researchpaper Haas & Höhn During the development of multicellular organisms,
cell shape changes drive the emergence of complex shapes by generating out-of-
plane stresses that cause tissues to fold, but quantifying such out-of-plane
stresses in curved tissues has remained challenging. Here, we develop a
framework combining microsurgery experiments and a mechanical model to
quantify these stresses and infer mechanical properties of the tissue based on
the unfurling of the tissue on being cut. We find such recoils in the alga
_Volvox_ and use our model to predict the stresses required to reproduce the
observed tissue shape sequence quantitatively, both before and after cutting.
Our results emphasise the effects of even small out-of-plane tissue stresses
in development. S.M.H.H. and P.A.H. designed the study. S.M.H.H. performed
experiments. P.A.H. derived the theoretical models. S.M.H.H. and P.A.H.
analysed and interpreted the data and wrote the manuscript. The authors
declare that they have no competing interests. 1P.A.H. and S.M.H.H.
contributed equally. 2Joint corresponding authors. E-mails<EMAIL_ADDRESS><EMAIL_ADDRESS>
# Cut it out: Out-of-plane stresses in cell sheet folding of _Volvox_ embryos

Pierre A. Haas  Max Planck Institute for the Physics of Complex Systems,
Nöthnitzer Straße 38, 01187 Dresden, Germany  Max Planck Institute of
Molecular Cell Biology and Genetics, Pfotenhauerstraße 108, 01307 Dresden,
Germany  Center for Systems Biology Dresden, Pfotenhauerstraße 108, 01307
Dresden, Germany  Stephanie S. M. H. Höhn  Department of Applied Mathematics
and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge
CB3 0WA, United Kingdom

###### Abstract

The folding of cellular monolayers pervades embryonic development and disease.
It results from stresses out of the plane of the tissue, often caused by cell
shape changes including cell wedging via apical constriction. These local
cellular changes need not however be compatible with the global shape of the
tissue. Such geometric incompatibilities lead to residual stresses that have
out-of-plane components in curved tissues, but the mechanics and function of
these out-of-plane stresses are poorly understood, perhaps because their
quantification has proved challenging. Here, we overcome this difficulty by
combining laser ablation experiments and a mechanical model to reveal that
such out-of-plane residual stresses exist and also persist during the
inversion of the spherical embryos of the green alga _Volvox_. We show how to
quantify the mechanical properties of the curved tissue from its unfurling on
ablation, and reproduce the tissue shape sequence at different developmental
timepoints quantitatively by our mechanical model. Strikingly, this reveals
not only clear mechanical signatures of out-of-plane stresses associated with
cell shape changes away from those regions where cell wedging bends the
tissue, but also indicates an adaptive response of the tissue to these
stresses. Our results thus suggest that cell sheet folding is guided
mechanically not only by cell wedging, but also by out-of-plane stresses from
these additional cell shape changes.

###### keywords:

cell sheet folding $|$ development $|$ tissue elasticity

This manuscript was compiled on .

The folding of tissues into three-dimensional shapes is a crucial part of
morphogenetic processes such as neurulation and gastrulation in natural
embryonic development (1, 2, 3, 4), associated birth defects such as spina
bifida (5), and synthetic morphogenesis in organoids (6, 7, 8, 9). These
dramatic shape changes are regulated by a complicated interplay of mechanical
forces and molecular signalling (10, 11, 12, 13, 14, 15, 16, 17). The latter
can be visualised by fluorescent tagging (18, 19, 20), but the mechanical
forces must be inferred indirectly and have therefore remained one of the
biggest mysteries in the life sciences.

These mechanical forces are often caused by cell shape changes, including
local cell wedging through apical or basal constriction (10, 3, 21), which
locally imparts a preferred, intrinsic curvature to the tissue. This intrinsic
curvature drives tissue bending by generating out-of-plane forces. Such forces
not only cause morphogenetic changes, but also elicit biochemical responses
(22). However, physically, the mechanics and, more biologically, the role in
development of different cell shape changes are still unclear: For example,
what is the relative contribution to tissue folding of areas of cell wedging
and adjacent areas with less pronounced cell shape changes (15)? Moreover, the
local intrinsic curvature driving tissue folding is not in general compatible
with the global geometry of the tissue. The folding of the tissue does not in
general resolve this geometric incompatibility, so it leads to persisting
stresses in the tissue termed residual stresses (23, 24). Now tissues are
known to have the ability to alleviate tensile (25), compressive, or bending
(26) stresses through cell movements or changes in cell adhesion or shape and
this adaptation can be crucial for maintaining tissue integrity (27). How the
mechanical state of the tissue is involved in this adaptation in general
remains, however, an open question. Addressing these outstanding problems
fundamentally relies on quantifying the spatio-temporal distribution of
mechanical forces in tissues.

Over the past decade, different techniques to infer mechanical forces in
tissues have therefore emerged, ranging from atomic force microscopy, traction
force microscopy, and fluorescence resonance energy transfer-based molecular
force sensors to optical tweezers, micropipette aspiration, magnetic beads,
and liquid droplets (28, 29). However, methods for quantifying out-of-plane
stresses in folding epithelia are still lacking. In fact, we are only aware of
a single quantitative study of out-of-plane stresses in cell sheets: Fouchard
_et al._ (26) measured the force needed to unfurl curled synthetic cell sheets
and went on to estimate the active torques and the bending modulus of
epithelial monolayers (30). In particular, while numerous studies have used
laser ablation to infer in-plane forces from quantifications of the ensuing
recoil of the tissue [see, e.g., Refs. (31, 32, 33, 34)], extending these
methods to out-of-plane forces in geometrically more complex curved dynamic
tissues has proved challenging because of the difficulty of imaging out-of-
plane recoils on ablation and the need for a mechanical model of these complex
deformations to link the measured recoil to the mechanical forces causing it.

Here, we present our framework combining orthogonal laser ablation (OLA, Fig.
1A) and a mechanical model to quantify out-of-plane stresses and infer
mechanical properties of curved cell sheets, which we apply to the
gastrulation-like inversion process of the microalga _Volvox_.

Inversion in _Volvox_ and the related volvocine algae (Chlorophyta) is an
emerging model developmental event, during which the spherical embryonic cell
sheet turns itself inside out through a programme of cell shape changes (35,
36, 37, 38, 39, 40, 41, 42, 43). Here, we focus on type-B inversion in _V.
globator_ (44, 37, 41, 45, Fig. 1B–G,B′–G′). We have previously described this
inversion with a mechanical model (45, 46, 47) in which the programme of cell
shape changes driving the process appears as variations of the intrinsic
curvatures and intrinsic stretches of an elastic shell (45, 46) and which
reproduces the inversion process quantitatively (47). These _in silico_
approaches have made a number of mechanical predictions (45, 47), but the
elastic framework underlying them has in fact remained untested.

