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corresponds to the state, of a collection of cells, with particular phys-
ical characteristics and subject to speci¬ed physical constraints, that
has the lowest energy, at least temporarily. As cells differentiate in
a regional fashion and change their physical properties the original
structure will no longer correspond to the lowest energy state of the
embryo or primordium. Time-dependent physical processes will be
triggered (i.e., shape changes) that will steer the system into a new
equilibrium (or steady) state, which corresponds to a new shape, again
temporary. The altered shape of the embryo, or portion thereof, may
in turn in¬‚uence the course of subsequent development.
We have already carried out such a program in Chapter 2 when
discussing cleavage and blastula formation. We used Monte Carlo

simulations to demonstrate how a model system representing the
early embryo changed from one equilibrium state to another. In the
case of a single cleavage event it was the astral signal (causing an
imbalance in pressure) that triggered the departure from the initial
spherical equilibrium state of the embryo and led eventually to the
¬nal equilibrium state with two daughter cells. In the case of blas-
tula formation it was the sequence of cleavages that resulted in new
equilibrium states and ¬nally in a hollow sphere.
As powerful as the methods of energy minimization are, Monte
Carlo simulation being one such, they have a severe drawback: the
¬nal equilibrium state of the system, which corresponds to the lowest
(free) energy under speci¬ed conditions is arrived at through postu-
lated displacements of its components (or in simple cases by explicit
minimization of the mathematical expression for the energy). In the
case of the Drasdo--Forgacs model, for example, the energy is calcu-
lated after computer-selected shape deformations of randomly chosen
cells. Such an approach provides no information about the true dy-
namics of the system, however, and makes no attempt to explain how
it reaches one equilibrium state from another.
The explicit time evolution of a process (e.g., epithelial folding)
characterized by a set of parameters x1 , . . . , xn is described by the set
of equations
= fi (x1 , . . . , xn ) (i = 1, . . . , n). (5.1)
In most physicochemical applications the parameters xi represent
amounts or concentrations of substances. To apply this formalism
to shape changes, the parameters should also include those char-
acterizing the geometric properties of the embryo (e.g., the linear
dimensions, radii of curvature, etc.). With appropriate choice of the
functions fi the process may drive the system to one or several steady
states, which are stable against perturbations and capable of being
reached from any initial state nearby (see the discussion on dynami-
cal systems in Chapter 3). The stable states correspond to local minima
in the energy landscape of the system.
In what follows we use both energy minimization and the dy-
namical approach to capture different aspects of epithelial folding
and rearrangement. Most of what we will present relates to gastru-
lation and neurulation, in keeping with our focus on tracking the
processes of early development in approximate temporal order. We
will begin, however, by discussing a model for epithelial folding and
shape change that, while motivated by the problem of leg morpho-
genesis in insects, is also applicable to the phenomena that concern
us here. To do this we ¬rst have to introduce the notion of the ˜˜work
of adhesion.”

Work of adhesion
Consider a unit interfacial area between two materials A and B im-
mersed in a medium, denoted by M (which in particular could be the

Medium Medium


Fig. 5.1 Schematic illustration of the creation of an interface between two materials A
and B immersed in a common medium. In the two-step process shown, the ¬rst free
interfaces of materials A and B are produced (middle panel), which requires the
separation of their corresponding subunits (molecules, cells) from one another and thus
involves the works of cohesion w AA and w BB . In the second step the free interfaces are
combined to form AB interfaces (by rearranging the A and B blocks). The separation of A
and B requires the work of adhesion w AB . If the cross-sectional area of columns A and B
is of unit magnitude then the operation shown results in two units of interfacial area
between A and B.

vacuum or, in the case of tissues, the extracellular matrix or tissue
culture medium). We de¬ne the work of adhesion w A B as the energy
input required to separate A and B across a unit area in medium M.
We can imagine such a unit area to be formed in the following way.
First, we separate a rectangular column of A and B to produce two
free unit surfaces of each substance (Fig. 5.1). This requires amounts of
work wAA and wBB , respectively. These quantities are called the works
of cohesion. (Note that the magnitudes of the works of adhesion and
cohesion depend on the medium. To avoid the use of clumsy nota-
tion, the explicit dependence on M is not indicated here.) We then
combine these pieces to end up with two unit interfacial areas be-
tween A and B, as shown in Fig. 5.1. Thus the total work, A B , needed
to produce a unit interfacial area between A and B is given by
= 1 (wAA + wBB ) ’ wAB . (5.2)
AB 2

