Probing Embryonic Tissue Mechanics with Laser Hole-Drilling
Xiaoyan Ma, Holley E Lynch, Peter C Scully and M Shane Hutson
Vanderbilt Institute for Integrative Biosystem Research & Education,
Department of Physics & Astronomy, Department of Biological Sciences,
Vanderbilt University, Nashville, TN 37235
Email: [email protected]
Abstract. We have used laser hole-drilling to assess mechanical changes in an embryonic
epithelium during development – in vivo and with subcellular resolution. This method
uses a single laser pulse to ablate a cylindrical hole (1 m in diameter, 5-7 m tall) clean
through the epithelium, and tracks the subsequent recoil of adjacent cells (with ms time
resolution). We investigate dorsal closure in the fruit fly with emphasis on apical
constriction of amnioserosa cells. We find that substantial in-plane tension is carried
across each cell surface and not just along cell-cell interfaces. In early phases of
constriction, the tension is 1.6-fold greater along cell-cell interfaces. Later, the two
tensions are indistinguishable. Other changes associated with constriction include a
decrease in the characteristic recoil times and an increased anisotropy. The smaller time
constants imply a more solid-like tissue. The anisotropy matches changes in the
underlying quasi-hexagonal cellular mesh. The results of these laser hole-drilling
experiments provide new avenues for evaluating extant computational models. Similar
experiments should be applicable to a wide range of morphogenetic events.
Keywords: morphogenesis; dorsal closure; viscoelasticity; rheology; laser ablation;
apical constriction
PACS codes: 87.19.lx, 87.17.Pq, 87.17.Rt, 83.85.Tz
1. Introduction
Developmental biology, in particular those aspects referred to as morphogenesis, is clearly both a
genetic and a mechanical process. The mechanical aspects are actually the proximate cause of the
main developmental observables – shape and form – but typically receive scant attention. When
developmental mechanics are considered, the normal course is for physical scientists to develop
computational models [1-6]. These models reproduce the shapes and forms of morphogenetic
episodes with varying degrees of success and complexity. Ideally, such models would be
challenged and refined with complementary experiments. In reality, such models have often
languished because the experimental tools of developmental biology are ill equipped to test
mechanical hypotheses. New tools are needed. In this light, we show that laser hole-drilling –
with quantitative analysis of the subsequent cellular recoils – can elucidate how stresses and
mechanical properties change during an episode of morphogenesis in living embryos and with
subcellular resolution.
The specific episode of morphogenesis investigated here is dorsal closure in the fruit fly. At the
beginning of dorsal closure, Bowne’s stage 13 [7], a one-cell thick epithelial tissue known as the
amnioserosa covers the embryo’s dorsal surface. As dorsal closure proceeds (stages 13-15), this
tissue dives into the embryo as an adjacent epithelial tissue (lateral epidermis) advances from
both sides and eventually seals it over. Dorsal closure has been extensively characterized in terms
of its genetics, its cell shape changes and its tissue-level mechanics [8-12]. Laser microsurgery
1
already played a large role in the latter and established that closure is driven by a combination of
three processes [10, 11]: an adhesive interaction between approaching flanks of epidermis;
contraction of a supracellular “purse-string” along the amnioserosa-epidermis boundary; and
apical constriction of amnioserosa cells. The latter contributes most of the force towards closure
and is the subject of our analysis here.
Another reason for choosing apical constriction is its widespread involvement in other
morphogenetic episodes from gastrulation to neurulation [13, 14]. In each case, a subgroup of
epithelial cells contracts their apical surfaces while expanding in the apical-basal direction. When
viewed in cross-section, the constricting cells eventually take on a wedge-shaped morphology.
These stereotypical movements have been reproduced by multiple models ranging from
phenomenological elongation-contraction coupling [1] to finite-element [2, 3] and hydrodynamic
formulations [6]. The most robust models consider several alternative mechanisms and their
different mechanical consequences. Beyond the specifics of apical constriction, there are more
general questions with regard to how an epithelium should be modeled. Most current models for
developmental mechanics follow the paradigm of differential adhesion [15] – with the dominant
tensions along cell-cell interfaces [4, 16]. The cell cytoplasm is then a passive, incompressible
medium. In both the general and specific cases, our goal as experimentalists is to test, validate
and eliminate some of these alternatives.
The experiments reported here are inspired by well established hole-drilling methods for
evaluating residual stress [17, 18]; however, those methods estimate stress from equilibrium
elastic deformation. For a viscoelastic material like a cell sheet, the primary information is the
recoil dynamics. Thus, for all of our experiments, hole-drilling consists of ablation with a single
laser pulse (at 355 nm, 2 threshold). Under these conditions, the longest-lived transients
associated with ablation itself, i.e. cavitation, are complete in less than 2 s [19]. Furthermore, the
immediate impact of ablation (again mediated by cavitation) extends out less than 5 m, and
perhaps much less; right after ablation, the damaged region is less than 1 m in diameter. Such
localized, impulsive ablation is critical for measuring the subsequent recoil dynamics on the mstimescale. These recoil dynamics are the key to evaluating how the stresses and mechanical
stiffness vary within an epithelium and between developmental stages.
2. Materials and Methods
2.1 Fly strains and microscopy.
The strain of Drosophila melanogaster used in this study was ubi-DE-Cad-GFP [20] obtained
from Kyoto Drosophila Genetic Resource Center (DGRC 109007). The ubiquitously-expressed
cadherin-GFP chimera effectively labels epithelial cell-cell junctions. Fly embryos were
dechorionated in 50% bleach solution, immersed in halocarbon oil 27 (Sigma-Aldrich, St. Louis,
MO), and sandwiched between a cover glass and an oxygen-permeable membrane (YSI, Yellow
Spring, OH) for live fluorescent imaging [21]. Images were captured on a Zeiss LSM410 laserscanning confocal microscope (inverted) with a 40, 1.3 NA oil-immersion objective and 488-nm
excitation. The scanning times were 2-8 s per frame and 15.74 ms per kymograph line.
