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0.3
0 10 20 30 40 50
SITES

Fig. 9.9 Two “snapshots” of a continuous Ca2+ wave in the FDF model of Dawson
et al. (1999). The concentration pro¬le is shown along a streak of release sites; 0 marks
the initial site. The green vertical lines indicate the sites that are simultaneously ¬ring at
the time of the ¬rst snapshot (solid red line). The broken line corresponds to a later
time. The wave is traveling with a velocity of 5.2 m/s (corresponding to parameter
values „ = 9 s, d = 3 m, D = 50 m2 /s, σ = 1.35 — 10’14 mol, [Ca2+ ]T = 0.4 M,
and [Ca2+ ]B = 0.3 M, taken from the literature on the fertilization wave in Xenopus
eggs (Fontanilla and Nuccitelli, 1998)). Note that the waveform travels without
observable deformation.




Surface contraction waves and the initiation
of development
We have seen that fertilization begins by the mechanical activity of
the sperm (its ¬‚agellar-driven motility) followed by a chemically in-
duced sperm exocytotic event (the acrosome reaction). This is followed
by a sequence of chemical excitations in the egg cytoplasm; the most
dramatic are traveling waves of Ca2+ concentration of varying fre-
quency and amplitude. These waves trigger subsequent events of fer-
tilization and zygote formation, prominent among which is the egg™s
own exocytotic event, the release of the cortical granule contents re-
sulting in the slow block to polyspermy. The appearance of rippling
waves on the egg surface (see Fig. 9.8), which immediately precedes
the cleavage process, takes us back to the point at which the zygote
began its developmental excursion.
In sea urchins, for example, successive waves of microvilli elon-
gation and stiffening (Cline et al., 1983) propagate with the same
directionality and speed as the initial calcium wave (Suzuki et al.,
9 FERTILIZATION: GENERATING ONE LIVING SYSTEM FROM TWO 243


1995). At about nine minutes after fertilization, actin ¬laments de-
tach from the cortex and translocate away from the surface into the
deeper regions of the cytoplasm (Terasaki, 1996). Within 15 minutes,
the egg cortex is transformed from a fairly ¬‚at, soft, layer studded
with short micropapillae (broad microvilli) into a stiff layer contain-
ing numerous surface microvilli and a new set of cortical vesicles, re-
placing the exocytosed cortical granules (Sardet et al., 2002). In Xenopus
eggs, within minutes after fertilization, dynamic actin ˜˜comet tails”
accumulate around intracellular vesicles, which then begin to move
through the cytoplasm (Taunton et al., 2000), possibly representing
the same phenomenon seen in the sea urchin.
In essentially all species, the postfertilization cortical micro¬la-
ment cytoskeleton (comprising actin) is reorganized and contracts in
a wave-like manner starting from the site of sperm entry. This me-
chanical motion of the egg cortex is based on its ability to undergo
cycles of calcium-dependent contraction and relaxation (Sardet et al.,
1998; Roegiers et al., 1995, 1999; Benink et al., 2000). The cortical re-
organizations and cytoplasmic ¬‚ows that occur between fertilization
and ¬rst cleavage -- primarily the translocation of cortical and subcor-
tical materials parallel to the plane of the plasma membrane (Eidne
et al., 1994) -- appear to be driven by interactions between the micro¬l-
ament and microtubule cortical cytoskeletons (Canman and Bement,
1997; Benink et al., 2000).
The cortex of the fertilized egg possesses altered mechanical and
viscoelastic properties. For one thing, it is generally thicker than the
unfertilized cortex, since the cores of the newly arising microvilli
have micro¬lament bundles that extend and intermingle with the mi-
cro¬lament meshwork underlying the membrane (Wong et al., 1997).
The cortical micro¬lament network contracts and relaxes during spe-
ci¬c phases of the meiotic and mitotic cell cycles, a process that ap-
pears to be regulated by the presence of microtubules (Mandato et al.,
2000).
Recall that the nature of the ˜˜astral signal,” proposed by White
and Borisy (1983) as triggering rearrangement of the cortical cyto-
plasm and eventually initiating cleavage, is unknown (see Chapter
2). It would be satisfying if one or more of the postfertilization
Ca2+ waves that sweep across the zygote served this function. That
things might be more complicated was suggested by the studies of
Wong and coworkers (1997), who used the drug cytochalasin D in sea
urchins to block dynamical changes in cortical actin organization
downstream of the Ca2+ waves and showed that cytokinesis for
the ¬rst cleavage occurred nonetheless. The establishment of the
contractile apparatus for cytokinesis of the ¬rst cleavage division (the
function of the putative astral signal) thus appears to be independent
of the Ca2+ -induced waves of surface contraction, though possibly
not of other effects of the calcium waves.
In Xenopus, these early surface contraction waves are apparently re-
quired for the cytoplasmic rearrangement leading to localization of
the germ plasm (Quaas and Wylie, 2002), the maternally synthesized
244 BIOLOGICAL PHYSICS OF THE DEVELOPING EMBRYO


