Insights & Perspectives
Hypotheses

Characterization of stem cells andcancer cells on the basis of gene
expression profile stability, plasticity,and robustness
Dynamical systems theory of gene expressions under cell-cell interaction explains
mutational robustness of differentiated cells and suggests how cancer cells emerge
Kunihiko Kaneko
Here I present and discuss a model that, among other things, appears able
to describe the dynamics of cancer cell origin from the perspective of stableand unstable gene expression profiles. In identifying such aberrant gene
expression profiles as lying outside the normal stable states attracted
through development and normal cell differentiation, the hypothesis explains
why cancer cells accumulate mutations, to which they are not robust, andwhy these mutations create a new stable state far from the normal gene
expression profile space. Such cells are in strong contrast with normal cell
types that appeared as an attractor state in the gene expression dynamicalsystem under cell-cell interaction and achieved robustness to noise
through evolution, which in turn also conferred robustness to mutation. In
complex gene regulation networks, other aberrant cellular states lacking
such high robustness are expected to remain, which would correspond tocancer cells.

.

Keywords:
attractor; cell-cell interaction; evolution; robustness; stem cell

DOI 10.1002/bies.201000153
Department of Basic Science and ResearchCenter for Complex Systems Biology, Universityof Tokyo, Komaba, Meguro, Tokyo, Japan*Corresponding author:
Kunihiko Kaneko
E-mail: kaneko@complex.c.u-tokyo.ac.jp
Abbreviation:
GRN, gene regulation network.

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Introduction
Robustness and plasticity are essential
features of all biological systems [1–8].
Robustness describes the insensitivity ofa system to external perturbations, andplasticity describes the changeability ofa system in response to changes in external conditions. In this sense, robustness
and plasticity tend to compete with eachother if they work at the same dimension.
Biological systems, however, somehow
strike a balance between the two, or
adopt each of the two separately depending on the situation.
Sources of external perturbations
have distinct time scales. On a fastertime scale, external perturbations ofnoise occur in developmental processes.
On a much slower time scale, changes ingenetic sequence may perturb developmental dynamics. Biological organisms
somehow achieve robustness at thesetwo different scales, developmental
and evolutionary [9–12].
In multicellular organisms, thedevelopmental process typically has anintermediate timescale – intracellulargene expression dynamics and a multicellular process involving the increase incell number [12–14]. Cells differentiate
into several types, while an organism
or tissue is shaped over a time scalewww.bioessays-journal.com

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K. Kaneko

based on the increase in the cell number, which is much slower. Multicellularorganisms achieve robustness to external perturbations for both the scales,that is, cell-type-specific stability ingene expression and stability against achange in the number distribution ofcells. Also, the path taken to reach thefinal cellular state is fairly stable.
Furthermore, in multicellular organisms, a multipotent cell generally existsthat has the potentiality to differentiate
into many cell types, depending on
conditions. In this sense, such cells
have plasticity, while robustness ofan organism at a global scale isachieved. Understanding multipotency
and robustness in developmental processes remains one of the basic problems
in developmental biology.
In this essay, I discuss the issue ofrobustness and plasticity in terms ofdynamical systems theory [15, 16].
Concepts in dynamical systems are central to understanding this mathematicalformalization of cell behavior, so that I
give a brief intuitive account on these:
The concept of an attractor:
Waddington put forward an image ofdevelopment as a ball rolling downalong a landscape with peaks and valleys [1]. The ball will finally come to restat the bottom of a valley, a local minimum of the particular landscape. If
there are many valleys and peaks, thefinal position may depend on the starting point of the ball. In the theory ofdynamical systems, the final destinationof the ball is the ‘attractor’, and each of
the domains of the valley separated bypeaks is known as the ‘‘basin’’ of therespective attractor.
Now imagine some perturbations
kick the ball. If the valley is deep, thefinal position of the ball is still at thesame bottom of the valley. Hence theattractor concept is important to understand the robustness of a system.
Essentially this attractor concept
can be applied to gene expression levels
resulting from a network of interacting
proteins – in this case, gene/protein
expression levels being the ball. With
time, the expression levels result in an
equilibrium between the components,
which corresponds to the bottom of
the valley in the afore-mentioned landscape. Perturbation on the ‘‘ball’’ – i.e.
changes in one or more protein levels,
caused by environmental change or