2.2 Laser microsurgery.
Ablations were performed with the 3rd harmonic (355 nm) of a Q-switched Nd: YAG laser (5-ns
pulsewidth, Continuum Minilite II, Santa Clara, CA). This laser was coupled into the Zeiss
LSM410 with independent beam steering for simultaneous ablation and imaging [21]. The pulse
energy was just high enough (2-3 threshold) to insure consistent single-pulse ablation. This
energy varied with embryo stage and tissue – 1.16 ± 0.25 J for amnioserosa cells in stage 13 and
3.33 ± 0.91 J for these cells in stage 14 – due to differences in depth below the embryo’s
vitelline membrane – 2.3 ± 1.7 versus 16.4 ± 6.1 m, respectively.
2
2.3 Image and data analysis.
Overlay and cross-section images were constructed using ImageJ (NIH, Bethesda, MD). Analysis
of the spatial dependence of recoil displacements was carried out using the plugin UnwarpJ [22].
The time-dependent cell edge positions were extracted from kymographs using a self-developed
ImageJ plugin based on the Lucas-Kanade algorithm [23].
The cell edge positions obtained as above were subject to non-linear regression with Mathematica
(Wolfram Research, Champaign, IL). The positions from one to two seconds before ablation and
up to two seconds after ablation were fit to a piecewise continuous function:
x ( t ) = d0 + m(t t 0 )
t < t0
x(t) = d0 + d1 (1 e( tt0 )/ 1 ) + d2 (1 e(tt 0 )/ 2 )
t t0
Eqn 1
for a total of seven adjustable parameters (t0, d0, m, d1, t 1, d2, t2). All graphs below have been
shifted so that d0 = 0 and t0 = 0. The initial recoil velocity, v0, was taken as the slope of the double
exponential function at t0. The fits were also used to estimate the recoil velocity averaged over the
first two seconds, <v0-2 s>.
For further analysis, the data was grouped into four categories: (i) stage 13 wounds to cell
centers; (ii) stage 13 wounds to cell edges; (iii) stage 14 wounds to cell centers; and (iv) stage 14
wounds to cell edges. For each group and each recoil parameter (generalized as q), we
constructed a probability density estimate
1
1 N
P (q) = e
N i=1 w i 2
( qq i )2
2w i2
Eqn 2
where N is the number of experiments in a group, qi is the parameter estimate from the ith
experiment and wi is the larger of the standard error of qi and a parameter-specific wmin. Each
parameter was assigned a wmin that was chosen just large enough to yield a smooth probability
density estimate (noted on the respective graphs). This wmin is analogous to the choice of bin
width in a histogram, but by using the standard error if it is larger, one can properly reduce the
contribution of individual experiments with poorly constrained parameter estimates. For each
group, the mean parameter values are reported as mean ± standard error of the mean. Standard
errors of the mean were calculated from the total variance, i.e. the sum of the variance among the
parameter estimates and the mean of the squared standard errors (from each experiment).
To evaluate whether the parameter distribution for two groups differed significantly, we used four
different statistical tests with complementary strengths and weaknesses [24]: (t1) the two-sample
student t-test – which assumes normal distributions with the same variance; (t2) Welch’s
modified t-test with independent variance estimates for the two groups; (W) the Wilcoxon twosample rank sum test – which relaxes the normality criterion, but assumes identically shaped
distributions; and (KS) the Kolmogorov-Smirnov two-sample procedure – which assumes
continuous distributions, but places no other constraint on distribution shape. Although the t1, t2
and W tests disregarded the standard error from each experiment, the KS procedure includes the
standard errors during construction of the probability distributions. The KS procedure is the most
stringent of these tests for significant differences, but it is also the most likely to erroneously
claim two distributions as indistinguishable. Statistical analysis was performed in Excel
(Microsoft, Redmond, WA).
3. Results and Discussion
We performed single-pulse laser ablation in the amnioserosa of stage 13 and 14 fruit fly embryos
– one ablation per embryo, but at different locations. No matter where this embryonic epithelium
3
is wounded, the surrounding cells always
recoil away from the wound site (Fig 1).
By 10 s after ablation, the nearest
neighbors of the wounded cell(s) recoil
about of a cell diameter (average
velocity ~ 1 m/s). The maximum recoil
of about a cell diameter is reached
within 20-60 s. The neighboring cells
then move back towards the ablation site
as wound healing begins.
In these experiments, the ablated region
is much smaller than a single cell. This is
most clearly seen via the hole created in
the embryo’s overlying vitelline
membrane (e.g. the dark region with a
hyper-fluorescent ring in Fig 1E). This
hole is bounded by a coverslip and the
glue holding the fly embryo onto the
coverslip. Thus, it expands little and does
not allow material to flow in or out. The
hole apparent in Fig 1E is elliptical with
semiminor and semimajor axes of 0.50
and 0.75 m – approximately one-tenth
the size of a typical cell.
To assess the extent of ablation in the zdirection, i.e. through the cell layer, we
imaged the one-cell-thick epithelium in
cross-section (Fig 1GH). The laser
wounds cut cleanly through the
epithelium (~6 m thick in stage 13), so
that the apical and basal surfaces both
move freely. As the cells recoil, they do
not shear. There is some apparent shear
in the first image after wounding (second
frame from top in Fig 1G), but this is just
a motion artifact. The apparent shear is
reversed when the z-scans are collected
from bottom-to-top (second frame from
bottom in Fig 1H).
