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work has a characteristic mesh size, set by the average chord length.
Below the percolation threshold no network forms. If the VEGF con-
centration is above its saturation limit then the cell receptors become
desensitized (Anderson and Chaplain, 1998) and no network forms
because cells are unable to follow the chemotactic gradient.
This two-dimensional model of Bussolino and coworkers is based
on the following assumptions.
(i) The cell population can be described by a continuous time-
dependent density ρ(x, y; t) and velocity vector v (x, y; t).
(ii) It is assumed that there is no endothelial cell proliferation
and so the cell number is conserved during network forma-
tion. Although this assumption may not be strictly true, net-
work formation is likely to take place independently of any
increase in cell number.
(iii) In the early stages of vascular assembly the cell population
can be modeled as a ¬‚uid of particles, which interact by way
of a chemoattractant (i.e., a form of VEGF) present at concen-
tration c(x, y; t).
(iv) The chemotactic agent is secreted by the cells themselves and
thus depends on cell density. It diffuses and degrades with
time.
(v) Cells are accelerated in their motion by gradients of the
chemoattractant and obey Newton™s second law. Note that
this assumption implies a linear force--acceleration relation
and thus it is not consistent with the linear force--velocity
relation that, as argued in Chapter 1, is biologically more re-
alistic for cells moving through viscous media. We comment
on this disparity at the end of this section.
Even though the model is two-dimensional, to simplify the mathe-
matics we will analyze the basic equations in one spatial dimension.
Let us then cast the above assumptions into mathematical equations.
In order to express the conservation of cell number, point (ii), con-
sider a small volume element V around a given point in space (a
small circle in two dimensions, a line segment in one dimension).
The number of ¬‚uid particles (corresponding to the cells, point (iii)
above) crossing the area A enclosing V in unit time is by de¬ni-
tion j A, where j = ρv is the particle current perpendicular to A.
The net change per unit time of the number of particles inside V is
on the one hand given by (ρ V )/ t; on the other hand it is equal
to the difference between the in-¬‚owing and out-¬‚owing particles
(per unit time) through the enclosing area, which can be expressed
as ’ ( j A) = ’( jin ’ jout ) A. (Note that by convention a current
component in the outward direction is taken as positive.) If particle
202 BIOLOGICAL PHYSICS OF THE DEVELOPING EMBRYO


number is conserved, it can change inside V only as a result of the
¬‚ow of particles across the enclosing area (i.e., no ˜˜sources” or ˜˜sinks”
of particles are present) and therefore (ρ V )/ t = ’ ( j A) must
hold. If V (and thus A ) is ¬xed, this relationship is equivalent to
ρ j
=’ .
V/ A
t
In one dimension V / A = x and, replacing the ¬nite differences
with in¬nitesimal ones, we ¬nally arrive at the conservation law
‚ρ ‚j
=’ . (8.3)
‚t ‚x
Next we consider the dynamics of cellular (i.e. ¬‚uid-particle) mo-
tion, which obeys Newton™s second law, mass times acceleration =
m total v/ t = net force, where total v is the total change in velocity
v in time t (see below). The net force, according to the model (see
point (v)) is provided by the gradient of the chemoattractant and is
thus proportional to c/ x. The velocity of a ¬‚uid particle depends
on both x and t. At a given point in space, v in general changes with
time and, at a given instant, v may be different at different locations.
The total change total v is therefore the sum of changes t v + x v due
to spatial and temporal variations respectively. We can write this as
tv xv
= t+
total v x,
t x
which in the limit of in¬nitesimal changes becomes
‚v ‚v
dv = dt + dx.
‚t ‚x
Dividing both sides of this equation by dt, we arrive at
‚v ‚v ‚c
+v =β . (8.4)
‚t ‚x ‚x
Here we have used dx/dt = v, and the constant β incorporates the
reciprocal of the mass and the proportionality constant due to the
relationship between force and chemoattractant gradient.
Finally, point (v) can be expressed mathematically as
‚c ‚ 2c c
= D 2 + ±ρ ’ . (8.5)
‚t ‚x „
In the above equation, D , ±, and „ are respectively the diffusion con-
stant, the rate of release, and the characteristic degradation time of
the chemoattractant.
Collectively, the two-dimensional analogues of Eqs. 8.3--8.5 de-
¬ne the dynamics of capillary network formation in the model of
Bussolino and coworkers. The authors performed a series of calcula-
tions on the time course of vascular network formation based on this
model. The similarity between the experimental patterns (Fig. 8.7,
upper row) and the model patterns (Fig. 8.7, lower row) is striking.
Thus it is reasonable to conclude that the physical mechanism that
has been described incorporates the most salient features of network
assembly during early capillary formation.
8 ORGANOGENESIS 203



