures for and against

genetic instability in cancer:

a time-dependent problem

NataliaL.Komarova1,2,*,AlexanderV.Sadovsky2,3andFredericY.M.Wan1,4

1Department of Mathematics,2Institute for Genomics and Bioinformatics,

3California Institute for Telecommunications and Information Technology, and

4Center for Complex Biological Systems, University of California, Irvine, CA 92697, USA

Genetic instability in cancer is a two-edge sword. It can both increase the rate of cancer

progression (by increasing the probability of cancerous mutations) and decrease the rate of

cancer growth (by imposing a large death toll on dividing cells). Two of the many selective

pressures acting upon a tumour, the need for variability and the need to minimize deleterious

mutations,affectthetumour’s‘choice’ofastableorunstable‘strategy’.Ascancerprogresses,the

balance of the two pressures will change. In this paper, we examine how the optimal strategy of

cancerouscellsisshapedbythechangingselectivepressures.Weconsiderthetwomostcommon

patterns inmultistage carcinogenesis: the activationofanoncogene (a one-step process) and an

inactivation of a tumour-suppressor gene (a two-step process). For these, we formulate an

optimalcontrolproblemforthemutationrateincancercells. Wethendevelopamethod tofind

optimal time-dependent strategies. It turns out that for a wide range of parameters, the most

successful strategy is to start with a high rate of mutations and then switch to stability. This

agrees with the growing biological evidence that genetic instability, prevalent in early cancers,

turns into stability later on in the progression. We also identify parameter regimes where it is

advantageous to keep stable (or unstable) constantly throughout the growth.

Keywords: multistage carcinogenesis; chromosomal instability; somatic evolution;

telomeres; optimization, bang-bang control; nonlinear control

1. INTRODUCTION

Geneticinstabilityisfoundinagreatmajorityofcancers,

in very small lesions as well as in relatively advanced

tumours.Twomaintypesofgeneticinstabilityhavebeen

described (Lengauer et al. 1998; Sen 2000). One type,

termed microsatellite instability (or MSI; Kinzler &

Vogelstein1996;Perucho1996),iscausedbyadeficiency

in the mismatchrepair systemand is characterizedby an

elevated rate of errors in the so-called microsatellites

(which are stretches of DNA in which a short motif,

usually one to five nucleotides long, is repeated several

times). The other broad type of instability is chromo-

somal instability or CIN. It is characterized by gross

chromosomal abnormalities in cancerous cells such as

lossesandgainsofchromosomes,translocations,etc.The

causes for MSI are relatively well understood: it is

triggered by an inactivation of one of the mismatch

repair genes such as hMSH1 and hMLH1. The origins of

CINarestillunknown.Severalgenecandidateshavebeen

found whose inactivation causes chromosomal aberra-

tions of cells (Cahill et al. 1998; Bardelli et al. 2001;

Nasmyth 2002; Yarden et al. 2002; Rajagopalan et al.

2004); another possible reason for CIN may be the

telomere crisis (Maser & DePinho 2002, 2004; Bailey &

Murnane 2006).

The question whether genetic instability is a driving

force or a consequence of cancer is as controversial today

as it was three decades ago (Loeb et al. 1974; Breivik &

Gaudernack 1999; Tomlinson & Bodmer 1999; Li et al.

2000; Shihetal. 2001; Marx2002; Breivik & Gaudernack

2004). Analytical and computational approaches have

been used to define the timing and other parameters of

genetic instability as well as to study its role in

carcinogenesis (Breivik & Gaudernack 1999; Breivik

2001,2005;Komarovaetal.2002,2003;Nowaketal.2002;

Little & Wright 2003; Michor et al. 2003, 2005; Nowak

etal. 2006).A commonmotif ofmany ofthese papers is a

microevolutionary nature of carcinogenesis.

All types ofgenetic instability are characterized by an

increased rate of change of the cell’s genome. This

produces at least two effects. One is a possibly increased

probability for a cell to experience an advantageous,

malignant mutation which can increase the cell’s fitness

and lead to further growth. The other is an increased

chance of deleterious, unwanted changes in the cell’s

genome which can reduce the cell’s fitness or lead to the

cell’s death. These considerations suggest that, in

principle, genetic instability can both increase the rate

of cancer progression (by increasing the probability of

J. R. Soc. Interface (2008) 5, 105–121

doi:10.1098/rsif.2007.1054

Published online 19 June 2007

*Author for correspondence (komarova@math.uci.edu).

Received 26 March 2007

Accepted 24 May 2007

105

This journal is q 2007 The Royal Society

Page 2

cancerous mutations) and decrease the rate of cancer

growth (by imposing a large death toll on dividing cells).

The question central to this paper is whether instability

helps cancer, in the sense of Darwinian microevolution

inside one organism.

The microevolutionary forces that act on cancer cells

during multistage carcinogenesis can be modelled by

taking into account these effects (Wodarz & Komarova

2005). In previous work, it was possible to recast the

problem by formulating the question from the view-

point of ‘selfish’ cancerous cells (Komarova 2004;

Komarova & Wodarz 2004). What is the optimal

level of instability which makes the cancer progress in

the fastest way? The mathematical problem is finding

the most efficient (from the point of view of cancer) rate

at which genetic changes occur in cells. It was shown

(Komarova & Wodarz 2004) that ‘too much’ instability

is detrimental for the cells due to an increased death

rate. ‘Too little’ instability also slows down the progress

because the basic rate at which cancerous mutations are

acquired is low. An optimal level of genetic instability

has been identified which maximizes the rate of

progression. This was quantified in terms of the

probability of chromosomal loss per cell division and

compared with available in vitro experimental

measurements of this rate. The mathematical result

turned out robust (it depended only logarithmically on

parameter values), and its order of magnitude was

consistent with the data (Lengauer et al. 1997).

In this paper, we take this model a step further and

study the temporal change of the level of instability.

As cancer progresses, the microevolutionary pressures

inevitablychange.Whatmighthavebeenagoodstrategy

atthebeginningofthegrowthmaybedetrimentalforthe

colonylateron.Thistimedependencefindsexperimental

support: a recent paper by Chin et al. (2004) argues that

the level of genetic instability in breast cancers first

increases,reachesapeakandthendecreasesasthecancer

passes throughtelomerecrisis.A paperby Rudolphetal.

(2001) reports data on intestinal carcinoma in mice and

humans which is consistent with a similar model:

telomere dysfunction promotes chromosomal instability

which drives carcinogenesis at early stages; and telomer-

ase activation restores stability to allow further tumour

progression.Themechanismoftelomeraseactivationand

subsequent prevention of chromosomal instability is also

described in the papers by Samper et al. (2001) and

Artandi & DePinho (2000). It is shown that short

telomeres can make mice resistant to skin cancer due to

anincreasedcelldeathrate(Gonzalez-Suarez etal.2000)

which also suggests that telomerase activation and

reduction in the level of chromosomal instability may

be a necessary step for cancer to develop.

Theideathatinstabilitymaybebeneficialforcancerat

an early stage and can become a liability later on is

developed in the present paper. We formulate the time-

dependent optimization problem to investigate the way

to maximize cancerous growth. To model growth and

mutations, we employ ordinary differential equations

(ODE) similar to quasispecies equations (Eigen &

Schuster 1979), widely used in modelling (micro-)

evolutionaryprocesses.Usingmethodsofoptimalcontrol

theory, we find strategies most advantageous for the

tumour’s growth. The degree of instability (the rate of

mutations) appears as an unknown function of time,

sought to maximize the growth of the mutants.

Mathematical theory of optimal control has been

used in many areas of biosciences (Sontag 2004;

Lenhart & Workman 2007). In biomedical applications,

control theory has usually been employed to design

treatment strategies by methods of optimization (Swan

1990; Kirschner et al. 1997; de Pillis et al. in press). In

this paper, we apply optimal control theory to studying

cancer in a very different way. We solve an optimi-

zation problem for the dynamics of cancerous growth in

order to understand why cancer behaves the way it

does. This approach is similar in spirit to the work of

Iwasa & Levin (1995) that analysed the optimal timing

of life strategies of breeding and migrating organisms.

In a sense, we study the ‘ecology’ of cancer, based on

our current knowledge of carcinogenesis, to see that the

observed behaviour of tumours is essentially a conse-

quence of the process of optimization.

Our main findings are as follows.

— For a wide range of parameters, the most successful

strategyistokeepahighrateofmutationsatfirstand

then switch to stability. This explains much of the

biological data (Rudolphetal. 2001; Chin etal. 2004).

—The time of the switching depends, to a small degree,

on the ‘target’ tumour size. It is independent of the

basic mutation rate or of the maximum rate of change

caused by the instability (as long as the latter is much

greater than the former, which is the biologically

relevant scenario). The time of the switching is

sensitive to the rate of growth of the mutants and is a

decaying function of this parameter.

