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point forces in the mechanism of Lubkin and Li (2002), vertebrate limb
development depends on one or more well-regulated mechanisms of
spatial pattern formation such as those discussed in Chapter 7.
The limbs take form from mounds of tissue (˜˜limb buds”), which
emerge from the body wall, or ¬‚ank, at four discrete sites -- two for the
forelimbs and two for the hindlimbs. The mesenchymal tissue of the
early limb bud, which gives rise to the skeleton and muscles, forms
a paddle-shaped tissue mass referred to as a ˜˜mesoblast,” surrounded
by a layer of simple epithelium, the ectoderm. The skeletons of all ver-
tebrate limbs develop in a proximodistal fashion: that is, the structures
closest (˜˜proximal”) to the body form ¬rst, followed, successively, by
structures more and more distant (˜˜distal”) from the body. For the
forelimb of the chicken, for example, this means that the humerus
of the upper arm is generated ¬rst, followed by the radius and ulna
of the mid-arm, the wrist bones, and ¬nally the digits (Fig. 8.13).
The bones of the limb skeleton do not take form directly as bone
tissue. The pattern is ¬rst laid out as cartilage, which is replaced by
bone at a later stage of embryogenesis in most, but not all, verte-
brate species. Some salamanders, for example, have limb skeletons
composed almost entirely of cartilage.
Before the cartilages of the limb skeleton form, the mesenchymal
cells of the mesoblast are dispersed in a hydrated ECM, rich in the
glycosaminoglycan hyaluronan. The ¬rst morphological evidence that
cartilage will differentiate at a particular site in the mesoblast is the
emergence of precartilage mesenchymal condensations. The cells at
these sites then progress to fully differentiated cartilage elements by
switching their transcriptional capabilities. As discussed in detail in
Chapter 6, condensation involves the transient aggregation of cells
within a mesenchymal tissue. This process is mediated ¬rst by the
local production and secretion of ECM glycoproteins such as ¬bro-
nectin (Tomasek et al., 1982; Kosher et al., 1982; Frenz et al., 1989b),






Fig. 8.13 The progress of chondrogenesis in the chick wing bud between four and
seven days of development. The limbs are shown as if transparent. The lighter gray
regions represent precartilage; the darker-gray regions represent de¬nitive cartilage.
(After Newman and Frisch, 1979). The single proximal element that forms ¬rst is the
humerus (the femur in the leg); the two elements of the mid-wing form next, the radius
and the ulna (the tibia and ¬bula in the leg); the distal-most, last-forming, elements are
the digits. The proximodistal, anteroposterior, and dorsoventral axes are indicated on an
illustration of a human hand.

which act to alter the movement of cells and trap them in spe-
ci¬c places. The aggregates are then consolidated by direct cell--cell
adhesion. For this to occur the condensing cells need to express, at
least temporarily, adhesion molecules such as cadherins (Oberlender
and Tuan, 1994; Simonneau et al., 1995).
Because all the precartilage cells of the limb mesoblast are capa-
ble of producing ¬bronectin and cadherins, but only those at sites
destined to form skeletal elements do so, there clearly must be
communication among the cells to divide the labor in this respect.

