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1.751 mN/m.
Cells are continuously exposed to mechanical forces whose origin
is in the external environment. Most of these forces are weak and
cause no drastic changes in the integrity of the cell. Often they are
only able to change the local curvature of the cell membrane with-
out changing the overall surface area. For example, experiments per-
formed with arti¬cial vesicles made of amphipathic molecules (which
contain a water-soluble, hydrophilic, part and a water-insoluble, hy-
drophobic, part) show that small changes in temperature (i.e., in ther-
mal energy) may lead to marked changes in shape without any change
in the surface area of the vesicle (see the review by Seifert, 1997).
This would also be the dominant mode of shape change for cells
with no hidden (convoluted) area. Such changes are controlled exclu-
sively by the bending rigidity or bending stiffness (yet another elastic
constant), i.e., the capacity of the membrane to resist changes in its
local curvature.
According to Helfrich (1973) the curvature energy Wc of a small
membrane surface A is given by
κ
Wc = (C ’ C 0 )2 A. (2.5)
2
The parameters κ, C , and C 0 are the bending rigidity (a material prop-
erty whose unit is the joule), the local average curvature introduced
earlier, and the spontaneous curvature, respectively. The spontaneous
curvature re¬‚ects the possible asymmetry of the membrane and char-
acterizes its shape in the state of lowest curvature energy. For an in-
herently ¬‚at membrane the spontaneous curvature is zero, whereas
for a vesicle with a preferred spherical shape of radius R it is 1/R.


Physical models of cleavage and blastula formation
So far in this chapter we have entertained the notion that, with re-
gard to their shapes, individual cells and in particular the zygote can
be modeled as liquid drops. When we reached the plausible limits of
this simple model we introduced additional parameters; this seemed
to be necessary for characterizing the physical shape of cells. Follow-
ing on these preliminary steps, we will now develop a quantitative
description of the ¬rst cleavages and blastula formation. Although
our discussion, with appropriate modi¬cation, can be applied to em-
bryos of various species, we will concentrate mainly on the sea urchin,
since experimental results in this system permit speci¬cation of most
of the parameters thus far introduced. Finally, while development can
proceed by relatively abrupt morphological changes, some of which
40 BIOLOGICAL PHYSICS OF THE DEVELOPING EMBRYO


will be analyzed later in this book, we will assume that the embryo
is ˜˜well behaved” as a physical system before and after such changes.
Thus, we consider the impact of variations in gene expression on the
relevant physical parameters to be gradual. Indeed, if this were not
the case, efforts to interpret morphogenesis in terms of such param-
eters would be largely futile.
As mentioned earlier, the initial divisions of the embryo of a mul-
ticellular organism are unusual in that they take place at constant
embryo mass. At this time, in most cases, including that of the sea
urchin, the embryo utilizes stored RNAs and proteins rather than syn-
thesizing them itself. Thus, divisions are rapid (of order 1.5 hours in
the sea urchin, re¬‚ecting the absence of G1 and G2 phases) and lead
to smaller cells with increasing nucleo-cytoplasmic ratio. Taking this
into account, we next present a simple surface-tension-based model
(where ˜˜surface tension” in fact represents the cortical tension) of the
early cleavage stages in the sea urchin (for comprehensive reviews, see
Rappaport 1986, 1996). The results of the model will also characterize
the ¬rst and later cleavage stages of other classes of organisms.




