# The physical basis of self-organization of the mammalian oocyte spindle

Colm P. Kelleher  Department of Molecular and Cellular Biology, Harvard
University, Cambridge, MA 02138  Daniel J. Needleman  Department of Molecular
and Cellular Biology, Harvard University, Cambridge, MA 02138  John A. Paulson
School of Engineering and Applied Sciences, Harvard University, Cambridge, MA
02138  Center for Computational Biology, Flatiron Institute, New York, NY
10010

###### Abstract

To prepare gametes with the appropriate number of chromosomes, mammalian
oocytes undergo two sequential cell divisions. During each division, a large,
long-lived, microtubule-based organelle called the meiotic spindle assembles
around condensed chromosomes. Although meiotic spindles have been intensively
studied for several decades, as force-generating mechanical objects, they
remain very poorly understood. In materials physics, coarse-grained theories
have been essential in understanding the large-scale behavior of systems
composed of many interacting particles. It is unclear, however, if this
approach can succeed in capturing the properties of active, biochemically
complex, living materials like the spindle. Here, we show that a class of
models based on nematic liquid crystal theory can describe important aspects
of the organelle-scale structure and dynamics of spindles in living mouse
oocytes. Using our models to interpret quantitative polarization microscopy
data, we measure for the first time material properties relating to stress
propagation in living oocytes, including the nematic diffusivities
corresponding to splay and bend deformations. Unlike the reconstituted
amphibian spindles that were previously studied in vitro, nematic elastic
stress is exponentially screened in the microtubule network of living
mammalian oocytes, with a screening length of order $1\,\mu\text{m}$. This
observation can be explained by the relatively high volume fraction of
embedded chromosomes in mammalian meiotic spindles, which cause long voids in
the microtubule network and so disrupt orientational stress propagation.

##  1 Introduction

In materials physics, coarse-grained theories have been essential in bridging
the conceptual gap between the properties of molecules and the mechanics of
large objects. Living materials challenge this paradigm: they are typically
composed of hundreds of chemically distinct species (proteins and other
biomolecules) whose interactions may be tightly regulated in space and time;
this makes it difficult to identify a tractable number of coarse-grained
variables and material parameters. They are typically far from local
thermodynamic equilibrium, and so lack the microscopic time-reversal symmetry
that, for passive materials, greatly constrains the set of allowed macroscopic
behaviors [29]. These conceptual difficulties are compounded by the
experimental challenge of measuring material properties in vivo. For example,
one of the best-studied structures in cell biology is the spindle, the self-
assembled organelle that separates chromosomes during cell division. The
spindle is composed primarily of microtubules (MTs) – long, rigid protein
polymers – as well as a variety of other proteins. In mammalian cells,
spindles typically have linear dimension $\sim$1-10 $\,\mu\text{m}$, and move
chromosomes over comparable distances. Extensive efforts over several decades
have elucidated the spindle’s molecular composition; it is believed we now
have a complete, or nearly complete, “parts list” for the mammalian spindle
[10, 24, 39, 49]. More recently, coarse-grained models have been shown to
explain many important aspects of the organelle-scale properties of amphibian
meiotic spindles reconstituted in vitro, for example their shapes and the
mechanisms whereby stresses propagate in the MT network [43, 8, 34, 11]. Up to
now, however, it has been unclear whether this approach will succeed in the
more complex environment of a living cell.

Here, we use quantitative polarization microscopy to study the structure and
dynamics of meiotic spindles in living mouse oocytes. As they prepare for
fertilization, mouse oocytes undergo nuclear envelope breakdown (which is
often called “germinal vesicle breakdown” or GVBD in this context), and then
two sequential, highly asymmetrical cell divisions. Each division requires the
formation of a relatively large spindle, 10-20 $\,\mu\text{m}$ in linear
dimension, called the metaphase I and metaphase II (MI and MII) spindles.
Whereas mitotic spindles typically remain in metaphase steady state for at
most tens of minutes, meiotic spindles in mammalian oocytes naturally remain
in metaphase for several hours. We take advantage of the long lifetimes of
these spindles to collect a large amount of data of fluctuations of the MT
network around a well-defined steady state. Using this data, we show that
coarse-grained physical models can describe the structure and dynamics of
spindles in living oocytes, and that important organelle-scale material
properties of these spindles, for example nematic diffusivities, can be non-
invasively measured.

##  2 Steady-State MT Orientation in Oocyte Spindles Follows Circle Arcs

We begin by characterizing the geometry of the MT network in steady-state MI
and MII spindles. To do this, we harvest immature (GV-stage) oocytes in the
presence of a drug cocktail designed to inhibit maturation (Methods 1). To
visualize MI and MII spindles various points through maturation, we wash out
the inhibitor cocktail, place the oocytes in a glass-bottom Petri dish, and
image them using LC-PolScope. PolScope is a non-invasive, label-free
polarization microscopy technique that measures both the optical retardance
$r$ and optical slow axis $\theta$ at each pixel of a 2D image [28]. For
structures composed of aligned microtubules, oriented approximately
perpendicular to the optical axis $\mathbf{\hat{o}}$, retardance is
proportional to the density of MTs integrated over $\mathbf{\hat{o}}$, and the
slow axis angle provides the MT orientation averaged over $\mathbf{\hat{o}}$
(Supplemental Information, SI 1). A typical maturation time course shows the
formation of fully assembled MI and MII spindles (Fig. 1(a)). When the spindle
axis $\mathbf{\hat{x}}$ is perpendicular to the optical axis, MTs in the
central region of the spindle are approximately horizontal (Fig. 1(b)). Over
the course of several hours, both MI and MII spindles often rotate through
large angles, and/or translate through tens of microns, inside the roughly
spherical oocyte (Movie 1). Spindle rotation complicates interpretation of
PolScope data, since in that case the constant of proportionality between
observed retardance and spindle retardance changes as a function of
$\mathbf{\hat{x}}$. To avoid this complication, we analyze PolScope movies of
only those spindles whose long axes remain approximately in the image plane,
$\mathbf{\hat{x}}\cdot\mathbf{\hat{o}}\approx 0$, for $\sim$30 mins. For these
spindles, we define a spindle-referenced coordinate system in which
$\mathbf{\hat{z}}=\mathbf{\hat{o}}$,
$\mathbf{\hat{y}}=\mathbf{\hat{z}}\times\mathbf{\hat{x}}$, and the origin is
at the center of the spindle (Fig. 1(b-d)). We perform all subsequent analysis
in the spindle frame of reference (Methods 2).

To characterize the shape and internal structure of oocyte spindles at
different times during maturation, we calculate their aspect ratios, as well
as the cross-sectional density and orientation of MTs in the spindle
interiors. To measure the aspect ratio, we find the spindle boundary via a
retardance threshold (Fig. 1(c&d)), and measure the belt radius $R_{0}$ and
half-length $x_{0}$ (Fig. 1(e), left). Consistent with previous observations,
we see that both MI and MII spindles lack the focused poles seen in most
metazoan mitotic spindles, and instead are barrel-shaped, with flattened poles
[23]. To quantify cross-sectional MT density, we calculate the average
retardance in a $2\,\mu\text{m}\times 2\,\mu\text{m}$ square in the center of
the spindle. Since the contribution of a single horizontal MT to the total
retardance can be estimated we can use a combination of the measured
retardance and spindle radius to estimate the average MT cross-sectional
density $\langle\rho\rangle$ in the spindle interior (Methods 3). We ignore
for now the presence of long-lasting spatial inhomogeneities in MT density,
evident in our retardance images as horizontal stripes $\sim 1-2\,\mu\text{m}$
in width and comparable in length to the spindle; we will return to these
these features in Sections 7 and 8. To characterize MT orientation, we
calculate the time-averaged orientation field inside the spindle boundary
(Methods 4 & 5). We fit this measured field to the predictions of a model we
call the circle arcs model. In this model, an orientation is defined
everywhere in space as the tangent angle to a family of circle arcs
intersecting at points $(\pm L,0)$ that lie along the spindle axis (Methods
6). This model has one adjustable parameter, $L$; given a measured field of
angles defined in some region of space, the best-fit value of $L$
characterizes the curvature of the MT orientation field in that region (Fig.
1(e)). For all spindles we have imaged, the points $(\pm L,0)$ are outside the
spindle boundary, so we call these features “virtual defects.” We observe
that, for both MI and MII spindles, both the aspect ratio $x_{0}/R_{0}$ and
virtual defect half-spacing $L$ reach roughly constant values $\lesssim$ 2 hrs
after spindle formation. Cross-sectional density, meanwhile, builds up 1-4 hrs
after GVBD for MI spindles, while MII spindles reach a steady-state density
more quickly than we can measure using our data, $<2$ hrs after completion of
anaphase I (Fig. 1(f)).

To further characterize the steady-state MT orientation field, we compare the
average orientation field to the predictions of the circle arcs model. To do
this, we first find the best-fit position of the virtual defects for each
spindle individually (Fig. 2). We then rescale coordinates of that spindle by
the defect half-spacing, $\mathbf{r}\rightarrow\mathbf{r}/L$, and average the
orientation of different spindles at corresponding values of $\mathbf{r}/L$.
For both MI and MII spindles, this procedure reveals close agreement between
our data and this simple, one-parameter model (Fig. 2). Remarkably, this
pattern was theoretically predicted within the framework of equilibrium
nematic liquid crystal theory [47, 35]. Since that time, it has been observed
in several experiments and simulations of nematic materials made up of wide
variety of molecular constituents, although all previous observations are
presumably at or near thermodynamic equilibrium [31, 19, 40, 25].