—It turns out that, depending on the concavity of the

functional form chosen to express the death rate as a

function of the mutation rate, the corresponding

optimal strategies are qualitatively different. If the

death rate is a linear or concave function of the

mutation rate, then the optimal strategy is an abrupt

(discontinuous) change from maximum instability to

maximum stability. If the function is convex, the

transition is more gradual.

—Forsomeparameterregimes,theoptimalstrategyisto

remain maximally unstable throughout the growth.

This occurs, for example, if the magnitude of

mutation-related death rate is small, while the gain

in the mutation rate due to instability is large. On the

other hand, a very large death rate and a small gain in

mutation rate make instability disadvantageous at all

times, and the best strategy then is to remain stable.

The rest of the paper is organized as follows. In §2,

we formulate the biologically based mathematical

model which describes cells’ growth and mutations.

Section 3 introduces the formalism of optimal control

theory and summarizes the maximum principle for the

optimal strategy. Section 4 is a detailed study of a

subset of biologically relevant parameters. Optimal

strategies are found for different choices of functional

form of the death rate of cells. Section 5 considers the

entire parameter space of the system and identifies in

which cases instability is advantageous. Section 6

106Optimal strategy of cancerous cells N. L. Komarova et al.

J. R. Soc. Interface (2008)

Page 3

contains conclusions and discussion; it also provides

suggestions for several experiments that may confirm or

refute our theory and guide us in further research of the

functional role of genetic instability in cancer.

2. THE MODELS

We model a birth and death process with mutations in a

hypothetical setting where the mutation rate can be set

to arbitrary (biologically admissible) values at each

instantoftime.Cellsgothroughasequenceofmutations

until an advantageous phenotype is achieved. Until this

happens, the colony is subject to a regulatory process

which keeps its size constant. The colony must escape

this regulation in order to initiate the first wave of

clonal expansion; it must ‘overcome selection barriers

in the race to tumorigenesis’ (Cahill et al. 1999). Once

advantageous mutants are produced, they have the

ability to overcome the regulation and spread.

The mutation rate can have two effects. On the one

hand, a large mutation rate leads to a high death toll in

the population thus reducing the fitness of the cells. On

the other hand, an increased mutation rate can lead to a

faster production of the advantageous mutants, thus

accelerating the net growth of the colony.

2.1. A one-step system

Let us suppose that a colony of cells is currently at a

constant population size near a selection barrier. The

growthisstalledandthecellularpopulationremainsnear

the ‘carrying capacity’ which is defined by the available

space, nutrients and the cells’ ability to divide and die.

Thisbarriercanbeovercomebytheoffspringofamutant

whose properties are different. For instance, the mutant

cells could have an activated oncogene and show an

increased division rate or a decreased death rate. We

assume that such transformed cells are created by means

of one molecular event, genetic or epigenetic.

Let us denote by x1the population of cells that have

not undergone cancerous mutations and by x2 the

mutated type. The probability for a cell to acquire an

inactivating mutation of a particular gene upon a cell

division is denoted by ? m; this quantity is called the

‘basic mutation rate’. The probability p is an additional

transformation rate resulting from genetic instability.

This quantity measures the degree of genetic instability

in cells. It is low in stable cells (cells without CIN), but

it can be highly elevated in chromosomally unstable

cells. Effectively, if p/ ? m, then there is no genetic

instability; p[ ? m means a genetically unstable cell

population. Both probabilities ? m andp are measured per

gene per cell division.

With these notations in mind, we can present the

processes of growth and mutations described above by

means of a mutation diagram, figure 1a. This mutation

diagram is of the same type as used in Nowak et al.

(2002) and Komarova et al. (2003). The probability p is

the parameter of optimization in our problem. We will

try to find a strategy (i.e. the function p(t)) that

maximizes the growth of cancer.

Before we go on, we would like to comment further on

the biologicalmeaningofthe quantityp.The mechanism

of mutations resulting from genetic instability, which is

quantified by p, can be the same as or different from that

ofthebasicmutations.Forinstance,onecanassumethat

the transformation is a small-scale change in the DNA

sequence,andthegeneticinstabilityischaracterizedbya

deficiency in the nucleotide repair system (Benhamou &

Sarasin2000),whichresultsinanincrease(byanadditive

term p) of the basic mutation rate.

In other scenarios, the molecular mechanism of

genetic instability can be different from that of the basic

mutations. This is typical for the initiating events of

familial colorectal cancers (familial adenomatous poly-

posis, FAP). Each gene is normally present in two

copies (or alleles), a paternal and a maternal one. FAP

patients are born with one of the copies of the

adenomatosis polyposis coli (APC) gene inactivated

in all cells. Then, an inactivation of the second copy

eventually causes early lesions and further disease

progression. The second copy of the APC gene can

become inactivated by a point mutation or by a loss-

of-chromosome event. The rate of chromosomal loss is

often greatly elevated in colon cancers as a result of

genetic (chromosomal) instability or CIN (Lengauer

et al. 1998). In this situation, ? m is the basic point

mutation rate and p is the rate of chromosomal loss.

The rate of chromosome loss, p, is a quantity which

depends on many factors. No single gene responsible for

chromosomal instability has been found; instead, a lot

(of the order of hundreds) of genes have been shown to

participate in various ways in the process of chromo-

some duplication, segregation, etc. A defect in any of

those genes can change the resulting probability of

chromosomal loss. Apart from that, the telomeres

(regions of highly repetitive DNA at the end of a linear

chromosome that function as disposable buffers) have

been shown to play a role in genomic stability.

Therefore, a change in any of these factors may be

responsible for a change in the level of chromosomal

instability and, therefore, can control the value of p.1

The dynamics of the cells is modelled here as follows.

Cells reproduce and die, and the rate of renewal is taken

to be 1 for the type x1. In the absence of dangerous

m+p

(a)

dd

growth

deathdeath

x1

x2

m+p

2m

(b)

d

dd

growth

death death death

x1

x0

x2

Figure 1. Mutation diagrams for (a) the one-step process and

(b) the two-step process.

1The inactivation of the TSG itself may actually be responsible for the

change in p directly. It has been suggested (Fodde et al. 2001) that the

inactivation of the APC plays a role in triggering genetic instability in

colon cancer. This scenario remains controversial and is not included

in the paper.

Optimal strategy of cancerous cells N. L. Komarova et al.107

J. R. Soc. Interface (2008)

Page 4

mutants, x2, the total number of cells, x1, is assumed to

obey the well-known logistic growth law, that is, near

the equilibrium the population stays constant, with a

positive growth rate for the number of cells below the

equilibrium, and a negative growth rate above the

equilibrium. The mutants x2expand at the rate aO1.

Mutation diagram in figure 1a translates into the

following ODEs describing the rate of change of the two

cell populations,

x0

x0

1Zð1K ? mKpKdðpÞÞx1Kfx1;

2ZðpC ? mÞx1Cað1KdðpÞÞx2Kfx2;

where ($)0Zd($)/dt, fZð1KdðpÞÞx1=N, and x1(0)Z

N, x2(0)Z0. The term f is similar to logistic growth in

the absence of mutants, and accounts for the homeo-

static control present in a system of x1-cells. x2-cells

break out of regulation and enter a phase of exponential

growth. The term with x1in the equation for x2is added

to represent a partial, non-symmetric, homeostatic

control that may play some role at the beginning of the

growth of x2 cells. Later on that term is simply a

correction to the growth rate of the x2cells. This way of

modelling the dynamics is not unique, and in fact the

f-term may be removed from the equation for x2.

A more detailed discussion of the robustness of the

model is presented in the context of the two-step model,

§§2.2 and 4.3.

ð2:1Þ

ð2:2Þ

2.2. A two-step system

In this section, we investigate another model where an

inactivation of a tumour suppressor gene (TSG) leads

to a clonal expansion. This is a two-step molecular

process, whereby the two alleles of the gene are

inactivated one at a time. The inactivation of just one

allele does not result in any phenotypic changes. The

inactivation of the second allele leads to the cells’

unchecked growth. These processes can be summarized

by a mutation diagram, figure 1b. There, x0 is the

population of TSGC/Ccells (i.e. cell with both copies

of the TSG intact), x1is the population of TSGC/K

cells, where one of the copies of the TSG has been

mutated, and x2is the population of TSGK/Kcells,

where the remaining copy of the TSG has been lost.

The process described by the mutation diagram in

figure 1b contains two steps: an inactivation of one, and

then the other allele of the TSG. Note that the

probabilities at which the two inactivation events

occur are not equal. In the diagram of figure 1b, ? m is

the basic mutation rate by which an allele can be

inactivated. Since there are two alleles of each gene, and

either of them can be inactivated first, the total

probability of the first inactivation event is 2? m. The

second inactivation event can happen by another

mutation (probability ? m). As in the case of colorectal

cancer and the APC gene, there is a different

mechanism by which the second copy of the TSG can

be turned off. This is a ‘loss-of-chromosome’ event,

which is known to be responsible for the inactivation of

a large percentage of TSGs in cancers (Kinzler &

Vogelstein 2002). As a result of this event, the whole

chromosome corresponding to the TSG in question

becomes lost (or, more commonly, is replaced by a copy

of the other chromosome where the TSG is mutated).

