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from or towards an enclosed observer. The average curvature at con-
vex or concave points is respectively positive or negative, whereas at
a saddle point -- where one of the radii of curvature is positive and
the other is negative -- it can have either sign.)

S1
S2
P Fig. 2.5 Illustration of the radius
of curvature. A point on a
two-dimensional surface (P) can be
S2
matched with two orthogonal
R2
S1 circles, whose radii determine the
radii of curvature of the surface at
R1
that point. The ¬gure also
illustrates by the broken arrows
that the surface or interfacial
tension is the force perpendicular
to any line of unit length within the
liquid surface or interfacial area.
34 BIOLOGICAL PHYSICS OF THE DEVELOPING EMBRYO



Box 2.1 Application of the equations of Laplace
and Young

1. Shape of a spherical drop under compression.
Consider Fig. 2.6, which illustrates the shape change of an originally spherical liquid
drop when compressed between parallel plates with a known constant force F (we
ignore the force due to gravity). When Eq. 2.2 is evaluated along the ¬‚at plates,
where the Laplace pressure is zero (R 1 = R 2 = in¬nity) one can determine »
since » = p e = F /(π R 3 ), where R 3 is the contact radius. Once the constant »
2

is known, applying Laplace™s equation at the equator (point P in Fig. 2.6, where
pe = 0) yields
1 1 F
σ + = . (B2.1a)
π R3
2
R1 R2
Here R 1 is the radius of the drop at its equator and R 2 is its radius of curvature
in a meridianal plane, as shown in Fig. 2.6. By measuring the three radii, Eq. B2.1a
provides the value of σ .
2. Immiscible liquid drops in contact
Let us consider the situation shown in Fig. 2.7, which depicts contacting drops of
immiscible liquids A and B embedded in a medium M. We assume that the shape
of each of the three interfaces can be approximated by a part-spherical cap, with
radii as indicated in Fig. 2.7, and we ask, what is the condition of equilibrium for
given values of the interfacial tensions? Applying Young™s equation to the vertical
components of the interfacial tensions at point P in Fig. 2.7 yields the relationship
between θAM , θBM , θAB and σAM , σBM , σAB :

σAM cos θAM + σAB cos θAB = σBM cos θBM . (B2.1b)

Equation B2.1b provides information on the location (but not the shape) of the
contact line and thus on the extent of the engulfment of drop A by drop B. When
phase B just fully envelops phase A (i.e., it completely wets the interface between
phase A and the medium) we have θAM = θBM = 0 and θAB = 180—¦ , and Eq. B2.1b
reduces to

σAB = σAM ’ σBM . (B2.1c)

This result will be used in Chapter 4 (see Fig. 4.5), in connection with the mutual
envelopment of tissues.
To gain information on the radius of curvature of the contact line (i.e., R AB ) we
combine Laplace™s equation for a spherical drop p = 2σ/R , discussed above in
the text, with the identity 0 = ( pA ’ pM ) ’ ( pA ’ pB )’ ( pB ’ pM ), where pA , pB ,
and pM are respectively the pressures in phase A, phase B, and the medium. This
results in
σAB σAM σBM
= ’ . (B2.1d)
R AB RA RB


Equation 2.2 is valid locally, i.e., at each point of the surface.
Whereas for an ideal liquid σ is a constant, the other quantities (ex-
cept for ») may vary along the surface. Thus in the absence of exter-
nal forces, pe = 0, the average curvature is the same at each point
2 CLEAVAGE AND BLASTULA FORMATION 35



Fig. 2.6 The typical shape of an
originally spherical liquid drop
F when compressed uniaxially, as
indicated, between parallel plates.
The three radii are used to
establish a relationship between
the compressive force and the
R3
surface tension in Example 1 in
Box 2.1.
P
R1
R2




F




of the drop™s surface. Consequently 1/R 1 + 1/R 2 = 2/R and the drop
assumes a spherical shape with radius R, as expected. The (convex)
spherical shape arises because the pressure inside the sphere, pi , is
larger than the external pressure, pe . At equilibrium, therefore, the
pressure difference across the spherical surface must be balanced by
the Laplace pressure, p = pi ’ pe = 2σ/R .