Cleavage: the White“Borisy model
The sea urchin zygote is initially spherical. Its cytoplasm has the
same composition throughout the cell cortex and it thus possesses an
isotropic surface tension (Yoneda, 1973). The only structure that can
select a direction and thus the actual position of the cleavage plane is
the mitotic spindle. Indeed, the cleavage plane is always perpendicu-
lar to this structure. If the spindle™s position is physically manipulated
(translated or rotated) before the furrow is fully established, the cell
assembles the contractile ring so as to retain its relative orientation
to the spindle. If the formation of the spindle is blocked (by the ad-
dition of chemicals such as colchicine, which disrupts microtubules),
no furrow is formed. If the spindle is disrupted or even completely
removed from the cell after the cleavage plane has fully been estab-
lished (Hiramoto, 1968), however, the separation of the two daughter
cells will progress to completion. The actual progression of the furrow
thus is not dependent on the presence of the mitotic spindle, which
seems to be needed only to set up the initial conditions for furrow
formation. This suggests that once furrowing is under way, it is au-
tonomous. Furthermore, the role of the asters of the mitotic spindle
in the initiation of cytokinesis is well established (Rappaport, 1966):
there is both a minimum and maximum separation distance of asters
from the cell cortex beyond which a furrow will not be induced.
On the basis of these experimental results, White and Borisy (1983)
proposed that cytokinesis is initiated by a signal emitted from the mi-
totic apparatus (the astral signal) and transported to the cell cortex
(see Fig. 2.9). The signal polarizes the initially uniform cortex by re-
laxing the surface tension in a distance-dependent fashion, resulting
in a gradient of the surface tension. This gradient ( σ/ x) can lead
2 CLEAVAGE AND BLASTULA FORMATION 41



Microfilaments
(contractile ring)
Cortex P




Pole
Pole
centrosome centrosome centrosome
centrosome




Equator
Chromosome Microtubules

Fig. 2.9 Illustration of the physical process underlying the White“Borisy model. A
putative biochemical cue, the astral signal, emanates from the centrosomes and at the
cortex modi¬es the zygote™s surface tension in a distance-dependent manner. Thus at
any point P (left panel), the surface tension decreases in proportion to the inverse
second or fourth power of the distances measured from the centrosomes. The resulting
gradient in the surface tension sets up a ¬‚ow of cortical elements from the pole region
towards the equator (right panel), which eventually results in a contractile ring and a
constricting cleavage furrow.



to shear and produce a ¬‚ow in the peripheral cytoplasm, according to
σ v
=· . (2.6)
x z
Here x and z measure the distance respectively along and perpendicu-
lar to the cell surface and .· is the viscosity of the cortical cytoplasm.
(Note that Eq. 2.6 is analogous to Eq. B1.1a in Chapter 1, since the
units of σ/ x are force per area (F /A).)
The ¬‚ow carries microscopic contractile (i.e., ˜˜muscular”) elements
from the polar region to the equator (see Fig. 2.9), where during
this rearrangement a spontaneous alignment of actin and myosin
¬laments into an organized ring takes place. The ring is capable of
exerting sustained circumferential tension of the order of 10’8 N
(Hiramoto, 1978), while progressively constricting in the form of a
groove. This moving cleavage furrow ultimately pinches the cell in
half via a ˜˜purse-string” action.
The above ˜˜standard model,” which is supported by a number of
experiments (see the overview by He and Dembo, 1997), provides a
good example of the interplay of generic and genetic processes. A
biochemical signal (the astral signal) leads to a change in the cell™s
surface tension. This generates a physical process (cytokinesis), which
eventually leads to renewed gene activity in the daughter cells.
The speci¬c assumptions White and Borisy make about the
astral signal are: (a) the stimulating activity of the asters is isotropic
and depends only on the distance (but not the direction) measured
from the asters; (b) the signals from the two asters combine in an
additive way; (c) the magnitude of the stimulus decreases with dis-
tance r away from the centrosomes according to 1/r 2 or 1/r 4 . This
42 BIOLOGICAL PHYSICS OF THE DEVELOPING EMBRYO