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noise – may shift the equilibrium to a
new one, corresponding to the ballbeing jostled out of one valley, and finding its way to a nearby depression in thelandscape.
Sometimes, the nature of interactions
between proteins can generate a bistablesystem where two very different equilibria – or attractors – exist, for example inthe case of a two-protein network where
protein A inhibits the synthesis of protein
B and protein B likewise inhibits the synthesis of protein A.
So far, I have talked about simple,
static, attractors. There are also attractors in which the ball (i.e. the expression
level) keeps changing position withtime. These correspond, for example,
to an oscillating state of protein levels,
such as that manifested by the following
simple network: protein A activatestranscription of protein B and itself,
and protein B inhibits transcription ofprotein A. Under appropriate setting
and parameter values, the very oscillatory behavior of the system is a stablestate, and hence can be considered as an
attractor. If more proteins are involved,
the oscillation may be complex, without
periodic repetition.
Flow in state space: Mathematically,
the above picture is represented by aflow in the state space. For simple illustration, consider a system consisting of
two proteins X and Y. Then the ‘state’ ofthis two-protein system is plotted as apoint in a two-dimensional space, withx and y axes as the expression levels(concentration) of X and Y, respectively.
In general, each protein influences thesynthesis or degradation of the otherprotein. Hence, depending on concentrations of the two proteins, theirincrease or decrease in time is determined (by the gene network, i.e. by
the rule of chemical reactions). In otherwords, ‘flow’ arises in the two-dimensional x–y plane, represented by arrows
in the plane. The expression levels (concentrations) change in time along thisflow, and this trail of points in the plane
is a trajectory. Now, with time they may
settle on a point where flow vanishes,
i.e. the reactions are still occurring, butthey balance each other out, and there isno net directionality.
This point corresponds to the bottom of the valley, where the flow to rolldown the ball vanishes. To compare thelandscape picture, ‘height’ is assigned.....

at each point in this state space, and theflow is assigned to the direction of
decreasing height. However, the ‘flow’
picture in the ‘state space’ is more general than the landscape, and it can cover
the case that the concentrations do notsettle to fixed values, but keep on oscillating. For example, if x and y settle tosimple sinusoidal oscillation, given byx ðtÞ ¼ r cosðtÞ þ x0 ; y ðtÞ ¼ r sinðtÞ þ y0 ,the trajectory forms a circle, with aradius r, centered at (x0 ; y0 ). In this casethe attractor is a circle, whereas in general the attractor corresponding to oscillatory state is a closed curve in the state
space (Fig. 1).
This picture is straightforwardly
extended to the case in which more genes(proteins) are involved. If three are
involved, the state space is three dimensional, and if k proteins are involved, thestate space is k-dimensional (see Fig. 1,
and Box 1).
In general, there is deviation in theconcentration change from the original
flow by gene regulation dynamics, since
all the chemical reactions are due torandom collisions of molecules. Thisrandom perturbation is called ‘‘noise’’(see Box 1).
Now, consider the effect of perturbation by noise on the state variables(concentrations) at an attractor. Oncethe cell state is on an attractor, its statevariables return to the original values by
the dynamics even if perturbed by noise,
as long as the state stays within thebasin of attraction. This means thatthe protein concentrations stay at a certain range of values, determined by an
attractor, even under the noise. Hence,
the attractor concept is essential tounderstanding robustness against noise[12, 16–18, 23].
Still, the notion of attractors does
not solve all the questions related torobustness. First, whether the cell statereaches an attractor globally from various initial values depends on the size ofbasin. During the dynamics to reach an
attractor, the trajectory may be perturbed by noise and the state may notreach the given attractor. Hence, therobustness depends on gene expression
dynamics. How are gene-expression
dynamical systems evolved to increase
robustness?
Second, stem cells in multicellularorganisms can both proliferate and differentiate. For proliferation, the cell state

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K. Kaneko

Box 1

Terminology in Dynamical Systems applied tocell behavior (see Fig. 1)
Consider a cell state that is specified in terms of theconcentrations of its components. For example, if there
are k proteins, the cell state is described by the concentrations of those proteins, i.e. (p1 ðtÞ; p2 ðtÞ; p3 ðtÞ; . . . ; pk ðtÞ)which may change through time, t. These are statevariables, whose number, k, is the degree of freedom of
the system. These state variables influence each other,and change across time. This gives rise to a flow in thestate space. Accordingly, the time course of the change
in state variables is represented as a trajectory in thestate space. For example, expression of a particular geneis either activated or repressed by expression of othergenes, and the concentrations of the resulting proteinschange in time in accordance with the gene regulatory
network.

Terminology in Developmental Biology usedin the present text
Noise (in gene expression dynamics)
On average, change in concentration of proteins is given byconcentrations of chemicals that constitute the reaction.
However, the reaction is due to molecular collisions, andthere are deviations from the average. This fluctuation isregarded as a noise term added to the dynamical system.
Note that this particular use of the term ‘noise’ is adopted inphysics and also in recent literature in systems biology, but
in evolutionary biology it is used in a different context.
Robustness to noise
Insensitivity of the phenotype (state variables of the attractor) to perturbation by noise during the (developmental)
dynamics.
Robustness to mutation (mutational robustness)

Figure 1. Schematic description of dynamical systems corresponding to the terms in Box 1.

should be stable, while for differentiation
it should be plastic. How are robustness
and plasticity compatible in stem cells?
Here I discuss these issues from a
dynamical-systems perspective, basedon my group’s earlier numerical simulations of cell differentiation and evolution of cells over a decade. Following
the discussion, we propose a hypothesis
on cancer state as an attractor that is not
visited in the normal course of development and lacks mutational robustness.