##  3 Orientational Fluctuations Reveal Anisotropic Material Properties

To study the forces controlling MT orientation in our spindles, we take
advantage of the fact that the orientation of MTs is not static. Rather, the
director field $\mathbf{\hat{n}}$ that represents MT orientation fluctuates
about its average value with a typical angular amplitude $\sim 0.1$ rad.
Moreover, similar fluctuations are correlated in both space and time (Fig.
3(a); Movie 2). These correlations are intimately related to the spindle’s
material properties. For amphibian spindles reconstituted in vitro, these
correlations were used to test theories for the mechanism(s) of orientational
stress propagation in the MT network of that system [8].

To quantify the patterns we observe and compare them to the predictions of
specific physical models, we calculate correlation functions. In regions near
the middle of the spindle (e.g. the white dashed box inset in Fig. 3(a)), the
time-averaged director field $\mathbf{\hat{n}_{0}}(\mathbf{r})$ is
approximately horizontal, $\mathbf{\hat{n}_{0}}\approx\mathbf{\hat{x}}$. In
that case, fluctuations take a particularly simple form, since
$\delta\mathbf{n}(\mathbf{r},t)=\mathbf{\hat{n}}(\mathbf{r},t)-\mathbf{\hat{n}_{0}}$
is perpendicular to $\mathbf{\hat{x}}$ (Fig. 3(b)). In our analysis of (2+1)D
PolScope movies, therefore, we only need to analyze the
$\mathbf{\hat{y}}$-component of the fluctuations, $\delta
n_{y}=\delta(\sin{\theta})\approx\delta\theta$, which we may treat as a
scalar. We use our PolScope movies to calculate the full (2+1)-D
spatiotemporal correlation function in Fourier space, as well as the
corresponding 2D equal-time correlation functions,

$C_{\text{nn}}(\omega,\mathbf{q})\equiv\delta\tilde{n}_{y}(\omega,\mathbf{q})^{*}\delta\tilde{n}_{y}(\omega,\mathbf{q});\qquad
S_{\text{nn}}(\mathbf{q})\equiv\langle\delta\tilde{n}_{y}(\mathbf{q})^{*}\delta\tilde{n}_{y}(\mathbf{q})\rangle,$
(1)

where $\delta\tilde{n}_{y}(\omega,\mathbf{q})$ is the Fourier transform of
$\delta n_{y}(\mathbf{r},t)$ and the angle brackets in the last equation
indicate a time average (Methods 7). While the true director field of the
spindle has three spatial dimensions, the projection over $\mathbf{\hat{z}}$
implicit in the PolScope measurement is equivalent to taking the
$q_{z}\rightarrow 0$ limit in Fourier space (SI 2). All experimental
correlation functions should be read in this limit.

To compare the correlation functions we measure in oocyte spindles with those
of the in vitro  spindles previously studied, we plot
$C_{\text{nn}}(\omega,\mathbf{q})$ and $S_{\text{nn}}(\mathbf{q})$ in several
long-wavelength limits (Fig. 3(c)). The spatiotemporal correlation function
$C_{\text{nn}}(\omega,\mathbf{q}\rightarrow 0)$ decays approximately as
$\omega^{-2}$, which is consistent with previous observations in vitro. In the
limit $q_{i}\rightarrow 0$, $S_{\text{nn}}(\mathbf{q})$ decays with increasing
$q_{j}$, where $i,j\in\\{x,y\\}$ and $i\neq j$. Qualitatively, these behaviors
are similar to those of in vitro spindles, and indicate that nearby MTs tend
to fluctuate in the same direction, while MTs that are farther apart are less
well coordinated. This is consistent with a picture where MTs are aligned by
local interactions, for instance those mediated via molecular motors, passive
cross-linkers, or steric repulsion. Quantitatively, however,
$S_{\text{nn}}(\mathbf{q})$ exhibits significant deviations from the $q^{-2}$
functional form that is observed in vitro and predicted by the simplest models
of nematic elasticity [16].

We can also use our measurements to characterize the mechanical anisotropy of
the MT network. In the equal-time correlation function
$S_{\text{nn}}(q_{x},q_{y}\rightarrow 0)$, for example, we observe a distinct
plateau around $q_{x}\approx 3\;\text{rad}\,\mu\text{m}^{-1}$ which is not
visible in $S_{\text{nn}}(q_{x}\rightarrow 0,q_{y})$ (Fig. 3(c&d)). While the
MT length distribution has not been measured in mammalian oocyte spindles, MTs
in other metazoan spindles have a typical length of around $2\,\mu\text{m}$
[50, 22]. This plateau we observe occurs at the $q_{x}$-value corresponding to
this length scale,

$q^{\text{MT}}\approx\frac{2\pi}{2\,\mu\text{m}}\approx
3\;\text{rad}\,\mu\text{m}^{-1}$

and so may indicate a breakdown of continuum behavior in our system. We also
observe anisotropy at the longest wavelengths we can probe: for values of
$q_{y}$ roughly corresponding to more than $1\,\mu\text{m}$, correlations in
the perpendicular direction decay more slowly than correlations in the
parallel direction. This implies that, at long wavelengths, coupling between
fluctuations along the direction perpendicular to the spindle axis is weaker
than coupling in the direction parallel.

##  4 Orientational Fluctuations Are Described By Screened Nematic Elasticity

By extending the physical models previously developed for in vitro spindles,
we can construct an empirical equation of motion that quantitively captures
the deviations from nematic-elastic behavior that we observe In those models,
spindles were described as active nematics [8, 34]. Quite generally, the
director field of an active nematic obeys the equation of motion

$\frac{\partial\hat{n}_{i}}{\partial
t}=v_{j}\nabla_{j}\hat{n}_{i}+M_{ij}\hat{n}_{j}-\frac{1}{\gamma}\frac{\delta\mathcal{F}_{\text{el}}}{\delta\hat{n}_{i}}+S_{i}.$
(2)

Here, the terms containing $\mathbf{v}$ and $M_{ij}$ represent flow-related
phenomena: $\mathbf{v}$ is the velocity field of the fluid surrounding the
MTs, and enters directly into the advection term $v_{j}\nabla_{j}\hat{n}_{i}$.
The matrix $M_{ij}$ is linear in derivatives of $\mathbf{v}$; the
corresponding term represents hydrodynamic torque on MTs in a flow gradient
(so-called “flow alignment” or“flow tumbling”) [13]. The functional derivative
$\frac{\delta\mathcal{F}_{\text{el}}}{\delta\hat{n}_{i}}$ encodes passive
stresses arising from from nematic elasticity, with the Frank free energy

$\mathcal{F}_{\text{el}}=\frac{1}{2}\int
d^{3}\mathbf{r}\Big{\\{}k_{1}(\nabla\cdot\mathbf{\hat{n}})^{2}+k_{2}(\mathbf{\hat{n}}\cdot\nabla\times\mathbf{\hat{n}})^{2}+k_{3}(\mathbf{\hat{n}}\times\nabla\times\mathbf{\hat{n}})^{2}\Big{\\}},$
(3)

where $k_{1}$, $k_{2}$, and $k_{3}$ are splay, twist and bend elastic
constants. The coefficient $\gamma$ is a rotational viscosity, while the last
term $S_{i}$ is a Gaussian noise term included as a source for random
fluctuations. In Eqn. (2), active stresses enter in two different ways:
indirectly, via a self-generated cytoplasmic flow field
$\mathbf{v}(\mathbf{r},t)$, and directly, via the noise term, which in active
materials like spindles is typically not of thermal origin [8]. For in vitro
spindles, it has been suggested that nematic elastic stresses dominate
stresses from fluid flow, and the equation of motion of fluctuations is Eqn.
(2) with $\mathbf{v}=0$ (and hence $M=0$) [8],

$\frac{\partial\delta n_{y}}{\partial
t}=-\frac{1}{\gamma}\frac{\delta\mathcal{F}_{\text{el}}}{\delta\delta
n_{y}}+S_{y}.$ (4)

Far from system boundaries, this equation predicts that
$S_{\text{nn}}(\mathbf{q})\sim q_{j}^{-2}$ in the limit $q_{i}\rightarrow 0$
(SI 2). This describes fluctuations in in vitro spindles very well, but is
inconsistent with our data from spindles in living oocytes (Section 3).

To empirically model the observed suppression of long-wavelength fluctuations,
we modify Eqn. (4) by adding a“screening” term,

$\frac{\partial\delta n_{y}}{\partial t}=-\frac{k}{\gamma\ell_{s}^{2}}\delta
n_{y}-\frac{1}{\gamma}\frac{\delta\mathcal{F}_{\text{el}}}{\delta\delta
n_{y}}+S_{y},$ (5)

where $k=(k_{1}+k_{3})/2$ is the average elastic constant and $\ell_{s}$ is an
extra fit parameter which we call the nematic screening length.111In general,
there are screening lengths corresponding to each elastic constant [16]; for
simplicity, we neglect this complication. In real space, the screening term
leads to exponential suppression of correlations over a characteristic
distance $\ell_{s}$ [16]. Fitting our data to the predictions of this model,
we find good quantitative agreement with all correlation functions for both MI
and MII spindles (Fig. 3(d-f)). (With the following exception: in these fits,
and for the remainder of this paper, we neglect the part of
$S_{\text{nn}}(q_{x},q_{y}\rightarrow 0)$ corresponding to $q_{x}>q_{x}^{MT}$,
since that part of the curve apparently arises from single-MT behavior.)

##  5 The Origin of Screening

The screening term on the right-hand side of Eqn. (5) can arise from several
distinct physical mechanisms. These include alignment in an external field
[16], screening from flow alignment (i.e. the term involving $M$ in Eqn. (2))
(SI 3), and “anchoring screening,” caused by the presence of nearby interfaces
with a well-defined anchoring condition for the nematic [6, 46], analogous to
Faraday screening in electrostatics. While we know of no external field that
could affect MT alignment in the spindle, both flow alignment and anchoring
are plausible candidates for the screening we observe experimentally.