FIGURE 1. Mechanical response of an embryonic epithelium to single-pulse laser ablation. The
epithelium was the amnioserosa in stage 13 fruit fly embryos expressing GFP-cadherin. (A-C) are a series
of confocal images taken before, 10 s and 20 s after ablation. The post-ablation images are overlays
comparing the cell border positions before (magenta) and after ablation (green). As denoted by the
crosshairs in A, the laser was targeted to the border between two cells. (D-F) are a similar series in which
the laser targets a single cell’s apical surface. For each image, anterior is to the right. (G-H) are each a
temporal series of four cross-sectional images through the amnioserosa (dorsal up). Cell borders appear as
nearly vertical bright lines. The crosshairs mark the location and time of ablation (t = 0 s). The crosssections in (G) were collected dorsal-to-ventral, so time increases as one goes down a single image and
from one image to the next. The cross-sections in (H) were collected ventral-to-dorsal, so time increases
as one goes up a single image.
4
Thus, in living fly embryos, we can drill a hole through the epithelium about 1 m in diameter
and 5-7 m deep. Such subcellular ablation provides very interesting clues as to how stress is
distributed. Most importantly, strong recoils occur regardless of whether the wound is targeted to
a cell edge (Fig 1A-C) or cell center (Fig 1D-F). Apparently – and in contrast to common
assumptions [4, 10, 25] – tension is not limited to the actin-rich apical belts around each cell.
Further evidence that tension is not limited to cell-cell borders comes from the pattern of local
recoil displacements. In general, the local pattern is radial – with some interesting anisotropy (see
below) – but with little dependence on the exact positions of the local cell edges. This is
particularly striking in the cell-edge wound of Fig 1C. The displacements of the triple junctions at
the ends of the wounded edge are not exceptionally large.
3.1 Spatial dependence of the relaxed displacements.
We next compare the overall pattern of recoil displacements and relaxed strains to those
generated by drilling a through-hole in a homogeneous, isotropic and linearly elastic thin sheet.
This special case underlies the standard engineering method for determining residual stress [18].
Under conditions of biaxial plane stress (x, y), the radial relaxed strain is given by [26, 27]
Eqn 3
r (r) = A(r)( x + y ) + B(r)( x y )cos2
in polar coordinates centered on the hole with measured from the direction of principal stress.
The dependence of the relaxed strain on distance from the hole is contained in the two
coefficients (for r R0):
A(r) =
2
(1+ ) R0 2E r and
B(r) =
2
R 4 (1+ ) 4 R0 3 0 r
2E 1+ r Eqn 4ab
where R0 is the radius of the hole, is Poisson’s ratio and E is Young’s modulus. The expected
displacements follow from the above with the addition of a possible rigid body translation (of
magnitude utr and direction tr):
ur (r) = A(r)( x + y ) + B(r)( x y )cos2 + utr cos( tr )
Eqn 5
where
A(r) =
1+ R02
2E r
and
B(r) =
1+ 4 R02 R04 3
2E 1+ r
r Eqn 6ab
We assessed the global pattern of relaxed strain in our experiments by finding a B-spline
approximation to the displacement field that would “warp” a post-ablation image to match a preablation image [22]. These displacements at 20 s after ablation are shown as the vector fields in
Fig 2. In any particular direction, the displacements fall off as approximately 1/r. This
dependence holds out to a distance of three cell diameters and implies relaxed strains that fall off
as -1/r2, roughly equivalent to the idealized case above. Beyond three cell diameters, the
displacements fall off faster than expected.
As for the angular dependence, the two experiments shown in Fig 2 clearly have patterns that
match the cos( - tr)-dependence of a rigid body shift to the right (anterior) between ±30º. A
larger sampling of experiments reveals that the matching shifts in these two examples are
coincidental. In general, the shifts do not strongly favor the anterior direction. In any case, when
the rigid body shift is removed by using fitted splines to calculate the local relaxed strain, one
finds a pattern close, but not exactly identical to a cos2-dependence. This dependence is most
closely matched for the cell-edge wound in Fig 2A-B. The polar plot of relaxed strain shows the
expected, nearly elliptical pattern in the first ring of neighboring cells (at r = 15.6 m); however,
the ellipse is not centered, so the angular span between directions of greatest (negative) relaxed
5
strain is not exactly 180º. Nonetheless, the closest match to a cos2-dependence-axis implies a
slight stress anisotropy with the direction of principle stress between 30-60º. Interestingly, this
direction does correspond to the orientation of the ablated cell edge. For the cell-center wound in
Fig 2A-B, the situation is a bit more complicated. At a distance of 15.6 m, the relaxed strain
pattern is again roughly elliptical, but with less anisotropy and a principle stress in the 50-80º
direction. For the same wound, but at the closer distance of 9.4 m (close to the boundaries of the
ablated cell), the relaxed strain actually follows a tri-lobed pattern. For a homogeneous material,
this pattern cannot be explained by any far-field stress distribution. We investigate this anisotropy
in more detail below and find that it correlates with the favored directions of the quasi-hexagonal
arrangement of cells.