We start by using our ablation framework to reveal persistent residual out-of-
plane stresses in the posterior hemisphere throughout inversion by showing how
the cell sheet unfurls at the boundary of a circular ablation at the posterior
pole. We use our mechanical model to show that the geometric incompatibilities
causing this recoil result from mismatched intrinsic curvatures of the cell
sheet. From quantifications of the recoil, we infer mechanical properties of
the posterior hemisphere during invagination and its intrinsic curvature,
which is consistent with the observed cell shapes changes. These results do
not only therefore provide proof-of-principle of our ablation framework, but
also test the elastic framework underlying our previous _in silico_
approaches. We go on to extend our elastic model (47, 48) to reproduce the
tissue shapes quantitatively at different stages of inversion. Strikingly, the
resulting fitted mechanical sequence and the ablation data show a clear
signature of out-of-plane stresses in the posterior hemisphere and suggest an
adaptive response to these stresses. This shows that the tissue dynamics of
_Volvox_ inversion rely not only on cell wedging, but also on out-of-plane
stresses from the additional cell shape changes in the posterior hemisphere.

### Type-B inversion in _Volvox globator_

We close this introduction with a more detailed description of type-B
inversion in _Volvox globator_ , summarising results that this paper relies
on. After completion of an initial cell cleaveage phase (49, 50), the embryos
of _V. globator_ consist of a spherical monolayer of $\sim 3000$ cells (37).
Each embryo is located within a fluid-filled embryonic vesicle that is
embedded in the extracellular matrix (ECM) of the parent (51, 52, 53, 54). The
embryonic cells are connected to each other by a network of cytoplasmic
bridges (CBs) resulting from incomplete cytokinesis (55). A ring of cells at
the anterior pole lacks CBs, leaving an opening, the phialopore. During the
morphogenetic process of inversion, the embryos turn themselves inside-out
through this phialopore within $\sim 60-80\,\text{min}$ in order to expose the
two cilia that grow at each apical cell pole (41).

During inversion the embryonic cells, which have a microtubule-based cortical
cytoskeleton, but neither cell walls nor ECM (56), remain connected to each
other by CBs, so the positions of the cells relative to their neighbours are
maintained (41), which suggests our elastic description of the cell sheet (45,
46, 47, 57, 48). The global deformations of the embryonic cell sheet during
inversion are driven by cell shape changes (Fig. 1B–G) that have previously
been described based on electron and confocal microscopy (41).

Prior to inversion, all embryonic cells are teardrop-shaped and connected at
their broadest point by CBs (Fig. 1B,B′). Type-B inversion begins (“early
invagination”) with the appearance of a circular bend region (BR) just below
the equator of the embryo, caused by cell-wedging and re-location of the CBs
to the thinner basal cell poles (41). Simultaneously, the cells in the
posterior hemisphere undergo actin-dependent thinning (58, 59) and become
spindle-shaped (Fig. 1C,C′). Our previous _in silico_ analyses (45, 47)
indicate that active contraction of the posterior cell sheet, associated with
this latter cell shape change, is necessary to explain the resulting “mushroom
shape” of the embryos. As inversion proceeds (“late invagination”), a wave of
cell wedging travels from the equator towards the posterior pole and the
posterior hemisphere moves into the anterior one (41). The cells at the
posterior pole retain their spindle shape (Fig. 1D,D′). The posterior
hemisphere then moves entirely into the anterior, but a “dimple” remains where
the cell sheet around the posterior pole has not yet inverted. The cells at
the posterior pole remain spindle-shaped while the basal cell poles in the BR
are less wedge-shaped than before, resulting in a relaxation of the curvature
in the BR and near the posterior pole (Fig. 1E,E′). By the time the posterior
hemisphere has fully inverted, the phialopore has widened and the anterior
hemisphere has started to peel over the inverted posterior. The cells in the
inverted posterior have adopted a pencil shape (Fig. 1F,F′) at this stage.
Closure of the phialopore marks the completion of inversion and the end of
embryogenesis in _Volvox_. At this stage, all cells are pencil-shaped with
pointy apical cell poles (41). Within $30-60\,\text{min}$ after inversion, all
cells shorten, widen, and adopt a columnar shape while the radius of the
juvenile spheroid increases (Fig. 1G,G′).

## Experiments

### Orthogonal laser ablation (OLA) in type-B _Volvox_ inversion

To quantify out-of-plane stresses in _Volvox_ inversion, we performed laser
ablation experiments on _V. globator_ embryos in consecutive stages of
inversion (_Materials and Methods_). In order to capture out-of-plane elastic
responses to these ablations, they were performed on the mid-sagittal cross-
sections of the axisymmetric _V. globator_ embryos. Cross-sections were imaged
using 2-photon microscopy (Fig. 1A and _Materials and Methods_), and
approximately circular holes were created at the posterior pole of the embryos
by laser ablation using a separate 2-photon laser (_Materials and Methods_).
The axisymmetry of the embryos and the ablations ensures that any subsequent
deformation of the cell sheet is approximately axisymmetric, too. The
dimensions of the resulting hole in the cell sheet were determined in three-
dimensional datasets recorded subsequently (_Materials and Methods_).

Figure 1: Experiments: orthogonal laser ablation (OLA) of inverting _Volvox
globator_ embryos. (A) Experimental setup for 2-photon microscopy and OLA,
illustrated for a surface rendering of a _V. globator_ embryo at an early
inversion stage. Samples were embedded in low-melting point agarose (LMA),
submerged in Standard _Volvox_ Medium (SVM), and imaged using 2-photon
microscopy. Holes were created using a separate 2-photon laser (_Materials and
Methods_) at the posterior pole of embryos before, during, and after
inversion. (A′) Overlay (white) of micrographs of the mid-sagittal cross-
section of the embryo in panel (A), just before ablation (red) and after
relaxing into its post-ablation equilibrium shape (cyan), showing the unfurled
edge of the cell sheet. (A′′) Tracing of the cell sheet (transparent area) and
its midline (dashed line) in the square magnified from panel (A′). (A′′′)
Mechanical hypothesis: posterior cell shape changes not only contract the
posterior hemisphere (45), but also induce a curvature mismatch in it that
causes out-of-plane stresses distant and distinct from those generated by the
wedge-shaped cells in the bend region. (B)–(G) Schematic cross-sections of _V.
globator_ embryos in consecutive developmental stages, modified from Ref.
(41), showing associated cell shapes, the network of cytoplasmic bridges (CBs,
red line), and the growing cilia. BR: bend region, W: wedge-shaped cells.
Insets show cell shapes (41) near the posterior pole of the embryo (T:
teardrop-shaped cells; S: spindle-shaped cells; P: pencil-shaped cells; C:
columnar cells). (B′)–(G′) Maximum intensity projections of z-stacks showing
lateral views of _V. globator_ embryos in the developmental stages in panels
(B)–(G). The anterior and posterior hemispheres are labelled in panel (C′).
(B′′)–(G′′) Posterior view of the embryos in panels (B′)–(G′) after laser
ablation. Dashed white line: outline of the hole created by laser ablation
after relaxation into its post-ablation equilibrium shape. (B′′′)–(G′′′)
Overlays (white) of micrographs of the mid-sagittal cross-section of the
embryos in panels (B′)–(G′) just before ablation (red) and after relaxing into
their post-ablation equilibrium shapes (cyan). Dashed lines: areas of laser
ablation. Thin dotted lines in panels (B′′′), (E′′′)–(G′′′): field of view in
which the laser ablation was performed. Times $t$ are given relative to the
start time $t=0$ of inversion. Scale bars: $20\,\text{\textmu m}$.

