Cortical microtubule nucleation can organise the cytoskeleton of Drosophila oocytes to define the anteroposterior axis
Philipp Khuc Trong, Hélène Doerflinger, Jörn Dunkel, Daniel St. Johnston, Raymond E. Goldstein
CCortical microtubule nucleation can organise the cytoskeleton of
Drosophila oocytes to define the anteroposterior axis a Philipp Khuc Trong , , H´el`ene Doerflinger , J¨orn Dunkel , ,Daniel St. Johnston , and Raymond E. Goldstein Department of Applied Mathematics and Theoretical Physics,University of Cambridge, Wilberforce Road,Cambridge CB3 0WA, United Kingdom Department of Physics, University of Cambridge,J.J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom Wellcome Trust/Cancer Research UK Gurdon Institute,Tennis Court Road, Cambridge CB2 1QN, United Kingdom and Department of Mathematics, Massachusetts Institute of Technology,77 Massachusetts Avenue E17-412 Cambridge, MA 02139-4307, USA (Dated: October 8, 2018)
Abstract
Many cells contain non-centrosomal arrays of microtubules (MT), but the assembly, organisationand function of these arrays are poorly understood. We present the first theoretical model for thenon-centrosomal MT cytoskeleton in
Drosophila oocytes, in which bicoid and oskar mRNAs becomelocalised 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 theobserved 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 anorganisation of the MT cytoskeleton that is more ordered than previously thought. Furthermore,modulating cortical MT nucleation induces a bifurcation in cytoskeletal organisation that accountsfor the phenotypes of polarity mutants. Thus, our three-dimensional model explains many featuresof the MT network and highlights the importance of differential cortical MT nucleation for axisformation. a This paper has been published: eLife , e06088 (2015). DOI: http://dx.doi.org/10.7554/eLife.06088 a r X i v : . [ q - b i o . S C ] S e p NTRODUCTION
Microtubules (MTs) are polar cytoskeletal filaments that can adopt different global net-work organisations to fulfil different functions. The vast majority of studies have focussed onunderstanding the architecture and function of MT arrays organised by centrosomes, such asradial 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 differentiatedcell types, such as neurons, epithelia, fission yeast and plants [1, 2]. In a variety of cell-types,non-centrosomal MT arrays play an essential role in directing the subcellular localisation ofmRNAs 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 cellbody [3, 4]. MTs and associated motor proteins were implicated in the transport of mR-NAs along dendrites [5], where their activity-dependent translation contributes to long termchanges in synaptic function, neuronal circuitry and memory [6–8]. Similar overlapping,bidirectional MT arrays in the Xenopus oocyte have been proposed to mediate the trans-port of Vg1 mRNA to the vegetal cortex, where it orchestrates germ layer patterning [9, 10].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 requiredfor the localisation of bicoid and oskar mRNAs to opposite ends of the cell [11]. Despite thelarge amount of work, however, the organisation of the non-centrosomal MT cytoskeletonunderlying this mRNA localisation is controversial [12–14], and its assembly and functionare not understood. In stage 9 oocytes, MTs grow from most parts of the cell cortex intothe volume [15, 16] thereby giving rise to a complex, three-dimensional MT network withoutpronounced polarity along the anterior-posterior axis [15, 17]. In contrast to this apparentdisordered organisation, bicoid and oskar mRNAs become reliably localised by Dynein andKinesin to the anterior corners and to the posterior pole of the oocyte, respectively, therebydefining the anterior-posterior axis [14, 18–21].The most pronounced feature of the MT cytoskeleton at stage 9 is a gradient of corticalMTs from the anterior to the posterior pole, where MT nucleation is suppressed by thepolarity protein PAR-1 [22, 23]. Live imaging of oskar mRNAs and direct measurements ofgrowing MTs showed that the MT network is mostly disordered with only a weak statisticalbias of about 8% more plus ends pointing towards the posterior pole [14]. This bias vanishes2n absence of PAR-1 when MTs nucleate from all over the cortex [16]. While these findingsestablished a directionality of the MT meshwork for the first time, several questions remainunanswered. For example, mRNA transport and localisation is highly reproducible, raisingdoubts 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 [17], which is difficult to reconcilewith a cytoskeleton that is simply biased towards the posterior everywhere. Similarly, thedifferent