determinant of the primordial germ cells discussed at the beginning
of this chapter. It is possible that cortical waves are involved in the lo-
calization of other cytoplasmic determinants of future pattern as well.
The mechanochemical coupling of Ca2+ traveling waves and
shape changes in the cortical cytogel occurs across all classes of
animals. Biologically, it represents an anticipatory link from the
stage immediately after the gametes join in fertilization to the stage
at which the gametes that will give rise to the next generation
begin to form. Because it also represents a physical phenomenon not
encountered until now in our presentation, we will conclude this
chapter with a model for this process.

Mechanochemical model for cortical activity in fertilized eggs
The generation and propagation of chemical waves (as described for
example by the FDF model) can be formulated in terms of reaction--
diffusion equations like the ones discussed in Chapter 7. In order to
produce morphogenetic modi¬cations of the embryo, chemical waves
must stimulate the mechanical rearrangement of cell or tissue com-
ponents. As we have seen, calcium waves in the zygote in fact trigger
mechanical waves of cortical expansion and contraction that precede
the ¬rst cleavage (Takeichi and Kubota, 1984; Deguchi et al., 2000;
Ducibella et al., 2002).
Living cells, such as the zygote, represent excitable media, which
respond actively to external perturbations. The typical outcome
is the formation and propagation of chemical or mechanical waves.
An example of mechanical excitability was discussed in Chapter 5 in
connection with gastrulation, and examples of chemical excitability
were discussed in Chapter 7 in relation to pattern formation and ear-
lier in this chapter in connection with the formation and propagation
of calcium waves. Here we encounter an example of cell excitabi-
lity that combines mechanical and chemical effects: cortical expan-
sion and contraction and eventually the cleavage of the zygote. To
model the Ca2+ -wave-generated shape changes of the embryo, the
coupling of chemical and mechanical waves is needed; this goes be-
yond the usual reaction-diffusion formalism discussed in Chapter 7.
Oster and coworkers (Cheer et al., 1987) proposed a mechanochemical
model for cortical activity in the postfertilized embryo. Later, Ballaro
and Reas (2000) extended the model to incorporate newer experimen-
tal ¬ndings.
Since cellular shape changes are driven by dynamical rearrange-
ment of the cytoskeleton, e.g., the polymerization--depolymerization
of the actomyosin network, the mechanochemical model must incor-
porate the concentrations of Ca2+ and of the molecules that control
its release, sequestration, and resequestration (e.g., IP3 , cAMP), as well
as the physical state of the cytoskeletal ¬laments with their cross-
linker proteins and of the motor proteins that can move these ¬l-
aments (e.g., myosin). Calcium in¬‚uences the cytoskeleton through
its interaction with the cross-linkers. At low Ca2+ levels actin re-
mains in a gel state but, as the Ca2+ level increases and cross-linkers
9 FERTILIZATION: GENERATING ONE LIVING SYSTEM FROM TWO 245


preferentially bind with calcium, actin ¬laments are broken down
and the gel changes into a sol. It is this Ca2+ -driven sol--gel transi-
tion that is responsible for the expansion--contraction waves in the
cortex. The modeling of such complex phenomena, involving numer-
ous molecular species, is not straightforward. Here (following Cheer
et al., 1987) we only outline how this program can be formulated.
Mathematical details can be found in the cited references.
As discussed in Chapter 7 (see Eqs. 7.5--7.7), the general equations
for the concentrations c i , i = 1, 2, . . . , N, of N molecular species whose
spatial distributions are governed by both chemical reaction and dif-
fusion are
‚c i ‚ 2c i
= D i 2 + F i (c 1 , c 2 , . . . , c N ). (9.10)
‚t ‚x
For the modeling of mechanochemical wave propagation in the cor-
tex of the embryo, depending on the complexity of the model the
c i may denote the concentrations of Ca2+ , IP3 , cAMP, the actomyosin
network, and the cross-linking, capping, and severing proteins (the
last three types of molecule are collectively called solation factors be-
cause their concentration determines whether the cytoskeletal net-
work is in the gel or sol phase). The rate constants of the various
molecular interactions determine the functions F i ; for speci¬c exam-
ples of such expressions, see Cheer et al. (1987) or Ballaro and Reas
(2000). The reaction--diffusion equations 9.10 now have to be coupled
to the mechanical properties of the cortex.
Following Cheer et al., let us consider a small volume element
V of the cortical actomyosin gel, which is essentially a network
of cross-linked polymer ¬bers (i.e., ¬lamentous F-actin). As the gel is
placed in solvent it swells owing to the difference in the osmotic pres-
sure, P O , between the solvent and the gel. Owing to the cross-linking
of ¬laments, the gel has elastic properties and thus resists swelling.
The total swelling pressure in the gel, P S , is the sum of the osmotic
pressure, which tends to expand it and the elastic stress, P E , which
restrains it: P S = P O + P E . (Note that in this expression P E should be
considered as a negative quantity since it acts in opposition to P O .)
The addition of calcium to V diminishes the concentration of sola-
tion proteins. Thus the gel weakens and the magnitude of its elastic
pressure decreases. This in turn leads to an increase in P S . As the
amount of calcium falls, the gel strengthens and P S decreases. A wave-
like periodic variation in calcium concentration, as described earlier,
thus modulates the sol--gel transition in the actin network, leading to
expansion--contraction cycles (a solation wave) and the movement of
the egg cortex. As the solation wave passes through a volume element
V , it displaces it. We denote by µ(θ, t) the tangential displacement of
the volume element V that is initially at latitude θ on the surface
of the spherically symmetric embryo (Fig. 9.10A). Its variation in time
is given by the following equation:
‚µ
· = PO + PE + PA. (9.11)
‚t
246 BIOLOGICAL PHYSICS OF THE DEVELOPING EMBRYO




