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further to embryonic mesenchymes and treated such tissues also as
liquids in their model. We saw in Chapter 6 that certain aspects of
mesenchymal morphogenesis indeed can be interpreted by attribu-
ting liquid-like behavior to these tissues.
The major assumptions and elements of the Lubkin--Li model are
as follows (see also Fig. 8.10).
(i) A branching rudiment is considered as a uniform epithelium
inside a uniform mesenchyme. Both tissues are modeled as
liquids of uniform density (in both space and time) and char-
acterized by their respective viscosities (·’ and ·+ ) and sur-
face tensions. For simplicity only a planar section of the epi-
thelium is modeled.
(ii) Epithelial growth and the formation of lumens (which are
generated as the epithelium matures) are ignored; this is con-
sistent with the model™s focus on cleft formation rather than
the growth of the branching rudiment.
(iii) Shape changes are driven by spatial variations in the inter-
facial tension γ , between the two tissues, which arise due to
point forces, fi , f j , . . . , acting at selected sites i, j, . . . along
the interface, denoted by in Fig. 8.10. The origin of these
forces, as discussed above, lies in the contractile properties
of epithelial cells and ¬broblasts and in the biomechanical
properties of the ECM and is not modeled explicitly.
(iv) Forces create stresses and deformations in the two tissue ¬‚u-
ids, which lead to spatial rearrangements and potentially to
cleft formation and branching.
As before, let us cast these assumptions into mathematical language.
The physical state of any liquid is described by its density ρ, a velocity
¬eld v, and a pressure ¬eld p (think of hydrostatic pressure as one of







Fig. 8.10 The geometry of the ¬‚uid components in the model of Lubkin and Li. The
inner ¬‚uid, the epithelium, has uniform viscosity ·’ . The outer ¬‚uid, of uniform viscosity
·+ , represents the mesenchyme. The quantities, f i , f j are point forces acting at the
interface in the directions indicated by the arrows. The normal and tangential directions
are denoted by n and „ , respectively. The surrounding region of ¬‚uid-mesenchyme is
large in relation to the size of the epithelial portion. After Lubkin and Li (2002).

the simplest such ¬elds). We will ¬rst set up the ˜˜equation of motion
of a liquid” in terms of the velocity and pressure ¬eld, using Newton™s
second law. This equation will govern the dynamical behavior of the
liquid, which in the present case is the hypothesized mechanism of
cleft formation.
The model of Lubkin and Li is two-dimensional (point (i)). More-
over, as discussed in the section on the viscous transport of cells in
Chapter 1, inertial forces can typically be neglected in early embryonic
processes because of the high viscosity of the materials involved.
Therefore the above equation simpli¬es considerably to (i = x, y)
‚p ‚ 2 vi ‚ 2 vi
fi ’ +· + = 0. (8.6)
‚i ‚ x2 ‚ y2
In the model of Lubkin and Li, Eq. 8.6 applies to both the epithelial
and mesenchymal ¬‚uids, which are distinguished by their respective
viscosities, ·’ and ·+ . In Eq. 8.6 f(x, y) is the (inward-directed, see
Fig. 8.10) clefting force, and Lubkin and Li assume that this force acts
at discrete points, with the same magnitude f0 , along the interface
(Fig. 8.10) between the two ¬‚uids.
Since the model does not incorporate growth (point (ii)), the ¬‚uids™
volumes are constant during cleft formation: they are assumed to be
incompressible. The condition of incompressibility (ρ is constant in
space and time) is easily obtained from the conservation law Eq. 8.3
generalized to two dimensions ( j = ρv)
‚vx ‚v y
+ = 0. (8.7)
‚x ‚y