behaviour of so-called conditioned bicoid mRNA, which localises specifically tothe anterior cortex irrespective of the injection site [17], has remained unexplained. Finally,motor-driven cytoskeletal transport is not the only transport mechanism in oocytes. Kinesinmoves only 13% of oskar mRNA at any given time, and the remaining 87% is subject todiffusion and slow cytoplasmic flows that are driven indirectly by Kinesin activity on theMT network [14]. This raises the question if cytoskeletal transport alone is sufficient toaccount for mRNA localisation [24, 25] and highlights the importance of distinguishing itscontribution to localisation from the contribution of flows and diffusion [26].Here, we present the first theoretical model for stage 9
Drosophila oocytes. Based onthe distribution of microtubule nucleation sites around the cortex, this model accuratelyreproduces the observed distribution of MTs and cytoplasmic flows in the oocyte. It revealsthat 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 organi-sation is sufficient to explain the localisation of oskar and naive bicoid mRNA. Finally, weshow that modulation of MT nucleation gradients causes a bifurcation in cytoskeletal or-ganisation that explains mutant phenotypes. Thus, our results explain many features of theassembly, organisation and function of the non-centrosomal MT array in
Drosophila oocytesand highlight the key role of differential MT nucleation or anchoring at the cortex [27].
RESULTS
MTs in the oocyte are nucleated or anchored at the cortex [15, 16] and grow from themembrane into the volume. The cortical MT density follows a steep gradient along theposterior-lateral cortex from high density at the anterior corners to low densities at theposterior pole (Fig. 1D, Materials and methods M2.1). In our model, we emulated these3eatures 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 ofa stage 9
Drosophila oocyte (Fig. 1A, Materials and methods M1). The density of seedingpoints decreases steeply along the posterior-lateral cortex to zero at the posterior pole, anddecreases weakly along the anterior cortex towards the anterior centre (Fig. 1A-B, Materialsand methods M2.1). Each seeding point nucleates a MT polymer that grows until it eitherhits a boundary or reaches a target length imposed by the aging of MTs before catastrophe[28] (Materials and methods M2.3). In total, we computed more than 55000 MTs for eachrealisation of a 3D wild-type cytoskeleton.A cross section through the computed MT meshwork (Fig. 1A,B) bears a striking resem-blance to confocal images of Tau- [29, 30] and EB-1- [16] tagged MTs, correctly showing thepronounced anterior-posterior gradient of MT density. Taking into account the orientationsof all MT segments in the 3D volume, it also reproduces the experimentally measured direc-tional bias [16] 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 theoocyte, ranging from 50.7% of posteriorly oriented MT segments at the cortex to 60.6% inthe mid plane (Fig. 1B, inset). This demonstrates that measurements in 2D confocal slicespoorly characterise the fully extended 3D system.Central to understanding mRNA localisation is the question of MT orientations. Cal-culating the local vectorial sum of MT segments on a coarse-grained grid gives the localMT orientations in the computed cytoskeleton. Local orientations of an individual realisa-tion 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. MTsdisappear within minutes in the presence of the MT-depolymerising drug colchicine thatblocks MT growth by sequestering free tubulin dimers, indicating that the whole networkturns over rapidly [15, 31] (V. Trovisco, personal communication). Thus, the oocyte sam-ples many tens to a hundred of independent MT organisations over the 6-9 hours of stage9. Summation of local orientations over an ensemble of 50 independent realisations of thecomputed MT network reveals a striking spatial partitioning of the cytoskeleton into severalsubcompartments (Fig. 1F). At the anterior, the mean orientation points posteriorly, whileMTs at the lateral sides on average point inwards toward the AP-axis. Counting the numberof MT segments that contribute to each grid box in the ensemble also shows the distribution4f MT density (Fig. 1H). In agreement with experiments (Fig. 1D), the MT density exhibitsa pronounced anterior-posterior gradient with highest values in the anterior corners, evenwhen MT seeding is uniform on the anterior surface.In the limit of a large ensemble, the MT polymers sample all possible initial directionsand possible lengths at every point along the cortex. In this limit, we can test the predictedcompartmentalised MT organisation