### Observations

Laser ablations were performed at the posterior pole of embryos in consecutive
developmental stages before, during, and after inversion (Fig.
1B′′–G′′,B′′′–G′′′).

Prior to inversion, embryos did not show any recoil on laser ablation (Fig.
1B′–B′′′), i.e. no deformation within at least $2\,\text{min}$ following
ablation ($N=5$ embryos), indicating that the cell sheet is not residually
stressed before the onset of inversion. However, during early invagination,
ablation led to outwards unfurling of the edges of the cell sheet and
relaxation into a new equilibrium shape (Fig. 1C′′,C′′′; $N=14$), showing that
out-of-plane residual stresses have appeared in the cell sheet. Ablations
during the late invagination stage again led to unfurling of the edges of the
cell sheet (Fig. 1D′′,D′′′; $N=5$). When ablations were performed at the
“dimple stage” (Fig. 1E,E′), the unfurling cell sheet adopted a curvature
closer to that in the BR (Fig. 1E′′,E′′′; $N=5$), indicating a mechanical
effect of the BR on the post-ablation equilibrium shape. Interestingly, for
ablations at the stage where the posterior hemisphere has fully inverted (Fig.
1F,F′), the direction of unfurling changed compared to earlier timepoints
(Fig. 1F′′,F′′′; $N=7$). Finally, for posterior ablations after the cells have
become columnar (Fig. 1G,G′), the cell sheet no longer shows any recoil within
at least $2\,\text{min}$ after laser ablation (Fig. 1G′′,G′′′; $N=4$),
indicating that the residual stresses have relaxed.

At all stages of inversion observed, unfurling of the edges of the cut upon
laser ablation and relaxation into the post-ablation equilibrium shape took
approximately $4-10\,\text{s}$. Within this time, the embryos otherwise
maintained their shape: Morphogenetic inversion movements, i.e. global
embryonic shape changes distinct from the local ablation response, were only
distinguishable after at least $1\,\text{min}$. While the hole caused by laser
ablation did not close up, the embryos did complete inversion after ablation,
unless the cut size exceeded about half the radius of the posterior hemisphere
in which case the anterior hemisphere failed to invert (not shown).

The observed outward unfurling of the cell sheet suggests that the preferred,
intrinsic curvature of the cell sheet differs from the curvature into which
its unablated spherical shape forces the cell sheet. We therefore hypothesise
that such a curvature mismatch gives rise to out-of-plane stresses in the
posterior hemisphere, including stresses distant and hence distinct from the
out-of-plane stresses in the bend region of wedge-shaped cells (Fig. 1A′′′).
The thinning of cells from teardrop to spindle shapes driving contraction of
the posterior hemisphere also changes its curvature (46, Fig. 1A′′′,B,C), so
would contribute to this mismatch of curvatures. The change of the direction
of unfurling with the change of the sign of the curvature of the posterior
hemisphere (Fig. 1B′′′,F′′′) is consistent with this hypothesis if the
intrinsic curvature of the tissue is smaller in magnitude than its actual
curvature.

## Origin of out-of-plane stresses in _Volvox_ inversion

### Mechanical toy problems

To test this hypothesis, we need to understand the mechanics of ablations in
curved tissues. For this purpose, we introduce three mechanical toy problems.

#### Toy problem 1: deformations of a spherical elastic shell with mismatched
intrinsic curvatures

Our first toy problem studies the effect of curvature mismatch on cell sheet
shapes before ablation: we consider a complete spherical shell of unit radius
and (relative) thickness $h\ll 1$. The intrinsic curvatures of the shell are
$\kappa^{0}=k\not=1$, different from the undeformed curvature of the shell.
This curvature mismatch causes the shell to deform; we assume that the shell
remains spherical, and denote by $f$ its deformed radius (Fig. 2A).

Figure 2: Origin of out-of-plane stresses in _Volvox inversion_ : mechanical
toy problems and analysis of experiments. (A) Toy problem 1: a spherical shell
of unit radius and thickness $h\ll 1$ deforms to a radius $f=1+O(h^{2})$ due
to mismatched intrinsic curvatures $\kappa^{0}=k\neq 1$. (B) Toy problem 2:
this mismatch of intrinsic curvatures causes a circular cut of angle
$2\varSigma$ and radius $c=2\sin{\varSigma}$ to open. Inset: definition of the
opening $o$, rotation $\varrho$, and displacement $d$ of the cut edge. (C) Toy
problem 3: anisotropic contraction also causes recoil on ablation. (D) Plot of
$o\sim h(1-k)$ against $c$. (E) Plot of asymptotic approximation for
$\varrho/\sqrt{h}(1-k)$ against $c/\sqrt{h}$. (F) Plot of asymptotic
approximation for $d/h(1-k)$ against $c/\sqrt{h}$. (G) Numerical solution of
toy problem 2: plot of cross-sections pre-ablation (light) and post-ablation
(dark), showing a recoil qualitatively consistent with experiments. (H)
Numerical solutions of toy problem 3: the recoil is much smaller than in
experiments despite the excessive anisotropy. (I) Fast localised relaxation
can also reproduce the experimental recoil qualitatively. (J) Estimated
dimensional intrinsic curvatures $K$, nondimensionalised with the average
posterior radius $\langle R\rangle$ during invagination. Estimates computed
from asymptotic results (open markers) and numerical solutions of toy problem
2 (filled markers) using measurements of $o$, $\varrho$, $d$ for $N=14$
_Volvox_ embryos at early invagination (inset). Arrows: off-scale values;
large markers and error bars: mean and standard deviation of estimates. (K)
Plot of the corresponding initial recoil velocity $V$ against its scaling.
Variables: elastic modulus $E$, effective viscosity $\mu$, (dimensional) cell
sheet thickness $H$, posterior radius $R$ during invagination, (dimensionless)
estimated intrinsic curvature $k$. A straight-line fit estimates the mean
$\langle E/\mu\rangle$. Inset: example measurement of the (dimensional) cut
displacement $D$ against time, showing the initial velocity and equilibrium
displacement $D_{\infty}$ used for estimates.

