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(complete cleavage) (incomplete cleavage)
Mesolecithal Telolecithal
Isolecithal Centrolecithal
(moderate, (dense yolk
(sparse, evenly distributed yolk) (yolk in egg's
vegetal yolk) throughout cell) center)

B. Spiral C. Bilateral D. Rotational
A. Radial A. Radial A. Bilateral B. Discoidal A. Superficial
(echinoderms, (annelids, (tunicates) (mammals, (amphibians) (cephalopods) (fish, (insects)
amphioxus) molluscs, nematodes) reptiles,
flatworms) birds)

Fig. 2.2 Shapes of blastulae and patterns of cleavage in various organisms. The
distribution of yolk, a dense, viscous material, is a major constraint in the pattern of
cleavage. In centrolecithal meroblastic cleavage (characteristic of the embryo of the
fruit-¬‚y Drosophila, see Chapter 10), unlike the other types pictured, the nuclei divide in a
common cytoplasm at the center of the egg during early development. This mode of
cleavage is completed as the nuclei move to the egg-cell periphery, where they become
separated from one another and the rest of the egg contents by the formation of
surrounding membranes. (After Gilbert, 2003.)

These materials can include nutrient yolk (typically composed of pro-
teins and lipids), messenger RNAs, and granules composed of several
kinds of macromolecules. Structural and compositional asymmetries
of the zygote will lead to unequal blastomeres, the cellular products
of cleavage.
Cleavage in which the zygotic mass is completely subdivided
(whether the initial blastomeres are equal or unequal) is called
holoblastic (Fig. 2.2). If only a portion of the zygote is subdivided, cleav-
age is referred to as meroblastic. Mammals and sea urchins exhibit
holoblastic cleavage, whereas cleavage in Drosophila embryos, where
just the nuclei divide at ¬rst, and only become incorporated into sep-
arate cells after they have migrated to the inner surface of the egg, is
meroblastic. Despite being vertebrates and therefore phylogenetically
closely related to mammals, ¬sh, reptiles and birds undergo meroblas-
tic cleavage, with the yolky portion of the egg cell failing to become
subdivided (Fig. 2.2).

Physical processes in the cleaving blastula
Embryonic development starts with a simple spherical or oblate fer-
tilized egg and produces complex shapes and forms. Shape will there-
fore be a major concern in what follows. Although organisms use the
complex machinery of DNA synthesis, mitosis, membrane biogene-
sis, and cytokinesis to subdivide the material of the fertilized egg
and become a blastula, the simplest physical model for this subdivi-
sion process is the behavior of a liquid drop. Such a description has
often been used to explain the behavior of individual cells exposed
to mechanical deformations (Yoneda, 1973; Evans and Yeung, 1989)
or to interpret the rheology of neutrophils during phagocytosis (van
Oss et al., 1975). Many biological functions such as mitosis and mem-
brane biogenesis are based on molecular interactions that are not
inherently three-dimensional and thus have no preferred geometry.
Since they occur in a context in which physical forces are also active,
it is reasonable to expect that in the ¬nal arrangement the physical
determinants will dominate.
The shapes of simple liquid drops are determined by one of their
physical properties, the surface tension (see below). Is this a good phys-
ical picture for the shape of a single cell, such as a fertilized egg?
We will quickly conclude that it is not -- the complexities of the cell™s
interior discussed in the previous chapter undermine its simple ¬‚uid
behavior. Most importantly its bounding layer, the plasma membrane,
does not have the extensibility that characterizes the surface of a uni-
form liquid. Nonetheless, it will be seen that when the cell™s complex
topography and the properties of the submembrane and extracellular
layers are factored in, the physics of surface tension gives a reasonable
¬rst approximation to the cell shape. In addition, as in our discus-
sion in Chapter 1, ˜˜generic” physical descriptions will be found to
reappear at a higher level of organization, that of the multicellular
Cells in suspension, or just prior to division, are spheroids. Eggs of
most multicellular organisms have this shape (Fig. 2.2). If the egg were
simply a drop of liquid, as early thinkers believed, its spheroid shape
would be the straightforward consequence of surface tension. But
the fact that some eggs are not spheres (see the asymmetric oblate
shape of the Drosophila egg in Fig. 2.2) indicates that more than surface
tension is determining the shape of these cells and thus, most likely,
that of spherical cells as well.
This being acknowledged, can the physics of surface tension and
related liquid phenomena tell us anything about how the fertilized
egg divides ¬rst into two and subsequently into a cluster of cells? In-
deed, theoretical analysis has been performed of ˜˜liquid drop cells”
that expand in size owing to the osmotic pressure exerted by the
synthesis and diffusion of macromolecules. Such objects have a ten-
dency to break up into smaller portions of ¬‚uid, each with a higher
surface-to-volume ratio than the original drop (Rashevsky, 1960).