This is mediated in part by secreted diffusible factors of the TGF-β
family of growth factors, which promote the production of ¬bronectin
(Leonard et al., 1991) and N-cadherin (Tsonis et al., 1994), though the ac-
tual CAM involved may be a different cadherin (Luo et al., 2005). Limb
bud mesenchyme also shares with many other connective tissues the
signi¬cant autoregulatory capability of producing more TGF-β upon
stimulation with this factor (Miura and Shiota, 2000b; Van Obberghen-
Schilling et al., 1988).
The limb bud ectoderm performs several important functions.
First, it is a source of ¬broblast growth factors (FGFs) (Martin, 1998).
Although the entire limb ectoderm produces FGFs, it is the particu-
lar mixture produced by the apical ectodermal ridge (AER), a narrow
band of specialized ectodermal cells running in the anteroposterior
direction (shown schematically in Fig. 8.14) along the tip of the grow-
ing limb bud in birds and mammals, that is essential to limb out-
growth and pattern formation. The AER keeps the precondensed mes-
enchyme of the ˜˜apical zone” (Fig. 8.14) in a labile state (Kosher et al.,
1979) and its removal leads to terminal truncations of the skeleton
(Saunders, 1948).
The FGFs produced by the ectoderm affect the developing limb
tissues through one of three distinct FGF receptors (Fig. 8.14). The
apical zone is the only region of the mesoblast containing cells that
express FGF receptor 1 (FGFR1) (Peters et al., 1992; Szebenyi et al.,
1995). In the developing chicken limb, cells begin to condense at a
distance of approximately 0.3 mm from the AER. In this, the mor-
phogenetically ˜˜active zone” (Fig. 8.14) FGFR1 is downregulated and
cells that express FGFR2 appear at the sites of incipient condensa-
tion (Peters et al., 1992; Szebenyi et al., 1995; Moftah et al., 2002).
Activation of these FGFR2-expressing cells by FGFs releases a lat-
erally acting inhibitor (that is, it acts in directions peripheral to
the condensations) of cartilage differentiation (Moftah et al., 2002).
Although the molecular identity of this inhibitor is unknown, its
behavior is consistent with that of a diffusible molecule. Finally,
differentiated cartilage in the more mature region, proximal to the
condensing cells, expresses FGFR3, which is involved in the growth
control of this tissue (Ornitz and Marie, 2002). (Since the pattern is
¬xed in this region it is referred to as the ˜˜frozen zone”; Fig. 8.14.)
The ectoderm, by virtue of the FGFs it produces, thus regulates the
growth and differentiation of the mesenchyme and cartilage.
The limb ectoderm performs one additional function. By itself
the mesenchyme, being an isotropic tissue with liquid-like proper-
ties, would tend to round up. Ensheathed by the ectoderm, however,
it assumes a paddle shape (see the upper part of Fig. 8.13). This is
evidently due to a biomechanical in¬‚uence of the epithelial sheet,
with its underlying basal lamina, along with the anteroposteriorly ar-
ranged AER, which has a less organized basal lamina (Newman et al.,
1981). There is no entirely adequate biomechanical explanation for
the control of limb bud shape by the ectoderm (but see Borkhvardt,
2000, for a review and suggestions). The shape of the limb bud will






FGFR2(+) cells

Fig. 8.14 Schematic representation of the biochemical genetic circuitry underlying
the pattern-forming instability described in the model of Hentschel et al. (2004),
superimposed on a two-dimensional representation of the 5-day limb bud shown in
Fig. 8.13. The positive autoregulation of TGF-β, the induction of ¬bronectin by TGF-β,
the promotion of precartilage condensation by ¬bronectin, and the FGF-dependent
elicitation of a lateral inhibitor of chondrogenesis from sites of condensation are all
supported by experimental evidence. The molecular identity of the inhibitor is unknown,
as is the factor or activity it inhibits. The model assumes that the inhibitor acts directly
on TGF-β. The colored rectangles represent the distribution of the densities of the
indicated cell types, de¬ned by the expression of the various FGF receptors in the
different zones. The apical zone contains a high density of cells expressing FGFR1
(green). In this zone, cell rearrangement is suppressed by the FGFs emanating from the
AER. The active zone is the site of the spatiotemporal regulation of mesenchymal cell
condensation (i.e., pattern formation). Pattern formation begins with the establishment
of populations of cells expressing FGF receptor 2 (red). The lower part of the ¬gure
gives an enlarged version of part of this zone. The curved arrows show positively
autoregulatory activators; the straight lines ending in circles show laterally acting
inhibitors. When condensed cells leave the proximal end of the active zone and enter
the frozen zone they differentiate into cartilage cells, which express FGFR3 (blue), and
their spatiotemporal pattern becomes ¬xed. At different stages of development the
active zone will contain different numbers of elements; eventually the frozen zone will
encompass the entire pattern. The length of the dorsoventral axis (normal to the plane
of the ¬gure; see Fig. 8.13) is collapsed to zero in this simpli¬ed model. PD,
proximodistal axis; AP, anteroposterior axis.

thus serve as an assumed boundary condition for the model of skele-
tal pattern formation that we will now consider.