latter assumption ensures that the maximum decrease in the surface
tension occurs at the poles, where the sum of the distances mea-
sured from the two asters is maximal, and progressively decreases
towards the equator. Since motion occurs only in the dense narrow
cortical-¬‚uid region, the rest of the cell™s interior is considered to be
more watery and static.
The originally spherical egg, with radius R 0 , is in a state of me-
chanical equilibrium because the uniform inward-directed force due
to the Laplace pressure, 2σ/R 0 , σ being constant along the cell sur-
face, is compensated by the elevated pressure inside the cell (due,
for example, to osmosis). According to the assumption of this model,
the astral signal breaks the uniformity of the surface tension and
leads to its decrease. If no simultaneous change occurs in the radii of
curvature then as the surface tension decreases so does the Laplace
pressure, which becomes smaller than the original internal pressure.
This leads to a force imbalance. Since the volume of the embryo is
constant, the embryo bulges outward at the poles (where the decrease
of the Laplace pressure is maximal), and ¬‚attens in the equatorial re-
gion (Fig. 2.9). White and Borisy demonstrated that mechanical equi-
librium can be reestablished if the embryo displaces its surface so
as to modify its radii of curvature. The length and velocity of the
displacement are respectively proportional to the magnitude of the
force imbalance and the viscosity of the cytoplasm. Under these cir-
cumstances, the distance between the asters is increased (in agree-
ment with the experimental results of Hiramoto, 1958). The radii of
curvature of the modi¬ed surface are then recalculated, and so is the
internal pressure. The pressure, considered to be uniform since the
embryo is modeled as a liquid drop, is obtained as the integrated
mean of the Laplace pressures at each point of the surface. The above
procedure is iterated, with the end result shown in Fig. 2.10 for an
axially symmetric cleavage con¬guration.
In the simplest formulation of the model the astral signal leaves
the surface tension forces isotropic, that is, not dependent on di-
rection along the surface of the cell. As a consequence the furrow
does not progress to completion. The reason is that, as the furrow
forms, the cell surface develops ˜˜saddle points” along the equator,
at which one of the radii of curvature gradually becomes negative.
This provides another reason, in addition to the astral signal, for the
Laplace pressure to decrease continuously. Thus, along the equator
the Laplace pressure and the intracellular pressure eventually bal-
ance each other locally: an equilibrium is reached and the furrow
stops elongating. This limitation of the model is removed if σ is al-
lowed to vary with orientation, as discussed above in connection with
the cortical tension. The anisotropy of σ in the present case is caused
by the movement and reorientation of the contractile elements as
they ¬‚ow to the equator to assemble the contractile ring (White and
Borisy, 1983).
Despite its success in providing a coherent description of cyto-
kinesis in a number of cleavage patterns, the standard model has a
2 CLEAVAGE AND BLASTULA FORMATION 43



B
A




C D




Fig. 2.10 Time evolution (A“D) of the shape of the zygote in the course of the ¬rst
cleavage, as described by the White“Borisy model. The sequence of events is driven by
the astral signal-induced changes in the surface tension and the subsequent changes in
shape to reestablish mechanical equilibrium, as explained in the main text. The length of
an arrow is proportional to the magnitude of the decrease in surface tension at a given
stage of the process and a given point on the cell surface. The horizontal line indicates
the distance between the centrosomes. (After White and Borisy, 1983.)




number of shortcomings (He and Dembo, 1997). The most obvious
is that it does not provide a molecular basis for the astral signal. It
arti¬cially divides the cytoplasm into two spatially distinct regions,
the gel-like or more viscous cortex and the watery interior ¬‚uid, but
nevertheless considers the cell as a liquid drop with uniform internal
pressure. Furthermore, the model does not regard chemical changes
in the composition of the cytoskeleton (the polymerization and de-
polymerization of cytoskeletal ¬laments) as being an essential part of
cytokinesis. In particular, it postulates that the assembly of the con-
tractile ring and its constriction result exclusively from circumferen-
tial tension, which is contrary to experimental ¬ndings (Schroeder,
1975). Moreover, it does not provide an explanation for the origin
of this tension or the mechanism of movement during furrowing.
These, and several other de¬ciencies of the standard model, were ad-
dressed by He and Dembo (1997). Starting from the analysis of White
and Borisy, these authors constructed a comprehensive, alternative,
mathematical model of the ¬rst cleavage division in the sea urchin
embryo. (The He--Dembo model is too complex to be discussed here,
but more advanced students are encouraged to work through their
paper.)

Blastula formation: the Drasdo“Forgacs model
As cleavage continues, the total number of cells in the embryo grows
exponentially (Gilbert, 2003). In addition to the complex processes
44 BIOLOGICAL PHYSICS OF THE DEVELOPING EMBRYO




F1



F2

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