# Cortical microtubule nucleation can organise the cytoskeleton of Drosophila
oocytes to define the anteroposterior axis111This paper has been published:
eLife 4, e06088 (2015). DOI: http://dx.doi.org/10.7554/eLife.06088

Philipp Khuc Trong1,2, Hélène Doerflinger3, Jörn Dunkel1,4, Daniel St.
Johnston3, and Raymond E. Goldstein1 1Department of Applied Mathematics and
Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3
0WA, United Kingdom  2Department of Physics, University of Cambridge, J.J.
Thomson Avenue, Cambridge CB3 0HE, United Kingdom  3Wellcome Trust/Cancer
Research UK Gurdon Institute, Tennis Court Road, Cambridge CB2 1QN, United
Kingdom  4Department of Mathematics, Massachusetts Institute of Technology, 77
Massachusetts Avenue E17-412 Cambridge, MA 02139-4307, USA

###### Abstract

Many cells contain non-centrosomal arrays of microtubules (MT), but the
assembly, organisation and function of these arrays are poorly understood. We
present the first theoretical model for the non-centrosomal MT cytoskeleton in
Drosophila oocytes, in which bicoid and oskar mRNAs become localised to
establish the anterior-posterior body axis. Constrained by experimental
measurements, the model shows that a simple gradient of cortical MT nucleation
is sufficient to reproduce the observed MT distribution, cytoplasmic flow
patterns and localisation of oskar and naive bicoid mRNAs. Our simulations
exclude a major role for cytoplasmic flows in localisation and reveal an
organisation of the MT cytoskeleton that is more ordered than previously
thought. Furthermore, modulating cortical MT nucleation induces a bifurcation
in cytoskeletal organisation that accounts for the phenotypes of polarity
mutants. Thus, our three-dimensional model explains many features of the MT
network and highlights the importance of differential cortical MT nucleation
for axis formation.

###### pacs:

numbers

##  I Introduction

Microtubules (MTs) are polar cytoskeletal filaments that can adopt different
global network organisations to fulfil different functions. The vast majority
of studies have focussed on understanding the architecture and function of MT
arrays organised by centrosomes, such as radial arrays or the mitotic spindle.
In contrast, much less is known about the organisation, assembly and function
of non-centrosomal MT arrays despite their ubiquity in differentiated cell
types, such as neurons, epithelia, fission yeast and plants Bartolini and
Gundersen (2006); Carazo-Salas and Nurse (2006). In a variety of cell-types,
non-centrosomal MT arrays play an essential role in directing the subcellular
localisation of mRNAs to spatio-temporally control gene expression. In
neuronal dendrites, for example, MTs form bidirectional arrays with MT plus-
ends both pointing away and towards the cell body Conde and Caceres (2009);
Kapitein and Hoogenraad (2011). MTs and associated motor proteins were
implicated in the transport of mRNAs along dendrites Bramham and Wells (2007),
where their activity-dependent translation contributes to long term changes in
synaptic function, neuronal circuitry and memory Miller  _et al._ (2002);
Sutton and Schuman (2006); Holt and Schuman (2013). Similar overlapping,
bidirectional MT arrays in the Xenopus oocyte have been proposed to mediate
the transport of Vg1 mRNA to the vegetal cortex, where it orchestrates germ
layer patterning Messitt  _et al._ (2008); Gagnon  _et al._ (2013). The
Drosophila oocyte is probably the best-studied example of mRNA transport along
non-centrosomal MTs. In this system, a diffuse gradient of MTs of mixed
polarity is required for the localisation of bicoid and oskar mRNAs to
opposite ends of the cell Becalska and Gavis (2009). Despite the large amount
of work, however, the organisation of the non-centrosomal MT cytoskeleton
underlying this mRNA localisation is controversial MacDougall  _et al._
(2003); Januschke  _et al._ (2006); Zimyanin  _et al._ (2008), and its
assembly and function are not understood. In stage 9 oocytes, MTs grow from
most parts of the cell cortex into the volume Theurkauf  _et al._ (1992);
Parton  _et al._ (2011) thereby giving rise to a complex, three-dimensional MT
network without pronounced polarity along the anterior-posterior axis
Theurkauf  _et al._ (1992); Cha  _et al._ (2001). In contrast to this apparent
disordered organisation, bicoid and oskar mRNAs become reliably localised by
Dynein and Kinesin to the anterior corners and to the posterior pole of the
oocyte, respectively, thereby defining the anterior-posterior axis Brendza
_et al._ (2000); Duncan and Warrior (2002); Januschke  _et al._ (2002); Weil
_et al._ (2006); Zimyanin  _et al._ (2008).

The most pronounced feature of the MT cytoskeleton at stage 9 is a gradient of
cortical MTs from the anterior to the posterior pole, where MT nucleation is
suppressed by the polarity protein PAR-1 Doerflinger  _et al._ (2006); Roth
and Lynch (2009). Live imaging of oskar mRNAs and direct measurements of
growing MTs showed that the MT network is mostly disordered with only a weak
statistical bias of about 8% more plus ends pointing towards the posterior
pole Zimyanin  _et al._ (2008). This bias vanishes in absence of PAR-1 when
MTs nucleate from all over the cortex Parton  _et al._ (2011). While these
findings established a directionality of the MT meshwork for the first time,
several questions remain unanswered. For example, mRNA transport and
localisation is highly reproducible, raising doubts about whether the
underlying cytoskeletal organisation can be mostly disordered. Moreover,
Dynein-dependent transport to MT minus ends localises injected, so-called
naive bicoid mRNA to the cortex closest to the injection site Cha  _et al._
(2001), which is difficult to reconcile with a cytoskeleton that is simply
biased towards the posterior everywhere. Similarly, the different behaviour of
so-called conditioned bicoid mRNA, which localises specifically to the
anterior cortex irrespective of the injection site Cha  _et al._ (2001), has
remained unexplained. Finally, motor-driven cytoskeletal transport is not the
only transport mechanism in oocytes. Kinesin moves only 13% of oskar mRNA at
any given time, and the remaining 87% is subject to diffusion and slow
cytoplasmic flows that are driven indirectly by Kinesin activity on the MT
network Zimyanin  _et al._ (2008). This raises the question if cytoskeletal
transport alone is sufficient to account for mRNA localisation Glotzer  _et
al._ (1997); Forrest and Gavis (2003) and highlights the importance of
distinguishing its contribution to localisation from the contribution of flows
and diffusion Serbus  _et al._ (2005).

Here, we present the first theoretical model for stage 9 Drosophila oocytes.
Based on the distribution of microtubule nucleation sites around the cortex,
this model accurately reproduces the observed distribution of MTs and
cytoplasmic flows in the oocyte. It reveals that the MT cytoskeleton is
compartmentalised and more ordered than previously thought. By modelling the
movement of mRNAs on this network, we also show that this MT organisation is
sufficient to explain the localisation of oskar and naive bicoid mRNA.
Finally, we show that modulation of MT nucleation gradients causes a
bifurcation in cytoskeletal organisation that explains mutant phenotypes.
Thus, our results explain many features of the assembly, organisation and
function of the non-centrosomal MT array in Drosophila oocytes and highlight
the key role of differential MT nucleation or anchoring at the cortex Lüders
and Stearns (2007).

##  II Results

MTs in the oocyte are nucleated or anchored at the cortex Theurkauf  _et al._
(1992); Parton  _et al._ (2011) and grow from the membrane into the volume.
The cortical MT density follows a steep gradient along the posterior-lateral
cortex from high density at the anterior corners to low densities at the
posterior pole (Fig. 1D, Materials and methods M2.1). In our model, we
emulated these features by selecting seeding points for MTs at random
positions along the oocyte cortex, comprised of two parabolic, rotationally
symmetric caps that capture the typical shape of a stage 9 Drosophila oocyte
(Fig. 1A, Materials and methods M1). The density of seeding points decreases
steeply along the posterior-lateral cortex to zero at the posterior pole, and
decreases weakly along the anterior cortex towards the anterior centre (Fig.
1A-B, Materials and methods M2.1). Each seeding point nucleates a MT polymer
that grows until it either hits a boundary or reaches a target length imposed
by the aging of MTs before catastrophe Gardner  _et al._ (2011) (Materials and
methods M2.3). In total, we computed more than 55000 MTs for each realisation
of a 3D wild-type cytoskeleton.

A cross section through the computed MT meshwork (Fig. 1A,B) bears a striking
resemblance to confocal images of Tau- Micklem  _et al._ (1997); Ganguly  _et
al._ (2012) and EB-1- Parton  _et al._ (2011) tagged MTs, correctly showing
the pronounced anterior-posterior gradient of MT density. Taking into account
the orientations of all MT segments in the 3D volume, it also reproduces the
experimentally measured directional bias Parton  _et al._ (2011) with 8.5%
more MT segments pointing posteriorly than anteriorly (Fig. 1A, inset). The
directional bias in a 2D slice varies depending on the depth of the slice in
the oocyte, ranging from 50.7% of posteriorly oriented MT segments at the
cortex to 60.6% in the mid plane (Fig. 1B, inset). This demonstrates that
measurements in 2D confocal slices poorly characterise the fully extended 3D
system.

Central to understanding mRNA localisation is the question of MT orientations.
Calculating the local vectorial sum of MT segments on a coarse-grained grid
gives the local MT orientations in the computed cytoskeleton. Local
orientations of an individual realisation of the cytoskeleton (Fig. 1A,B) show
poor global network order (Fig. 1C). However, cargo localisation in the oocyte
does not involve only a single cytoskeletal realisation. MTs disappear within
minutes in the presence of the MT-depolymerising drug colchicine that blocks
MT growth by sequestering free tubulin dimers, indicating that the whole
network turns over rapidly Theurkauf  _et al._ (1992); Zhao  _et al._ (2012)
(V. Trovisco, personal communication). Thus, the oocyte samples many tens to a
hundred of independent MT organisations over the 6-9 hours of stage 9.
Summation of local orientations over an ensemble of 50 independent
realisations of the computed MT network reveals a striking spatial
partitioning of the cytoskeleton into several subcompartments (Fig. 1F). At
the anterior, the mean orientation points posteriorly, while MTs at the
lateral sides on average point inwards toward the AP-axis. Counting the number
of MT segments that contribute to each grid box in the ensemble also shows the
distribution of MT density (Fig. 1H). In agreement with experiments (Fig. 1D),
the MT density exhibits a pronounced anterior-posterior gradient with highest
values in the anterior corners, even when MT seeding is uniform on the
anterior surface.

In the limit of a large ensemble, the MT polymers sample all possible initial
directions and possible lengths at every point along the cortex. In this
limit, we can test the predicted compartmentalised MT organisation by
constructing a second model in which MTs are represented as straight rods. The
net orientation of the cytoskeleton at a given point inside the oocyte is then
computed as weighted sum of MT contributions from each point along the
boundary (Materials and methods M3.1).

With a shorter MT target length to compensate for the effectively longer
length of straight rods compared to curved polymers, computation of net MT
orientations in the rod model shows a topology (Fig. 1G) that confirms the
mean topology in the polymer model (Fig. 1F). The three compartments of a
posterior-pointing anterior section and two inwards-pointing lateral sections
are effectively bounded by separatrices (Fig. 1G, orange arrows). Cargo
molecules that are transported on MTs to their plus ends move along the arrows
and converge at the separatrices, eventually leading to the posterior pole as
the sole point of attraction in the entire oocyte volume (the attractor). By
contrast, cargo that is transported to MT minus ends moves opposite to the
arrows and diverges away from the separatrices.

This mean topology of the MT cytoskeleton is insensitive to the exact choice
of the MT nucleation probability and to the choice of the MT length
distribution (Materials and methods M2.1, M2.3). It also remains unchanged in
a differently shaped oocyte geometry in both the polymer model and the rod
model (Figure 1 - figure supplement 1). In summary, both models show that even
disordered non-centrosomal MT arrays can feature well-defined mean
organisations, and that the MT cytoskeleton in oocytes is organised in a
compartmental fashion.

The suitably scaled local vectorial sum of MT segments (Fig. 1C) for an
individual realisation of the polymer model (Fig. 1A-B) is a vector field
$\bm{v}_{m}$ which represents active Kinesin-driven transport on the
cytoskeleton (Materials and methods M2.5). In vivo during stage 9, the oocyte
cytoplasm undergoes slow cytoplasmic flows that are abolished in kinesin heavy
chain mutants. This indicates that flows are driven by kinesin-dependent
transport of an unknown cargo through the viscous cytoplasm Palacios and St
Johnston (2002); Serbus  _et al._ (2005), thus making cytoplasmic flows a
secondary read-out of cytoskeletal organisation Khuc Trong  _et al._ (2012).
Therefore, we next tested if our computed polymer MT cytoskeleton can produce
flows that are consistent with observed cytoplasmic streaming.

Measurements of speeds of autofluorescent yolk granules in live stage 9
oocytes showed that mean flow velocities are slow (Fig. 2F). The physics of
slow incompressible fluid flows $\bm{u}$ driven by forces $\bm{f}$ is
described by the Stokes equations

$0=-\bm{\nabla}p+\mu\bm{\nabla}^{2}\bm{u}+\bm{f}~{},\quad\quad\bm{\nabla}\cdot\bm{u}=0~{}.$
(1)

We make the simplest possible assumption that the forces $\bm{f}$ are
proportional to the motor-velocity field, and use experimentally measured flow
speeds to calibrate the scalar factor of proportionality. By solving the
Stokes equations, we then computed the full 3D fluid flow field (Fig. 2A)
corresponding to an individual realisation of the MT cytoskeleton (Fig. 1A),
and 2D cross sections through the 3D field (Fig. 2B) were compared to in vivo
flow patterns visualised by particle image velocimetry (PIV, Fig. 2C).

Despite large variability, both computed and in vivo flows show very similar
patterns, generally being strongest in the anterior half of the oocyte, and
weaker in the posterior half (Fig. 2B-C). Computed flows occasionally reach
further into the posterior half than typically seen in PIV flow fields,
thereby slightly overestimating the range of flows. However, except in rare
cases, computed flows do not reach the posterior pole. This result is largely
independent of the presence of the oocyte nucleus which occupies less than
2.5% of the oocyte volume, even for small oocytes at the beginning of stage 9
(Figure 2 - figure supplement 1, Materials and methods M5.2, M5.3). Thus,
computed flows appear consistent with observed slow cytoplasmic streaming.