This radius is determined by the competition of the strains $E=f-1$ due to the
stretching of the shell and the bending strains $K=1/f-k$ resulting from the
difference of the actual and intrinsic curvatures of the shell. The elastic
energy of the shell is the sum of its stretching and bending energies (60), so
is proportional to

$\displaystyle
E^{2}+\dfrac{h^{2}}{12}K^{2}=(f-1)^{2}+\dfrac{h^{2}}{12}(1/f-k)^{2}.$ (1)

Assuming $k=O(1)$, this is minimised for

$\displaystyle f\sim 1-\dfrac{h^{2}}{12}(k-1).$ (2)

As expected, the shell grows ($f>1$) or shrinks ($f<1$) if its preferred
curvature is flatter ($k<1$) or larger (${k>1}$) than its undeformed
curvature, but the deformations are asymptotically small compared to the shell
thickness.

There is an interesting mechanical subtlety of this calculation: Though this
argument gives the correct $O(h^{2})$ scaling for the deformations of the
shell, the smallness of the deformations is beyond the realm of validity of
the shell theory assumed in writing down Eq. 1. For this reason, the correct
prefactor has to be calculated within bulk nonlinear solid mechanics; the
calculation is given in _SI Appendix_ , where we also discuss some of the
suprising behaviour revealed by that prefactor which shows how even seemingly
innocuous problems in nonlinear mechanics, such as this one, can break our
intuition.

#### Toy problem 2: circular ablation of a spherical elastic shell with
mismatched intrinsic curvatures

Next, we study the mechanics of a circular ablation in a shell with mismatched
intrinsic curvatures. The mismatched intrinsic curvatures and the no-torque
boundary condition at the rim of the cut cause the shell to deform near the
cut. For an ablation of angle $2\varSigma$, i.e. radius $c=2\sin{\varSigma}$
(Fig. 2B), asymptotic solution of the equations of shell theory for $h\ll 1$
(_SI Appendix_) shows that the cut opens by an amount (Fig. 2D)

$\displaystyle o\sim h(1-k)\sin{\varSigma}=\dfrac{h}{2}c(1-k).$ (3)

Importantly, this is asymptotically larger than the deformation without
ablation described by Eq. 2. Asymptotic approximations can also be found (_SI
Appendix_) for the rotation $\varrho$ of the rim of the cut and its
displacement $d$ due to the ablation (Fig. 2E,F).

#### Toy problem 3: circular ablation of a spherical elastic shell with
anisotropic contraction

Opening of an ablation does not require intrinsic curvature mismatches,
however: As explained in more detail in _SI Appendix_ , geometry requires the
stretches of an axisymmetric spherical shell to be equal at its poles, so
anisotropic contraction will cause stresses there and hence result in a recoil
on ablation (Fig. 2C).

### Discussion

These results allow us to test our hypothesis, that there is a curvature
mismatch during _Volvox_ inversion. First, toy problems 1 and 2 emphasise that
ablation experiments are needed: The unablated deformations are too small to
be visualised in experiments; only the deformations from ablations are
comparable to the thickness of the cell sheet, so allow quantification of
curvature mismatches that cannot be quantified from the unablated cell sheet
only.

#### The observed recoil during Volvox invagination results from a curvature
mismatch

Next, solving for post-ablation shapes in toy problems 2 and 3 numerically
(_Materials and Methods_), we observe that the recoil for an intrinsically
flat shell ($k=0$) agrees with experiments qualitatively (Fig. 1B′′′, Fig.
2G). By contrast, the recoil resulting from anisotropic contraction is much
smaller than the experimentally observed one, even for blatantly excessive
anisotropies (Fig. 1B′′′, Fig. 2H). This does not yet allow the conclusion
that the observed recoil results from a curvature mismatch: Indeed, fast
biological processes could _a priori_ be triggered close to the ablation site
to lead to a fast active relaxation of the shell there. Indeed, this could
result in a geometric incompatibility leading to a recoil qualitatively
consistent with experiments (Fig. 2I and _Materials and Methods_). If however
this mechanism underlied the experimental observations, we would expect to
observe a recoil for pre- and post-inversion ablations, too, which we do not
(Fig. 1B′′′,G′′′). We therefore discard this possibility, and conclude that
the observed recoil is only consistent with a curvature mismatch that existed
before ablation.

#### Quantitative estimates of the curvature mismatch during Volvox
invagination are consistent with the observed cell shape changes

To make this analysis more quantitative, we measured the recoil on ablation in
$N=14$ _Volvox_ embryos during early invagination (Fig. 1C–C′′′ and _Materials
and Methods_), and extracted estimates of intrinsic curvatures (Fig. 2J) from
the measurements of the cut opening $o$, rotation $\varrho$, and displacement
$d$ using the asymptotic approximations (Fig. 2D–F) and numerical solutions of
toy problem 2. The hemispherical shape of the posterior at these early
inversion stages (Fig. 2J, inset) justifies using toy problem 2 for this
inference. The estimates based on asymptotic approximations have a much larger
variance than those based on numerical solutions (Fig. 2J). This suggests
that, while the asymptotic estimates are useful to understand the mechanical
basis of the recoil upon ablation, the cell sheet of _Volvox_ is too thick for
them to yield good quantitative estimates. Meanwhile, we observe that the
estimate, based on numerics, of mean intrinsic curvature from $o$ has a larger
standard deviation than the estimates from $\varrho$ and $d$ (Fig. 2J).
Moreover, the latter are within one standard deviation of each other, while
the former is not (Fig. 2J). This is perhaps not unexpected, as measuring $o$
requires quantifying smaller deformations. We therefore estimate the mean
intrinsic curvature of the cell sheet based on $\varrho$ and $d$ only, finding
$\langle k\rangle\approx 0.05\ll 1$. Importantly, this estimate is consistent
with the observed cell shape changes: The spindle-shaped cells in the
posterior during early _Volvox_ invagination are symmetric, and connected at
their midplane by cytoplasmic bridges (Fig. 1C, inset), suggesting $k=0$. By
contrast, the teardrop-shaped cells in preinversion embryos (Fig. 1B, inset)
suggest a positive intrinsic curvature, and indeed the absence of recoil
observed preinversion (i.e. the absence of residual stresses) requires $k=1$.

