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