Theory of Robustness of Irreversible Differentiation in a...

J. theor. Biol. (2001) 209, 395}416doi:10.1006/jtbi.2001.2264, available online at http://www.idealibrary.com on

Theory of Robustness of Irreversible Di4erentiation in a Stem Cell System:Chaos Hypothesis

CHIKARA FURUSAWA* AND KUNIHIKO KANEKO

Department of Pure and Applied Sciences,;niversity of ¹okyo, Komaba, Meguro-ku, ¹okyo 153, Japan

(Received on 14 December 2000, Accepted in revised form on 11 January 2001)

Based on an extensive study of a dynamical systems model of the development of a cell society,a novel theory for stem cell di!erentiation and its regulation is proposed as the &&chaoshypothesis''. Two fundamental features of stem cell systems*stochastic di!erentiation of stemcells and the robustness of a system due to regulation of this di!erentiation*are found to begeneral properties of a system of interacting cells exhibiting chaotic intra-cellular reactiondynamics and cell division, whose presence does not depend on the detail of the model. It isfound that stem cells di!erentiate into other cell types stochastically due to a dynamicalinstability caused by cell}cell interactions, in a manner described by the Isologous Diversi"ca-tion theory. This developmental process is shown to be stable not only with respect tomolecular #uctuations but also with respect to the removal of cells. With this developmentalprocess, the irreversible loss of multipotency accompanying the change from a stem cell toa di!erentiated cell is shown to be characterized by a decrease in the chemical diversity in thecell and of the complexity of the cellular dynamics. The relationship between the division speedand the loss of multipotency is also discussed. Using our model, some predictions that can betested experimentally are made for a stem cell system.

( 2001 Academic Press

1. Introduction: some Biological Factsregarding Stem Cells

Stem cells are de"ned as cells that are capable ofproliferation, self-maintenance of their popula-tion, production of di!erentiated cells, and regen-eration of the tissue in concern (Potten & Loe!er,1990). In mammals, stem cells sustain tissuescomposed of cells that have a limited lifespan andlose the ability of self-renewal (e.g. blood andepidermis cells), by continuously supplying newdi!erentiated cells. The importance of stem cellsystems, however, is not restricted only to main-taining such kinds of tissues. For example, inprimitive multicellular organisms such as hydra

*Author to whom correspondence should be addressed.E-mail: [email protected]

0022}5193/01/080395#22 $35.00/0

(Barnes et al., 1988) and planarian (Baguna et al.,1989), the regulated proliferation and di!erenti-ation of stem cells both serve to replace di!erenti-ated cells that die through the normal course ofaging and allows for the regeneration of parts ofthe organisms damaged through injury and theasexual reproduction of the organism itself. Inplants, development and regeneration are also sus-tained by stem cells at the apical meristem (Albertset al., 1994). These facts suggest that the functionof stem cells can be regarded as fundamental increating and maintaining cellular diversity, andthat they play a most important role in sustainingthe reproduction process of multicellular organ-isms, including development, regeneration andordinary replacement of di!erentiated cells.

( 2001 Academic Press

396 C. FURUSAWA AND K. KANEKO

An important feature of a stem cell system isthe robustness of the system with regard to cellpopulation, as exempli"ed by regeneration.When tissue is damaged, active stem cells re-appear through the activation of quiescent stemcells or the de-di!erentiation of neighboring cells.The proliferation and di!erentiation of thesestem cells is regulated, depending on the environ-ment in such a way that the damaged tissue canbe repaired. For example, in a hematopoieticsystem, when the number of terminal di!erenti-ated cells of a particular type (e.g. red blood cells)is decreased due to some externally applied in#u-ence, the production of this cell type throughdi!erentiation is increased, and the originalpopulation distribution is recovered (Albertset al., 1994).

Although the behavior of stem cells has beenstudied intensely because of their biological andmedical importance, the problem of how stemcells determine their own fates (either self-renewalor di!erentiation) is still unsolved. Concerningthe &&decision'' of a stem cell to di!erentiate, twomajor hypotheses have been proposed. One ofthese is the cell-intrinsic hypothesis, according towhich stem cell di!erentiation is triggered by anintra-cellular program. The other is the inductivehypothesis, according to which di!erentiation iscaused by inductive signals received from outsidethe system, which causes a change in some stemcells that leads to di!erentiation.

A most important model representing thecell-intrinsic hypothesis is the stochastic di!er-entiation model, proposed to describe thehematopoietic stem cell system by Till et al. (1964).They observed heterogeneity in the number ofstem cells in spleen colonies which originate froma single stem cell. They explained this heterogen-eity in terms of the hypothesis that each decisionof a stem cell to either undergo self-renew ordi!erentiation is stochastic, with a given "xedprobability. Nakahata et al. (1982) extended thisstochastic determination hypothesis to the com-mitment of multipotential progeneitor cells toindividual lineages in a hematopoietic system.They prepared two daughter cells derived froma single multipotential stem cell and cultivatedthem independently under identical conditions.Repeating this experiment a number of times,they found that the distributions of two such

colonies generated by paired daughter cells oftenturned out to be di!erent from each other. Theseexperiments clearly demonstrate that micro-scopic #uctuations of an intra-cellular state canbe ampli"ed, and led to stochastic behavior ofstem cells. Following these experiments, thestochastic di!erentiation model is extended toother stem cell systems, such as those of thecentral nervous system (Douarin & Dupin, 1993),hydra (David & MacWilliams, 1978), and inter-stitial crypts (Loe%er et al., 1993).

In experiments on several stem cell systems,spontaneous di!erentiation has been observed,and in terms of the observables considered inthose studies, it appears that it occurs ina stochastic manner. Although external signalmolecules, such as growth factors, are required tomaintain these systems, it has been found that thedi!erentiation of stem cells does not require anysignals from the outside providing inhomogene-ity among the state of stem cells to govern theirdecision to di!erentiate or not. From these re-sults, it can be concluded that the inductive hy-pothesis of stem cell di!erentiation is incorrect,and that diversity comes from inside the system.

It should be noted that the cell-intrinsic hy-pothesis does not exclude the possibility thatextrinsic factors can a!ect a stem cell's behavior.Certainly, signal molecules can regulate the be-havior of cells in a stem cell system. For example,experiments using hematopoietic stem cells bothin vivo and in vitro suggest that the presence ofsignal molecules, such as colony stimulating fac-tors (CSFs) and interleukins, is essential for thesurvival and proliferation of the cells (Ogawa,1993). In vivo, these signal molecules, includingdi!usive chemicals and cell surface proteins, arereleased by cells of several types, such as stemcells, di!erentiated cells, and neighboring cells,like stromal cells in bone marrow. These signalssustain the complex interactions among thesecells in the stem cell system. These complex inter-action are responsible for the robustness of suchsystems that is generally observed.

An interesting and important underlying ques-tion here is whether the nature of this di!erenti-ation depends on these cell}cell interactions. Inother words, it is yet unknown whether the prob-abilities involved in di!erentiation are controlledextrinsically. The original form of the stochastic

*See also Appendix A for a brief introduction to thebackground of dynamical systems.

IRREVERSIBLE DIFFERENTIATION IN A STEM CELL SYSTEM 397

model assumes that the probability of di!erenti-ation is "xed, and that cell population isregulated by controlling the proliferation andsurvival of cells. However, there are some experi-ments which suggest that this probability can becontrolled by extrinsic factors. For example,when a single hematopoietic stem cell is trans-planted into a mouse whose hematopoietic sys-tem has been destroyed through irradiation, thiscell can reconstitute the hematopoietic systemwith high probability (Osawa et al., 1996). Thisresult implies that, the probability of di!erenti-ation is externally controlled at least in the casewhere there are only a small number of stem cells.The regulation of this probability has also beenreported in the spontaneous di!erentiation ofhydra stem cells (David & MacWilliams, 1978).These experimental results suggest that the exist-ence of regulation of di!erentiation keeps the sys-tem stable with respect to external perturbations.

Summarizing the discussion of the previoussection, the following features characterize a stemcell system.

