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Fig. 2.15 Typical evolution of the cellular pattern in a computer simulation of the
Drasdo“Forgacs model with speci¬c, experimentally determined values of the model
parameters. A dynamical instability in the shape of the blastula sets in at around the
64-cell stage, which in a spherical embryo would correspond to about 2000 cells. With
further increase in cell number the folding of the blastula becomes more pronounced.
Individual cells are indicated as circles up to the 64-cell stage.




values for the basic model parameters (Davidson et al., 1995, 1999). The
results of typical simulations are shown in Fig. 2.15. The number of
cells grows exponentially and after a series of cell divisions a hollow
spherical blastula forms. With further divisions the spherical symme-
try disappears: the blastula becomes unstable and folds. Within the
model this instability is generic and shows up over a wide range of
parameter values. We will return to the question of this instability in
Chapter 5 in our discussion of models of gastrulation.
50 BIOLOGICAL PHYSICS OF THE DEVELOPING EMBRYO



Perspective
Simple models, based on realistic physical properties of cells and their
interactions, can accurately reproduce many aspects of early cleavage
and blastula formation. For instance, while a true liquid surface ten-
sion per se is inapplicable to living cells, its analogue, cortical ten-
sion, has explanatory power in this realm and, within limits set by
the cell itself, obeys the same physical laws. Purely generic mecha-
nisms, however, have their limitations. A physical process that leads
to a spherical blastula, if allowed to proceed beyond a certain point
leads to morphological instability and ultimately to the derailment
of normal development. It is the multileveled determination of cells
and embryos which ensures that each driving force is constrained
by other driving forces and that the whole complex of forces is sub-
ordinated to the survival and propagation of the organism. Central
to the integration of the various physical determinants are the com-
plex biochemical signaling processes taking place in each individual
cell, and across the blastula and later-forming cell aggregates, which
control the values of cellular physical parameters and with them the
consequences of the physical interactions.
Chapter 3




Cell states: stability, oscillation,
differentiation

In the previous chapter we considered the forces leading to subdi-
vision of the fertilized egg and the large-scale morphological conse-
quences of successive cleavages. We followed the processes leading
to the generation of the blastula, an important developmental step.
In our treatment, however, the blastula was represented as a mass
of identical cells. In reality, at the blastula stage not all cells of the
embryo are identical: the ˜˜totipotent” zygote (with the potential to
give rise to any cell type) ¬rst generates cells that are ˜˜pluripotent”--
capable of giving rise to only a limited range of cell types. These cells,
in turn, diversify into cells with progressively limited potency, ulti-
mately generating all the (generally unipotent) specialized cells of the
body. At the same time, in the life of each dividing cell, regardless of
its level of developmental potency, there are transitions from state to
state as the cell performs various functions of the cell cycle.
It is the purpose of this chapter to provide a framework for un-
derstanding how genetically identical cells can change their physical
states in reliable and stable ways in time and space. We will also ex-
plore how these states, in turn, can provide a physical basis for a cell™s
performance of different tasks at different phases of its life cycle and
for its descendants™ performance of different specialized functions in
different parts of the developing and adult organism.
The transition from wider to narrower developmental potency is
referred to as determination. This stage of cell specialization generally
occurs with no overt change in the appearance of cells. Instead, subtle
modi¬cations, only discernable at the molecular level, set the altered
cells on new and restricted developmental pathways. A later stage of
cell specialization, referred to as differentiation, results in cells with
vastly different appearances and functional modi¬cations -- electri-
cally excitable neurons with extended processes up to a meter long,
bone and cartilage cells surrounded by solid matrices, red blood cells
capable of soaking up and disbursing oxygen, and so forth. Cells
have generally become determined by the end of blastula formation;
successive determination increasingly narrows the fates of their
progeny as development progresses. When the developing organism
requires speci¬c functions to be performed, cells will typically un-
dergo differentiation.
52 BIOLOGICAL PHYSICS OF THE DEVELOPING EMBRYO