We tested next if our computed cytoskeleton in combination with the derived
cytoplasmic flows and diffusion can account for dynamic mRNA transport and
localisation. We described the mRNA distributions as continuous concentration
fields. oskar mRNA is assumed to reside in either one of two states: the
Kinesin-bound state with concentration $c_{b}$, in which cargo is transported
actively on the cytoskeleton-derived motor-velocity field $\bm{v}_{m}$ (Fig.
1C), or the unbound state with concentration $c_{u}$, in which cargo is
transported by the cytoplasmic flows $\bm{u}$ (Fig. 2A) and diffuses with
diffusion constant $D$. Cargo can exchange between both states by chemical
reactions, thereby resulting in the reaction-advection-diffusion equations:

$\displaystyle\partial_{t}\,c_{b}+\bm{\nabla}\cdot\left(\bm{v}_{m}\,c_{b}\right)$
$\displaystyle=$ $\displaystyle k_{b}\,c_{u}-k_{u}\,c_{b}$ (2)
$\displaystyle\partial_{t}\,c_{u}+\bm{\nabla}\cdot\left(\bm{u}\,c_{u}\right)$
$\displaystyle=$ $\displaystyle-
k_{b}\,c_{u}+k_{u}\,c_{b}+D\bm{\nabla}^{2}c_{u}$

Parameter values were constrained with experimentally measured values
(Materials and methods M4.2). Furthermore, throughout the simulation we cycled
through the pairs of fluid flow $\bm{u}$ and motor-velocity fields
$\bm{v}_{m}$ (Fig. 3) to account for the dynamic nature of the MT network
Parton  _et al._ (2011) and flow patterns, which change over time scales of
minutes compared to the 6-9 hours during which oskar mRNA is localised
(Materials and methods M5.1).

Starting from an initial diffuse cloud of oskar mRNA in the centre of the
oocyte (Fig. 3A,D,E), simulations show that the mRNA quickly concentrates in
the centre before forming a channel from the centre of the oocyte towards the
posterior pole (Fig. 3E). Formation of this channel reflects the locally
inwards orientation of MTs in the posterior half of the oocyte (Fig. 1E,F).
This transient state closely resembles transient patterns of fluorescently-
tagged oskar mRNA during the transition between stages 8 and 9 (Fig. 3B),
thereby supporting the notion that our computed MT cytoskeleton correctly
captures key aspects of the in vivo MT network.

Upon reaching the posterior cortex, oskar mRNA is translated to produce Long
Osk Vanzo and Ephrussi (2002), which is involved in anchoring oskar mRNA.
However, oskar mRNA localises normally at stage 9 when anchoring is disrupted
Vanzo and Ephrussi (2002); Micklem  _et al._ (2000). Anchoring is therefore
not necessary at stage 9, and we did not include it in our model at this
point. Despite the lack of any anchoring, oskar mRNA forms a posterior
crescent after 1.5 hours and becomes highly concentrated by the end of the
simulation (Fig. 3F), correctly capturing observations of in vivo localisation
(Fig. 3C). A 4$\mu$m thick slice at the posterior pole contains 12.5% of total
cargo in the oocyte (Fig. 3F), showing that continual transport combined with
slow diffusion is sufficient to reach and maintain high concentrations of
mRNA.

In our description of transport (Eq. (2)), cargo acts as an unspecific passive
tracer. Passive tracers merely follow the transport fields $\bm{v}_{m}$ and
$\bm{u}$, and these two fields must contain all information about the
localisation sites. We therefore asked which field contains most information
about localisation. We first tested localisation in the absence of cytoplasmic
flows by setting the fluid flow field identical to zero $\bm{u}\equiv 0$. This
situation is similar to but more severe than in slow Kinesin mutants Serbus
_et al._ (2005). Starting from the central oskar mRNA cloud (Fig. 3D), we
found that oskar mRNA still localises to and forms a concentrated crescent at
the posterior pole (Fig. 3G). A 4$\mu$m thick slice off the posterior pole
contains 11.3% of total available oskar mRNA cargo, only marginally less than
localisation with cytoplasmic flows present. This suggests that the
cytoskeletal transport alone localises the majority of mRNA. Cargo
localisation in the absence of cytoskeletal transport can be tested by either
setting the motor-velocity field identical to zero $\bm{v}_{m}\equiv 0$, or by
setting the binding constant $k_{b}$ to zero. In either case, cargo disperses
throughout the oocyte without forming a central channel and completely fails
to produce any posterior crescent. We further asked whether the flow could
localise cargo to wild-type levels if an anchor captured cargo at the
posterior. To this end, we replaced the bound state in the model by an
anchored state to which cargo can bind only at the very posterior boundary.
Binding to the anchor occurs with a reaction rate constant that is ten times
larger than the reaction rate constant $k_{b}$ used for oskar mRNA binding to
the MT cytoskeleton (Materials and methods M4.2). Moreover, anchored cargo
cannot be released, thus making this anchor a perfect sink with stronger
trapping properties than any realistic anchor. Under these idealised
conditions, oskar mRNA accumulates slightly at the anchor (Fig. 3K). However,
a 4$\mu$m thick posterior slice contains only 2.7% of total cargo in the
oocyte, about 78% less cargo than in the wild-type simulations, and only ≈25%
more cargo compared to the purely diffusive case, which localises 2.2% of
cargo. Thus, even under the most favourable conditions, the cytoplasmic flows
cannot localise mRNA to wild-type levels, arguing against a mixing and
entrapment mechanism for oskar mRNA localisation at stage 9. Despite the small
fraction of actively transported cargo (13%), motor-driven transport is both
necessary and sufficient to account for oskar mRNA localisation.

In contrast to oskar mRNA, bicoid mRNA is believed to be transported by Dynein
Duncan and Warrior (2002); Januschke  _et al._ (2002); Weil  _et al._ (2006),
but other parameters such as the fraction of bound cargo and the active
transport speeds are similar to oskar mRNA Belaya (2008) (V. Trovisco,
personal communication). To test whether the proposed cytoskeletal
organisation also captures transport and localisation of injected bicoid mRNA,
we inverted the directions of the motor-velocity fields (Fig. 1C) to account
for the minus end directed transport by Dynein instead of the plus end
directed transport by Kinesin. Other parameters including cytoplasmic flow
fields remained unchanged.

Closely matching experiments with injected naive bicoid mRNA Cha  _et al._
(2001), simulations show accumulation of bicoid mRNA at both posterior-lateral
sides of the oocyte when placed initially in the posterior half (Fig. 3H,
inset), or accumulation at the anterior and ventral cortex when initially
placed in the anterior-ventral area (Fig. 3I, inset). Localisation of bicoid
mRNA is also virtually identical in the absence of cytoplasmic streaming.
Thus, splitting of a bulk amount of injected bicoid mRNA occurs when the RNA
is placed on the border (separatrices) between two diverging subcompartments
of the MT cytoskeleton, each one transporting part of the cloud towards the
adjacent cortex. The agreement between simulations and experiments therefore
further supports an on average compartmentalised MT cytoskeleton, and naive
bicoid mRNA, like oskar mRNA, unspecifically traces out the MT cytoskeleton.

In contrast to naive bicoid mRNA, endogenous bicoid mRNA is transported into
the oocyte after being transcribed in neighbouring nurse cells where it also
becomes modified in presence of the protein Exuperantia. Endogenous bicoid
mRNA can be approximated by so-called conditioned bicoid mRNA that is first
injected into the nurse cells, then sucked out and injected into the oocyte.
Conditioned bicoid mRNA localises specifically to the anterior surface within
30 minutes, even if this surface is not closest to the injection site Cha  _et
al._ (2001). In the model, starting from an initial distribution at the
anterior surface where endogenous bicoid mRNA enters the oocyte, cargo quickly
moves to the anterior cortex before concentrating in the anterior corners
after several hours of simulated time (Fig. 3J). This resembles the anterior
ring-like localisation of bicoid mRNA in wild-type stage 9 oocytes St Johnston
_et al._ (1989); Weil  _et al._ (2006), and occurs independently of
cytoplasmic flows. However, the model does not reproduce transport
specifically to the anterior surface when injection is further away from the
anterior. This suggests that Exuperantia activity somehow either enables
bicoid mRNA to move along an unidentified population of MTs Cha  _et al._
(2001); MacDougall  _et al._ (2003) that are not included in the computed
cytoskeleton, or that it allows localisation of the mRNA by an unknown MT-
independent mechanism.

We have shown so far that cortical gradients of MT nucleation are sufficient
to assemble a functional, compartmentalised MT cytoskeleton that successfully
localises oskar mRNA and naive bicoid mRNA. Next, we asked if cortical MT
nucleation can also produce cytoskeletal organisations that explain mutants
with partial or complete polarity defects. Mutations interfering with the
posterior follicle cells that surround the oocyte, for example, can disrupt
the proper positioning of mRNAs Gonzalez-Reyes  _et al._ (1995); Roth  _et
al._ (1995). Specifically, follicle cell clones (ras${}^{\Delta\text{C40b}}$)
that are adjacent to only one side of the oocyte posterior repel cargo from
that side of the cortex (termed clone adjacent mislocalisation Poulton and
Deng (2006)), likely due to an altered MT organisation. To mimic this
phenotype, we used an ensemble of wild-type cytoskeletons and artificially
added MT nucleation sites to a patch on one side of the posterior pole.
Simulations of cargo transport with diffusion, motor transport and flows show
that oskar mRNA is repelled from this patch and localises to the adjacent
posterior boundary, in agreement with experiments (Figure 4 - figure
supplement 2).

Mislocalisation of mRNAs also occurs in mutants of the polarity protein PAR-1,
which acts to suppress MT nucleation at the posterior pole of the oocyte
Shulman  _et al._ (2000); Doerflinger  _et al._ (2010). In strong par-1
hypomorphs with strongly reduced PAR-1 expression, MT minus ends occupy the
whole cortex including the posterior pole Parton  _et al._ (2011), thereby
causing oskar mRNA to mislocalise to a dot in the centre of almost 90% of
mutant oocytes Doerflinger  _et al._ (2006). To test this phenotype in the
model we distributed MT seeding points uniformly on the posterior surface
(Fig. 4G, inset) while keeping the seeding density on the anterior surface
constant. The anterior-posterior gradient of MT density then vanishes and the
directional bias of MT segments evens out (Fig. 4G, ensemble-averaged 3D-bias:
50.1%:49.9%). The ensemble-averaged local orientations of MTs show that MTs
point towards a single focal point in the centre of the oocyte (Fig. 4G) that
acts as a stable fixed point of the system. Computation of 3D flow fields for
each realisation in the new ensemble (3D mean: 13.5 nm/s; 2D mean: 14.5 nm/s,
N=50) and simulations of transport with and without cytoplasmic flows show
oskar mRNA concentrating in a cloud in the centre of the oocyte (Fig. 4C).
oskar mRNA accumulation to a central dot also remains unchanged if the
posterior MT seeding density is decreased slightly (Fig. 4H, inset) as might
be expected for hypomorphs that do not abolish PAR-1 expression completely.

Interestingly, bicoid mRNA simulated either with slightly decreased (Fig. 4H)
or with uniform (Fig. 4G) posterior MT nucleation not only localises to the
anterior corners as in wild-type but also enriches at the posterior pole (Fig.
4D, arrowheads), despite injection close to the anterior (Fig. 4D, inset).
This matches experiments showing bicoid mRNA mislocalisation to both anterior
and posterior in gurken, torpedo and cornichon mutants (EGFR mutants) in which
posterior follicle cells fail to signal a MT reorganisation in the oocyte
Gonzalez-Reyes  _et al._ (1995); Roth  _et al._ (1995).

Failure of the external signal to the oocyte is thought to have the same
effect as par-1 hypomorphs, but previous experiments with bicoid mRNA in par-1
hypomorphs resulted in ambiguous localisations Shulman  _et al._ (2000);
Benton  _et al._ (2002). Here, using confocal fluorescence in-situ
hybridisation (FISH), our experiments show that bicoid mRNA localises
primarily to the anterior corners and to the posterior pole in strong par-1
hypomorphs (Fig. 4L), thereby mirroring EGFR mutants and agreeing with the
model predictions. Thus, (near-)uniform MT nucleation along the posterior and
lateral cortex is sufficient to explain the polarity phenotypes observed for
oskar and bicoid mRNAs in gurken, torpedo and cornichon mutants and in strong
par-1 hypomorphs.

In 10% of strong par-1 hypomorphs and 30% of weak par-1 hypomorphs with weakly
reduced PAR-1 expression, oskar mRNA is only partially mislocalised, combining
a wild-type-like posterior crescent with a centrally mislocalised dot
Doerflinger  _et al._ (2006). We therefore modulated the MT nucleation along
the posterior-lateral cortex to test if this is sufficient to produce
intermediate cytoskeletal organisations between wild-type (Fig. 4F) and strong
par-1 hypomorphs (Fig. 4H).

Increasing MT nucleation laterally towards the posterior pole (Fig. 4G, inset)
again creates a stable focus in the centre of the oocyte. However, its domain
of attraction does not cover the entire oocyte. Instead, a small basin of
attraction towards the posterior pole persists (Fig. 4G, red), separated from
the basin of attraction of the stable focus by an unstable saddle-node point.
Further increasing the seeding density towards the posterior pole shows that
the pair of stable and unstable fixed points move further apart (Figure 4 -
figure supplement 1) until the unstable point no longer resides inside the
oocyte geometry, thereby giving rise to the strong par-1 hypomorph topology
(Fig. 4H, I, J). Cargo simulated on cytoskeletons with stable and unstable
fixed points and their corresponding flow fields can split into two separate
accumulations, first at the posterior pole and second in a dot along the AP-
axis (Fig. 4B). This agreement with experimental results suggests that
intermediate levels of PAR-1 at the posterior lead to a cytoskeleton with a
potential barrier between the oocyte centre and its posterior pole. It also
shows that modulation of MT nucleation along the posterior-lateral cortex
alone is sufficient to capture this. Because the potential barrier only
affects the bound state, diffusion and flows can aid posterior localisation in
this context by pushing unbound cargo across the unstable point and increasing
the amount of oskar mRNA that reaches the posterior pole. In summary,
variations of MT nucleation along the cortex not only explain mRNA
localisations in wild-type but can also account for the mislocalisations of
bicoid and oskar mRNAs in polarity mutants.

