Emergence of Rules in Cell Society: Differentiation, Hierarchy, and Stability

Abstract
A dynamic model for cell differentiation is studied, where cells with internal chemical reaction dynamics interact with each other and replicate. It leads to spontaneous differentiation of cells and determination, as is discussed in the isologous diversification. Following features of the differentiation are obtained: (1)Hierarchical differentiation from a ``stem'' cell to other cell types, with the emergence of the interaction-dependent rules for differentiation; (2)Global stability of an ensemble of cells consisting of several cell types, that is sustained by the emergent, autonomous control on the rate of differentiation; (3)Existence of several cell colonies with different cell-type distributions. The results provide a novel viewpoint on the origin of complex cell society, while relevance to some biological problems, especially to the hemopoietic system, is also discussed.

arXiv:adap-org/9802002v1 13 Feb 1998
Emergence of Rules in Cell Society:
Diﬀerentiation, Hierarchy, and Stability
Chikara Furusawa and Kunihiko Kaneko
Department of Pure and Applied Sciences
University of Tokyo, Komaba, Meguro-ku, Tokyo 153, JAPAN
Abstract
A dynamic model for cell diﬀerentiation is studied, where cel ls with
internal chemical reaction dynamics interact with each oth er and repli-
cate. It leads to spontaneous diﬀerentiation of cells and det ermination,
as is discussed in the isologous diversiﬁcation. Following features of
the diﬀerentiation are obtained: (1)Hierarchical diﬀerenti ation from a
“stem” cell to other cell types, with the emergence of the int eraction-
dependent rules for diﬀerentiation; (2)Global stability of an ensemble
of cells consisting of several cell types, that is sustained by the emer-
gent, autonomous control on the rate of diﬀerentiation; (3)E xistence of
several cell colonies with diﬀerent cell-type distribution s. The results
provide a novel viewpoint on the origin of complex cell socie ty, while
relevance to some biological problems, especially to the he mopoietic
system, is also discussed.
1 Introduction
A multicellular organism is an ordered clone of a fertilized egg. All the
cells contain the same genome set but are specialized in diﬀerent ways . The
emergence of diﬀerent cell types is determined rather precisely, w hile the
developmental process of the cells, viewed as a cell society, has ro bustness
against perturbations.
In molecular biology, the diﬀerentiation processes are often regar ded as
on-oﬀ switching processes. Switch depends on inputs by signal mole cules,
which leads to a variety of cell types as outputs. A large number of r eac-
tions between inputs and outputs are represented as a “cascade ”, where the
reactions are assumed to be approximately independent of the oth er reac-
tion processes in the cell. The switching behavior (given by the sigmoid al
function) is assumed to be generated from a chain of these reactio ns. With
this viewpoint, one can decompose the diﬀerentiation by successive local el-
ementary processes. It enables us to elucidate the diﬀerentiation processes
by experimental methods, where several signal molecules and ess ential genes
for diﬀerentiations are identiﬁed.
Of course, the development progresses through cooperation of several pro-
cesses. Successive diﬀerentiation processes are often express ed as “canaliza-
tion”, where diﬀerentiations are captured as a result of dynamics o f complex

chemical networks, in contrast with a linear combination of simple pat hways
of chemical reactions. The pioneering study of Kaufmann (1969) d emon-
strated that the Boolean network of genes gives a variety of ﬁnal states de-
pending on the initial conditions, and he has suggested that each ﬁn al state
corresponds to each cell type. However, needless to say, a single initial state
embedded in a fertilized egg can produce several diﬀerent cell type s. Thus,
the following questions remain unanswered about the gene network : How
do the diﬀerent initial states leading to the diﬀerent cell types arise in the
process of development? How does a selection of speciﬁc initial cond itions
lead to precise rules of diﬀerentiations?
It should be noted that in the gene network picture cellular interact ions
are not explicitly taken into account, which should be important in the course
of development. A pioneering study for the pattern formation is du e to
Turing (1952), where dynamic instability by the cellular interactions le ads to
the pattern formation (Newman, 1990). However, it remains still u nsolved
how such cell-to-cell interactions are incorporated with internal d ynamical
complexity including the gene networks (see also Bignone, 1993; Mjo lsness
et al., 1991; and Thomas et al, 1995).
Hence it is necessary and important to consider a system of interna l dy-
namics with suitable cell-cell interactions. One of the authors (KK) a nd
Yomo have performed several simulations of interacting cells with int ernal
biochemical networks and cell divisions that lead to the change in the num-
ber of degrees of freedom. The “isologous diversiﬁcation theory” is proposed
as a general mechanism of spontaneous diﬀerentiation of replicatin g biologi-
cal units (Kaneko & Yomo, 1994, 95, 97). In the theory, the follow ing three
points are essential.
• Spontaneous diﬀerentiation: The cells, which have oscillatory chem-
ical reactions within, diﬀerentiate through interaction with other c ells.
This diﬀerentiation is provided by the separation of orbits in the phas e
space. The dynamics of cells ﬁrst split into groups with diﬀerent phas es
of oscillations, and then to groups with diﬀerent compositions of che m-
icals. These diﬀerentiations are not caused by speciﬁc substances , but
are triggered by the instability brought about by nonlinear systems .
The background of this lies in the dynamic clustering in globally cou-
pled chaotic systems (Kaneko, 1990, 91, 92).
• Inheritance of the diﬀerentiated state to the oﬀspring: Each
diﬀerentiated state of a cell is preserved by the cell division and tra ns-
mitted to its oﬀspring. Chemical composition of a cell is recursively
kept with respect to divisions. Thus a kind of “memory” is formed,
through the transfer of initial conditions (e.g., of chemicals). By re pro-
duction, the initial condition of a cell is chosen so that the same cell
type is produced at the next generation.
• Global stability: Multicellular organism often shows a robustness
against some perturbation, such as somatic and other mutations. An
extreme example is seen in a mutation to triploid in newt, where the cell
size becomes three times, but the total cell number is reduced to o ne

third, and the ﬁnal body remains not much aﬀected by the mutation
(Fankhauser, 1995).
The distribution of cell types obtained is robust against external p ertur-
bations. For instance, when the number of one type of cells is decre ased
by external removal, the distribution is recovered by further diﬀe renti-
ations to generate the removed cell-type. In this theory, althoug h the
instability triggers the diﬀerentiations, the cell society as a whole is
stabilized through cell-to-cell interactions.
In the present paper we extend these previous studies (Kaneko & Yomo,
1997) to incorporate the formation of a complex cell society. We ex tend our
model to allow for complex internal dynamics, in particular, focusing on the
following three topics.
• (i) The hierarchical organization and the emergence of stoc has-
tic rules; The cell diﬀerentiation process in nature follows a hierar-
chical organization. For example, the pluripotent cells like stem cells
give rise to committed cells, which further diﬀerentiate to terminally
diﬀerentiated cells. Here the rules of diﬀerentiation are written as e x-
pressions of DNA in principle, but it should be noted that the diﬀer-
entiation is often interaction dependent. Furthermore, in the hem opoi-
etic system, the diﬀerentiation process appears to be stochastic , and
the probability of each choice seems to depend also on the distribu-
tion of cell types (Ogawa, 1993). Hence it is interesting how such
interaction-dependent rules of hierarchical diﬀerentiation are fo rmed
naturally through the interplay between internal dynamics and cell- to-
cell interactions.
• (ii) Stability of cell types and cell groups; Cells belonging to the
same cell-type also slightly diﬀer each other. Hence discretization of
states to types and their continuous change coexist among cells. T he
diﬀerentiation rules of cell types are written for the discrete type s.
When cell diﬀerentiation is determined, memory of the discrete stat e is
stable against cell division. Then, the state has to be dynamically sta -
ble (like an attractor), while for stem cells or undetermined cells, the ir
state must have both stability and variability (diﬀerentiability) by divi-
sions. Here we are interested in how such stability and diﬀerentiability
are compatible as a state of dynamical systems. Besides the stabilit y of
a cellular state, stability about the distribution of cell types has to b e
attained through the developmental process. For example, the d istri-
bution of cell types in the hemopoietic system is robust against exte rnal
perturbations. Here we try to answer the question of stability thr ough
an interplay of internal dynamics and cell-cell interaction, that lead s to
modulation of internal cellular states and to stability of distribution o f
the cells at the ensemble level.
• (iii) Diﬀerentiation of cell colony; In an organism, there often
appears a higher level of diﬀerentiations, leading to several distinc t
types of tissues. They consist of diﬀerent types of cells and/or diﬀ erent
distribution of cell types. Indeed in the hemopoietic system, sever al

