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

112 BIOLOGICAL PHYSICS OF THE DEVELOPING EMBRYO

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

5 EPITHELIAL MORPHOGENESIS: GASTRULATION AND NEURULATION 113

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

114 BIOLOGICAL PHYSICS OF THE DEVELOPING EMBRYO

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

follows:

d2 L dL

= ’k(L ’ L 0 ) ’ f + F load .

m (5.7)

dt 2 dt

Initial

equilibrium shape

Apical contractile

actin network

Imposed

stretch

Apical surface

Final

equilibrium shape

Basal surface

C

B

A

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.)

5 EPITHELIAL MORPHOGENESIS: GASTRULATION AND NEURULATION 115

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-

tions:

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

116 BIOLOGICAL PHYSICS OF THE DEVELOPING EMBRYO

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

cells.

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

cells.

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

processes.

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