1. The diversity of cell types resulting from di!er-entiation from a stem cell emerges without anyin#uence from the outside of the system. Thisdi!erentiation occurs stochastically.

2. The proliferation and di!erentiation of cells ina system is regulated by cell}cell interactions.This regulation keeps the system stable withrespect to external perturbations.

Note that, the origin of stochastic di!erenti-ation is yet unknown in the context of molecularbiology. A simple explanation for this could berandom collisions of speci"c chemical substancesin a cell. For example, such stochasticity is ex-pected in the case where di!erentiation is causedby the reaction of a DNA molecule and a regula-tor protein through a chance encounter.

If such uncontrollable random collisionswere the sole cause of stochastic di!erentiation,however, it would be very di$cult to imagine amechanism to regulate the probability itself. Ifdi!erentiation were caused by a speci"c stochasticreaction of this kind, this reaction should becontrolled by other stochastic chemical reactions.These chemical reactions may form a complexreaction network, in which each reaction often is

accompanied by a large #uctuation arising fromstochastic reactions among a small number ofchemical substances. Now, exercising control tochange the rate of some speci"c reaction in sucha network would probably be very di$cult.

Here, we note that chaotic dynamics providea plausible mechanism for such a control of theprobabilistic behavior involved in di!erentiation,because a set of deterministic equations that de-scribe biochemical reaction networks in a cell canyield stochastic behavior in the form of chaoticdynamics. As long as information concerning theinternal cellular state is not precisely known tous, di!erentiation must appear stochastic. On theother hand, since this &&stochastic'' behavioris generated by a deterministic mechanism, itis possible that chaotic dynamics of the bio-chemical reaction networks in cells providea mechanism for the self-regulation of di!erenti-ation.

In the present paper, we propose this &&chaoshypothesis'' of stem cell di!erentiation, in whichthe nonlinear dynamics of intra-cellular chemicalreactions and cell}cell interactions are essential.Indeed, the authors and T. Yomo have carriedout several numerical experiments modeling in-teracting cells with internal biochemical net-works and cell divisions, and proposed theIsologous Diversi"cation theory as a generalmechanism of the spontaneous di!erentiation ofreplicating biological units (Kaneko & Yomo,1994, 1997, 1999; Furusawa & Kaneko, 1998a, b).The background of this theory lies in the dynam-ics clustering in coupled dynamical systems(Kaneko 1990, 1994).* According to this theory,cells exhibiting oscillatory chemical reactions dif-ferentiate through interaction with other cells.These di!erentiations are not caused by speci"csubstances, but, rather, are brought about byorbital instabilities inherent in the underlyingnonlinear dynamics. The cell society includingvarious distinct cell types is stable with respect toexternal perturbations up to a certain degree,including the removal of some cells.

Here, we attempt to capture characteristicdynamics of stem cell systems using the abovetheoretical framework. The chaos hypothesis for

-Precisely speaking, this should be called tensor.


stem cell dynamics is proposed, according towhich stochastic di!erentiation is a result ofchaotic dynamics of the cell system arising fromintra-cellular chemical reactions and cell}cell in-teractions. These di!erentiations are controlledin a manner that depends on the distribution ofcell types. This control results in a system whoseoverall features are stable even without thepresence of an external regulation mechanism.Irreversible di!erentiation leading to a loss ofmultipotency is characterized theoretically.

2. Model

First, we summarize our standpoint for design-ing the model of a stem cell system.

Due to the complexity of cellular dynamicsin real organisms, it is almost impossible toconstruct a model that gives results in preciseagreement with the quantitative behavior of realbiological processes. Of course, one could goabout constructing a complicated model with thesole purpose of precisely reproducing experi-mental data obtained in cell biology studies andin this way imitate the behavior of living cells.However, such mimicry would get us no closer tounderstanding the essence of cellular systems,because any similar behavior to real biologicalprocesses could be obtained by adding complic-ated mechanisms. With such a model, we canneither identify the essential features of stem cellsnor extract universal features that are present inall stem cell systems. By contrast, we start witha simple model containing only the essential fea-tures of biological systems, and with this we at-tempt to capture the universal behavior exhibitedby all cell societies.

To investigate cell di!erentiation and therobustness of cell societies, the authors andT. Yomo have considered simple modelsconsisting only of the following basic cellfeatures.

f Internal dynamics consisting of a biochemicalreaction network within each cell.

f Interactions between cells (inter-cellulardynamics).

f Cell division and cell death.f Molecular #uctuations in the intra-cellular

reaction dynamics.

2.1. INTERNAL CHEMICAL REACTION DYNAMICS

Within each cell, there is a network of bio-chemical reactions. This network includes notonly a complicated metabolic network but alsoreactions associated with genetic expressions, sig-naling pathways, and so forth. In the presentmodel, a cellular state is represented by the con-centrations of k chemicals. The dynamics of theinternal state of each cell is expressed by a set ofvariables Mx(1)

i(t),2, x(k)

i(t)N representing the con-

centrations of the k chemical species in the i-thcell at time t.

As the internal chemical reaction dynamics, wechoose a catalytic network among the k chem-icals. Each reaction from some chemical i to somechemical j is assumed to be catalysed by a thirdchemical l, which is determined randomly. Torepresent the reaction matrix we use the nota-tion-Con (i, j, l), which takes the value 1 when thereaction from chemical i to chemical j is catalysedby l, and 0 otherwise. Each chemical acts asa substrate to create several enzymes for otherreactions and there are several paths to otherchemicals from each chemical. Thus, these reac-tions form a complicated network. The matrixCon (i, j, l) is generated randomly before a simula-tion, and is "xed throughout this simulation.

We denote that the rate of increase of x(m)i

(t)[and decrease of x(j)

i(t)] through a reaction from

chemical j to chemical m catalysed by l asex(j)

i(t)(x(l)

i(t))a, where e is the coe$cient for the

chemical reaction and a is the degree of catalysa-tion. For simplicity, we use identical values ofe and a for all paths. In this paper, we adopta"2, which implies a quadratic e!ect ofenzymes. This speci"c choice of a quadratice!ect is not essential in our model of cell di!eren-tiation.

Furthermore, we take into account the changein the volume of a cell, which varies as a result ofthe transportation of chemicals between the celland the environment. For simplicity, we assumethat the total concentration of chemicals in a cellis constant, +

mx(m)i

"const. It follows that thevolume of a cell is proportional to the sum of thequantities of all chemicals in the cell. The volumechange is calculated from the transport, as dis-cussed below.


2.2. CELL}CELL INTERACTION

Cells interact with each other through thetransport of chemicals into and out of the sur-rounding medium. The &&medium'' here is notmean external to the organism, but, rather, itrefers to the interstitial environment of each cell.Here, the state of the medium is expressed bya set of variables MX(1)(t),2, X(k) (t)N, whoseelements represent the concentrations of thek chemical species in the medium.

We assume that the medium is well stirred byignoring the spatial variation of the concentra-tion, so that all cells interact with each otherthrough an identical environment. Of course, thisis a simplifying assumption that introduces a lim-itation to our model, since spatial factors, such asthe gradient of signal molecules, play an essentialrole in morphogenesis. However, experimentsin vitro show that a diversity of cell types in stemcell systems can appear even in a homogeneousenvironment, without any spatial structures.Thus, the appearance of a diversity of cell typesdoes not require some kind of spatial in-homogeneity, which is often introduced fromoutside the system in the modeling of stem cellsbehavior. Instead, the diversity of cell typesemerges through a processes within the system.

In this model, we consider only indirectcell}cell interactions in the form of the di!usionof the chemical substances in the system, asa minimal form of interaction. Thus, the term inthe equation for each dx(m)

i(t)/dt describing the

transport from the medium into the i-th cell form-th chemical is given by D(X(m)(t)!x(m)

i(t)),

where D is a transport coe$cient.In general, the transport (di!usion) coe$cients

should be di!erent for di!erent chemicals. Here,we consider a simple situation in which there aretwo types of chemicals, those which can penetratethe membrane and those which cannot. This dis-tinction is made in the equation of motion by theparameter p

m, which is 1 if the m-th chemical is

penetrating and 0 if it is not.