Skeletal pattern formation: the model of Hentschel et al.
The vertebrate limb is clearly the most elaborate of the organs whose
development we are modeling in this chapter. It is a truly multi-
dimensional structure within a structure: a complex pattern of bones
embedded in a paddle-shaped tissue mass. Not only are the overall
external dimensions of this structure changing in time but so also are
those of the various interior zones, namely, the noncondensed zone at
the distal end, beneath the AER, the subapical zone of condensation,
and the more mature proximal zone.
Skeletal patterning in the limb involves cell movement and differ-
entiation: mesenchymal condensation is followed by chondrogenesis
(i.e., cartilage differentiation). It is dependent on the interaction of
epithelial and mesenchymal tissue types. The spatiotemporal evolu-
tion of the skeletal pattern is controlled by several classes of morpho-
genetic growth factors and their receptors, ECM molecules, and cell
adhesion molecules. The above list is far from being complete. It does
not take into consideration, for example, the complex but subsidiary
questions of the patterning of the limb muscles and nerves (see the
references given in Newman, 1988) and the replacement of the car-
tilage skeleton by bone tissue (processes that we will not explicitly
consider here).
A physical model that incorporates all the above ingredients must
involve a large number of coupled reaction--diffusion and domain-
growth equations, which are nonlinear (owing to complicated feed-
back mechanisms and differentiation, see Chapter 3) and which must
be solved under moving boundary conditions, since the entire limb
and its internal domains are constantly changing in size. The model
of Hentschel et al. (2004) is a synthesis of cell-biological and molecular
genetic knowledge in this area and draws on concepts presented in
previous chapters as well as earlier attempts to model avian limb de-
velopment (e.g., Newman and Frisch, 1979; Newman et al., 1988; Miura
and Shiota, 2000b; Miura and Maini, 2004). Rather than present this
model in elaborate detail, in the spirit of this book we will simply
sketch out the approach taken by Hentschel and coworkers in setting
up a biologically motivated mathematical formalism for the multi-
faceted problem of vertebrate limb development and then present
some results.

Geometry of the developing avian limb
The developing limb has a smooth, but nonstandard, geometric shape
that changes over time. Moreover, different processes take place in dif-
ferent parts of the changing structure. Hentschel and coworkers made
the following geometric approximation based on the simpli¬ed bio-
logical model shown in Fig. 8.14. The limb bud was considered as
a parallelepiped of time-dependent proximodistal length, L (t) along
the x axis, and ¬xed length l y along the anteroposterior (thumb to

little ¬nger) direction (the y axis). The dorsoventral (back to front)
width (the z axis) was collapsed to zero in this simple model. L (t)
was considered to consist of three regions, as described above: an api-
cal zone of size l apical (t), at the distal tip of the bud and consisting
of noncondensing mesenchymal cells, followed by an interior active
zone of length l x (t), which contains differentiating and condensing
cell types, and the proximal frozen zone of cartilage cells of length
l frozen (t); L (t) = l apical (t) + l x (t) + l frozen (t). The division of the distal part
of the limb into two zones re¬‚ects the activity of the AER in sup-
pressing differentiation of adjacent mesenchyme (see above). This is
assumed in the model to result from the distribution of FGF, which
is highest under the AER and lower at some distance from it. The ac-
tive zone, therefore, is where differentiating cells, morphogens, and
growth factors interact dynamically, giving rise to a time-dependent
pattern of condensations. As will be seen below, the length of the
active zone, l x (t), serves as a ˜˜control parameter” that determines the
character of the condensation pattern. Cells from the proximal end
of the apical zone are recruited into the active zone as they divide
and move away from the in¬‚uence of the AER. The active zone loses
cells in turn to the proximal frozen zone, the region where cartilage
differentiation has occurred and a portion of the de¬nitive pattern
has already formed.

Variables: cell types and molecules


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