E

A

F

B

G

C

H

D

Fig. 5.14 Neural tube formation in the Kerszberg“Changeux model in a

cross-sectional view similar to that of Fig. 5.13. (A) The initial epithelial sheet of cells

breaks up into a central portion, the prospective neural plate, and lateral non-neural

ectoderm (see Fig. 5.13), in response to the ¬rst morphogen signal. Five green blobs,

nuclei expressing Notch; blue envelopes, cell membranes; small red lines, adherens

junctions between adjacent cells. (B) Delineation of the neural plate in response to the

second, neuralizing, morphogen signal. Three yellow blobs, nuclei expressing both Notch

and its ligand Delta; central light blue envelopes, membranes displaying the Delta ligand.

Two patches form, corresponding to the two ridges in the neural plate. The neural

ectodermal cells begin to thicken, forming the neural plate. (C) Invagination of the neural

plate due to constriction of the apical surfaces of its cells. (D) Surface epithelial cells

grow over the neural cells, forming the neural folds. (E) Neural tube folding continues

under the joint effects of neural cells pulling downward and epithelial cells dividing and

pushing the neural folds inward. (F) Cells at the folds deform due to mechanical forces.

(G) Folding is almost complete. (H) Closure of the neural tube. (From Kerszberg and

Changeux, 1998, with slight modi¬cations. Used by permission.)

128 BIOLOGICAL PHYSICS OF THE DEVELOPING EMBRYO

Perspective

Physical models of a complex phenomenon such as epithelial folding

are bound to rely on simplifying assumptions. With improved experi-

mental input and increasing computational power, increasingly real-

istic models can be constructed. Since simpli¬cations will still have

to be made, it is important to consider such physical models in an ap-

propriate perspective. Any model must lead to testable experimental

predictions if it is to provide a useful explanation for a system™s beha-

vior. Most models discussed in this chapter indeed lead to speci¬c

predictions: the scaling relation in the Mittenthal--Mazo model, the

eventual instability of the blastula in the Drasdo--Forgacs model, and

the triggering of invagination by the excitable apical actin network in

the model of Odell and coworkers. Different physical properties and

phenomena (differential adhesion, energy minimization, balance of

forces) are focused on in the various models. Certain models, such

as that of Kerszberg and Changeux, are explicitly nonequilibrium.

But even models such as those of Mittenthal and Mazo, Drasdo and

Forgacs, and Odell et al., which invoke equilibrium considerations

(i.e., energy minimization and mechanical force balance) assume an

underlying excitability of tissues -- different cell types must be gener-

ated, the cytoskeleton must be mechanically excitable, and so forth.

In the model for convergent extension of Zajac and coworkers, planar

polarization, an active property of cells with no obvious counterpart

in the nonliving world, is invoked, along with the passive physical

mechanism of differential adhesion. Any comprehensive account of

epithelial morphogenesis must therefore take into consideration the

multiple properties of cell sheets, prominent among which are a wide

variety of generic physical mechanisms.

Appendix: Linear stability analysis

To illustrate the method of linear stability analysis we consider the

following dynamical system:

dx

= ’a(x ’ y), (A5.1)

d„

(4x ’ 3)2 (4y ’ 3)2

dy 1

=b ’ xy + ’1 . (A5.2)

d„ 16 2 2

These equations can be considered as the nondimensional analogues

of Eqs. 5.8a, b, which arose in our discussion of the force-balance

model of gastrulation earlier in this chapter. In the above equations

x and y are dynamical variables (dependent on the dimensionless

˜˜time” variable „ ), while a and b are positive constants. Equations A5.1

and A5.2 have three steady-state solutions, x1 = y1 = 1, x2 = y2 = 1/4,

and x3 = y3 = 1/2, which make the right-hand sides vanish. We show

below that the ¬rst two solutions are stable whereas the third is

5 EPITHELIAL MORPHOGENESIS: GASTRULATION AND NEURULATION 129

unstable. To accomplish this we will perform linear stability analysis

around the solutions. The ¬rst step in this method is to represent

the functions x(„ ) and y(„ ) as x(„ ) = xi + µi („ ) and y(„ ) = yi + ·i („ ),

i = 1, 2, 3, where xi , yi denote any of the above solutions and the

functions µi („ ) and ·i („ ) stand for small deviations from the solutions.