Insensitivity of the phenotype (state variables of the attractor) to the change of the rule of (developmental) dynamics
introduced by genetic change.

Robustness to noise arises
naturally when evolution ofgene networks is modeled
on a background ofmoderate noise
Let us begin the discussion on robustness at the single-cell level. As surveyed
in the introduction, I represent the phenotype of a cell by state variables thatare a set of protein concentrations. The

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temporal change in cell state is determined by complex gene expression
dynamics governed by a GRN. These
dynamics are subjected to noise due
to molecular fluctuations in chemical
reactions [24–28]. Under noise, as
shown in Fig. 2A, the orbit in the state
space, i.e. the time course of protein
concentrations, required to reach therelevant phenotype might be easily
diverted to phenotypes that lose thedesired function.

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Terminology in dynamical systems withapplications to biological systems

With time, the state variables reach, and stay at, a certain
range of values. This region of state space in which the statevariables can occupy after sufficient time is called an
attractor. The attractors can be stationary states (fixed
points), periodic oscillation (closed curve in the state
space), aperiodic irregular oscillation (chaotic behavior).
There can be several attractors that are reached depending on the initial values of the variables. The region of suchinitial values that from which the system will ‘‘gravitate’’towards each attractor is called its basin of attraction. The
boundary of the basin divides the state space, with regards towhich attractor the flow of state variables reaches.
Consider N cells, each of which has k state variables thatinteract with each other. Then the flow (temporal change) of
k variables of each cell is modified following the states ofother cells. With this cell-cell interaction, the state variables
may be kicked away from the original attractor, while novel
attracting states that did not exist in the single-cell dynamics may be generated (stability induced by interaction).

Insights & Perspectives.....

Hypotheses

K. Kaneko

Figure 2. Schematic representation of dynamics of a cellular state. A–C: Non-robust case.
D–F: Robust case evolved under a sufficient noise level in gene expression dynamics. A,D:
Flow structure of the cellular state. B,E: Illustration of gene expression dynamics as the falling
motion of a ball in a landscape. C,F: Robustness against mutational change.

The author studied gene expression
dynamics with randomly chosen regulation networks that control activation
and repression among proteins, andfound that most dynamics have such
weakness against perturbations [11,12]. Even if a network happened to resultin the phenotype concerned, only a very
narrow path existed to reach the functional phenotype (‘‘x’’ in Fig. 2A and B).
GRNs were then evolved numerically byusing a genetic algorithm to increase afitness function [11, 29–31] that was
defined to achieve functional phenotypes, i.e. to generate a given gene
expression pattern. When gene expression dynamics were subjected to a sufficient level of noise, it could not
generate the functional phenotype stably if the dynamics were of the typegiven in Fig. 2A and B. Rather, the orbitof the dynamics would be kicked awayfrom the original trajectory, and thegenerated phenotype would be different
from it, as shown as a, b, c,... in Fig. 2A
and B. However, when the network
evolved, they were found to attain406

robustness to noise, as shown in
Fig. 2D and E. The orbit required togenerate the requested phenotype was
not easily perturbed by noise, and evenfrom perturbed gene expression states,
the orbits as well as the final expression
pattern converged to the original phenotype. Hence, robustness to noise wasevolved naturally (see also [32, 33]).

Robustness to noise
results in robustness tomutation in gene
expression patterns
Let us represent the attraction to thefinal gene expression pattern as the
motion of a ball in a landscape. Fromnumerical simulations, it was foundthat gene expression dynamics frommost GRNs chosen randomly was
represented by a rugged landscape as
in Fig. 2B, whereas, after evolution
under a sufficiently high noise level, it
exhibited a rather smooth landscapewith global attraction to the phenotype,
as in Fig. 2E. In contrast, when the gene
expression dynamics evolved under alower noise level (or without noise),
the GRN remained in the form ofFig. 2A and B.
Now, consider mutation to the GRN.
(Here I use the term mutation rather
naively, to include any genetic change
to modify the GRN.) Now there are two
distinct concepts on robustness, that tonoise and to mutation (genetic change).
The former concerns the stability of acellular state against noise in gene
expression dynamics, for cells sharing
the same genes, while the latter concerns
the stability against genetic change thatslightly alters dynamical systems itself,
by modification of GRN (see Box 1).
From extensive simulations, it was
found that robustness of cellular state to
mutation differs distinctly between theabove two types of dynamics given in
Fig. 2A and D [29–31]: the GRN withoutrobustness to noise (Fig. 2A and B) also
lacked robustness to mutation (Fig. 2C).
Genetic changes perturbed GRNs andthe flow in the state space would be
perturbed. Then, the generated gene
expression pattern no longer gave riseto the requested phenotype. In dynamics with a rugged landscape as in Fig. 2A
and B, we can naturally expect that a