colonies consisting of diﬀerent cell types appear from same stem ce lls
(Nakahata et al., 1982). It is an important question how a single cell
can form such diﬀerent cell colonies. This is a higher level question
than cell diﬀerentiation, since the population of cell types has to be
diﬀerentiated.
In the present paper we study the above three problems by exten ding the
previous model of cell diﬀerentiation, to allow for complex internal d ynamics.
Here the cellular states are given by a set of chemical concentratio ns, while
the internal dynamics is given by mutually catalytic reaction network s. In
contrast with the previous model, the internal dynamics allows for c haos and
also coexistence of multiple attractors. Interaction among cells is g iven by
the diﬀusive transport of chemicals between each cell and a homoge neous
environment. Cell volume is increased through the transport of ch emicals
from the environment, which leads to the cell division when it is larger t han
a given threshold.
By allowing complex dynamics at the internal cell level, we will show
that the above three problems are answered from our standpoint . First, hi-
erarchical diﬀerentiation of several cell types is formed. There a ppears a cell
type that plays the role of “stem cell”, from which diﬀerent cell type s are
diﬀerentiated. The probabilistic switch of cell types is given through the in-
ternal dynamics, whose rate is dependent on the interaction, and accordingly,
on the distribution of other cell types. Second, the stability of cell types is
given as a “partial attractor” (to be discussed) of the internal d ynamics, sta-
bilized through interactions. Third, stochastic population dynamics of cell
types emerges as a higher level. It is found that this dynamics has se veral
attracting states, which supports diﬀerent stable cell colonies (t issues).
The organization of the paper is as follows. In §2, our model is pre-
sented. Although a speciﬁc type of catalytic reaction network is ad opted
in the present paper, it should be noted that the results are gener ally seen
in a variety of reaction networks. Although the interaction we adop t here is
global, in the sense that all cells interact with each other, our main co nclusion
on the diﬀerentiation and the formation of cell society is invariant ev en if the
“spatial” eﬀect is explicitly taken into account as local diﬀusion proce ss. In
§3, we will show numerical results of the evolution of cell society from a single
cell, where the emergence of distinct cell types is given. Diﬀerentiat ion of
these cells is found to obey a speciﬁc rule, that emerges as a higher le vel than
the chemical reaction rules we have adapted. The mechanism of disc retiza-
tion of states, and the formation of (interaction-dependent) ce ll memory is
discussed in §4. The rule of “stochastic” diﬀerentiation from a stem-type
cell is studied in §5, where the stability of the population of cell types is
noted. In §6, the diversity of cell colonies is shown in relation with several
attracting states of a higher-level dynamics, i.e., the population dy namics of
cell types. Summary and discussions are given in §7 and §8, where relevance
of our results to cell biology is discussed, which covers origin of stem cells, in
particular stochastic branching, stability and diversity of cell colon ies in the
hemopoietic system, and the origin of multicellular organism.

2 Model
Our model for diﬀerentiation consists of
• Internal dynamics by biochemical reaction network within each cell
• Interaction with other cells through media: Inter-cellular dynamics
• Cell division
The basic strategy of the modeling follows the previous works (Kane ko &
Yomo, 1997), although we take diﬀerent dynamics for each of the a bove three
processes. In essence we assume a network of catalytic reaction s for internal
dynamics that allows for a periodic and/or chaotic oscillations of chem icals,
while the interaction process is just a diﬀusion of chemicals through m edia.
We represent the internal state of a cell by k chemicals’ concentrations as
dynamical variables. Cells are assumed to be in surrounding media, wh ere
the same set of chemicals is given. Hence the dynamics of the interna l state
is represented by a set of variables x(m)
i (t), the concentration of the m-th
chemical species at the i-th cell, at time t. The corresponding concentration
of the species in the medium is represented by a set of variables X (m)(t). We
assume that the medium is well stirred by neglecting the spatial varia tion
of the concentration, so that all cells interact each other throug h identical
environment.
2.1 Internal chemical reaction
Within each cell, there is a network of biochemical reactions. The net work
includes not only a complicated metabolic network but also reactions a ssoci-
ated with genetic expressions, signaling pathways, and so on. In th e present
model, a cellular state is represented by the concentrations of k chemicals.
As internal chemical reaction dynamics we choose a catalytic netwo rk
among the k chemicals. Each reaction from the chemical i to j is assumed
to be catalyzed by the chemical ℓ, which is determined randomly. To repre-
sent the reaction-matrix we adopt the notation Con(i, j, ℓ) which takes unity
when the reaction from the chemical i to j is catalyzed by ℓ, and takes 0
otherwise. Each chemical has several paths to other chemicals, w hich act
as a substrate to create several enzymes for other reactions. Thus these
reactions form a complicated network. This matrix is generated ran domly
before simulations, and is ﬁxed throughout the simulation. We use th e same
reaction-matrix throughout a series of simulations in this paper (se e also §3
and §7 for dependence on the reaction-matrix).
Usually, chemical kinetics with enzymes is solved under some approxi-
mations, like Michaelis-Menten form. In this paper, we assume quadr atic
eﬀect of enzymes. Thus the reaction from the chemical m to ℓ aided by the
chemical j leads to the term e1x(m)
i (t)(x(j)
i (t))2, where e1 is a coeﬃcient for
chemical reactions, which is taken identical for all paths. The quad ratic ef-
fect of enzymes is not essential to our scenario of cell diﬀerentiat ions. Several
other forms on the internal dynamics lead to qualitatively the same b ehavior,
as long as nonlinear oscillation is included. The scenario of the diﬀerent iation

which we propose here is independent of the details of this speciﬁc ch oice of
biochemical dynamics.
Besides the change of chemical concentrations, we have to take in to ac-
count the change of the volume of cell. The volume is now treated as a
dynamical variable, which increases as a result of transportation o f chem-
icals into the cell from the environment. Of course, the concentra tions of
chemicals are diluted according to the increase of the volume of the c ell.
For simplicity, we assume that the volume of cell is proportional to th e sum
of chemicals in the cell. Under this assumption, the operation which co m-
pensates the concentration of chemicals with the volume change is id entical
to imposing the restriction ∑
ℓ x(ℓ)
i = 1, namely normalizing the chemical
concentrations at each step of the calculation, while the volume cha nge is
calculated from the transport as will be given later.
2.2 Interaction with other cells through media
Each cell communicates with its environment through transport of chemicals.
Interactions between cells, thus, occur through the environmen t. Here, the
environment does not mean external environment for individual or ganism,
but is intended as interstitial environment of each cell. In this model, we
consider only diﬀusion process through the cell membrane. Thus, t he rates
of chemicals transported into a cell are proportional to diﬀerence s of chemical
concentrations between the inside and the outside of the cell. Of co urse, the
transport through the membrane is not so simple, including several mecha-
nisms such as channel proteins and endocytosis. We omit these com plicated
mechanisms for simplicity.
The transportation or diﬀusion coeﬃcient should be diﬀerent for diﬀ erent
chemicals. Here we assume that there are two types of chemicals, t hose which
can penetrate the membrane and which can not. We use the notatio n σm,
which takes 1 if the chemical x(m)
i is penetrable, and 0 otherwise.
To sum up all these process, the dynamics of chemical concentrat ion in
each cell is represented as follows:
dx(ℓ)
i (t)/dt = δx(ℓ)
i (t) − (1/k)
k∑
l=1
δx(ℓ)
i (t) (1)
with
δxℓ
i(t) =
∑
m,j
Con(m, ℓ, j) e1 x(m)
i (t) (x(j)
i (t))2
−
∑
m′,j′
Con(ℓ, m′, j′) e1 x(ℓ)
i (t) (x(j′)
i (t))2
+σℓD(X (ℓ)(t) − x(ℓ)
i (t)) (2)
where the term with
∑
Con(· · ·) represents paths coming into ℓ and out of
ℓ respectively. The term δx(ℓ)
i gives the increment of chemical ℓ, while the
second term in eq.(1) gives the constraint of ∑
ℓ x(ℓ)
i (t) = 1 due to the growth
of the volume. The third term in eq.(2) represents the transport b etween
the medium and the cell, where D denotes a diﬀusion constant, which we
assume to be identical for all chemicals. Since the penetrable chemic als in

