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fragment sorting) to account for convergent extension the model of
Zajac and coworkers had to combine differential adhesion with an
active, ˜˜nongeneric,” property of living cells: their ability to undergo
planar polarization.

In animals with backbones, convergent extension occurs in several tis-
sue primordia along the primary embryonic axis. As noted above, this
axis is ¬rst de¬ned by the coordinated convergence and extension of
the broad somitic mesoderm and central notochord (Fig. 5.11A). Once
this has occurred, the ectoderm overlying the notochord, induced
by signaling molecules arising from the latter, ¬‚attens and thickens.
This neural plate then undergoes a further set of morphogenetic move-
ments, including convergence and extension of its own and also the
elevation of two ridges to either side of the underlying notochord.
The portion of the plate between the ridges sinks downward and
the ridges fuse at their outermost regions. This sealing of the dorsal
(back) surface of the embryo causes the neural plate to assume the
form of a hollow cylinder -- the neural tube -- between the surface
ectoderm and the notochord (Fig. 5.13). As noted at the beginning of
this chapter, the neural tube is the primordium of the spinal cord of
the mature animal, and this entire sequence of events is referred to
as neurulation (see Colas and Schoenwolf, 2001, for a comprehensive
Neurulation is a process of differentiation (see Chapter 3) and
pattern formation (see Chapter 7). The spinal cord arises as a distinct
morphological structure that also consists of a distinct population of
cells -- neurons, in a particular arrangement. The generation of this
new cell type is intimately tied to the change in shape and form of the
neural plate. During neurulation the interplay of activators (termed
˜˜proneural” gene products) and inhibitors of neurogenesis determine
precisely which cells of the dorsal ectoderm (the majority of which
are capable of adopting a neuronal fate) differentiate into neurons
and which adopt different fates (Brunet and Ghysen, 1999; Bertrand
et al., 2002).
Several of the morphogenetic phenomena in epithelial sheets de-
scribed earlier are utilized during neurulation. Other neurulation-
related changes are novel, though potentially explicable on the basis
of familiar physical principles. The cells of the neural plate, for in-
stance, undergo an elongation of their apical--basal axes, which can be
understood by differential adhesion. The cells adhere on their basal
surfaces to an underlying basal lamina. Their initially cuboidal shape
indicates that the adhesion forces between the cells and the basal
lamina are comparable with the homotypic forces between the cells
on their lateral aspects.
Induction of the neural plate by the underlying tissues can lead
to a thickening of the epithelium by one of several alternative mech-
anisms. The induced epithelial cells could increase the effective

Neural plate Columnar thickening
A of neuroepithelium

B Elevation of
neural folds



Convergence of
neural folds

Neural tube

Fusion of neural folds

of neural crest

Fig. 5.13 Schematic representation, viewed as a cross-section perpendicular to the
anteroposterior axis, of the successive morphogenetic changes involved in neurulation in
some vertebrate species. (A) Cuboidal cells of the dorsal ectoderm along the midline of
the embryo (which is elongating in the anteroposterior direction, see Fig. 5.11A) become
columnar, forming the neural plate. Cells lateral to the neuroepithelium composing this
plate remain cuboidal. (B) While basal portions of the central deep cells of the neural
plate remain attached to the notochord (which is derived from the axial mesoderm, see
Fig. 5.11A), the lateral regions of the neural plate become elevated in two parallel ridges,
the neural folds. (C) The neural folds begin to converge, meeting each other at a line
parallel to the notochord. As this occurs, the underlying epithelium invaginates to form a
cylinder, the neural tube, which begins to pinch off from the overlying neuroepithelium.
(The neural tube will subsequently develop into the spinal cord.) (D) With complete
fusion of the surface ectoderm, the neural tube becomes a separate structure. Cells
originally at the crests of the neural folds (the “neural crest”) end up between the surface
ectoderm and the neural tube. These cells detach from both structures and disperse
through speci¬c pathways as a mesenchymal population (see Chapter 6). The formation
of the neural tube occurs in the fashion described along much of the central body axis of
birds and mammals. In some regions of the axis of these organisms, and in other forms
such as ¬sh, the mechanics of neural tube formation is somewhat different (Colas and
Schoenwolf, 2001; Lowery and Sive, 2004). (After Wallingford and Harland, 2002.)