The creation of stable and unstable fixed points in the transition from wild-
type to strong par-1 hypomorph topology constitutes a classical saddle-node
bifurcation (Fig. 4K-M, Materials and methods M3.3) with the cortical MT
seeding density as the bifurcation parameter. Interestingly, in both the
polymer model and the rod model, the mean length of MTs acts as a second
bifurcation parameter. For example, for sufficiently short MT filaments (Fig.
4K), few MTs from the anterior can reach and contribute to orientations in the
posterior half of the oocyte. In the posterior half, MTs from the lateral side
either point towards the anterior, thereby combining with anterior MTs to
create a stable node, or towards the posterior, to form a domain of attraction
at the posterior pole. Therefore, for sufficiently weak contributions from
anterior MTs, MTs from the lateral sides create the unstable tipping-point.

To understand more generally how the MT lengths and the posterior-lateral
distribution of seeding points together influence the three different
cytoskeletal topologies (Fig. 4E-G) we computed the parameter space of the
straight rod model (Fig. 5). If MTs are absent at the posterior pole, the
wild-type topology (Fig. 5D i, vi) covers a large fraction of parameter space
(Fig. 5A, red) for sufficiently long MTs (large $\epsilon$) and cortical MT
nucleation gradient (large $k_{P}$). The creation of two fixed points splits
the oocyte into different domains of attraction (Fig. 5A, blue, D i-ii) for
either shorter MTs (Fig. 5A, vertical dashed arrow; Figure 4 - figure
supplement 1 I,N) or for increasingly uniform MT nucleation (Fig. 5A,
horizontal dashed arrow; Figure 4 - figure supplement 1 I,H). For almost
uniform nucleation along the entire posterior-lateral cortex, the unstable
fixed point exits the geometry, leaving behind only a single stable focus
(Fig. 5A, green, arrowhead, D iii-iv). If some MT nucleation is allowed at the
posterior pole this region of strong par-1 hypomorph topology expands
substantially (Fig. 5B, C, green). Interestingly, if MTs are sufficiently
long, MTs from the lateral and anterior cortex can reach and overpower those
from the posterior pole, thus restoring wild-type-like cytoskeletal topology
that allows oskar mRNA localisation. Therefore, a low level of posterior
nucleation of MTs does not necessarily lead to mislocalisation of oskar mRNA.
Instead, and in addition to the distribution of cortical MT nucleation, the
length of MT filaments emerges as another key regulator for polarisation and
function of the non-centrosomal network.

##  III Discussion

Non-centrosomal MT networks represent a large, yet poorly understood class of
MT arrangements that often fulfil specialised functions Keating and Borisy
(1999). Non-centrosomal MTs are frequently aligned in parallel, thereby
forming linear arrays as in epithelia or neurons Bartolini and Gundersen
(2006); Conde and Caceres (2009); Kapitein and Hoogenraad (2011). The non-
centrosomal MT cytoskeleton in Drosophila oocytes was also first hypothesised
to form a highly polarised linear array with MTs growing from the anterior
surface towards the posterior pole Clark  _et al._ (1994); Brendza  _et al._
(2000). Instead, the MT cytoskeleton is a complex network without a visibly
discernible polarity along the anterior-posterior axis Theurkauf  _et al._
(1992); Cha  _et al._ (2001, 2002), and its organisation remained ambiguous
MacDougall  _et al._ (2003); Januschke  _et al._ (2006); Zimyanin  _et al._
(2008); Parton  _et al._ (2011).

We showed that in two different theoretical models, in which MTs grow from the
oocyte cortex into the volume, the cytoskeleton features an on average
compartmental organisation and is therefore more ordered than previously
thought. In combination with cytoplasmic flows and diffusion, this
cytoskeletal organisation successfully reproduces localisations of oskar and
naive bicoid mRNAs in wild-type oocytes. Such a spatially varying,
compartmental organisation suggests that a single statistical measure of
polarity, derived by averaging data from the entire oocyte Zimyanin  _et al._
(2008); Parton  _et al._ (2011), may not be a sufficient metric to
characterise mRNA localisation.

Cytoplasmic flows may generally contribute to mRNA transport. Flows at later
stages of oogenesis are fast and well-ordered, and occur concurrently with a
late-phase enhancement of oskar mRNA accumulation Sinsimer  _et al._ (2011).
This is consistent with a contribution of flows to mRNA localisation Glotzer
_et al._ (1997); Forrest and Gavis (2003). At stage 9 of oocyte development,
which is the focus of the work presented here, flows are slow and chaotic. At
this stage, mutants with reduced flows showed oskar mRNA localising normally
Serbus  _et al._ (2005), thus suggesting that slow and chaotic flows may not
play the same role as in late oogenesis. However, flows in these mutants were
merely reduced rather than abolished completely, and the measurements
underestimated flow speeds Serbus  _et al._ (2005), hence leaving the
interpretation of flows unclear. We here find that the effects of slow
cytoplasmic flows on mRNA transport are negligible, and that cytoskeletal
transport alone is sufficient for localisations of oskar and naive bicoid
mRNAs. In this view, slow cytoplasmic flows arise primarily as inevitable
physical byproduct of active motor-driven transport on the cytoskeleton rather
than as an evolutionarily selected trait. This appears to mirror findings in
the C. elegans zygote in which P-granules segregate by dissolution and
condensation rather than via transport by cytoplasmic flows Brangwynne  _et
al._ (2009).

Interestingly, neither MT-based transport alone nor combined with cytoplasmic
flows and diffusion is sufficient to reproduce the anterior localisation of
nurse cell conditioned bicoid mRNA irrespective of the position of injection
into the oocyte Cha  _et al._ (2001). This Exuperantia-dependent mechanism may
rely either on an unobserved population of MTs in the oocyte that can be
specifically recognised by conditioned bicoid mRNA Cha  _et al._ (2001);
MacDougall  _et al._ (2003), or on an unknown MT-independent mechanism.

The central finding of our work is that gradients of cortical MT nucleation
are sufficient for the assembly of a functional compartmentalised MT
cytoskeleton in wild-type oocytes. While many non-centrosomal MT arrays are
linear and emphasise questions about establishment and maintenance of parallel
filament orientations Ehrhardt (2008); Lindeboom  _et al._ (2013), our result
stresses the need to understand gradients in MT nucleation as an alternative
strategy for the assembly of functional non-centrosomal arrays. Whether MTs in
Drosophila oocytes are differentially nucleated along the cortex itself or
created elsewhere and then differentially anchored at the cortex remains an
interesting open question, but both scenarios are compatible with our model.
Understanding how this gradient is established will therefore depend on
discovering how PAR-1 regulates MT interactions with the cortex.

Cortical MT gradients cannot only account for the wild-type cytoskeletal
configuration but also for the phenotypes observed in a hierarchy of par-1
hypomorphs. Modulation of MT gradients along the posterior-lateral cortex
alone are sufficient to explain the splitting of oskar mRNA between the centre
and the posterior pole of the oocyte Doerflinger  _et al._ (2006) via a
saddle-node bifurcation, suggesting that the anterior and posterior-lateral
surfaces of the oocyte are functionally decoupled. Generally, bifurcations in
temporal behaviour govern important qualitative transitions in many biological
systems, such as the lactose network in E. coli Ozbudak  _et al._ (2004), cell
cycle in yeast Charvin  _et al._ (2010), and collapses of bacterial
populations Dai  _et al._ (2012). In Drosophila, one important qualitative
change is the temporal transition between stages 7/8 and 9. During this
transition the MT cytoskeleton reorganises from uniform nucleation around the
cortex and oskar mRNA in the centre to the anterior-posterior MT nucleation
gradient with oskar mRNA at the posterior. It is therefore tempting to
speculate that the sequence of PAR-1 mutants and the underlying bifurcation
represent static snapshots of this dynamic developmental transition in wild-
type.

In conclusion, the present work provides a model that describes the assembly,
organisation and function of the non-centrosomal MT array in Drosophila
oocytes and directs future attention to the molecular mechanisms that enable
differential MT nucleation or anchoring at the cortex.

##  IV Material and Methods

###  IV.1 M1 Coordinates and oocyte geometries

For calculation of the microtubule (MT) cytoskeleton, the anterior-posterior
(AP) axis is aligned with the $z$-axis of a cartesian coordinate system, and
results are later shifted and rotated to align with the $x$-axis for
visualization, for subsequent computations of cytoplasmic flows and
simulations of cargo transport. Dimensional coordinates cover the ranges
$x\in[-L,L]$, $y\in[-L,L]$ and $z\in[0,z_{0}\,L]$ with length scale
$L=50\,\mu$m. After nondimensionalization with scale $L$, coordinates are
ranged as $x^{\prime}\in[-1,1]$, $y^{\prime}\in[-1,1]$ and
$z^{\prime}\in[0,z_{0}]$. Similarly, nondimensionalization of the shape
parameter $k_{P}$ in the MT seeding density leads to $k_{P}^{\prime}=k_{P}/L$,
and all primes will be dropped subsequently. Methods figures are shown in
nondimensional spatial coordinates, and reported parameter values are
nondimensional unless noted otherwise.

The 3D geometry of a typical stage 9 Drosophila oocyte is defined as two
parabolic, rotationally-symmetric caps (Fig. 1). Anterior ($i=A$) and
posterior ($i=P$) parabolic caps are parameterized in cylindrical coordinates
$\rho=\sqrt{x^{2}+y^{2}}\in[0,1]$ and $\phi\in[0,2\pi)$ as

$\displaystyle\bm{\sigma}_{i}(\rho,\phi)=\rho\,\bm{\hat{e}}_{\rho}+z_{0}^{i}\left(1-\rho^{2}\right)\bm{\hat{e}}_{z}\,,$
(3)

where $\bm{\hat{e}}_{\rho}=(\cos(\phi),\sin(\phi),0)$ and
$\bm{\hat{e}}_{z}=(0,0,1)$ are the unit vectors in radial and $z$-direction,
respectively. The line element along the parabola is calculated as

$\mathrm{d}s_{i}=\left\|\frac{\partial\bm{\sigma}_{i}}{\partial\rho}\right\|\mathrm{d}\rho=\sqrt{1+(2z_{0}^{i}\,\rho)^{2}}\mathrm{d}\rho$

with arclength

$s_{i}(\rho)=\frac{\rho}{2}\sqrt{1+(2z_{0}^{i}\,\rho)^{2}}+\frac{1}{4z_{0}^{i}}\sinh^{-1}(2z_{0}^{i}\,\rho)$
(4)

defined such that $s_{i}(\rho=0)=0$ denotes the tip of the parabolic cap while
$s_{i}(\rho=1)=s_{0}^{i}$ denotes the distance from the tip to the anterior
corners (Fig. 6 A,G), and hence $0\leq s_{i}(\rho)\leq s_{0}^{i}$. The surface
element for parabolic caps (in units of $L$) is given by

$\mathrm{d}\Sigma_{i}=\left\|\frac{\partial\bm{\sigma}_{i}}{\partial\rho}\times\frac{\partial\bm{\sigma}_{i}}{\partial\phi}\right\|=\rho\,\sqrt{1+\left(2z_{0}^{i}\rho\right)^{2}}\mathrm{d}\rho\,\mathrm{d}\phi\,,$
(5)

with total surface area

$\Sigma_{i}=\frac{\pi}{6\left(z_{0}^{i}\right)^{2}}\left(\left[1+(2z_{0}^{i})^{2}\right]^{3/2}-1\right)\,.$
(6)

The inward pointing normals for posterior and anterior caps are given by

$\displaystyle\bm{\hat{n}}_{P}(\rho,\phi)$ $\displaystyle=$
$\displaystyle-\frac{2z_{0}^{P}\,\rho\,\bm{\hat{e}}_{\rho}+\bm{\hat{e}}_{z}}{\sqrt{1+(2z_{0}^{P}\rho)^{2}}}$
(7) $\displaystyle\bm{\hat{n}}_{A}(\rho,\phi)$ $\displaystyle=$
$\displaystyle\frac{2z_{0}^{A}\,\rho\,\bm{\hat{e}}_{\rho}+\bm{\hat{e}}_{z}}{\sqrt{1+(2z_{0}^{A}\rho)^{2}}}\,.$

A special case arises for $z_{0}^{A}=0$ when the anterior parabolic cap
becomes a flat disc. In this case, the arclength (eq. (4)) reduces to
$s_{A}(\rho)=\rho$, and the surface area (eq. (6)) simplifies to
$\Sigma_{A}=\pi$. The inwards pointing normals for this case are given by
$\bm{\hat{n}}_{P}$ (eq. (7)) for the posterior cap and
$\bm{\hat{n}}_{A}=\bm{\hat{e}}_{z}$ for the anterior disc, respectively.

To investigate the robustness with respect to changes in oocyte shape, we test
our model for the cytoskeleton, cytoplasmic flows and mRNA transport in two
different geometries for a stage 9 Drosophila oocyte. Geometry-1 is used as
standard geometry, and is comprised of two parabolic caps (eq. (3)) with
$z_{0}^{P}=1.48$ for the posterior cap and $z_{0}^{A}=0.2$ for the anterior
cap (see Fig. 1). This results in a length of the anterior-posterior axis (AP-
axis) of $z_{0}^{P}-z_{0}^{A}=1.28$ with an aspect ratio of $1.56$ that
qualitatively resembles a typical stage 9 oocyte. Geometry-2 is tested as
alternative geometry, and consists of a posterior parabolic cap with
$z_{0}^{P}=1$, and an anterior flat disc with $z_{0}^{A}=0$ (Fig. 1 - figure
supplement 1), giving an AP-axis length of $1$ and aspect ratio of $2$. We
find that results are robust with respect to this change in geometry.