Gene expression and biochemical state
Since each cell of the organism (except for the egg and sperm and
their immediate precursors) contains an identical set of genes, a fun-
damental question of development is how the same genetic instruc-
tions can produce different types of cell. This question pertains to
both determination and differentiation. Since these two kinds of cell
specialization are formally similar and probably employ overlapping
sets of molecular mechanisms, we will refer to both as ˜˜differentia-
tion” in the following discussion, unless confusion would arise. Mul-
ticellular organisms solve the problem of specialization by activating
only a cell-type-speci¬c subset of genes in each cell type.
The biochemical state of a cell can be de¬ned as the list of all the
different types of molecules contained within it, along with their
concentrations. During the cell cycle the biochemical state changes
periodically with time. If two cells have the same complement of
molecules at corresponding stages of the cell cycle then they can
be considered to be of the same differentiated state. (The cell™s bio-
chemical state also has a spatial aspect -- the concentration of a given
molecule might not be uniform throughout the cell -- but we will ig-
nore this complication for the time being). Certain properties of the
biochemical state are highly relevant in understanding developmen-
tal mechanisms. The state of differentiation of the cell (its type) can
be identi¬ed with the collection of proteins that it is capable of mak-
ing. This, in turn, comes down to which genes are potentially active
(transcribable or ˜˜template competent”). Another important aspect of
determination and differentiation that is less well understood, and
will not speci¬cally enter into our discussion, concerns the gener-
ation of different proteins from a given active gene by ˜˜alternative
splicing” of the gene™s RNA transcript (see Maniatis and Tasic, 2002).
The cell type is determined by the structural properties of the
DNA--protein complex in the cell nucleus known as chromatin. Genes in
the transcribable subset must be in particular physical states, and this
is accomplished by variations in the packaging of different stretches
of DNA by proteins. If a gene is template competent, the stretch of
DNA that contains it is in a conformation referred to as ˜˜open.” In
cell types in which the same gene is inactive it is in a ˜˜closed” con-
formation (Gregory et al., 2001; Cunliffe, 2003).
Although having different sets of template-competent genes de-
¬nes cells as differentiated from one another, it is important to dis-
tinguish between two classes of these transcribable genes. Of the es-
timated 20 000--25 000 human genes (International Human Genome
Sequencing Consortium, 2004), a large proportion is constituted by
the ˜˜housekeeping genes,” involved in functions common to all or
most cell types (Lercher et al., 2002). In contrast, the template compe-
tency of a relatively small number of genes -- more than a hundred,
but possibly fewer than a thousand -- speci¬es the type of determined
or differentiated cell; these genes are developmentally regulated (i.e.,
3 CELL STATES: STABILITY, OSCILLATION, DIFFERENTIATION 53


turned on and off in a stage- and position-dependent fashion) during
embryogenesis. The generation of the approximately 250 different
cell types of the adult human body is accomplished by packaging
this special group of developmentally regulated genes into appropri-
ate chromatin states in distinct cell populations. Some more familiar
genes that are developmentally regulated are beta-globin in the retic-
ulocyte (the red blood cell precursor), keratins in skin, and insulin in
the pancreatic beta cell.
It is also important to recognize that template-competent genes
in a differentiated cell type are not constantly being transcribed. The
open chromatin state of template-competent genes is structural or
conformational and is essentially irreversible when the cell remains
in its natural context. The products of a special class of template-
competent genes are involved in maintaining established gene ex-
pression patterns across cell generations (Brock and Fisher, 2005).
The foregoing discussion implies that the biochemical state of a
cell has at least two hierarchically organized aspects -- the differentiated
state, de¬ned by the stable set of template-competent, developmentally
regulated genes it contains, and the functional state, de¬ned by the
(potentially varying) levels of housekeeping and developmentally reg-
ulated proteins that it produces. Differentiated cell types are among
the end-products of development. Moreover, the range of functional
states that a given cell can assume is limited by its state of differen-
tiation. Signi¬cantly, though, development itself is implemented by
variations in the functional states of emerging cell types at the early
and intermediate developmental stages.