2.3. CELL DIVISION AND CELL DEATH

Each cell receives penetrating chemicals fromthe medium as nutrients, while the reaction in thecell transforms them into non-penetrating chem-icals which comprise the body of the cell. As

a result of these reactions, the volume of the cell isincreased. In this model, the cell divides into twoalmost identical cells when the volume of the cellis doubled.

During this division process, all chemicals arealmost equally divided, with tiny random #uctu-ations [e.g. &10~6 x(m)

i]. Although the existence

of imbalance is essential to the di!erentiation inour model and in nature, the degree of imbalanceitself is not essential to our results to be discussed.In our model simulations, this tiny di!erence canbe ampli"ed, resulting from an intrinsic instabil-ity of the internal dynamics.

Penetrating chemicals can penetrate the cellmembrane in both directions, and these chem-icals may #ow out of a cell. As a result, thevolume of the cell can become smaller. In ourmodel, a cell dies when its cell volume becomesless than a given threshold.

2.4. MOLECULAR FLUCTUATIONS

In a real biological system, the number ofa speci"c type of molecules in a cell is often small.For example, some signal molecules can causea change of the cellular state even when thenumber of molecules is in the order of 1000. Withnumber of molecules on this order, it follows thatthe cellular dynamics always exhibit non-negli-gible molecular #uctuations. Thus, a noise termshould be included to take into account (thermal)#uctuations. Considering #uctuations of JN fora reaction involving N molecules, we add a noiseterm proportional to g(t)Jx(m)

i(t), corresponding

to a Langevin equation, where g(t) denotes Gaus-sian white noise satisfying Sg(m)

i(t)g(m{)

i(t@)T"

p2d(t!t@), with p as the amplitude of the noise.Summing up all of the processes described

above, the dynamics of the chemical concentra-tion in each cell is represented as follows:

dx(l)i

(t)/dt"dx(l)i

(t)!x(l)i

(t)k+

m/1

dx(m)i

(t) (1)

with

dx(l)i

(t)"+m,j

Con(m, l, j) e x(m)i

(t) (x(j)i

(t))a

! +m{,j{

Con(l,m@, j@) e x(l)i

(t) (x(j{)i

(t))a

#plD(X(l)(t)!x(l)i

(t))

#g(t)Jx(l)i

(t). (2)


Here, the term dx(l)i

is the increment of chemicall, while the second term in eqn (1) introduces theconstraint of +lx(l)

i(t)"1 due to the growth of

the volume.Since the penetrating chemicals in the medium

are consumed through their #ow into the cells, wemust supply these chemicals (nutrition) into themedium from the outside. Denoting the externalconcentration of the l-th such chemical by X(l)and the transport coe$cient of the #ow by f, itsdynamics are described by

dX(l)(t)/dt"fpl(X(l)!X(l)(t))

!(1/<)N+i/1

plD(X(l)(t)!x(l)i

(t)), (3)

where N denotes the number of the cells in theenvironment, and < denotes the volume of themedium in units of the initial cell size.

2.5. REQUISITE FOR INTRA-CELLULAR DYNAMICS

The dynamics of the chemical concentrationsdepend on the choice of the reaction network.They can be simple, like "xed-point or limit-cycledynamics, or they can be complex, like chaoticdynamics. Here we assume that the intra-cellulardynamics for a stem cell includes some dynamicalinstability, which typically leads to oscillatorybehavior including chaos, as depicted in Fig. 1(a).

In our model simulations, when the number ofpaths in the reaction matrix is small, in mostcases, the cellular dynamics fall into a steady statewithout oscillation, in which a small number ofchemicals are dominant, while most other chem-icals' concentrations vanish. On the other hand,when the number of reaction paths is large, inmost cases, all chemicals are generated by otherchemicals, and chemical concentrations come torealize constant value (which are almost equal formany chemicals). When there is an intermediatenumber of reaction paths do non-trivial oscilla-tions of chemicals appear with higher probabilityas seen in Fig. 1(a). Another factor relevant to thenature of the internal dynamics is the number ofauto-catalytic paths. We "nd that the probabilityof the appearance of oscillatory dynamicsincreases as the number of auto-catalytic pathsincreases. It is important to note, however, thateven by using appropriate parameter values

the cellular dynamics resulting from randomlychosen reaction networks are rarely oscillatory.We have performed simulations with our modelusing thousands of di!erent reaction networks,and only about 10% of these networks resulted inoscillatory dynamics.

The reasons for studying networks that displayoscillatory dynamics are as follows.

First, as shown throughout this paper, we havefound that robust developmental processes withspontaneous di!erentiation of stem-type cellsemerges commonly if and only if the individualcells exhibit oscillatory dynamics. Second, as dis-cussed in Section 7 and in Furusawa & Kaneko(2000), a cell system characterized by oscillatorydynamics has a higher growth speed as an en-semble, and for this reason it is expected to beselected through evolution.

In real biological systems, oscillations are ob-served in some chemical substrates such as Ca2`,NADH, cyclic AMP, and cyclins. These oscilla-tions are sustained by networks of nonlinearchemical reactions, often including positive feed-back reactions. Experimental results show that instem cell systems, reactions among signal mol-ecules form a complex network with positive andnegative feedback (Ogawa, 1993). Thus, it is natu-ral to postulate the existence of such oscillatorydynamics in our model system. The importanceof oscillatory dynamics in cellular systems hasbeen pointed out by Goodwin (1963).

3. Numerical Results

In this section, we present numerical resultsdemonstrating development with stem-cell-typebehaviors obtained from several simulations.Here, we consider numerical experiments em-ploying a particular reaction network with thenumber of chemicals k"32 and 9 connectionsfor each chemical. This case represents typicalbehavior of the developmental process. The re-sults presented in this section were obtainedwithout the use of molecular #uctuations. As wediscuss in Section 4, however, they are stable withrespect to noise up to a particular thresholdstrength. The parameters ware set as e"1.0,D"0.001, f"0.02, X(l)"0.2 for all l, and<"10. Chemicals x(l)(t) for l(10 are penetrat-ing (i.e. pl"1), and others are not.


3.1. DIFFERENTIATION PROCESS

As the initial state, we consider a single cellwhose chemical concentrations x(l)

iare deter-

mined randomly in such a way to satisfy theconstraint +lx(l)

i"1. In Fig. 1(a), we have plot-

ted a time series of the concentrations of thechemicals in such a single, isolated cell. The at-tractor of the internal chemical dynamics depic-ted here is chaotic. Here, most of the chemicalsare generated by other chemicals and coexistwithin the cell. We call cells in such a state &&type-0'' in this paper. This state appears dominantlywith randomly chosen initial conditions fora single, isolated cell; in other words, the basin ofattraction for this state is very large.?

Now, considering the case with di!usion, ex-ternal chemicals #ow into the cell, because oflower concentration of penetrating chemicals inthe cell by reactions transforming penetratingchemicals into non-penetrating chemicals. This#ow leads to the increase of the cell volume. Ifthis volume becomes su$ciently large, the celldivides into two, with almost identical chemicalconcentrations. The chemicals within the twodaughter cells oscillate coherently, with the samedynamical behavior as their mother cell (i.e.type-0 dynamics). As the number of cellsincreases by a factor of two (i.e. 1P2P4P8P2) with further divisions, although theinternal dynamics of each cell correspond to thesame attractor, the coherence of the oscillationsamong individual cells is easily lost due to thechaotic nature of the dynamics in these type-0cells. The microscopic di!erences introduced ateach cell division are ampli"ed to a macroscopiclevel, and this destroys the phase coherence.