by constructing a second model in which MTs arerepresented as straight rods. The net orientation of the cytoskeleton at a given point insidethe oocyte is then computed as weighted sum of MT contributions from each point alongthe boundary (Materials and methods M3.1).With a shorter MT target length to compensate for the effectively longer length of straightrods compared to curved polymers, computation of net MT orientations in the rod modelshows 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 arrowsand converge at the separatrices, eventually leading to the posterior pole as the sole point ofattraction in the entire oocyte volume (the attractor). By contrast, cargo that is transportedto 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 theMT nucleation probability and to the choice of the MT length distribution (Materials andmethods M2.1, M2.3). It also remains unchanged in a differently shaped oocyte geome-try in both the polymer model and the rod model (Figure 1 - figure supplement 1). Insummary, both models show that even disordered non-centrosomal MT arrays can featurewell-defined mean organisations, and that the MT cytoskeleton in oocytes is organised in acompartmental fashion.The suitably scaled local vectorial sum of MT segments (Fig. 1C) for an individualrealisation of the polymer model (Fig. 1A-B) is a vector field v m which represents activeKinesin-driven transport on the cytoskeleton (Materials and methods M2.5). In vivo duringstage 9, the oocyte cytoplasm undergoes slow cytoplasmic flows that are abolished in kinesinheavy chain mutants. This indicates that flows are driven by kinesin-dependent transportof an unknown cargo through the viscous cytoplasm [26, 32], thus making cytoplasmic flowsa secondary read-out of cytoskeletal organisation [33]. Therefore, we next tested if our5omputed polymer MT cytoskeleton can produce flows that are consistent with observedcytoplasmic streaming.Measurements of speeds of autofluorescent yolk granules in live stage 9 oocytes showedthat mean flow velocities are slow (Fig. 2F). The physics of slow incompressible fluid flows u driven by forces f is described by the Stokes equations0 = − ∇ p + µ ∇ u + f , ∇ · u = 0 . (1)We make the simplest possible assumption that the forces f are proportional to the motor-velocity field, and use experimentally measured flow speeds to calibrate the scalar factor ofproportionality. By solving the Stokes equations, we then computed the full 3D fluid flowfield (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 patternsvisualised 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 typicallyseen in PIV flow fields, thereby slightly overestimating the range of flows. However, except inrare cases, computed flows do not reach the posterior pole. This result is largely independentof 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, Materialsand methods M5.2, M5.3). Thus, computed flows appear consistent with observed slowcytoplasmic streaming.We tested next if our computed cytoskeleton in combination with the derived cytoplasmicflows and diffusion can account for dynamic mRNA transport and localisation. We describedthe mRNA distributions as continuous concentration fields. oskar mRNA is assumed toreside in either one of two states: the Kinesin-bound state with concentration c b , in whichcargo is transported actively on the cytoskeleton-derived motor-velocity field v m (Fig. 1C),or the unbound state with concentration c u , in which cargo is transported by the cytoplasmicflows u (Fig. 2A) and diffuses with diffusion constant D . Cargo can exchange between bothstates by chemical reactions, thereby resulting in the reaction-advection-diffusion equations:6 t c b + ∇ · ( v m c b ) = k b c u − k u c b (2) ∂ t c u + ∇ · ( u c u ) = − k b c u + k u c b + D ∇ c u Parameter values were constrained with experimentally measured values (Materials andmethods M4.2). Furthermore, throughout the simulation we cycled through the pairs offluid flow u and motor-velocity fields v m (Fig. 3) to account for the dynamic nature of theMT network [16] and flow patterns, which change over time scales of minutes compared tothe 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 forminga channel from the centre of the oocyte towards the posterior pole (Fig. 3E). Formation ofthis 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 supportingthe notion that our computed MT cytoskeleton correctly captures key aspects of the in vivoMT network.Upon reaching the posterior cortex, oskar mRNA is translated to produce Long Osk [34],which is involved in anchoring oskar mRNA. However, oskar mRNA localises normally atstage 9 when anchoring is disrupted [34, 35]. 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 theend of the simulation (Fig. 3F), correctly capturing observations of in vivo localisation (Fig.3C). A 4 µ 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 andmaintain 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 v m and u , and these two fields must containall information about the localisation sites. We therefore asked which field contains mostinformation about localisation. We first tested localisation in the absence of cytoplasmicflows by setting the fluid flow field identical to zero u ≡