To investigate the physical origin of screening in our spindles, we incubate
oocytes in several spindle-perturbing drugs at doses low enough to preserve
spindle bipolarity but high enough to significantly affect spindle shape
and/or density (SI 4; Fig. 4). We first investigate the contribution of flow.
The best-known large-scale flow in spindles is “poleward flux,” a steady flow
of MTs away from the center of the spindle, at a speed of around 0.5
$\,\mu\text{m}\,\text{min}^{-1}$ [14]. Poleward flux may be associated with
entrained cytoplasmic flow, although this has not been directly observed; if
so, this could in principle be sufficient to cause flow-induced screening (SI
3). To disrupt poleward flux, we administer monastrol, a inhibitor of the
kinesin-family motor protein Eg5. At doses sufficient to reduce MT flow by a
factor of about 10 [14], monastrol increases the screening length by roughly a
factor of two. We will show in Section 7 that this is consistent with a
monastrol-driven increase in spindle radius, and does not require a model that
invokes flow directly. We next investigate the role of the actin cables that
have recently been shown to run through the whole length of mouse oocyte
spindles [30]. Via actin-MT interactions, these cables might act as internal
boundaries that anchor the MT network, and thus screen fluctuations. To test
whether this occurs, we incubate MII spindles in the presence of cytochalasin
D (CytoD), an actin polymerization inhibitor that has been shown to greatly
reduce the mass of actin cables in mouse oocyte spindles [30]. We observe no
significant difference between CytoD-treated oocytes and the control group,
and conclude that actin is not responsible for screening.

We then tested the effects of two drugs that alter MT turnover dynamics: taxol
and nocodazole [45, 17, 36]). We find that taxol increases the screening
length by a factor of about three, while nocodazole reduces it slightly. These
effects are somewhat surprising, because the direct target of these drugs is
the polymerization dynamics of individual MTs, rather than their interactions,
either with each other or with other cellular structures. Moreover, at these
concentrations, neither drug significantly affects MT density (SI 4). However,
both taxol and nocodazole indirectly affect the spindle’s geometry: nocodazole
decreases spindle volume, while taxol decreases the spindle’s aspect ratio.
This suggests that boundary anchoring effects may be the source of the
screening we observe. In Section 7, we investigate this hypothesis in detail.

##  6 Bend/Splay Ratio is Consistent with Hard-Rod Nematic Behavior

Since they affect MT density, turnover dynamics, and interactions, our
molecular perturbations have the effect of changing both nematic diffusivities
as well as the screening length. Plotting the value of $K_{3}$ against the
value of $K_{1}$ for all conditions allows us to estimate the bend/splay ratio
over roughly an order of magnitude in each variable. We find that
$k_{3}/k_{1}=K_{3}/K_{1}\approx 5.8\pm 0.2$ (Fig. 4(c)). This is consistent
with the expected behavior of a nematic liquid crystal composed of long, hard
rods, in which elasticity originates primarily in steric repulsion between the
rods [27].

##  7 Screening Length is Consistent with a Model Where Inclusions Anchor MTs

Motivated by the hypothesis that the screening we observe occurs via anchoring
of MTs to system boundaries, we next investigate the relationship between
screening length and spindle geometry. We consider in particular two models,
the Outer Surface Model and the Inclusions Model. In the Outer Surface Model,
anchoring at the spindle’s surface leads to screening in the interior. This
model predicts that screening length is proportional to the spindle belt
radius, $\ell_{s}\propto R_{0}$, which is not consistent with our observations
(Fig. 5(c)). In the Inclusions Model, screening arises from the presence of
long, rod-like inclusions (for example chromosomes and associated structures)
uniformly distributed throughout the spindle’s interior, and aligned with the
spindle’s long axis. In this model, length is proportional to the average
perpendicular spacing between inclusions, $\ell_{s}\propto s$. The Inclusions
Model predicts a non-zero $R_{0}$-intercept, $R_{0}^{*}$, in a plot of
$\ell_{s}$ vs $\propto R_{0}$, corresponding to the radius of a spindle where
$s=0$ and inclusions are close-packed in the metaphase plate (SI 5). (In fact,
this model predicts a finite screening length even when $s=0$, from MTs in the
interstitial regions of close-packed chromosomes, but this effect is
unmeasurably small via our techniques and we assume that $\ell_{s}\approx 0$
when $s=0$). This prediction is consistent with our observations, and enables
us to calculate that close-packing of inclusions would occur when
$R_{0}=(4.5\pm 0.2)\,\mu\text{m}$. Assuming further that the inclusions are in
fact the mouse spindle’s $N=20$ chromosomes (and associated structures)
hexagonally packed in the metaphase plate, this model predicts that the
average chromosome has diameter $d_{0}=(1.9\pm 0.1)\,\mu\text{m}$.

##  8 Chromosomes Cause Long Voids in the MT Network

To explore whether MT anchoring to chromosomes and associated structures could
give rise to the observed screening of elastic interactions, we probe the
spindle’s internal structure with confocal microscopy (Methods 8). To label
chromosomes, we inject mRNA coding for a fluorescently tagged histone protein
into GV-stage oocytes prior to releasing them from meiotic arrest. To label
the MT network, we incubate oocytes in SiR-tubulin, a live-cell microtubule
dye (Methods 9). To take advantage of the maximal optical resolution available
in the confocal plane, we image spindles in two different orientations: (i)
with their long axes perpendicular to the optical axis (Fig. 6(a)); and (ii)
with their long axes parallel to the optical axis (Fig. 6(b)). Micrographs of
spindles in these two orientations reveal that the MT network is a contiguous
material perforated by structures that we call voids – long regions of low MT
density centered on an embedded chromsomes. Using classical image analysis
techniques, we quantify the distributions of distances to the nearest boundary
(with a void or with the spindle’s outer surface) for MTs in MI and in MII
spindles (Fig. 6(c)). This reveals that the both mean and standard deviation
of the MI distance distribution is significantly larger than those of the MII
distance distribution, consistent with observed difference in screening length
(Fig. 3). Moreover, these parameters are of order $1\,\mu\text{m}$ –
comparable to the elastic screening length measured from fluctuations and
consistent with the predictions of the Inclusions Model.

To more precisely quantify the shape of the voids, we return to our PolScope
retardance data. Since retardance is proportional to integrated MT density, we
can use these measurements to infer the average MT cross-sectional density in
each slice along the spindle long axis (Fig. 6 (d&e)). For both MI and MII
spindles, the average cross-sectional density is lowest at the metaphase plate
and highest near the poles (Fig. 6 (f)). By assuming all of this “missing”
density is due to $N$ equally-sized voids, we can infer the void profile as a
function of position $x$ along the spindle axis. We find that the average void
profile is well-approximated by a tactoid, the shape generated by revolving a
circular arc about its chord (Fig. 6 (g)). Tactoid-shaped holes in nematic
liquid crystals have been observed previously, and are known as “negative
tactoids” [4, 31]. Voids in MI and MII spindles have similar waist diameters:
$d_{0}=(1.7\pm 0.2)\,\mu\text{m}$ for MI versus $(2.1\pm 0.3)\,\mu\text{m}$
for MII. These values are consistent with the value predicted by the
Inclusions Model, $d_{0}=(1.9\pm 0.2)\,\mu\text{m}$. (The latter figure was
obtained from a data set that included both MI and MII spindles.) For both MI
and MII spindles, void length equals ($(11\pm 1)\,\mu\text{m}$, comparable to
length of the spindle itself. This is consistent with a picture in which void
length is controlled by the system size – a generic feature of voids caused by
compact inclusions in confined nematics (Fig. 6 (h)).

##  9 Conclusions

At length scales comparable to the dimensions of the spindle, the orientation
of MTs in both MI and MII mouse oocyte spindles is well-described by a simple
geometrical model with just one fitting parameter, the position of the virtual
defect $L$. In passive nematic droplets, this geometrical model arises from a
physical model in which $L$ is determined by a balance of elastic energy and
surface energy [35]. In the spindle, these forces may play a similar role,
with the possible addition of active stress generated by MT sliding [34]. In
living cells, these parameters are difficult to measure independently, so it
may be quite complicated to derive $L$ directly from the relevant material
properties. Nevertheless, considering that the circle arcs model was
originally developed to describe simple one-component molecular systems at
thermodynamic equilibrium, the quantitative agreement with the observed MT
orientation in living spindes is quite remarkable.

At length scales of single microns, fluctuations in MT orientation are
screened. Screening is most pronounced for perpendicular fluctuations, as can
be seen by the onset of a plateau in $S_{\text{nn}}$ for values of $q_{y}$
corresponding to distances larger than about a micron (Fig. 3(c&d)), and is
more pronounced for MII spindles than for MI. To investigate the origin of the
observed screening, we incubated MII oocytes in the presence of several
spindle-perturbing drugs (Fig. 4). Our observations suggest a model of spindle
organization in which the MT network behaves like a traditional nematic liquid
crystal at sub-micron scales (but with non-thermal noise). Condensed
chromosomes, each 1-2 $\,\mu\text{m}$ in linear dimension, are impermeable to
MTs [41] and create voids in the MT network that extend along the spindle long
axis. These voids act like colloidal inclusions in a molecular liquid crystal,
and create a significant amount of internal surface area with which the MT
network must interact. This interaction takes the form of anchoring screening,
in which fluctuations can only propagate freely in regions far from void
boundaries. If the voids are themselves aligned by interactions with adjacent
MTs – as appears to be the case, since they are parallel with the spindle long
axis – nematic elasticity will be recovered on length scales of tens of
microns, albeit with larger elastic constants [42]. While we do not have
direct evidence of long-wavelength recovery of elastic behavior in our system,
such an effect could reconcile the organelle-scale structure observed in Fig.
2, which is predicted by an unscreened theory, with the micron-scale screening
demonstrated in Figs. 3 & 4. Our model may be also able to account for the
unscreened correlations observed in vitro: Assuming that reconstituted
$Xenopus$ spindles contain a similar number and volume of chromosomes to our
mouse spindles, their significantly larger belt radii would predict screening
lengths $\ell_{s}\gg 1\,\mu\text{m}$, which would be difficult to observe via
correlation functions. While the volume and spacing of DNA in each spindle was
not measured, prominent “stripes” of locally high MT density, such as those
visible in Fig. 1(c), are notably absent, which my indicate an absence of
voids in that system.