When the number of cells exceeds some thre-shold value, some cells begin to display di!erenttypes of dynamics. The threshold number de-pends on the nature of the networks and theparameter values used in our model. In the situ-ation we consider presently, two types of cellswith distinct internal dynamics begin to emergefrom type-0 cells when the total cell number be-comes 64. In Fig. 1(b) and (c), the time series ofthe chemical concentrations in these new types of

? In the case of single-cell dynamics considered here, thereare very rare initial conditions that results in a "xed-pointattractor, as shown below.

cells are plotted. We call these &&type-1'' and&&type-2'' cells, respectively. In Fig. 1(e), orbits ofthe chemical concentrations at the transitionfrom type-0 cell to type-1 cell is plotted in thephase space. This "gure shows that each of theseattractors occupies a distinct region in the phasespace, and that each can be clearly distinguishedas a distinct state. These transitions of the cellularstate are interpreted as di!erentiation.

Note that this di!erentiation is not induceddirectly by the tiny di!erences introduced at thecell divisions. Also, the transition from one celltype to another does not occur at the time of celldivision, rather later through the interactionamong the cells. The phenomenon of di!erenti-ation observed here is caused by an instability inthe entire dynamical system consisting of all thecells and the medium. It is due to this instabilitythat tiny di!erences between two daughter cellscan become ampli"ed to a macroscopic levelthrough the intra-cellular dynamics and interac-tions between cells. Only when the strength ofthis instability exceeds the threshold, does di!er-entiation occur. Then, the emergence of new celltypes stabilizes the dynamics of the other celltypes. The cell di!erentiation process in ourmodel is due to the ampli"cation of tiny phasedi!erences through orbital instability (transientchaos), while the coexistence of di!erent cell typesstabilizes the system as a whole.

The important point here is that the spontan-eous di!erentiation from the type-0 cells is&&stochastic''. The choice of a type-0 cell to eitherreplicate or di!erentiate appears stochastic if weconsider only the cell type. Of course, if one coulddetermine completely the chemical compositionand the division-induced asymmetry, then thefate of the cell would also be determined, sinceour model is deterministic. Here, however, itwould be practically impossible to make sucha determination because it would necessitate theinternal cell state to arbitrary precision, given thechaotic nature of its dynamics.

As the cell number increases further, anothertype of cell appears through di!erentiation fromtype-0 cells, which we call &&type-3'' cells [seeFig. 1(d)]. In this example, eventually four typesof cells appear through the developmentalprocess. The cell lineage diagram is shown inFig. 2.

FIG. 1. (a) Overlaid time series of x(m)(t) for the type-0 cell. The vertical axis represents the concentration of chemicals andthe horizontal axis represents time. In this "gure, we have plotted the time series of only 6 of the 32 internal chemicals, forclarity. The lines designated by the numbers m"1, 16, 19, 25, 27, and 28 represent the time series of the concentrations of thecorresponding chemicals x(m)(t). ( ) x1 ; ( ) x16 ; ( ) x19; ( ) x25 ; ( ) x27; ( ) x28. (b)}(d) Time series of x(m) (t)in a cell, representing the course of di!erentiation to type-1 and 2, and type-3 cells, respectively. (e) Orbit of the internalchemical dynamics in the phase space projected onto [x(28), x(6)], corresponding to (b). The orbits of type-1 cells fall onto the"xed point.


It should be noted that these di!erentiated celltypes do not necessarily correspond to attractorsof the internal dynamics of a single cell. For the

interaction matrix and parameter values used inthe simulation described above, for a single-cellsystem, there is no attractor corresponding to

FIG. 2. Cell lineage diagram. The vertical axis represents time, while the horizontal axis represents a cell index, which isde"ned only for the practical purpose of avoiding line crossing. In this diagram, each bifurcation of lines through thehorizontal segments corresponds to the division of the cell, while each vertical line represents an existing cell, with thefollowing lines indicating its cell type. ( ) Type-0; ( ) Type-1; ( ) Type-2; ( ) Type-3.


type-2 dynamics. In a multiple-cell system, thestability of such internal dynamics is sustainedonly through interactions among cells. Thismeans that when a single type-2 cell is trans-planted into a medium containing no other cells,this cell is de-di!erentiated back into a type-0cell. On the other hand, type-1 and type-3 dy-namics do exist as attractors of the single-cellsystem, like type-0 dynamics. However, these twoattractors are not observed when the initial con-ditions (i.e. the initial concentrations of chemicalsin the "rst cell) are chosen randomly. This revealsthat the basins of attraction for these states aremuch smaller than that of the type-0 state.

The transitions between di!erent cell typesthrough di!erentiation follows speci"c rules.These rules originate in a constraint on the tran-sient dynamics exhibited between the attractorstates corresponding to each cell type. In thisexample, type-0 cells are regarded as stem cellsthat have the potential both to reproduce them-selves and to di!erentiate into other cell types (i.e.

AOf course, existing multicellular organisms (such asmammals) often have hundreds of cell types with morecomplicated hierarchical rules of di!erentiation. The num-ber of cell types that appears in our dynamical di!erenti-ation model does not show a clear increase with the numberof chemical substances. We believe that the existence of onlya few types of cells is due to the random nature of thechemical reaction network and the fact that the rate con-stants for all reactions are equal. In a real biological system,the chemical reaction network is more organized, possibly ina hierarchical manner, through evolution.

0P1, 2, 3), while the di!erentiated cells have lostsuch potential and only reproduce the same type.

In the present example, four distinct cell typesare observed. The number of cell types appearingin the cell society depends on the nature of thenetwork reaction network. In Fig. 3, some otherexamples of the dynamics of distinct cell typesand their di!erentiation rules are displayed,where di!erent reaction networks have beenemployed. We see that the number of cell typesappearing is often larger than four, as in Fig. 3(a),and the rules of di!erentiation are often hier-archical, as in Fig. 3(b).A


Next, we consider the existence of multiple cellcolony states. For example, in the system corre-sponding to Fig. 3(b), not all of the six types ofcells coexist in a developed cell society. Cellgroups consisting of only two or three cell typesappear, depending on the initial condition. Forexample, there is a cell colony consisting of onlytype-0, 1, and 2 cells and of only type-0, 2, and4 cells. In this example, we found that four typesof cell colonies, i.e. four stable distributions of celltypes, appear through developmental processeswith di!erent initial conditions (i.e. initial con-centrations of chemicals in the "rst cell), eventhough the process always starts from a singletype-0 cell. The selection of one cell colony isdetermined by the distribution of cell types at anearly stage of development, when the "rst di!er-entiations from &&0'' to &&1'' and &&2'' occur stochas-tically [see Furusawa & Kaneko (1998a) for de-tails]. It is important to note that each stabledistribution of cell types is stable with respectto macroscopic perturbations, including theremoval of cells, as is shown in Section 4.

4. Robustness of a Cell Society

An important implication of our chaos hy-pothesis for stem cell systems is that a cell societywith various cell types that develops according tothis hypothesis is stable with respect to certaintypes of perturbations. In this section, we discussthis robustness on two di!erent levels, at themolecular (microscopic) level and at the cell}population (macroscopic) level.

4.1. STABILITY WITH RESPECT TO MICROSCOPIC

PERTURBATIONS

In our model, a cell society with a variety ofcell types is stable with respect to molecular #uc-tuations, even though the process of di!erenti-ation is based on the dynamical instability caused

$&&&&&&&&&&&&&&&&&&&&&&&&&

FIG. 3. Some other examples of dynamics of distinct celltypes and rules of di!erentiation. Each "gure is obtained byadopting a di!erent reaction network, generated randomly.In each node, the time series of chemical concentrations ina cell are shown as in Fig. 1, representing the course ofdi!erentiation to the corresponding cell type. Each arrowbetween the nodes represents the potentiality to di!erenti-ation to the corresponding cell type.

by cell}cell interactions. This cell society is stablewith respect to perturbations in the form of anexternal in#uence that alters the concentrationsof metabolites. As long as such a perturbation isnot too large, the same set of cell types, a similardistribution of cell types, and similar cell lineagesare obtained.