###  IV.2 M2 Polymer model for MT cytoskeleton

####  IV.2.1 M2.1 MT nucleation probability

The first step in the computation of the MT cytoskeleton is the generation of
MT seeding points that are randomly positioned along the oocyte membrane. We
define the following probability density along the arclength $s_{i}(\rho)$ of
the anterior and posterior parabolic caps

$p_{\Sigma}(s_{i}(\rho)|h_{0}^{i},k_{i})=\frac{1}{A}\,\tilde{p}_{\Sigma}(s_{i}(\rho)|h_{0}^{i},k_{i})\,,$
(8)

with normalization over the oocyte surface

$\displaystyle A$ $\displaystyle=$
$\displaystyle\int_{\Sigma_{A}}\\!\mathrm{d}\Sigma_{A}\,\tilde{p}_{\Sigma}(s_{A}(\rho)|h_{0}^{A},k_{A})$
(9) $\displaystyle+$
$\displaystyle\int_{\Sigma_{P}}\\!\mathrm{d}\Sigma_{P}\,\tilde{p}_{\Sigma}(s_{P}(\rho)|h_{0}^{P},k_{P})~{},$

where the surface elements were specified in eq. (5) and the arclength was
given in eq. (4). For the functional form of
$\tilde{p}_{\Sigma}(s(\rho)|h_{0},k)$ we use the expression

$\tilde{p}_{\Sigma}(s(\rho)|h_{0},k)=h_{0}+(1-h_{0})\left[1+\left(\frac{k}{s_{0}}\right)^{2}\right]\frac{s(\rho)^{2}}{k^{2}+s(\rho)^{2}}\,,$
(10)

wherein the index $i$ was dropped for simplicity. Note that for the maximum
arclength $s=s_{0}$ it holds that $\tilde{p}_{\Sigma}(s=s_{0}|h_{0},k)=1$.

$h_{0}\in[0,1]$ and $k\in(0,\infty)$ represent two parameters governing the
shape of the probability density. For small values $k\ll s_{0}$ the nucleation
density approaches a Hill function with Hill coefficient $n=2$ and depth
$h_{0}$

$\lim_{k\ll
s_{0}}\tilde{p}_{\Sigma}(s(\rho),|h_{0},k)=h_{0}+(1-h_{0})\frac{s(\rho)^{2}}{k^{2}+s(\rho)^{2}}\,.$
(11)

In the opposite limit $k\gg s_{0}$, the probability density approaches a
parabola with depth $h_{0}$ (Fig. 6B,H, red curve)

$\lim_{k\gg
s_{0}}\tilde{p}_{\Sigma}(s(\rho),|h_{0},k)=h_{0}+(1-h_{0})\left(\frac{s(\rho)}{s_{0}}\right)^{2}\,,$
(12)

thus creating a pronounced gradient of MT nucleation from the pole ($s=0$) to
the corners ($s=s_{0}$).

MT nucleation along the arclength can increase from the gradient to a
homogeneous nucleation along the arclength in two different ways: (i) either
by increasing nucleation from the corners laterally towards the posterior
pole, or (ii) by increasing nucleation uniformly at the posterior pole. Each
shape parameter $h_{0}$ and $k$ accounts for one of these possibilities.
Increasing the parameter $h_{0}$ leads to shallower gradients by increasing
the probability for nucleation at the pole from 0 for $h_{0}=0$ to the value
for homogeneous nucleation for $h_{0}=1$ (Fig. 6H). Contrary, decreasing the
parameter $k$ leads to narrower regions of low nucleation probability by
decreasing its width laterally (Fig. 6B).

To compare to experimental data, we stained $\alpha$-tubulin in fixed stage 9
wild-type Drosophila oocytes and extracted fluorescence intensity profiles
along the cortex from the posterior pole to the anterior corners (Fig. 7A,B).
An ensemble of smoothed fluorescence profiles shows steep increases in
cortical MT density from posterior to anterior, albeit with high variability.
Still, a quadratic fit of the ensemble shows that a parabolic function is a
plausible representation of the increasing MT density (Fig. 7C). Several
oocytes show fluorescent profiles that exhibit a local maximum at the
posterior. This likely stems from out of plane fluorescence due to the
geometric shape of the oocyte pole, and we neglect this minor effect here. At
the anterior surface, MT density tends to be more homogeneous with only
shallow increases in cortical MT density towards the corners. In summary, for
the posterior cap we choose a probability density with parabolic shape that
reaches zero at the posterior pole $h_{0}^{P}=0,k_{P}=20$ (Fig. 7D, bottom),
whereas for the anterior cap we chose a density with parabolic shape but only
shallow depth $h_{0}^{A}=0.8,k_{P}=20$ (Fig. 7D, top).

####  IV.2.2 M2.2 Generation of seeding points

Discrete seeding points that are randomly positioned along the oocyte membrane
according to the probability density $p_{\Sigma}(s_{i}(\rho)|h_{0}^{i},k_{i})$
(eq. (8)) can be achieved by inverse transform sampling. To this end, the
cumulative distribution function on the anterior ($i=A$) or posterior ($i=P$)
cap can be computed as

$\mathcal{C}(\rho^{\prime}|h_{0}^{i},k_{i})=\int_{0}^{2\pi}\int_{0}^{\rho^{\prime}}\mathrm{d}\Sigma_{i}\,p_{\Sigma}(s_{i}(\rho)|h_{0}^{i},k_{i})~{}.$
(13)

Note that $\mathcal{C}(\rho^{\prime}=1|h_{0}^{i},k_{i})\neq 1$ due to
normalization across the total oocyte surface rather than across each
individual cap. Therefore, random seeding points on each cap can be drawn by
applying the inverted cumulative distribution function to a random sample that
is uniformly distributed over the interval
$[0,\mathcal{C}(\rho^{\prime}=1|h_{0}^{i},k_{i})]$. In this formulation, the
total number of seeding points is $N=N_{A}+N_{P}$ (see Fig. 6). The ratio of
anterior ($N_{A}$) and posterior ($N_{P}$) seeding points to be drawn must
equal the ratio of total probability for a point to fall on the anterior and
posterior caps, respectively. For a fixed value $N_{A}$, this results in

$N_{P}=N_{A}\,\frac{\int\mathrm{d}\Sigma_{P}\,p_{\Sigma}(s_{P}(\rho)|h_{0}^{P},k_{P})}{\int\mathrm{d}\Sigma_{A}\,p_{\Sigma}(s_{A}(\rho)|h_{0}^{A},k_{A})}~{}.$
(14)

Here, we generated seeding points in an equivalent way by renormalizing the
probability density (eq. (8)) on each cap individually as

$\mathcal{P}_{\Sigma}(s_{i}(\rho)|h_{0}^{i},k_{i})=\frac{1}{\mathcal{A}}\,\tilde{p}_{\Sigma}(s_{i}(\rho)|h_{0}^{i},k_{i})\,,$
(15)

wherein the new normalization factor $\mathcal{A}$ is obtained by integration
over only one parabolic cap

$\mathcal{A}(h_{0}^{i},k_{i})=\int\mathrm{d}\Sigma_{i}\,\tilde{p}_{\Sigma}(s_{i}(\rho)|h_{0}^{i},k_{i})\,.$
(16)

Hence, the corresponding cumulative distribution function

$\mathcal{C}(\rho^{\prime}|h_{0}^{i},k_{i})=\int_{0}^{2\pi}\int_{0}^{\rho^{\prime}}\mathrm{d}\Sigma_{i}\,\mathcal{P}_{\Sigma}(s_{i}(\rho)|h_{0}^{i},k_{i})$
(17)

covers the range $[0,1]$ and seeding points are obtained as
$\mathcal{C}^{-1}(u)$ for a random sample $u$ with uniform distribution
$u\in[0,1]$.

For parabolic caps, the normalization (eq. (16)) can not be calculated
analytically. Therefore, the probability density (eq. (15)) and its cumulative
distribution function (eq. (17)) were evaluated and inverted numerically. For
the special case of a flat anterior ($i=A$) disc in the alternative
geometry-2, the normalization can be solved in closed form as

$\displaystyle\mathcal{A}$
$\displaystyle(h_{0}^{A},k_{A})=\int_{0}^{2\pi}d\phi\int_{0}^{1}d\rho\,\rho\,\tilde{p}_{\Sigma}(\rho,h_{0}^{A},k_{A})=$
$\displaystyle\pi\left[(1-h_{0}^{A})k_{A}^{2}+1-(1-h_{0}^{A})k_{A}^{2}(k_{A}^{2}+1)\ln(1+1/k_{A}^{2})\right]\,.$

The corresponding cumulative distributions function evaluates to

$\displaystyle\mathcal{C}$
$\displaystyle(\rho^{\prime})=\int_{0}^{2\pi}d\phi\int_{0}^{\rho^{\prime}}d\rho\,\rho\,\mathcal{P}_{\Sigma}(\rho,h_{0}^{A},k_{A})=$
$\displaystyle\frac{\rho^{2}\left[(1-h_{0}^{A})k_{A}^{2}+1\right]-(1-h_{0}^{A})k_{A}^{2}(k_{A}^{2}+1)\ln[1+(\rho/k_{A})^{2}]}{(1-h_{0}^{A})k_{A}^{2}+1-(1-h_{0}^{A})k_{A}^{2}(k_{A}^{2}+1)\ln[1+(1/k_{A})^{2}]}\,,$

which was numerically inverted to generate randomly positioned seeding points
on the anterior disc.

Due to the new individual normalization, the values of anterior and posterior
probability densities differ at the contact line between anterior and
posterior caps. Consider two rings with areas $\mathrm{d}A_{i}$ along the
contact line ($s=s_{0}^{i}$) on the anterior ($i=A$) and posterior ($i=P$) cap
which contain
$\mathrm{d}N_{i}=N_{i}\,\mathcal{P}_{\Sigma}(s=s_{0}^{i}|h_{0}^{i},k_{i})\,\mathrm{d}A_{i}$
seeding points. Setting the number of anterior points fixed, and enforcing the
point densities to be equal in the corner rings
$\mathrm{d}N_{A}/\mathrm{d}A_{A}=\mathrm{d}N_{P}/\mathrm{d}A_{P}$ for equal
ring areas $\mathrm{d}A_{A}=\mathrm{d}A_{P}$ allows to compute the number of
posterior seeding points as

$\displaystyle N_{P}$ $\displaystyle=$ $\displaystyle
N_{A}\frac{\mathcal{P}_{\Sigma}(s=s_{0}^{A}|h_{0}^{A},k_{A})}{\mathcal{P}_{\Sigma}(s=s_{0}^{P}|h_{0}^{P},k_{P})}$
(20) $\displaystyle=$ $\displaystyle
N_{A}\frac{\mathcal{A}(h_{0}^{P},k_{P})}{\mathcal{A}(h_{0}^{A},k_{A})}~{},$

where the second line follows because
$\tilde{p}_{\Sigma}(s_{i}=s_{0}^{i}|h_{0}^{i},k_{i})=1$. Inserting the
expression for $\mathcal{A}(h_{0}^{i},k_{i})$ (eq. (16)) shows that forcing
seeding point densities on the anterior and posterior caps to be equal at the
anterior corners (eq. (20)) is identical to chosing points according to the
ratio of total probabilities on the anterior and posterior caps (eq. (14)).
Thus, chosing points from $p_{\Sigma}$ normalized over the entire surface (eq.
8) or from $\mathcal{P}_{\Sigma}$ normalized over each cap (eq. 15) is
equivalent.

##### Alternative seeding densities

Our model is not sensitive to the exact choice of the probability density for
MT seeding points, and here we define alternative seeding densities that are
mathematically tractable. Specifically, as randomly positioned seeding points
are generated by operating the inverse CDF on a uniformly distributed sample
$u\in[0,1]$ and $\phi\in[0,2\pi)$, it is convenient to chose the seeding
probability such that the CDF can be inverted analytically.

We first note that writing out explicitly the CDF for a rotationally-symmetric
probability density function $p(s_{i}(\rho))$ along the arclength (eq. (13))
leads to

$\mathcal{C}(\rho^{\prime})=\int_{0}^{2\pi}\mathrm{d}\phi\int_{0}^{\rho^{\prime}}\mathrm{d}\rho\,p(s_{i}(\rho))\,\rho\,\sqrt{1+(2z_{0}^{i}\rho)^{2}}~{}.\quad$
(21)

For the case of a gradient of MT seeding points along the arclength of a
parabolic cap we can define the density to be

$p(s_{i}(\rho))\equiv
p_{n}(\rho|n,z_{0}^{i})=\frac{(n+1)\,\rho^{n-1}}{2\pi\,\sqrt{1+(2z_{0}^{i}\rho)^{2}}}\,.$
(22)

From eq. (21) this results in a CDF
$u\equiv\mathcal{C}_{n}(\rho^{\prime})=\rho^{\prime n+1}$ which can be
analytically inverted as $\rho^{\prime}=\mathcal{C}_{n}^{-1}(u)=u^{1/(n+1)}$.

For the case of a uniform distribution of seeding points along the arclength,
the probability density is constant and given by the inverse surface area of
the parabolic caps (eq. 6)

$p(s_{i}(\rho))\equiv p_{u}(\rho|z_{0}^{i})=\frac{1}{\Sigma_{i}}~{}.$ (23)

For $p_{u}(\rho|z_{0}^{i})$ we find the CDF

$u\equiv\mathcal{C}_{u}(\rho^{\prime})=\frac{\left(1+(2z_{0}^{i}\rho^{\prime})^{2}\right)^{3/2}-1}{\left(1+(2z_{0}^{i})^{2}\right)^{3/2}-1}\,,$
(24)

which can again be inverted analytically to give

$\mathcal{C}_{u}^{-1}(u)=\frac{1}{2z_{0}^{i}}\sqrt{\left(u\left[\left(1+(2z_{0}^{i})^{2}\right)^{3/2}-1\right]+1\right)^{2/3}-1}\,.$
(25)

For the special case of a flat disc, the uniform distribution of seeding
points simplifies the CDF to

$u\equiv\mathcal{C}_{u}=\int_{0}^{2\pi}\mathrm{d}\phi\int_{0}^{\rho}\mathrm{d}\rho\,\frac{1}{\pi}\,\rho=\rho^{2}$
(26)

with inverse $\rho=\mathcal{C}_{u}^{-1}(u)=\sqrt{u}$.