A




B
500




300




100
0 200 400



Fig. 8.8 (A) Model networks simulated with different values of the interaction range
ξ . (B) Characteristic mesh size l vs. ξ . The values of l were calculated as the average
node-to-node distance in the simulation. The mesh size in the model networks of Serini
et al. (2003) is the analogue of the average chord length in the experiments. (A reprinted
from Serini et al., 2003, with permission. B after Serini et al., 2003.)




Equations 8.3--8.5 contain a natural length scale ξ = D „ , which
represents the effective range of interaction between cells mediated
by the chemoattractant, and a natural time scale (D /(±β)2/3 in one
dimension), representing the characteristic time for network forma-
tion. Simulations of network formation performed with different val-
ues of ξ are shown in Fig. 8.8A. The plot in Fig. 8.8B shows that
ξ is proportional to the characteristic mesh size l of the assembled
model networks, the analogue of the average chord length in the ex-
periments (cf. the upper and the lower rows in Fig. 8.7). Performing
additional experiments, Bussolino and coworkers found that, above
the percolation threshold, the average chord length is independent
of the cell-plating density, in the range ρ0 = 100--200 cells per square
millimeter. This is consistent with the ¬nding shown in Fig. 8.8B,
according to which the model network™s characteristic mesh size
depends on the range of cell interaction, set by independent param-
eters (such as D and „ ).
As noted above, the model of Bussolino and coworkers makes the
biologically implausible assumption that cells accelerate through the
204 BIOLOGICAL PHYSICS OF THE DEVELOPING EMBRYO


ECM in response to the chemotactic gradient. Merks and coworkers,
using a cellular automata model based on the scheme of Serini et al.
(2003) found that the effect of dropping the acceleration assump-
tion was that disconnected island-like patterns were produced, rather
than realistic networks (Merks et al., 2004). Merks and coworkers
also showed, however, that realistic patterns could be obtained when
various combinations of cell adhesion, contact-inhibition of motility
(resulting from, e.g., crowding), and cell elongation (which in their
model could be controlled independently) were substituted for the ac-
celeration assumption. Which of these possibilities, if any, represents
the biological reality needs to be tested. This demonstrates that the
continuum and discrete versions of a given model may permit the
manipulation of different aspects of the simulated developmental
process, thereby suggesting alternative experimental strategies.


Branching morphogenesis: development of the
salivary gland
Up to now we have largely considered the morphogenetic and pattern-
forming capacities of initially uniform epithelial or mesenchymal
cell populations. (An exception was the interaction between the no-
tochord and the surface epithelium during neurulation, discussed in
Chapter 5). In most cases of organogenesis, however, interaction and
cooperation between distinct cell populations is the norm. A com-
mon theme is the involvement of both epithelial and mesenchymal
components.
Glandular organs, which include the salivary and mammary
glands, the pancreas, and anatomically similar structures such as
the lung and kidney, form initially from a mass of mesenchymal tis-
sue surrounding a hollow, unbranched, epithelial tube. One or more
clefts appear at the tip of the epithelial tube, causing it to split into
two or more lobules. Mesenchymal cells deposit ECM and condense
(see Chapter 6) around the clefts and stalks of the lobules, while the
epithelium continues to proliferate. When the lobules have grown
suf¬ciently large, additional clefting and bifurcations occur, leading
to a highly branched structure (Fig. 8.9).
The submandibular salivary glands of rodents have been the sub-
ject of many studies devoted to uncovering the mechanisms of branch-
ing morphogenesis (reviewed in Hieda and Nakanishi, 1997, and Mel-
nick and Jaskoll, 2000). Treatment with X-rays or chemical inhibitors
of DNA synthesis showed that while branching requires the prolifera-
tion of epithelial cells, clefting does not (Nakanishi et al., 1987). Sali-
vary gland epithelia separated from their mesenchymes branch nor-
mally when recombined with mesenchyme of the same organ type
but abnormally or not at all when recombined with mesenchyme
from other organs (Spooner and Wessells, 1972; Ball, 1974; Lawson,
1983). These and a number of additional key experimental ¬nd-
ings, along with insights from the physical analysis of epithelial and
8 ORGANOGENESIS 205