This is a gross chromosomal change, whose probability,

p, is our optimization parameter.

Before we goon, wewouldliketoaddress the question

of the asymmetry between the first and the second

inactivationevents.Inprinciple,thefirstallelecanalsobe

inactivatedbyaloss-of-chromosomeevent.However,the

fitnessofacellwithamissingchromosomeandoneactive

copyoftheTSGisverylow.Suchcellswillquicklydieout

and will make no difference for the present analysis. On

the other hand, a cell with one chromosome missing and

both copies of the TSG inactivated has a selective

advantage, because we assume that the inactivation of

the TSG leads to an increase in the cell’s growth rate.

Such cells are produced by the sequence of events

depicted in figure 1.

A system of ODEs that describes all these processes

is as follows:

x0

0Zð1K2? mKdðpÞÞx0Kfx0;

ð2:3Þ

x0

1Z2? mx0Cð1K ? mKpKdðpÞÞx1Kfx1;

x0

ð2:4Þ

2ZðpC ? mÞx1Cað1KdðpÞÞx2;

ð2:5Þ

where

f Zð1KdðpÞÞðx0Cx1Þ

N

:

ð2:6Þ

Cells reproduce and die, and the rate of renewal is taken

as 1 for types x0and x1. In the absence of dangerous

mutants, x2, the total number of cells x0and x1stays

constant, i.e. the sum of equations (2.3)–(2.5) with

x2Z0. The mutants x2expand at rate aO1.

This formulation for the two-step process is not

unique. We will refer to system (2.3)–(2.5) and (2.6) as

Model I. One possible modification is to replace the

expression for f, equation (2.6), with

f Z1KdðpÞðx0Cx1Þ

N

;

ð2:7Þ

system (2.3)–(2.5) and (2.7) will be referred to as Model

II. Also, we may include the f-term in the last equation,

that is, replace equation (2.5) with

x0

2ZðpC ? mÞx1Cað1KdðpÞÞx2Kfx2:

We will name systems (2.3), (2.4), (2.8), (2.6) and (2.3),

(2.4), (2.8), (2.7) Model I?and Model II?, respectively.

Such changes in the model equations will lead to

quantitative changes in the outcome, but, as we will

demonstrate below, the results remain qualitatively

robust with respect to such modifications.

Note that our formulation is similar to the well-

studied quasispecies model (Eigen & Schuster 1979) in

that the competition among cells is captured by means

of an additive term (f). This type of a description is

widely used in cancer modelling (Wodarz & Komarova

2005) and other areas of mathematical biology. If we

suppose aZ0 (i.e. that the double-mutants, x2, do not

reproduce), then the first two equations, (2.3) and (2.4),

are quasispecies equations. The inclusion of the non-

zero growth term for the mutants that are not subject

ð2:8Þ

108 Optimal strategy of cancerous cellsN. L. Komarova et al.

J. R. Soc. Interface (2008)

Page 5

to competition reflects the escape of the biological

system from the homeostatic control.

2.3. The death rate parameterization

The death rate, d(p), is a function of the mutation rate,

p. If p is small then chromosome losses do not happen,

and if p is large a cell often loses chromosomes which

results in an increased death rate. Therefore, in general,

the function d(p) will be a monotonically increasing

function of p.

Here we present an example of a parameterization of

the death rate as a function of p. It is convenient to

introduce a normalized rate of chromosome loss, u, and

express the death rate in terms of this parameter

dðuÞ Zdmð1Kð1KuÞaÞ;u Z

pKpmin

pmaxKpmin

;

pmin%p%pmax;

aO0:

ð2:9Þ

The motivation for this particular dependency is as

follows. Let us suppose that a cell dies if it loses one of a

essential chromosome copies (out of the total of 2!23

copies in a human). Then, if we set pminZ0 and pmaxZ1,

the death rate (2.9) can be written in a form

dðuÞ Zdm!½Probability of cell death by chromosome loss?:

The constant dmdefines the magnitude of the death

rate, and is taken to be in the interval 0%dm%1. In this

paper, we use general values for pminand pmaxsuch that

0%pmin!pmax%1; these quantities define a biologically

relevant range of the mutation rate, u. We allow a, the

exponent in equation (2.9), to be a real positive

number. In particular, we investigate the influence of

the concavity of this function on the optimal solution

(cases a!1 and aO1). The special case aZ1 yields a

control problem where the controls enter linearly, a case

much studied in the optimal control literature and rich

with analytical results.

The variables and parameters used in our model, as

well as their scaled versions employed in the next

sections, are summarized in table 1.

2.4. Formulation of the optimization problem

In this paper, we adopt the theoretical framework

where it is possible to set the rate of genetic instability

to an arbitrary (but meaningful) value at each moment

of time. Mathematically, the above framework means

specifying the instability rate, p(t), as a function of

time. Every choice of such a function determines a

growth process of the tumour. We shall seek the choice

of p(t) that allows the cancerous population to reach a

given size, M, in the shortest possible time. In the

terminology of optimization and control theory, the

population size M is called the target, the possible

values of genetic instability rate p are called ‘admissible

controls’, and each choice of the function p(t) is called a

strategy. A strategy steering the system to the target

faster than any other strategy is said to be an ‘optimal

strategy’ or ‘optimal control’. In this terminology, we

seek a strategy for controlling the system to reach the

target as soon as possible. A meaningful qualitative

comparison between two strategies is now possible: the

‘better’, or ‘more advantageous’, strategy is the one

allowing the system to reach the target sooner. Thus, an

‘advantageous strategy’ is advantageous for the cancer

in the sense of Darwinian microevolution in an

individual organism.

To find the optimal strategy, we consider the

quantity, T, which is the solution of the equation

x2ðTÞ ZM;

wherex2isthesolutionofsystem(2.1)and(2.2)or(2.3)–

(2.5).The growthtime, T, dependsonall the parameters

of the system, including the time-dependent mutation

rate, p. The optimal strategy is the one that minimizes

the value of T.

Inthesimplestcase,werestricttheclassofadmissible

controls, p(t), to constant functions. Then, the result of

the optimization problem is a single value, popt, which

will depend on the parameters of the system. A similar

problem was solved in Komarova & Wodarz (2004).

However, better growth times can be achieved if we

allow p to be a function of time. It seems intuitive and is

Table 1. Variables, model parameters and their definitions.

notationbiological interpretation

x0(t), x1(t)

x2(t)

? m

p(t)

pmin, pmax

um

N

M

s

T

m

u(t)

A

d(u)

a

dm

pre-cancerous cell populations

malignant cell population which escapes homeostatic control

basic mutation rate of genetically stable cells (probability of mutation per cell division)

the additional mutation rate resulting from genetic instability

the range of p(t)

pmaxKpmin

the population size in the absence of malignant cells

the target population size of a growing tumour colony

M/N

the time it takes the malignant colony to reach size M

? mCpmin

(pKpmin)/(pmaxKpmin), the scaled rate of genetic instability

the exponential growth rate of malignant cells

the death rate of cells

the exponent in the definition of the death rate as a function of the rate of instability, formula (2.9)

the magnitude of the death rate, formula (2.9)

Optimal strategy of cancerous cells N. L. Komarova et al.109

J. R. Soc. Interface (2008)

Page 6

evidentfromexperimentalresultsthathigherinitialand

lower subsequent values of p will facilitate the growth.

3. MATHEMATICAL APPARATUS

In this section, we develop a mathematical framework

for the one-step process. Similar calculations lead to the

corresponding formulation for the two-step problem to

be presented in appendix A.

3.1. Statement of the one-step problem

3.1.1. Equations of state. Let us define the following

parameter combinations: sZM/N, umZpmaxKpmin.

Introducingthescaled

x?

we can rewrite system (2.1) and (2.2) as

x0

hg1ðx1;x2;uÞ;

2Z1

sðmCumuÞx1C½1KdðuÞ?ðaKx1Þx2

hg2ðx1;x2;uÞ:

The death rate, d(u), is given by equation (2.9). Recall

that u is the normalized gross chromosomal change rate

with 0%u%1. As we show below, the three cases aO1,

aZ1 and a!1 may have to be treated separately.

quantitiesx?