For a wild-type cytoskeleton, we impose a uniform MT density
$p_{u}(\rho|z_{0}^{i})$ along the anterior cap and a gradient of MTs
$p_{n}(\rho|n,z_{0}^{i})$ along the posterior cap. At the contact line between
anterior and posterior caps, the seeding densities need to be matched by
adjusting the total number of posterior seeding points $N_{P}$. A ring on the
anterior cap at the anterior corners contains
d$N_{A}=\left(N_{A}/\Sigma_{A}\right)\mathrm{d}A_{A}$ seeding points, while a
ring on the posterior cap at the anterior corners contains
d$N_{P}=N_{P}\,p_{n}(\rho=1|n,z_{0}^{P})\,\mathrm{d}A_{P}$ points. Using eq.
(22) and eq. (6) and forcing densities to be equal
d$N_{A}/\mathrm{d}A_{A}=\mathrm{d}N_{P}/\mathrm{d}A_{P}$ results in

$\displaystyle N_{P}$ $\displaystyle=$
$\displaystyle\frac{N_{A}}{\Sigma_{A}\,p_{n}(\rho=1)}$ (27) $\displaystyle=$
$\displaystyle\frac{3\left(2z_{0}^{A}\right)^{2}N_{A}}{n+1}\,\frac{\sqrt{1+(2z_{0}^{P})^{2}}}{\left(1+(2z_{0}^{A})^{2}\right)^{3/2}-1}\,.$

For the alternative geometry-2 comprised of a posterior parabolic cap and flat
anterior disc, matching of seeding densities results in

$N_{P}=\frac{2\,N_{A}}{n+1}\,\sqrt{1+(2z_{0}^{P})^{2}}\,.$ (28)

Using these alternative seeding densities does not qualitatively change the
behavior of the model.

####  IV.2.3 M2.3 MT growth

We describe MTs as persistent random walks, i.e. a chain of straight segments
of constant length $\lambda$ that show some flexibility in their relative
orientations. The orientation $\bm{\hat{\mu}}_{i}$ of the $i$-th segment is
drawn from a von Mises-Fisher probability distribution on a 2D sphere

$f(\bm{\hat{\mu}}_{i}|\bm{\hat{\mu}}_{i-1},\kappa)=\frac{\kappa}{4\pi\sinh(\kappa)}\,e^{\kappa\,\bm{\hat{\mu}}_{i}\cdot\bm{\hat{\mu}}_{i-1}}\,,$
(29)

where $\kappa$ is the concentration parameter around the orientation
$\bm{\hat{\mu}}_{i-1}$ of the previous segment (see sec. 2.5 for parameter
values). The first orientation $\bm{\hat{\mu}}_{1}$ is drawn from the uniform
angular distribution on a 2D sphere ($\kappa=0$) where orientations are
rejected if pointing locally outwards of the geometry
$\bm{\hat{n}}_{A,P}(\rho,\phi)\cdot\bm{\hat{\mu}}_{1}<0$.

Polymers stop growing if they either encounter a boundary, or if they reach a
pre-defined target length $l$ drawn from a probability distribution. The
probability density is based on the experimental finding that MTs in-vitro
undergo a three-step aging process leading to catastrophe, resulting in
catastrophe lengths ($l_{c}$) that follow a Gamma distribution Gardner  _et
al._ (2011). From this catastrophe length distribution, the length
distribution observed in an ensemble of growing MTs without shrinkage was
calculated as

$\phi_{\Gamma}(l|n,\Lambda)=\frac{\Gamma(n,l/\Lambda)}{n\,\Lambda\,\Gamma(n)}\,,$
(30)

where $\Gamma(n,l)$ is the incomplete Gamma function, $\Gamma(n)$ is the Gamma
distribution, $n=3$ is the number of steps to reach catastrophe and $\Lambda$
is the step length (equivalent to a transition time per step for constant
growth velocity, see Gardner  _et al._ (2011)). We set the expectation value
$E\left[\phi_{\Gamma}(l|n=3,\Lambda)\right]=2\,\Lambda\equiv\mathcal{N}_{\Gamma}^{-1}$
as fraction $\epsilon$ of the anterior-posterior axis length, hence
$\Lambda=\epsilon\left(z_{0}^{P}-z_{0}^{A}\right)/2$.
Pseudo-random numbers distributed according to this probability density (eq.
(30)) were generated by inverse transform sampling of the cumulative
distribution function

$\Phi_{\Gamma}(l|n=3,\Lambda)=1-\frac{6\,\Lambda^{2}+4\,l\,\Lambda+l^{2}}{6\,\Lambda^{2}}\,e^{-l/\Lambda}\,.$
(31)

Similar to the MT seeding densities, the model is not sensitive to the exact
choice of the MT length distribution. For example, using the one-parameter
exponential distribution

$\phi_{e}(l|\Lambda)=\frac{1}{\Lambda}\,e^{-l/\Lambda}$ (32)

as an alternative length distribution and again regulating the expectation
value
$E\left[\phi_{e}(l|\Lambda)\right]=\Lambda=\epsilon\left(z_{0}^{P}-z_{0}^{A}\right)$
of MT lengths via a parameter $\epsilon$ does not qualitatively change our
results.

####  IV.2.4 M2.4 MT persistence length

The stiffness of a polymer is commonly characterized by its persistence length
$P$, which is defined as the decay length of tangent-tangent correlations. To
calculate the persistence length of MT random walks, we first define the end-
to-end vector for a polymer with $N_{s}$ segments of length $\lambda$ and
given initial orientation $\bm{\hat{\mu}}_{1}$ as

$\bm{R}(N_{s}|\bm{\hat{\mu}}_{1})=\lambda\sum_{i=1}^{N_{s}}\bm{\hat{\mu}}_{i}\,.$
(33)

Orientations are drawn from a von Mises-Fisher distribution (eq. (29)) with
mean

$\left<\bm{\hat{x}}|\bm{\hat{\mu}}\right>=\int\bm{\hat{x}}\,f(\bm{\hat{x}}|\bm{\hat{\mu}},\kappa)d\bm{\hat{x}}=\left(\frac{1}{\tanh(\kappa)}-\frac{1}{\kappa}\right)\bm{\hat{\mu}}\equiv\sigma\,\bm{\hat{\mu}}\,.$
(34)

In order to calculate the persistence length of these polymers, we first
consider the mean orientations of the second and third MT segments

$\displaystyle\left<\bm{\hat{\mu}}_{2}|\bm{\hat{\mu}}_{1}\right>$
$\displaystyle=$
$\displaystyle\int\bm{\hat{\mu}}_{2}\,f(\bm{\hat{\mu}}_{2}|\bm{\hat{\mu}}_{1},\kappa)d\bm{\hat{\mu}}_{2}=\sigma\,\bm{\hat{\mu}}_{1}\,,$
$\displaystyle\left<\bm{\hat{\mu}}_{3}|\bm{\hat{\mu}}_{1}\right>$
$\displaystyle=$
$\displaystyle\int\bm{\hat{\mu}}_{3}\prod_{i=2}^{3}f(\bm{\hat{\mu}}_{i}|\bm{\hat{\mu}}_{i-1},\kappa)d\bm{\hat{\mu}}_{i}=\sigma^{2}\,\bm{\hat{\mu}}_{1}\,,$

and we find in general the relation

$\left<\bm{\hat{\mu}}_{N_{s}}|\bm{\hat{\mu}}_{1}\right>=\int\bm{\hat{\mu}}_{N_{s}}\prod_{i=2}^{N_{s}}f(\bm{\hat{\mu}}_{i}|\bm{\hat{\mu}}_{i-1},\kappa)d\bm{\hat{\mu}}_{i}=\sigma^{N_{s}-1}\,\bm{\hat{\mu}}_{1}\,.$
(35)

Hence, using eq. (35) and shifting indices, the expectation value of the end-
to-end vector $\left<\bm{R}(N_{s}|\bm{\hat{\mu}}_{1})\right>$ is given by

$\lambda\sum_{i=1}^{N_{s}}\left<\bm{\hat{\mu}}_{i}|\bm{\hat{\mu}}_{1}\right>=\lambda\sum_{n=0}^{N_{s}-1}\sigma^{n}\bm{\hat{\mu}}_{1}=\lambda\frac{1-\sigma^{N_{s}}}{1-\sigma}\,\bm{\hat{\mu}}_{1}\,.$
(36)

For stiffness parameter $\kappa\rightarrow 0$ ($\sigma\rightarrow 0$), the von
Mises-Fisher distribution becomes a uniform distribution on a 2D sphere. In
this case, the polymers curl up with mean end-to-end distance of a single
segment

$\lim_{\kappa\rightarrow
0}\left<\bm{R}(N_{s}|\bm{\hat{\mu}}_{1})\right>=\lim_{\sigma\rightarrow
0}\left<\bm{R}(N_{s}|\bm{\hat{\mu}}_{1})\right>=\lambda\,\bm{\hat{\mu}}_{1}\,.$
(37)

In the opposite limit $\kappa\rightarrow\infty$ ($\sigma\rightarrow 1$), the
polymers become stiff rods and are extended to their maximum possible length

$\lim_{\kappa\rightarrow\infty}\left<\bm{R}(N_{s}|\bm{\hat{\mu}}_{1})\right>=\lim_{\sigma\rightarrow
1}\left<\bm{R}(N_{s}|\bm{\hat{\mu}}_{1})\right>=\lambda\,N_{s}\,\bm{\hat{\mu}}_{1}\,,$
(38)

thereby showing that the polymers interpolate between an undirected random
walk and completely straight rods.

The persistence length $P$ is defined as the spatial decay length of the
correlations of polymer segment orientations
$e^{-L/P}=\left<\cos(\theta)\right>\equiv\left<\bm{\hat{\mu}}_{N_{s}}\cdot\bm{\hat{\mu}}_{1}\right>$.
For large $L=N_{s}\,\lambda$, we have

$\displaystyle\frac{1}{P}$ $\displaystyle=$
$\displaystyle-\lim_{L\rightarrow\infty}\frac{1}{L}\,\ln(\left<\bm{\hat{\mu}}_{N_{s}}\cdot\bm{\hat{\mu}}_{1}\right>)$
(39) $\displaystyle=$
$\displaystyle-\lim_{N_{s}\rightarrow\infty}\frac{1}{\lambda\,N_{s}}\ln(\sigma^{N_{s}-1})$
$\displaystyle\approx$ $\displaystyle-\frac{1}{\lambda}\ln(\sigma)\,.$

For $\kappa\ll 1$, it holds that
$\sigma\approx\kappa/3+\mathcal{O}\left(\kappa^{3}\right)$ and hence

$P\approx\frac{\lambda}{\ln(3/\kappa)}\,,$ (40)

whereas for $\kappa\gg 1$ we find

$P\approx\lambda\,\kappa\,.$ (41)

Therefore, the persistence length is approximated as segment length $\lambda$
multiplied by concentration parameter $\kappa$.

####  IV.2.5 M2.5 Motor velocity field

From each realization of the MT cytoskeleton in the polymer model, we derive a
nondimensional vector field termed motor-velocity field $\bm{v}^{\prime}_{m}$
by computing the local vectorial sum of MT segments. To this end, we define a
coarse-grained cubic grid with nondimensional side length d$G=0.04$ ranging
from $-1+\mathrm{d}G/2$ to $1-\mathrm{d}G/2$ in x- and y-direction, and from
$z_{0}^{A}+\mathrm{d}G/2$ to $z_{0}^{P}-\mathrm{d}G/2$ in z-direction. The
center point for each MT segment is computed, and the orientations for all MT
segments whose center points fall within the same grid volume are vectorially
summed to give the average local MT orientation. By normalizing the resulting
vector field to a mean vector magnitude of 1, we finally construct the motor-
velocity field $\bm{v}^{\prime}_{m}$. $\bm{v}^{\prime}_{m}$ is used
subsequently to compute the cytoplasmic flow field, and to simulate active
motor-driven cargo transport on the cytoskeleton.

####  IV.2.6 M2.6 Parameter values

For all computations of the MT cytoskeleton, we used $N_{A}=25000$ anterior
seeding points, giving a total number of points between $N=55764$ for wild-
type cytoskeleton and $N=84953$ for par-1 null mutants.

The length of an individual MT segment $\lambda$ needs to be short compared to
the system length $L$, yet long enough to allow simulations of even long MTs
with feasible computational effort. Here, we chose $\lambda=0.015\,\mu$m. The
maximum number of segments per MT is set to $N_{s}^{\text{max}}=200$. Care was
taken that few MTs reach the maximum length $\lambda\,N_{s}^{\text{max}}$.

Experimental measurements of MT persistence lengths $P$ are on the order of
millimeters van Mameren  _et al._ (2009). However, the effective persistence
lengths of MTs in oocytes is likely shorter due to effects of cytoplasmic
flows as well as high density of yolk granules and other obstacles in the
cytoplasm that induce MT bending. MTs in the oocyte are neither seen to be
completely straight (Fig. 8, right), nor strongly curled up (Fig. 8, left).
Therefore, we chose an intermediate value of $\kappa=18$, corresponding to an
effective persistence length (eq. (41)) of $P\approx 13.5\,\mu$m (Fig. 8,
center).

Due to their curved nature, the reach of a MT polymers is effectively shorter
than the length of a straight rod composed of the same number of segments. For
example, for the chosen values of $\kappa$ and $\lambda$, the end-to-end
distance of a MT polymer with $N=25$ segments (eq. (36)) on average 54% as
long as a fully extended polymer or straight rod. Therefore, to compare the MT
polymer model to a model in which MTs are represented as straight rods
necessitates an adjustment in the mean MT length.

###  IV.3 M3 Rod model for the MT cytoskeleton

####  IV.3.1 M3.1 Model setup

In addition to the polymer model for MTs, we test the net orientation of the
cytoskeleton by comparing with a second model in which MTs are nucleated
continuously around the cell membrane and act as straight rods with a given
length distribution. A single MT from a point $\bm{\sigma}(\rho,\phi)$ on the
boundary that hits an observation point $\bm{x}$ in the volume contributes an
orientation vector pointing from $\bm{\sigma}$ to $\bm{x}$ to the observation
point. Heuristically, this contribution is expected to be weighted by three
different factors: (i) the probability density of MT nucleation
$p_{\Sigma}(\bm{\sigma})$ at boundary point $\bm{\sigma}(\rho,\phi)$, (ii) the
probability that the nucleated MT rod is oriented in a direction such that it
hits the observation point $\bm{x}$, and (iii) the probability that the MT rod
is at least long enough to reach from $\bm{\sigma}$ to $\bm{x}$ across a
distance $r_{\sigma}=\|\bm{x}-\bm{\sigma}\|$. Summing up all the weighted
orientational contributions from the entire boundary eventually gives the net
orientation at a given observation point inside the oocyte.