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for differentiation from stem cells[37, 38]. During the developmental process, differentiation occurs from stem
cells to several cell types. All the cells
converge to one of these types, while
stem cells can remain, and have potentiality both for proliferation and differentiation [39, 40].

Waddington’s epigenetic
landscape accounts forthis channelingTo understand robustness in these cell
types as well as over the temporal
course of development, Waddington
adopted the landscape picture as a
metaphorical picture for developmental
processes [1]. We could interpret thispicture as follows (Fig. 3). At each time
point there are several valleys, whichcorrespond to each cell type; in termsof dynamical systems as applied to gene
expression, each is regarded as an
attractor, as discussed in the last section(Fig. 2). In fact, several attempts havebeen made to represent such a landscape from the gene expression dynamics [41] and experimental data [42].
Note, to support several cell types, gene
expression dynamics must have several

attractors that correspond to several valleys, as also demonstrated in several
GRN models [18, 23].

However, the landscape mustchange over time to explain
fundamental features of tissues
If the landscape is temporally fixed,
however, we cannot answer the following three basic questions. First, why aresome cells located in a basin of one
valley, and others in a different valley?
How are such initial conditions for eachcell type determined? Second, the number distribution of cells at each valley
should not be arbitrary, but rather kept
within some range, for a tissue or amulticellular organism to function normally. Indeed, the organism or tissue
consists of several cell types whose
number ratios are determined to somedegree. How is such proportional regulation achieved? Third, how is the
temporal course of differentiation, i.e.
at what stage each cell type appears,
determined in a robust manner?
Indeed, in the landscape drawn by
Waddington, the hills and valleys are
not fixed, but also develop over time.
Hence, two courses of events exist. OverMultipotency anddifferentiation can be
explained by modeling gene
expression dynamics andcell-cell contacts together
Cell-cell interactions channel
differentiation towards stable types
In a multicellular system, developmental dynamics are more complex; they
involve both intracellular gene expression dynamics and cell-cell interaction.
The latter influences the internal
expression dynamics and depends on
the surrounding cells, whose number
increases over development. The
importance of cell-cell interaction was
experimentally demonstrated as thecommunity effect [36], and discussed

Figure 3. Epigenetic landscape shaped with the increase in cell number and cell-cell interaction. Schematic drawing. The springs between cells at different valleys are schematic
representations of cell-cell interaction. (In the present biology, epigenetics often refer tohistone modification and DNA methylation. The term epigenetics, however, was originallycoined by Waddington, as a developmental process by which interaction among genes aswell as with the environment produce a phenotype.)

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slight change easily destroys the attraction to the original valley. Hence, deleterious mutations would often appear.
In contrast, it was found that the
GRN evolved under a sufficiently highnoise level (Fig. 2D and E) had robustness to mutation (Fig. 2F). The attraction
to the original path in Fig. 2D was sostrong that most perturbations in flowcaused by mutations in GRNs still gave
rise to the original phenotype. Hence,
most mutations were neutral. In asmooth landscape with a global basinof attraction, changes in dynamics
caused by mutations are expected not
to damage the attraction to the original
gene expression pattern, which explains
the robustness to mutation.
Different robustness to mutation
between the above two cases werenumerically confirmed [29–31]. In the
GRN evolved under noise, orbits fromdifferent initial values took a similarpath, as postulated by Waddington as
homeorhesis [1]. Evolution of robustness
to noise and mutation was also confirmed in catalytic reaction networks[34] and a toy protein-folding model[35]. Such converging orbits were also
observed in the gene expression dynamics of yeast [33].
To summarize, evolution to achieverobustness to noise was found to introduce robustness to mutation. Hence,cells under selection pressure for somefunction are expected to achieve these
two types of robustness.

K. KanekoHypotheses

K. Kanekoa faster scale, cells are attracted into
each valley, while the landscape itself
grows over a slower time scale of development. This change in the landscapeoccurs through the increase in cell number. With this increase, the gene expression dynamics of each cell can be
modified by the changes in cell-cellinteractions. This change will introduce
the change of the landscape (Fig. 3).

