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provide a biologically realistic description.

Modeling sea urchin gastrulation: an approach based
on energy minimization
In Chapter 2 we presented a model of blastula formation in the sea
urchin embryo (Drasdo and Forgacs, 2000) based on energy consid-
erations. The model incorporated several key energy contributions
characteristic of the developing embryo, and the resulting cellular

pattern corresponded to the minimum of the total energy. In the de-
scription of development from the zygote to the hollow spheroidal
blastula, the latter eventually became unstable and its folding un-
predictable. The folding inwards corresponds to the appropriate bio-
logical outcome -- invagination of the vegetal plate. The folding
outward, in contrast, corresponds to ˜˜exogastrulation,” which exper-
iments have shown to be also in the physical repertoire of the living
embryo (Hoshi, 1979; Kamei et al., 2000). The origin of this instability
is easy to understand. The growth and division of cells require that
either individual cells or entire cell layers (epithelial sheets) be capa-
ble of migrating. Since in the simulation the irregular active motion
of each cell within the layer was mimicked by random displacement,
small stochastic differences in cell translocation must exist. These
lead to undulations in the shape of the sheet. Such undulations are
unfavorable because they increase the bending energy of the sheet
and, as long as they remain small, are eliminated (since the Monte
Carlo protocol is designed to drive the system toward lower energies).
When cell proliferation is introduced, both the number and the ex-
tent of the undulations increase. Eventually these become too large to
be controlled by the bending energy: the spheroidal cell arrangement
ceases to correspond to the lowest energy con¬guration and folding
takes place.
In the energy-minimization model, the direction of folding of the
blastula, i.e., whether normal gastrulation or exogastrulation takes
place, is subject to the in¬‚uence of alterations in the local physical
parameters of the embryo. Recent experiments have suggested that
the suppression of the tendency to exogastrulate is under genetic
control (Kamei et al., 2000). One way in which this might be imple-
mented in the Drasdo--Forgacs model is through the assumption that
the increase in bending energy leads to local changes in gene activity
along the blastula surface, which in turn result in the modi¬cation
of the cells™ physical properties. The authors thus postulated that a
preexisting nonuniformity in the distribution of one or more gene
products causes the sheet of cells near the vegetal pole to acquire
nonzero spontaneous curvature (see the discussion following Eq. 2.5).
This was suf¬cient to account for the observation, described above,
that near the vegetal pole a distinct group of primary mesenchy-
mal cells ingresses into the blastocoel shortly after the formation of
the blastula. ˜˜Snapshots” of the developing system governed by the
model, as implemented through Monte Carlo simulations, are shown
in Fig. 5.8.
The Drasdo--Forgacs model describes morphogenesis in the sea
urchin embryo from the ¬rst cleavage until the completion of gas-
trulation. It provides an explicit example of how an important set
of morphological changes in early development can potentially be
accounted for by an interplay between genetic and generic physical
mechanisms. The most serious limitation of the model is that no in-
formation exists at this point on how to relate changes in the value
of spontaneous curvature to the speci¬c gene activity accompanying

Fig. 5.8 A representation of early development in the Drasdo“Forgacs model. Events
in the upper row illustrate the ¬rst cleavages leading to blastula formation and were
discussed in Chapter 2. The lower row shows the simulation of gastrulation. The nine
cells pictured in green develop a spontaneous curvature different from that of the
remaining cells. It is this change that drives invagination in the model. Note that these
simulations are performed with constant cell number, corresponding to the fact that cell
division halts during gastrulation in the sea urchin. (After Drasdo and Forgacs, 2000.)

the onset of gastrulation (even though most of the other parame-
ters have been measured, Davidson, L. A., et al., 1999; see Chapter 2).
The Drasdo--Forgacs model, similarly to the Mittenthal--Mazo model,
describes morphogenetic transformations as processes that generate
global equilibrium shapes corresponding to the minimum of some en-
ergy expression containing competing contributions. It is implicitly
assumed that the system can arrive at these energy minima (instead
of being locked into long-lived metastable states), a hypothesis that
requires experimental veri¬cation. Other characteristic cell and tis-
sue properties underlying gastrulation, such as excitability or shape
transformations, have also been neglected so far and will be incorpo-
rated in the model presented below.

