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be in contact with one another throughout gastrulation (the set
of rearrangements that establish the body™s main tissue layers, see
Chapter 5) in order to develop into terminally differentiated mus-
cle (Gurdon, 1988; Standley et al., 2001); (ii) if the cells in the
70 BIOLOGICAL PHYSICS OF THE DEVELOPING EMBRYO


neuron-containing layer of the retina (the neural retina or NR cells)
of the developing chicken eye are placed in culture they ˜˜dediffer-
entiate” away from the neuronal type and can ˜˜transdifferentiate”
into a different cell type, that of the pigmented layer of the retina
(the pigment retina or PR cells). PR cells do not arise individually
but as small random groups within sheets of NR cells. If the PR cells
that form in this fashion are isolated and exposed to the appropriate
molecular signals they will transdifferentiate back into NR cells (Opas
et al., 2001).
The need for cells to act in groups in order to acquire new iden-
tities during development has been termed the ˜˜community effect”
(Gurdon, 1988). This phenomenon is a developmental manifestation
of the general property of cells and other dynamical systems of as-
suming one or another of their possible internal states in a fashion
that is dependent on inputs from their external environment. But in
the two cases noted above the external environment consists of other
cells of the same type, and in discussions of the community effect
there is the strong implication that dynamical interactions among
similar cells, and not just exposure to secreted factors, is responsible
for the emergence of new cell types (Carnac and Gurdon, 1997).
Kaneko, Yomo, and coworkers (1994; 1997; 2003, and references
therein) described a previously unknown physical process, termed
˜˜isologous diversi¬cation,” by which replicate copies of the same dy-
namical system (e.g., cells of the same type) can undergo stable dif-
ferentiation simply by virtue of exchanging chemical substances with
one another. This differs from the Keller model that we have just con-
sidered in that the ¬nal state achieved in the Kaneko--Yomo model
exists only in the state space of the collective ˜˜multicellular” system.
Whereas the distinct local states of each cell within the collectivity
are mutually reinforcing, these local states are not necessarily attrac-
tors of the dynamical system representing the individual cell, as they
are in Keller™s model. The Kaneko--Yomo system thus provides a model
for the community effect.
The general scheme of the model, as described in Kaneko and
Yomo (1999), is shown in Fig. 3.7A. Here we present the model in its
simplest form (Kaneko and Yomo, 1994) to illustrate how dynamical
system analysis at the multicellular level can describe differentiation.
Improvements and generalizations of the simple model do not change
its qualitative features.
Kaneko and Yomo considered a system of originally identical
model cells (called ˜˜cells” in what follows) with intracell and in-
tercell dynamics that incorporate cell growth, cell division, and cell
death. The dynamical variables (spanning the state space of the sys-
tem) are the concentrations of molecular species (˜˜chemicals”) inside
and outside the cells. The criterion by which differentiated cells are
distinguished is the average of the intracellular concentrations of
these chemicals (over the cell cycle). As a vast simpli¬cation, only
three chemicals, P, Q, R with respective time-dependent intracellu-
Q
lar concentrations in the ith cell (xiP (t), xi (t), xiR (t) and intercellular
3 CELL STATES: STABILITY, OSCILLATION, DIFFERENTIATION 71


Active transport
A P
P
xXii

Q
xi
P
xi
Diffusion



R
xi

ith cell



Cell division




X
Interaction
through
metabolites


B

A
Ei
S
E A B
S
DNA
xi xi
xi
B
Ei



Fig. 3.7 (A) General scheme of the model of Kaneko and Yomo. The biochemical state
of a cell is characterized by chemicals P , Q , R , etc., with corresponding intracellular
concentrations x iP , x iQ , x iR , etc., which vary from cell to cell (cell i is shown in the
upper center). Chemicals may exit cells (chemical P in the example), their extracellular
concentrations being XiP , XiQ , XiR , etc., and enter other cells either via passive diffusion
or active transport; this establishes interactions between cells (as shown at the bottom
for two daughter cells). The concentration of any chemical inside a given cell depends on
chemical reactions in which other chemicals are precursors or products (solid arrows)
or cofactors (e.g., enzymes; broken-line arrow). A cell divides when the synthesis of
DNA (one of the “chemicals”) exceeds a threshold. (B) Schematic representation of the
intracellular dynamics of a simple version of the Kaneko“Yomo model. Red arrows
symbolize catalysis. The variables x iA (t), x iB (t), x iS (t) and E iA , E iB , E S denote
respectively the concentrations of chemicals A, B, and S and of their enzymes in the ith
cell, as described in the text. Note that the letters P , Q , R are used for the general
scheme of the Kaneko“Yomo model whereas the letters A , B , S are used for the
speci¬c mathematical representation given in the text.
72 BIOLOGICAL PHYSICS OF THE DEVELOPING EMBRYO