More broadly, our work underscores the surprising relevance of coarse-grained
physical modeling in cell biology. As material objects, spindles are far more
complex than the systems traditionally studied by physicists: they are
composed of hundreds of interacting species of biomolecules that self-assemble
into a hierarchically organized, force-generating structure that must
continuously consume energy in order to maintain its mechanical integrity.
Despite this, important aspects of the mechanical behavior of spindles can be
described using simple models very similar to those used to describe synthetic
liquid crystals, and which invoke only a small number of physically meaningful
parameters. By adjusting these parameters, our models are flexible enough to
describe both MI and MII spindles, as well as the effects of drug
perturbations at doses high enough to significantly disrupt spindle structure,
morphology, and function. Because of this flexibility, and the insight our
non-invasive measurements yield into the spindle’s physical properties, the
techniques we have developed may be of significant clinical interest: in in
vitro fertilization (IVF) clinics, it is critical to accurately identify which
oocytes have the greatest developmental potential [37]. The meiotic spindle is
thought to play a crucial role in determining developmental potential; recent
work suggests that micron-scale defects in spindle structure may lead to
chromosome mis-segregation, which is the largest contributor to developmental
failure in humans [20]. Moreover, human spindles in IVF clinics are already
routinely imaged with PolScope, but the diagnostic benefit of the current mode
of clinical use, where individual retardance images are used to estimate gross
morphological features of the spindle, is unclear [9]. Our approach for
quantifying the physical properties of the spindle from orientation data could
potentially make better use of this existing technology.

Figure 1:  LC-PolScope reveals the formation of steady-state MI and MII
spindles. (a) Left image: Polarization microscopy image of several immature
(germinal-vesicle-stage) oocytes. Right image series: Simultaneous
epifluorescence and PolScope retardance images of oocyte maturation. Times are
relative to release from meiotic arrest. A fully assembled MI spindle is
visible 7 hrs after meiotic release, and an MII spindle is visible 16 hrs
after release. (b) Aligned MTs are birefringent, and so convert incident light
with polarization $\mathbf{p_{1}}$ to transmitted light with polarization
$\mathbf{p_{2}}$. For spindles with horizontal long axis
($\mathbf{\hat{o}}\cdot\mathbf{\hat{x}}=0$), retardance is proportional to the
MT density and the optical slow axis gives MT orientation. If the spindle long
axis is not in the image plane ($\mathbf{\hat{o}}\cdot\mathbf{\hat{x}}\neq
0$), PolScope data is more difficult to interpret. (c) & (d) Snapshots and
time averages (500 frames, 40 mins) of the retardance $r$ and slow axis
$\theta$ for an MII spindle with its long axis in the image plane. Scale bar
$5\,\mu\text{m}$. Spindle boundary is found from a retardance threshold
(Methods 4). (e) The time-averaged orientation field can be fitted to a family
of circle arcs that intersect at $(\pm L,0)$. For all spindles, $L$ is greater
than the spindle half-length $x_{0}$ and we call the points of intersection
“virtual defects.” (f) The aspect ratio $x_{0}/R_{0}$, MT cross-sectional
density $\langle\rho\rangle$, and virtual defect position $L$ as a function of
time after release from meiotic arrest. The quantity $\langle\rho\rangle$ is
calculated by averaging the time-averaged retardance over a
2$\,\mu\text{m}\times$ 2$\,\mu\text{m}$ square in the center of the spindle.
The grey regions indicate the typical timing of GVBD and anaphase I for our
oocytes. For MI spindles, the dashed curves are fits to a sigmoid (middle row)
and decaying exponentials (top and bottom rows); for MII spindles, the dashed
lines in all plots indicate the average. Details of these fits, including
best-fit parameters, are given in Methods 10.

Figure 2:  Microtubule orientation in MI and MII spindles follows circle arcs.
(a) & (c) Rescaling of experimental data by defect spacing,
$\mathbf{r}\rightarrow\mathbf{r}/L$, allows averaging over spindles, and so
enables comparison of the average orientation field with the predictions of
the circle arcs model. This model predicts that rescaled data is well-
described by a family of arcs that intersect at $(\pm 1,0)$, indicated by
stars in all images. Boundaries are averaged as described in Methods 4. (b) &
(d) Horizontal and vertical line profiles of MI and MII spindles. Faint curves
show the line profiles of individual spindles, which are binned to generate
the points with error bars; error bars indicate the standard error of the mean
in a given bin. The predictions of the circle arcs model, which contain no fit
parameters, are shown as bold curves.

Figure 3:  Orientational fluctuations in MI and MII spindles are screened at
long wavelengths. (a) Fluctuations are calculated by subtracting time-averaged
values of the MT orientation from instantaneous values. The MII spindle shown
here is the same as that shown in Fig. 1(c) & (d). Correlation functions are
calculated in the white dashed box in the middle of the spindle in the
fluctuation image, where $\mathbf{\hat{n}_{0}}\approx\mathbf{\hat{x}}$
(Methods 7). The scale bar in the same image corresponds to 5 $\,\mu\text{m}$.
(b) To lowest order in $\delta\theta$, fluctuations in $\mathbf{\hat{n}}$ are
perpendicular to $\mathbf{\hat{n}_{0}}$, and near the middle of the spindle
$\delta\mathbf{n}\cdot\mathbf{\hat{x}}=0$. (c) Director autocorrelation
functions $C_{\text{nn}}(\omega,\mathbf{q}\rightarrow 0)$,
$S_{\text{nn}}(q_{x},q_{y}\rightarrow 0)$, and $S_{\text{nn}}(q_{x}\rightarrow
0,q_{y})$ for several MI spindles. Correlation functions for individual
spindles are plotted in color. Because of the projection over
$\mathbf{\hat{z}}$ implicit in the PolScope measurement, all correlation
functions are in the $q_{z}\rightarrow 0$ limit. We obtain binned data points
by averaging the curves of individual spindles; error bars indicate the
standard error of the mean in a given bin (calculated in log space). In all
panels, dashed lines represent a power law decay with exponent -2. The equal-
time correlation function $S_{\text{nn}}(q_{x},q_{y}\rightarrow 0)$ plateaus
at values of $q_{x}$ that roughly coincide with the finite length of MTs in
$q$-space, $q_{x}^{\text{MT}}\sim(2\pi)/(2\,\mu\text{m})\approx 3$
rad$\,\mu\text{m}^{-1}$ (middle plot). At long wavelengths (low $q$’s),
$S_{\text{nn}}(q_{x}\rightarrow 0,q_{y})$ flattens more dramatically than
$S_{\text{nn}}(q_{x},q_{y}\rightarrow 0)$, indicating that fluctuations
decouple over shorter distances in the direction perpendicular to the spindle
axis (middle vs right plot). (d) Average correlation functions for MI and MII
spindles, and corresponding fits to the screened nematic model. To avoid the
finite-MT effects visible in $S_{\text{nn}}(q_{x},q_{y}\rightarrow 0)$, we
exclude data points with $q_{x}>q_{x}^{\text{MT}}$ from our fits. (e)
Predictions of the screened nematic model in the limits plotted in (c) and
(d). (f) Best-fit values corresponding to the fits shown in (d). Error bars in
these parameters are obtained by fitting the correlation functions of
individual spindles to the screened nematic model, and comparing the mean of
the distribution of individual best-fit values to the best-fit value of the
average correlation function shown in (d).

Figure 4:  Spindle-targeting drugs affect screening lengths and nematic
diffusivities. (a) Equal-time correlation functions
$S_{\text{nn}}(q_{x}\rightarrow 0,q_{y})$, rescaled, as indicated in the axes
labels, by the best-fit values of the screened nematic model. Plots correspond
to treatment conditions where: the kinesin-family molecular motor Eg5 is
inhibited (Monastrol Vs Control); actin is depolymerized (CytoD Vs Control);
MTs are stabilized (Taxol Vs Control); and MTs are destabilized (Nocodazole Vs
Control). Insets show PolScope retardance images for each condition. All
images are shown at the same magnification; scale bars correspond to 5
$\,\mu\text{m}$. (b) Model parameters corresponding to all the conditions
shown in (a). (c) Plot of bend diffusivity $K_{3}$ vs splay diffusivity
$K_{1}$ for all conditions shown here, as well as for the untreated spindles
shown in Fig. 3. In all plots, binned data points are obtained by averaging
the curves of individual spindles (light curves); error bars indicate the
standard error of the mean in a given bin (calculated in log space).