Box 8.2 Dynamics of viscous liquids: the Navier“Stokes

We consider the mass of liquid, m = ρ V , within a small volume element
V = x y z. The liquid inside V experiences stresses (stress = force/area)
due to the surrounding liquid. Let the stresses acting on the faces y z at x = 0
and at x = x be respectively σ0 and σ x (Fig. 8.11); σ0 is seen to have only a
normal component, whereas σ x has both normal and tangential components. The
magnitudes of the normal components of the stress forces are simply σ0 y z =
’ p(0) y z and σ n x y z = ’ p( x) y z, where p is the pressure (the mi-
nus sign is due to the fact that the pressure force, by de¬nition, is compressive, thus
opposite to the indicated direction of the stresses), whereas the tangential compo-
nents provide shear (see Chapter 1). Let us ¬rst assume that the liquid in V is in an
equilibrium state such that its acceleration and velocity are both zero everywhere.
In this case no shear forces can be present (remember, shear can be maintained
only in ¬‚owing liquids, see Chapter 1) and therefore the in-plane components of
the stresses must be zero everywhere. If an external force per unit volume f acts on
the small volume element then the total force acting in the x direction on the small
mass m, which must be zero, is f x x y z ’ [ p( x) ’ p(0)] y z = 0 ( f x
is the x component of the force f ). Dividing this equation by V and taking the
limit of in¬nitesimal changes, the equilibrium condition for the liquid is expressed
by f x ’ ‚ p/‚ x = 0. Clearly, similar equations are obtained in the y and z direc-
tions. Since V is arbitrary, these equations are valid anywhere in the liquid, so
long as it is in equilibrium. If, however, the velocity and acceleration are not zero
then the shear forces cannot be ignored. The terms that take shear forces into
account involve the density, velocity ¬eld, and viscosity. Their derivation is beyond
the scope of this book and we only quote the result (for details see for example
Tritton, 1988). Finally, then, the dynamical behavior of a liquid, as a consequence
of Newton™s second law, is governed by the equation (i = x, y , z)
‚v i ‚vi ‚vi ‚vi ‚p ‚ 2 vi ‚ 2 vi ‚ 2 vi
ρ + vx + vy + vz = fi ’ +· + + .
‚t ‚x ‚y ‚z ‚i ‚ x2 ‚y2 ‚z2


Note that here the quantities ρ, v i , f i , p in general depend on all three coordi-
nates x, y , z. The above equation is known as the Navier“Stokes equation. The
expression in parentheses on the left-hand side is the total acceleration and is the
generalization of the result in Eq. 8.4, which, although introduced in a different
biological context “ the ¬‚ow of cells in a chemotactic model for angiogenesis “ is
similarly based on Newton™s second law of motion.

Equations 8.6 and 8.7 provide the mathematical basis for the ana-
lysis of Lubkin and Li. These equations need to be supplemented by
˜˜boundary conditions,” physically justi¬ed relations between the fun-
damental quantities at the interface between the two ¬‚uids (as well
as at the physical boundary of the entire system). Since the two ¬‚uids
are different, the two dynamical quantities v (x, y) and p(x, y) are


σ∆ x ∆ y ∆ z
σ 0 ∆ y∆ z y


0 ∆x x

Fig. 8.11 Illustration of the various stresses acting on a small volume element in a
viscous liquid, used for clarifying the basis of the Navier“Stokes equation. Forces acting
on the volume element are drawn outwards by convention. In general, the force acting
on a given face is not perpendicular to the face. The normal and tangential components
are related respectively to pressure and shear; thus the force acting at x = 0 gives rise
only to pressure whereas the one acting at x provides both pressure and shear. For
more details see Box 8.2.