Inserting these forms into Eqs. A5.1 and A5.2 and retaining terms only

up to ¬rst order in µi („ ) and ·i („ ) (i.e., linearizing the equations) we

obtain

dµi

= ’a(µi ’ ·i ), (A5.3)

d„

d·i

= ’bi (µi + ·i ), (A5.4)

d„

with b1 = 15 b, b2 = 16 b, and b3 = ’ 16 b. The second step is to look for

3 3

4

the solutions of these equations in the form

µ(„ ) = Ae ’»1 „ + B e ’»2 „ , (A5.5)

·(„ ) = C e ’»1 „ + D e ’»2 „ . (A5.6)

(Each quantity here should carry the index i, but for clarity we have

omitted it.) For the time being the quantities A, B , C , D , »1 , and »2 are

unknown. (The mathematically sophisticated reader will recognize

that »1 and »2 are the eigenvalues of the 2 — 2 matrix constructed

from the coef¬cients of µ and · on the right-hand side of Eqs. A5.3

and A5.4.) To determine the stability of a given solution of the original

Eqs. A5.1 and A5.2, it is suf¬cient to calculate »1 and »2 . If »1 and »2

are both positive then the corresponding solution is stable, since as

the system evolves in time both µ and · eventually vanish. To calculate

»1 and »2 we insert the trial solutions given in Eqs. A5.5 and A5.6 into

Eqs. A5.3 and A5.4. Performing the differentiation (remembering that

de ’»„ /d„ = ’»e ’»„ ) and rearranging the equations (still omitting the

index i), we arrive at

[»1 A ’ a(A ’ C )] e ’»1 „ + [»2 B ’ a(B ’ D )] e ’»2 „ = 0, (A5.7)

[»1 C ’ b(A + C )] e ’»1 „ + [»2 D ’ b(B + D )] e ’»2 „ = 0. (A5.8)

These equations can be satis¬ed only if the coef¬cients of the two

exponential factors in each equation separately vanish. (We assume

that »1 = »2 , which needs to be veri¬ed once these quantities have

been determined.) It is easy to see that the two equations containing

»1 have nontrivial solutions for A and C (i.e., different from zero) if

and only if »1 satis¬es

»2 ’ »1 (a + b) + 2ab = 0. (A5.9)

1

(The above equation is equivalent to setting equal to zero the 2 — 2

determinant constructed from the coef¬cients of A and C in Eqs. A5.7

and A5.8.)

A similar analysis for the equations containing »2 reveals that for

B and D to be nonzero, »2 must satisfy the same equation as »1

130 BIOLOGICAL PHYSICS OF THE DEVELOPING EMBRYO

above. Solving the quadratic equation Eq. A5.9 leads to (reinserting

the index i)

2

a + bi a + bi

(i)

»1,2 = ± ’ 2abi . (A5.10)

2 2

Here the subscripts 1 and 2 correspond respectively to the plus and

minus in front of the square root. If both a and bi are positive, as is

the case for i = 1, 2 (see after Eq. A5.4), then it is obvious that both

»1 and »2 must be positive since the second term on the right-hand

side of Eq. A5.10 is smaller than the ¬rst (a and b in Eqs. A5.1 and

A5.2 can be chosen so that the expression under the square root is

positive for each case, i = 1, 2, 3). For i = 3, since b3 < 0 we have that

»2 is negative. Thus the steady-state solutions x1 = y1 = 1 and x2 =

y2 = 1/4 are stable, whereas the solution x3 = y3 = 1/2 is unstable.

Chapter 6

Mesenchymal morphogenesis

During both gastrulation and neurulation certain tissue regions that

start out as epithelial -- portions of the blastula wall and the neural

plate -- undergo changes in physical state whereby their cells detach

from one another and become more loosely associated. Tissues con-

sisting of loosely packed cells are referred to as ˜˜mesenchymal” tis-

sues, or mesenchymes. Such tissues are susceptible to a range of physical

processes not seen in the epithelioid and epithelial tissues discussed

in Chapters 4 and 5. In this chapter we will focus on the physics of

these mesenchymal tissues.

We encountered this kind of tissue when we considered the ¬rst

phase of gastrulation in the sea urchin, the formation of the primary

mesenchyme, in which a population of cells separate from the vege-

tal plate, and from one another (except for residual attachments by

processes called ¬lopodia), and ingress into the blastocoel (Fig. 5.7).