Figure 5:  Two models for anchoring screening in oocyte spindles. (a) In the
Outer Surface Model, screening arises from MT anchoring at the spindle
boundary, and screening length is proportional to spindle belt radius $R_{0}$.
Bold curves indicate the presence of a specific anchoring condition for MTs.
(b) In the Inclusions Model, screening is caused by MT anchoring to
inclusions, such as chromosomes, embedded in the spindle network. In this
case, screening is proportional to the perpendicular space $s$ between
inclusions. Box Inset: In the special case where the inclusions are close
packed ($s=0$), screening becomes smaller than we can observe experimentally
($\ell_{s}\approx 0$), and the spindle belt radius is determined only by the
number $N$ and diameter $d_{0}$ of the inclusions. (c) The observed dependence
of screening length $\ell_{s}$ on spindle radius is consistent with the
Inclusion Model, but not with the Outer Surface Model. Shaded region indicates
a 95% confidence interval for a linear fit to the data. Bold arrow indicates
the value of the intercept $R_{0}^{*}=(4.5\pm 0.2)\,\mu\text{m}$, which
corresponds to an inclusion diameter $d_{0}=(1.8\pm 0.2)\,\mu\text{m}$
(assuming $N=20$). Error bars indicate the standard errors of the means in
each condition. See Figs. 3 and 4 for sample numbers and details of each
condition.

Figure 6:  Chromosomes create long, tactoid-shaped voids in the MT network of
MI and MII spindles. (a) Confocal micrographs of living MI and MII oocytes,
taken with the spindle long axis in the confocal plane,
$\mathbf{\hat{o}}\cdot\mathbf{\hat{x}}=0$. MTs are labeled with the live-cell
stain SiR-tubulin, and chromosomes are labeled with fluorescent histone
protein H2B-EGFP. The scale bar in all micrographs corresponds to
$5\,\mu\text{m}$. (b) Micrographs taken with the spindle long axis
perpendicular to the confocal plane,
$\mathbf{\hat{o}}\cdot\mathbf{\hat{x}}=1$. (c) MTs in MI spindles are further
from void boundaries than MTs in MII spindles. Using adaptive binarization of
images like those shown in (b), we construct masks of the MT-dense region in
the metaphase plate (i.e. the $x=0$ plane). From these masks we calculate the
distance transform, in which pixel color indicates distance to nearest edge,
which may be the outer boundary of the spindle or a void boundary. From the
distance transform data, we make histograms of the distribution of distances
to the nearest boundary. Both the average (black dashed line in each plot) and
the standard deviation of this distribution are larger for MI than for MII
spindles, $(0.5\pm 0.4)\,\mu\text{m}$ for MI versus $(0.3\pm
0.2)\,\mu\text{m}$ for MII. (d) Semi-elliptical PolScope retardance profiles
can be fitted to find the spindle radius and density using the formula
$r(y)=2A_{0}\rho_{yz}(R(x)^{2}-x^{2})^{1/2}$. Here,
$\rho_{yz}\equiv\langle\rho(x,y,z)\rangle_{y,z}$ is the MT cross-sectional
density averaged over $y$ and $z$, $R(x)$ is the spindle radius at position
$x$ along the long axis, and the constant $A_{0}\approx 7.9$ nm2 characterizes
the birefringence of a single MT (Methods 3). If some of the cross-sectional
area is occupied by voids, the measured value of $\rho_{yz}$ will be lower.
(e) Fits of measured retardance profiles to $r(y)$ at three different
positions $x$ along the spindle axis, for both MI and MII spindles. (f) Fitted
values of $\rho_{yz}$ as a function of $x$ for both MI and MII spindles. (g)
By assuming that the “missing” density in the middle of the spindle is due to
the presence of voids, we can infer the average void profile, which is well
fit by the arc of a circle. To aid visualization of the physical shape of the
void, the fitted arc is also shown as a dashed curve reflected in the
$x-$axis. Error bars in (f) and (g) indicate the standard error of the mean
for the data points in a given bin. (h) The formation of long voids around
compact inclusions can be seen in everyday nematic materials. One example is
provided by cylindrical lip gloss tubes, $\sim 1$ cm in diameter, inserted
into a straw broom head whose loose end is wrapped with a hair tie. A US
quarter dollar coin is included for scale.

Methods for “The physical basis of self-organization of the mammalian oocyte
spindle ”

##  Methods 1 Oocyte Collection & In Vitro Maturation, Drug Treatments

We obtain oocytes from B6C3F1/J mice using standard protocols [3]. Briefly, we
inject female mice, aged 8-12 weeks (Jackson Labs [2]), with 5 I.U. pregnant
mare serum gonadotrophin (ProspecBio), and harvest germinal vesicle (GV) stage
oocytes 44-48 hrs later. We preform ovary dissection and oocyte collection in
commercially available Advanced KSOM culture medium (EmbryoMax) to which we
add 4 g/L bovine serum albumin (“BSA,” MilliporeSigma) and and meiotic
inhibitor cocktail that has recently been shown to minimize damage to in vitro
matured oocytes [48]: 10 $\mu$M 3-Isobutyl-1-methylxanthine (MilliporeSigma)
and 50 $\mu$M N6,2’-O-Dibutyryladenosine 3’,5’-cyclic monophosphate sodium
salt (Sigma). We remove cumulus cells by gentle pipetting through a
100$\,\mu\text{m}$ Stripper tip (Origio) that we attach to a mouth pipette
using UV-cure glue (Norland). We rinse the oocytes through several droplets of
inhibitor-free medium (Advanced KSOM + 4 g/L BSA), and place them into a Petri
dish containing a single droplet of maturation medium. Prior to use, all media
is pre-equilibrated in a 5% CO2 atmosphere at 37∘C for at least 1 hr. To
prevent evaporation, droplets that are exposed to air for longer than 15 mins.
are covered with several mm of LiteOil (EmbryoMax). Oocyte collection and
manipulation is carried out on a dissection microscope with a stage heated to
37∘C. To perturb MII oocytes with small molecule drugs, we first prepare 200x
v/v stock solutions of the drug dissolved in dimethyl sulfoxide (“DMSO,”
MilliporeSigma). We add the drug to the oocyte culture media 1-4 hrs prior to
imaging. Table 1 shows where we obtain the drugs.

Drug | Company | Product No.
---|---|---
Monastrol | Selleck | S8439
Taxol | Selleck | S1150
Nocodazole | Selleck | R17934
Cytochalasin D | MilliporeSigma | C2618
Table 1:  Ordering information for the drugs used in Section 5 of the Main
Text.

##  Methods 2 PolScope Microscopy & Image Registration

The LC-PolScope hardware (Cambridge Research Instruments) is mounted on a
Nikon TE2000-E microscope equipped with a 100$\times$ NA 1.45 oil immersion
objective and a 0.5$\times$ air condenser lens. We control the PolScope
hardware and analyze the images we obtain using the OpenPolScope software
package. Prior to imaging, we manually search for a spindles whose long axis
lies in or near the image plane. To confirm that this is the case, we take a
z-stack, and visually check that the poles are symmetrical to within $\sim
2\mu$m of each other in the direction perpendicular to the image plane; this
ensures that $\tan{\phi}\equiv\mathbf{\hat{x}}\cdot\mathbf{\hat{o}}\lesssim
0.1$. Since the measured spindle retardance $r_{\text{meas.}}$ scales with the
intrinsic retardance $r$ as $r_{\text{meas.}}=r\cos^{2}{\phi}$, the maximum
error between measured and intrinsic spindle retardance is $\lesssim 1\%$. Our
movies typically consist of $\sim$500 frames acquired over $\sim$30 mins.
During image acquisition, spindles often translate and rotate through several
microns. We discard data from those spindles that rotate or translate out of
the image plane, and, for the remainder, use a custom-written image
registration algorithm, implemented in the Mathematica programming language,
to ensure that all subsequent analysis is performed in the rest frame of the
spindle.

Our registration algorithm comprises four steps. First, we manually crop a
region of interest in the camera frame that includes all positions of the
spindle over the course of the entire movie. Second, we apply a long-pass
filter to find the approximate center of brightness of each spindle retardance
image. Using this center of brightness, we translate both the retardance and
slow axis images such, in all frames of the movie, that the center of the
spindle corresponds with the center of the image. Third, we use the
orientation image to rotate the spindle such that its average orientation is
horizontal. Finally, we align each retardance image with a reference
retardance image generated by applying a moving average to the neighboring
frames, with window width $\sim 50$ frames. To perform the alignment, we apply
a gradient filter to both the image for registration and the reference image.
Our code is available at [1]. None of our later results depend sensitively on
the details of the image registration; for instance, we can entirely omit the
gradient alignment step without significantly changing the results presented
in the Main Text.