The quantity AB is called the interfacial energy and Eq. 5.2 is known
as the Dupr© equation (Israelachvili, 1991). If the interface is formed
by two immiscible liquids then Eq. 5.2 can readily be expressed in
terms of liquid surface and interfacial tensions. By de¬nition the sur-
face tension is the energy required to increase the surface of the
liquid by one unit of area (see Chapter 2). Since an amount of work
wAA + wBB creates two units of area, we obtain
σAB = σAM + σBM ’ wAB , (5.3)

where σAM , σBM , and σAB are respectively the surface tensions of liquids
A and B and their mutual interfacial tension (Israelachvili, 1991).
(Whereas for solids AB depends on the amount of interfacial area
between A and B, for liquids, σAB does not; see Chapter 2.) Note that
the immiscibility of A and B implies that σAB > 0. If, on the contrary,
σAB ¤ 0 then it is energetically more bene¬cial for the molecules of
liquid A to be surrounded by molecules of liquid B, and vice versa;
that is, A and B are miscible.

Fig. 5.2 Geometric con¬gurations of immiscible liquids A and B, and the
corresponding relations between the works of cohesion and the work of adhesion. It is
assumed that A is more cohesive than B, so that w AA > w BB and σAM > σ BM (M denotes
the surrounding medium).

If we now invoke the liquid-like behavior of tissues, we can apply
Eqs. 5.2 or 5.3 to obtain the conditions for sorting in terms of the ws
or the σ s. We imagine the cells of tissues A and B to be initially ran-
domly intermixed and surrounded by tissue culture medium and we
then allow them to sort, as discussed in Chapter 4. Let us assume that
tissue A is the more cohesive. This implies that wAA is the largest of
the three quantities on the right-hand side of Eq. 5.2. In the energet-
ically most favorable con¬guration at the end of the sorting process,
cells of tissue A form a sphere, the con¬guration in which they have
minimal contact with their environment and maximal contact with
each other. Then, depending on the relative magnitudes of the ws, the
sphere of tissue B may completely or partially envelope the sphere of
tissue A, or the two spheres may separate (Fig. 5.2) (Steinberg, 1963,
1978; Torza and Mason, 1969).
When liquid B just fully envelopes liquid A, σAB = σAM ’ σBM (see
Chapter 2, Eq. B2.1c). Combining this result with Eq. 5.3 and the rela-
tionships σAM = wAA /2, σBM = wBB /2 yields wAB = wBB . Thus, complete
envelopment takes place when B adheres more strongly to its partner
than to itself (Fig. 5.2, panel on the left). Partial envelopment oc-
curs when the less cohesive liquid, B, adheres more strongly to itself
than to its partner, and therefore tries to minimize its contact with A
(Fig. 5.2, middle panel). To sum up, partial or complete envelopment
corresponds respectively to wBB > wAB and wBB ¤ wAB . When wAB = 0,
there is no energy gain from the contact of the two liquids and they
separate (Fig. 5.2, panel on the right).
Applying Eqs. 5.2 and 5.3 to actual tissues results in quantitative
expressions for the extent of differential adhesion and its effect on
cell sorting.

The Mittenthal“Mazo model of epithelial shape change
A particularly simple and elegant quantitative model of epithelial
morphogenesis was constructed by Mittenthal and Mazo (1983). The
model accounts for the generic shape changes that occur when an
anchored epithelial sac (an ˜˜imaginal disc” on the surface of a fruit