FIGURE 2. Displacement and relaxed
strain patterns at 20 s after ablation. (A,C)
are displacement vector fields from Bspline warps (light arrows) or from
tracking specific cellular triple junctions
(bold arrows). The ablated spot is denoted
by (crosshairs) with concentric rings at
distances of 9.4 m (dashed) and 15.6 m
(solid). The scale bar in A applies to both
panels. (B,D) are corresponding polar plots
of the relaxed strains for r = 9.4 m
(dashed) and 15.6 m (solid). The relaxed
strain scale ranges from +0.2 at the center
of the plots to -0.4 at the outer ring. Note
that (A-B) correspond to Fig 1A-C and (CD) correspond to Fig 1D-F.
3.2 Temporal dependence of the relaxed displacements.
One clear problem with the comparison above is the fact that cells are viscoelastic. By 20 s after
ablation, the recoils have slowed dramatically, but they are not in a true static equilibrium. In fact,
expansion of the ablated hole only pauses transiently before wound healing commences. Thus,
recoil patterns on the >10-s time scale may be contaminated by secondary effects, i.e. biologically
regulated changes in the cytoskeleton. More direct information with regard to the epithelium’s
viscoelastic properties and its local stresses is contained in the short-time recoil kinetics.
Our full-frame confocal images are too slow to measure these kinetics (>2 s per frame), so we
limited our measurements to repeated scans along a single line (one scan every 15.7 ms). These
line scans are used to construct a recoil kymograph (Fig 3B). Before ablation, the kymograph has
a series of bright vertical bands. Each corresponds to a nearly immobile cell edge. Immediately
after ablation, the bands fan out as the cells recoil away from the ablation site.
For the cell edges closest to a wound (not counting the wounded edge itself), the recoil
displacements show multiphasic behavior (Fig 3C). During the first two seconds after ablation,
the recoil displacements, x(t), are well fit by a double exponential (Eq 1). The first exponential
generally has a time constant between 40-80 ms and accounts for <1 m of recoil. The second
exponential generally has a time constant between 1-2 s and accounts for 1-3 m of recoil. We
will take a closer look at the fit parameters below, but will first make some model-independent
recoil comparisons.
6
Figure 3. Line-scan kymographs for
measuring the fast recoil after laser ablation.
(A) is a confocal fluorescent image of
amnioserosa cells before ablation. The
targeted cell edge is marked with crosshairs
and the dashed line marks the line that will
be repeatedly scanned. The repeated line
scans are used to build the kymograph (B).
The horizontal axis is position along the
marked line; the vertical axis is time. The cell
borders crossed by the line scan appear as
bright bands that are nearly vertical before
ablation. After ablation (marked in time and
space by the crosshairs), these bands fan out
as the cells recoil. The cell border marked
with an asterisk was tracked to yield the
displacement versus time plot (C). The postablation recoil was fit with the piecewise
function described in Eqn 1 (solid line).
Figure 4. Dependence of the dynamic
recoils on wound location, developmental
stage, and direction. (A-E) are from stage
13 embryos in early dorsal closure; (A-E)
are from stage 14 embryos in late dorsal
closure. (A,A) Confocal fluorescent
images of amnioserosa cells (anterior to the
left). Note the large changes in cell shape
during dorsal closure. (B,B) Average
dynamic recoils for cell-edge wounds
(grey-solid) or cell-center wounds (reddashed). The lightly shaded regions
represent the standard deviations; the
darker regions represent the standard errors
of the means. (C,C) Histograms of cell
edge orientation at each developmental
stage. Angles are defined as shown in A.
(D,D) The initial recoil velocities are
plotted versus the tracked recoil direction
for cell-edge ablation. The tracked
direction was always parallel to the ablated
edge. Each point represents a single
experiment with error bars determined via
confidence intervals for the best-fit
parameters. (E,E) Similar plot of initial
recoil velocities versus tracked direction
for cell-center wounds.
7
These comparisons are designed to assess the sub-cellular distribution of stress in the
amnioserosa of fruit fly embryos. We first measured recoil kymographs in stage 13 embryos
(early dorsal closure as pictured in Fig 4A) following 51 cell-center wounds and 58 cell-edge
wounds. The average recoil kinetics are compared in Fig 4B. At this developmental stage, the
cell-edge wounds recoil to a greater extent (~1.6). One is tempted to attribute this difference to
the fact that a cell-edge wound damages two cells; however, when the same experiment is
conducted in stage 14 embryos (late dorsal closure as pictured in Fig 4A´, about an hour later in
development), we measure no significant difference between cell-edge and cell-center wounds.
As one can see by comparing Fig 4BB´, the major change from stage 13 to 14 is a large reduction
in the recoil extent for cell-edge wounds.
To interpret these results, one must remember that the recoil extent is determined both by the preablation stress at the location of the hole and the post-ablation mechanical properties of the
surrounding epithelium. In our experiments, the hole is a very small fraction of the entire
epithelium, ~10-4. The post-ablation mechanical properties of the surrounding tissue should be
nearly independent of the hole’s subcellular location. This supposition is supported by model fits
to the recoil data (below). Thus, within a single stage, the relative amount of recoil directly
reports on the subcellular stress distribution. Our results suggest an uneven distribution in early
dorsal closure with ~1.6 more stress along cell edges. As closure progresses, this distribution
evolves into a uniform loading. Unfortunately, one cannot say how this evolution takes place.
Stress could decrease along cell edges or increase across cell centers. The ambiguity arises
because the mechanical properties could also change with developmental stage. Without further
information, the best one can do is build scenarios for how the stresses and mechanical properties
coordinately change.