Fig. 8.9 Branching morphogenesis in a salivary gland. On the left, a bud of epithelioid
tissue with a simple unbranched shape protrudes into a mass of mesenchyme. The
cut-away portion of the bud shows the tightly adhering cuboidal epitheliod cells in its
interior. The loose mesenchymal cells exterior to the bud are also seen. Middle, the
epithelial bud ¬‚attens slightly and splits into two or more lobules by the formation of
clefts. The epithelioid cells also begin to rearrange into a single layer surrounding an
interior lumen. As a cleft deepens, mesenchyme in and near it condenses and deposits
new ECM. When the young lobules have grown suf¬ciently large, further branching
occurs followed by extension and mesenchymal condensation. This process continues
until a highly branched structure has formed. (After Lubkin and Li, 2002, and Sakai et al.,
2003.)




mesenchymal tissues, provide the basis for a physical model of sali-
vary gland morphogenesis.


Modeling branching morphogenesis: the analysis of
Lubkin and Li
Most studies on branching morphogenesis have concentrated on bio-
chemical aspects. However, during this process tissues grow, move,
and most importantly change shape; this cannot take place without
physical forces. This is particularly evident during cleft formation,
a process which is not possible to understand without taking into
account biophysical and biomechanical considerations.
On the one hand a number of experiments suggest that ECM and
growth-factor components of the mesenchyme by themselves, rather
than any mechanical action or support, are suf¬cient to promote
branching morphogenesis of the epithelium (Nogawa and Takahashi,
1991; Takahashi and Nogawa, 1991). These experiments form the
basis of the epithelial theory of branching morphogenesis (Hardman
and Spooner, 1992), which assumes that the forces necessary to drive
cleft formation originate from within the epithelium, in particular
from the contraction of actin ¬laments. This theory is supported
by experiments in which cultured salivary glands were treated with
cytochalasin B (an F-actin disrupting agent), which resulted in the
abolishment of clefting (Spooner and Wessells, 1972). Similar results
206 BIOLOGICAL PHYSICS OF THE DEVELOPING EMBRYO


have been obtained with lung epithelium (Nogawa and Ito, 1995;
Miura and Shiota, 2000a).
On the other hand it is known that a wide variety of cell types,
including ¬broblasts, generate traction forces within the extracellular
matrix (Stopak and Harris, 1982; Vernon et al., 1992, 1995) that result
in deformation and, under certain conditions, pattern formation. The
mesenchymal theory of branching morphogenesis (Hieda and Nakanishi,
1997) places the origin of cleft formation in the contractile behavior
of ¬broblasts in the mesenchyme.
Lubkin and Li (2002) proposed that neither the epithelial nor the
mesenchymal theory of branching morphogenesis alone can fully ac-
count for cleft formation and that branching observed in in vitro
mesenchyme-free experiments is not mechanically equivalent to cleft
formation in mechanically intact rudiments or in vivo. They con-
structed a biomechanical model of cleft formation based on the me-
chanical properties of both the epithelium and the mesenchyme.
Lubkin and Li adopted the earlier proposition by Steinberg that em-
bryonic epithelia in many respects mimic the behavior of liquids
(Steinberg and Poole, 1982; Steinberg, 1998), a notion we have already
encountered in Chapters 4 and 5. They generalized Steinberg™s ideas

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