How physics describes the behavior
of a complex system
The biochemical state of a living cell does not remain ¬xed. Amino
acids and nucleotides are constantly being consumed to make pro-
teins and nucleic acids. These and other small molecules are con-
tinually being synthesized from precursors or being brought into the
cell from the outside. Usually, the cell is also involved in transporting
functionally useful products and waste materials across its boundary.
Given all this, it would be surprising if there were not wide swings
in the concentrations of many of the cell™s internal components. As
mentioned above, however, the cell™s state of differentiation provides
limits to these swings.
When physicists have to deal with a complex system containing
many interacting components -- a ˜˜dynamical system” -- they typically
represent its state as a point in a multidimensional space. Let us imag-
ine that such a system contains only two interacting components. The
system then resides in a two-dimensional ˜˜state space,” also called the
˜˜phase plane,” de¬ned by pairs (A, B), where A is the concentration of
the ¬rst component, measured along the x axis and B is the concen-
tration of the second component, measured along the y axis.
54 BIOLOGICAL PHYSICS OF THE DEVELOPING EMBRYO


The ¬rst question we can ask about such a representation is: at
a given instant of time, where will we ¬nd the system on the phase
plane? It is important to emphasize that all the dynamical systems
we will be considering are open systems, that is, they can exchange
chemicals and energy with their environment. Indeed, these are the
only kinds of system that can exhibit interesting behaviors, including
the characteristics of being alive. All closed chemical systems (ones
that have no interaction with their environment) eventually settle
into a dead state, determined by the initial concentrations, referred
to as ˜˜chemical equilibrium.” Open systems that consume energy and
operate far from thermodynamic equilibrium are known as dissipative
systems. As we will see, such systems can exhibit interesting types of
spatiotemporal organization of their constituents.
Since our system of interest is open, we are at liberty to prepare
it in any initial condition we choose. If, like a cell, it is bounded by a
membrane, then we can simply inject enough of the two components
to bring the system to any desired point. But will it stay there? In
fact, dynamical systems are governed by physical relationships among
their components. In a simple example, if the components are two
types of molecule and the system™s chemistry dictates that the ¬rst is
reversibly transformed into the second then there is a constraint be-
tween their concentrations A and B: when one goes up the other goes
down. The laws of chemical transformation also dictate the direction
of change: if we add more of the ¬rst component to the system, for
instance, some of it will be transformed into the second. Thus the
system will not remain in any arbitrary initial state.
The dynamical interactions in such systems are usually repre-
sented by sets of coupled differential equations. We saw in Chapter 1
that a derivative represents the rate of change of one variable as a
function of another variable. (In dynamical systems the derivatives of
the state variables with respect to time are of particular concern). In a
differential equation the derivative of one variable (say the concentra-
tion A of the ¬rst component in the system discussed above) is equated
to an expression containing other variables and possibly including
that variable itself. A system of coupled differential equations repre-
sents the time dependence of each state variable expressed in this
fashion.
To gain some insight into how dynamical systems are analyzed
mathematically, consider, as an example of a two-component dynam-
ical system, a chemical system governed by the following set of cou-
pled differential equations (Leslie, 1948; Maynard Smith, 1978):
dA
= 4A ’ A 2 ’ A B ,
dt
(3.1)
B2
dB
=B’ .
dt A

(Remember the meaning of d/dt explained in Box 1.1). Wherever the
system may start out on the phase plane, Eqs. 3.1 assign an arrow to
that point indicating where the system will move to next. As time
3 CELL STATES: STABILITY, OSCILLATION, DIFFERENTIATION 55


progresses, the arrows thus outline a trajectory along which the sys-
tem moves in time in the phase plane. The map of all possible sets
of arrows (i.e., trajectories) on the phase plane is called a ˜˜vector
¬eld.” At certain points in this state space the time derivatives of
both variables vanish simultaneously. Any such solution is called a
˜˜¬xed point” or ˜˜stationary state” or ˜˜steady state.” Setting both time
derivatives in Eqs. 3.1 to zero we ¬nd ¬xed points at (0, 0) and (2, 2).
Not all ¬xed points are equivalent, however. In some cases the
¬xed point will be stable against perturbations, i.e., if the system
is displaced to a nearby point then changes in reaction rates will
be induced that will return it to the original steady state (think of a
marble sitting at the bottom of a round-bottomed cup). In other cases
a tiny change in state will be suf¬cient to drive the system away from
the ¬xed point (think of a marble at the top of an inverted round-
bottomed cup). In the case of the two stationary states of Eqs. 3.1, (2, 2)
is stable and (0, 0) is unstable (see Fig. 3.1). (The general mathematical
technique to study the nature of a ¬xed point, whether it is stable or
unstable, is called linear stability analysis. In the Appendix to Chapter
5 we demonstrate how this method works.)
Of course, our dynamical system may have other forms. Consider,
for example,

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