#### Recoil velocity measurements yield a bound on the elastic modulus of the
cell sheet

The (dimensional) recoil velocity $V$ is set by the interplay between the
elastic force driving it and viscous dissipation in the tissue and surrounding
fluid. A scaling argument (_SI Appendix_) yields

$\displaystyle V\sim\dfrac{E}{\mu}\dfrac{H^{5/2}}{R^{3/2}}|1-k|,$ (4)

in which $E$ is the elastic modulus of the cell sheet, $\mu$ is an effective
viscosity, and $H$ and $R$ are the dimensional thickness and posterior radius
of the cell sheet. We obtain $V$ from measurements of the initial recoil
velocity of the cell sheet (Fig. 2K). We stress our use of this initial
velocity for these dynamic measurements, by contrast with the above matching
of (quasi-)equilibrium quantifications of the recoil to a static toy problem
(Fig. 2K, inset). From the scaling relation, we obtain a mean value $\langle
E/\mu\rangle\approx 0.6\,\mathrm{s}^{-1}$ (Fig. 2K). With the lower bound
$\mu\gtrsim 0.8\,\mathrm{mPa\,s}$ corresponding to the viscosity of algal
growth medium (61), so neglecting dissipation within the tissue, this yields
$E\gtrsim 0.5\,\mathrm{mPa}$, much lower than the values reported for
confluent tissues, yet perhaps appropriate for an extremely floppy, non-
confluent tissue. With $H\approx 11\,\text{\textmu m}$ (41), the bending
modulus $K=EH^{3}\gtrsim 7\,\cdot 10^{-19}\,\mathrm{J}$, well above the energy
of thermal fluctuations, provides a sanity check of this estimate.

Figure 3: Effect of out-of-plane stresses in _Volvox_ inversion: fits of the
elastic model to average experimental cross-sections (47) and numerical
ablations. (A) Overlay of experimental cross-sections (white) before (red) and
after (cyan) a posterior ablation at an early invagination stage (Fig. 1B′′′).
(B) Analogous plot for an ablation at a late invagination stage (Fig. 1C′′′).
(C) Analogous plot for an ablation at a mid-inversion stage (Fig. 1D′′′).
Outside the dotted rectangle, only pre-ablation shapes are plotted (white).
(D) Fit, for zero posterior intrinsic curvatures
$\smash{\kappa^{0}_{\mathrm{post}}=0}$, of the elastic model to the average
experimental cross-section at an inversion timepoint corresponding to panel
(A) (red), overlaid with equilibrium shape after numerical ablation (cyan),
showing a recoil on ablation. (E) Analogous plot, for posterior intrinsic
curvatures $\smash{\kappa^{0}_{\mathrm{post}}=R^{-1}}$, equal to the
undeformed curvature of the embryo. The recoil is reduced. The dashed box
highlights thinning of the cell sheet in the fitted shape. (F) Analogous plot,
for posterior intrinsic curvatures
$\smash{\kappa^{0}_{\mathrm{post}}=r_{\mathrm{post}}^{-1}}$, equal to the
inverse contracted intrinsic radius of the posterior. There is almost no
recoil. (G) Fit, for $\smash{\kappa^{0}_{\mathrm{post}}=0}$, of the elastic
model to the average experimental cross-section at an inversion timepoint
corresponding to panel (B), and numerical ablation. Inset: comparison of
fitted shape and average experimental cross-section. (H) Analogous plot for
$\smash{\kappa^{0}_{\mathrm{post}}=R^{-1}}$. The fitted shape reproduces the
experimental shape better than that in panel (G). (I) Analogous plot for
$\smash{\kappa^{0}_{\mathrm{post}}=r_{\mathrm{post}}^{-1}}$. The fit is again
better than that in panel (G). (J) Fit, for
$\smash{\kappa^{0}_{\mathrm{post}}=0}$, of the elastic model to the average
experimental cross-section at an inversion timepoint corresponding to panel
(C), and numerical ablation. (K) Analogous plot with
$\smash{\kappa^{0}_{\mathrm{post}}=R^{-1}}$. The recoil on ablation is
qualitatively similar. (L) Analogous plot with
$\smash{\kappa^{0}_{\mathrm{post}}=r_{\mathrm{post}}^{-1}}$. (M) Illustration
of the numerical ablation plots using the plots in panel (D): the
superposition of the fitted shape before numerical ablation (red) and that
after numerical ablation (cyan) defines the overlaid shape (white). (N) Plot
of the fit energy $E_{\mathrm{fit}}$ (arbitrary units) against inversion
(pseudo)time for fits with
$\smash{\kappa^{0}_{\mathrm{post}}\in\\{0,R^{-1},r_{\mathrm{post}}^{-1}}\\}$
and for a fit in which
$\smash{\kappa^{0}_{\mathrm{post}}},\smash{r_{\mathrm{post}}}$ are among
additional fitting parameters. $E_{\mathrm{fit}}=0$ would indicate a perfect
fit; the horizontal dashed line indicates the value of $E$ at which fitting
was cut off. Pseudotimepoints used for the fits in panels (D)–(L) are
highlighted. (O) Plots of fitted $\smash{\kappa^{0}_{\mathrm{post}}}R$ and
$\smash{(r_{\mathrm{post}}/R)^{-1}}$ against inversion (pseudo)time. Inset:
plot of fitted $\smash{r_{\mathrm{post}}}/R$ against inversion pseudotime. The
constant values used for the fits with
$\smash{\kappa^{0}_{\mathrm{post}}\in\\{0,R^{-1},r_{\mathrm{post}}^{-1}}\\}$
are highlighted.

## Effect of out-of-plane stresses in _Volvox_ inversion

Having thus shown that out-of-plane stresses resulting from a curvature
mismatch persist during _Volvox_ invagination, we ask: What is the effect of
these out-of-plane stresses on the tissue shape sequence during inversion?

### Quantitative model of _Volvox_ inversion

The toy models in the previous section can describe the shapes of _Volvox_
embryos during early invagination locally, close to the posterior ablation,
but cannot capture the global embryo shapes and mechanics. We therefore
extended our previous detailed morphoelastic model of the cell shape changes
of inversion (47) and fitted parameters encoding the cell shape changes
driving inversion (41) to the average cell sheet midlines at various
timepoints of inversion (47), as described in detail in _Materials and
Methods_ and _SI Appendix_. We then performed numerical ablations on these
fitted shapes and compared _in vivo_ and _in silico_ results (Fig. 3A–M).

#### Results for early invagination stages are consistent with the observed
cell shape changes and the estimates from ablation recoils

We first considered an early invagination timepoint (Fig. 3A), for which we
fitted the average embryo shape for three different values of the posterior
intrinsic curvature $\kappa_{\mathrm{post}}^{0}$ (Fig. 3D–F):
$\kappa_{\mathrm{post}}^{0}=0$, corresponding to the flat posterior intrinsic
curvature predicted by the quantitative estimates of the previous section and
the observed cell shape changes in the posterior;
$\kappa_{\mathrm{post}}^{0}=R^{-1}$, equal to the inverse radius of the
undeformed embryo, i.e. the value suggested at pre-inversion stages by the
lack of recoil on ablation;
$\kappa_{\mathrm{post}}^{0}=r_{\mathrm{post}}^{-1}$, equal to the inverse
intrinsic contracted radius of the posterior resulting from the cell shape
changes to spindle-shaped cells (Fig. 1C). With this last value, the
contracted shape of the uninverted posterior is unstressed. The model can
reproduce the average embryo midlines well for all three values of
$\kappa_{\mathrm{post}}^{0}$ (Fig. 3D–F), although the fit parameters inferred
from the cell sheet midlines for the latter two values imply a thinning of the
anterior hemisphere of the cell sheet (highlighted in Fig. 3E,F) that is only
observed experimentally at slightly later stages of invagination, when it
associated with the formation of pancake-shape cells there (41, Fig. 1D).
However, numerical ablations reveal that these three fitted shapes are
mechanically different (Fig. 3D–F): Only the first case leads to a sizeable
recoil on ablation, qualitatively consistent with experiments. The fitting
thus suggests that $\kappa_{\mathrm{post}}^{0}=0$ and is therefore consistent
with the estimates of the previous section and the observed cell shape
changes.