1Zx1=N,

2Zx1=M, and dropping the asterisks for simplicity,

1ZKðmCumuÞx1C½1KdðuÞ?ð1Kx1Þx1

ð3:1Þ

x0

ð3:2Þ

3.1.2. Boundary conditions. The two ODEs (3.1) and

(3.2) are subject to the following three auxiliary

conditions:

x1ð0Þ Z1;

where T is the time when the dangerous mutant cell

population reaches the target size.

x2ð0Þ Z0;x2ðTÞ Z1;

ð3:3Þ

3.1.3. Problem. Choose the control function u(t) to

minimize the time T needed to reach the target

population size of the dangerous mutants subject to

the inequality constraint,

0%u%1;

ð3:4Þ

on the control u(t) and non-negativity constraints on

the cell populations,

x1R0;x2R0:

ð3:5Þ

3.1.4. Restatement of the optimal control problem. To

apply the usual Hamiltonian system approach, we

recast the problem described in §2.4 by choosing the

control function u(t) to minimize the performance index

ðT

subject to the equations of state (3.1) and (3.2), the

boundary conditions (3.3) and the inequality constraints

(3.4) and (3.5) with TO0 as a part of the solution.

J Z

0

1dt;

ð3:6Þ

3.2. The maximum principle

3.2.1. The Hamiltonian and adjoint variables. Optimal

control problems can be effectively analysed through the

Pontryagin maximum principle and its associated

Hamiltonianformalism(Pontryaginetal.1962;Bryson&

Ho 1969; Wan 1995). In this section, we develop

componentsoftheHamiltonianformalismforoursystem.

The Hamiltonian for our problem is

H Z1Cl1ðtÞg1Cl2ðtÞg2;

ð3:7Þ

where l1and l2are the two continuous and piecewise

differentiable adjoint (or costate) variables for the

problem chosen to satisfy two adjoint ODEs,

?

l0

vx2

l0

1ZK l1

vg1

vx1

vg1

Cl2

vg2

vx1

vg2

vx2

?

?

;

ð3:8Þ

2ZK l1

Cl2

?

;

ð3:9Þ

and (for the given auxiliary conditions on the state

variable x1and x2) one transversality condition

l1ðTÞ Z0:

ð3:10Þ

Note that (3.1), (3.2), (3.8) and (3.9) form a

Hamiltonian system for the Hamiltonian given in

(3.7). (More generally, the Hamiltonian should be

taken in the form

H Zl0Cl1ðtÞg1Cl2ðtÞg2;

for some non-negative constant l0. But with the

terminal time definitely having a role in the optimal

control problem, the additional (constant) adjoint

variable l0 does not vanish. We can rescale the

Hamiltonian and simplify it to (3.7).)

With the set of the admissible controls specified

by (3.4), an optimal strategy u(t) is continuous or

has finite jump discontinuities in [0, T ]. This follows

from the way how u(t) appears in the state and

adjoint ODE, (3.1), (3.2), (3.8) and (3.9), that the

state and adjoint variables are continuous in (0, T),

including instances of jump discontinuities in the

optimal control.

3.2.2. Formulation of the maximum principle. The

optimal solution of our minimum terminal time

problem requires the optimal control function ? uðtÞ to

satisfy the following necessary conditions, known as the

maximum principle (Pontryagin et al. 1962; Gelfand &

Fomin 1963; Wan 1995).

(i) Four continuous and piecewise differentiable

functions f? x1; ? x2;?l1;?l2g exist and satisfy the

four differential equations (3.1), (3.2), (3.8) and

(3.9), and four auxiliary conditions in (3.3) and

(3.10) for the admissible control ? uðtÞ.

(ii) The minimum terminal time T obtained with

uZ ? uðtÞ, satisfies a free end condition

½H?tZTZ½1C?l2? g2?jtZTZ0;

with

? g2Zg2ð? x1; ? x2;?l1;?l2; ? uðtÞÞ;

where the adjoint boundary condition (3.10) has

been used to simplify the expression for H.

ð3:11Þ

110 Optimal strategy of cancerous cellsN. L. Komarova et al.

J. R. Soc. Interface (2008)

Page 7

(iii) For all t in [0, T ], the Hamiltonian achieves its

minimum for uZ ? uðtÞ, i.e.

Hð? x1ðtÞ; ? x2ðtÞ;?l1ðtÞ;?l2ðtÞ; ? uðtÞÞ

Zinf

v

Hð? x1ðtÞ; ? x2ðtÞ;?l1ðtÞ;?l2ðtÞ;vÞ

??; ð3:12Þ

for all v in the set of admissible controls

restricted by (3.4).

(iv) If there should be a change in the control ? u at the

instance Tsthat involves a finite jump disconti-

nuity in the value of the control, optimality

requires that the Hamiltonian be continuous at

Ts(Bryson & Ho 1969; Wan 1995)

½H?tZTsC

Given the admissible controls as specified by (3.4), the

optimal control ? uðtÞ can have finite jump discontinu-

ities in (0, T). It follows from the way u(t) appears in

the state and adjoint ODE that the state and adjoint

variables are continuous in (0, T), including instances

of a control jump discontinuity.

tZTsKZ0:

ð3:13Þ

3.2.3. The interior control. Suppose the optimal control

strategy ? uðtÞ satisfies 0% ? uðtÞ%1 and the equation

vH

vu

uZ? u

??

Z0;

ð3:14Þ

with

vH

vuZl1fK ½umCd$ðuÞ?x1Cd$x2

Cl2

sfumx1Ksd$x2ðaKx1Þg

l2

sKl1

Cl2x2ðaKx1Þgd$:

Then we call ðx1;x2;l1;l2; ? uÞ an interior solution for

which the optimal control ? uZuintðtÞ is an extremum

(or, more correctly, an extremal) of the Hamiltonian.

The stationary condition (3.14) can be written as

fl1x1ð1Kx1ÞCl2x2ðaKx1Þgd$Zum

1g

Zumx1

??

Kfl1x1ð1Kx1Þ

ð3:15Þ

sx1fl2Kl1sg:

ð3:16Þ

Using the expression for the death rate (2.9) with d$Z

a(1Ku)aK1, we obtain from condition (3.17) the

following formula for the interior solution,

uðtÞ ZuintðtÞ

Z1K

umx1ðl2Ksl1Þ

asfl1x1ð1Kx1ÞCl2x2ðaKx1Þg

?? 1

aK 1:

ð3:17Þ

It can be shown (Wan et al. in preparation) that the

interiorsolutionaboveviolatestheinequalityconstraints

(3.2) in some part of the solution domain for some range

of system parameter values. Consequently, some com-

bination of the upper corner solution (u(t)Z1), the lower

corner solution (u(t)Z0) and the interior solution has to

beconsideredfortheoptimalsolution.Wheneveracorner

control is applicable, the adjoint variables (and the

corresponding adjoint ODE and auxiliary conditions)

may or may not play a role in the solution process since

the control variable u(t) is completely specified (and not

determined by the stationary condition (3.14)).

3.2.4. A vanishing Hamiltonian,H(t)Z0. Foranautono-

mous control problem, the Hamiltonian is constant for

the optimal solution ? uðtÞ (Wan 1995; Wan et al. in

preparation). The free-end condition (3.11) and the

continuity condition (3.13) then require H(t)Z0. This

result(togetherwiththeformulafortheinteriorsolution,

equation(3.17))willbeusedtofindanoptimalcontrol.It

can also be used to check how far a candidate control

function is from the actual optimal control.

4. THE OPTIMAL RATE OF INSTABILITY

In this section, we consider the special case, dmZ1, see

equation (2.9). We will examine the corresponding

problem in some detail and then turn to the analysis of

the general problem in the following section.

The case dmZ1 corresponds to fixing the death rate

(relative to the cell division rate) at its highest possible

value. The other parameter in expression (2.9) remains

unspecified, such that aZ1 corresponds to the linear

dependence of the death rate on the mutation rate u,

aO1 gives a concave dependence and a!1 a convex

dependence. It turns out that the concavity of the

function d(u) is critical to the qualitative shape of the

optimal control function. We will examine the cases

aR1 and 0!a!1 separately.

4.1. Non-convex death rates, aR1

4.1.1. The case aZ1. In this special case, the control

variable, u, enters system (2.1) and (2.2) linearly.

It follows from the Pontryagin maximum principle

(Boltyanskii et al. 1962) that the optimal control in this

caseisbang-bang(Wan1995;Wanetal.inpreparation),

i.e. piecewise constant switching between the values,

uminZ0 and umaxZ1. In that case, every such control

u(t) is completely determined by its initial value, u(0),

and the switching times s1;s2;.2ð0;TÞ at which u(t)

experiences a switching, i.e. a discontinuous change

of value.

For the problem stated in §§3.1 and A.1, the optimal

control is even simpler. It is possible to show (Wan

et al. in preparation) that the optimal control starts

with umaxZ1 for a period 0%t!Tsand then switches to

uminZ0 for the rest of the growth process, Ts!t!T;

there is only one switching in the optimal control

function ? uðtÞ. Biologically, in order to maximize the

growth, it is reasonable to take u as large as possible,

namely, uZ1 at the start, such that some pool of

mutants is created quickly, and then to switch to uZ0

later on, to take advantage of the exponential growth of

x2(t). Therefore, the optimal solution has the form

? uðtÞ ZqðTsKtÞ;

where q is the Heaviside function and Ts is some

switching time, 0!Ts!T, to be determined as a part of

the solution process.