discontinuous at the interface. These discontinuities can be deter-
mined from the condition that no uncompensated forces can arise
along the interface (such forces would result in in¬nite acceleration).
For an interface with a complicated shape, such as the one in Fig. 8.10,
the mathematical expressions for the boundary conditions are com-
plicated, because the direction of the normal and tangent vectors
(along the n and „ directions in Fig. 8.10 respectively) in general do
not coincide with that of the main coordinate axis. Therefore we will
not dwell on the explicit form of these conditions. Instead, we use
simple physical considerations, based on the material of earlier chap-
ters, to illustrate the main points. (For more details see for example
Leal, 1992.)
Let us ¬rst consider the normal (perpendicular, n-directed) stresses
at the interface. If the two liquids are in equilibrium with v (x, y) = 0
everywhere then the shape of the interface is determined by the
Laplace equation 2.2: p = pe ’ pm = γ /R . Here p is the discon-
tinuity (jump) in pressure across the interface, pe and pm denote re-
spectively the local pressure (the normal stress) on the epithelial and
mesenchymal side of the interface, and R is the radius of curvature
at the same location (since the interface is a line, there is only one
radius of curvature). When the velocity is not zero it provides an ad-
ditional stress; therefore an extra term, containing the discontinuity
(across the interface) of the viscosity times the velocity gradient in the
normal direction, arises in the above Laplace equation. (Stress is pro-
portional to the gradient of the velocity; see Eq. B1.2a in Chapter 1).
Next we consider the tangential (in-plane) stresses. We recall (see
Eq. 2.6) that an interfacial tension which is spatially varying (along
the interface) provides a shear stress proportional to the gradient of
the velocity component in the direction of the normal. In conclusion,
the boundary conditions along the normal and tangential directions



0 5 10 15 20 25 t/T
Fig. 8.12 Effect of the viscosity ratio ± on the evolution of clefting, as a function of
the nondimensional time t/T (where t is the real time and T is the characteristic time),
from an initially three-lobed epithelial rudiment, in the model of Lubkin and Li (2002).
The directions of the clefting forces are indicated by arrows. Their magnitude • = 2.5,
whereas the nondimensional surface tension is β = 0.01. When the epithelium is
embedded in a material more viscous than itself, it takes longer to form the same depth
of cleft than when it is embedded in a material of the same viscosity.

both involve the interfacial tension and the gradient of the velocity
in the normal direction.
The model of Lubkin and Li is thus fully de¬ned by Eqs. 8.6,
8.7, and the equations expressing the boundary conditions. These
equations can be rewritten in terms of nondimensional parameters
± = ·+ /·’ (the viscosity ratio), β = γ0 T /(·’ L ) (the nondimensional
surface tension) and • = f0 / (γ0 L ) (the nondimensional clefting force),
where L and T are characteristic length and time scales, respec-
tively. The uniform interfacial tension γ0 characterizes the bound-
ary between the epithelium and the mesenchyme in the absence of
any clefting force. The clefting force makes the interfacial tension
nonuniform. The magnitudes of L and T re¬‚ect the spatial and tem-
poral characteristics of salivary globules and the branching process
(L ≈ 100 µm is the typical diameter of a salivary globule, T ≈ 8 hr is
the typical time between branchings). All the other quantities were
estimated on the basis of experimental results (Table 1 in Lubkin and
Li, 2002).
Lubkin and Li applied their model to an initially three-lobed rudi-
ment by numerically solving the equations governing cleft formation.
They tested the effects of the viscosity ratio, clefting force, and inter-
facial tension. Figure 8.12 shows the time evolution of cleft formation
within the model and the sensitivity of this process to the magnitude
of the nondimensional parameter ±.

Vertebrate limb development
The vertebrate limb is one of the most intensively studied systems
in developmental biology, both experimentally and theoretically.

Because many of the major molecular determinants of limb skeletal
pattern formation have been described, and because viable models
for this process invoke a variety of physical mechanisms acting in a
concerted, integrated, fashion, we will devote the remainder of the
chapter to a discussion of this system.
Like the salivary glands, limb morphogenesis involves the co-
operation of epithelial and mesenchymal components. In contrast
with the glands, however, the pattern of structural elements in the
limb is determined with great precision. A salivary gland can perform
its function -- secretion of its enzyme products into the oral cavity via
a hierarchy of tubular ducts -- with variable numbers of clusters of
secretory cells, lobules, and ducts. Its pattern is not entirely determi-
nate. In contrast, all normal limbs of a given type (e.g., the forelimb
of a bird, the leg of a frog) have a ¬xed and precisely arranged set of
structural elements, i.e., bones, muscles, and nerves, and the coordi-
nation of these elements is integral to the limb™s optimal functioning.
Thus, unlike the branching morphogenesis of salivary glands, which
may depend on random cues such as are provided by the action of


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( 66 .)