¬‚y larva) is transformed into a series of tubular structures, such as leg
segments in the adult insect. Since these authors base their analysis
on tissue liquidity and elasticity, notions with which we are already
familiar, we will start our discussion of the physics of epithelial sheets
with this model. The Mittenthal--Mazo analysis can be summarized
by the following points.
1. In the course of transformation of the two-dimensional leg imag-
inal disc into a series of elongated three-dimensional leg segments
(which takes place with negligible cell division), the leg segments tele-
scope out from the disc, leading to a progressive narrowing and elon-
gation of the structure. (Note that the disc™s thickness is the height
of an epithelial cell.) As Mittenthal and Mazo noted, there are two
limiting mechanisms that could contribute to the reshaping of an
epithelial sheet such as the imaginal disc. In one mechanism, cells
would deform without rearrangement as an elongated segment forms
through the bulging out (i.e., evagination or eversion) of the sheet. In
the second mechanism, the reshaping of the tissue takes place by the
rearrangement of its cells; speci¬cally, the cells exchange neighbors
without changing their own shape. In the ¬rst case the epithelial
sheet behaves as an elastic medium (Fig. 5.3, panel on the left), in
the second case as a liquid (Fig. 5.3, panel on the right). The ¬rst
basic assumption of the Mittenthal--Mazo model is that the epithe-
lium of the imaginal disc, the hypodermis, is entirely liquid in the
tissue plane, exactly as we have assumed in the Steinberg differential
adhesion model in Chapter 4.
2. If the two-dimensional imaginal disc were an ideal liquid it
would resemble a soap ¬lm. Certain experimental ¬ndings, however
(Fristrom and Chihara, 1978), led Mittenthal and Mazo to make a sec-
ond basic assumption: unlike ordinary two-dimensional liquids such
as soap ¬lms, in which the two faces are mechanically equivalent, em-
bryonic epithelia have a globally distributed elastic component that
resists bending outside the plane of the tissue. As noted earlier in this
chapter, the cells of epithelia are polarized and their basal surfaces
adhere to basal laminae consisting largely of dense sheets of collagen-
ous protein (Yurchenco and O™Rear, 1994). These planar structures pro-
vide a resistance to the bending of the tissue sheet. Mittenthal and
Mazo thus treated the epithelium as a fluid elastic shell, ¬‚uid-like in its
capacity for in-plane rearrangement of cells but resembling an elastic
sheet when bending.
3. It was further assumed that the imaginal disc contains sev-
eral epithelial cell types, cells of each type having different adhesive
properties. When Steinberg™s differential adhesion hypothesis for cell
sorting in a binary mixture is applied to multiple cell types on a two-
dimensional sheet, the predicted pattern is reminiscent of a planar
bulls-eye target consisting of several concentric bands, as shown in
Fig. 5.4A. In a two-dimensional system, this pattern is a straightfor-
ward consequence of the conditions discussed in connection with the
complete envelopment of tissue A by tissue B (Fig. 5.2, left-hand panel).



Fig. 5.3 Deformation of an epithelial sheet in the Mittenthal“Mazo model. (A) If a
square patch of more cohesive tissue (e.g., imaginal disc tissue, shown as green) is
grafted onto a less cohesive host substratum (e.g., the surface of a fruit ¬‚y larva, shown
as pink), it will tend to minimize its contact region (i.e., perimeter) with the host. This
can be accomplished by a rounding, or, if the patch is more elastic than liquid, by a
bulging out from the host. In the course of bulging the individual cells in the patch must
change their shape. (B) If two patches, which are more liquid than elastic, from different
donor sites and thus of differing cohesive properties are grafted adjacent to each other
at a third host site, they assume one of the con¬gurations expected from the differential
adhesion hypothesis for sorting: (top to bottom) complete envelopment, partial
envelopment, or separation (compare with Fig. 5.2). In these con¬gurations the
individual cells may retain their original shape. (After Mittenthal and Mazo, 1983.)

Thus if the bands are numbered sequentially with numbers ( j = 1,
2, . . .) increasing toward the center, the bulls-eye pattern is stable if

> 0, (5.5a)
j, j+1

w j+1, j > w j, j , (5.5b)
w j,k > w j+1,k k < j.
for (5.5c)

Conditions 5.5a--c ensure, respectively, the stable segregation of cell
types, the complete envelopment of each band by its outer neighbor,
and the maintenance of a particular sequence of bands. (In Eq. 5.5a we
have used the more general quantity j, j+1 instead of the interfacial
tension σ j, j+1 in order to emphasize the mixed elastic--liquid nature
of the epithelium.)

A B C Fig. 5.4 Deformation of a planar
bull™s-eye target into a hollow
cone. (A) Each annulus, proceeding
toward the center, contains
successively more cohesive tissue.
(B, C) Successive stages of
deformation. (After Mittenthal and
Mazo, 1983.)


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