#### Results for late inversion stages are consistent with the observed
expansion of the region of cell wedging

Next, we considered a late inversion timepoint (Fig. 3C), around the time of
phialopore opening. Again, all three values of posterior intrinsic curvature
could reproduce the observed cell sheet midlines (Fig. 3J–L) and the fitted
shapes even reproduce the observed thinning of the anterior hemisphere (41)
even though the fits are based on the cell sheet midlines only. Moreover,
numerical ablations lead to similar recoils in all three cases (Fig. 3J–L).
This is consistent with the observed expansion of the wave of cell wedging
towards the posterior pole (41): The recoil magnitude is set no longer by the
geometric incompatibilities of the few remaining spindle- or pencil-shaped
cells (Fig. 1E, inset), but by that of the wedge-shaped cells in the bend
region (Fig. 1C).

### Discussion

These results thus provide a check of the fitting and suggest that the
mechanics of out-of-plane stresses in _Volvox_ inversion can be captured
correctly by this fitting. In turn, this indicates that we can use the fit
results to understand out-of-plane stresses at other stages of inversion.

#### The fitted mechanics at midinversion stages predict changing posterior
out-of-plane stresses

We thus turn to a midinversion stage next (Fig. 3B), again fitting the model
to the experimental data for the three different values of
$\kappa_{\mathrm{post}}^{0}$ (Fig. 3G–I). Interestingly, the quality of the
fit to the average experimental inversion shape is considerably worse for
$\kappa_{\mathrm{post}}^{0}=0$ (Fig. 3G, inset) than for the other two values
(Fig. 3H–I, insets). Interestingly, the fit parameters inferred from the cell
sheet midlines in the case $\kappa_{\mathrm{post}}^{0}=R^{-1}$ imply a
thickness gradient in the anterior hemisphere (Fig. 3H) that is consistent
with the formation of pancake-shaped cells starting in the anterior fold and
propagating towards the phialopore, as described previously (41, Fig. 1D). All
of this suggests that the intrinsic curvature of the spindle-shaped cells and
hence the out-of-plane stresses in the uninverted posterior change at
midinversion stages. The fact that a better fit (Fig. 3H–I) is obtained for
positive values of $\kappa_{\mathrm{post}}^{0}$ that are associated with
reduced geometric incompatibilities in the uninverted posterior begs the
question: Could this change constitute an adaptive response of the tissue to
the stresses resulting from the geometric incompatibility generated by the
spindle-shaped cells in the posterior?

#### The fitted mechanical sequence hints at an adaptive response to out-of-
plane stresses

To provide a hint of an answer to this question, we considered fits of our
model to the experimental data for more developmental timepoints before
phialopore opening111We did not fit our model beyond the onset of phialopore
opening because that is associated with poorly understood cell rearrangements
near the phialopore (47), the mechanics of which cannot be described by our
elastic model. and for the three values of $\kappa_{\mathrm{post}}^{0}$. While
the fit energies (Fig. 3N) are comparable at early and late stages of
inversion, those at mid-inversion stages are lowest, by some margin, for the
fit with $\kappa_{\mathrm{post}}^{0}=r_{\mathrm{post}}^{-1}$. We have already
noted above that the contracted shape of the uninverted posterior is
unstressed in this case. This therefore hints that
$\kappa_{\mathrm{post}}^{0}$ changes in such a way as to relieve the geometric
incompatibilities in the uninverted posterior, and hence at an adaptive
response to the stresses. To substantiate this hint, we performed a further
fit in which $\kappa_{\mathrm{post}}^{0},r_{\mathrm{post}}$ are among
additional fitting parameters (_SI Appendix_). The resulting fit energy (Fig.
3N) is not substantially lower than that of the more constrained fit with
$\kappa_{\mathrm{post}}^{0}=r_{\mathrm{post}}^{-1}$ at mid-inversion stages.
Moreover, the fitted values of $\smash{\kappa_{\mathrm{post}}^{0}}$ (Fig. 3O)
increase from zero at the start of invagination to values comparable to
$r_{\mathrm{post}}^{-1}$ at mid-inversion stages. All of this lends further
support to the hypothesis of an adaptive response to stresses in the tissue.
Interestingly, the fitted values of $\kappa_{\mathrm{post}}^{0}$ (Fig. 3O)
decrease and those of $r_{\mathrm{post}}$ (Fig. 3O, inset) decrease before
increasing again at mid-to-late inversion stages, when the fit energy of this
fit becomes rather smaller than that of the other fits (Fig. 3N). All of this
could be a signature of the pencil-shaped cells (41, Fig. 1E, inset) that
begin to form at these stages of inversion (and another adaptive response):
Indeed, the position of cytoplasmic bridges in these cells may suggest
$\kappa_{\mathrm{post}}^{0}R<1$, and they are narrower than spindle-shaped
cells (41), indicating reduced $r_{\mathrm{post}}$. We discuss these results
further in _SI Appendix_.

## Conclusion

We have combined ablation experiments and a detailed elastic model to
establish the origin and analyse the mechanical effects of out-of-plane
stresses during _Volvox_ inversion. Strikingly, our results show how changes
of posterior intrinsic curvature—albeit small compared to the intrinsic
curvature imposed by the wedge-shaped cells in the bend region—contribute to
the observed tissue shape sequence of _Volvox_ inversion. In particular, the
theory predicts an intriguing adaptive response of the geometric
incompatibilities in the cell sheet to its out-of-plane stresses.

Tissue bending is not therefore all about apical constriction: Rather, out-of-
plane stresses from other cell shapes changes also contribute mechanically to
_Volvox_ inversion. This is possible for biological tissues are not infinitely
thin: If they were, the asymptotic separation of stretching and bending
energies (60) would imply that only the large bending deformations of apical
constriction (30, 48) that break this asymptotic separation would have an
effect comparable to that of in-plane stretching deformations. The question
now becomes: Do these out-of-plane stresses that impact the shapes of
inverting _Volvox_ embryos and their adaptive response predicted by theory
have a mechanical function? Since inversion is still possible for all three
scenarios in Fig. 3D–L, they are not required mechanically for inversion,
unlike the formation of wedge-cells or contraction, which is known to be
required mechanically for the closely related type-A inversion in _V. carteri_
(58, 59, 57). However, one might speculate that these stresses or equivalently
the stored elastic energy contribute to the mechanical robustness of _Volvox_
inversion. Also, these stresses and their predicted adapation during inversion
could have a function related to mechanosensing and hence to the unknown
signalling processes that orchestrate inversion. Testing such hypotheses will
require again close integration of experiment and theory and, in particular,
dynamic quantification of cell shapes to link “microscopic” cell-level changes
to geometric incompatibilites and hence stresses at the “macroscopic” tissue
level.