the medium can be consumed with the ﬂow to the cells, we need some ﬂo w
of chemicals (nutrition) into the medium from the outside. By denotin g the
external concentration of these chemicals by X and its ﬂow rate per volume
of the medium by f , the dynamics of penetrable chemicals in the medium is
written as
dX (ℓ)(t)/dt = f σℓ(X (ℓ) − X (ℓ)(t)) − (1/V )
N∑
i=1
σℓD(X (ℓ)(t) − x(ℓ)
i (t)) (3)
where N denotes the number of the cells in the environment, and V denotes
the volume of the medium in the unit of a cell.
2.3 Cell division
Each cell takes penetrable chemicals from the medium as the nutrien t, while
the reactions in the cell transform them to unpenetrable chemicals which
construct the body of the cell such as membrane and DNA. As a res ult of
chemical ﬂow, the volume of the cell is increased by the factor (1 +
∑
ℓ δxℓ
i (t))
per dt. In the present paper, the cell is assumed to divide into two almost
identical cells when the volume of the cell is doubled. 1 2
The concentrations of chemicals in the daughter cells are almost equ al
to the concentrations of the mother cell. “Almost” here means tha t the
concentrations of chemicals in a daughter cell are slightly diﬀerent f rom the
mother’s. Each cell has (1 + ǫ)x(l) and (1 − ǫ)x(l) respectively with a small
“noise” ǫ, a random number with a small amplitude, say over [ − 10−6, 10−6].
Although the existence of imbalance is essential to the diﬀerentiatio n in our
model and in nature, the degree of imbalance itself is not essential t o our
results to be discussed. The important feature of our model is the ampliﬁ-
cation of microscopic diﬀerences between the cells through the inst ability of
the internal dynamics.
2.4 Internal dynamics in single cell
Before studying the dynamics of cell society, we demonstrate a ty pical be-
havior of our model by taking only one cell and medium. In our theory , the
fundamental assumption is that the internal dynamics of chemicals in the
cell shows oscillation as in Fig.1. In real biological systems, oscillations are
1In other words, the cell divides at the time t when
∫ t
tb
exp(1 +
∑
ℓ
δxℓ
i (t′))dt′ = 2 (4)
is satisﬁed since the previous division time tb.
2 Embryos fall into two general categories: those in which cell division is accompanied
by growth of the cells back to their former volume (as mammals and bir ds); those in which
cell division results in cells 1/2 the previous volume (as amphibians). Alt hough our model
here adopts the division process as in mammals and birds, we have also conﬁrmed that
the present diﬀerentiation mechanism also holds for a model with amp hibians-like rules,
where cell division makes cell volume 1/2, and each cell interacts with neighborhood cells
like gap junctions.

observed in some chemical substrates such as Ca, NADH, cyclic AMP , and
cyclins (Tyson et al., 1996; Hess et al., 1971; Alberts et al., 1994). He nce it is
natural to postulate such oscillatory dynamics to our model. The imp ortance
of oscillatory dynamics in cellular systems has been pointed out by Goo dwin
(1963).
The nature of internal dynamics by eqs.(1)-(2) depends on the ch oice of
the reaction network, in particular on the number of paths in the re action
matrix. When the number of reaction paths is small, cellular dynamics f alls
into a steady state without oscillation, where a small number of chem icals is
dominant while other chemicals’ concentrations vanish. On the othe r hand,
when the number of reaction paths is large, many chemicals generat e each
other. Then chemical concentrations take constant values (whic h are of-
ten almost equal). Only for medium number of reaction paths, non-t rivial
oscillations of chemicals appear as in Fig.1. We use such network for ou r
simulation. It is not easy to estimate the number of paths in real bioc hem-
ical data, although they may suggest the medium number (3-6) of p aths as
required in our simulation.
Furthermore, the behavior of dynamics depends on the number of pene-
trating chemicals. The number of penetrating chemicals is another c ontrol
parameter for the capacity of the oscillation or diﬀerentiation. Whe n the
number of penetrating chemicals is small, e.g., only one, the rate of ra n-
domly chosen reaction networks which show oscillatory dynamics is sm all.
On the other hand, when the number of penetrating chemicals is too large,
it is also diﬃcult to obtain the network with oscillatory dynamics.
Another relevant factor to the nature of internal dynamics is the frequency
of auto-catalytic paths. Indeed, the oscillatory dynamics is rathe r common
as the number of auto-catalytic paths is increased (see §7).
3 Diﬀerentiation Process: Numerical Results
We have performed several simulations of our model with diﬀerent c hemical
networks and diﬀerent parameters. Since typical behaviors are r ather com-
mon, we present our results by taking a speciﬁc chemical network w ith the
number of chemicals k = 20. 3
As an initial condition, we take a single cell, with randomly chosen chem-
ical concentrations of x(ℓ)
i satisfying ∑
ℓ x(ℓ)
i = 1. In Fig.1, we have plotted
a time series of concentration of the chemicals in a cell, when only a sing le
cell is in the medium. This attractor of the internal chemical dynamic s is a
limit cycle, whose period is longer than the plotted range in Fig.1. We call
this state “attractor-0” or “type-0” in this paper. This is the only attractor
that is detected from randomly chosen initial conditions 4.
3 In this example, we do not choose reaction paths equivalently among all chemicals,
but select two class of reactions randomly. One class of reactions is paths from penetrable
to any other chemicals, and another is paths from any of chemicals t o penetrable ones. The
purpose of this selection is to enhance auto-catalytic reaction loop , and to get oscillatory
reaction dynamics easily. Of course, reaction networks chosen eq uivalently and randomly
can also show the same type of behavior to be discussed in this paper . As for relationship
between auto-catalytic reactions and our scenario for diﬀerentia tion, see also §7.
4As will be seen later, there is another attractor as a single cell stat e. However, this