3.3 Initial recoil velocities.
Previous experiments have focused on the initial recoil velocities as probes of cell and tissue
mechanics [11, 28, 29], but the recoils were tracked with non-impulsive ablation and much
slower time resolution. Previous analyses of the amnioserosa used multiple-cell cuts with 10-100
pulses at 10 Hz [11, 28, 29]. Images were taken every 2-8 s and the measured “initial” recoil
velocities were just 1-3 m/s. A different analysis of cells in the fly wing disk did ablate single
cell edges, but did so with 30 pulses at 100 Hz [30]. In this case, images were taken every 5 s and
the “initial” recoil velocities were still less than 1 m/s. Here, we’ve increased the time resolution
by two orders of magnitude and observed a much faster recoil phase with initial velocities 10100 faster.
To assess the initial recoil velocities, we fit the cell edge displacements to a piecewise function
(Eqn 1): linear for t < t0; and a double-exponential for t t0. We found initial recoil velocities, v0,
that ranged from 1.8 to 52 m/s. The mean v0 for each experimental category (cell-edge or cellcenter wounds in stage 13 or 14) is listed in Table I. In early dorsal closure, the recoil velocities
were ~1.5 faster for cell-edge wounds. This difference is statistically significant at p < 0.01 by
three of four tests. The only exception is the KS procedure (for which p = 0.19). In late dorsal
closure, the recoil velocities for the two wound types are not significantly different by any of the
four tests. As with the general mechanical properties, the viscous damping coefficient should be
independent of the wound’s subcellular location. Thus, on balance, the recoil velocities do
suggest that cell-edge and cell-center stresses are different in stage 13, but not in stage 14.
In terms of stage-dependence, it is only cell-center wounds that have a significant change in v0 – a
1.8 increase. Does this mean that the stresses on cell centers increase during dorsal closure? Not
necessarily. The implication would follow if the viscous damping coefficient for cell motion was
stage-independent, but there is no reason to assume such a priori constancy. In fact, if one
8
assumed constancy of all the mechanical properties, then the stage-dependence observed for the
recoil extents and that for the recoil velocities would be in conflict. As discussed further below,
the initial recoil velocities are just one constraint on potential mechanical scenarios.
3.4 Correlations with cellular geometry.
We also looked for correlations between the recoil parameters and several geometric factors.
Most gave negative results. For example, we found no correlation between the initial recoil speed
and any of the following: length of the ablated edge; area of the ablated cell(s); perimeter of the
ablated cell(s); and angles at the triple-junctions adjacent to an ablated edge.
The only geometric factor with a significant correlation to the initial recoil speed was the
direction in which recoil was tracked (with respect to the embryo’s long axis as in Fig 4AA). For
any one ablation, we obtained the initial recoil speed in a single direction; however, a clear
pattern emerges from tracking different recoil directions for many different wounds. This pattern
points to an anisotropic stress. The strongest anisotropy is evident in late dorsal closure (Fig
4DE) where both cell-center and cell-edge wounds recoil faster in the 20-30° and 150-160°
directions. Interestingly, similar peaks are seen in the stage-14 distribution of cell edge
orientations (Fig 4C). Such anisotropy is less evident in earlier, stage 13 embryos. The recoil
speeds for cell-center wounds are essentially isotropic (Fig 4E), and those for cell-edge wounds
have only weakly indicated peaks (Fig 4D). As in stage 14, these peaks roughly align with peaks
in the distribution of cell edge orientations (Fig 4C). Thus, the discrepancies in angulardependence between the measured recoils and those expected for a homogeneous thin sheet are
related to the quasi-hexagonal arrangement of cells.
3.5 Correlations on longer time scales.
As noted above, previous measurements of ablation-induced recoil velocities were conducted on
much longer time scales [11, 28-30]. To compare our results to those, we used the doubleexponential fits to simulate the expected results for slower imaging. We calculated the average
recoil velocity during the first two-seconds after ablation, <v0-2 s>, and found speeds comparable
to those reported elsewhere (~1 m/s), but much lower than the true initial recoil velocity. In
comparisons of <v0-2 s> for cell-edge and cell-center wounds, one finds the same shifts as above: a
significant difference in stage 13 (1.41 ± 0.09 versus 0.88 ± 0.06 m/s); and no difference in
stage 14 (0.91 ± 0.05 versus 0.95 ± 0.08 m/s). On the other hand, the simulated <v0-2 s> have no
particular dependence on cell edge orientation (Supporting Fig S1). Thus, the faster recoil
measurements still provide a much richer analysis.
3.6 Fitting the recoil kinetics with mechanical equivalent circuits.
To build quantitative scenarios for the mechanics of dorsal closure, we compiled the doubleexponential fit parameters (Eqn 1) from all of the individual experiments. In all cases, the fit was
good. As an example, the recoil fits for stage 14 cell-edge wounds had R2 that ranged from 0.906
to 0.995. In most cases, the fit parameters were also reasonable; however, in about 10% of the
fits, one of the time constants and its corresponding coefficient would become unreasonably large
(with an extremely large standard error). This relatively common error occurred when the two
seconds of experimental data were not enough to constrain the second time constant. In these
cases, the second exponential term approximated a straight line with slope dL/L. When
calculating the parameter means and evaluating statistical significance, we excluded such fits
with L > 10 s. The parameter compilations are shown as the mean ± standard error of the mean in
Table I and as probability density estimates in Fig 5. The corresponding conventional histograms
are available as supplemental Fig S2. Note that the fit parameters all have a large degree of
variation – with standard deviations comparable to the means.