##  Methods 3 Calculation of MT Density from Observed Retardance

MTs are birefriengent primarily because they are long cylinders of high
refractive index [38]. We can use a model based on this effect to relate the
observed retsardance to the average 2D cross-sectional density in the
direction perpendicular to the MTs $\langle\rho\rangle$. To do this, we
calculate the average retardance in a $2\,\mu\text{m}\times 2\,\mu\text{m}$
square in the center of the spindle. Assuming that light remains collimated as
it passes through the spindle (SI 1), we follow reference [32] to calculate
$\langle\rho\rangle$ from the measured retardance $r$ and the sample thickness
$2R_{0}$. By definition, the retardance of a sample depends on its thickness
and the refractive index mismatch $\Delta n=n_{\text{slow}}-n_{\text{fast}}$
between light polarized along the fast and slow optical axes,

$r=2R_{0}\Delta n.$

For materials like MTs, whose birefringence arises from their rod-like shape
rather than their intrinsic chemical anisotropy, $\Delta n$ is related to
their volume fraction $f$ and refractive index $n_{\text{MT}}$, as well as the
refractive index of the surrounding cytoplasm $n_{\text{Cyto}}$,

$\Delta
n=f\frac{(n_{\text{Cyto}}^{2}-n_{\text{MT}}^{2})^{2}}{2n_{\text{Cyto}}(n_{\text{Cyto}}^{2}+n_{\text{MT}}^{2})}.$

This expression is equivalent to Eqn. 5 in reference [32] with
$\epsilon=n^{2}$, and assumes $f\ll 1$. For a bulk material composed of
aligned MTs, it is the also case that

$f=\Big{(}\frac{\pi d_{MT}}{2}\Big{)}^{2}\langle\rho\rangle,$

where $d_{MT}$ is the diameter of a single MT. Putting these equations
together, we obtain the equation we use to relate measured retardance $r$ to
the MT cross-sectional density $\langle\rho\rangle$,

$\langle\rho\rangle=\frac{r}{2A_{0}R_{0}},$

where the so-called retardance area $A_{0}$ characterizes the contribution of
a single MT to the measured retardance,

$A_{0}\equiv\big{(}\frac{\pi
d_{MT}}{2}\Big{)}^{2}\frac{(n_{\text{Cyto}}^{2}-n_{\text{MT}}^{2})^{2}}{2n_{\text{Cyto}}(n_{\text{Cyto}}^{2}+n_{\text{MT}}^{2})}.$

The MT-related quantities $d_{MT}$ and $n_{\text{MT}}$ have been previously
measured, and should not depend on cell type: $d_{MT}\approx 24$nm and
$n_{\text{MT}}\approx 1.512$ [33, 38]. To our knowledge, the refractive index
of mouse oocyte cytoplasm has not been measured. For other metazoan cell
types, $n_{\text{Cyto}}$ typically lies in the range 1.36-1.40 [18]. Assuming
that $n_{\text{Cyto}}=1.38\pm 0.02$, we obtain $A_{0}=(7.9\pm 0.2)$ nm2, which
is consistent with direct measurements of this quantity in vitro [32].

##  Methods 4 Finding Spindle Boundary from Retardance Threshold

To find the spindle boundaries, we use the registered images found in Methods
Methods 2. We first find the boundaries of each frame of the movie
individually. To do this, we (further) crop the registered movie such that all
frames have an area $\sim 2\times-5\times$ the area of the spindle. We next
apply a Gaussian blur filter with width $\sim 1\,\mu\text{m}$ to all images.
We find the intensity value $r_{\text{cent.}}$ in a $\sim 1\,\mu\text{m}^{2}$
region in the center of each image, and apply a retardance threshold
$r_{\text{cent.}}/2$ to the blurred image. This yields the edge shown in Fig.
6(d, upper panel); the results obtained from this algorithm are not sensitive
to parameters such as cropped area, Gaussian blur width, etc. To average the
boundaries found in all the frames of the movie, we express the boundary
curves $\mathbf{b}_{t}$ in polar coordinates (with origin at the center of the
image),

$\mathbf{b}_{t}=b_{t}(\phi)\\{\cos{t},\sin{t}\\},$

where $t$ indicates the frame of the movie, $\phi$ is the azimuth, and
$b_{t}(\phi)$ is the radial coordinate of the boundary curve. The average
boundary curve $\mathbf{b}$ may be defined as follows,

$\mathbf{b}\equiv\langle b_{t}(\phi)\rangle_{t}\\{\cos{t},\sin{t}\\}.$

This procedure produces the curves such as that shown in Fig. 1(d). The same
technique can be used to average the boundaries of different spindles (either
in rescaled or unrescaled coordinates).

##  Methods 5 Averaging Nematic Variables

MT orientation must be averaged in a way that respects its nematic symmetry:
since $\theta=0$ and $\theta=\pi$ are equivalent, their average is not $\pi/2$
but 0 or $\pi$. To do this, we average a set of orientations
$\\{\theta_{j}\\}_{j=1}^{N}$ using the formula

$\langle\theta\rangle\equiv\frac{1}{2}\arg{\sum_{j=1}^{N}e^{2i\theta_{j}}}.$

##  Methods 6 Calculation of Virtual Defect Position

To find the best-fit value of the defect half-spacing $L$ from an averaged
orientation image such as Fig. 1(d), we first center the image by fitting
$\sin{\theta}$ to a hyperbolic paraboloid,

$\sin{\theta}=C_{1}((x-x_{1})^{2}-(y-y_{1})^{2}),$

and subtracting the vector $(x_{1},y_{1})$ from all coordinates. In the upper
half-plane $y>0$, the angle field generated by a field of circle arcs with
(virtual) defects at $(\pm L,0)$ is given by

$\theta_{\text{arcs}}(x,y)=\text{atan2}\big{(}y+\frac{L^{2}-x^{2}-y^{2}}{2y},-x\big{)}.$
(6)

We find the best-fit value of $L$ by minimizing the $L^{2}$-norm between the
observed angle field and the field generated by (6),

$||\theta(x,y)-\theta_{\text{arcs}}(x,y)||\equiv\int_{\Omega}dA|e^{2i\theta}-e^{2i\theta_{\text{arcs}}}|,$

where the domain of integration $\Omega$ is the area enclosed by the
boundaries of the spindle.

##  Methods 7 Calculation of Experimental Correlation Functions

To avoid finite-size and spindle-boundary effects, we calculate correlation
functions in a series of regions of interest (RoI’s), the smallest around
$1\mu$m in side length, and the largest (such as that shown in the inset to
Fig. 3(a)) around 10 $\mu$m. To ensure that the approximation
$\mathbf{\hat{n}_{0}}\approx\mathbf{\hat{x}}$ holds, we ensure that, in all
boxes, $|\mathbf{\hat{y}}\cdot\mathbf{\hat{n}_{0}}|$ is at most 0.5, roughly
corresponding to $|\theta|<30^{\circ}$ everywhere inside the box. The average
anglular deviation is typically much smaller: in the largest box (with the
greatest amount of angular deviation from the spindle axis),
$\langle|\mathbf{\hat{y}}\cdot\mathbf{\hat{n}_{0}}|\rangle\approx 0.1$,
equivalent to $\langle|\theta|\rangle\approx 6^{\circ}$. For each point in the
RoI, we calculate the mean value of the director
$\langle\mathbf{\hat{n}}\rangle$, and thus the fluctuation $\delta
n_{y}(x,y,t)=\hat{n}_{y}(x,y,t)-\langle\hat{n}_{y}\rangle$ at each instant in
time. Using the software package Mathematica, we calculate the (2+1)D fast
Fourier transform of the fluctuations $\delta\tilde{n}_{y}(\mathbf{q},\omega)$
inside this region, and thus the correlation functions
$C_{\text{nn}}(\mathbf{q},\omega)$ defined in Eqn. 1 of the Main Text. Equal
time correlation functions are calculated using the formula

$S_{\text{nn}}(\mathbf{q})=\frac{1}{2\pi}\int_{-\infty}^{\infty}C_{\text{nn}}(\omega,\mathbf{q})\text{d}\omega$

Using a technique similar to that used in soft matter simulations, we
extrapolate the measured correlation functions to their infinite-system-size
limits (SI 6). At the highest frequencies, $\omega>0.05$ rad s-1, the
correlation function $C_{\text{nn}}(\omega,0)$ is dominated by white noise,
presumably arising from camera shot noise, high frequency mechanical
vibrations of the microscope, etc. In this regime, we find that

$C_{\text{nn}}(\omega,0)\approx A,$

where $A$ is a positive constant. To correct for this effect, we subtract the
constant $A$ from $C_{\text{nn}}(\omega,0)$ for all values of $\omega$,

$C_{\text{nn}}(\omega,0)\rightarrow C_{\text{nn}}(\omega,0)-A,$

and only fit our corrected data in the range $\omega<0.05$ rad s-1. At the
highest wavenumbers, the equal time correlation functions
$S_{\text{nn}}(\mathbf{q})$ are slightly affected by artificial correlations
due to the diffraction-limited $xy-$resolution of the PolScope. To correct for
this effect, we perform Fourier-space deconvolution of these correlation
functions,

$S_{\text{nn}}(\mathbf{q})\rightarrow\exp{\frac{-\mathbf{q}^{2}}{2q_{0}^{2}}}S_{\text{nn}}(\mathbf{q}),$

where $q_{0}=2\pi/(0.5\,\mu\text{m})$ is the diffraction limit in Fourier
space, and only fit these functions for $|\mathbf{q}|<q_{0}$.

##  Methods 8 Live Cell Staining and mRNA Microinjection

To image DNA and tubulin, we use the live-cell stains SiR-DNA and SiR-tubulin
(Cytoskeleton Inc.) We add these stains to the culture media at least 30 mins
prior to imaging, at a concentration of 0.5 $\mu$M. To minimize the exposure
of the oocytes to DMSO, we perform intermediate dilutions into aliquots of
maturation media rather than DMSO. We do this immediately prior to adding the
stain to the culture medium. To simultaneously image tubulin and DNA, we
synthesize mRNA coding for the fluorescently tagged histone protein H2B-EGFP,
using the HiScribe T7 Arca kit (New England Biolabs) according to the
manufacturer’s instructions. Using a previously described technique, we inject
mRNA solution into GV oocytes at volumes approximately equal to 1% of the
oocyte volume [15]. For the injections, we use the Narishige micromanipulation
system (Narishige International US, Inc.) for oocyte micromanipulation; a
PV850 Microinjector (World Precision Instruments) for mRNA delivery; and an
IE-251A Intracellular Electrometer (Warner Instruments) for delivering the
capacitive pulse that helps the oocytes survive the injection. This equipment
is mounded on a Nikon TE2000-E microscope mounted with a 10x objective lens
and equipped with crossed polarizers to increase contrast. Injections take
place in medium containing the miotic inhibitor cocktail and are performed at
room temperature. After injecting mRNA, we release the GV oocytes for meiotic
arrest, and allow them to mature as described in the previous section.