0. This situation is similar to but7ore severe than in slow Kinesin mutants [26]. Starting from the central oskar mRNA cloud(Fig. 3D), we found that oskar mRNA still localises to and forms a concentrated crescent atthe posterior pole (Fig. 3G). A 4 µ m thick slice off the posterior pole contains 11.3% of totalavailable oskar mRNA cargo, only marginally less than localisation with cytoplasmic flowspresent. 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 themotor-velocity field identical to zero v m ≡
0, or by setting the binding constant k b to zero.In either case, cargo disperses throughout the oocyte without forming a central channel andcompletely fails to produce any posterior crescent. We further asked whether the flow couldlocalise 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 onlyat the very posterior boundary. Binding to the anchor occurs with a reaction rate constantthat is ten times larger than the reaction rate constant k b used for oskar mRNA binding tothe MT cytoskeleton (Materials and methods M4.2). Moreover, anchored cargo cannot bereleased, thus making this anchor a perfect sink with stronger trapping properties than anyrealistic anchor. Under these idealised conditions, oskar mRNA accumulates slightly at theanchor (Fig. 3K). However, a 4 µ m thick posterior slice contains only 2.7% of total cargoin the oocyte, about 78% less cargo than in the wild-type simulations, and only 25% morecargo compared to the purely diffusive case, which localises 2.2% of cargo. Thus, even underthe most favourable conditions, the cytoplasmic flows cannot localise mRNA to wild-typelevels, arguing against a mixing and entrapment mechanism for oskar mRNA localisationat stage 9. Despite the small fraction of actively transported cargo (13%), motor-driventransport 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 [19–21], but other parameters such as the fraction of bound cargo and the active transport speedsare similar to oskar mRNA [36] (V. Trovisco, personal communication). To test whetherthe proposed cytoskeletal organisation also captures transport and localisation of injected bicoid mRNA, we inverted the directions of the motor-velocity fields (Fig. 1C) to accountfor the minus end directed transport by Dynein instead of the plus end directed transportby Kinesin. Other parameters including cytoplasmic flow fields remained unchanged.Closely matching experiments with injected naive bicoid mRNA [17], simulations showaccumulation of bicoid mRNA at both posterior-lateral sides of the oocyte when placed8nitially in the posterior half (Fig. 3H, inset), or accumulation at the anterior and ventralcortex 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 theborder (separatrices) between two diverging subcompartments of the MT cytoskeleton, eachone transporting part of the cloud towards the adjacent cortex. The agreement betweensimulations and experiments therefore further supports an on average compartmentalisedMT cytoskeleton, and naive bicoid mRNA, like oskar mRNA, unspecifically traces out theMT cytoskeleton.In contrast to naive bicoid mRNA, endogenous bicoid mRNA is transported into theoocyte after being transcribed in neighbouring nurse cells where it also becomes modifiedin presence of the protein Exuperantia. Endogenous bicoid mRNA can be approximated byso-called conditioned bicoid mRNA that is first injected into the nurse cells, then sucked outand injected into the oocyte. Conditioned bicoid mRNA localises specifically to the anteriorsurface within 30 minutes, even if this surface is not closest to the injection site [17]. In themodel, starting from an initial distribution at the anterior surface where endogenous bicoid mRNA enters the oocyte, cargo quickly moves to the anterior cortex before concentratingin the anterior corners after several hours of simulated time (Fig. 3J). This resemblesthe anterior ring-like localisation of bicoid mRNA in wild-type stage 9 oocytes [21, 37], andoccurs independently of cytoplasmic flows. However, the model does not reproduce transportspecifically to the anterior surface when injection is further away from