concentrations (X P (t), X Q (t), X R (t)) are considered. One of those (P )
serves as the source for the others (see Fig. 3.7A). The model has the
following features.
1. Intracellular dynamics. The source chemical, denoted by S in the
simpli¬ed scheme of Fig. 3.7B, is catalyzed by a constitutive (always
active) enzyme (concentration E S ) to produce chemical A, which in
turn is catalyzed by a regulated (˜˜inductive”) enzyme (concentration
E iA ) to produce chemical B. Chemical B on one hand is catalyzed by
its own inductive enzyme (concentration E iB ) to produce A and on the
other hand controls the synthesis of DNA. This sequence of events is
shown schematically in Fig. 3.7B. The concentration of the constitu-
tive enzyme is assumed to have the same constant value E S in each
cell, whereas those of the inductive enzymes in the ith cell, E iA and
E iB are both taken to be proportional to the concentration xiB of the
chemical B in that cell (and therefore to be dependent on time), so
that E iA = e A xiB and E iB = e B xiB (e A and e B are constants). Thus, in terms
of chemicals A and B the intracellular dynamics is described by

dxiA
= e B xiB xiB ’ e A xiB xiA + E S xiS ,
dt
(3.5)
dxiB
= e A xiB xiA ’ e B xiB xiB ’ kxiB .
dt

Here the factor k accounts for the decrease in B due to its role in
DNA synthesis (see Fig. 3.7B). Note the nonlinear character of these
equations. (Parentheses are used to indicate the inductive enzymes.)
2. Intercellular dynamics. Cells are assumed to interact with each
other through the changes in the intercellular concentrations of the
chemicals A and B. Chemicals are transported in and out of the cells.
The rate of transport of a chemical into the cell is proportional to its
concentration outside. However, it also depends on the internal state
of the cell, which we have characterized in terms of the intracellular
concentrations of the chemicals A and B. This dependence is typically
complicated. Kaneko and Yomo assumed that the rate of import of
chemical M (i.e., A, B, or S) into the ith cell, denoted by TranspiM , has
the form

3
TranspiM (t) = p xiA + xiB X M. (3.6)

Here p is a constant. As long as the dependence of Transp on the in-
tracellular concentrations is nonlinear, any choice of exponent (taken
to be 3 above) leads to the same qualitative result.
Besides the mechanism of active transport described by Eq. 3.6,
chemicals also enter the cells by diffusion through the membrane.
The corresponding rate is taken as

Diff iM (t) = D X M (t) ’ xiM (t) , (3.7)

where D is a (diffusion) constant.
3 CELL STATES: STABILITY, OSCILLATION, DIFFERENTIATION 73


Combining intracellular (Eq. 3.5) and intercellular (Eqs. 3.6 and 3.7)
dynamics, the rate equations for the intracellular chemicals become

dxiS
= ’E xiS + TranspiS + Diff iS ,
dt
dxiA
= e B xiB xiB ’ e A xiB xiA + E xiS + TranspiA + Diff iA ,
dt
dxiB
= e A xiB xiA ’ e B xiB xiB ’ kxiB + TranspiB + Diff iB . (3.8)
dt

It is further assumed that only the source chemical is supplied by
a ¬‚ow from an external tank to the chamber containing the cells.
Since it must be transported across the cell membrane to produce
chemical A (see Eqs. 3.5), the intercellular dynamics of the source
chemical is described by

N
dX S
= (X S ’ X S ) f ’ TranspiS + Diff iS . (3.9)
dt i=1



Here X S is the concentration of the source chemical in the external
tank, f is its ¬‚ow rate into the chamber, and N is the total number of
cells in the system.
3. Cell division. Kaneko and Yomo considered cell division to be the
result of the accumulation of a threshold quantity of DNA. DNA is
synthesized from chemical B and therefore the ith cell, born at ti0 ,
will divide at ti0 + T (T de¬nes the cell-cycle time), when the amount
of B in its interior (proportional to xiB ) reaches a threshold value.
t 0 +T
(Mathematically this condition is expressed as t 0i xiB (t)dt ≥ R in the
i
model, R being the threshold value.)
4. Cell death. To avoid in¬nite growth in cell number, a condition
for cell death also has to be imposed. It is assumed that a cell will
die if the amount of chemicals A and B in its interior is below the
˜˜starvation” threshold S, which is expressed as xiA (t) + xiB (t) < S.
Simulations based on the above model and its generalizations
using a larger number of chemicals (Kaneko and Yomo, 1997, 1999;
Furusawa and Kaneko, 2001), lead to the following general features,
which are likely to pertain also to real, interacting, cells:

(i) The state of a cell, de¬ned as a point in its chemical state
space (see Fig. 3.1), tends to recur over time in a periodic or
quasi-periodic fashion, analogous to the cell cycle.
(ii) As cells replicate (by division) and interact with one another,
eventually multiple biochemical states corresponding to dis-
tinct cell types appear.
(iii) The different types are related to each other by a hierarchical
structure in which one cell stands at the apex, cells derived
from it stand at subnodes, and so on (Fig. 3.8). Such pathways
of generation of cell types, which are seen in real embryonic
systems, are referred to as developmental lineages.
74 BIOLOGICAL PHYSICS OF THE DEVELOPING EMBRYO


300




250




TIME STEPS (thousands)
200




150




100


Type 3
Type 2
50
Type 1

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