Insights & Perspectives

Modeling gene expression
dynamics in concert with cellcell interactions recapitulates
stem cell and differentiated cell
identities
Based on the above picture, Furusawa
and the author studied a class of models[14, 43–46], in which a cell state wasrepresented by a set of concentrations ofBox 2
A bit more on intra-inter-cellular
dynamics theory for cell differentiations(a) Differentiation of phases of oscillations
Furusawa and the author chose a gene regulation network(GRN) that showed temporal oscillation in concentrationsof some proteins at the single-cell level (Fig. 4A). With theincrease in cell number by divisions, the degree of cell-cellinteraction increases, which causes these oscillations todesynchronize among cells. This is because only somechemical species diffuse, and homogenization of concentrations works only for some chemicals. With intracellular
feedback reactions [13, 16–19], cell-to-cell difference in theconcentration for other non-diffusible chemicals is not
reduced but is amplified. Thus, the phase of oscillations
differs across cells. This difference in phases, under the
influence of the cell-cell interactions, leads to irregularity
in oscillation, which causes differences in protein concentrations between cells to be amplified.

(b) Differentiation in composition of proteins
This amplification reaches a stage at which the composition of proteins is distinct between cells, leading to differentiation to a new cell type (Fig. 4B). Some proteins indifferentiated cells have much lower or higher concentrations than in the original cell type. The cell state leaves
the basin of attraction of the original cell type: a gate opens,allowing a cell to exit from the original valley. Here, theinstability for differentiation is not directly coupled withthe cell division event, but occurs through the increase incell number.

(c) Stabilization of a differentiated state bycell-cell interaction
The new stationary cell state is stabilized by the interaction
with the original cell type. In other words, a new valley isgenerated. In this differentiated cell type, the concentration
oscillation that exists in the original cell type is lost or
weakened (Fig. 4B).
These dynamics of differentiation can be understood as
follows. The trajectory of the original cell state is modulated

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proteins (gene expression levels). The
model consisted of (i) intracellular reaction (gene expression) dynamics; (ii)cell-cell interactions mediated by the
diffusion (or transport) of some chemicals; and (iii) increase in cell number
over time as a result of cell division.
Protein concentrations changed due tointracellular reactions and were also
influenced by the cell-cell interaction,from the single-cell attractor by the cell-cell interaction, sothat it goes across the basin boundary of the original attractor. Then, the state of such cells moves to a new state,
implying cell differentiation into another type. The stabilityof the differentiated type depends on the cell-cell interaction,
and accordingly, on the number ratio of the two cell types.

(d) Regulation between proliferation anddifferentiation leading to macroscopic
robustness
In the above it is found that stem cells in the model lose
their stability to sustain their original state, due to the cellcell interaction. However, the increase in the number ofdifferentiated cells changes the interaction, and theoriginal cell state restores stability; thus, the attractor
and its basin boundary no longer touch (Fig. 4C). With
the proliferation of the original cell type, its attractor andits basin boundary touch again, and the differentiation
occurs. In this way, stem cells in the model keep on either
proliferating or differentiating, depending on the number
ratio of the cell types. If the ratio of stem cell is too large(small), the ratio of differentiation (proliferation) increases,
so that the number distribution of two cell types remains
within a certain range, implying (macroscopic) robustnessof the number ratio of cells of different types (see [20] for ademonstration of such regulation of cell number and [21]for possible relationship with the present theory. Also see[22] for a bifurcation analysis on the stabilization of a newcell type by cell-cell interaction.).

(e) Generality and limitations
Thus, characterization of stem cells, as well as the loss ofpotentiality of further differentiation as a result of celldifferentiation, is represented in terms of dynamical systems of interacting cells. As long as such gene expression
dynamics allows for oscillation between on and off states ofgene expression, this course of differentiation appears
universally.
Of course, there will be limitations to the theory here.
When isolated and put into the original culture, differentiated cells in the model investigated here often revert
to the original stem cell state. The differentiation does notreach the stage of rigid fixation, for which molecular processes such as DNA methylation might be necessary.

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Insights & Perspectives(1) Let us first assume that concentrations of some proteins at thesingle-cell level showed temporaloscillation (Fig. 4A). As the numberof cells increased, the influence ofcell-cell interaction increased, andaccordingly, these oscillations lostsynchrony among cells. With intracellular feedback reactions [13, 16–19], cell-to-cell difference in the concentration was amplified, until some
proteins were highly expressed insome cells, whose state deviatedfrom the original attractor (Fig. 4B).