###  Methods 8.1 Confocal Microscopy & Image Analysis

The confocal stacks shown in Fig. 6 of the Main Text are acquired on an
inverted Nikon Ti microscope equipped with a Yokogawa CSU-X1 confocal scanner,
with solid state 100 mW 488 nm and 633 nm lasers, using a 60x 1.4 NA plan Apo
oil immersion objective lens, and a Hamamatsu ORCA R-2 Digital CCD Camera. To
perform the analysis shown in Fig. 6(c), we use a custom program written in
Mathematica. To identify chromsome/void edges in the metaphase plate, we first
average over several confocal slices within $\pm 2\,\mu\text{m}$ of the
metaphase plate $x=0$. We binarize the resulting image by setting to zero all
points that are more than one standard deviation below the mean brightness
value. We next apply Mathematica’s inbuilt LocalAdaptiveBinarize[] function,
with the radius parameter set to the expected chromosome diameter,
$2\,\mu\text{m}$. We next apply a Gaussian blur with radius one-half the
expected chromosome diameter, and again binarize the resulting image. To
obtain the final binarized images shown in Fig. 6(c), we discard any connected
features (black regions on white background or white regions on black
background) that have a smaller area than 0.1 the expected chromosome cross-
sectional area, $1.26\,\mu\text{m}^{2}$. To create the distance-transformed
images shown in Fig. 6(c), we use a linear resampling algorithm to re-grid our
binarized images in order to avoid pixellation artifacts; resampled images
have pixels that are $\approx 0.02\mu$ in linear dimension. Finally, we apply
Mathematica’s inbuilt DistanceTransform[] function to our resampled images.
This program takes no additional parameters. Throughout, our algorithm uses
only physically motivated image analysis parameters, for instance expected
chromosome diameter, and the results highlighted in the Main Text do not
depend sensitively on the choices of these parameters.

##  Methods 9 Fitting Time Evolution of Spindle Aspect Ratio, Virtual Defect
Position, and Density

In Fig. 1(f), we fit the development of spindle aspect ratio $x_{0}/R_{0}$, MT
cross-sectional density $\langle\rho\rangle$, and the half-distance between
virtual defects $L$. We use the following ad-hoc functional forms:

$\displaystyle x_{0}/R_{0}$ $\displaystyle=A_{1}-A_{2}t$
$\displaystyle\langle\rho\rangle$
$\displaystyle=B_{1}(1+\tanh{(t-B_{2})/B_{3}})$ $\displaystyle L$
$\displaystyle=C_{1}-C_{2}t.$

The fit parameter values are given in Table 2.

$A_{1}$ | 2.08
---|---
$A_{2}$ | 0.0565 hr-1

$B_{1}$ | 17.7 $\,\mu\text{m}^{-2}$
---|---
$B_{2}$ | 3.08 hrs
$B_{3}$ | 0.607 hrs

$C_{1}$ | $17.0\,\mu\text{m}$
---|---
$C_{2}$ | 0.426 $\,\mu\text{m}$ hr-1

Table 2:  Parameter values for the fits shown in Fig. 1(f) of the Main Text.

Supplemental information for “The physical basis of self-organization of the
mammalian oocyte spindle ”

##  SI 1 PolScope Data Represent Simple Averages Over Optical Axis

In the main text, we treat all PolScope data as if it represented a simple
average over the optical axis $\mathbf{\hat{o}}$. This is equivalent to
treating the light passing through the spindle as perfectly collimated, or
assuming that the optical section depth of the PolScope is much larger than
the spindle diameter. This is consistent with previous measurements in a
quantitative polarization microscopy system similar to PolScope [sato1975]. In
that work, it was shown that measured retardance values do not depend on the
details of the optical train, in particular on the NAs of the objective and
condenser lenses, which is consistent with the assumption of collimated light.
However, we use immersion objectives with significantly higher NA than in the
previous work, and, to our knowledge, there is not a simple way to estimate
the optical section depth of“semi-coherent” microscopy like PolScope [44]. To
check whether our results depend on the details of the optical train, we
measure the retardance profiles of three MI spindles and three MII spindles
with three different objectives, whose NAs vary from 0.3 (magnification 4x) to
1.4 (magnification 100x). Consistent with the results of the previous study,
we find that our retardance measurements do not depend strongly on the NA or
magnification of the objective lens used, and conclude that transmitted light
is approximately collimated in all cases (Fig. 7). In all experiments, we use
a condenser lens with NA 0.5, with the iris in the half-closed position such
that the effective NA is $\sim 0.25$. Since the NA of the condenser lens is
lower than that of the objective in all cases, we did not investigate the
effect of changing this component.

Figure 7:  (a) The same MI spindle, imaged with three different objective
lenses: 4x (NA 0.3); 10 x (NA 0.7); 100x (NA 1.4). Scale bars
$10\,\mu\text{m}$. (b) Retardance profiles through the center of three MI
spindles (top rop) and three MII spindles (bottom row), in the direction
perpendicular to the spindle axis (indicated by dashed lines in (a)). (c) The
maximum measured retardance is similar in all cases, as is the maximum
retardance relative with the background retardance values subtracted. In all
cases, error bars indicate the standard error of the mean.

##  SI 2 Calculation of Theoretically Predicted Correlation Functions

###  SI 2.1 Fourier-Space Equations of Motion of Fluctuating Fields, Nematic
Model

For small fluctuations perpendicular to the director axis $\mathbf{\hat{x}}$,
fluctuations may be decomposed into the two perpendicular directions,

$\delta\mathbf{n}=\delta n_{y}\mathbf{\hat{y}}+\delta n_{z}\mathbf{\hat{z}},$

and the corresponding free energy is given by Eqn. (3.72) of reference [16],

$\mathcal{F}_{\text{el}}=\frac{1}{2}\int
d^{3}\mathbf{r}\Big{\\{}k_{1}\Big{(}\frac{\partial\delta n_{y}}{\partial
y}+\frac{\partial\delta n_{z}}{\partial
z}\Big{)}^{2}+k_{2}\Big{(}\frac{\partial\delta n_{y}}{\partial
z}-\frac{\partial\delta n_{z}}{\partial
y}\Big{)}^{2}+k_{3}\Big{(}\frac{\partial\delta n_{y}}{\partial
x}^{2}+\frac{\partial\delta n_{z}}{\partial x}^{2}\Big{)}\Big{\\}},$

In Fourier space, this can be written

$\tilde{\mathcal{F}}_{\text{el}}=\frac{1}{2V}\sum_{\mathbf{q}}k_{1}|\delta\tilde{n}_{z}(\mathbf{q})q_{z}+\delta\tilde{n}_{y}(\mathbf{q})q_{y}|^{2}+k_{2}|\delta\tilde{n}_{z}(\mathbf{q})q_{y}-\delta\tilde{n}_{y}(\mathbf{q})q_{z}|^{2}+k_{3}q_{x}^{2}(|\delta\tilde{n}_{z}(\mathbf{q})|^{2}+|\delta\tilde{n}_{y}(\mathbf{q})|^{2}),$
(SI 1)

where $\delta\tilde{n}_{y}$ and $\delta\tilde{n}_{z}$ are the Fourier
transforms of $\delta n_{y}$ and $\delta n_{z}$ respectively. In equilibrium
systems, the equipartition theorem can be applied to (SI 1) to calculate the
spectrum of fluctuations. In non-equilibrium systems like the spindle, we can
recover an analogous result by constructing an equation of motion and applying
the principles of fluctuating hydrodynamics [12]. To do this, we first write
the equation of motion for fluctuations (Eqn. 4 in the Main Text) explicitly
in terms of the components,

$\frac{\partial\delta n_{y}}{\partial t}=-K_{3}\frac{\partial^{2}\delta
n_{y}}{\partial x^{2}}-K_{2}\frac{\partial}{\partial
z}\Big{(}-\frac{\partial\delta n_{z}}{\partial y}+\frac{\partial\delta
n_{y}}{\partial z}\Big{)}-K_{1}\frac{\partial}{\partial
y}\Big{(}\frac{\partial\delta n_{z}}{\partial z}+\frac{\partial\delta
n_{y}}{\partial y}\Big{)}+S(\mathbf{x},t),$

with a similar expression for $\partial\delta n_{z}/\partial t$. Taking the
Fourier transform of this expression, we get

$-i\omega\delta\tilde{n}_{y}=K_{3}q_{x}^{2}\delta\tilde{n}_{y}+K_{2}q_{z}(-q_{y}\delta\tilde{n}_{z}+q_{z}\delta\tilde{n}_{y})+K_{1}q_{y}(q_{z}\delta\tilde{n}_{z}+q_{y}\delta\tilde{n}_{y})+\tilde{S}(\mathbf{q},\omega),$

###  SI 2.2 PolScope Measurements; Projection Slice Theorem

In PolScope measurements, we do not measure the full 3D orientation fields.
Rather, we measure the orientation field averaged over the optical axis
$\mathbf{\hat{z}}$. According to the projection-slice theorem of Fourier
analysis, this is equivalent to taking a long wavelength limit in the
$\mathbf{\hat{z}}$ direction, i.e. $q_{z}\rightarrow 0$ [7]. In that case, the
equation of motion for $\delta\tilde{n}_{y}$ becomes

$-i\omega\delta\tilde{n}_{y}=(K_{3}q_{x}^{2}+K_{1}q_{y}^{2})\delta\tilde{n}_{y}+\tilde{S}(\mathbf{q},\omega),$

which no longer contains the twist diffusivity $K_{2}$.