To demonstrate this robustness, we have intro-duced noise in our simulations to represent mo-lecular #uctuations, described by a Langevinequation. In Fig. 4, we have plotted the temporalaverage of the concentrations of two chemicals[x(19) and x(27)] when the number of cells is 200.In this "gure, each point represents the state ofa cell. Without the noise, the cells split into fourcell types as shown in Fig. 1, which correspond tothe distinct clusters of points in Fig. 4. As shownin Fig. 4, these four distinct clusters are preservedas long as the amplitude of noise p is less than0.01. For larger p, the states of di!erentiatedcell types (i.e. types 1, 2, and 3) start becomingdestabilized, and all cells fall into type-0 dyna-mics. It is thus seen that the region occupiedby the orbit of type-0 becomes much broader,as the strength of the noise increases [seeFig. 4(d)].

This result may be expected when one con-siders that the di!erentiated cell types havesmaller basins of attraction than the stem-typecell. Note, however, that the basin of attraction ofeach cell type depends on the cell}cell interac-tions. For example, the measure of the basin ofthe attraction of type-2 dynamics is zero whena single cell is in the medium, and it increases asthe number of type-0 cells increases beyond thethreshold.

From several simulations using di!erent reac-tion networks, the following common featuresare found:

(i) The dynamics of a stem-type cell generallyhave a larger basin of attraction thanthose of di!erentiated cell types.

(ii) The nature of the stem-type cell canchange signi"cantly due to a tiny changein the cell}cell interactions, and has po-tency to di!erentiate into other cell types.

(iii) In spite of the instability with respect tochange in interactions, the stem cell hasa larger basin of attraction in the presence

FIG. 4. The temporal averages of the concentrations of x(19)i

(t) and x(27)i

(t) for 20 000 time steps. In the "gures, each pointrepresents the state of a cell. In the situation considered here, the number of cells is 200. The noise amplitudes p are 0.0003 in(a), 0.001 in (b), 0.003 in (c), and 0.01 in (d). Some of the points are overlaid and may be invisible since the plotted value of thecells are rather close to each other.


of noise, and its dynamics are stable withrespect to noise.

These three features are reminiscent of globallystable Milnor attractors, found in a class of coupleddynamical systems (Kaneko, 1998) (see also Appen-dix A). In such dynamical systems, there exists acertain type of attractor (Milnor attractor) which iseasily altered by small structural perturbations butmaintains globally attractive nature, which is oftenenhanced by noise. Thus, there are examples in the

study of dynamical systems of the type of stem}celldynamics that we have been discussing here.

As is discussed in Kaneko & Yomo (1999), itmay be possible to estimate the minimal numberof molecules in a cell required for a robust stemcell system from the value of the threshold noisepthr

. As a rough estimate [see Kaneko & Yomo(1999) for details], the minimal number ofmolecules necessary for robust development isaround 100}1000, which is consistent with obser-vations in experimental biology.

FIG. 5. The rate of the di!erentiation from type-0 to othercell types. In this simulation, the total cell number was "xedto 200 by disallowing cell division. We take the initialdistribution of the number of cell types as (n

0, n

1, n

2, n

3)"

(n0, 130!n

0, 35, 35), where n

krepresents the number of

type-k cells. Starting the simulation with this initial distribu-tion, the number of di!erentiations from type-0 cells and the"nal number of each cell type are plotted as functions of theinitial number of type-0 cells, n

0, when the system settles

down to a stable distribution of cell types. The negativevalue of the number of di!erentiations from type-0 cellsre#ects the occurrence of the de-di!erentiation process fromdi!erentiated cells to type-0 cells, which never occurs in thedevelopmental process from a single cell. ( ) Type-0;( ) Type-1; ( ) Type-2; ( ) Type-3.


4.2. STABILITY WITH RESPECT TO MACROSCOPIC

PERTURBATIONS

In addition to the stability discussed abovewith respect to molecular #uctuations, our cellsociety is also stable with respect to macroscopicperturbations, for example, the removal of somecells.

As mentioned above, stochastic di!erentiationfrom stem cells obeys speci"c rules, as discussedin Section 3.1. When there are multiple choicesfor a di!erentiation process (as &&0''P&&0'',&&0''P&&1'', &&0''P&&2'', &&0''P&&3''), the probabilityto select each path is neither "xed nor random,but depends on the number distribution of celltypes in the system. This regulation of di!erenti-ation probabilities from stem-type cells intothe various di!erentiated cell types sustain therobustness of a cell society with regard to thepopulation of each cell type. As an example,let us consider the case with four cell types(&&0'', &&1'', &&2'', &&3'' in Fig. 1). When type-1 cells areremoved, decreasing their population, the prob-ability of di!erentiation from a stem cell (type-0)to a di!erentiated cell of type-1 increases, andthe original cell-type distribution is roughly re-covered.

In Fig. 5, the rate of di!erentiation from thetype-0 cell to others is plotted, as a function of thenumber of type-0 cells in the cell ensemble. In thissimulation, to display the dynamics of the rela-tive number of each cell type more clearly, thetotal number of cells in the medium is "xed (toN"200 in the present case), by disallowing celldivision. As the initial conditions, N cells areplaced in the medium and the concentrations ofthe chemicals in each cell are selected so that theygive type-0, 1, 2 or 3 cells. The switch of cell typeswere measured when the system settled down toa stable distribution of cell types. The simulationswere repeated, changing the initial distribution ofcell types to obtain the plot in Fig. 5 of thenumber of switches from type-0 cells to theothers, where the "nal number of cells for eachtype is also plotted. As shown in Fig. 5, thefrequency of switches from type-0 cell increasesalmost linearly with the initial number of thatcell type. Through these switches, the cell distribu-tion approaches approximately (n

0, n

1, n

2, n

3)"

(60, 50, 50, 40), where nkrepresents the number of

type-k cells.

To maintain the robustness of a cell society,cells must obtain information concerning the dis-tribution of cell types in the system. According toour result, this information is embedded in theinternal dynamics of each cell. Each attractingstate of internal dynamics, corresponding toa distinct cell type, is gradually modi"ed in thephase space as the number distribution of thefour cell types changes. However, these modi"ca-tions of internal dynamics are much smaller thanthe di!erences among the dynamics of the di!er-ent cell types. Thus, there are two types of in-formation contained in the internal dynamics:&&analog'' information, re#ecting the global distri-bution of cell types, and &&digital'' informationde"ning the cell type. This analog informationcontrols the rate of di!erentiation, because theprobability of di!erentiation from the stem cell toother cell types depends on the modi"cation ofinternal dynamics in the stem cell. On the otherhand, a change in the distribution of cell types,resulting from di!erentiation, brings abouta change in the analog information. As a result of


this interplay between the two types of informa-tion, a higher-level dynamics emerges, which con-trols the probability of stochastic di!erentiationand sustains the robustness of the entire system.

It should be stressed that, in our model, thedynamical di!erentiation process from stem cellsis always accompanied by this kind of regulationprocess, without any sophisticated programs im-plemented in advance. In this process, di!erenti-ations occur when the instability of the systemexceeds some threshold through the increase ofthe cell number, and the cellular dynamics ofsome cells are destabilized and they di!erentiateinto other states. Then, the emergence of di!eren-tiated cells stabilizes the system. Therefore,a large perturbation, such as the removal of cells,causes the system to become unstable again, butas a result, di!erentiations that lead the systemback toward the stable state occur. In otherwords, in the systems that we have studied, onlycell types that possess this regulation mechanismto stabilize the coexistence with other cell typescan appear through di!erentiations from thestem cells. Here, the stability with respect to theexternal perturbations is sustained by the dynam-ical instability in the entire system, even thoughthis relationship seems paradoxical at a "rstglance.

5. Decrease of Complexity in Cellular Dynamicswith the Developmental Process

In the normal development of cells, there is aclear irreversibility, resulting from the successiveloss of multipotency. ES cells that can create alltypes of cells di!erentiate into several types ofstem cells with multi-potency, which producea given set of cell types, and then further di!eren-tiations from these stem cells follow to providethe determined cells, that can produce only thesame type of cells. How is this irreversibility char-acterized in our dynamical systems model of stemcells?