Our experimental and theoretical approach sets out a framework for inferring
the mechanical state and properties of curved tissues based on their unfurling
on ablation, which we expect to be applicable to a broad range of problems in
cell sheet folding. However, it has also revealed key differences between the
analysis of ablations in curved and flat tissues: First, the standard measure
of recoil on ablation in flat tissues, i.e. what we termed the cut opening
above, may not be appropriate for analysing ablations in curved tissues,
because it requires quantification of smaller deformations than the measures
of cut rotation and cut edge displacement that we introduced here, and is
therefore more prone to experimental noise. Second, the inference of cell
sheet properties from the deformations on ablations is more complex in curved
tissues: It may require numerical solution of a “toy problem” rather than use
of a closed-form expression because tissues are not asymptotically thin, as
already noted above: The tissue thickness, which is the natural small
parameter for asymptotic solution of such toy problems, may not be
sufficiently small for closed-form asymptotic expressions to enable
quantitative inference.

Finally, from a more physical point of view, it is striking that the “simple”
tissue shapes of early _Volvox_ invagination did not constrain the mechanical
state of the cell shape to the same extent as the more complex shapes at mid-
inversion stages: At early invagination stages, knowledge of the unfurling of
the tissue on ablation was required to distinguish between the three
mechanical scenarios discussed in Fig. 3D–F, while, at later stages, we could
distinguish between these possibilities based on the shapes of cell sheet
alone (Fig. 3G–I). It is tempting to ask whether this observation generalises:
Do more complex tissue shapes intrinsically contain more information about the
mechanical state of the tissue? Of course, this begs a more basic question:
What is the right way of quantifying tissue shape complexity? Addressing these
question both in simple physical model problems and in simple biological
systems like _Volvox_ or synthetic, organoid models will allow us to take the
first steps towards addressing, too, the much more fundamental biological
problem that is the relation between tissue shape (complexity) and biological
function.

### Model organism cultivation

Wild-type strain  _Volvox globator_ Linné (SAG 199.80) was obtained from the
Culture Collection of Algae at the University of Göttingen, Germany (62), and
cultured as described previously (63, 45) in liquid Standard _Volvox_ Medium
(SVM) with a cycle of $16\,\text{h}$ light at $24^{\circ}\text{C}$ and
$8\,\text{h}$ dark at $22^{\circ}\text{C}$. _V. globator_ cultures in the
asexual life cycle of these monoecious microalgae were used.

### 2-photon microscopy and laser ablation experiments

_Volvox_ spheroids containing embryos undergoing inversion were embedded in
$2\,\text{\textmu l}$ of $1\%$ low-melting-point agarose (LMA), covered with
SVM, and imaged using a Trim Scope 2-photon microscope (LaVision). A laser
line at $\lambda=1040\,\text{nm}$ was used for imaging and a separate laser
line at $\lambda=900\,\text{nm}$ for performing laser ablations. As a single
chloroplast mostly fills each _Volvox_ cell, chlorophyll-autofluorescence was
detected at $\lambda>647\,\text{nm}$, and used for visualising the embryonic
cell sheet. Z-stacks were recorded before and after each laser ablation
experiment, with a z-step of $2\,\text{\textmu m}$. To maximise acquisition
speed and capture the elastic response to the laser cut, a single plane was
imaged during ablation experiments, which removed cells within a radius of
$3-6\,\text{\textmu m}$ of the posterior pole of the cell sheet. Videos of the
mid-sagittal plane were recorded at $1-4.5\,\text{fps}$ for at least
$10\,\text{s}$ before and $30\,\text{s}$ after laser ablation. Videos used for
determining velocities of elastic recoils were recorded at the maximal speed
possible for the field of view size used, corresponding to
$2.5-4.5\,\text{fps}$.

### Analysis of experimental data

Outlines of the posterior hemisphere were traced manually on recordings of
midsagittal embryo cross-sections using Fiji (64). To trace the midline of the
cell sheet, the line width was set to fit the thickness of the cell sheet.
Recoil quantification was then performed using custom Matlab (The MathWorks,
Inc.) scripts; in particular, the fit in Fig. 2K uses the polyfitZero function
by M. Mikofski, obtained from the Matlab file exchange (file 35401). Posterior
cell sheet radii were estimated by fitting a circle to the cross-sections in
Fiji. Cell sheet thicknesses were estimated by scaling the chloroplast
thicknesses measured in Fiji so that the resulting mean value matched that
reported previously (41).

### Morphoelastic shell theory

To describe _Volvox_ morphogenesis in terms of the axisymmetric deformations
of an elastic shell with varying intrinsic stretches and curvatures, we use
the shell theory derived in the biologically relevant limit of “large bending
deformations” in Ref. (48): For an axisymmetric elastic shell of relative
thickness $h$, we denote by $s$ and $S$ the respective arclengths of the
undeformed and deformed cross-sections of the axisymmetric shell. These cross-
sections are described by their distances $\rho,r$ from the axis of symmetry
and and the tangent angle $\psi$ of the deformed cross-section. The meridional
and circumferential stretches and curvatures of the deformed shell are thus

$\displaystyle
f_{s}(s)=\dfrac{\mathrm{d}S}{\mathrm{d}s},\;f_{\phi}(s)=\dfrac{r(s)}{\rho(s)},\quad\kappa_{s}(s)=\dfrac{1}{f_{s}}\dfrac{\mathrm{d}\psi}{\mathrm{d}s},\;\kappa_{\phi}(s)=\dfrac{\sin{\psi(s)}}{r(s)},$
(5)

and we denote by
$f_{s}^{0}(s),f_{\smash{\phi}}^{0}(s),\kappa_{s}^{0}(s),\kappa_{\smash{\phi}}^{0}(s)$
their intrinsic, preferred values. The differences between the actual
stretches and curvatures and their intrinsic values define the shell strains