Modeling sea urchin gastrulation: an approach based
on force balance
Cells of the developing embryo exert forces on each other. In doing
so they undergo changes in both position and shape. Models of epi-
thelial morphogenesis that are based on energy minimization, such
as those of Mittenthal and Mazo (1983) and of Drasdo and Forgacs
(2000), typically track pattern development by postulating changes
in cell arrangement and comparing the energies of the original and
modi¬ed patterns. The system reaches equilibrium when it arrives at
the global energy minimum. Such models therefore do not consider
the origin of cell-shape modi¬cations explicitly.
According to Newton™s second law, at equilibrium all the forces
and all the torques acting in a system must balance at each point.
This law also speci¬es how each volume element of the system must
move to reach equilibrium. Thus combining such an approach with

modern techniques that allow the following of individual cells during
development (Czirok et al., 2002; Kulesa and Fraser, 2000, 2002) may
provide further insight into the mechanisms of early morphogenesis.
In a series of papers, Oster and coworkers (Odell et al., 1981; Davidson,
L. A., et al., 1995, 1999) used this approach to distinguish between sev-
eral potential mechanisms of primary invagination in the sea urchin
embryo. Here we focus on their model, based on apical constriction
(Odell et al., 1981), the major assumptions and properties of which
are the following.
1. The cells making up the epithelial sheet of the sea urchin
blastula undergo a shortening of their apical circumference by the
contractile activity of actin ¬laments just beneath their plasma
membranes (Fig. 5.9A). This constriction proceeds in a ˜˜purse-string”
manner along the blastula similarly to the progression of the cleavage
furrow in cytokinesis (see Fig. 2.9) and must overcome the intracel-
lular viscous forces and the tractions exerted by neighboring cells.
To model these viscoelastic forces, we represent each face and inter-
nal diagonal of a cell by a viscoelastic element (Fig. 5.9B). The linear
characteristics, denoted by L , such as the circumference of a face or
the length of a diagonal vary according to Newton™s second law as
d2 L dL
= ’k(L ’ L 0 ) ’ f + F load .
m (5.7)
dt 2 dt

equilibrium shape
Apical contractile
actin network
Apical surface

equilibrium shape

Basal surface


Fig. 5.9 Assumptions concerning the mechanism of cell-shape change in the apical
constriction model of gastrulation. (A) Schematic representation of the contractile apical
actin network. (B) The mechanical analogue of a trapezoidal cell in the blastula. External
faces and internal diagonals are represented by viscoelastic elements, which are indicated
by connected small rectangles. The apical element is excitable and thus differs from the
others. (C) Illustration of the excitability of the apical surface: depending on the
magnitude of its deformation it is capable of changing its equilibrium geometry. (After
Odell et al., 1981.)

Here the term on the left-hand side is the inertial force (mass times
acceleration), whereas the ¬rst and second term on the right-hand
side represent, respectively, the elastic and viscous restoring forces
acting on a face or along the internal diagonal of a cell. L0 is the
equilibrium circumference of a face or the equilibrium length of an
internal diagonal and k and f are material parameters characterizing
the elastic and viscous properties (see Chapter 1, Eqs. 1.5 and 1.7). The
third term on the right-hand side is the force exerted by the neigh-
boring cells, and m is the net mass moved due to the change in L .
The diagonal elements model the cells™ internal viscoelastic properties
and thus, in the corresponding equations, F load = 0.
2. Equation 5.7 is valid for each face and internal diagonal. The
apical face, however, is special in that the underlying contractile ¬l-
aments constitute an active, excitable, system. If an apical ¬ber is
stretched by a small amount (which happens when the vegetal plate
starts ¬‚attening) it behaves as an elastic material: upon release of the
stretch it returns to its original length (L0 ). If, however, the stretch
exceeds a critical value, it elicits an active response: the contractile
system ˜˜¬res” and does not return to its original length (Beloussov,
1998): it freezes in a new contracted state, with a changed equilib-
rium length resulting in an apical surface area smaller than before
(Fig. 5.9C). Thus the apical viscoelastic element differs from the others
(Fig. 5.9B) and the corresponding version of Eq. 5.7 has to be supple-
mented by another equation describing the variation of the equilib-
rium length (see below).
3. Inertial forces are insigni¬cant. As discussed in Chapter 1, di-
mensional analysis indicates that for typical embryonic processes in-
volving the motion and shape changes of cells such forces can be
neglected. Thus Eq. 5.7 simpli¬es, and the evolution of the apical
surface of an isolated cell can be described by the following equa-
dL k
= ’ (L ’ L 0 ), (5.8a)
dt f
(4L /a ’ 3)2 (4L 0 /a ’ 3)2
dL 0 a 1 LL0
= ’2 + ’1 . (5.8b)