ð4:1Þ

Optimal strategy of cancerous cellsN. L. Komarova et al. 111

J. R. Soc. Interface (2008)

Page 8

4.1.2. The case aO1. Next, we turn to strictly concave

death rates. For nonlinear control problems, the

optimal control is generally not bang-bang. One

normally attempts to solve equations (3.1), (3.2),

(3.8) and (3.9) with boundary conditions (3.3), (3.8),

and the conditions (3.11) and (3.13) at the terminal

time and at the switch points, Ts, by u(t), respectively,

with the control variable expressed in terms of the state

and adjoint variables by (3.17). For our problem, the

solution obtained by this method, however, violates the

constraint 0%u(t)%1, and thus is not applicable, at

least, initially. Therefore, an optimal solution is to

start with a corner control. It can be shown (Wan et al.

in preparation) that optimal controls are again bang-

bang, as in the case aZ1, starting with uZ1 and

switching to uZ0. In short, the nonlinear case aO1 of

the control problem is characterized by a bang-bang

solution of the form (4.1) just like the linear case aZ1.

Formula (4.1) shows that an optimal control is

completely determined by the value Tsof the switching

time. In theory, this value is determined by H(ts)Z0

which involves the solution for the adjoint variables. It is

simpler to find Tsusing the following approach. Varying

Tsover a suitable interval [r1, r2], compute for each Ts

value the corresponding terminal time, T, by solving the

initial-value problem (3.1)–(3.3). This process yields a

function T(Ts), r1%Ts%r2. The value Tsat which T(Ts)

attains a minimum specifies an optimal control.

Figure 2 illustrates this procedure for a particular set

of parameters and shows that the function T(Ts) has

one minimum, given by the value TsZ?Tsz1:04. This

value of Tsminimizesthe time to target,and the control

function, ? uðtÞ with TsZ?Ts, equation (4.1), is the

optimal control for the set of parameter values shown in

the legend of figure 2.

We have also investigated how the switching time

depends on various parameters of the system. The

following are the results.

— Tsis a monotone, very gradually increasing, concave

function of s for sO1 (recall that sZM/N defines

the target colony size relative to the normal colony

size, N),see figure3a. The relative switch time, Ts/T,

is a decreasing function of s (not shown).

—Tsis a monotone, decreasing, convex function of the

parametera,thegrowthrateofmutants,seefigure3b.

—The value of Tsis a decreasing function of m. As m

decreases below the value um, the function Tsreaches

saturation and does not change much with the

value of m.

—Tsisanincreasingfunctionofum,suchthatforum/0,

Ts/0. However, as umbecomes greater than m, Ts

reaches saturation. Therefore, for the biologically

relevant regime of um[m, Tsis essentially indepen-

dent of um.

We can see that as long as um[m (a biologically

relevant parameter regime), the switching time of the

optimalcontroliseffectivelyindependentofthemutation

rates m and um. The switching time also does not depend

strongly on s (it is a slowly increasing function of s).

Hence,theswitchingtimeessentiallydependsonlyonthe

growthrateofthemutantcells,a.Thefasterthemutants

grow, the sooner the switch happens.

4.2. Convex death rates, 0!a!1

We now turn to the remaining case of 0!a!1. The

death rate d(u) is then a convex function of u. Optimal

controls are not bang-bang in this case, as shown in

Wan et al. (in preparation). This can also be seen from

the following numerical experiment. Let us first assume

that u(t) is given by formula (4.1) and find the value Ts

which minimizes the time it takes for x2(t) to grow to

size 1. The corresponding time, T, tells us how well a

bang-bang control does. Now, let us expand the class of

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

TS, the switch time

T, the time to target

12345

Figure 2. The minimum time to target, T, for different values

of the switch time, Ts. The parameters are aZ2, sZ10, aZ2,

umZ1, mZ10K7.

1.6(a)

1.0

1.1

1.2

1.3

1.4

1.5

s

TS, the switch time

200 40 80120 160

510

a

1520

1

2

3

4

5

TS, the switch time

(b)

Figure 3. The switch time, Ts, as a function of parameters (a)

s and (b) a. The other parameters are aZ2 and umZ1 in (a),

sZ10 and umZ10K1in (b), aZ1.5, mZ10K5.

112Optimal strategy of cancerous cellsN. L. Komarova et al.

J. R. Soc. Interface (2008)

Page 9

possible control functions to a two-parametric family,

u1ðtÞ Z1K1

2

1CtanhtKTs

w

??

;

ð4:2Þ

where Tsis the characteristic time of the ‘transition’

from high to low values of u, and w is the width of this

smooth transition. In fact, it is enough to leave only one

free parameter, w, and fix Tsto the ‘best’ switching

time obtained for the bang-bang control.

Now, let us vary the parameter w and calculate the

time T needed for the colony of mutants to grow to size

M. We will obtain a function T(w). The value w

corresponding to the minimum T gives the best

performance of family (4.2). If the best width is wZ0,

then we can conclude that the sharp, bang-bang-type

transition cannot be improved by smoothing it out.

However, as figure 4 illustrates for a particular set of

parameters, the best control for family (4.2) corre-

sponds to non-zero w. That is, a smooth transition can

do better than the best of the bang-bang family. This

means that the optimal control is not bang-bang.2

The next question is finding the actual optimal

control for a!1. Solving the boundary value problem

with the interior solution again does not work. In fact,

one can prove that (unless a is very small) the interior

solution, (3.17), fails at and near the terminal time

(Wan et al. in preparation).

In order to find the optimal control, we have

designed the following method.

(i) Startfromanycontrol,0%u0(t)%1,e.g.u0(t)Z1.

(ii) Solve the initial value problem (3.1)–(3.3) for

0%t%T1such that x2(T1)O1.

(iii) Findthe solution,tZT0,ofthe equationx2(t)Z1.

This is the zeroth approximation to the best

terminal time.

(iv) Solvetheboundaryvalueproblem(3.8)–(3.11)on

0%t%T0with the functions x1(t), x2(t) known

from the previous step.

(v) Use the obtained state and adjoint variables to

calculate the function u(t) by formula (3.17). Call

this function ? u1ðtÞ.

(vi) Take u1ðtÞZ ? u1ðtÞqð? u1ðtÞÞ (again, q is the Heavi-

side step-function). That is, replace all negative

valuesofthecontrolbyzero.Thisgivesusthenext

approximation to the optimal control, u1(t).

(vii) Go to step (ii) and repeat all the operations.

Numerical evidence suggests that this method con-

vergestoafixedpointwhichisindeedtheoptimalcontrol.

Toverifytheoptimality,wecanevaluatethevalueofthe

Hamiltonian, equation (3.7), at each step. According to

themaximumprincipleforourproblem,theHamiltonian

must vanish for all t, see §3.2.4. A sample run of the

algorithm is presented in figure 5a, where we plot the

logarithm of the value jjHijjZÐTi

converges to H(t)Z0, and therefore the corresponding

limiting function uN(t) is the optimal control for the

0jHðui;tÞjdt for

consecutive iterations. We can see that the Hamiltonian

problem. Figure 5b shows the values of the time to target

aftereachiteration. Figure 6 shows someexamplesofthe

optimal control functions, ui(t) for large i, found by this

method for different a values.

We note that a control found by simply maximizing

x2(t) at each t is not optimal. This control is obtained

by solving vg2=vuZ0, see equation (3.2). This gives the

value of u maximizing the time-derivative of x2

?

We have solved the initial value problem obtained from

(3.1)–(3.3) with the above expression for u(t). The

solution x1(t), x2(t) can be inserted in the expression

(4.3). The resulting control is compared with the

optimal control in figure 7; we can see that the two

functions differ from each other. The performance of the

optimal control obtained from the maximum principle

is found to be better, as expected.

Wehavealsoemployedsequentialquadraticprogram-

ming (SQP) algorithm (Gill et al. 2005) with direct

collocation and automatic differentiation (implemented,

respectively, in the software packages SNOPT v. 7

(Gill et al. 2005), DIRCOL v. 2.1 (Von Stryk 2000) and

ADIFOR v. 2.0 (ADIFOR 1994)) to find an accurate

approximation to the optimal solution. A sample of such

numerical solutions is shown in figure 8a. We see that as

a/1 from below, the transition from uZ1 to uZ0

becomes sharper and sharper. The optimal control

becomes bang-bang control as a/1 from below.

u Z1C

umx1

asðaKx1Þx2

? 1

aK 1:

ð4:3Þ

0.20.40.60.8 1.0

1.46

1.50

1.52

1.54

1.56

w

T

(a)

0.20.4 0.6 0.81.0 1.21.4

0.2

0.4

0.6

0.8

1.0(b)

t

u(t)

bang–bang

smooth

Figure 4. Does bang-bang work in the a!1 case? (a) The time

totarget, T,as afunction of parameterw informula (4.2). The

other parameters are aZ2, sZ10, aZ0.5, umZ1, mZ10K1.