Fig. 2.7 Illustration of the
system of contacting drops A and B
considered in Example 2 in Box
2.1. The red arrow is drawn
through the centers (1, 2) of the
sphere B and the part-sphere A
and meets the surface of A at 3.
For further details see the text in
Box 2.1.
36 BIOLOGICAL PHYSICS OF THE DEVELOPING EMBRYO



Cortical tension
No cell, including the egg or the zygote, is purely liquid. Even
intracellular water is only partly osmotically active. The water of
hydration (composed of water molecules tightly bound to proteins
and intracellular organelles), in particular, is osmotically inactive.
And, of course, the interior of the cell is structured by ¬brous pro-
teins, with elastic as well as viscous properties. The cell is bounded
by a more or less rigid lipid membrane. It has no bulk reservoir of
free-¬‚oating lipid molecules that could be mobilized to increase the
area of the membrane by transporting material from the interior of
the cell to its surface. As a consequence, one cannot de¬ne a pure
liquid surface tension, as described above, for the cell. Nevertheless,
a physically analogous quantity, the cortical tension, turns out to be
useful for characterizing certain shape changes in living cells.
The membranes of most cells can be represented by a highly con-
voluted two-dimensional surface decorated by spiky protrusions (mi-
crospikes, microvilli, ¬lopodia), sheet-like extensions (lamellipodia),
and invaginations (Fig. 2.8). When exposed to mechanical forces (e.g.,
distension or compression), therefore, a cell may mimic liquid drop
behavior, its apparent surface area undergoing rapid increase or de-
crease by opening and closing the ˜˜wrinkles.” (Speci¬cally, the pool
of invaginated membrane can be considered the analogue of the bulk
reservoir of liquid molecules.) How easy it is to accomplish this area
change strongly depends on the density of the cortical network of
actin ¬laments underlying and attached to the plasma membrane
(Tsai et al., 1994, 1998), which must locally be separated from the
membrane and resealed every time the cell surface is changed. Thus
the energy needed to increase the area of a cell by one unit (its ef-
fective surface tension) has two contributions (Sheetz, 2001): (a) the
membrane--cytoskeleton adhesion energy, which must be overcome
to separate the membrane from the cytoskeleton; and (b) the energy
needed to increase the area of the lipid membrane itself (which can be
considered as a sheet of liquid within which the lipid molecules can
¬‚ow freely, as will be discussed in more detail in Chapter 4; see Fig.
4.3). In a typical cell, about 75% of the apparent membrane tension
is due to cytoskeletal adhesion (Sheetz, 2001).
Evans and Yeoung (1989) introduced the notion of ˜˜cortical ten-
sion” to make simultaneously an analogy and a distinction between
pure liquid surfaces and the plasma membrane. Cortical tension can
be measured by micropipette aspiration (Needham and Hochmuth,
1992) or pulling tethers (membrane nanotubes) from the plasma
membrane (Sheetz, 2001). In the former case the behavior of a single
cell is followed as it is aspirated into a micropipette narrower than
the cell™s diameter (Fig. 2.8). The shape of the deformed cell in equi-
librium is considered as if it were purely liquid. The cortical tension,
similarly to the purely liquid surface tension, is then determined from
Laplace™s equation applied to this shape. Here, despite the molecular
and structural complexity of the actual biological system, its mor-
phology can be accounted for by a physical law that governs a set of
generic properties. Therefore, in what follows we will retain the term
2 CLEAVAGE AND BLASTULA FORMATION 37




Fig. 2.8 Topographic specializations of the cell surface. The ¬bers underlying the
membrane and extending into the microspikes and lamellipodia are typically made of
actin ¬laments or microtubules. The ¬gure also illustrates schematically how a small
portion of the cell membrane (circled in the lower right panel) is aspirated into a
micropipette to measure the cortical tension.



