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