With the diﬀusion term, external chemicals ﬂow into the cell, which lead s
to the increase of the volume of the cell. Thus the cell is divided into tw o,
with almost identical chemical concentrations. Chemicals of the two daughter
cells oscillate coherently, with the same dynamical behavior as the mo ther
cell (i.e., attractor-0). Successive cell divisions occur simultaneou sly, and the
cell number increases as 1 − 2 − 4 − 8 · · ·, up to some threshold number. At
this stage, internal dynamics of each cell belongs to the same attr actor (i.e.,
attractor-0), but the oscillations are no longer synchronized. Th e microscopic
diﬀerences introduced at each cell division are ampliﬁed to a macrosc opic
level through the interaction, which destroys the phase coheren ce.
When the number of cells exceeds this threshold value, some cells sta rt
to show a diﬀerent type of dynamics. The threshold number depend s on
the parameters of our model. In the present example, 2 cells start to show
a diﬀerent dynamical behavior (as plotted in Fig.2(a)), when the tot al cell
number becomes 16. In Fig.2(a), the time series of the chemicals in th is cell
are plotted. We call the state as “partial attractor-1” (or “typ e-1” cell). We
do not call it an attractor, since the state does not exist as an att ractor of
internal dynamics of a single cell. As will be discussed later, the stabilit y
of the state is sustained only through the interaction. In Fig.3(a), orbits of
chemical concentrations are plotted in the phase space during the transition
form type-0 to type-1. It shows that each attractor occupies d istinct regimes
in the phase space. These two types of cells are clearly distinguishab le as
digitally distinct states. Hence we interpret this phenomenon as diﬀe rentia-
tion.
As the cell number further increases, another type of cell appea rs, which
we call type-2 here. It is again diﬀerentiated from the type-0 cell ( see Fig.2(b)
and Fig.3(b)). The type-0 cells have potentiality to diﬀerentiate to e ither “1”
or “2”, while some of the type-“0” cells remain to be of the same type by the
division.
For some simulations (i.e., for some initial conditions), the diﬀerentiat ion
process stops at this stage, and only three types of cells coexist. In many
other simulations, however, the diﬀerentiation process continues . At this
stage, hierarchical diﬀerentiation occurs. The cell type “1” furt her diﬀeren-
tiates into either of three groups represented as “3”, “4”, or “5 ”. The time
series of these three types are shown in Fig.2(c)-(e). The interna l dynamics of
each type is plotted in a projected phase space in Fig.4 . The orbit of t ype-1
cell itinerates over the three regions corresponding to “3”, “4”, and “5”. For
example, Fig.3(c) shows a switch from type-1 to type-3 in the phase space
by taking a projection diﬀerent from that in Fig.3(a)(b) (note the d iﬀerence
of scales). It is also noted that the diﬀerence by cell types is more c learly
distinguishable by chemicals with lower concentrations.
In the normal course of cell diﬀerentiation process (without exte rnal oper-
ation), cells of the types “2” and “1” reproduce themselves or fur ther diﬀer-
entiate to the other cell types, but the oﬀspring never go back to the type-0
cell. Besides the cell type-2, the cell types-3, 4, and 5 reproduce themselves
without any further diﬀerentiation. Among these three types, on ly the cell
attractor is not observed when the initial condition is randomly chos en; in other words
the basin volume for it is very small.

type-5 is an attractor by itself, while others replicate only under th e pres-
ence of diﬀerent types of cells. Indeed, the type-5 is rather spec ial, whose
appearance destabilizes the cell society consisting of “0”, “1”, an d “2”. Once
the type-5 cell appears, all the cells will ﬁnally be transformed to th is type.
Whether the type-5 cell appears or not depends on the initial cond ition, while
the cell society without the type keeps diversity of cell types (see §6).
At this stage the diﬀerentiation is determined, and cellular memory is
formed as is ﬁrst discussed in (Kaneko and Yomo, 1997). According ly we
can draw the cell lineage diagram as shown in Fig.5, where the division
process with time is represented by the connected line between mot her and
daughter cells while the color in the ﬁgure shows the cell type.
The switch of types by diﬀerentiations turns out to obey a speciﬁc r ule.
In Fig.6, we write down an automaton-like representation of the rule of dif-
ferentiation. The node “0” has three paths; one to itself, and the others to
the nodes “1” and “2”. The path to itself means replication of the sa me cell
type through division, while the other paths give the diﬀerentiation t o the
corresponding cell types. Fig.6 represents the potentiality of the se diﬀeren-
tiations.
Note that this diﬀerentiation is not induced directly by the tiny diﬀer-
ences introduced at the division. The switch from one cell-type to an other
does not occur simultaneously with the division, but occurs later thr ough
the interaction among the cells. This phenomenon is caused by dynam ical
instability in the total system consisting of all cells and medium. The tin y
diﬀerence between two daughter cells is ampliﬁed to yield macroscopic diﬀer-
ence through the interaction. Our results show that these trans itions are not
accompanied by the cell division but occur through cell-to-cell inter actions.
This conclusion is consistent with experimental data, where the ons et of new
gene expression is not always accompanied by the cell division. Accor ding
to our theory and simulations, the time lag between the cell division an d
the onset of new gene expressions depends on the cell-to-cell inte raction, i.e.,
the surrounding cells. On the other hand, change in the number of d egrees
of freedom by division ampliﬁes the instability in the dynamics of the tot al
system. When the instability exceeds some threshold, the diﬀerent iations
start. Then, the emergence of another cell type stabilizes the dy namics of
each cell again. The cell diﬀerentiation process in our model is due to the
ampliﬁcation of tiny diﬀerences by orbital instability (transient chao s), while
the coexistence of diﬀerent cell types stabilizes the system.
4 State Discretization, Hierarchical Organi-
zation and Dual Memory
One might wonder that our deﬁnition of types is rather ambiguous an d is
not clearly deﬁned. Indeed one can clearly distinguish them by plottin g and
comparing the time series and check how these orbits are separate d. To
conﬁrm that the state in each type is clearly separated, we introdu ce the
distance between cells in the k-dimensional phase space.
Since a cell’s state is determined by chemical concentrations in the pr esent
model, the cellular state is represented by an orbit in the k-dimensional phase

space. Here we ﬁrst consider the average position of an orbit for s implicity;
x(ℓ)
i = (1/T )
∫
x(ℓ)
i (t)dt (5)
As the diﬀerence between two cells we adopt the Euclid distance
Di,j ≡
√ ∑
ℓ
(x(ℓ)
i − x(ℓ)
j )2 (6)
The distance between two cell types is plotted in Table I. Note that t here
remains some diﬀerence in the same type of cell as mentioned. Howev er,
this diﬀerence is clearly much smaller than that between diﬀerent cell types.
This demonstrates that the diﬀerentiated cell types (from “0” to “5”) are
well-deﬁned as “digitally” distinct states. Then one might suspect th at these
diﬀerent states may be just a diﬀerent attractor in each dynamics . This is not
the case. Except the type-0 and type-5 cells, the state of diﬀere ntiated cells
is unstable by itself. When we start the simulation of a single cell with th e
state of cell type “1”,“2”,“3”,or “4” with the same media (but withou t any
other cells), the cell is de-diﬀerentiated back to the attractor-0 . The states
for types “1”,“2”,“3”, and “4” are stabilized only through the inter action
among other cells. For example, the existence of type-0 cells is nece ssary to
keep the stability of cell types “1” and “2”.
It is also interesting to compare the bifurcation rule of cell types (in Fig.6)
with the distance. If the history of cell lineage reﬂects on the dista nce of cell
features, it is expected that for j = 3 , 4, 5 D1,j < D 0,j or D1,j < D 2,j since
the types 3,4,5 are derived from the type-1 cell. This is not necessar ily true
in Table I. The reason for this discrepancy is due to the insuﬃciency in the
representation for the distance measured after taking the aver age. As is seen
in Fig.4, the orbit of the type-1 cell itinerates over the states close to the
type 3,4, and 5. Hence it is useful to deﬁne the minimal distance by
Dmin
i,j ≡ mint


√
∑
ℓ
(x(ℓ)
i (t) − x(ℓ)
j (t))2

 (7)
where mint means the minimum over time. The distance is given in Table II,
where one can clearly see the hierarchical organization of cell type s according
to the bifurcation rule of Fig.6. The distance between two of cell typ es
“0”,“1”, and “2” is smaller than that between “0” or “2” and “3”,“4” ,or “5”.
The distance between “1” and “3”,“4”, or “5” is much smaller than ot hers.
Let us reconsider the form of memory using the distance. First, th e
memory of cell types is sustained in the internal dynamics modulated by
the interaction. The memory corresponds to a partial attractor stabilized
by the interaction. Here, the information on the distribution of cell types is
embedded in each internal dynamics.
For example, each internal dynamics is gradually modiﬁed with the cha nge
of distribution of other cells. In Fig.7 we have studied how the dynamic s of
the type-2 cell changes when the rate of type-0 cell is varied. In t he sim-
ulation, we choose (a stable) cell society consisting of types “0”, “ 2”, and
“3”, and successively replace a cell of type-0 by type-3. To avoid t he per-
turbation due to the change in the number of cells, we remove the ru le of