dt 16 a 2 2
Here a and „ are positive constants (with units of length and time
respectively) and the speci¬c expression on the right-hand side of
Eq. 5.8b has been chosen for illustrative purposes. As discussed by
Odell et al. (1981), the system™s behavior can be derived from certain
of its qualitative features. Thus, if L0 is a constant (i.e., dL 0 /dt = 0)
then a stretched apical face eventually returns to its original size.
(The solution of Eq. 5.8a is L = L 0 + (L i ’ L 0 )e’kt/µ , where L i is the
initial circumference (at t = 0) of the stretched surface.) However,
if L 0 varies with time then Eqs. 5.8a, b represent a simple dynami-
cal system like those discussed in Chapter 3. It has two stable ¬xed
points, L = L 0 = a and L = L 0 = a/4, at which the left-hand sides of
Eqs. 5.8a, b vanish; small deviations from these stationary points re-
turn the system to the same points. We can identify the untriggered

equilibrium length with L 0 = a and the rest length after ¬ring with
L 0 = a/4. A separatrix that divides the space in the LL0 -plane into
the basins of attraction of the stable ¬xed points (Chapter 3) passes
through the point L = L 0 = a/2, which is an unstable ¬xed point of
the system de¬ned in Eqs. 5.8a, b. This example illustrates how an ex-
citable biological material can be modeled by a dynamical system. In
the Appendix at the end of this chapter we use Eqs. 5.8a, b to demon-
strate how the mathematical method of linear stability analysis can
be employed to study the ¬xed-point structure of dynamical systems
de¬ned by differential equations.
4. Taking advantage of the near-spherical symmetry of the sea
urchin blastula, Odell and coworkers (1981) performed their calcu-
lations in two dimensions. Thus, L in Eqs. 5.7 and 5.8a, b denotes
the linear sizes (e.g., height or diagonals) of trapezoidal cells that
are initially arranged along the perimeter of a circle, similarly to
the Drasdo--Forgacs model (Fig. 5.8). Force and torque equilibrium
is assumed along each face and corner of the initially tension-free
Invagination is initiated in the model by the ¬ring of a single
cell at the middle of the vegetal pole. The apical contraction of this
cell dilates the apical surface of its neighbors, an effect which, if
suf¬ciently large, evokes their ¬ring and subsequent apical contrac-
tion. This sequence of events leads to a spreading wave of contrac-
tion, which eventually generates an invagination in the cell layer,
as illustrated in Fig. 5.10. The buckling of the vegetal plate, in this
model, crucially depends on the unique excitable nature of these
In contrast with the Drasdo--Forgacs model, on the one hand the
model of Odell et al. postulates no biological basis for the unique be-
havior of vegetal plate cells. Its starting con¬guration is the spherical
blastula, which it treats in isolation. It does not consider the preced-
ing developmental processes. In particular, it does not address the
issue of the folding instability disclosed by the energy-minimization-
based Monte Carlo analysis. On the other hand, the dynamical nature
of the model of Odell et al. permits it to capture certain characteristic
shape changes of cells and tissue sheets associated with invagination.
As can be seen, the various approaches to modeling sea urchin gas-
trulation by their very nature have restricted applicability. They can
be viewed, however, as complementary and their eventual combina-
tion may provide a more comprehensive account of morphogenetic
Fig. 5.10 Representation of
gastrulation in the model of Odell
and coworkers. Here invagination
is triggered by changes in cell
shape resulting from the excitable
nature of the apical contractile


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