In our model simulations, the dynamics of astem-type cell exhibit irregular oscillations withorbital instability leading to chaos and involvea variety of chemicals. Stem cells with these com-plex dynamics have the potential to di!erentiateinto several distinct cell types. Generally, the dif-ferentiated cells always possess simpler cellular

dynamics than the stem cells, for example, "xed-point dynamics and regular oscillations (seeFig. 1 as an example). The di!erentiated types ofcells with simpler dynamics do not have the abil-ity to di!erentiate, and they simply reproduce thesame type of cells (or stop dividing). Here, the lossof multipotency is accompanied by a decrease inthe diversity of chemicals and in the complexityof intra-cellular dynamics. Although we havenot yet succeeded in formulating the irreversibleloss of multipotency in terms of a singlefundamental quantity (analogous to thermo-dynamic entropy), we have heuristically deriveda general law describing the decrease of twoquantities in all our numerical experiments, usinga variety of reaction networks. These are dis-cussed below.

5.1. LAW OF IRREVERSIBLE DEVELOPMENT:

WORKING HYPOTHESIS

In the normal developmental process startingfrom a stem cell from which arise various di!er-entiated cell types, the loss of multipotency isaccompanied by the following:

(i) A decrease in the diversity of chemicals.(ii) A decrease in the degree of orbital instabil-

ity (&&chaos'').

First, we study the diversity of chemicals in thecellular dynamics. In our model, a variety ofchemicals participate in the irregular oscillationsof the stem-type cell's dynamics, as shown inFig. 1(a). On the other hand, in di!erentiated celltypes, the concentrations of some chemicals fallapproximately to zero, and the dynamics consistof #uctuations of a smaller set of chemicals. Inthis sense, the number of dimensions of the e!ec-tive phase space of the intra-cellular chemicaldynamics become smaller. To study the decreaseof chemical diversity quantitatively, we de"nethis diversity for the i-th cell by

Si"!

k+j/1

pi( j) log p

i( j) (4)

with

pi( j)"T

x(j)i

+km/1

x(m)iU , (5)

where S2T represents the temporal averagetaken between successive cell divisions.

FIG. 6. Examples of the change in the diversity ofchemicals through the developmental process. In each "g-ure, the vertical axis represents the diversity of chemicalsde"ned by Eq. (5), while the horizontal axis represents thetime. Each point gives the average chemical diversity ofa cell at a cell division, where the average is taken from themost recent division time. Figs (a), (b), and (c) display theresults obtained by using the examples in Figs 1, 3(a), (b),respectively.


In Fig. 6, we display some examples of thechanges in the chemical diversity accompanyingthe developmental process. These correspond tothe examples in Figs 1 and 3(b). In the "gures,each point corresponds to a cell division, and thechemical diversity within that cell, averaged fromits most recent division time, is plotted as a func-tion of time. During the "rst stage of develop-ment [time(10 000 in Fig. 6(a)], cells have notyet di!erentiated, and the dynamics of cells arethose with a high diversity of chemicals. At a latertime, di!erentiated cells begin to appear from theinitial cell type. In the case with hierarchicaldi!erentiation depicted in Fig. 3(b), the diversityof chemicals decreases successively with the de-velopment, as shown in Fig. 6(b). These "guresclearly show the tendency towards a decrease inthe diversity of chemicals along the direction ofdevelopment.

This decrease in the diversity of chemicals isaccompanied by a decrease in the diversity of useof reaction paths in cellular dynamics. In Fig. 7,the amounts of use of each reaction path duringa certain period are plotted for each cell type. Inthis "gure, the data are arranged in the orderof the magnitude of the use of each reactionpath, computed as the average of the termCon (m, l, j)x(m)

i(t)(x(j)

i(t))a in eqn (2). This "gure

shows that, in type-0 dynamics with complexoscillation and diversity of chemicals, more di-verse reaction paths are used than simple dynam-ics of di!erentiated cell types, in which a smallnumber of reaction paths are used dominantlywhile other paths are hardly used.

Next, to con"rm the decrease in complexityof the dynamics along with the loss of multi-potency, we have measured the &&local'' Kol-mogorov}Sinai (KS) entropy. The KS entropyis known to be a good measure of the variety oforbits and the degree of chaos (see Appendix A).Although the KS entropy is de"ned using alldegrees of freedom in the system, includingall cells and the medium, here we measure onlythe complexity of the intra-cellular dynamics byneglecting the instability arising from cell}cellinteractions. We do this by computing the &&localKS entropy'' of the intra-cellular dynamics, de-"ned as the sum of positive Lyapunov exponents,by restricting the set of possible perturb-ations (the tangent space) to those of only the

FIG. 8. Relation between the division speed and thechemical diversity within a cell. Figs (a), (b), and (c) displaythe results obtained by using the examples in Figs 1, 3(a), (b),respectively. In the "gures, the data points show how thedivision speed depends on the diversity in the case wherethere are 200 total cells. Some of the points are overlaid andmay be invisible since the plotted value of the cells are ratherclose to each other.

FIG. 7. Use of reaction paths in dynamics of each celltype. The amounts of use of each reaction path mPl are com-puted by the average of the term Con(m, l, j)x(m)

i(t)(x(j)

i(t))a

over 20 000 time steps. The amount of the use of eachreaction path is plotted in the order of its magnitude, as thehorizontal axis representing the order. Each line corres-ponds to each cell type shown in Fig. 1.


intra-cellular dynamics of each cell. We havefound that this local KS entropy of di!erentiatedcells is always smaller than that of stem-typecells before di!erentiation. For example, the KSentropy of the stem-type cell in Fig. 1 is approx-imately 1.2]10~3, while it is 2.0]10~5 for type-2 cell, and less than 1.0]10~6 for type-1 andtype-3 cells. In the case with hierarchical di!eren-tiation, the local KS entropy decreases success-ively with each di!erentiation. These resultssuggest that the multipotency of a stem-type cellis due to the complexity of its intra-cellulardynamics.

6. Change in Growth Speed of Cells with theDevelopmental Process

In our model simulation and in nature, thegrowth speed of each cell depends on its cell typeand certain characteristics of the cell society envi-ronment, such as the number distribution of celltypes therein. In our simulations, during the "rststage of the development, the stem-type cells (e.g.the type-0 cells in Fig. 1) grow and divide rapidlywhen the number of cells is small and only stem-type cells exist in the system. However, as thenumber of cells increases and other types of cellsappear through di!erentiation, the growthspeeds of all cells start to depend on their types.

For example, some cell types maintain a highgrowth speed and undergo successive divisions,while some others start to lose chemicals into themedium and begin to shrink their volume.

In general, the growth speed of stem-type cellsis found to be smaller than those of di!erentiatedcells, when they coexist in the system. In Fig. 8,we display some examples of the relationshipbetween the growth speed of a type of cell and thechemical diversity within the cell, which is a good


measure of the degree of di!erentiation, as dis-cussed in Section 5. In the "gures, the data pointsshow how the division speed depends on thediversity in the case where there are 200 totalcells. Note that the growth speed is measured bythe speed for division in our model. Each "gurecorresponds to a di!erent reaction network.Here, clusters of points correspond to di!erentcell types, and the cluster with the highest diver-sity corresponds to stem-type cells. These exam-ples clearly show the increase of cellular growthspeed with the decrease of chemical diversity thatprogresses along the direction of di!erentiation.It should be noted that the growth speed of stem-type cells decreases only when the di!erentiatedcell types begin to coexist with them. That is, thegrowth speed of an isolated stem-type cell is notalways smaller than that of an isolated di!erenti-ated cell.

The decrease of the growth speed of stem-typecells can be explained in terms of a change inthe requirement for nutrients. A stem-type cellwith high chemical diversity absorbs a variety ofnutrients from the medium. On the other hand,a di!erentiated cell type with simple internaldynamics and lower chemical diversity is special-ized in the use of only a few kind of nutrients. Asshown in Fig. 7, in such cell with simple internaldynamics, a small number of reaction paths areused dominantly and they often form a simpleauto-catalytic reaction network. Since each reac-tion rate is nonlinear with respect to the involvedchemical concentration, the internal reactiondynamics of such a simple cell progresses faster.Then, required nutrients are transformed intonon-penetrating chemicals much faster thanthose in stem-type cell with complex cellular dy-namics. When the stem-type cells and di!erenti-ated types of cells coexist, di!erentiated cells cane$ciently obtain the nutrients they require, whilestem-type cells cannot do the same, or they mayeven release necessary nutrients, because the con-centration of these chemicals in the medium isoften lower than that in the stem-type cells.