Possibly this inherent tendency is mobilized during the division or
cleavage of real cells and zygotes. But, as we will see, the basis of
apparent surface tension is not straightforward in a living cell; its
possible mobilization during division will be correspondingly com-
Further, as the embryo undergoes consecutive cleavages and grad-
ually becomes an aggregate of cells, it is reasonable to ask what
physics can tell us about how the cells become arranged in such aggre-
gates. Here we must consider the physical properties (e.g., elasticity)
of the acellular materials -- the hyaline layer, the zona pellucida --
that variously surround the blastulae of different species. Because
the blastula eventually ˜˜hatches” from such enclosing structures, we
must also consider cell--cell adhesive forces, without which the di-
vided cells would simply drift apart. (Indeed these adhesive forces are
known to be present already in the prehatching blastula; see Chap-
ter 4.) We will ¬nd that under experimentally con¬rmed assumptions
about these adhesive forces, cells in an aggregate will easily move past
one another in analogy to molecules moving randomly in a liquid.
Thus, as we progress to the higher level of organization represented
by the multicellular embryo, we will ¬nd that the liquid drop again
becomes a relevant physical model, and we can make many strong in-
ferences about the shapes and behaviors of cell aggregates from the
physics of surface tension.

Surface tension
A liquid drop, left alone, in the absence of any force (in particu-
lar, gravity) will adopt a perfectly spherical shape. In other words,
it will minimize the interfacial area with its surroundings in order
to achieve a state with minimal surface or interfacial energy. (For a
given volume the sphere is the shape with minimal area.) The sur-
face or interfacial energy per unit area of a liquid is called its surface
or interfacial tension and is denoted by σ . (Surface tension, strictly
speaking, is the term reserved for the case when the liquid is in a
vacuum but it is customarily used when the liquid is surrounded by
air. The term ˜˜interfacial tension” is used when the liquid adjoins
another liquid or a solid phase.)
To see surface tension in action one can prepare a small wire frame
with one movable edge (of length L), as shown in Fig. 2.3 and wet the
frame with a soap solution. The area of the wire frame (and thus of
the liquid surface) can be increased by pulling on the movable edge
with a force F by x in the x direction. According to the de¬nition
of σ , we have W = F x = σ A, where W is the work performed to
increase the area by A. Since A = L x (Fig. 2.3), F = σ L , which
provides another way of de¬ning the surface tension: it is the force
that acts within the surface of the liquid on any line of unit length.
The surface of a liquid is thus under a constant tension which acts
tangentially to the surface in a way as to contract the surface as
much as possible. (A simple demonstration of this is to carefully place
a small light coin on the surface of water in a cup; the coin will

Fig. 2.3 Schematic representation of surface tension σ . The wire frame, whose original
area isA = L x , carries a soap solution ¬lm. The surface tension is the energy required to
increase the area of the ¬lm by one unit. The work F x done by the force to increment
the original area by L x is also equal to σ L x . From the equality of these two
expressions, it is evident that σ can also be interpreted as a force acting perpendicularly
to the contact line (along the handle) between the liquid and the surrounding medium
and directed into the liquid, thus opposing the effect of the external force F.