(b) The control function, u1(t), corresponding to the best

value of w. Also, the best bang-bang control for these

parameters is shown. The time to target for the bang-bang

control is Tz1.57. For the function u1(t), it is Tz1.49.

2If we perform the same operations in the aR1 case, we obtain that

wZ0 gives the best result. This of course does not prove that the

optimal controls are bang-bang, but it is consistent with the

conclusions of §4.1.2.

Optimal strategy of cancerous cellsN. L. Komarova et al.113

J. R. Soc. Interface (2008)

Page 10

4.3. The two-step problem

Most of the qualitative conclusions obtained for the

one-step problem also hold for the two-step problem.

Namely, for aR1 we have a bang-bang optimal control,

and for 0!a!1 we have an interior solution for part of

the domain [0, T ]. This can be demonstrated again by

comparing the best bang-bang solution with a family of

continuous functions u(t) with a finite width. Note that

an improvement on the bang-bang control for 0!a!1

may not be found in the class of functions defined by

equation (4.2). In some instances, we have used the

function (4.2) multiplied by some small power of t. In all

cases, a non-zero width wO0 gives a better performance

than the bang-bang control.

The iterative method developed above for approxi-

mating the optimal control for the one-step model does

not work for the two-step problem. There, we do not

observe a convergence of the algorithm to a fixed point.

Instead, we used SQP algorithm to find the solution. An

example is shownin figure 8b, where the optimalcontrol

is found numerically for different values of a!1. As in

the case of a one-step process, the control becomes

steeper as a/1.

0.20.40.6

0.8

1.0 1.21.4

0.2

0.4

0.6

0.8

1.0

u(t)

t

a = 0.1

a = 0.2

a = 0.3

a = 0.4

a = 0.5

Figure 6. The optimal control found by the iterative method

for a!1. The optimal functions for five values of a are

presented. The optimal time to target is TZ1.238 for aZ0.1,

1.330 for aZ0.2, 1.395 for aZ0.3, 1.445 for aZ0.4 and 1.484

for aZ0.5. The parameters are aZ2, sZ2, umZ1, mZ10K1.

0.20.4 0.60.8

t

1.01.21.4

0.2

0.4

0.6

0.8

1.0

u(t)

optimal

control

maximizing

growth of

mutants

Figure 7. The optimal control found by the iterative method,

compared to the function u(t) found from formula (4.3), which

maximizes the growth of x2(t). The parameters are aZ2,

aZ0.5, sZ2, umZ1, mZ10K1.

0.51.0 1.52.0 2.5

t

a = 0.9

(a)

(b)

a = 0.9

a = 0.1

a = 0.1

a = 0.3

a = 0.3

a = 0.5

a = 0.5

a = 0.7

a = 0.7

a = 0.99

a = 0.99

0.2

0.4

0.6

0.8

1.0

u(t)

0.2

0.4

0.6

0.8

1.0

u(t)

12345

Figure 8. The optimal control found by the SQP method,

for different values of a!1, for (a) a one-step process and

(b) a two-step process. The other parameters are aZ2, sZ10,

umZ1, mZ10K1.

10 20 30

–6

–5

–4

–3

–2

iterations

log ||Hi||

(a)

10 2030 40

1.45

1.50

1.55

1.60

1.65

1.70

1.75

iteration number

Ti

(b)

Figure 5. The iterative algorithm to find the optimal control

in the a!1 case. (a) The value HiZÐTi

time to target, T. The parameters are aZ2, sZ2, aZ0.5,

umZ1, mZ10K1.

0jHðui;tÞjdt for

consecutive iterations. (b) The iterations of the minimum

114Optimal strategy of cancerous cellsN. L. Komarova et al.

J. R. Soc. Interface (2008)

Page 11

Robustness of the model has been checked. We have

compared the qualitative shapes of the optimal controls

for Models I, II, I?and II?, formulated in §2. For all the

models,theoptimalcontrolwithaR1isofthebang-bang

type, and for a!1 it is a continuous monotonically

decaying function of time, starting at uZ1 at tZ0.

Figure 9 presents the optimal controls found by SQR

algorithm for the four models, for aZ0.5 (and all other

parameters taken the same for all models). The optimal

control(ofthebang-bangtype)foraZ1.5ispresentedfor

the four cases in table 2. We can see that there are

quantitative differences, but the qualitative behaviour is

the same.

5. THE OPTIMAL STRATEGY FOR CANCER

In what follows, we drop the restriction dmZ1 in

formula (2.9) and investigate the two-dimensional

parameter subspace of the problem, (um, dm). The

magnitude umtells us the maximum mutation rate (due

to genetic instability) that is possible in the system.

The value dmcharacterizes the maximum magnitude of

the death rate associated with the instability.

Let us first take a!1, fix a value um!1 and find an

optimal control ? uðtÞ for several values of the magnitude

of the death rate, dm. Figure 10 presents the functions

? uðtÞ obtained by means of the method described in §4.2

We have also used the SQP algorithm to double-check

the results. Figure 10a shows several runs for aZ0.1,

and figure 10b shows several runs for aZ0.9. We

observe the following trend: as dmbecomes smaller, the

optimal controls become larger. In other words, for

small values of the death rate, optimal strategies tend

to favour large values of u(t) for a longer initial period of

time. For example, in the case aZ0.1 the transition to

uZ0 does not happen for dm!0.001. On the other

hand, for large values of dm, optimal controls decrease

continuously from their maximum value to zero very

early in the growth process. Note that increasing dmto

values larger than 1 will lead to an even sharper

decrease in ? uðtÞ.

Next, we turn to the values aR1. In this case, the

results do not depend on the actual value of a; optimal

controls were found (Wan et al. in preparation) to be

bang-bang with exactly one switching. In other words,

the result that no feasible interior solution exists for

aR1 extends to the case dms1. The function ? uðtÞ

assumes only one or both of the values uZ1 and uZ0

(with no more than one switching), and the value of the

death rate, dð? uÞ, satisfies d(0)Z0 and d(1)Z1 for all

aR1. This simplifies the problem because the form of

optimal controls for all aR1 is the same, and it is

enough to perform simulations for just one value aR1,

say, aZ1.

We therefore take u(t) to be of the bang-bang form,

given by equation (4.1), and find the optimal switching

time, Ts, for various points in the (um, dm) plane.

Figure 11 presents a two-dimensional density plot of

the quantities Ts/T, the relative switching time, for

various pairs (um, dm). The lighter shades correspond

to smaller values of the relative switching time. The

corresponding diagram for aR1 in the (um, dm)

parameter subspace for the two-step model looks

qualitatively the same and is not presented here.

As dmchanges, the behaviour of the optimal controls

in the aR1 case follows the same trend as in the case

a!1. Namely, for large values of dm, the period of time

1234

5

0.2

0.4

0.6

0.8

1.0

II*, T = 5.52

I*, T = 5.21

II, T = 3.29

I, T = 3.29

Figure 9. A comparison of the optimal controls obtained for

models I, II, I?and II?, with parameters aZ2, sZ10, umZ1,

mZ0.1, aZ0.5. The optimal control is plotted as a function

of time.

Table 2. Optimal controls (of the bang-bang type) for Models

I, II, I?and II?, for parameters aZ2, sZ10, umZ1, mZ0.1 and

aZ1.5.

modelI III?

II?

TS

T

0.75

3.36

0.70

3.37

1.36

5.40

1.10

5.68

0.2

0.4

0.6

0.8

1.0

u(t)

u(t)

dm = 1

dm = 0.1

dm = 0.01

dm = 0.001

dm = 0.0001

1234

0.2

0.4

0.6

0.8

1.0

time (t)

dm = 1

dm = 0.01

dm = 0.1dm = 0.0001

dm = 0.001

(b)

(a)

Figure 10. The optimal controls ? uðtÞ for a!1 and various

values of dm. (a) aZ0.1, the optimal times to target are 4.37,

4.35, 4.28, 4.27 and 4.27 for dZ1; 0.1; 0.01; 0.001 and 0.0001,

respectively. (b) aZ0.9, the optimal times to target are 4.36,

4.31, 4.28, 4.27 and 4.27. The other parameters are aZ2,

uZ0.1, sZ10 and umZ0.01.

Optimal strategy of cancerous cellsN. L. Komarova et al.115

J. R. Soc. Interface (2008)

Page 12

where the maximum value of u is advantageous

becomes shorter. We now present a mathematical

explanation together with a biological interpretation

of the various regions in the parameter space of

figure 11, with aR1.

(i) Small values of um and large values of dm

correspond to Ts=T/1, the regime where

genetic instability is never advantageous, except

in a very short time-interval at the beginning of

the growth (the black cells in figure 11). This is

because the small gain in the mutation rate (um)

is not worth risking the penalty (the large death

rate), and the cells are better off without the

instability. Mathematically, there is one switch-

ing from uZ1 to uZ0 in the function ? uðtÞ, but

this switching happens so early in the growth

that, biologically, the initial, unstable, period of

time (with uZ1) is negligible, and the cells are

characterized by low mutation rates at all times.