9
s (s)
L (s)
0.065 ± 0.009
0.074 ± 0.010
0.038 ± 0.006
0.041 ± 0.006
1.73 ± 0.22
1.81 ± 0.24
1.16 ± 0.07
1.88 ± 0.34
v0 (m/s)
13.4 ± 1.5
20.0 ± 2.0
24.7 ± 2.6
19.4 ± 2.1
1.65 ± 0.18
(0.004 KS)
St14/St13: center wounds
0.67 ± 0.04
(0.050 KS)
1.84 ± 0.19
(0.018 KS)
St14/St13: edge wounds
0.74 ± 0.07
(0.012 KS)
0.69 ± 0.13
(0.025 t2)
0.59 ± 0.09
(0.017 KS)
0.56 ± 0.11
(0.030 KS)
Category
Stage 13: center (51,46)
edge (58,54)
Stage 14: center (33,32)
edge (37,31)
ds(m)
0.55 ± 0.04
0.87 ± 0.07
0.71 ± 0.07
0.64 ± 0.04
dL(m)
1.75 ± 0.13
2.88 ± 0.23
1.45 ± 0.14
2.00 ± 0.35
Edge/center ratio in stage 13
1.60 ± 0.17
(0.001 KS)
Edge/center ratio in stage 14
TABLE I. Summary of recoil parameters using a double exponential fit. The top half of the table reports
each parameter as its mean ± standard error of the mean. For each experimental category, the numbers in
parentheses are the total number of edges tracked and the number used for calculating the means (after
excluding those with L > 10 s). The bottom half of the table reports the parameter ratios between category
pairs. The ratios are only reported when the parameter estimates for the two categories are significantly
different (p 0.05 for all four statistical tests). For each reported ratio, the numbers in parentheses are the
highest p-value and the test for which this p-value occurred (t2 = Welch’s modified t-test with
independent variance estimates for the two groups; KS = the Kolmogorov-Smirnov two-sample
procedure).
The distributions of these parameters enable further comparisons of cell-center versus cell-edge
wounds and early versus late dorsal closure (stage 13 vs. 14). Such comparisons show quite
different behavior for the two coefficients (ds and dL) and the two time constants (s and L). In
stage 13, the coefficients depend on wound location – larger for cell-edge wounds. In stage 14,
this dependence disappears. The only coefficients that stand out as different are for cell-edge
wounds in early dorsal closure. Such behavior is identical to that noted in the model-independent
comparison. On the other hand, the
time constants are independent of
wound location, and yet stagedependent. The shorter time constant,
s, clearly gets smaller as closure
progresses. The longer time constant,
L, behaves more erratically. It clearly
decreases for cell-center wounds, but
not for cell-edge wounds. It also
appears to depend on wound-location
in stage 14; however, this difference
does not meet statistical significance
by all four tests (p <0.025 for the t1,
Figure 5. Probability density estimates
for the cellular recoil parameters. The
legend in B applies to all panels:
center-wounds (red); edge-wounds
(black); stage 13 (solid); stage 14
(dashed). Each panel has distributions
for a single primary or derived
parameter using the noted minimum
Gaussian width (wmin).
10
t2 and W tests, but p = 0.19 for the KS test). As noted above, L is generally the least constrained
parameter and we had to make an ad hoc decision on which fits to include in our subsequent
analysis. If all the fits were included, no L comparisons would meet the requirement for
statistical significance.
To interpret the stage and location dependence of these parameters, we note that a doubleexponential recoil is equivalent to a mechanical circuit with two Kelvin-Voigt elements in series
(a Kelvin-Voigt element being a spring with modulus G and dashpot with viscosity in parallel).
The fit parameters are related to the properties of the mechanical equivalent circuit by:
/Gs,L = s,L ds,L /r0
Eqn 7
s,L /Gs,L = s,L
Eqn 8
where is the stress removed by hole drilling, s,L is a relaxed strain coefficient and r0 is the
initial distance from the hole to the tracked edge. For completeness, we can also relate the
mechanical equivalent properties to derived quantities like the initial strain rate, 0, and initial
recoil velocity, v0.
Eqn 9
/s + /L = /eff = 0 v 0 /r0
In Eqn 7 and 9, we approximate the strains from the displacements. When the displacements fall
off as u 1/r , then the relaxed strains are = u /r . This approximation makes almost no
difference for intra-stage comparisons, but impacts inter-stage comparisons because the mean r0
gets smaller in late dorsal closure. The mean values of r0 for each stage and wound location are
compiled in Table II.
The relationship of the time constants to mechanical equivalent properties (Eqn 8) yields two
important results. First, the fact that the time constants are independent of the wound location
supports the supposition that the mechanical properties of the surrounding tissue are also
independent of the wound location. This in turn strengthens the conclusion that stress is
somewhat concentrated along cell edges in early dorsal closure, but more uniformly distributed in
late dorsal closure. Second, the fact that the time constants decrease in late dorsal closure implies
that the amnioserosa becomes more solid-like – i.e. more stiff or less viscous or some
combination of the two.
We can also use the fit parameters and Eqn 7-9 to derive allowable scenarios for dorsal closure in
terms of ratios of ratios of mechanical properties. For any general mechanical property q, we
define Rq = qSt14 /qSt13 . The ratios of such ratios are compiled in Table II. For example, RC/RE =
1.60 ± 0.39 details how cell-edge and cell-center stresses are coupled. We cannot say which stress
individually increases or decreases, but we can describe scenarios that are consistent with our
experimental results.
TABLE II. Ratios of ratios summarizing how the stresses and mechanical properties of amnioserosa
cells coordinately change during dorsal closure. The first column reports the mean distances from the
ablation site to the tracked edges.