We denote the joint probability that a MT rod from the surface element
d$\Sigma$ around a point $\bm{\sigma}$ reaches a volume element d$V$ around
$\bm{x}$ by $p(\bm{x},\bm{\sigma})\mathrm{d}\Sigma\,\mathrm{d}V$.
$p(\bm{x},\bm{\sigma})$ is normalized over the entire oocyte surface area
$\Sigma$ and volume $V$ as

$\int_{\Sigma}\mathrm{d}\Sigma\int_{V}\mathrm{d}V\,p(\bm{x},\bm{\sigma})=1~{}.$
(42)

The marginal distribution $p_{\Sigma}(\bm{\sigma})$ given by

$p_{\Sigma}(\bm{\sigma})=\int_{V}\mathrm{d}V\,p(\bm{x},\bm{\sigma})$ (43)

defines the probability density of MT nucleation on the oocyte surface.
Conversely, the marginal distribution $p_{V}(\bm{x})$

$p_{V}(\bm{x})=\int_{\Sigma}\mathrm{d}\Sigma\,p(\bm{x},\bm{\sigma})$ (44)

is proportional to the total density of MTs that reach the observation point
$\bm{x}$ from the entire boundary. Each individual MT rod contributes an
orientation unit vector
$\bm{\hat{e}}_{x\sigma}=(\bm{x}-\bm{\sigma})/\|\bm{x}-\bm{\sigma}\|$. Thus,
the net MT orientation at observation point $\bm{x}$ corrected for the density
of contributing MTs can be calculated as

$\bm{o}(\bm{x})=\frac{1}{p_{V}(\bm{x})}\int_{\Sigma}\mathrm{d}\Sigma\,\bm{\hat{e}}_{x\sigma}\,p(\bm{x},\bm{\sigma})\,.$
(45)

It is convenient to split the joint probability $p(\bm{x},\bm{\sigma})$ into
the conditional probability and the probability for MT nucleation at the
surface

$p(\bm{x},\bm{\sigma})=p(\bm{x}|\bm{\sigma})\,p_{\Sigma}(\bm{\sigma})~{}.$
(46)

For the nucleation of MTs on the oocyte surface $p_{\Sigma}(\bm{\sigma})$, we
use the same probability density as in the polymer model (eq. (8)). To specify
the remaining conditional probability $p(\bm{x}|\bm{\sigma})$, first note that
inserting eq. (46) into eq. (43) imposes the normalization condition

$1=\int_{V}\mathrm{d}V\,p(\bm{x}|\bm{\sigma})~{}.$ (47)

Consider a local spherical polar coordinate system
$(r_{\sigma},\theta_{\sigma},\phi_{\sigma})$ centered at a given point
$\bm{\sigma}$ on the oocyte surface with the $z_{\sigma}$-axis identical to
the locally inwards pointing normal $\bm{\hat{n}}_{\sigma}$. MT rods are
assumed to be oriented in any direction into the positive half space
$z_{\sigma}>0$ with equal probability. To determine the general structure of
the conditional probability density function $p(\bm{x}|\bm{\sigma})$, we first
assume MT rods of cross sectional area $A$ and fixed length $l$ corresponding
to a length distribution

$\phi_{\delta}(r_{\sigma})=\delta(l-r_{\sigma})~{},$ (48)

with cumulative distribution function

$\displaystyle\Phi_{\delta}(r_{\sigma})$ $\displaystyle=$
$\displaystyle\int_{0}^{r_{\sigma}}\\!\mathrm{d}r\,\delta(l-r)=\Theta(r_{\sigma}-l)$
(49) $\displaystyle=$ $\displaystyle 1-\Theta(l-r_{\sigma})~{}.$

The volume $V_{c}=A\,l$ of such a cylindrical MT rod can be approximated as
integral over the positive half space by

$\displaystyle V_{c}$ $\displaystyle=$ $\displaystyle
A\int_{0}^{l}\mathrm{d}r_{\sigma}=A\int_{0}^{\infty}\mathrm{d}r_{\sigma}\,\Theta(l-r_{\sigma})$
$\displaystyle=$ $\displaystyle
A\int_{0}^{2\pi}\mathrm{d}\phi_{\sigma}\int_{0}^{\pi/2}\mathrm{d}\theta_{\sigma}\,\sin(\theta_{\sigma})\int_{0}^{\infty}\mathrm{d}r_{\sigma}\,r_{\sigma}^{2}\frac{\Theta(l-r_{\sigma})}{2\pi
r_{\sigma}^{2}}$ $\displaystyle\Leftrightarrow 1$ $\displaystyle=$
$\displaystyle\int_{V}\mathrm{d}V\,\frac{\Theta(l-r_{\sigma})}{2\pi
r_{\sigma}^{2}l}~{},$ (50)

where the last step resulted from division by $A\,l$. Comparing this result
with eq. (47) and using eq. (49), we identify the conditional probability
density function for randomly oriented MTs rods of fixed length as

$p(\bm{x}|\bm{\sigma})=\mathcal{N}_{\delta}\,\frac{1}{2\pi
r_{\sigma}^{2}}\left[1-\Phi_{\delta}(r_{\sigma})\right]~{}.$ (51)

Herein, the factor $\mathcal{N}_{\delta}=1/l$ corresponds to the mean MT
length, the term $(2\pi r_{\sigma}^{2})^{-1}$ represents the uniform angular
distribution on a half sphere, and the expression
$\left[1-\Phi(r_{\sigma})\right]$ is the probability that MT rods are at least
$r_{\sigma}$ long.

To account for the experimentally measured MT length distribution, eq. (51) is
finally generalized by replacing $\Phi_{\delta}(r_{\sigma})$ by the cumulative
distribution used in the polymer model $\Phi_{\Gamma}(r_{\sigma}|n=3,\Lambda)$
(eq. (31)) with $\mathcal{N}_{\Gamma}=1/(2\Lambda)$. Thus, the complete joint
probability distribution $p(\bm{x},\bm{\sigma})$ (eq. (46)) for a MT emanating
from $\bm{\sigma}(\rho)$ and contributing to the net orientation at
observation point $\bm{x}$ in the oocyte volume is defined as

$p(\bm{x},\bm{\sigma})=p_{\Sigma}(s|h_{0},k)\,\mathcal{N}_{\Gamma}\,\frac{1}{2\pi
r_{\sigma}^{2}}\left[1-\Phi_{\Gamma}(r_{\sigma}|3,\Lambda)\right]~{}.$ (52)

Note that MT straight rods always cover the distance between nucleation and
observation point in a straight line. If the anterior surface is curved
inwards, these straight lines can temporarily pass outside of the oocyte
geometry before reaching the observation point inside the volume again. The
rod model does not prevent such unphysical contributions (though the polymer
model does prevent this). However, this error does not occur in a geometry
with a flat anterior boundary. Calculation of the cytoskeleton in a geometry
with flat anterior (Fig. 1 - figure supplement 1G) does not change the
observed topology, suggesting that in practice this error is negligible.

####  IV.3.2 M3.2 Parameter space and comparison to polymer model

The continuum description of the rod model allows one to follow the saddle-
node bifurcation in the MT cytoskeleton precisely throughout parameter space.
The creation of stable and unstable fixed points always occurs along the
anterior-posterior (AP) axis because the system is rotationally symmetric. We
therefore compute the uncorrected net orientation
$\bm{o}(\bm{x})p_{V}(\bm{x})$ (eq. (45)) for observation points $\bm{x}$ only
along the symmetry axis as a function of cortical nucleation parameter $k_{P}$
and mean MT length $\epsilon$.

For a cytoskeletal topology without any fixed point, the $x$-component
$o_{x}(\bm{x})p_{V}(\bm{x})$ will be pointing towards the posterior pole
everywhere along the AP axis. A positive value $o_{x}(\bm{x})p_{V}(\bm{x})>0$
everwhere along the central axis therefore defines the regions of a wild-type
cytoskeletal topology. If a pair of a stable and a saddle-node is present,
however, $o_{x}(\bm{x})p_{V}(\bm{x})$ will be negative at at least one point
along the AP axis, while at the same time it will be positive at the grid
point closest to the posterior pole $\bm{x}_{P}$ where the unstable node
creates a limited domain of attraction. Combination of both criteria therefore
defines the region of parameter space in which the oocyte is split into two
different regions of attraction. Finally, a negative value at the posterior
pole $o_{x}(\bm{x}_{P})p_{V}(\bm{x}_{P})<0$ is the signature of a cytoskeleton
with only a stable point whose domain of attraction covers the entire volume
(see Fig. 5).

In the ensemble averages of the polymer model, we find that the bifurcation
point occurs for small values of $k_{P}$ if $\epsilon$ is large, but shifts to
larger values of $k_{P}$ if $\epsilon$ becomes smaller. Furthermore, the
bifurcation point can be crossed solely by varying $\epsilon$ while keeping
$k_{P}$ fixed (Fig. 4 - figure supplement 1). In summary, this confirms that
the polymer model exhibits the same parameter space structure as the rod model
(Fig. 5), thus further supporting the similarity of the two models. Moreover,
as in the rod model, this parameter space structure is again insensitive to
the oocyte shape.

####  IV.3.3 M3.3 Bifurcation normal form

The saddle-node bifurcation normal form in two dimensions is given by

$\displaystyle\begin{pmatrix}v_{x}\\\
v_{y}\end{pmatrix}=\begin{pmatrix}x^{2}+\lambda\\\ -y\end{pmatrix}~{},$ (53)

wherein $\lambda$ is the bifurcation parameter. For negative $\lambda$, stable
and unstable fixed points exist and are located at
$x_{\pm}=\pm\sqrt{-\lambda}$. The critical bifurcation point occurs at
$\lambda=0$ (Fig. 4 - figure supplement 1P-T).

###  IV.4 M4 Additional Methods

####  IV.4.1 M4.1 Nondimensionalization and mathematical methods

In dimensional units, the system of transport equations for bound and unbound
cargo fractions $c_{b}$ and $c_{u}$ is defined as

$\displaystyle\partial_{t}\,c_{b}+\bm{\nabla}\cdot\left(\bm{v}_{m}\,c_{b}\right)$
$\displaystyle=$ $\displaystyle k_{b}\,c_{u}-k_{u}\,c_{b}$ (54)
$\displaystyle\partial_{t}\,c_{u}+\bm{\nabla}\cdot\left(\bm{u}\,c_{u}\right)$
$\displaystyle=$ $\displaystyle-
k_{b}\,c_{u}+k_{u}\,c_{b}+D\,\bm{\nabla}^{2}c_{u}\,,$

while the Navier-Stokes-Equations and volume forces are defined by

$\displaystyle\rho\left(\partial_{t}\,\bm{u}+\left(\bm{u}\cdot\bm{\nabla}\right)\bm{u}\right)$
$\displaystyle=$
$\displaystyle-\bm{\nabla}p+\mu\,\bm{\nabla}^{2}\bm{u}+\bm{f}$ (55)
$\displaystyle\bm{f}$ $\displaystyle=$ $\displaystyle a\,\bm{v}_{m}\,,$ (56)

with no-slip boundary conditions on the oocyte surface. Note that cytoplasmic
streaming is present in oocytes even when oskar mRNA is not expressed Palacios
and St Johnston (2002), showing that some cargo other than oskar mRNA is
transported by kinesin and responsible for driving flow. This justifies that
the parameter $a$ in eq. (56) is independent of the concentration of the bound
cargo $c_{b}$.

Computations of the fluid flow field and simulations of cargo transport are
performed using a nondimensionalized version of eqs. (54-56). For
nondimensionalization, we select a length scale $L=50\,\mu$m for scaling of
space $\bm{x}=L\,\bm{x}^{\prime}$, and an advection scale derived from the
active motor-driven transport $V=0.5\,\mu$m/s Zimyanin  _et al._ (2008) for
scaling of velocities $\bm{v}_{m}=V\,\bm{v}_{m}^{\prime}$, implying an
advection time scale $\tau=L/V=100\,s$ for scaling of time
$t=\tau\,t^{\prime}$. $\bm{v}_{m}^{\prime}$ denotes the coarse-grained motor-
velocity field computed from each realization of the MT polymer model (Sec.
S2.4). Defining the mean raction rate constant $K=\left(k_{b}+k_{u}\right)/2$
and a nondimensional reaction parameter $\beta=k_{b}/(2\,K)$ we find the
nondimensional version of eqs. (54)

$\displaystyle\partial_{t^{\prime}}c_{b}^{\prime}+\bm{\nabla}^{\prime}\cdot\left(\bm{v}_{m}^{\prime}\,c_{b}^{\prime}\right)$
$\displaystyle=$ $\displaystyle 2\mathrm{Da}\left(\beta
c_{u}^{\prime}-(1-\beta)c_{b}^{\prime}\right)$ (57)
$\displaystyle\partial_{t^{\prime}}c_{u}^{\prime}+\bm{\nabla}^{\prime}\cdot\left(\bm{u}^{\prime}\,c_{u}^{\prime}\right)$
$\displaystyle=$ $\displaystyle 2\mathrm{Da}\left(-\beta
c_{u}^{\prime}+(1-\beta)c_{b}^{\prime}\right)$
$\displaystyle+\mathrm{Pe}^{-1}\bm{\nabla}^{\prime 2}c_{u}^{\prime}\,,$

where $\mathrm{Da}=L\,K/V$ is the Damköhler number and $\mathrm{Pe}=L\,V/D$ is
the Peclet number.