division in the present simulation, to ﬁx the number. As a result of cha nge
of the distribution of cell types (i.e. the fraction of type-0 cells), t he dy-
namics of each cell (e.g., of the type 2) is modulated. We have plotted the
distance D2,20 where the 2 0 denotes the cell when the distribution of cells
satisﬁes ( n0, n2, n3) = (23 , 50, 27), the condition at the left-end point of the
axis, where nk represents the number of the type- k cell. The distance D2,20
increases (roughly linearly) with the decrease of n0, until the further decrease
destabilizes the cell society and the switching of cells to type-5 star ts. The
gradual change of D2,20 means that the internal cell state varies according to
the cell distribution. Hence the global information on the cell distrib ution is
embedded in the internal cellular state. We note that this informatio n adopts
“analogue” representation, instead of digital one adopted for th e distinct cell
type. Hence our cellular system has both analogue and digital memor ies.
5 Interaction-dependent rules and stability of
cell society
In Fig.6, we have shown that the automaton-like rule has emerged wit hout
explicit implementation. The rule is not solely determined by its cell type .
When there are multiple choices of diﬀerentiation process (as in “0” → “0”,
or “0” → “1”, and “0” → “2”) the rate of each path is neither ﬁxed nor
random, but depends on the number distribution of cell types in the system,
embedded in the internal dynamics. This implies that a higher level dyn amics
emerges, which controls the rate of cell division and diﬀerentiation a ccording
to the number of each cell type. In other words, the dynamics on t he number
of each cell type n0,n1,.., and n5 can be represented by {nk} (k = 0 , · · ·, 5).
(This dynamics should be stochastic, since we have neglected the inf ormation
on each cellular state and reduced it to only the number of cell types ).
This dynamics allows for stability at the level of ensemble of cells. The
variety and the population distribution of cell types are robust aga inst ex-
ternal perturbations. As an example, let us consider the case with three cell
types (“0” ,“1” ,“2” in Fig.6). When the type-2 cells are removed to de crease
their population, events of diﬀerentiations from “0” to “2” are enh anced, and
the original cell-type distribution is recovered.
In Fig.8, the rate of diﬀerentiation from the type-0 cell to others is plotted.
In this simulation, to capture the dynamics of the number of each ce ll type,
the total number of cells in the medium is ﬁxed (to N=100 in the present
case), by removing the division rule. As the initial condition, N cells are
placed in the medium, where the concentration of chemicals in each ce ll is
selected so that they give type-0, 1, or 2 cell. The switch of cell typ es is
measured when the system settles down to a stable distribution of c ell types.
The simulations are repeated by changing the initial distribution of ce ll types
(n0, n1, n2), to plot the number of the switches from 0 to others, while the
ﬁnal number of cells for each type is also plotted. As in Fig8, the freq uency
of switches from the cell type-0 increases almost linearly with n0 when it
is larger than approximately 40%. With this switch, the stability of cell
distribution around approximately ( n0, n1, n2) = (40 , 30, 30) is attained.
This kind of robustness at an ensemble level is expected from our iso lo-

gous diversiﬁcation theory, since the stability of macroscopic char acteristics
is attained in coupled dynamical systems (Kaneko 1992, 94). In our case,
the macroscopic stability is sustained by the change of the rate of d iﬀeren-
tiation from “0” to other types. Recall that, the diﬀerentiations f rom “1”
or “2” to “0” does not occur (see Fig.6), even if some of the type “0 ” cells
are removed 5. In the hierarchical structure represented in Fig.6, the cell
at an upper node behaves as a stem cell, and regulates the distribut ion of
the cells at a lower node. This type of regulation system is often adop ted in
the real multicellular organism (e.g. in the hemopoietic system)(Scho ﬁeld et
al., 1980). An important point of our result is that the dynamical diﬀe ren-
tiation process always accompanies this kind of regulation process, without
any sophisticated programs implemented in advance. This robustne ss pro-
vides a novel viewpoint to understand how the stability of the cell so ciety is
maintained in the multicellular organism.
6 Diﬀerentiation of Colonies
The automaton rule of Fig.6 does not necessarily mean that all of the se six
types of cells coexist in a cell society emerged in the course of the de velop-
ment. Cell groups consisting only of two or three cell types can app ear: For
example, cell groups only of “0”,“1”, and “2” types and of “0”, “2” , and “4”
types are observed.
This implies that the dynamics on the number of cells of each type has
also several stable attractors due to the autonomous control o f the rate of
diﬀerentiation. They correspond to stable distributions of cell typ es in each
cell group. In other words, there are several possible distributio ns of cell
types when cells are developed from a single cell. To conﬁrm it, we have
performed the following simulations. First, we initially put one cell whos e
internal chemical concentrations are chosen randomly. Then the cell society
is evolved following the rules of the present model, until the total ce ll number
reaches a given threshold value, when we stop the simulation and mea sure
the distribution of cell types. We have repeated this course of simu lations
for hundred times, starting from diﬀerent initial conditions.
In Fig.9, the number of initial conﬁgurations leading to a cell-type dist ri-
bution with a given range of n2 is plotted as a histogram, where the number
of type-2 cells n2 is measured when the total cell number has reached 300.
Four peaks are clearly visible at n2 = 0 , ∼ 100, ∼ 150, and ∼ 220, which
correspond to possible distinct sets of cell distributions. As mentio ned, the
possible set of cell types (from “0” to “5”) and the temporal orde ring of
their appearance (e.g., 0 → 1 → 2) are independent of the initial conditions.
However, at the later stage, several types of cell groups emerg e depending on
the initial conditions.
The most relevant factor to the choice of cell groups is the ratio of the
numbers of diﬀerentiated cells (i.e. type-1 and type-2 cells) to undiﬀ erenti-
ated cells (i.e. type-0 cells) at an early stage of development, when t he ﬁrst
5Transformation from type-2 to type-0 cells occurs as a transient process to type-5 cell,
which is seen only in the case when the type-5 cell appears and start s to dominate the
society.

diﬀerentiations from “0” to “1” and “2” occur. Thus the fate of ce ll groups
is determined at a rather early stage.
Recall that the diﬀerentiation rate of cells (each arrow in Fig.6), and
accordingly the higher-level dynamics of nk depend on the distribution of
cells nk. The result of Fig.9 implies that there are several attractors on th is
higher-level dynamics on nk. As discussed in §5, an “attractor” of this higher
level dynamics is stable against perturbations to change the numbe r of cells
of each type.
In Fig.10 we have shown the ﬂow chart of the change of ( n0, n1, n2), where
the direction of change of n0 and n2 is represented by the arrow, starting
from the initial distribution given by ( n0, n1, n2) of the corresponding site.
To draw the ﬁgure, we adopt the same rules as in Fig.8, where the tot al
cell number ( n0 + n1 + n2) is ﬁxed to 100, and the division rule is removed.
From the 2-dimensional plane, the number of cells of types 1,3,4, and 5 are
given by 100 − n0 − n2. The chart shows that cell colonies on the cell-
type distribution {n0, · · ·n5} have at least 5 stable states around ( n0, n2)
=(0,0), (38,32), (30,50), (18,58), and (0,78) respectively. Each s tate has a
basin of attraction, and the corresponding cell-type distribution is stable
against external perturbations, as is supported by the higher-le vel dynamics
on {n0, · · ·n5}.
The ﬁxed point at ( n0, n2) = (0 , 0) (“A” in the ﬁgure) corresponds to
a colony consisting only of type-5 cells, while the ﬁxed point denoted b y
“B” corresponds to a colony only of 0,1,2, “C” of 0,2,3, and “D” of 0,2,4 ,
respectively. Indeed, these cell-type distributions correspond t o the peaks of
Fig.9, respectively.
Still there is a clear diﬀerence between the developmental process from
one cell (Fig.9) and the present simulation (Fig.10) with a ﬁxed cell num ber.
Some region in the plane of Fig.10 cannot be reached by the simulation f rom
a single cell. For example, the state “E” consisting of types 2 and 4 ca nnot be
obtained from the developmental process from a single cell. Furthe rmore, the
state “B”, which does not have a large attraction volume in Fig.10, ha s the
largest probability to be reached from the developmental process (see Fig.9).
This discrepancy is caused by the conjunction of cell number chang e with
the population dynamics of cell types. Through the change of the n umber of
cells, the population dynamics shifts from one ﬂow chart of Fig.10 to a nother
with diﬀerent number of cells. The organized cell colony from a single c ell
has such developmental constraints.
Now the coexistence of several stable cell colonies is clear. Depend ing
on the initial cell condition, diﬀerent cell colonies are obtained. The r esult
here means that several types of tissues can appear through th e interactions
among cells. This kind of diversity is often observed in a cultivation sys tem
of a colony of blood cells starting from a stem cell (Nakahata et al., 19 82).
7 Summary
In the present paper, we have studied a dynamical model to show t hat a
prototype of cell diﬀerentiation occurs as a result of internal dyn amics, in-
teraction, and division. We have made several simulations choosing s everal