Accordingly, the number density of stem cellsgenerally decreases as the total cell number in-creases. Let us further consider the case in whichthe total number does not increase inde"nitely,due to probabilistic deaths of each cell type.Then, depending on the network structure, it is

expected that either the stem cells continue toconstitute a small fraction of the total number ofcells and maintain a slow division speed, or thestem cells disappear in the steady state. In theformer case, the spontaneous regulation of di!er-entiation from the stem cells insures the stabilityof the cell number distribution with respect tochanges in the death rates of each cell type, whilein the latter case, the cell number distributionof di!erentiated cell types is vulnerable to suchchanges.

7. Discussion

To con"rm the validity of our chaos-basedmodel of stem-cell systems, we have carried outnumerical experiments using several di!erent setsof parameter values and choosing thousands ofdi!erent reaction networks. As a result, we havefound that the scenario discussed in precedingsections is commonly observed for some fractionof randomly generated reaction networks. Withthe parameter values used in the example con-sidered in Fig. 1, approximately 10% of random-ly chosen reaction networks result in oscillatorybehavior, while others converge to "xed points.Furthermore, approximately 20% of these oscil-latory dynamics are destabilized through cell div-ision. In these cases, the diversity of cell typesemerges as discussed above.

One may ask why we select such complexoscillatory dynamics to describe general mecha-nisms for the dynamics of stem cells, while onlya small fraction of randomly chosen reactionnetworks lead to such oscillatory dynamics. Inanticipation of this question, we have studied thegrowth speed of an ensemble of cells to determinewhat kind of reaction networks can possibly ap-pear through evolution. As is shown in Furusawa& Kaneko (2000), if the internal dynamics can bechaotic and if the cell ensemble contains a varietyof cell types di!erentiated from stem-type cells,the cell society can attain a larger growth speedas an ensemble by realizing a cooperative use ofresources, in comparison with an ensemble withhomogeneous cells and simpler internal dynam-ics. This suggests that the emergence of astem}cell system with complex cellular dynamicsand a diversity of cell types is a necessary coursein the evolution of multicellular organisms, as


aggregates of cells compete for "nite resourcesnecessary for growth and reproduction. Fromthis reasoning, we "nd that the use of reactionnetworks leading to complex oscillatory dynam-ics for the description of stem cells is also sup-ported by evolutionary consideration.

Note that the results found in this paper arechanged little when we change the model para-meters. They are essentially unchanged as long asthe magnitudes of the internal reaction term andthe cell}cell interaction term are of comparableorder, and the set of equations describing theinternal reaction are su$ciently complex to ex-hibit nonlinear oscillatory dynamics that in somecase can be chaotic. Such complex internaldynamics are observed with some fraction ofrandomly generated reaction networks, when thenumber of chemicals is su$ciently large (say fork'10) and the number of the reaction paths is inthe intermediate range. For example, whenk"32, there must be approximately 6}12 pathsfor each chemicals. If there are fewer or more, theintra-cellular dynamics converge to a "xed pointfor most cases, without di!erentiation. The re-sults are also independent of other details of themodel. For example, they are unchanged if wechange the catalysation degree a to 1 or 3.

8. Predictions

We believe that our results are universal ina class of dynamical systems satisfying minimalrequirements of the developmental process.Although some of these universal features arenot yet examined experimentally, we make somepredictions here as general features commonlysatis"ed in real stem}cell systems. To concludeour paper, we summarize the predictions we canmake using our model, and discuss the possibilityof experimental veri"cation.

8.1. CHAOTIC REACTION DYNAMICS IN A STEM CELL AS

A SOURCE OF STOCHASTIC BEHAVIOR

In our theory, the stochastic behavior of stemcell di!erentiation results from nonlinear, chaoticchemical reaction dynamics existing within thestem cell. Although such complexity of intra-cellular dynamics has not yet been observed, webelieve that they are essential in the behavior of

stem cells. It was recently reported that the Ca2`oscillation in a cell depends on the cellular state,and that can change the pattern of gene expres-sions (Dolmetsch et al., 1998). One possible wayto check our conjecture is simply to study theoscillatory reaction dynamics, such as Ca2` os-cillation, in a cell and check how the irregularityin the oscillation di!ers between a stem cell anda di!erentiated cell. Studying the dynamicchange of gene expression patterns is also a pos-sible way to check our theory.

Plasticity in the states of stem cells has beenfound experimentally. For example, neural stemcells were found to be capable of re-constitutingthe hematopoietic system in irradiated adult mice(Bjornson et al., 1999). These observations sug-gest that the behavior of stem cells can change#exibly according to the environment, while dif-ferentiated cells have no such #exibility. In ourtheory also, the variability of cellular behaviordecreases from a stem to di!erentiated cells. In-deed, in our numerical experiments, when theconcentrations of nutrients are changed evenslightly in the medium, stem cells often undergosigni"cant variation, and as a result new types ofdi!erentiated cells can appear. In our theory, this#exibility of stem cells results from the complex-ity of their internal dynamics with a variety ofchemical species, and we expect that this is truealso in real stem}cell systems.

8.2. INTERACTION-BASED DIFFERENTIATION

In our theory, the di!erentiation of a stem cellis induced by interactions between cells in thesystem. Thus, we believe that in real systems, thebehavior of each cell in a stem}cell system de-pends strongly on the existence of and the statesof surrounding cells. For example, we expect thatif a stem cell is maintained in an isolated environ-ment by continuously removing divided cells, itwill not be able to di!erentiate and will onlyreplicate itself.

Recall that in our model, di!erentiation ofa stem cell is not necessarily concurrent with itscell division. Rather, this di!erentiation occurswhen the instability of the cellular dynamics ex-ceeds some threshold, and this is caused by theincrease of cell number through cell division. Webelieve that similar behavior can be found in real


systems, in particular when the di!erentiationrate of a stem cell can be varied by changing thedistribution of other cell types and that this dif-ferentiation is not necessarily coincident with celldivision.

8.3. REGULATION OF THE DIFFERENTIATION

PROBABILITY LEADING TO THE ROBUSTNESS

OF THE CELL SOCIETY

As mentioned in previous sections, the prob-abilities governing di!erentiation of a stem cellare not "xed in our theory, but are controlled bythe distribution of surrounding cell types. Animportant point here is that this regulation ofdi!erentiation occurs in such a manner that thee!ect of an external perturbation is compensated,in analogy to the Le Cha( telier principle inthermodynamics. This regulation is that whichcreates the robustness of the system.

By controlling the cell-type distribution, onecan examine experimentally if the rate of di!eren-tiation of stem cells is regulated so as to compen-sate for external change. For example, if thedeath rate of one type of cell is increased, or cellsof this type are continually removed, the di!eren-tiation rate from the stem cell into that type ofcell should be increased.

8.4. IRREVERSIBLE LOSS OF MULTIPOTENCY

CHARACTERIZED BY DECREASE OF COMPLEXITY

IN CELLULAR DYNAMICS

In our numerical experiments, stem cells,which have the potential to di!erentiate, possesscomplex cellular dynamics with irregular oscilla-tions and a high diversity of chemical species,while di!erentiated cell types always possesssimpler dynamics, for example, those with "xed-points or regular oscillations, with a low diversityof chemical species. We have characterized thischange of cellular dynamics by the decrease ofchemical diversity, decrease of diversity in use ofreaction paths, and the decrease of instabilityof the dynamics, measured by the KS entropy.The results of the present study suggest that, thischange in dynamics along with development isuniversal in systems composed of cells with intra-cellular dynamics and cell}cell interactionsamong these cells.

