8.4 Gastrulation
439
left). During Step 2, these cells shorten radially and invaginate to create a pit
surrounded by ectoderm (Fig. 8.23b, right). In experiments, Sherrard et al. (2010)
found that these two steps are driven by apical and apico-basal (radial) contraction,
respectively. These authors also used a 2D computer model to demonstrate the
physical plausibility of this mechanism.
To explore how well this mechanism works with spherical geometry, consider
a finite-element model for Step 2. In the initial configuration, the model is a solid
hemisphere composed of an outer layer of ectoderm (modulus c = 1), an inner
region of endoderm (c = 10), and a relatively thin apical layer covering the
flat lower surface (c = 20) (Fig. 8.23c, left). For simplicity, stresses left at the
end of Step 1 are ignored, although the apical layer is assumed to remain in its
contracted (stiffened) state during Step 2. The relative values of the specified moduli
correspond to the amount of stiffening caused by contraction.
Setting G r = 0.7 and G = 1 in the endoderm yields an invagination that agrees
qualitatively with images of the sea squirt Phallusia mammillata (Fig. 8.23b, c,
right). This mechanism, i.e., apical followed by radial contraction, is essentially
the same one used to obtain a closed mesodermal tube in the 2D model for ventral
furrow formation in Drosophila (see Fig. 8.19d).
8.4.5 The Mystery of Volvox
Volvox, a type of green algae, is a spherical shell composed of a single layer of cells
bound together by flexible cytoplasmic bridges. During gastrulation, the organism
turns itself completely inside out, with the inner surface becoming the outer surface
and vice versa. This transformation brings flagella from inside to outside, where
they can be used for propulsion. Hoop stress makes inverting a spherical shell an
extremely challenging problem in the mechanics of morphogenesis, and Volvox
has developed a remarkable sequence of highly coordinated cell-shape changes and
shifting cytoplasmic bridges to accomplish this feat (Viamontes et al. 1979; Hohn
and Hallmann 2011; Matt and Umen 2016).
Researchers have described two main strategies that Volvox use to invert themselves (Fig. 8.24). Type A inversion involves outward bending of the wall that begins
at the top and gradually works its way to the bottom of the embryo. During Type
B inversion, bending begins near the equator and pulls the bottom hemisphere
upward until snap-through buckling inverts the bottom half. A wave of bending
then completes the process. Investigators have described the mechanical challenges
in qualitative terms (Viamontes et al. 1979; Hohn and Hallmann 2011; Matt and
Umen 2016), and Haas et al. (2018) have recently published a model for type B
inversion based on shell theory.
After gastrulation, the embryo gets busy constructing specialized tissues and
organs. The next few sections consider organogenesis.
439
left). During Step 2, these cells shorten radially and invaginate to create a pit
surrounded by ectoderm (Fig. 8.23b, right). In experiments, Sherrard et al. (2010)
found that these two steps are driven by apical and apico-basal (radial) contraction,
respectively. These authors also used a 2D computer model to demonstrate the
physical plausibility of this mechanism.
To explore how well this mechanism works with spherical geometry, consider
a finite-element model for Step 2. In the initial configuration, the model is a solid
hemisphere composed of an outer layer of ectoderm (modulus c = 1), an inner
region of endoderm (c = 10), and a relatively thin apical layer covering the
flat lower surface (c = 20) (Fig. 8.23c, left). For simplicity, stresses left at the
end of Step 1 are ignored, although the apical layer is assumed to remain in its
contracted (stiffened) state during Step 2. The relative values of the specified moduli
correspond to the amount of stiffening caused by contraction.
Setting G r = 0.7 and G = 1 in the endoderm yields an invagination that agrees
qualitatively with images of the sea squirt Phallusia mammillata (Fig. 8.23b, c,
right). This mechanism, i.e., apical followed by radial contraction, is essentially
the same one used to obtain a closed mesodermal tube in the 2D model for ventral
furrow formation in Drosophila (see Fig. 8.19d).
8.4.5 The Mystery of Volvox
Volvox, a type of green algae, is a spherical shell composed of a single layer of cells
bound together by flexible cytoplasmic bridges. During gastrulation, the organism
turns itself completely inside out, with the inner surface becoming the outer surface
and vice versa. This transformation brings flagella from inside to outside, where
they can be used for propulsion. Hoop stress makes inverting a spherical shell an
extremely challenging problem in the mechanics of morphogenesis, and Volvox
has developed a remarkable sequence of highly coordinated cell-shape changes and
shifting cytoplasmic bridges to accomplish this feat (Viamontes et al. 1979; Hohn
and Hallmann 2011; Matt and Umen 2016).
Researchers have described two main strategies that Volvox use to invert themselves (Fig. 8.24). Type A inversion involves outward bending of the wall that begins
at the top and gradually works its way to the bottom of the embryo. During Type
B inversion, bending begins near the equator and pulls the bottom hemisphere
upward until snap-through buckling inverts the bottom half. A wave of bending
then completes the process. Investigators have described the mechanical challenges
in qualitative terms (Viamontes et al. 1979; Hohn and Hallmann 2011; Matt and
Umen 2016), and Haas et al. (2018) have recently published a model for type B
inversion based on shell theory.
After gastrulation, the embryo gets busy constructing specialized tissues and
organs. The next few sections consider organogenesis.