###  SI 2.3 Fourier-Space Correlation Functions, Nematic Model

To explicitly calculate correlation functions, we assume Gaussian-distributed
noise. In real space, $S(\mathbf{x},t)$ is completely characterized by the
equations

$\langle S(\mathbf{x},t)\rangle=0,\qquad\langle
S(\mathbf{x},t)S(\mathbf{x}^{\prime},t^{\prime})\rangle=S_{0}^{2}\,\delta(\mathbf{x}-\mathbf{x}^{\prime})\delta(t-t^{\prime}).$

The above equations are equivalent to the Fourier space expressions

$\langle\tilde{S}(\mathbf{q},\omega)\rangle=0,\qquad\langle\tilde{S}(\mathbf{q}_{1},\omega_{1})\tilde{S}(\mathbf{q}_{2},\omega_{2})\rangle=S_{0}^{2}\,\delta(\mathbf{q}_{1}+\mathbf{q}_{2})\delta(\omega_{1}+\omega_{2}),$

Using these expressions, we may calculate the full spatiotemporal correlation
function

$C_{nn}^{\text{nem}}(\mathbf{q},\omega)\equiv\delta\tilde{n}_{y}(\mathbf{q},\omega)^{*}\delta\tilde{n}_{y}(\mathbf{q},\omega)=\frac{S_{0}^{2}}{(\omega^{2}+(K_{3}q_{x}^{2}+K_{1}q_{y}^{2})^{2})}.$
(SI 2)

The $\mathbf{q}\rightarrow 0$ limit can be obtained directly from this
expression; the equal time correlation function can be calculated via

$S_{nn}^{\text{nem}}(\mathbf{q})\equiv\langle\delta\tilde{n}_{y}(\mathbf{q})^{*}\delta\tilde{n}_{y}(\mathbf{q})\rangle=\frac{1}{2\pi}\int
C_{nn}^{\text{nem}}(\mathbf{q},\omega)d\omega,$ (SI 3)

equivalent to evaluating the inverse Fourier transform (in the time domain
only) of $C_{nn}^{\text{nem}}(\mathbf{q},\omega)$at $t=0$. From expressions
(SI 2) and (SI 3), we can calculate the limits shown in Table SI 3.

$C_{\text{nn}}(0,\omega)$ | $\frac{S_{0}^{2}}{\omega^{2}}$
---|---
$S_{\text{nn}}(q_{x},0)$ | $\frac{S_{0}^{2}}{2K_{3}q_{x}^{2}}$
$S_{\text{nn}}(0,q_{y})$ | $\frac{S_{0}^{2}}{2K_{1}q_{y}^{2}}$
Table 3:  Correlation functions predicted by unscreened nematic elasticity.

###  SI 2.4 Fourier-Space Correlation Functions, Screened Nematic Model

In the screened nematic model, the equation of motion is given by Eqn. 5 in
the Main Text. Repeating the calculation in the previous for this expression
gives

$C_{nn}^{\text{scr.}}(\mathbf{q},\omega)\equiv\langle\delta\tilde{n}_{y}(\mathbf{q},\omega)^{*}\delta\tilde{n}_{y}(\mathbf{q},\omega)\rangle=\frac{S_{0}^{2}}{(\omega^{2}+(K\ell_{s}^{-2}+K_{3}q_{x}^{2}+K_{1}q_{y}^{2})^{2})},$
(7)

and hence the limits presented in Fig. 3(e) of the Main Text.

##  SI 3 Nematics with Flow Alignment

###  SI 3.1 Equation of Motion with Flow Alignment

Active systems like spindles can generate coherent, steady-state flow (of MTs
and/or cytoplasm). This can lead to “flow alignment” torques, or hydrodynamic
torques on MTs that compete with nematic elastic torques [26]. The relative
importance of these torques can be quantified by comparing the magnitude of
the terms on the right-hand side (RHS) of Eqn. 2 in the Main Text. In the one-
constant approximation, the elastic term has magnitude $K_{3}\nabla^{2}n_{i}$,
where $K_{3}$ is the larger of the two measured elastic constants. The flow
contribution involves terms of the form $v_{j}\nabla_{j}n_{i}$ and
$n_{j}\nabla_{j}v_{i}$, in some cases multiplied by a constant of order unity.
All these terms have units of 1/time, and may be interpreted as quantifying
the duration required for a global deformation of the spindle to relax to the
steady state configuration. We donate by $\tau_{\text{el}}$ the timescale
associates with elastic relaxation, and $\tau_{\text{flow}}$ the timescale
associated with flow alignment.

###  SI 3.2 Comparison of Flow Alignment Stress with Nematic Stress

To estimate the magnitudes of these two terms, we use the larger elastic
constant $K_{3}$ for MII spindles, and take the MII spindle radius,
$R_{0}\approx 6\,\mu\text{m}$, as the length scale of all gradients. To our
knowledge, the steady state cytoplasmic flow speed in the spindle interior has
not been directly measured. Instead, we use the MT poleward flux speed,
$v_{0}\approx 0.5\,\,\mu\text{m}\,\text{min}^{-1}$. This gives

$\tau_{\text{el}}^{-1}\sim\frac{K_{3}}{R_{0}^{2}}\sim
10^{-4}s^{-1};\qquad\tau_{\text{flow}}^{-1}\sim\frac{v_{0}}{R_{0}}\sim
10^{-3}s^{-1}.$

For these parameter values, the term associated with flow alignment is of a
similar order of magnitude to the elastic term.

##  SI 4 Properties of Spindles Perturbed with Drugs

Figure 8: Physical properties of spindles perturbed with drugs compared to
control condition. The properties plotted are MT cross-sectional density
$\langle\rho\rangle$ (top row), spindle volume, and spindle aspect ratio
$x_{0}/R_{0}$. Error bars indicate standard errors in the means. Stars
indicate $p$-values for rejection of the null hypothesis that the observed
values are drawn from distributions with equal means. One star (*) indicates
$0.01<p<0.05$; two stars (**) indicate $p<0.01$ (t-test, equal variances).

##  SI 5 Model for Spacing $s$ Between Inclusions

In the metaphase plate (i.e. the $x=0$ plane), we can find a purely
geometrical relationship between the spindle radius $R_{0}$, the number $N$ of
inclusions, the diameter $d_{0}$ of inclusions, and spacing $s$ between
inclusions (Fig. 9). We make the following simplifying assumptions:

  * •

In the slice of interest, all inclusions are identical disks;

  * •

Inclusions are arranged in a hexagonal lattice with lattice spacing $a$;

  * •

The spacing between the spindle boundaries and the nearest inclusions is
similar to the spacing between inclusions, i.e. we do not treat the boundary
explicitly.

The density $\sigma$ of inclusions is

$\sigma=\frac{N}{\pi R_{0}^{2}}.$ (SI 4)

Neglecting the presence of boundaries, for a hexagonal lattice, the
relationship between density and lattice spacing is given by

$\sigma=\frac{2}{a^{2}\sqrt{3}},$ (SI 5)

and the lattice spacing, inclusion diameter and spacing between inclusions are
related by

$a=s+d_{0}.$ (SI 6)

Combining equations (SI 4), (SI 5), and (SI 6) gives the desired relation,

$s=R_{0}\sqrt{\frac{2\pi}{N\sqrt{3}}}-d_{0}.$

Figure 9:  Model for the spacing between inclusions.

##  SI 6 Extrapolating Experimental Correlation Functions to Infinite System
Size

To minimize artifacts arising from the finite volume of the region where we
calculate correlation functions, we use an extrapolation procedure similar to
that previously used in a variety of experiments and simulations of condensed
matter systems [5, 51, 21]. In many of these systems, the limiting form of the
extrapolating curve can be rigorously calculated. Since this is not the case
here, we use ad hoc linear and quadratic fits to extrapolate our measured
correlation functions to the limit of infinite system size (Fig. 10). We
estimate the true limiting value by taking the average of these two
extrapolated values, and use their difference to estimate the uncertainty
associated with this method. In our linear fits, we discard the points
corresponding to the two smallest boxes. Altering the specifics of the
extrapolation technique, for example by using different fitting functions for
extrapolation, can systematically change our estimates of spindle material
properties (elastic constants and screening lengths) by up to $\sim 30\%$.
However, none of the qualitative results presented in the main text are
sensitive to the details of the extrapolation procedure.

Figure 10:  Extrapolation of experimental correlation functions to infinite
system size. (a) Correlation functions calculated directly from data are shown
as thin colored curves. Each color corresponds to a different box; red
$\rightarrow$blue corresponds to increasing box size. The corresponding
extrapolated function is shown as a bold black curve. (b) The boxes
corresponding to the curves shown in (a). The largest box has edge length
$X_{0}\approx 10\,\mu\text{m}$. Scale bar $5\,\mu\text{m}$.(c) For the
correlation function $S_{\text{nn}}(q_{||}\rightarrow 0,q_{\perp})$, and for a
specific wavenumber $q_{\perp}^{*}$, the infinite-system-size limit is found
by plotting the measured (finite-size) values of
$S_{\text{nn}}(q_{||}\rightarrow 0,q_{\perp}^{*})$ as a function of inverse
box size $X_{0}/X$. Each data point in this plot corresponds to the
intersection of the dashed line in the rightmost panel of (a) with the
measured (colored) curves. The extrapolated value of
$S_{\text{nn}}(q_{||}\rightarrow 0,q_{\perp}^{*})$ is found by fitting the
measured points to a linear (blue) and quadratic (yellow) curve, and
evaluating these fits at $X_{0}/X=0$, which corresponds to $X=\infty$. In the
linear fit, the two data points corresponding to the smallest boxes are
neglected. This procedure is repeated for all value of $\omega$, $q_{||}$, and
$q_{\perp}$ to generate the bold curves shown in (a). Gray areas in (a)
indicate uncertainty calculated from the difference between linear and
quadratic fits.

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