number of cell adhesion molecules (CAMs) on their lateral surfaces
and thus increase their lateral surface area at the expense of their
common interface with the basal lamina. Alternatively, they may de-
crease the effective number of substratum adhesion molecules (SAMs)
on their basal surfaces. Either mechanism (or both) would lead to
elongation along the apical--basal axis and epithelial thickening, a
phenomenon utilized during later organogenesis as well (see Chap-
ter 8) and known generally as ˜˜placode formation.” CAMs and SAMs,
such as cadherins and integrins, are differentially regulated as neu-
rulation begins (Levi et al., 1991; Espeseth et al., 1995; Joos et al., 1995;
Lallier et al., 1996) but it is not yet clear whether a simple differential
adhesion mechanism, or some more complicated process (e.g., one in-
volving polarized microtubule assembly, Colas and Schoenwolf, 2001),
drives neural plate formation.
The morphogenetic events that follow neural plate formation are:
(i) convergent extension, which is mechanically independent of the
convergent extension of the underlying mesodermal tissues (Keller et
al., 1992a, b) but utilizes similar cellular mechanisms; (ii) bending or
buckling of the neural plate, which involves elevation of the neural
ridges and formation of the trough-like neural groove; and (iii) closure
of the neural groove (Colas and Schoenwolf, 2001). We will focus on
just one of these events here, the inward bending (invagination) of
the neural plate to form a groove and eventually a detached cylinder.
The invagination of the neural plate was modeled by Odell et al.
(1981) along the same lines as those used to model the vegetal plate in
the course of sea urchin gastrulation. Since their analysis has similar
strengths and limitations in each case, we will present here instead
the approach followed by Kerszberg and Changeux (1998), who used
˜˜cellular automata” (CA), an increasingly widely employed method
for modeling morphogenetic processes (Alber et al., 2003). Cellular au-
tomata are computer programs made up of interacting ˜˜cells,” each
of which acts like a computer programmed with a set of rules (an ˜˜au-
tomaton”). Such automata de¬ned by simple rules sometimes give rise
to surprisingly complex structures (see, for example, Wolfram, 2002).
In such cases it is dif¬cult to ascertain whether a pattern has formed
for the same reasons as in a biological system. At the other extreme,
the rules can be made suf¬ciently complex that a given CA program
is indistinguishable from the standard models of physics employing
dynamical laws and ¬eld concepts, like those we have considered up
to now. A convenient middle strategy is to employ CA rules which
are simple enough to take advantage of the method™s computational
speed and the ease with which parameters can be revised, but which
contain suf¬cient biological speci¬city that the patterns generated
can be attributed to realistic properties (Kiskowski et al., 2004). The
Kerszberg--Changeux model for neurulation represents an example of
this middle strategy. Since it deals with epithelial folding, like the
models discussed earlier in this chapter, it can serve to highlight the
differences between the CA framework and those that employ me-
chanical and ¬eld-based concepts.