In terms of molecular biology, the decrease indiversity of chemicals and diversity of use ofreaction paths in cellular dynamics can be inter-preted as the decrease of the number of expressedgenes. This decrease of the number of expressedgenes, indeed, has been observed in real systems.For example, multipotent progenitor cells in he-matopoietic systems are found to co-express lin-eage-speci"c genes of several di!erentiated celltypes, such as erythroid and myeloid cells (MingHu et al., 1997). It should be noted that the levelsof this co-expression in multipotent cells aresubstantially lower than those seen in maturelineage-committed cells. It is suggested byexperimental "ndings that this low-level co-expression in the multipotent cell is sustainedby complex reaction networks including positive-and negative-feedback reactions. Here, our con-jecture in this regard is supported by existingexperimental observations. Considering the gen-erality of our conjecture, it is relevant to checkhow the diversity of gene expressions changes inthe course of development from a stem cell.Furthermore, we conjecture that not onlythe &&static'' diversity but also the &&dynamic''complexity in intra-cellular dynamics decreasesalong with the developmental process froma stem cell.

While during the normal course of develop-ment, this loss of multipotency is irreversible, it ispossible to recover the multipotency of a di!eren-tiated cell through perturbation, by changing thediversity of chemicals or the complexity of thedynamics. For example, by expressing a varietyof genes compulsively in di!erentiated cells, theoriginal multipotency may be regained. Notethat, according to our model simulations, thebasin of attraction of the stem cell is much largerthan that of di!erentiated cells. This implies thatby adding a large perturbation that results in thepresence of a variety of chemicals in a cell, the cellde-di!erentiates back into a stem cell.

8.5. SMALLER GROWTH SPEED OF STEM

CELLS THAN DIFFERENTIATED CELLS

As mentioned in Section 6, in our simulation,the growth speed of stem cells is generally smallerthan that of di!erentiated cells, when the stemand di!erentiated cells coexist. In many cases, the


stem cells stop growing or begin to shrink afterthe number of cells di!erentiated from them hasincreased to a certain level. In general, there isa negative correlation between the chemical di-versity (or the diversity in gene expressions) andthe growth speed, which can be experimentallyexamined.

Of course, in a real biological system, theactivation or suppression of cell growth anddivision involves several factors, including signalmolecules called &&growth factors'', in addition tothe condition for nutrients. Still, we believe thatthis smaller growth speed of stem cells is a gen-eral characteristic of the diversi"cation process instem}cell systems. In our numerical simulations,the speed of di!erentiated types is greater be-cause they are specialized in obtaining certainspeci"c nutrients, in contrast with the stem cells,which must obtain a larger variety of nutrients.We suspect that this may re#ect the situation inreal systems in which stem cells require diversegrowth factors for their growth, while di!erenti-ated cells are more specialized in some speci"cfactors.

In real multicellular systems, stem cells, suchas, neural or hematopoietic stem cells, often entera quiescent state (i.e. a G

0state) without division.

These stem cells resume division actively whensome external condition is changed, for example,an injury that decreases the population of di!er-entiated cells. On the other hand, di!erentiatedcells, such as progenitor cells in the hema-topoietic system and interstitial crypt, generallyhave a faster growth speed. Here, as an experi-mentally veri"able proposition, we conjecturethat the growth of stem cells generally stops ifthey are surrounded by a su$cient number ofdi!erentiated cells (that are produced by thatstem cell), while the decrease of the number ofsurrounding cells leads to a resumption of thedivision and di!erentiation.

8.6. POSITIVE CORRELATION BETWEEN GROWTH

SPEED OF CELL ENSEMBLE AND CELLULAR DIVERSITY

As shown in a previous work (Furusawa &Kaneko, 2000), an ensemble of cells with a varietyof cell types that is sustained by di!erentiationsfrom stem-type cells generally has a largergrowth speed than an ensemble of homogeneous

cells, because of the greater capability of the for-mer to transport and share nutritive chemicals.This positive correlation between the growthspeed and the diversity of cell types (at the cellensemble level) can be experimentally veri"ed inany stem}cell system. For example, if a stem cellloses the ability to di!erentiate through muta-tion, tissue consisting only of the homogeneouscells has smaller growth speed as a whole thantissue of the wild type with various cell types. Infact, this correlation is suggested by studies ofcultures of the hematopoietic system (Suda et al.,1984). Further study of this correlation will alsobe important to understand the evolutionaryprocess of multicellular organisms.

The authors are grateful to T. Yomo, T. Ikegami,and S. Sasa for stimulating discussions. The work issupported by Grant-in-Aids for Scienti"c Researchfrom the Ministry of Education, Science and Cultureof Japan (11CE2006; Komaba Complex Systems LifeProject; and 11837004). One of the authors (CF) issupported by research fellowship from Japan Societyfor Promotion of Science.

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APPENDIX A

Brief Introduction to Dynamical Systems Study

Theoretical study of cell di!erentiation ascoupled dynamical systems has been put forwardsince (Kaneko & Yomo, 1994), and a model exhi-biting stem-cell-type behavior is presented inFurusawa & Kaneko (1998a). In these dynamicalsystems study, a cell type is given as an attractingstate of intra-cellular dynamics, stabilizedthrough cell}cell interaction, while a di!erenti-ation is represented by a dynamical switch

from one state to another. Theoreticalbackground of such coupled dynamical systemslies in the study of coupled map lattices(Kaneko, 1989) and globally coupled maps(Kaneko, 1990). In this appendix, we brie#y ex-plain some terms in dynamical systems in thepresent paper, for those who are not familiar withdynamical systems. See textbooks on chaos ornonlinear dynamics for details (Ott, 1993; Bergeet al., 1984).

Chaotic attractor. In dynamical systems, thechange of a state is represented by an orbit in thephase space. The region whose neighborhood anorbit recurrently returns to is called an attractor.Roughly speaking, the attractor is the region towhich the orbit is attracted to as time passes. Anattractor can be a "xed point or a cycle, whilethere is a chaotic attractor that is neither periodicnor represented by a combination of severalcycles.

¸yapunov exponents. In a chaotic attractor,small di!erence in initial condition is amplifed.¸yapunov exponents characterize this orbital in-stability (i.e. the ampli"cation or contraction rateof a tiny di!erence around a orbit) in a dynamicalsystem. Roughly speaking, the distance betweentwo orbits with slight initial di!erence increasesas exp(jt) for a chaotic attractor, and this j givesthe (maximal) Lyapunov exponent.

Stochasticity from deterministic dynamics. Fora chaotic attractor, the future course of an orbitis unpredictable as long as our precision ofobservation is "nite, even though the dynamicsare deterministic. Then, the dynamic processis characterized only by introducing probability.Hence, chaos gives a prototype of how adeterministic system produces a stochasticbehavior, which is relevant in understandingthe two faces of stem cell dynamics discussed inSection 1.

Basin of attraction. In some dynamical systems,there are several attractors. Depending on theinitial condition, orbits are attracted to one ofthem. The set of initial conditions that areattracted to a given attractor is called basin ofattraction.


Kolmogorov}Sinai (KS) entropy. It is a measureof the time rate of creation of informationas an orbit evolves. In a chaotic orbit,small initial di!erence is ampli"ed, and aslong as one observes the state with a "niteprecision, new information on the initialcondition is gained with some rate, which isthe KS entropy. Roughly speaking, thisentropy gives a measure of a variety of orbits.In general, the sum of the positive Lyapunovexponents gives an upper bound of the KS en-tropy, while the two are proven to coincide forsome cases.

Milnor attractor. Milnor de"ned an attractor asan attracting set whose basin has a positivemeasure. In many cases, this de"nition agreeswith our naive picture on attractor; i.e. orbits thatare attracted to it and remain stable within. Insome type of attractors like the de"nition of Mil-nor, however, a "nite measure of initial points areattracted to the attractor, but any small per-turbation to the orbit can kick it out of theattractor. [For example, consider the case thatthe boundary of the basin of the attractor touchesthe attractor itself at some point(s).] Such &&weak''attractor is called Milnor attractor.

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