The secondary mesenchyme forms later, at the tip of the archenteron,

and these newly differentiated cells help the tube-like archenteron to

elongate by sending their own ¬lopodia to sites on the inner surface

of the blastocoel wall. In other forms of gastrulation, such as that

occurring in birds and mammals, there is no distinction between pri-

mary and secondary mesenchyme: cells originating in the epiblast,

the upper layer of the pregastrula, ingress through a pit (the ˜˜blasto-

pore”) and the ˜˜primitive streak” that forms behind it as the mound

of tissue (˜˜Hensen™s node”) surrounding the blastopore moves anteri-

orly along the embryo™s surface (Fig. 6.1). The cells stream into the

extracellular-matrix (ECM)-¬lled space between the epiblast and the

underlying hypoblast, whereupon some of them are transformed into

the embryo™s mesenchymal middle layer, the mesoblast. Other cells of

the epiblast, after displacing the hypoblast cells, form the embryo™s

epithelial endodermal layer.

In contrast with the epithelioid and epithelial tissues, in which

cells are directly adherent to one another over a substantial portion

of their surfaces, mesenchymes and their mature counterparts in the

adult body, the connective tissues, consist of cells suspended in an

ECM. Often these tissues contain more ECM than cellular material

by volume. Thus there exists a set of morphogenetic processes that

132 BIOLOGICAL PHYSICS OF THE DEVELOPING EMBRYO

Fig. 6.1 Schematic view of a gastrulating chicken embryo. The cross-section through

the anteroposterior axis is shown; the direction away from the viewer is anterior. The

stage of development illustrated precedes that in the top panel of Fig. 5.13 and is roughly

equivalent to the stage of frog development shown in Fig. 5.11A. The arrows show

pathways of ingression and dispersion of cells entering the blastopore and primitive

streak to form the mesoblast. See the main text for an additional description.

Gastrulation in mammals is organized in a similar fashion.

occur in mesenchymal, but not epithelioid tissues, which depend on

the physical properties of the ECM, changes in the distance between

cells, the effects of cells on the organization of the ECM, and the

effects of the ECM on the shape and cytoskeletal organization of cells

(reviewed in Newman and Tomasek, 1996).

We will see later, in such phenomena as the separation and emer-

gence of the limb bud from the ¬‚ank or body wall of a developing

vertebrate embryo (Chapter 8), that, just as in epithelioid tissues, do-

mains of immiscibility (i.e., tissue compartments) can occur in mes-

enchymal tissues. This may seem surprising, since differential adhe-

sion per se is not relevant to cell populations in which cells do not

contact one another directly.

We will describe in this chapter ˜˜model” mesenchymes consisting

of isolated ECM components mixed with cell-sized particles. Such sys-

tems have been used to demonstrate that a characteristic feature of

mesenchymal-tissue ECMs -- namely, the presence of large numbers

of extended ¬bers that can arrange themselves in ˜˜paracrystalline”

arrays or form random networks (Meek and Fullwood, 2001; Ushiki,

2002) -- can provide the basis for mesenchymal tissue domains that

behave as distinct phases. In some mature connective tissues, such as

the cornea of the eye (Linsenmayer et al., 1998), tendons, and bones

(Cormack, 1987), highly regular arrangements do occur. These resem-

ble ˜˜liquid crystals” (Bouligand, 1972; Gaill et al., 1991; Giraud-Guille,

1996) and could potentially be analyzed in terms of the physics of

these materials, which are capable of undergoing well-de¬ned phase

transitions (de Gennes and Prost, 1993). The mesenchymal tissues

6 MESENCHYMAL MORPHOGENESIS 133

typically found in early embryos, however, contain ECMs with ran-

domly arranged ¬bers and, as we shall see, the physical state of such

a tissue can be drastically affected by the number density and aspect

(length-to-diameter) ratio of such ¬bers, which despite their random

arrangement can also form networks that undergo phase transitions.

The viscoelastic properties of ECMs are of clear importance in de-

termining the capacity of embryonic tissues to undergo rearrange-

ment and shape change. Equally important is the role of the ECM in

providing a communication medium between cells. The cells of the

epithelial and epithelioid tissues, discussed in earlier chapters, be-

ing in direct contact can signal each other by exchanging molecules

through gap junctions (see Chapter 4) or by surface-bound receptor--

ligand pairs (˜˜juxtacrine” signaling; see Chapter 7). In mesenchymal

and connective tissues, if cells communicate with one another then

they must do so over relatively long distances, through ECMs. They