chemical networks, also with a diﬀerent number of chemical species , and the
same scenario for cell diﬀerentiation is obtained. Under the same pa rameters
used in the previous example, approximately 5% of randomly chosen c hem-
ical networks show oscillatory behavior, while others fall into ﬁxed p oints.
Furthermore, approximately 20% of these oscillatory dynamics are destabi-
lized through the cell division, where some of the cells diﬀerentiate fo llowing
a speciﬁc rule like Fig.6.
Some may cast a question why we can select such oscillatory dynamics
to draw a general mechanism for diﬀerentiation, even if only a few ra ndomly
chosen chemical networks are oscillatory. One reason why only a fe w reaction
networks are oscillatory is that we choose reaction paths randomly and with
the identical coeﬃcients. On the other hand, the chemical reactio n network of
the real biological system is more sophisticated through evolutiona ry process.
For example, there are positive and negative feedback reactions u biquitously.
This feedback mechanism, in particular auto-catalytic reaction, is im portant
to provide oscillatory dynamics which are observed in the real biologic al
system (e.g. Ca oscillation).
In the present model with randomly chosen networks, only few rea ctions
have auto-catalytic eﬀects. By increasing the rate of auto-cata lytic reaction
paths, the probability of the network with oscillatory dynamics and d iﬀer-
entiation gets much higher. For comparison, we have also studied a c lass of
models where each chemical can catalyze a reaction to generate its elf from
another chemical, besides the ordinary reaction paths determined randomly.
By sampling several reaction networks, we have found that 40% of the reac-
tion networks has oscillatory dynamics and more than 20% of these d ynamics
are destabilized to show cell diﬀerentiation by cell divisions.
Then, why are such auto-catalytic reactions common? To make rep lica-
tions eﬃciently, some mechanism to amplify reaction by its product is g ener-
ally expected at the ﬁrst stage of life (Eigen and Schuster, 1979). Also, auto-
catalytic reactions are necessary to add new metabolites in the met abolic
network through the evolutionary process. Indeed, when novel chemicals
are included in the evolutionary process of metabolic network, their con-
centrations must be ampliﬁed by the reactions. This implies that thes e new
chemicals must constitute an auto-catalytic set (see Appendix of K aneko and
Yomo, 1997).
Let us summarize the consequences of our simulations. First, we ha ve
provided a further support for isologous diversiﬁcation previously proposed.
Cells are diﬀerentiated through the interplay between intra-cellular chemical
reaction dynamics and the interaction among cells through media. As the
cell number is increased, the oscillatory dynamics in each cell is desta bilized
and loses synchrony. Then, some of cells change their internal dyn amics,
which form a group with a diﬀerent stable dynamics. Discrete, diﬀere ntiated
states appear, which are transmitted to their daughter cells as a m emory.
We interpret this phenomenon as (determined) diﬀerentiation.
The diﬀerentiation in our theory is caused by the instability in internal
dynamics triggered by cell-to-cell interactions. Microscopically spe aking in
biological terms, this may be regarded as a switching process followin g signal
molecules from outside of the cell. Our theory is not inconsistent with such
biological knowledge, but the point in our theory lies in that such local

transition of internal states has also the information on macrosco pic states,
i.e., the distribution of cell types. With this, the robustness of cell s ociety
emerges in spite of instability in each internal dynamics.
Besides this further support for the isologous diversiﬁcation, we h ave
demonstrated the hierarchical cell diﬀerentiation, generation of interaction-
dependent rules, and the existence of distinct cell groups.
1) hierarchical organization
Diﬀerentiation from a stem cell to two diﬀerent types, and then to t hree
types from one of them are observed. Hierarchical rule of diﬀeren tiation is
thus generated. Although the number of cell types and the rule of diﬀerenti-
ation depend on the choice of chemical networks, generation of a h ierarchical
rule (written by the tree-type diagram as in Fig.6) is generally observ ed.
2) generation of rules and internal memory reﬂecting on the
environment (that is the distribution of other cell types)
These diﬀerentiations obey a speciﬁc rule, which emerges from inter - and
intra- dynamics. It is often believed that the rules of the diﬀerentia tion,
which determine when, where, and what type of a cell appears in a mu lticel-
lular organism, should be pre-speciﬁed as the information on DNA. We do
not deny such role of DNA, but it should be stressed that the rules o f diﬀer-
entiation and the higher level dynamics emerge through interaction of cells
with internal dynamics. As a consequence of our interaction-base d approach,
the diversity of cells and the stability of cell society naturally follow.
The rate of diﬀerentiation and reproduction vary with the distribut ion
of cell types. The global stability of the whole system is obtained, wh ich is
sustained by regulating the rates of the diﬀerentiations.
As a coupled dynamical system, the memory of cell types is given in a
state stabilized by interactions. This state is not necessarily an att ractor
as a single cell dynamics, but is a “partial attractor” stabilized only in the
presence of suitable interactions provided by the distribution of ot her cells.
Through the cell divisions and the evolution of the cell society, the c ells
choose suitable interactions so that the memory of their types is pr eserved.
This is the mechanism of how the recursivity of cell types is attained, while
the global stability of cell society is assured through the interactio n. Indeed
such partial attractors lose stability and switch to other cell type s when the
interaction by the cell distribution is not “suitable”.
It should be noted that two types of memory coexist, analogue and digital.
The former gives information on the cell society, i.e., the distribution of cell
types, while the latter gives a distinct internal state on cell diﬀeren tiation.
We believe that such dual memory structure is a general feature in a biological
system. In cell biology, the “analogue” diﬀerence reﬂecting on the interaction
is known as modulation (Alberts et al., 1994).
3) formation of higher level dynamics and diversity of cell g roups
The rule of diﬀerentiation depends on the number distribution of oth er cell
types, for which stochastic dynamics at a higher level is formed. Th e result
provides the ﬁrst example that a 2-step higher-level dynamics is fo rmed, that
is a colony level from cellular one, that is formed from a chemical netw ork
level. Here the dynamics of a colony level (i.e., the change of the numb er
of each cell type) is “stochastic”, because the information on the number of
cell types is not complete, where the lower-level information on the internal

state (of chemical concentrations) is discarded. It is interesting to note that
the macroscopic ﬂow chart on the number of cell types is formed in s pite of
the stochasticity.
Our result shows that there are several attracting states for t his higher-
level dynamics. In biological term, this corresponds to the existen ce of several
cell colonies, distinguishable by the number distribution of cell types . These
diverse colonies appear from a single stem cell. Each cell colony is stab le, in
the sense that the original distribution is recovered after pertur bations (of
not too large size) are added on the cell colony, such as elimination of few
cells of one type.
8 Discussion
Of course, there has been preceding theories for cell diﬀerentiat ion. The idea
to regard the diﬀerentiation as the transition in cellular multi-station arity is
traced back to Delbr¨ uck (1949), who proposed a simple bistable reaction net-
work with two metabolic chains that are cross-inhibited by their prod ucts.
Indeed the epigenetic transmission of such stationary states hav e been re-
ported in unicellular organisms (Novick and Wiener, 1957; Sonneborn , 1964).
In general, this multi-stationarity results from positive and negativ e feed-
backs in metabolic reaction networks. This leads to the viewpoint tha t each
diﬀerentiated cell state is represented by an attractor of intra- cellular dynam-
ics, as has been demonstrated by Kauﬀman (1969) in his Boolean net work. It
leads to a variety of stable states (attractors), depending on th e initial condi-
tions. Here, each cell type corresponds to an attractor of inter nal networks,
while an external mechanism is required to have the transition betwe en these
attractors.
Such mechanism is supported by cell-to-cell interaction. Indeed, a mech-
anism of interaction-induced diﬀerentiation has been proposed by T uring’s
pioneering study (Turing, 1952). Now, a well-known mechanism of ex ter-
nal regulation is gradient of morphogen, in which the transition depe nds
on the concentration of chemical substances. (See, however, K aneko and
Yomo (submitted), for instability due to stochastic ﬂuctuation in th e thresh-
old mechanism on the gradient of chemicals). Another possible mecha nism
for interaction-induced diﬀerentiation is proposed by Gordon, whe re the me-
chanical wave transmits among the cells and controls the cell state (Gordon
et al., 1994).
Then, combination of the multi-stationary reaction network and th e ex-
ternal regulation mechanism might be relevant to explain the local diﬀ er-
entiations. However, to understand the complexity and the stabilit y of cell
society, an important question still remains: How is such external r egulation
mechanism regulated? Is another external mechanism required?
Our results provide a distinct, and plausible standpoint for this prob lem.
Noticing the interplay between intra-cellular dynamics and interactio n, we
have proposed a novel concept “partial attractor”, which is sta bilized only
by cell-to-cell interaction. Thus, a cell state is not always determin ed by
the attractor of internal dynamics, but it also depends on the oth er cells.
An important consequence of our results is that there is no distinct ion be-