(ii) The opposite situation arises when instability is

advantageous and it does not become a liability

for the entire duration of the growth (until the

colony reaches size M, see white cells in

figure 11). This happens when umis large and

dmis small: a small penalty for a large gain in the

mutation rate. In this regime, j1KTs=Tj/1,

that is, for most, or all, of the time the optimal

control is ? uðtÞZ1.

(iii) The grey cells in figure 11 correspond to

intermediate values of Ts, such that Ts/T and

[1KTs/T ] are not too small. In this regime, the

optimal strategies are one-switch bang-bang

controls with an unstable strategy advantageous

at first and then becoming disadvantageous,

prior to reaching the target. The switching

occurs at some intermediate time, neither at

the very start nor close to the end of the growth.

This regime characterizes the middle portion of

the parameter space.

(iv) Finally, we have the white rectangle in figure 11

which corresponds to small values of dmand um.

In such cases, the shape of optimal controls is

bang-bangwithanintermediateswitch,similarto

case (iii) above. However, in this regime, changes

in u(t) affect the time to target, T, only very

slightly. No matter what the shape of u(t) is, the

changes in the mutation rate and the death rate

are bounded by very small umand dm, respect-

ively, and cannot significantly alter the solutions

oftheODEs.Insidethewhiterectangleinfigure11,

the difference between the maximum and the

minimum time to target, T, as a function of Ts, is

less than 1%. In this (biologically irrelevant)

region of the parameter space, genetic instability

is unimportant. The results for this region are

included for mathematical completeness only.

A more quantitative picture for optimal controls in

the aR1 case is presented in figure 12. There, we plot

the optimal time to target, T (thin black lines), and the

corresponding switching time, Ts(thick grey lines), as

functions of um, for different values of dm. The lines

corresponding to T are monotonically increasing,

concave functions of log um; the different lines corre-

spond to different values of dm; the direction in which

dm increases is marked by an arrow. The lines

corresponding to Tshave a one-hump shape; again,

different lines correspond to different dm.

For large values of umand small values of dm, the

lines for T and Tscoincide. In this case, no switching is

observed and instability is advantageous at all times.

As umdecreases, the value Tstends towards zero. This

means that stability throughout the growth is observed.

Intermediate values of um and dm correspond to a

switching sometime in the course of growth, not too

close to tZ0 or tZT.

6. DISCUSSION

6.1. Summary

We have examined optimal strategies for a cancerous

colony with respect to the magnitude of the mutation

rate, as the colony acquires carcinogenic mutations and

enters a phase of a clonal expansion. Biologically, a

large mutation rate corresponds to genetic instability

0

1

log dm

log um

–1

–2

–3

–4

–5

–6

–7

–10 –2–3 –4

Figure 11. The relative switching time, Ts/T, for the optimal

strategy, depending on the parameters umand dm. Black

corresponds to Ts/TZ0 (an immediate switching) and white

to Ts/TZ1 (no switching). The parameters are aZ2, aR1,

sZ10, mZ10K4.

–1 –2–3–4 –5–6 –7

2

4

6

8

10

time

T

Ts

dm increases

dm increases

log um

Figure 12. The optimal time to target, T (black lines), and the

corresponding switching time, Ts(grey lines), as functions of

umfor different values of dm. The values of log dmvary from

K4 to 0, the direction of the increase of dmis indicated. The

other parameters are the same as in figure 11.

116 Optimal strategy of cancerous cellsN. L. Komarova et al.

J. R. Soc. Interface (2008)

Page 13

and a small mutation rate to stability of cancerous cells.

The two types of carcinogenic mutations that we

considered are activation of an oncogene (the one-step

model) or inactivation of a TSG (the two-step model).

In order for a cancer to progress, a cell colony first

has to generate carcinogenic mutants and then to grow.

Genetic instability may expedite the first of these

processes and slow down the second. Therefore, this

process can be examined as an optimization problem.

Genetic instability is ‘blind’, i.e. it does not

necessarily ‘hit’ the exact genes necessary for cancer

progression. It may cause defects in other genes thus

creating deleterious cells. The question is whether the

gain in progression speed due to the increased mutation

rate would outweigh the losses suffered by the cells as a

result of spurious, deadly mutations created by the

instability. In order to model this, we introduce two

parameters: um, the maximum rate of mutations; and

dm, the magnitude of the death rate. Small values of um

mean a small gain in creating cancerous mutations.

Large values of dmmean a large penalty paid by the

colony as a result of many mutation-related deaths.

First we examined in detail a subset of the (um,dm)

parameter space, namely, dmZ1. We found that large

mutation rates at first andlower mutation rates later on

constitute the optimal strategy. The exact shape of the

optimal mutation strategy depends on the concavity of

the function d(u), the instability-dependent death rate.

We distinguish two cases. For non-convex death rates

(§4.1), the best performance is achieved if the mutation

rate jumps (in a discontinuous, abrupt fashion) from

maximum to minimum. For convex death rates (§4.2),

the transition in an optimal control is gradual. In both

cases, having the highest possible mutation rate is

advantageous at first; later on, it pays off to switch to a

lower mutation rate.

We also performed numerical simulations to find

optimal strategies for all biologically reasonable values

of umand dm. We found three qualitatively different

forms of optimal strategies. In one scenario, the

instability makes a minimal contribution to creating

carcinogenic mutations (small um), but significantly

increases the death rate (large dm). Consequently, it

does not pay to be unstableat any stage of the growth in

this case. At the other extreme (large umand small dm),

instability is useful and it ‘comes cheap’; in other words,

the death toll paid by the affected cells is small.

Therefore, the colony is better off being unstable at all

times. Finally, between these two extremes, optimal

controls start at the maximum admissible mutation

rate and then drop to the lowest possible value at some

time during the dynamics. This corresponds to genetic

instability being advantageous at the beginning and

becoming a liability later. This explains the growing

experimental literature suggesting that tumours switch

from genetic instability to stability some time in the

course of cancer progression.

The advantage of our approach is that several results

can be obtained analytically. These first simple models

can be extended in many ways to include more

information about the biological reality. For instance,

the models have a stochastic version. That is, instead

of average quantities, one could use probability

distributions. More details about specific mutations

can be included if one chooses to focus on a particular

case study. Various additional constraints on the

strategies can be imposed, reflecting the dependence

of the mutation rate on other characteristics of the

system, e.g. the system size. Finally, the parameteriza-

tion of the death rate as a function of the instability can

be generalized. However, the general trends found in

the simple models are biologically intuitive and

experimentally supported. With possible non-principal

modifications, they are likely to persist in certain more

sophisticated scenarios.

6.2. Does cancer solve an optimization problem?

Not literally, of course. However, by solving this

problem, one can obtain valuable information about

the growth of cancer. This is similar to the general

philosophy of the evolutionary game theory, as, for

example, in Maynard Smith (1982). In that paper,

different strategies are played against each other to see

which one wins. In principle, one could design an ‘ideal’

strategy which leads to the maximal pay-off in the

game. The game can be a model of something that

happens in nature, for instance the behaviour of

animals in different situations or adaptations of cells

in various environments. The ideal (optimal) strategy

may not even be realistic (there are many constraints in

nature which escape modelling, but can make a strategy

impossible). What occurs in reality, however, tends to

approximate an optimal strategy. Finding the

‘evolutionary stable strategy’ or the ‘Nash equilibrium’

in the system helps us understand the general

experimentally observed trend. A plausible explanation

for the survival of those animals is that they have won

the evolutionary game against other animals that used

an inferior strategy.

In the present paper, we use similar ideas. We solve

the optimization problem for cancerous growth and find

optimal strategies. Does cancer always use an optimal

strategy? Probably not. One obstacle to optimality is

that cancer is unable to adjust its level of instability

instantaneously throughout the entire population to

optimize the growth. However, a cancer which follows

the general trend, i.e. a strategy close to an optimal one,

will grow faster. These are the colonies that ‘succeed’

causing a disease. Other pre-cancerous colonies of cells

might exist in any organ of an organism, but if they do

not use a strategy sufficiently close to optimal, they do

not succeed with further growth and are not observed.

6.3. Implications for somatic evolution of cancer

In this paper, we formulate the problem of cancer

growth from the point of view of cancerous cells, in

order to find the most optimal mutation strategy. Here

we discuss our model in the context of somatic

evolution of cells.

Selection in this problem takes place on two levels.

One level is the level of individual cells, where normal

cells are competing for space and nutrients (this is one

of the mechanisms of homeostatic control), and

cancerous cells escape this control to enter a phase of

Optimal strategy of cancerous cellsN. L. Komarova et al.117

J. R. Soc. Interface (2008)

Page 14

exponential growth. The forces of selection are mani-

fested in the nonlinearities of the basic ODEs. The

second level of selection is the level of cell colonies.

