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 <