tween internal dynamics that determines the cell state and the re gulation
mechanism of diﬀerentiation. Rather, the mechanism for regulation is spon-
taneously accompanied by the multi-stationarity, because the num ber of cells
with each partial attractor is found to depend on the circumstanc e of cell so-
ciety which sustains them.
A consequence of our theory is ‘relativity’ of determination of a diﬀe r-
entiated cell. Since our cellular state reﬂects the interaction, the r ule of
diﬀerentiation as well as the recursivity may be aﬀected by the cells a round
the cell in concern. A suitable experiment system to distinguish our t heory
from previous theories is provided by a hemopoietic stem cell system , where
spatial pattern mechanism a l’a Turing no longer works. Now we will disc uss
brieﬂy relevance of our results to the cell biology.
Application to biological problems
Since our model reaction process does not have one-to-one corr espondence
to any existing biochemical network, one cannot make a detailed pre diction
on a speciﬁc example in biology. However, the present scenario shed s a new
light on some open questions in biology, by providing a coherent viewpo int
on them. Let us discuss two of them.
Since our results provide hierarchical diﬀerentiation, it is interestin g to
compare them with such an example in cell biology. A well known example is
a hemopoietic system (Ogawa, 1993). The blood contains many type s of cells
with diﬀerent functions, while a pluripotent stem cell in the bone marr ow
gives rise to all classes of blood cells. The hemopoietic system can be v iewed
as a hierarchy of cells, where pluripotent stem cells diﬀerentiate to p rogenitor
cells determined as ancestors of one or a few terminal diﬀerentiate d blood cell
types. In general, these terminal blood cells have limited lifespans an d are
produced throughout the life of the animal. Thus, to keep a variety of blood
cells, it is important to control the diﬀerentiation and proliferation o f the
stem cell at the higher level of hierarchy. However, because of th e diﬃculty of
identifying the stem cells in the bone marrow, the behavior of the plur ipotent
stem cells in vivo remains especially elusive. In the experimental result in
vitro, even if the cells have been selected to be as homogeneous as p ossible,
there is a remarkable variability in the sizes and often in the characte rs of
the developed colonies (Nakahata et al., 1982). Even if two sister ce lls are
taken immediately after a cell division and cultured apart under ident ical
conditions, they frequently give rise to colonies that contain diﬀere nt types
of blood cells or diﬀerent distribution of the types of cells.
It is often interpreted that the diﬀerentiation of a hemopoietic ste m cell is
stochastic whose probability is controlled by some other control me chanism,
by which the multicellular system as a whole regulates the distribution o f
cell types(Till et al., 1964; Ogawa, 1993). Results of our model pro vide a
novel interpretation of these experiments. First, the rules of diﬀ erentiation
are generated through interactions. Second, the stochastic diﬀ erentiation of
the cells and regulation of the probability of the diﬀerentiations natu rally
emerge from the cell-to-cell interaction, without imposing any rand om event
or external regulation mechanism. Third, the diversity of the colon ies which
have developed from a same type of cell is a natural consequence, as a multi-
stability of higher-level dynamics. We note that a single cell with a sligh tly
diﬀerent initial condition can lead to a diﬀerent colony in our simulation.

Through the chemical substrates such as the Interleukins, the c omplicated
interactions among the blood cells are observed experimentally. It is plausible
to assume the dynamical interaction adopted in our theory.
Here, we propose an experiment on the hemopoietic system to make some
predictions. As mentioned, one of the important consequences of our results
is that the states of cells are not always determined by the attract or of inter-
nal dynamics, but often are sustained by interaction with other ce lls. This
implies that some types of cells in the hemopoietic system do not corre spond
to a stable attractor of internal dynamics, but are stabilized by ot her cells.
The pluripotent stem cell and the terminal diﬀerentiated cell, the to p and
bottom of the hierarchy respectively, seem to correspond to a st able attrac-
tor, because they can stand independently of other blood cells. On the other
hand, the progenitor cells can be observed only in the colonies of bloo d cell.
We expect that the internal dynamics of these progenitor cells is re presented
by a partial attractor. Then, these cells can diﬀerentiate back to the cell of
higher hierarchy when these cells are separated and cultured indep endently.
To conﬁrm this hypothesis, diﬀerences between an isolated blood ce ll and
that surrounded by other cells should be tested experimentally. We predict
that the potentiality of blood cells to diﬀerentiate and to proliferate is quite
diﬀerent these two situations, which conﬁrms the importance of th e cell-cell
interaction in the hemopoietic system.
In general, there is a level of diﬀerentiations in cell. The determined d if-
ferentiation keeps the memory even if a cell is transplanted, while so me cells
can be transdiﬀerentiated (Alberts et al., 1994). In our dynamical systems
representations, such diﬀerence can be expressed as the distinc tion between
an attractor by the cell itself and the partial attractor stabilized by the inter-
action. The merit in our approach is that such levels of diﬀerentiation appear
without external implementation, which is important when one consid ers the
origin of multicellular organisms.
There are several types of cells in a multicellular organism. In particu lar,
almost all organisms have distinction between the germ cells and the s omatic
cells. The appearance of these two types are controlled elaborate ly by com-
plicated interaction among cells in the contemporary multicellular orga nisms.
However, it is hard to postulate that such a mechanism appeared at the same
time with the emergence of the multicellular organisms. Our theory pr ovides
one possible solution to this problem. According to our results, diﬀer enti-
ation in a group of identical cells occurs through the dynamical inter action
among cells, as long as the intra-cellular reaction dynamics can show n onlin-
ear oscillations. The diﬀerentiation, as well as the stability of such div erse
cells, is a rather natural consequence of interacting cells. At the n ext stage in
the evolution, more complex cell society must have appeared, wher e several
types of tissues exist as a higher hierarchy, which interact each ot her. Our re-
sult about the diversity of cell group shows potentiality that sever al types of
tissues appear at this stage, based on the dynamical interaction a mong cells.
In our model, we do not take into account of the spatial variation. S election
of each cell group, thus, depends on the choice of diﬀerent initial c onditions.
On the other hand, when the spatial information is included in this sys tem,
it is possible that several types of cell group coexist at a (spatially) diﬀerent
region.