(2) With the interaction with theoriginal cell type, the trajectory ofsuch deviated cells went beyond thebasin of attraction to the original
attractor and was attracted to a
new stationary state. Differentiation into a new cell type occurred,in which the concentration oscillation was lost (Fig. 4B).
(3) Under the presence of (a sufficientnumber of) differentiated cells, theflow of the cell state was shifted
towards the original attractor, due
to the interaction with them. The
original state recovered stability(Fig. 4C), leading to its proliferation.
In this way, this cell type, the stem
cell in the model, differentiated if its
number ratio was large, and proliferated if it was small. The two cellstates remained stable as attractors
only if the ratios of two cell types fellwithin a limited range. This gives apossible explanation of the macroscopic robustness of multicellular
systems.

The model explains decreasing
variation as differentiation
proceeds
For complex GRNs, hierarchical differentiation progressed as in Fig. 5. Here,the initial cell type had potentiality
to produce any other type including
itself, while each type at a lower hierarchy produced only itself and thetypes downward. Thus, multipotency
decreased. This decrease in multipotency was characterized by:

Figure 4. Schematic representation of the cellular differentiation process from multipotent
stem cells, drawn by gene expression dynamics in the state space of protein concentrations.
A: Oscillatory dynamics of a single stem cell, and illustration of the protein concentration ofsuch a cell. Arrows represent the flow in the state space. B: With the increase in cell number,the orbit, the temporal course of protein concentrations, touches the boundary of its basin of
attraction, so that some cells go out of the original attractor. Note that there are many directions in the state space. The original attractor remains attracted from most directions and theorbit goes out only through a limited direction at a specific timing. After escape from theoriginal attractor, such cells are attracted to a different state in which the protein concentration is no longer oscillatory. C: Whether the attractor of the original cell is contacted withits basin boundary changes depending on the number ratio between the two cell types.

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(i) Oscillations of protein concentrations decreasing as cells become
more differentiated: the multipotent
cells in the model presented here
demonstrated temporal oscillationin the concentration of some
proteins. This oscillation was not
regular, and the cellular state itinerated over a few quasi-stablestates. As differentiation progressed, this temporal variation
decreased, so that the range ofstates that the cell could visit
decreased with the developmental
course (Fig. 5).
(ii) Cell-cell variation decreasing fromstem to differentiated cell: this
temporal variation was rather slow,

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which changed with the increase in cellnumber. Furusawa and the author carried out extensive simulations of thesemodels and confirmed the following picture (see Box 2 for details):

K. Kaneko

K. Kaneko

Insights & Perspectives.....

Hypotheses

Unfortunately, from the above theoretical argument, we cannot predict
which genes should be activated torestore pluripotency. Recalling, however, that oscillatory gene expression,
generated as a result of feedbacks ingene regulation, is required for multipotency in our theoretical framework, it
is expected that genes responsible forsuch oscillation should be activated torestore plasticity.

Evolutionary robustness ofa multicellular system
Figure 5. Differentiation in the normal developmental process and the existence of otherattractors corresponding to cancer state. The normal developmental course starts from thestem-cell attractor, and progresses with the increase in the cell number and the following
cell-cell interaction. Besides these normal attractors, a cancer attractor(s) exists that isreached when a large perturbation is applied to deviate from the normal developmental
course. The stability of a cancer attractor could be increased by the genetic change.
Schematic drawing in the state space.

so that at each snapshot in time,
concentrations of some proteinsdiffered distinctively across cells. By
comparing cells at a given time, thecell-cell variation was large for multipotent stem cells and decreased as
differentiation progressed.
Point (i) refers to the temporal variationof a single cell. Indeed, recent singlecell measurements of Hes1 protein
concentrations using an imaging technique demonstrated the existence ofoscillatory gene expression in embryonic stem cells within a period of afew hours [47]. This oscillation disappeared in differentiated cells [48].
Also, measurement of stem cell marker
Sca-1 revealed the existence of slow
transitory dynamics over metastablestates for hematopoietic progenitor cells[49]. Point (ii), on the other hand, isconcerned with the cell-to-cell variation.
Heterogeneity in expressions of Stella,
Nanog, and Hes1 expressions wasrecently observed in embryonic stem
cells [50, 51]. To further examine thispoint, measurement of the variance ofgene expressions over cells is important(as has been extensively carried out inbacteria and yeast [52, 53]). Decrease incell-cell variations from embryonic stem
cells to committed cells must be quantitatively confirmed.

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Why differentiated cells canrecover multipotency by changing
expression of a few genes
In the simulations, the range of expression levels that the cellular state varied
was larger for multipotent cells than forcommitted cells (see Fig. 3). In the stem
cells in the model presented here,
expressions of many genes mutually
regulated each other. This complex
regulation led to oscillatory dynamics
with transitions over metastable states.
In committed cell types, expressions
were biased to fewer genes, which gave
rise to stationary states. Thus, it is naturally expected that differentiated cells
could recover the multipotency by perturbing expressions of a few genes, say
by activating some genes whose expression is suppressed.
In fact, successful regaining of multipotency in animal cells was firstachieved by nuclear transplant [54].
Furthermore,
inducedpluripotent
stem cells recently demonstrated that
pluripotency is restored by activatinga few genes from differentiated skincells, in agreement with the above discussion [55]. (There could be other ways
to perturb gene expressions. Even byremoving some gene, such reprogramming might be possible, as reported
in [56].)