<r13>: center
edge
<r14>: center
edge
7.55 ± 1.60
7.64 ± 1.59
5.24 ± 1.35
5.82 ± 1.35
Relative Changes (Stage 13 vs 14) in Stresses and Mechanical Equivalents
RC / RE 1.60 ± 0.39
Rs / RE 0.52 ± 0.11
RGs / R E 0.91 ± 0.10
RL / R E 1.09 ± 0.39
RGL / R E 1.27 ± 0.11
Reff / RE 0.66 ± 0.10
11
A scenario is fully described by specifying how any single property changes with developmental
stage. As just one example, consider the implications for no change in the viscosity s. Under this
scenario, the stresses along cell edges and cell centers would both increase (by 1.9 and 3.1
respectively) and the tissue would become stiffer (by 1.7 and 2.4 for Gs and GL). The range of
consistent scenarios places serious constraints on models for dorsal closure (discussed below), but
we cannot tighten the constraints to a single scenario without independent estimates of one or
more mechanical properties.
3.7 Comparison to previous microrheology.
The experiments above have a great deal in common with cell microrheology [31-34]. In those
experiments, a pulse of constant stress is applied to a cell or cell-attached bead, and the
subsequent time-dependent strain (t) is used to measure a creep compliance, J(t) = (t)/ , which
characterizes a cell’s time-dependent mechanical properties. For small stresses, J(t) is
independent of stress. In our experiments, a laser pulse is used to drill a 1-m hole in a cell sheet
– a very localized removal of stress. This removal is equivalent to the sudden application of an
extra local stress of opposite sign, so the measured recoil displacement/strain is related to the
creep function, but multiplied by the removed stress, i.e. un-normalized.
To be more specific about the relationship, we can start with the homogeneous elastic solution for
the relaxed strains after hole-drilling (Eqn 3, 4). This elastic solution can be converted to a
linearly viscoelastic solution by applying the elastic-viscoelastic correspondence principle [35].
For impulsive removal of stress at the hole and sufficient damping to ignore inertial terms, the
viscoelastic solution takes the exact same form as the elastic solution, but with the inverse
modulus 1/E replaced by the creep function J(t). Note that in both microrheology and laser holedrilling, the creep function is only valid out to a few seconds because the stresses do not remain
constant and because cells are an active medium; their mechanical properties certainly change at
longer times (e.g. the wound healing response).
In some cases, the creep functions measured via microrheology have been described with
equivalent mechanical circuits [31, 32]. Although different mechanical circuits were used, we can
compare our measured parameters with this previous microrheology to generate order-ofmagnitude estimate for forces and stresses in the amnioserosa. The creep functions measured with
microrheology have an initial exponential transient followed by a region of constant slope. In
[31], the mean time constant of the initial transient was 90 ms – just slightly larger than the
shorter time constant measured here. Thus, although we don’t have a region with truly constant
slope, we can draw a rough equivalence between the constant slope region and the initial slope of
our second exponential, i.e. dL/L. In our experiments, the mean value of dL/L for each category
ranges from 1-2 m/s. The constant slope region from [31] corresponds to an effective viscosity
of 0.031 Pa·s·m. When combined with our parameters, this yields estimated forces of 30-60 nN at
the perimeter of the ablated cell(s). Using typical amnioserosa cell sizes (~5 m thick with
perimeters of ~80 m), these forces correspond to stresses in the 10-100 Pa range. These are each
reasonable values, but we really need complementary microrheology data from the amnioserosa
in living embryos. This is a technical problem that we have not yet solved.
In many other cases, the measured creep functions are not describable via mechanical equivalent
circuits and instead display power-law behavior [33, 34]. The time-dependent strains measured
here can be fit quite well with a power-law from ~100 ms to 2 s with exponents in the range of
0.2-0.4. We are currently expanding the temporal range of our measurements to see if this powerlaw behavior holds over more than one decade.
3.8 Implications for models of epithelia in general.
Certainly, the most surprising finding here is the subcellular stress distribution within an
embryonic epithelium. Like many epithelia, the amnioserosa cells at this stage of development
12
have their actin cytoskeletons organized cortically, in so-called circumferential microfilament
bundles. This organization is ubiquitously assumed to yield concentrated tensile forces along cellcell interfaces. Whether this assumption is stated explicitly [25] or not [10], it is often used to
estimate relative cortical tensions based on the angles at cell triple-junctions. Our results suggest
that this assumption is questionable and triple-junction analysis should be viewed very cautiously.
Unfortunately, it is not clear what structure(s) carry the cell-center stress. Under the conditions
used here, laser ablation cuts clean through the one-cell thick epithelium and overlying vitelline
membrane. We have performed additional experiments in which only the vitelline is cut. If the
ablated hole is in a region surrounded by glue and coverslip, as in all the experiments here, no
material can flow in or out and the underlying cells do not move. On the other hand, if the hole is
drilled in a region not covered with glue, cells are pushed out through the hole, which suggests
that the epithelial layers are under tension to constrain inflation pressure from the underlying
tissues (e.g. the yolk sac). Nonetheless, the cell-center stress could be carried by structures
anywhere from the apical surface to the basal surface and its loosely organized extracellular
matrix. At times, one can see transient accumulations of actin under the apical surfaces of
amnioserosa cells in GFP-moe labeled embryos [10]. These actin-rich apical regions are
accompanied by myosin accumulations, especially in those cells that dive out of the epithelial
plane early [29, 36]. We do not yet know if these accumulations correlate with the cell-center
stresses, but this question deserves further investigation.