Fig. 9.10 (A) Notation used in the mechanochemical model of Cheer et al. (1987).
The displacements shown in B and C are measured along meridians as indicated here for
one case by the segment with arrows. (B) The experimentally measured displacements
(in mm) along meridians of carbon beads on the surface of a “fertilized” Xenopus egg,
after the induction of Ca2+ waves by pricking the embryo between the animal pole and
the equator. The ¬gure shows displacement vs. time for each of four particles. The
displacement is de¬ned as their instantaneous distance along the meridian from the
pricking point, marked as 0. (C) The theoretical displacements along meridians of the
volume element shown in A in the mechanochemical model of Cheer et al. (1987)
for the Xenopus egg. The four curves correspond to angular displacements in radians
(0.3 radians = 17—¦ ) at regular intervals measured from a ¬xed pricking point (marked as
0). Note that, since the pricking points in the experiments and model do not coincide,
the four curves should not be compared in their minute details. Furthermore, the
experimental curves extend to over six minutes, whereas the model curves are shown
only up to about four minutes. (A and C after Cheer et al., 1987; B after Takeichi and
Kubota, 1984.)
9 FERTILIZATION: GENERATING ONE LIVING SYSTEM FROM TWO 247


Here · is the viscosity of the cytoplasm and P A denotes the metaboli-
cally regulated active stresses generated by the actomyosin gel. Cheer
et al. (1987) provided speci¬c expressions for P O , P E , and P A in terms of
the elastic moduli of the cortex. (Note that since µ and the pressures
depend both on space and time variables, we need to use partial dif-
ferentiation.) From the above discussion it follows that both P O and
P E depend on the concentration of calcium in V , c Ca , whose vari-
ation in time is given by the reaction--diffusion equation 9.10 with an
appropriately chosen reaction function F Ca (for its speci¬c form see
Cheer et al., 1987). The coupled equations Eq. 9.10 and 9.11 provide
a mechanochemical model for cortical waves. The results of such an
analysis and corresponding measurements on the amphibian egg are
shown in Figs. 9.10B and C.


Perspective
Fertilization closes the circle begun in Chapter 2, where the zy-
gote departed on its developmental trajectory. There the physical
forces bringing about multicellularity were considered. The interven-
ing steps of development (as part of the generation of body form and
organs) accomplished the all-important task of producing individual
cells with specialized biological properties. Signi¬cantly, some differ-
entiated cell types make use of particular physical mechanisms in a
more exaggerated fashion than does the ˜˜general” cell discussed in
Chapter 1. The sperm and egg are vivid examples of cells with promi-
nent physics-based specializations. The sperm (in the short term) is a
virtually autonomous organism with unique, ¬‚agellum-based, motile
properties. The egg, the body™s largest cell, is also unique in being
capable of sustained chemical and mechanical excitations. The biolo-
gical outcome of the uni¬cation of these entities is a new individual:
the evolutionary goal of fertilization. It also represents an example of
two distinct complex physical systems, which, when combined, yield
a new system with entirely novel physical properties.
Chapter 10




Evolution of developmental
mechanisms

The view of embryonic development presented in the preceding nine
chapters is rather different from accounts to be found in other mod-
ern developmental biology textbooks. We have focused on the phe-
nomena of transitions between cell types, changes in the shape of
tissues, and the generation of new arrangements of cells and have
approached them as problems in physics. In contrast, when these sub-
jects are dealt with in most contemporary treatments of development
it is primarily as problems in regulated differential gene expression.
While we have by no means ignored the roles of gene products and
gene regulatory systems in our account of development, and while
the notion of the embryo as a physical system is not entirely absent

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. 48
( 66 .)



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