For nondimensionalization of eqs. (55-56), we define scales for the pressure
$\mathbb{P}=\mu\,V/L$, force density $\mathbb{F}=\mu\,V/L^{2}$ and force-
velocity scaling $\mathbb{A}=\mathbb{F}/V$ to obtain

$\displaystyle\mathrm{Re}\left(\partial_{t}^{\prime}\,\bm{u}^{\prime}+\left(\bm{u}^{\prime}\cdot\bm{\nabla}^{\prime}\right)\bm{u}^{\prime}\right)$
$\displaystyle=$
$\displaystyle-\bm{\nabla}^{\prime}p^{\prime}+\bm{\nabla}^{\prime
2}\bm{u}^{\prime}+\bm{f}^{\prime}$ (58) $\displaystyle\bm{f}^{\prime}$
$\displaystyle=$ $\displaystyle a^{\prime}\,\bm{v}_{m}^{\prime}$ (59)

wherein $p^{\prime}=p/\mathbb{P}$, $\bm{f}^{\prime}=\bm{f}/\mathbb{F}$,
$a^{\prime}=a/\mathbb{A}$, and $\mathrm{Re}=\rho\,V\,L/\mu$ is the Reynolds
number. Note that the Reynolds number is defined using the scale $V$ of the
active transport field because the flow velocity field $\bm{u}$ is derived
from it and hence varies. To estimate an upper bound on the Reynolds number,
we use the viscosity of water $\mu=10^{-3}$ Pa s and ten times the density of
water $\rho=10^{4}\mathrm{kg}/\mathrm{m}^{3}$ as lower and upper bounds for
the viscosity and density of the cytoplasm in the oocyte. This results in an
upper bound $\mathrm{Re}=2.5\times 10^{-4}$, thus justifying to neglect the
inertia terms in eq. (58).

####  IV.4.2 M4.2 Parameter values

The unbinding constant $k_{u}$ can be estimated as ratio of the mean active
transport velocity and the mean track length of oskar mRNA in stage 9 oocytes
as $k_{u}=0.17\,s^{-1}$ Zimyanin  _et al._ (2008). The nondimensional
parameter $\beta$ determines the fraction of cargo that resides in the bound
state. Given the fact that 13% of oskar mRNA are bound at any given time we
set $\beta=0.13$ Zimyanin  _et al._ (2008). The unbinding constant $k_{u}$ and
$\beta$ can be used to calculate $k_{b}=0.0255\,s^{-1}$. Therefore, the mean
reaction rate constant and the Damkoehler number are $K=0.0978$ and
$\mathrm{Da}=9.775$, respectively. Diffusion of mRNP particles in the ooplasm
is widely considered to be very slow. Experimentally measured diffusion
constants for mRNA transcripts of sizes similar to oskar mRNA (2.9 kB) in
mammalian cells are in the range of $D=0.02\,\mu$m/s or lower Mor  _et al._
(2010). With this value, the Peclet number evaluates to $\mathrm{Pe}=1250$.

The nondimensional scaling parameter $a^{\prime}$ (eq. (56)) which converts
the active transport velocities into forces acting on the cytoplasmic fluid
absorbs several unknown quantities including the size of the cargo that drives
fluid flow, drag forces on the fluid with local viscosity, the density of such
cargo-motor complexes on MTs and any collective effects. The value of
$a^{\prime}$ determines the mean nondimensional and dimensional fluid flow
speeds $\left<\left|\bm{u}^{\prime}\right|\right>$ and
$V\left<\left|\bm{u}^{\prime}\right|\right>$. We regard $a^{\prime}$ as a
macroscopic, phenomenological parameter. The value $a^{\prime}=45$
($a^{\prime}=55$ for alternative geometry-2) is calibrated such that the
resulting mean dimensional fluid flow speeds
$V\left<\left|\bm{u}^{\prime}\right|\right>$ match the average fluid flow
speeds measured experimentally for stage 9 Drosophila oocytes (see Fig. 2)
(compare to similar approach in He  _et al._ (2014)).

To test the anchoring mechanism, the bound state is replaced by an anchored
state to which cargo can only bind at the extreme posterior pole. For the
anchoring state we set $k_{u}^{\mathrm{anch}}=0$ s-1 and
$k_{b}^{\mathrm{anch}}=10\,k_{b}$, resulting in $\beta^{\mathrm{anch}}=1$ (no
unbinding), $K^{\mathrm{anch}}=0.1275$, and
$\mathrm{Da}^{\mathrm{anch}}=12.75$.

####  IV.4.3 M4.3 Computational methods

For computation of the cytoplasmic flow field and subsequent transport
simulations, the motor-velocity field $\bm{v}^{\prime}_{m}$ and forces
$\bm{f}^{\prime}$ from the MT polymer model are shifted and rotated such that
the AP axis aligns with the $x$-axis and the origin of the coordinate system
is located in the oocyte center. To compute the cytoplasmic flow field we use
the open source FreeFem++ finite element solver Hecht (2012). Forces $\bm{f}$
defined on a 3D cubic grid are interpolated to the unstructured 3D grid used
by FreeFem++, and resulting flow velocity vectors are interpolated back to the
cubic grid and derived staggered grids.

Cargo simulations of eqs. (57) are performed using custom written Matlab code
in finite volume formulation on staggered grids as described in Versteeg and
Malalasekera (2007). As nondimensional time step we used $\Delta t=0.005$.
Each simulation cycles through many pairs of active transport fields
$\bm{v}^{\prime}_{m}$ and their corresponding cytoplasmic flow fields
$\bm{u}^{\prime}$ to account for a dynamically remodelling cytoskeleton and
temporally changing cytoplasmic flow fields. Each pair of
$\bm{v}^{\prime}_{m}$-$\bm{u}^{\prime}$-fields is active for 432 time steps,
corresponding to 3.6 minutes of simulated real time.

####  IV.4.4 M4.4 Fly stocks and experimental methods

Oocyte microtubules were stained as described by Theurkauf et al. Theurkauf
_et al._ (1992) using a FITC-coupled anti-alpha-tubulin antibody at 1:200
(Sigma). The general shape and size of the oocyte was imaged by recording the
fluorescence from a par-1 protein trap (PT) GFP line Lighthouse  _et al._
(2008). Homozygous or heterozygous par-1 PT flies were also used for
measurements of cytoplasmic flows by imaging movements of autofluorescent yolk
granules. The allelic combination par-1W3/par-16323 was used for imaging
cytoplasmic flows in strong par-1 hypomorphs Shulman  _et al._ (2000). In all
cases of flow measurements, flies were yeast fed overnight, dissected under
Voltalef 10S oil and imaged at 40x magnification in a confocal microscope.
Cytoplasmic flows were recorded with a 405 nm laser, 4 $\mu$s pixel dwell
time, collecting 13 frames in $25s$ intervals for a total movie duration of
5.17 minutes.

For the test of oskar mRNA localization under wild-type conditions and under
sub-physiological temperatures we used an oskMS, MS2-GFP line Belaya and St
Johnston (2011). For cold experiments, flies were kept at room temperature
(22C), yeast fed overnight at 25C, and subsequently kept at 25C, 7C and 4C
degrees for 6 hours before dissection and fixation. In situ hybridizations and
imaging of living tissue were performed according to standard methods Palacios
and St Johnston (2002). The bicoid mRNA probe was labeled with Digoxigenin-UTP
(Boehringer Mannheim).

####  IV.4.5 M4.5 Data analysis

To compare topologies of in-vivo cytoplasmic flows with topologies of
numerically computed cytoplasmic flow fields, we used particle image
velocimetry (PIV) to generate vector fields capturing the instantaneous
direction and magnitude of particle movements. PIV was performed using the
open source code PIVlab Thielicke and Stamhuis (2010), and all PIV vector
fields obtained from pairs of consecutive movie frames were averaged over the
movie duration. For each movie of cytoplasmic streaming, the mean flow speeds
were calculated as the average over all PIV vector magnitudes. Flow speeds
obtained from PIV were confirmed by automatic particle tracking using open
source code by Blair and Dufresne Blair and Dufresne . Only particles that
could be tracked in at least three consecutive frames were accepted. Note that
instantaneous speeds were computed as ratio of particle distance traveled
between frames and frame interval. This method avoids underestimation of flow
velocities as was pointed out by authors of Serbus  _et al._ (2005), thereby
ensuring that our conclusions about the neglibible contribution of flows to
transport remain sound.

###  IV.5 M5 Flow fields and oocyte nucleus

####  IV.5.1 M5.1 Autocorrelation function

Cytoplasmic flow patterns not only show a high variability between different
oocytes, but also change over time in any individual oocyte. To estimate the
rate of change of individual flow fields we calculate the unbiased, discrete
vector autocorrelation over time for the sequence of PIV flow fields in each
movie according to the expression

$C(k)=\left<\frac{1}{N-k}\sum_{i=x,y}\sum_{n=0}^{N-k}\left(u_{i}(n+k)\,u_{i}(n)\right)\right>\,$
(60)

where angular brackets denote the average across all points of the PIV grid,
and the peak value of $C(k)$ is normalized to one. Autocorrelation functions
from $N=48$ movies exhibit a large spread across generally low correlation
values, and an exponential fit gives a decay time constant of 4.4 minutes.

####  IV.5.2 M5.2 Nucleus to oocyte volume ratio

In 2D confocal images, the oocyte nucleus occasionally appears to cover a
significant fraction of oocytes, particularly in young oocytes up to stage 8.
To quantify the size of the nucleus compared to the size of the oocyte, we
therefore estimate the fraction of the oocyte volume that is covered by the
nucleus. We first extract the entire boundary of the oocyte using a custom
written macro in Fiji Schindelin  _et al._ (2012). Averaging the boundary
shape from below and above the AP axis results in a symmetrized parametric
curve denoted by $(x(t),y(t))$ (Fig. 2 - figure supplement 1C, blue curve).
After subdividing the boundary into anterior ($x_{A}(t),y_{A}(t)$) and
posterior ($x_{P}(t),y_{P}(t)$) curve, the volume of the respective solids of
revolutions can be computed as

$V_{i}=\pi\int\mathrm{d}t\left(y_{i}(t)\right)^{2}\frac{\mathrm{d}x_{i}}{\mathrm{d}t}\,.$
(61)

While the anterior surface is typically curved inwards for stage 9 oocytes,
the anterior surface in young oocytes up to stage 8 can be outwards or inwards
curved. Hence, depending on the shape of the anterior the volumes of the
anterior $V_{A}$ and $V_{P}$ have to be suitably added or subtracted.

The parabolic caps used in the model geometry (eq. (3)) can be parameterized
along the $z$-direction, and in dimensional units their volume is then
determined by

$\displaystyle V_{i}^{\sigma}$ $\displaystyle=$
$\displaystyle\pi\int_{0}^{L\,z_{0}^{i}}\mathrm{d}z\left(L\sqrt{1-z/\left(L\,z_{0}^{i}\right)}\right)^{2}$
(62) $\displaystyle=$ $\displaystyle\pi L^{3}z_{0}^{i}/2\,.$

Given that the anterior surface in the standard geometry-1 is curved inwards,
the total oocyte volume is given by the difference between between the
posterior and the anterior cap volumes $V_{P}^{\sigma}-V_{A}^{\sigma}=\pi
L^{3}\left(z_{0}^{P}-z_{0}^{A}\right)/2$.

To approximate the volume of the nucleus, we select its shape and fit a circle
to the selection in Fiji. Using the circle radius, we calculate the volume of
the nucleus as $4/3\,\pi\,r^{3}$. We find that the nucleus generally covers
less than $2.5\%$ of the oocyte volume (Fig. 2 - figure supplement 1), thereby
showing that the nucleus volume is negligible even in young stage 9 oocytes,
and that 2D confocal images convey a poor impression of 3D volumes. Given the
small volume of the nucleus and its location at the anterior of wild type
oocytes, the nucleus is unlikely to influence oskar mRNP transport towards the
posterior, and we therefore neglect it in simulations of cargo transport.

####  IV.5.3 M5.3 Impact on flow field

The nucleus may be expected to disturb the cytoplasmic flow field by means of
adding an additional no-slip boundary condition inside the volume. We tested
this effect by including a spherical excluded volume at the oocyte anterior
surface in the solution of the 3D Stokes equations (Fig. 2 - figure supplement
1A,F,H). The nucleus leads to a clear local disturbance of the flow field in a
cross section that includes the nucleus compared to the flow field without
(Fig. 2 - figure supplement 1E,F). However, taking a perpendicular cross
section through the flow fields in which the nucleus is not visible shows that
the nucleus rarely changes the flow field (Fig. 2 - figure supplement 1G,H).
This low sensitivity with respect to excluded internal volumes is due to the
fact that flows are driven by volume rather than by boundary forces. Note that
the flow fields shown in Fig. 2 - figure supplement 1 still likely
overestimate impact of the nucleus because it generally is not spherical and
often seen to squeeze deeply into the anterior corners, therefore reaching far
less into the oocyte volume than the idealized sphere used here.

##  V Acknowledgements

This work was supported in part by the Boehringer Ingelheim Fonds and EPSRC
(P. K. T.), core support from the Wellcome Trust [092096] and Cancer Research
UK [A14492] (H. D.), the MIT Solomon Buchsbaum Award (J. D.), a Wellcome Trust
Principal Research Fellowship [080007] (D. StJ.), the Leverhulme Trust, and
the European Research Council Advanced Investigator Grant [247333] (R. E. G.).