So far I have reviewed the studies on
the evolution of robustness and celldifferentiation. Let us now discussrobustness of the multicellular systemthrough evolution. As the gene expression in a cell is noisy, the discussion
concerning robustness can also be
applied here. First, the initial stem
cell type will be stable at the single-celllevel and remain so under a certain
degree of cell-cell interaction introduced by the increase in cell number.
The differentiated cell type must also
be stable against perturbations, tomaintain proliferation. These cellstates must have achieved robustness
against noise in gene expression
dynamics, if they are necessary for survival of the multicellular organism, and
thus
under
selectionpressure.
Following the discussion in the earlier
section, this robustness is also expected
to imply robustness to mutation.
Thus, each cell type that appears overnormal development will be robust to
mutation.
Here recall that in the simulations
described in the last section, we foundeach cellular state, as well as the number distribution of each cell type, wasfairly stable against noise. Since thedifferentiation process here was basedon cell-cell interaction, each cell type
and the distribution would stabilize
each other: robustness to noise at thesingle-cell and ensemble levels will consolidate each other. Recalling therelationship between robustness tonoise and to mutation, the number
distribution of cells of each type is also
expected to be robust to genetic
changes.

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Insights & Perspectives

Finally, I propose a hypothesis for cancer cells from the study on robust
dynamical systems of development.
Generally, GRNs are complex, involving
a huge number of proteins that mutually
influence the expression of others. Many
attractors generally exist for such highdimensional dynamical systems (as confirmed by simulations of a random
GRN). The number of attractors might
generally decrease as robustness
increased through evolution [29], but
considering the huge number of
involved protein species, some otherattractors in gene expression dynamics,
besides those visited during normal
development, may remain (Fig. 5).
Indeed, in the earlier simulations by
Furusawa and the author, such an
attractor, which did not appear overnormal development, was also identified [43].
Such states, as they are attractors,
are stable to small perturbations.
However, they are not generated as aresult of normal cell-cell interaction,and not stabilized in the presence ofother cell types, nor do they help thestate of others to reach stability. These
additional attractors would be selfish inthat they can grow without forming
stable relationships with others. This
is in strong contrast to normal cells in
our model, which were stabilized bycell-cell interaction and form mutuallystabilizing relationships.

Far from a stable attractor: Whycancer cells are particularly
sensitive to mutations
Recall that most attractors in a randomly chosen GRN were not robust to
mutations as long as it was sufficiently
complex, while robustness to noise andto mutation discussed previously was a
result of evolution to produce the functional phenotype. Since the aberrant
cell types here are not functionallynecessary, and thus not under a selection pressure to preserve their state,
they need not achieve high robustness
against noise or mutation, throughevolution. Accordingly, through divisions of such cells, their state would
be more changeable by somatic
mutation. Since the state is not yet optimized for the growth speed or robustness, they can be increased by suchmutation. Hence, some mutations
would increase the population of suchcells. Accordingly, such cellular states
would easily accumulate mutations.
This is in contrast with the normal cell
types that are expected to have alreadyachieved robustness to noise andmutation through evolution, where
somatic mutation can increase neither
robustness nor growth speed.
To summarize, the cell types discussed here will be characterized asfollows:(i) They do not appear over the courseof normal development, but
are generated as a single-cellattractor when somehow perturbed
sufficiently.
(ii) They are selfish and do not form
stabilizing relationships with cellsof other types.
(iii) Their phenotypes are vulnerable tonoise or to changes in cell-cellinteraction.
(iv) They are not robust to mutation.
(v) They can easily accumulate
mutations, with which robustness
can be increased.
(vi) They are in differentiated statescompared with pluripotent cells,
but are not terminally committed
cell types.
Cells with these properties may fit withobservations of cancer cells (or cancer
stem cells). Indeed, the hypothesis ofcancer cells as attractors was proposedby Kauffman [57] and recently put forward by Huang et al. [58, 59]. Here, myproposal concerns intra-intercellular
dynamics rather than single-cell gene
expression dynamics, and distinguishes
cancer cells from normal cell types withregard to robustness. This distinction
leads to the above six characteristic
points of cancer-type cells.
According to point (i), a sufficiently
large external perturbation may trigger
the production of the cancer cell. Point(ii) might explain how cancer cells are
not useful for other normal cell types. In
addition, because the normal developmental course is a result of cell-cell

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interaction in the theory presented here,
the appearance of cancer-type cells is
also expected to depend on cell-cellinteraction [14, 43].
The relatively large variation in cancer cell phenotypes observed [60–62]
may reflect points (iii) and (iv). Their
phenotype, even though they might bedifferentiated, could be changed moreeasily than the normal committed cells.
Furthermore, their phenotype willchange depending on interaction withother cells and mutations. The mostplastic type will correspond to cancer
stem cells [63].