Different colonies are characterized by different func-

tions u(t). They are not assumed to be in direct

interaction with one another. The competition in this

case manifests itself in whether agiven colony will reach

a cancerous state quickly and become observable. The

mathematical problem we solve is to find the optimal

strategy u(t), such that the colony with this strategy

will be the first one to ‘make it’.

Several other papers have proposed evolutionary

models of genetic instability. An important component

of many such models is costs and benefits of cell repair,

(Breivik & Gaudernack 1999, 2004; Breivik 2001;

Komarova & Wodarz 2003; Breivik 2005). It is argued

that an evolutionary reason for instability is the fact

that cell repair is costly, and under some circumstances

it may pay off to avoid repair, which leads to genomic

instability. In this paper, we do not focus on the

detailed analysis of costs and benefits of cell repair;

instead, we concentrate on their effect on the choice of

the most advantageous strategy with respect to the rate

of chromosome loss. However, the model includes the

costs of repair in the following indirect way. Cells with

efficient repair have a disadvantage because repair is

costly (having to enter cell cycle arrest in order to

repair the genome decreases the growth rate of the

population). Cells with inefficient repair have a

disadvantage of creating deleterious mutations. The

difference between the two, that is, the relative

disadvantage of instability, is captured in the quantity

d(u). The function d(u) reduces the fitness of unstable

cells with respect to stable cells. We assume that this

quantity is positive, due to the large damage inflicted

by losses of ‘wrong’ chromosomes.

6.4. Suggestions for experiments

The theory developed here would greatly benefit from

further experiments that would aim at quantification of

the processes of tumour growth and mutation. We

propose two types of experiments.

The rate of instability as a function of the tumour

stage. Despite many reports that the degree of genetic

instabilitygoesdownasthetumourprogresses(Rudolph

et al. 2001; Chin et al. 2004), a quantification of this

phenomenon is still lacking. The rate of chromosomal

loss has been measured by Lengauer et al. (1997). In a

similar way, a thorough study of the ‘natural history’ of

genetic instability can be performed. A systematic study

of cells harvested at different stages of tumorigenesis

would yield a curve, where the mutation rate is a

function of the tumour age. This study can be done in a

controlled manner with animal models.

The death rate of cells as a function of the mutation

rate. In our mathematical model, we postulate that the

death rate of cells is a function of the cells’ mutation

rate. In the presence of instability, we assume that the

death rate is elevated; this can be quantified experimen-

tally. In the last several years, much work has been

done to understand the role instability plays in

tumours, in particular, in the way it affects the death

of cells (Lowe et al. 2004; Bartkova et al. 2005; Chen

et al. 2005; Gorgoulis et al. 2005; Deng 2006). However,

a detailed measurement of the death rate as a function

of a mutation rate has not been performed. One way to

do this is as follows, similar in spirit to previous work

(Marder & Morgan 1993; Nagar et al. 2003; Smith et al.

2003) on cells treated by radiation. As the amount of

radiation increases, the mutation rate increases and so

does the cell death. The dependence of the cell death on

the mutation rate could serve to improve our model of

the function d(u), and help us reason about optimal

strategies of tumours. Another way to quantify the

dependence of death on the degree of genetic instability

is to measure the death rate together with the degree of

instability in a series of experiments with cells

harvested at different stages of tumour growth.

N.L.K.’s research was supported by the Sloan Fellowship, and

NIH grants 1R01AI058153-01A2, 1R01 CA118545-01A1 and

R01GM075309. A.S.’s research was supported by the NIH

National Research Service Award 5 T15 LM007443 from the

National Library of Medicine, awarded to P. Baldi. The

research of F.Y.M.W. was supported by NIH grants

R01GM067247, R01GM075309 and UC Irvine research

grant 445861 (the two NIH R01 grants were awarded through

the Joint NSF/NIGMS Initiative to Support Research in the

Area of Mathematical Biology). We thank P. Gill for help

with obtaining and using SNOPT v. 7.0, M. Fagan for help with

obtaining ADIFOR v. 2.0 and O. Von Stryk for help

with obtaining DIRCOL v. 2.1. Thanks are also due to

A. Vladimirsky for a helpful discussion of computational

approaches to finding time-optimal controls.

APPENDIX A. THE TWO-STEP PROBLEM: THE

MATHEMATICAL FORMALISM

A.1. Statement of the problem

A.1.1. Equations of state. Let x0(t) be the population

size of normal cells and x1(t) be the population size of

harmless mutants with only one functioning copy of its

TSG, both normalized by the initial normal cell

population size N. Let x2(t) be the population size of

dangerous TSG mutant cells normalized by its final

target population size M. With pminZ0, the time

evolution of the three cell populations is governed by

the following three first-order ODEs:

x0

0ZK 2mx0Cx0ð1KdðuÞÞð1Kx0Kx1Þ

hg0ðx0;x1;x2;uÞ;

ðA 1Þ

x0

1Z2mx0KðmCumuÞx1Cx1ð1KdðuÞÞð1Kx0Kx1Þ

hg1ðx0;x1;x2;uÞ;

x0

sðmCumuÞx1Cx2ð1KdðuÞÞðaKx0Kx1Þ

ðA 2Þ

2Z1

hg2ðx0;x1;x2;uÞ;

again with ðÞ0ZdðÞ=dt;sZM=N[1;aO1;0!um%1,

0%m/1; and the death rate (2.9) where the non-nega-

tive normalized gross chromosomal change rate u

is limited by the inequality constraint (3.4). System

(A 1)–(A 3)correspondstosystem(2.3),(2.4)and(2.8)of

§2.2 of the main text.

ðA 3Þ

118Optimal strategy of cancerous cells N. L. Komarova et al.

J. R. Soc. Interface (2008)

Page 15

A.1.2. Boundary conditions. The three equations of

state (A 2)–(A 4) are subject to the following four

auxiliary conditions:

x0ð0ÞZ1;x1ð0ÞZ0;x2ð0ÞZ0;x2ðTÞZ1;

ðA4Þ

where T is the time when the dangerous mutant cell

population reaches the target size.

A.1.3. Problem. Choose the control function u(t) to

minimize the time T needed to reach the target

population size of the dangerous mutants subject to

the inequality constraint (3.4), on the control u(t) and

the non-negative constraints

xkR0;

on the cell populations. The problem can be restated as

choosing the control function u(t) to minimize the

performance index J defined in (3.6) subject to the

equations of state (A 1)–(A 3), the boundary conditions

(A 4), the inequality constraints (3.4) and (A 5) with

TO0 as a part of the solution.

k Z0;1;2;

ðA 5Þ

A.2. The Hamiltonian and adjoint variables

The Hamiltonian for the two-step problem is

H Z1Cl0g0Cl1g1Cl2g2;

where lkare the three adjoint variables for the problem

described by three adjoint ODEs,

vg0

vxk

vxk

ðA 6Þ

l0

kZK l0

Cl1

vg1

Cl2

vg2

vxk

??

;

ðk Z0;1;2Þ;

ðA 7Þ

and (for the given auxiliary conditions on the state

variable (xk)) two transversality conditions

l0ðTÞ Zl1ðTÞ Z0:

For the optimal solution of our minimum terminal time

problem, it is customary to investigate first the

possibility of an interior solution, by seeking

(i) a control uZuint(t) to satisfy the optimality

condition (3.14),

(ii) six quantities fxiðtÞ;ljðtÞg to satisfy the six

differential equations (A 1)–(A 3) and (A 7) and

six auxiliary conditions in (A 4) and (A 8), and

(iii) the terminal timeT to satisfy a free end condition

whichagainbecomes(3.11)aftersimplificationby

the adjoint boundary condition (A 8).

ðA 8Þ

For the state equations (A 1)–(A 3), the three

adjoint differential equations (A 7) take the form

l0

0ZKl0fð1K2x0Kx1Þð1KdÞK2mg

Kl1f2mKx1ð1KdÞgCl2x2ð1KdÞ;

1Zl0ð1KdÞx0Kl1fð1Kx0K2x1Þð1KdÞ

KðumuCmÞgKl2

ðA 9Þ

l0

sfðumuCmÞKsx2ð1KdÞg;

ðA 10Þ

ðA 11Þ

l0

2ZKl2ðaKx0Kx1Þð1KdÞ:

Consider the interior control u(t) determined by the

optimality condition (3.15), vH=vuZ0, with

vH

vuZumx1

s

ðl2Ksl1ÞKfðl0x0Cl1x1Þð1Kx0Kx1Þ

Cl2x2ðaKx0Kx1Þgd$;

where d$Zað1KuÞaK 1for the death rate (2.9). We have

from (3.14)

ðA 12Þ

d$Zað1KuÞaK1

Z

x1fl2Kl1sg

smfðl0x0Cl1x1Þð1Kx0Kx1ÞCl2x2ðaKx0Kx1Þg:

ðA13Þ

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