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actin network (compare with the
Drasdo“Forgacs model in Fig. 5.8).
(After Odell et al., 1981.)

Convergence and extension
During the subsequent course of sea urchin development the prim-
itive gut or ˜˜archenteron” formed by the invaginating blastula wall
elongates and narrows. This is an example of the phenomenon of
convergence and extension mentioned at the beginning of this chap-
ter. This effect occurs across a wide range of animal phyla. In insects,
for example, the surface ectodermal cells of the gastrula, along with
the underlying mesoderm, elongate posteriorly in a process termed
˜˜germ-band extension.” In chordates, the wider group of organisms
that includes vertebrates, the forming mesoderm separates into two
distinct subtissues, the broad somitic mesoderm that gives rise to the
backbone and musculature and the notochord, a stiff central rod of
mesoderm that ¬rst de¬nes the axial skeleton and then induces the
formation of the vertebral column and spinal cord. Both of these pri-
mordia also undergo convergence and extension (Keller et al., 2000;
Keller, 2002; see Fig. 5.11A).
On the basis of our earlier discussion of tissue behavior, these
characteristic movements of late gastrulation might seem physically
counterintuitive. In Chapter 4, for example, we saw how the liquid-like
properties of tissues led to their ˜˜rounding up” to attain minimum
surface-to-volume ratios. Because tissues undergoing convergence and
extension violate this expectation, it was at one time thought that
their movements might be dependent on external forces from adhe-
sive substrata provided by adjoining tissues in the embryo. This was
proved incorrect by experiments in which the dorsal sectors of two
frog gastrula (that is, the back regions, containing the forming meso-
derm) were cut out and their inner, deep-cell, surfaces sandwiched
together. The composite tissue converged and extended purely on the
basis of internally generated forces, with no mechanical or adhesive
assistance from the rest of the embryo (Keller et al., 1985; Keller and
Danilchik, 1988).
What then, is the origin of these shaping forces? Some hints can
be gained from the models described earlier in this chapter. In the
Mittenthal--Mazo model, for example, epithelial tissues, using the pas-
sive mechanisms of differential adhesion and elastic response, bulged
out and elongated into cones and tubes, reminiscent of convergent
extension rather than rounding up. This effect arose because the
cell sheet had a nonuniform distribution of cell adhesivity, a prop-
erty presumed to depend on the active, nonequilibrium, processes
of cell differentiation discussed in Chapter 3. However, because the
frog mesoderm studied by Keller and coworkers consists of an essen-
tially uniform population of cells this ˜˜adhesive prepattern” mecha-
nism cannot by itself account for the classic cases of convergence and
The model of Odell and coworkers (1981) accounted for non-passive
tissue behaviors by invoking the dynamical properties of individual
cells, such as apical constriction and mechanical excitability. That


A mesoderm

Bottle cells




B intercalation

Fig. 5.11 Convergence and extension movements during gastrulation in Xenopus. (A)
A view of the dorsal aspect of the gastrulating embryo. Both the mesodermal cells (red)
and the ectodermal cells (blue) of the anteroposterior body axis ¬rst undergo radial
intercalation (B, on the left, not shown in A), the rearrangement of several layers of deep
cells along the radius of the embryo (normal to the surface) to form tissue masses having
fewer cell layers; next, these same tissues undergo mediolateral intercalation. AP, animal
pole; VP, vegetal pole. (B) On the right, the rearrangement of multiple rows of cells
along the mediolateral axis (indicated by the small horizontal arrows in A) to form
narrower tissue masses that are elongated along the anteroposterior axis (indicated by
the long arrows in A and B). (A, after Gilbert, 2003; B, after Keller et al., 2000.)