##  VI References

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##  VII Figures

Figure: 1\. Models for the MT meshwork show local order in the cytoskeleton.
A) 3D geometry of a stage 9 Drosophila oocyte (grey, anterior to the left,
posterior to the right) containing more than 55000 MT seeding points. From the
corners to the centre, MT seeding density decreases weakly along the anterior
($k_{A}=1000\mu$m, $h_{0}^{A}=0.8$) and strongly along the posterior-lateral
cortex ($k_{P}=150\mu$m, $h_{0}^{P}=0$, see Materials and methods M2.1).
Nucleated MT polymers are stiff random walks, initially pointing in a random
direction. Only MT segments in a cross section are shown (green) to emulate
confocal images. MT target lengths are chosen from a probability distribution
that accounts for the MT aging process. The mean target length is set to a
fraction $\epsilon$ of the AP-axis length, here $\epsilon$=0.5 (Material and
methods M2.3). The inset shows the 3D angular distribution of 0.5% of all MT
segments with 3D statistical bias. B) Cross section through the MT
cytoskeleton shown in A with 2D directional bias (top right). The inset shows
2D posterior bias (in percent) as function of depth (bottom right). C) Local
vector sum of MT segments from the cross section in panel B on a coarse-
grained grid shown as streamlines that visualize local directionality. D)
Staining of $\alpha$-tubulin (green) shows MT density distribution in a fixed
stage 9 oocyte. Nuclei in blue (DAPI), scale bar is $30\mu$m. E) Schematic
detailing the work flow in the model and comparisons to experiments. F) Local
directionality of MT cross section as in panel C for an average over 50
independent realizations of the cytoskeleton. G) Local directionality computed
from the rod model with the same parameters as in panels A-B but shortened MT
lengths $\epsilon=0.25$ (Material and methods M2.5). Orange arrows show the
separatrices between subcompartments. H) MT density distribution computed from
50 realizations of the polymer model. Figure: 1 - figure supplement 1.
Compartmentalization of the MT cytoskeleton is robust to changes in oocyte
geometry. A) Alternative geometry for a stage 9 oocyte comprised of a
posterior parabolic cap and an anterior disc. MT segments intersecting a cross
section in one realization of the polymer model with nondimensional seeding
density parameters $k_{P}=150\mu$m, $h_{0}^{P}=0,k_{A}=1000\mu$m,
$h_{0}^{A}=0.8$ and MT length parameter $\epsilon=0.5$ from more than 55000
seeding points are shown in green. Inset shows directionality of 0.5% of all
MT segments with posterior bias. B) Cross section through 3D MT cytoskeleton
shown in panel A with 2D directional bias (top right). Inset shows 2D
posterior bias (in percent) as function of depth (bottom right). C) Local
vectorial sum of MT segments visualized as a streamplot. One individual
realization of the MT cytoskeleton shows poor spatial order. D)
$\alpha$-tubulin staining of an early stage 9 oocyte. Scale bar is $25\mu$m.
E) Schematic of the work flow in our model. F) Streamplot of the local
vectorial sum of 50 realizations of the polymer MT cytoskeleton with
parameters as in A,B. G) Local net orientation of straight rod MTs in the rod
model with parameters as in A,B but reduced mean MT target length
$\epsilon=0.28$, showing the same compartmentalization of the oocyte as the
average in the polymer model. H) MT density distribution in the ensemble of 50
realization of the polymer MT cytoskeleton. Figure: 2\. Computed cytoplasmic
flow fields capture key elements of in vivo flows. A) Streamlines (light blue
lines) visualize the 3D cytoplasmic flow field computed from the realization
of the cytoskeleton shown in Fig. 1A. The horizontal plane shows a 2D cross-
section through the 3D field. Anterior to the left, posterior to the right. B)
Cross-section through the 3D field shown in panel A with arrows indicating
flow directions and colouring indicating flow speeds. C) Confocal image of a
live stage 9 oocyte. Arrows show the flow field computed from PIV of streaming
yolk granules and averaged over $\approx 5$ minutes. Scale bar is $25\mu$m. D)
Same as A, but showing the mean flow organization for an average of 100
individual 3D flow fields, analogous to the mean organization of the MT
cytoskeleton in Fig. 1F. E) Same as B for the average in panel D. F) Mean
fluid flow speeds were obtained by PIV (red, $13.7\pm 0.8$ nm/s, mean $\pm$
sem) and automatic particle tracking (orange, $15.3\pm 0.7$ nm/s, mean $\pm$
sem) from 48 oocytes. Experimentally measured flow speeds were used to
calibrate the forces f in the Stokes equations Eq. (2) such that the computed
mean speeds in 3D (blue, mean: $14.5$ nm/s) or in 2D cross sections (green,
mean: $14.8$ nm/s) match the measured values. The larger spread in
experimental flow speeds may reflect greater variability of motor activity,
cytoplasmic composition, geometry or age in vivo. Figure: 2 - figure
supplement 1. The oocyte nucleus covers a negligible fraction of the oocyte
volume and disturbs the flow field only locally. A) Streamlines visualizing
the flow field computed in the 3D oocyte geometry with additional no-slip
boundary condition on a sphere representing the nucleus. Same forces as in
Fig. 2A-B, main text. The nucleus occupies 2.2% of the oocyte volume.
Horizontal plane shows a cut through the flow field. B) Young stage 9 oocyte
stained with DAPI (blue), phalloidin (red) and showing oskar MS2 GFP (green)
in the process of reaching the posterior pole. Scale bar is $25\mu$m. C)
Extracted and rotated shape of the oocyte in panel B (red), and symmetrized
geometry resulting from averaging the shape above and below the AP-axis
(blue). Black circle indicates the nucleus. D) Ratio of nucleus volume to
oocyte volume computed for $N_{e}=9$ early (red) and $N_{m}=7$ mid stage 9
oocytes (blue). Arrowhead marks the oocyte shown in panel B. E-F) Cross
sections through the 3D flow fields for the same forces as in panel A with (F)
and without (E) nucleus. G-H) Same as E and F, but vertical cross section that
does not contain the nucleus. Figure: 3\. The model recapitulates oskar and
bicoid mRNA transport, implying dominance of cytoskeletal transport. A-C)
Shown are fixed oocytes with oskar MS2 GFP (green) and stained with DAPI
(blue) and Phalloidin (red). oskar mRNA forms a central cloud at late stage 8
(A), a collimated channel while moving to the posterior (B), and a posterior
crescent at stage 9 (C). Scale bars are $25\mu$m. D) 3D oocyte showing the
initial oskar mRNA cargo distribution. E-F) Cross sections through a
simulation of oskar mRNA transport with diffusion, motor-transport and flows
showing the distribution of total cargo $c_{b}+c_{u}$ at the indicated time
points. No posterior anchor is present. Simulations of 3 (6) hours cycle
through 50 $\bm{v}_{m}$-$\bm{u}$-pairs in random order once (twice, see Fig.
4). For simulations of 1.5 hours, 25 $\bm{v}_{m}$-$\bm{u}$-pairs were chosen
at random. Compare to experimental observations in panels B and C. G) Same
simulation as in D-F, but without cytoplasmic flows, showing largely identical
localization as in F. H) Simulation of bicoid mRNA transport shows the mRNA
quickly accumulating at the nearest cortex (arrowheads) when injected in the
posterior (inset), corresponding to the behavior of naive bicoid. I) Same as
in H for bicoid mRNA injection at the anterior-dorsal region (inset). J) Same
as in H for injection at the anterior middle (inset), showing bicoid mRNA
localization to the anterior corners (arrowheads). Localization to the
anterior depends on sufficient proximity of the injection site as observed for
naive bicoid mRNA. K) Same simulation for oskar mRNA as in D-F, but with a
posterior anchor (arrowhead) and without active motor-driven transport.
Compare to panels F and G. Figure: 4\. The cortical MT seeding density
determines sites of mRNA localisations and gives rise to a bifurcation in the
cytoskeleton. A-E) Simulations of oskar (A-C) and bicoid (D) mRNA transport
with diffusion, cytoskeletal transport and flow for the cytoskeletal
architectures shown in panels E-H, and their corresponding flow fields. For
oskar mRNA, simulations reproduce wild-type localization (A, same as Fig. 3F),
and partial (B) or complete mislocalisation (C). Initial condition as in Fig.
3D. Simulation times were occasionally increased to 6h (B) to rule out
transient concentration patterns. For bicoid mRNA, simulations capture
mislocalisation to both anterior and posterior as in gurken/torpedo/cornichon
mutants and in strong par-1 hypomorphs (D, arrowheads). Time points as
indicated. The inset in D shows initial condition. E-H) Average local MT
orientations in ensembles of 50 realizations of the polymer model for varying
MT seeding densities along arclength $s$ (see panel E) of the posterior-
lateral cortex (insets). A MT seeding density that increases from a wild-type
gradient (E, $h_{0}^{P}=0,k_{P}=150\mu$m) laterally towards the posterior (F,
$h_{0}^{P}=0,k_{P}=17.5\mu$m) to a near-uniform distribution (G,
$h_{0}^{P}=0,k_{P}=1\mu$m) shows a saddle-node bifurcation by creating a pair
of stable (green) and unstable (red) fixed points. H) A MT seeding density
that is slightly lower at the posterior pole ($h_{0}^{P}=0.7,k_{P}=40\mu$m)
produces mean MT orientations virtually indistinguishable from uniform seeding
density (compare to G). I-K) Vector field of the mathematical normal form of
the 2D saddle-node bifurcation (Materials and methods M3.3). Values of the
bifurcation parameter $\lambda$ as indicated. Fixed points are located at
positions $x=\pm\sqrt{\lambda}$. L) Fluorescence in-situ hybridization to
bicoid mRNA in a strong par-1 hypomorph. The mRNA (red) localises around the
cortex, with most accumulation at the anterior corners and at the posterior
pole (compare to panel D, arrowheads). Scale bar is $25\mu$m. Figure: 4 -
figure supplement 1. Either the MT seeding density or the MT length can act as
bifurcation parameter. A-J) Each panel shows the mean local MT orientations
for an ensemble of 50 realizations of the polymer MT cytoskeleton under
variations of the seeding density parameter $k_{P}$ and MT length parameter
$\epsilon$ as indicated. Approximate location of the stable fixed point is
shown in green, with region of attraction in blue, and domain of attraction of
the posterior pole in red. The point on the AP-axis between red and blue arrow
region marks the unstable fixed point. Percentages indicate the the ensemble-
averaged 3D directional bias. Experimentally, the directional bias was
measured as $57.97\%:42.03\%$ Parton  _et al._ (2011). Cytoskeleton in panel I
was used as wild-type cytoskeleton (Fig. 1F, main text). Anterior seeding
density is unchanged in all panels $k_{A}=1000\mu$m, $h_{0}^{A}=0.8$. K-O)
Same as panels A-J but for ensembles with 100 realizations of the cytoskeleton
to ensure reliable visualization of vector field even when fewer MTs reaching
the oocyte center. P-T) Vector fields computed from a saddle-node bifurcation
normal form. The critical bifurcation point is $\lambda=0$, and for
$\lambda<0$, fixed points are located at $x_{\pm}=\pm\sqrt{-\lambda}$. Figure:
4 - figure supplement 2. Lateral MT growth produces the clone-adjacent-
mislocalization phenotype (see main text). A-D) Addition of lateral MTs
reproduces the clone-adjacent-mislocalization phenotype. A) Cross section
through one realization of the wild-type MT cytoskeleton in the polymer model
($h_{0}^{A}=0.8,k_{A}=1000\mu$m, $h_{0}^{P}=0,k_{P}=150\mu$m). Additional MTs
have been added dorsal to the posterior pole (black arrow) to mimick posterior
follicle cell clones ($RAS^{\Delta C40b}$ MARCM) that overexpress dystroglycan
Poulton and Deng (2006). B) Density of MTs for 50 realizations of the
cytoskeleton shows enrichment on one side of the posterior pole (white arrow).
C) Streamplot showing the local vectorial orientation averaged over 50
realizations of the cytoskeleton. Note the upwards tilt away from the central
posterior pole. D) Simulations of oskar mRNA transport by diffusion,
cytoskeletal transport and corresponding cytoplasmic flows show that cargo is
repelled from the site of additional MT nucleation on one side of the
posterior pole, thereby capturing the CAM phenotype Poulton and Deng (2006).
Figure: 5\. The parameter space of the rod model shows the relation and
interconversion between all three distinct cytoskeletal architectures. A) The
regions corresponding to wild-type (red), weak par-1 hypomorph (blue) and
strong par-1 hypomorph topologies (green, arrowhead at far left) are shown as
a function of the mean MT length $\epsilon$ and extent of the posterior
seeding density $k_{P}$. The bifurcation line between wild-type and weak par-1
hypomorph topologies can be crossed by either changing the seeding density
laterally (horizontal dashed arrow), or by shortening the MTs (vertical dashed
arrow). B-C) Parameter spaces as in A for increasing MT nucleation at the
posterior pole (B: $h_{0}^{P}=0.03,C:h_{0}^{P}=0.1$). Inset in panel C shows a
magnification of the triple point at which small changes in parameters can
convert each cytoskeletal architecture into any other. D) Local net
orientations of MT rods for parameter values indicated in the inset of panel C
(yellow circles). In all panels, parameter values for anterior MT nucleation
were kept constant ($h_{0}^{A}=0.8,k_{A}=1000\mu$m). Figure: 6\. Seeding point
density in the polymer model. Shown are $N$ randomly drawn seeding points on
the posterior cap of the standard oocyte geometry, according to eq. (15)
distributed either (near-)uniformly on the cap (A, G) or in a parabolic
gradient from the posterior pole to the anterior corners (F, L). Between
uniform and parabolic distribution, the seeding density can vary either by
laterally reducing the density (B, C-E, parameter $k_{P}$), or by reducing the
density at the posterior pole (H, I-K, parameter $h_{0}^{P}$). Total number of
points is calculated with a fixed number of anterior points $N_{A}=4000$ (eq.
(20)). For actual computations of wild-type cytoskeletons
($k_{P}=3,h_{0}^{P}=0,k_{A}=20,h_{0}^{A}=0.8$, see main text for dimensional
parameters), more than $55000$ seeding points are used. Figure: 7\. The
cortical MT density increases approximately parabolically from the posterior
pole towards the anterior corners. A) $\alpha$-tubulin staining of a stage 9
oocyte (green) with DAPI staining of DNA (blue). Arrows indicate arclengths
$s_{P}$ from the posterior pole to the anterior corners and analogously
$s_{A}$ at the anterior. B) Fluorescence intensity profile (green) with moving
average (red) extracted from a 10 pixel wide line along $s_{P}$ indicated in
panel A. Arclength was normalized to one, and the minimum intensity of the
moving average was subtracted from both fluorescence profiles. C) Average
intensity profiles (red) of cortical MT density as in panel B from $N=15$
oocytes. The minimum intensity value across all profiles was subtracted from
each. Blue line shows a parabolic fit. D) Probability densities for the
distribution of MT seeding points along anterior arclength (top) and posterior
arclength (bottom) used for the wild-type cytoskeleton (blue lines). Black
dashed line in bottom panel shows exact parabola. Arclengths are
nondimensional lengths with scale $L=50\mu$m. Figure: 8\. von Mises-Fisher
parameter $\kappa$ determines MT stiffness. Shown are three cross sections
through MT cytoskeletons for indicated values of $\kappa$. MTs become stiffer
with increasing values of $\kappa$. Insets show angular distribution for
$3\times 10^{3}$ points drawn from von Mises-Fisher distribution around mean
direction $\bm{\hat{\mu}}=\bm{\hat{e}}_{z}$.