Of course, contemporary multicellular organisms such as mammal of ten
have hundreds of cell types, though our results in this paper can s how coex-
istence with only few cell types. The number of cell types in our dyna mical
diﬀerentiation model does not show clear increase with the number o f chem-
ical substances. We suppose that the reason for these few type s of cells is
due to our random choice of chemical reaction network, where the reaction
paths are chosen equivalently. In real biological system, the chem ical reaction
network is more organized, possibly in a hierarchical manner.
To choose such suitable network, evolutionary aspect of chemical reaction
network should be taken into account for our model. This problem als o con-
cerns with the emergence of gene expression system, and is under our current
investigation.
acknowledgements
The authors are grateful to T. Yomo, T. Ikegami, S. Sasa, and T. Yamamoto
for stimulating discussions. The work is partially supported by Grant -in-Aids
for Scientiﬁc Research from the Ministry of Education, Science, an d Culture
of Japan.
References
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Figure caption
Fig.1: Overlaid time series of x(m)(t) of a single cell in medium, obtained
form a network with 16 chemicals and three connections in each chem ical.
Only the time series of 5 chemicals are plotted out of 16 internal chem icals.
Each line with the number m=2,9,10,11,12 gives the time series of the con-
centrations of the corresponding chemical x(m)(t). This oscillatory behavior
is a limit cycle, whose period T is longer than the plotted range of the ﬁg-
ure ( T ∼= 16000 time steps). The parameters are set as e1 = 1, D = 0 .01,
f = 0 .01,X (ℓ) = 0 .1 for all ℓ, and V = 100. Chemicals x(ℓ)(t) for m ≤ 3 are
penetrable(i.e., σℓ = 1), and others are not. The reaction network Con(i, j, ℓ)
is randomly chosen initially, and is ﬁxed throughout the simulation resu lts
of the present paper.
Fig2: Time series of x(m)(t), overlaid for the 5 chemicals (as given in Fig.1)
in a cell. (a)-(e) represent the course of diﬀerentiation to type-1 , 2, 3, 4, and
5 cells respectively. The diﬀerentiation to type-3, 4, and 5 cells alway s occurs
from type-1 cells.
Fig.3: Orbits of internal chemical dynamics in the phase space. (a) a nd (b)
show the orbits of chemical concentrations for a switching proces s from type-0
to type-1,2 cells, respectively, plotted in the projected space ( x(2)(t), x(13)(t)).
Fig.3(c) gives a plot of ( x(1)(t), x(8)(t)), which shows a switch from type-1 to
type-3 cells. (Note the diﬀerence of scales.) Each cell type is clearly distinct
in the phase space.
Fig.4: Orbits of internal dynamics for each cell type. The dynamics o f
each cell type is plotted in the same projected space as in Fig.3(a),(b ), i.e.,
(x(2)(t), x(13)(t)). Color corresponds to each cell type.
Fig.5: Cell lineage diagram. Diﬀerentiation of cells (whose indices are giv en
by the horizontal axis) is plotted with time as the vertical axis. In th is dia-
gram, each bifurcation of lines through the horizontal segments c orresponds
to the division of the cell, while the color indicates the cell type ( red fo r
type 0, green for type 1, blue for type 2 respectively.). We have plo tted up
to the stage of the diﬀerentiation to three types, while the bifurca tion to 3,4,
and 5 from 1 occurs at a later course.
Fig.6: Automaton-like representation of the rule of diﬀerentiation. The path
returning to the node itself represents the reproduction of its ty pe, while
the paths to other nodes represent the potentiality to diﬀerentia tion to the
corresponding cell types. The dotted line from type-2 to type-0 g ives an ex-
ceptional case: Indeed the diﬀerentiation from “2” to “0” never o ccurs when
several types of cells such as “0”,“1” and “2” coexist. It occurs e xceptionally
only if “5” cells dominate the system, when all cells are ﬁnally diﬀerentia ted
to type-”5”. In this case the type-“2” cells de-diﬀerentiate to “0 ” ( and ﬁ-
nally to “5”).
Fig.7: Variation of dynamics of type-2 cells with the change of the rat e

of type-0 cell. The distance D2,20 of eq. (6) is measured between two
type-2 cells from the conditions ( n0, n2, n3) = (23 , 50, 27) and ( n0, n2, n3) =
(n0, 50, 50 − n0),as one of type-3 cells is successively switched to type-0 ex-
ternally by 2 × 104 step. Besides the distance, the number of type-0 cells is
plotted against time.
Fig.8: Rate of the diﬀerentiation from type-0 to other cell types. T he total
cell number is ﬁxed to 100 (without division process), while we take th e ini-
tial cell distribution of three types as ( n0, n1, n2) = ( n0, 30, 70 − n0). Starting
the simulation with this initial condition, the ﬁnal number of each cell t ype,
as well as the number of diﬀerentiations from type-0 to others, is p lotted, as
a function of the initial number of type-0 cells ( n0). Within this range of cell
type distribution, none of type-3,4,5 cells appear(see section 6).
Fig.9: Histogram about the number of type-2 cells. Starting from a s ingle
cell with randomly chosen chemical concentrations, the simulation is carried
out until the total cell number reaches 300, when the number n2 of type-2
cells is measured. Repeating the runs 347 times, we have counted th e num-
ber of such initial conditions that n2 falls onto a given bin (with the size 5).
The histogram of n2 is obtained from the count. There are four peaks at
n2=0,100,150,220, each of which corresponds to a stable distribution o f the
cell colony.
Fig.10: Flow chart of the change of ( n0, n1, n2). We have carried out the
simulations starting from the initial condition at each ( n0, 100 − n0 − n2, n2)
by ﬁxing the total cell number to 100 (by removing the cell division pr ocess).
Change of the number of cell types is measured from simulations, fr om which
the direction of changes of ( n0, n2) is shown as an arrow in the ( n0, n2) space.
As is seen, there are 5 ﬁxed points, each of which corresponds to t he stable
population distribution of cell types.
Table Caption
Table I: The average distance in phase space between each cell typ e: Di,j
in eq.(6) is estimated by taking the average over 5 × 104 time steps. Each
cell type is sampled from a course of the evolution starting from one cell.
Table II: Minimum distance in phase space between each cell type. Dmin
i,j
in eq.(7) is estimated from 5 × 104 time steps. Each cell type is sampled from
a course of the evolution starting from one cell.

0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
94000 96000 98000 100000 102000 104000 106000
concentration
time
x2
x9
x10
x11
x12

0.1
0.2
0.3
0.4
0.5
0.6
0.7
100000 120000 140000 160000 180000 200000
concentration
time
type-0 type-1

0.1
0.2
0.3
0.4
0.5
0.6
0.7
110000 115000 120000 125000 130000 135000 140000 145000 150000 155000
concentration
time
type-2type-0

0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
100000 110000 120000 130000 140000 150000 160000 170000 180000 190000
concentration
time
type-0 type-1 type-3

0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
100000 110000 120000 130000 140000 150000 160000 170000
concentration
time
type-0 type-1 type-4

type-0 type-1 type-5
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
100000 150000 200000 250000 300000
concentration
time

0.05
0.1
0.15
0.2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
chemical13
chemical2
type-0
type-1

0.05
0.1
0.15
0.2
0 0.1 0.2 0.3 0.4 0.5 0.6
chemical13
chemical2
type-0
type-2

0.001
0.002
0.003
0.004
0.005
0.006
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008
chemical8
chemical1
type-1
type-3

0.05
0.1
0.15
0.2
00.10.20.30.40.50.60.7
chemical13
chemical2
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
00.10.20.30.40.50.60.7
type-0
type-1
type-2
type-5
type-4
type-3
expansion

1 2

200000 250000 300000 350000 400000 450000 500000 550000 600000 650000200000 250000 300000 350000 400000 450000 500000 550000 600000 650000
distance
type-0
number of type-0 cell
time
distance

-10
10 20 30 40 50 60 70
number
initial number of type-0 cell
type-2
type-1
type-0
differentiation

0 50 100 150 200 250 300
trial
number of type-2 cell

20 40 60 800
A
B
C
D
E
number of type-0 cell
number of type-2 cell

/99/101/108/108/45/48 /99/101/108/108/45/49 /99/101/108/108/45/50 /99/101/108/108/45/51 /99/101/108/108/45/52 /99/101/108/108/45/53
/99/101/108/108/45/48 /49 /58 /51 /2 /49/48
/49
/49 /58 /56 /2 /49/48
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/49 /58 /54 /2 /49/48
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/99/101/108/108/45/52 /42 /42 /42 /42 /42 /50 /58 /57 /2 /49/48
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/99/101/108/108/45/53 /42 /42 /42 /42 /42 /42