such dynamical properties might enter into the physics of conver-
gence and extension is suggested by the fact that cells within the
epithelial sheets undergoing these shape changes (Fig. 5.11A) mani-
fest intercalation (Keller, 2002) in which they interdigitate among one
another, ¬rst along the radius of the embryo, normal to its surface
(Fig. 5.11B), and then along its mediolateral axis (e.g., from the center
to the edges) (Fig. 5.11B) to produce a narrower, longer, and thicker
array (Keller et al. 1989; Shih & Keller, 1992).
Recent evidence indicates that these intercalation movements de-
pend on a dynamical property of cells termed ˜˜planar polarity”
(Mlodzik, 2002). This phenomenon differs from the longer-known
apical--basolateral polarity described earlier in this chapter. Planar
polarization is a dynamical response of cells to extracellular effectors
and homophilic adhesive interactions. In particular, it leads, via an
intracellular signaling cascade, to cytoskeletal rearrangements and
consequent elongation and ¬‚attening (Mlodzik, 2002), activities not
speci¬cally associated with apical--basal polarity. Once the planar po-
larization response ˜˜¬res,” the nonuniform localization of the intra-
cellular components of several signaling pathways means that the
cells acquire differing adhesive properties on their different surfaces.
Zajac and coworkers (Zajac et al., 2000, 2003) have developed a physi-
cal model for convergence and extension that makes use of precisely
such triggered anisotropies in cell properties.

The Zajac“Jones“Glazier model of convergent extension
As described above, convergent extension is a morphogenetic process
that transforms an epithelial sheet composed of cuboidal cells into
a tissue that contains a more or less ordered array of elongated cells
and has a characteristic shape, with the length along one direction
considerably greater than that along the others (Fig. 5.12). Zajac and
coworkers (2000, 2003) constructed a model for a cell population un-
dergoing convergent extension based on differential adhesion. In this
model, the cellular pattern arrived at through convergent extension
corresponds to the energetically most favorable con¬guration of the
system. The model is based on the following assumptions.
(i) During convergent extension, cell division is negligible and
the volume of individual cells is unchanged.
(ii) Cells form a closely packed array with no internal empty
spaces (Fig. 5.12).
(iii) The original cuboidal cells (Fig. 5.12A) are ˜˜triggered” into
elongated shapes by the acquisition of planar polarity men-
tioned earlier, with the result that differential adhesion oc-
curs along the various faces (Fig. 5.12B).
We ¬rst show that under these assumptions the elongated cells
prefer to arrange themselves in an ordered state in which they
preferentially adhere along their similarly deformed surfaces (i.e.,
elongated--elongated and narrowed--narrowed, Fig. 5.12C). Since con-
vergent extension can be manifested in the behavior of a cell sheet,




Fig. 5.12 Representation of convergent extension in the model of Zajac and
coworkers. (A) Epithelial sheet, made of cuboidal cells, before the onset of convergent
extension. (B) Top: Schematic representation of the epithelial sheet after the cells have
elongated, become aligned, and begun to intercalate. Bottom: Representation of
intercalation in the two-dimensional plane spanned by axes along the long sides l and the
short sides s of rectangular model cells. (C) Top: Schematic representation of the cellular
arrangement after the completion of convergent extension. Tissue elongation has
occurred in the direction perpendicular to the long axes of the cells. Bottom and right:
Two possible ordered cellular arrays predicted by the model. Under rather general
conditions (see the main text) it is the taller pattern, on the right (resembling the cellular
arrangement that results from convergent extension), that corresponds to the lowest
energy con¬guration. (After Zajac et al., 2000.)