Genetic instability in cancer:
A result, rather than cause,of phenotypic instability?
According to point (v), genetic instability in cancer cells would not be a cause
but a result of phenotypic instability indynamics – a trait of the weak attractor.
In this sense, genetic instability in cancer cells may be regarded as a genetic
response to increase reproducibility ofthe phenotype under the environment
in concern, a kind of genetic assimilation [1] in Waddington’s sense.
Phenotypic change as a result of gene
expression dynamics would then inducechanges in DNA methylation pattern[63], and later be fixed to genetic
changes.
The question remains as to how cancer cells are quantitatively characterized. In the previous section, I
proposed possible measures for multipotency. In the examples of earlier
numerical studies [14, 43], chemical
concentrations in such cancer-type cells
did not show oscillatory change in time,
but demonstrated larger variability
across cells, compared with committed
cells. This finding is consistent with thepoint (iii), weakness in phenotypicrobustness. It was also found numerically that the chemical composition was
more biased than that of the original
stem cell. In summary, cancer-type cells
are expected to have smaller temporalvariation than embryonic stem cells,with cell-cell variation much larger thanthat of normally differentiated cell
types. The chemical diversity of a cancer-type cell is expected to be smaller
than that of embryonic stem cells andprobably at the level of normally differentiated cell types.

411

Hypotheses

Cancer: An attractor that is
not reached from thenormal developmental pathand lacks mutational
robustness

K. Kaneko

K. Kaneko

Hypotheses

Consequences of the hypothesis
The hypothesis presented here will leadto several consequences. According to
the present picture, mutation would not
be a direct cause of cancer cells, andgeneration of cancer cells would bestrongly dependent on cell-cell interactions and environmental variation.
In fact, density dependence of the frequency of tumor cells was reported by
Rubin [64, 65]. Also, several experimental reports have shown that transplantation of cancer cells into some othertissue, which modifies their interaction
with other cells, makes the cancer cells
normalize [66–69].
From the view here, it is expectedthat cancer cells do not accumulate
mutations when they are first generated.
At this stage, they could revert tooriginal multipotent cells by appropriate operations. In fact, such reprogramming has already been observed, bynuclear transplantation and nuclear
cloning [70, 71]. If the change in cellularstates can be observed by single-cell
imaging, the present hypothesis could
be confirmed by combining with themicroarray analysis.

Conclusions
Cellular state is generated by gene
expression pattern. Cells functioning
in a given environment are robust to
external perturbations. I proposed thatrobustness of the gene expression pattern to noise also leads to robustness tomutation, through evolution of suchcells. Cells under selection pressure
for some function are expected toachieve these two types of robustness.
In multicellular organisms, differentiation from stem cells to committed
cells is also robust to perturbations overdevelopment. Here, stem cells havepotentiality both to proliferate anddifferentiate. The former requires
stability to keep the cellular state overcell divisions, and the latter requires
plasticity to allow for change in the cellular state. Numerical simulations so far
demonstrated that oscillatory gene
expression dynamics switching between
on and off make the robustness compatible with the plasticity. Cell-cell interactions, then, caused different cell

412

Insights & Perspectives

types to stabilize each other, achievingrobustness to external perturbations.
Since both the stem and differentiated cell states are necessary for a multicellular organism, robustness of eachcell type to noise has to be achieved,
which, in turn, would confer robustness
to mutation.
In complex gene expression dynamics with many genes, however, there canbe some other cellular states that are
attractors but do not appear over thenormal developmental course. In contrast to normal cell types, these cells do
not necessarily form stabilizing relationships with other cell types. Since thesecell types are not under selection pressure through evolution, their phenotype
would not necessarily be robust to
mutation, and the robustness of phenotypes to perturbations could increase by
mutations. Hence, somatic mutations
will easily be accumulated through cell
divisions. I hypothesize that these cell
types have something in common withcancer cells. This dynamical systems
viewpoint provides a novel perspective
on cancer cells, which can be verified
experimentally.

Acknowledgments
I would like to thank Sui Huangfor stimulating discussions, Chikara
Furusawa and Narito Suzuki for collaborative studies, Taeko Kobayashi and
Kaoru Sugimura for critical reading of
the manuscript, and Motoko Miura fordrawing figures. This work is partially
supported by the people of Japan, via a
Grant-in-Aid for scientific research(No.21120004) from MEXT, Japan.

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