Another deserving question is the source of the anisotropic recoil velocities. The directions with
higher recoil velocities imply directions of higher stress. Even more intriguing, the cell borders
preferentially align in these directions. Previous cell-based constitutive models for epithelia have
noted a correspondence between the overall direction of principle stress and the average
alignment of cells [37], but our finding goes a step beyond this relationship. We see a local effect
with two or three peaks in the angular pattern of recoil velocity that correspond to the underlying
quasi-hexagonal arrangement of cells. Is this anisotropy imposed on the amnioserosa via its
boundary conditions or generated by individual amnioserosa cells? Other embryonic epithelia are
known to accumulate non-muscle myosin II along preferentially aligned cell borders [38, 39] – an
accumulation regulated by the planar cell polarity (PCP) pathway. Although mutations in PCP
genes can disrupt dorsal closure [40], their exact role is not clear and there have been no reports
of anisotropic myosin accumulations in the amnioserosa.
3.9 Implications for apical constriction models.
All previously reported models for apical constriction either consider elements that represent
multiple cells [2, 3], or only consider 2D cross-sections of cells [1]. Thus, the comparison of cellcenter versus cell-edge wounds is not really applicable. On the other hand, we can compare our
observations to model predictions for stage and cell-shape dependencies. As a first example, the
elongation-contraction model predicts a bimodal dependence of stiffness and stress on the size of
cells [1]. Within the range of scenarios consistent with our results, it is possible for amnioserosa
cells to become stiffer and carry more stress as they shrink apically. On the other hand, within a
single developmental stage, the recoil parameters have no apparent dependence on cell size. As a
second example, finite-element (FE) models of sea urchin invagination posit an initial strain that
drives constriction as it relaxes [3] – i.e. a constant elastic modulus, but temporally decreasing
stress. Such models are not consistent with our experiments. If we invoke a scenario for which Gs
is constant, then the stresses on cell edges and centers must increase during closure by 1.1 and
1.8. If GL is constant, then the stress on cell edges does decrease (by 0.8), but the stress on cell
centers still increases by 1.3. In addition, other sea urchin FE models consider mechanisms that
produce constriction through compressive buckling. The buckling mechanisms are inconsistent
with our observation that the constricting tissue is clearly under tension. This discrepancy might
be alleviated if the models explicitly added an inflation pressure from the enclosed yolk. As a
13
third and final example, FE models of neurulation drive apical constriction by positing a locally
higher interfacial tension [2] – which yields more realistic cell shapes if the tension increases with
time. This is possible with our scenarios, but implies simultaneous increases in the cell-center
stresses and tissue stiffness. Unfortunately, the time dependence of these parameters was not
reported. Note that each of these extant models produced simulated cell shapes that matched
observations, but our results offer more stringent mechanical tests. We hope that next generation
models will include explicit predictions for hole-drilling experiments.
4. Conclusions and Outlook
We have used laser hole-drilling to evaluate the mechanics of apical constriction during dorsal
closure. Substantial in-plane tension is carried across each cell in this embryonic epithelium and
not just along cell-cell interfaces. The two-dimensional patterns of recoil displacements are
similar to what one would expect for a homogenous thin sheet, but with some very interesting
discrepancies. In early dorsal closure, the tension is mainly along cell-cell interfaces, but is later
distributed more uniformly throughout each cell. Despite this redistribution in late dorsal closure,
the stress becomes remarkably anisotropic – with preferred directions of greater stress that match
preferred cell edge orientations. The experiments reported here cannot provide a unique
description of how this redistribution occurs, but they do place quantitative constraints on how
the stresses coordinately change with the epithelium’s mechanical properties. Together, these
results show that laser hole-drilling can be a quantitative tool for assessing embryonic tissue
mechanics in vivo.
Acknowledgements
This work supported by the National Science Foundation (IOB-0545679) and the Human Frontier
Science Program (RGP0021/2007C). The authors would like to thank G.W. Brodland, of U.
Waterloo for many helpful discussions.
Glossary
amnioserosa – a one-cell thick embryonic epithelium that covers most of the dorsal surface of
fruit fly (Drosophila) embryos in the latter half of embryogenesis
apical constriction – type of morphogenetic event in which a subset of cells in an epithelium
contract their apical surfaces (the ones facing the medium or lumen) to adopt a wedge-shaped
morphology which leads to a local bending or invagination of the tissue
laser hole-drilling – process in which a tightly focused, high peak-power laser pulse is targeted
onto a surface to locally destroy the surface’s mechanical integrity; typically occurs via local
plasma generation, subsequent confined boiling, bubble expansion and collapse
relaxed strain – the change in strain after mechanical modification of a surface; typically after
cutting or hole-drilling
relaxed displacement – field of displacements that occurs as strain relaxes after mechanical
modification of a surface
creep compliance – function that characterizes the entire time-dependent strain response of a
material to a unit step change in stress
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SUPPLEMENTARY INFORMATION
Figure S1. Dependence of the dynamic recoils
on direction as measured by the average recoil
velocity during the first two seconds after
ablation <v0-2 s>. (A-D) are from stage 13
embryos in early dorsal closure; (A-D) are
from stage 14 embryos in late dorsal closure.
(A,A) Confocal fluorescent images of
amnioserosa cells (anterior to the left). (B,B)
Histograms of cell edge orientation at each
developmental stage. Angles are defined as
shown in A. (C,C) <v0-2 s> is plotted versus the
tracked recoil direction for cell-edge ablation.
The tracked direction was always parallel to the
ablated edge. Each point represents a single
experiment with error bars determined via
confidence intervals for the best-fit parameters.
(E,E) Similar plot of <v0-2 s> versus tracked
direction for cell-center wounds.
Figure S2. Histograms of the cellular recoil
parameters. The legend in B applies to all
panels: center-wounds (red); edge-wounds
(grey). Each panel has distributions for a single
primary or derived parameter. (A-F) Unprimed
panels are for stage 13 embryos; (A-F) primed
panels are for stage 14.
17