we will construct the model in two dimensions. According to assump-
tion 3, each elongated cell is represented by a rectangle with its longer
and shorter sides denoted by l and s, respectively (Fig. 5.12B; the depth
of the cells, perpendicular to the ¬gure is added for illustrative pur-
poses). Adjoining cells may in principle contact each other along their
lengths (ll contact), along their width (ss contact), or form a mixed in-
terface (ls contact). We denote the total contact length of each type as
L ll , L ss and L ls . The total contact length between the l and s sides and
the surrounding medium at the boundaries is represented, respec-
tively, by Sl and Ss . For N cells in the array the total contour length
along the l and s sides (each cell having two l sides and two s sides)
is 2Nl and 2Ns, respectively, which can be expressed in terms of the
above quantities as
2N l = 2L ll + L ls + Sl , (5.9a)
2N s = 2L ss + L ls + Ss . (5.9b)
If we denote the works of adhesion along the ll, ss, and ls contacts
respectively by wll , wss , and wls (here they are energies per unit length,
not per unit area as in Eq. 5.2), the total work necessary to disassemble
an array with a given con¬guration (speci¬ed by the values of L ll , L ss ,
and L ls ) is W = L ll wll + L ss wss + L ls wls . The most stable con¬guration
clearly corresponds to that set of L ll , L ss , and L ls which maximizes
W. Using Eqs. 5.9a, b we eliminate L ll and L ss from W and arrive at
W = ’L ls + N (lwll + s wss ) ’ 1 (Sl wll + Ss wss ) . (5.10)
ls 2

where ls = 1 (wll + wss ) ’ wls is the work needed to create a contact
of unit length between the long and short sides of two cells, a quan-
tity analogous to the interfacial energy introduced in Eq. 5.2. For
close-packed cellular arrays (such as those shown schematically in √
Fig. 5.12A, B, C) the boundary lengths are proportional to N ; thus
for large N the third term in the above equation is negligible. Equa-
tion 5.10 then reduces to W = ’L ls ls + C , where C is a constant
(it denotes the second term on the right-hand side of Eq. 5.10, which
contains only ¬xed model parameters). Finally, assuming that ls > 0,
W is maximal if L ls = 0, which corresponds to the case of ordered
con¬gurations of rectangular cells such as those shown in Fig. 5.12C.
The various works of adhesion clearly are related to the densi-
ties of binding sites c l and c s for the l and s sides respectively, and
(since an adhesive bond contains two adhesion molecules) can be
expressed as wll = c l c l , ws s = c s c s , and wsl = c l c s . (To make these equa-
tions dimensionally correct, a factor with units of energy times length
should be incorporated into each one.) Using these forms we obtain
ls = 2 (c l ’ c s ) > 0. Thus ordered arrays correspond to energetically
1 2

favorable cellular patterns under rather general conditions. (Note that
the above conclusions are valid for any non-equal values of c l and c s .)
In Fig. 5.12C two different ordered rectangular arrays, each con-
taining the same number of cells, are shown. To analyze the differ-
ence between their stability we have to return to Eq. 5.10 and con-
sider the last term on the right-hand side. Since the arrays contain
a ¬nite number of cells (12), N cannot be assumed to be arbitrarily
large, therefore this term cannot now be ignored. For the two ordered
rectangular arrays shown in Fig. 5.12C, we have (Nl /2) — (N s /2) = N ,
where Nl and N s are the numbers of cells on the array boundaries
along the directions of elongation and narrowing respectively. Since
L ls = 0 for both con¬gurations, we now have to ¬nd the maximum
value of W = ’ 1 (Sl wll + Ss wss ) + C , or equivalently the minimum of
Sl wll + Ss wss = Z with Sl = Nl l and Ss = N s s, for ordered rectangu-
lar arrays. (Nl = 8 and 4, N s = 6 and 12, respectively for the 3 — 4 and
6 — 2 ordered arrays in Fig. 5.14C.) The minimization of Z as a
function of the boundary lengths then leads to the relationship
Ss /Sl = wll /wss . (The mathematically astute reader will notice that
minimization is carried out by ¬rst expressing Z in terms of Nl (or
N s ) only by using the above relationship between Nl , N s , and N , and
then equating the derivative of Z with respect to Nl (or N s ) to zero.)
If we now employ the fact that l > s (we have not used this anywhere
up to now) then we may safely assume that wll > wss . The above min-
imization procedure then leads to Ss > Sl , which corresponds to the
6 — 2 array. This resembles the array that results from convergent
This model of convergent extension suggests that differential ad-
hesion may underlie morphogenetic processes whose outcomes ap-
pear different from what would normally be expected from the clas-
sic differential adhesion hypothesis. It should be noted however, that
(in contrast with the explanations given for tissue rounding-up or


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