˜˜cellular surface tension”. It will need to be remembered, however,
that its origin lies in the interaction of the cortical actin network
with the cell membrane.
A true liquid is isotropic: its physical properties do not depend on
orientation. In particular, the liquid surface tension is the same along
the two orthogonal directions de¬ned by the two radii of curvature
in Laplace™s equation, Eq. 2.2. For a viscoelastic material such as a
cell, Hiramoto (1963) proposed a generalization of the Laplace pres-
sure to σ1 /R 1 + σ2 /R 2 . Experiments of Yamamoto and Yoneda (1983)
suggested that whether a cell can be described in terms of isotropic
or anisotropic surface tension depends on its position in the cell
38 BIOLOGICAL PHYSICS OF THE DEVELOPING EMBRYO


cycle. This may be a further indication for the role of the cortical
actin network, as will be seen in the discussion of cleavage below.
Those familiar with the properties of cells will recognize that
surface material (in the form of cell membrane) can be added and
subtracted from the cell in other ways than that described above. In
a growing cell, for example, the increase of the cytoplasmic mass is
coordinated with the biosynthesis and insertion of new membrane
in the smooth endoplastic reticulum (Fig. 1.1) and its transport and
insertion into the cell surface. If this did not happen the membrane
would rupture and the cell contents would spill out. Unlike the con-
tinuous, reversible exchange between the bulk and the surface in a liq-
uid, however, such an expansion of the plasma membrane is mainly a
one-way affair: cells do not reverse their growth in the normal course
of events. Thus, in contrast with the cortical tension, which mobilizes
and demobilizes a reserve of cell surface, the properties of the cell
membrane are not a good formal analogue to surface tension.
Another way material can be added to or removed from the cell
surface is by the fusion of vesicles from the interior with the plasma
membrane during the process of exocytosis or their removal by en-
docytosis. This material exchange between the surface and the bulk
could, indeed, potentially serve as an analogue of surface tension,
but in fact it also is not a good one. The reason is that exocytosis
and endocytosis are regulated by the physiological needs and func-
tions of the cell rather than being responsive to mechanical stresses
as are the surface tension of a pure liquid and the cortical tension
described above. Although endocytosis has been observed in the cells
of early-stage mammalian embryos (Fleming and Goodall, 1986), exo-
and endocytosis are probably not extensive enough in most types of
embryos (such as that of the sea urchin -- see below) to contribute
substantially to surface forces during cleavage.


Curvature energy
It is clear that the notion of constant cortical tension can give at
best only a crude approximation to the shape of the cell membrane.
The extent to which a convoluted cell membrane can be smoothened
depends not only on the composition of the cortical actin network
but also on the composition of the membrane itself. If the membrane
is fully distended, no increase in the cell surface area can be achieved
without pulling individual lipid molecules apart from one another. As
the liquid drop model breaks down, elastic effects are bound to show
up. Even before this extreme state is attained it is quite possible that
the increase in the membrane area would depend on the total area.
For example, stretching actin ¬bers could depend on the extent to
which they have already been stretched. (This would, of course, violate
the assumption of liquidity.) Indeed, as discussed by Needham and
Hochmuth (1992) and Derganc et al. (2000) the energy of a neutrophil
membrane Wn is well approximated by

K
Wn = γA + (A ’ A 0 )2 . (2.4)
2A 0
2 CLEAVAGE AND BLASTULA FORMATION 39


Here we write the constant part of the cortical tension as γ , to dis-
tinguish it from the true liquid surface tension σ ; K is the area
expansivity modulus (another elastic parameter), whose units are J/m2
(or equivalently N/m) and A0 is the area of the free spherical neu-
trophil. According to the measurements of Needham and Hochmuth
(1992) the membrane tension and area modulus of neutrophils
are in the range 0.01 mN/m < γ < 0.08 mN/m and 0.015 mN/m < K <

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