The Kerszberg“Changeux model of neurulation
The folding neural plate resembles an elongated analog of the invagi-
nating vegetal plate that we encountered in sea urchin gastrulation.
It is therefore not surprising that some of the preceding discussion is
applicable to this morphogenetic process as well (Odell et al., 1981; Ja-
cobson et al., 1986). Like several other models discussed in this book,
the computational model of Kerszberg and Changeux (1998) explic-
itly uses the interplay of genetic and generic mechanisms to describe
a complex developmental process, drawing in this case on the large
body of data available on neurulation (reviewed in Colas and Schoen-
wolf, 2001). Below we summarize the general features of the model.
1. The model simulations are carried out on a two-dimensional lat-
tice of pixels, which represents a thin slice transverse to the embryo.
Each cell occupies a certain number of pixels, some being reserved
for the nucleus; this number varies as cells grow and migrate. There
are numerous genes that are activated during neurulation. Kerszberg
and Changeux considered some explicitly, the effect of others be-
ing incorporated through model parameters. Two extracellular signal-
ing molecules (BMP, i.e., the Bone Morphogenetic Proteins BMP-2 and
BMP-4, and Sonic hedgehog, Shh) referred to as ˜˜morphogens,” with
predetermined concentration gradients, i.e., ˜˜morphogenetic ¬elds”,
resulting from a prior pattern-forming mechanism (see Chapter 7), act
on the initially homogeneous peripheral epithelial cells. These cells
are held together at all times by intercellular adherens junctions; in
the simulation each cell has two.
2. Depending on the local concentrations of the morphogens,
which are determined by the morphogenetic ¬elds, two sets of au-
toregulatory transcription factors (constituting transcription factor
networks like those discussed in Chapter 3), act as genetic switches
that regulate the expression level of the genes Notch and Delta. The
products of these genes are widely employed positional mediators of
early-cell-type determination (see Chapter 7). Notch, a neuroectoder-
mal or neural plate gene, is activated ¬rst and later Delta, a proneural
gene, is induced. Since the products of Notch and Delta are a receptor
and its ligand, respectively, through their presence on adjoining cells
they eventually help to de¬ne a precise topographic assignment of
individual cells to a neuronal fate.
3. A major assumption of the Kerszberg--Changeux model is that
differential adhesion and cell motility during neural tube formation
are coupled and are under strict genetic control: neuroepithelial and
presumptive neuronal cells express a particular homophilic adhesion
molecule whose membrane concentration is proportional to the ac-
tivities of the regulatory genes Notch and Delta (themselves set by the
morphogenetic ¬elds). Cells change shape by movement and growth.
In the simulation a cell grows by the addition of pixels to its surface
at randomly chosen locations. If this results in a contact with another
cell that displays a suf¬cient concentration of the adhesion molecule,
the probability of effectively adding the pixel to the ¬rst cell at this

particular site will be increased relative to that of adding the pixel
at a location that does not result in a contact. The net outcome is
an ˜˜effective adhesion force” that brings and keeps cells together.
Cellular automata are well suited to modeling this kinetic aspect of
cell adhesivity, which is not considered explicitly in purely equilib-
rium analyses such as the differential adhesion hypothesis (DAH; see
Chapter 4).
4. As the differentiation pattern sharpens, morphogen distribu-
tions decay and cells remain committed to their acquired fate. As
they continue to move and grow and eventually divide they deform
under the mechanical forces to which they are subject (by contact
interactions with neighboring cells). The direction of movement is
biased by the adhesive forces between cells and by their cohesive in-
teractions with the basal lamina and extracellular matrix. Division
occurs with a probability that is a function of cell size and type, and
these in turn depend on the expression of Notch and Delta. The pixels
previously occupied by the mother cell are split into two sets belong-
ing to each of two daughter cells, which have nuclear transcriptional
states identical to that of the mother cell.
The formation of the neural tube in the simulations is depicted in
Fig. 5.14. The Kerszberg--Changeux CA model differs in a number of
ways from the approaches to morphogenesis discussed earlier. It is a
non-equilibrium model: the cellular pattern does not evolve towards
an energy minimum (as in the Mittenthal--Mazo or Drasdo--Forgacs
models) or to a state with balanced forces (as in the model of Odell
and coworkers). It is a discrete model: the time evolution of the con-
centration pro¬les of morphogens and gene products is not governed
by differential equations, as are the dynamical systems discussed in
Chapter 3. Instead, the movement of individual cells is dictated by
local rules set by the authors, which are chosen on the basis of ex-
perimental results and are intended to re¬‚ect biological reality. These
rules de¬ne effective forces, the variations in the expression levels of
regulatory genes (i.e., Notch, Delta) and the local concentrations of
molecules that mediate cell--cell interactions (i.e., adhesion proteins).
The appealing feature of such CA is that they explicitly demon-
strate how simple local rules may lead to complex global patterns.
Thus, they may reveal information on the hierarchical organization
of molecular circuits governing cellular processes, in particular mor-
phogenesis. Clearly their success depends on the ability to choose
the ˜˜right” local rules. Because of the ¬‚exibility (and arbitrariness)
of such models they can be modi¬ed and tweaked until they work,
although this is not always a satisfying way of gaining a fundamental
understanding of a system™s behavior and can represent a drawback to
such models. For example, an energy minimization principle such as
that employed by the DAH may indeed be fundamental to epithelial
folding, and the kinetic approach of Kerszberg and Changeux may be
incomplete in this respect. The local rules of their model can, in prin-
ciple, be designed to incorporate global effects, though not without
sacri¬cing some of the elegance and facility of the CA approach.


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