not sink. If the coin is placed on its edge, however, it will sink: the
pressure it exerts on the liquid surface in this con¬guration exceeds
that of the surface tension.)
The unit of σ is thus either J/m2 or equivalently N/m. The surface
tension of water is approximately 72 mN/m. Since σ is exclusively
the property of the liquid, it does not depend on the magnitude of
F applied to increase the area in Fig. 2.3. (Note that for a solid the
surface or interfacial energy both depend on F and the initial area;
see also below.)
The origin of surface and interfacial tension is easy to trace. When
a liquid molecule is forced to leave the bulk and move to the surface,
it is in an energetically less favorable state if its interaction energy
with the dissimilar molecules at the surface is higher than with the
molecules like itself that surround it in the liquid™s interior. In the
interior or bulk, the energy of a molecule in equilibrium with its sur-
roundings is the cohesive energy E C ; at the surface it is the adhesion

energy E A (E C and E A refer to one molecule). The change in the
molecule™s energy when it is transferred from the bulk to the surface
is E = E A ’ E C . According to its de¬nition, the surface tension is
σ = N E , where N is the number of molecules per unit area of the
surface or interface. Thus, if E = E A ’ E C < 0 then it is energeti-
cally more favorable for the liquid molecule to be at the surface, and
the liquid is driven to increase its interface with the surroundings
as much as possible. Under these circumstances the liquid is said to
˜˜wet” the interface; in principle, it can eventually thin to a layer of
molecular dimension.
The second de¬nition of surface tension (i.e., that it is a force) al-
lows one to establish a relationship between σ and the contact angle.
Consider a liquid drop L placed on a solid substratum S in medium M
(Fig. 2.4). The equilibrium shape of the drop near the line of contact
with the substratum is characterized by the contact angle ±, whose
value can be determined from the condition of mechanical equilib-
rium (a special case of Newton™s second law) along the contact line
between L, S, and M. Equating the sum of forces acting on the unit
length of the contact line along the x direction to zero (see Fig. 2.4),
we obtain

σ S M = σ L S + σ L M cos ±, (2.1)

where σ S M , σ L S , σ L M are the respective interfacial tensions between
the three phases. Equation (2.1) is known as Young™s equation (see
for example Israelachvili, 1991). The cases ± > 0 and ± = 0 (cos ± = 1)
correspond respectively to partial and to complete wetting of the solid
substratum by the liquid.
Liquid molecules are mobile whereas atoms or molecules in a solid
are not. Therefore a liquid can increase its surface area by exporting
molecules from the bulk while retaining the same surface density.
Solids can increase their surface area only by stretching the distance
between surface atoms, which results in a decrease in the surface
density. As a consequence, the surface tension of a liquid drop is
independent of the drop™s area at any time, whereas the analogous
quantity for a solid, the surface or interfacial energy per unit area,
increases with area since it becomes progressively more dif¬cult to
pull atoms apart.

Fig. 2.4 Illustration of Young™s
equation. At mechanical M LM
equilibrium the shape of the liquid
drop L near the contact line with
the solid S and the medium M is
characterized by the contact angle LS
±. Its magnitude is determined SM
from Newton™s law (see the main
text for details). S

Laplace™s equation
Living cells in most cases are not perfectly spherical. This does not, by
itself, make surface tension or related concepts irrelevant to under-
standing cell shape, since liquid drops, when exposed to external
forces, are typically also nonspherical. The shape of the liquid drop
shown in Fig. 2.4 could be the result of gravity, for example. Under
such circumstances the full equilibrium shape of an ideal incompress-
ible liquid drop is determined by Laplace™s equation (Israelachvili,
1 1
σ + + pe = ». (2.2)
R1 R2
Here R 1 , R 2 are the radii of curvature, σ(1/R 1 + 1/R 2 ) = σ C is the
Laplace (or curvature) pressure, C being the average curvature, and pe
is the pressure due to external forces. (Equation 2.2 is valid for any
liquid surface, not only drops.) The radius of curvature is the radius
of the circle that matches a curve or a surface at a given point. (A sur-
face can be matched by circles along two orthogonal directions; see
Fig. 2.5.) Equation 2.2 is simply the consequence of Newton™s second
law under the condition of incompressibility. Incompressibility en-
ters the equation through the constant », a Lagrange multiplier. (La-
grange multipliers are used to incorporate speci¬c constraints into
mathematical equations. In the present case the constraint is due to
the incompressible nature of the liquid drop: its volume is constant
while its shape can change.) Note that for a ¬‚at surface (with in¬nite
radii of curvature) the Laplace pressure is zero, whereas for a convex
closed surface it produces an inward-directed force. (A closed surface
is convex or concave according to whether it respectively bends away


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