$\displaystyle E_{s}=\dfrac{f_{s}-f_{s}^{0}}{f_{s}^{0}},$ $\displaystyle
E_{\phi}=\dfrac{f_{\phi}-f_{\phi}^{0}}{f_{\phi}^{0}}$ (6a) and curvature
strains $\displaystyle
K_{s}=\dfrac{\tilde{f_{s}}\kappa_{s}-f_{s}^{0}\kappa_{s}^{0}}{(f_{s}^{0})^{2}f_{\phi}^{0}},$
$\displaystyle
K_{\phi}=\dfrac{f_{\phi}\kappa_{\phi}-f_{\phi}^{0}\kappa_{\phi}^{0}}{f_{s}^{0}(f_{\phi}^{0})^{2}}$
(6b)

that appear in the elastic energy

$\displaystyle\mathcal{E}=2\textrm{p}\int_{\mathcal{C}}{e(s)\rho(s)\,\mathrm{d}s},$
(7a) where the integration is along the cross-section $\mathcal{C}$ of the
shell, and the energy density is $\displaystyle e$
$\displaystyle=\dfrac{C}{2}\left\\{h\left[\alpha_{ss}E_{s}^{2}+(\alpha_{s\phi}+\alpha_{\phi
s})E_{s}E_{\phi}+\alpha_{\phi\phi}E_{\phi}^{2}\right]\right.$
$\displaystyle\qquad\quad+h^{2}\left(\beta_{ss}E_{s}K_{s}+\beta_{s\phi}E_{s}K_{\phi}+\beta_{\phi
s}E_{\phi}K_{s}+\beta_{\phi\phi}E_{\phi}K_{\phi}\right)$
$\displaystyle\qquad\quad+\left.h^{3}\left[\gamma_{ss}K_{s}^{2}+(\gamma_{s\phi}+\gamma_{\phi
s})K_{s}K_{\phi}+\gamma_{\phi\phi}K_{\phi}^{2}\right]\right\\},$ (7b)

in which $C$ is a material parameter and
$\alpha_{ss},\alpha_{s\smash{\phi}},\alpha_{\smash{\phi}s},\alpha_{\smash{\phi}\smash{\phi}},\beta_{ss},\beta_{s\smash{\phi}}$,
$\beta_{\smash{\phi}s},\beta_{\smash{\phi}\smash{\phi}},\gamma_{ss},\gamma_{s\smash{\phi}},\gamma_{\smash{\phi}s},\gamma_{\smash{\phi}\smash{\phi}}$
are functions of $\eta=\kappa_{s}^{0}h/2f_{s}^{0}f_{\smash{\phi}}^{0}$ only,
the (complicated) explicit expressions for which are given in Eqs. (58) and
(64) of Ref. (48).

To find the deformed shape of the shell resulting from the imposed intrinsic
stretches and curvatures numerically, we solve the boundary value problem
associated with Eqs. 7 derived in Ref. (48) using the bvp4c solver of Matlab.

### Numerical solution of the toy problems

We solve these equations for a circular cross-section $\rho(s)=\sin{s}$ for
$\varSigma\leqslant s\leqslant\textrm{p}$. At $s=\textrm{p}$, we impose the
boundary conditions $r=\smash{\psi}=0$; if $\varSigma>0$, we impose no-force
and no-torque conditions at $s=\varSigma$ which, from Ref. (48), take the form
$\alpha_{ss}E_{s}+\alpha_{s\smash{\phi}}E_{\phi}+h(\beta_{ss}K_{s}+\beta_{s\smash{\phi}}K_{\phi})=0$,
$\beta_{ss}E_{s}+\beta_{\smash{\phi}s}E_{\phi}+h(\gamma_{ss}K_{s}+\gamma_{s\smash{\phi}}K_{\phi})=0$;
if $\varSigma=0$, the boundary conditions at $s=0$ are instead $r=\psi=0$.

For toy problem 2, we take $\smash{f_{s}^{0}=f_{\smash{\phi}}^{0}=1}$,
$\smash{\kappa_{s}^{0}=\kappa_{\smash{\phi}}^{0}}=k$. For toy problem 3, we
take $f_{s}^{0}=1+f_{0}=1/f_{\smash{\phi}}^{0}$, where $f_{0}$ is a constant
that expresses the anisotropy of contraction, and
$\kappa_{s}^{0}=\kappa_{\smash{\phi}}^{0}=1$. Finally, to describe a possible
fast active deformation of the tissue on cutting, we take
$f_{s}^{0}=f_{\smash{\phi}}^{0}=1+F_{0}(\varSigma+s_{0}-s)/s_{0}$ if
$\varSigma\leqslant s\leqslant\varSigma+s_{0}$ for some length scale $s_{0}$,
and $f_{s}^{0}=f_{\smash{\phi}}^{0}=1$ otherwise, as well as
$\kappa_{s}^{0}=\kappa_{\smash{\phi}}^{0}=1$; results do not seem to depend
strongly on the form of the decay of $f_{s}^{0},f_{\smash{\phi}}^{0}$ away
from $s=\varSigma$ (not shown). The parameter values for Fig. 2G–I are
$h=0.3$, $\varSigma=0.3$, $k=0$, $f_{0}=\pm 0.1$, $F_{0}=0.35$, $s_{0}=0.2$.

### Fitting the morphoelastic model to the experimental data

For a quantitative description of _Volvox_ inversion, similarly to Ref. (47),
we consider again a circular cross-section $\rho(s)=\sin{s}$, but now for
$\varSigma\leqslant s\leqslant\textrm{p}-P$, where $P$ is the phialopore size.
We set $P=0.3$ and $h=0.15$, as in Ref. (47). The boundary conditions are
those of the toy problems discussed above, except for $s=\textrm{p}-P$, where
we impose no-force and no-torque conditions.

We define functional forms for the intrinsic stretches and curvature functions
$\smash{f_{s}^{0}(s),f_{\phi}^{0}(s),\kappa_{s}^{0}(s),\kappa_{\phi}^{0}(s)}$
that allow a minimal representation of the cell shape changes observed during
_Volvox_ inversion (41), described in detail in _SI Appendix_. These
functional forms define a number of fitting parameters, which, with
$\varSigma=0$, were fitted to the average inversion shapes (47), by minimising
a fit energy $E_{\mathrm{fit}}$ that measures the difference between the
average and fitted shapes using the Matlab function fminsearch, similarly to
Ref. (47). This fitting is described in detail in _SI Appendix_.

For numerical ablations, the value of $\varSigma$ was increased from
$\varSigma=0$ in these fitted shapes, with boundary conditions as in the toy
problems above.

### Analytical calculations

Details of the analytical calculations for and additional discussion of the
three mechanical toy problems, and the scaling argument for the recoil
timescale are given in _SI Appendix_.

The authors are very grateful to Raymond E. Goldstein, in whose research group
their collaboration on _Volvox_ inversion mechanics started, for many
discussions, mentoring, and invariably helpful scientific advice. The authors
also thank members of the Cambridge Advanced Imaging Centre, particularly
Martin Lenz and Kevin O’Holleran, for their advice and support. The authors
gratefully acknowledge funding from the Max Planck Society (P.A.H.), and the
Wellcome Trust and the John Templeton Foundation (S.S.M.H.H.).

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