the anterior. Thissuggests that Exuperantia activity somehow either enables bicoid mRNA to move along anunidentified population of MTs [12, 17] 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 afunctional, compartmentalised MT cytoskeleton that successfully localises oskar mRNA andnaive bicoid mRNA. Next, we asked if cortical MT nucleation can also produce cytoskeletalorganisations that explain mutants with partial or complete polarity defects. Mutationsinterfering with the posterior follicle cells that surround the oocyte, for example, can disruptthe proper positioning of mRNAs [38, 39]. Specifically, follicle cell clones (ras ∆C40b ) that areadjacent to only one side of the oocyte posterior repel cargo from that side of the cortex(termed clone adjacent mislocalisation [40]), likely due to an altered MT organisation. To9imic this phenotype, we used an ensemble of wild-type cytoskeletons and artificially addedMT nucleation sites to a patch on one side of the posterior pole. Simulations of cargotransport with diffusion, motor transport and flows show that oskar mRNA is repelled fromthis 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, whichacts to suppress MT nucleation at the posterior pole of the oocyte [41, 42]. In strong par-1hypomorphs with strongly reduced PAR-1 expression, MT minus ends occupy the wholecortex including the posterior pole [16], thereby causing oskar mRNA to mislocalise to adot in the centre of almost 90% of mutant oocytes [22]. To test this phenotype in the modelwe distributed MT seeding points uniformly on the posterior surface (Fig. 4G, inset) whilekeeping the seeding density on the anterior surface constant. The anterior-posterior gradientof 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 MTsshow that MTs point towards a single focal point in the centre of the oocyte (Fig. 4G) thatacts as a stable fixed point of the system. Computation of 3D flow fields for each realisationin the new ensemble (3D mean: 13.5 nm/s; 2D mean: 14.5 nm/s, N=50) and simulations oftransport with and without cytoplasmic flows show oskar mRNA concentrating in a cloud inthe centre of the oocyte (Fig. 4C). oskar mRNA accumulation to a central dot also remainsunchanged if the posterior MT seeding density is decreased slightly (Fig. 4H, inset) as mightbe expected for hypomorphs that do not abolish PAR-1 expression completely.Interestingly, bicoid mRNA simulated either with slightly decreased (Fig. 4H) or withuniform (Fig. 4G) posterior MT nucleation not only localises to the anterior corners as inwild-type but also enriches at the posterior pole (Fig. 4D, arrowheads), despite injectionclose to the anterior (Fig. 4D, inset). This matches experiments showing bicoid mRNAmislocalisation 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 theoocyte [38, 39].Failure of the external signal to the oocyte is thought to have the same effect as par-1hypomorphs, but previous experiments with bicoid mRNA in par-1 hypomorphs resultedin ambiguous localisations [41, 43]. Here, using confocal fluorescence in-situ hybridisation(FISH), our experiments show that bicoid mRNA localises primarily to the anterior corners10nd to the posterior pole in strong par-1 hypomorphs (Fig. 4L), thereby mirroring EGFRmutants and agreeing with the model predictions. Thus, (near-)uniform MT nucleation alongthe 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-1hypomorphs.In 10% of strong par-1 hypomorphs and 30% of weak par-1 hypomorphs with weaklyreduced PAR-1 expression, oskar mRNA is only partially mislocalised, combining a wild-type-like posterior crescent with a centrally mislocalised dot [22]. We therefore modulatedthe MT nucleation along the posterior-lateral cortex to test if this is sufficient to produceintermediate cytoskeletal organisations between wild-type (Fig. 4F) and strong par-1 hypo-morphs (Fig. 4H).Increasing MT nucleation laterally towards the posterior pole (Fig. 4G, inset) againcreates a stable focus in the centre of the oocyte. However, its domain of attraction doesnot cover the entire oocyte. Instead, a small basin of attraction towards the posterior polepersists (Fig. 4G, red), separated from the basin of attraction of the stable focus by anunstable saddle-node point. Further increasing the seeding density towards the posteriorpole 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 simulatedon cytoskeletons with stable and unstable fixed points and their corresponding flow fields cansplit into two separate accumulations, first at the posterior pole and second in a dot alongthe AP-axis (Fig. 4B). This agreement with experimental results suggests that intermediatelevels of PAR-1 at the posterior lead to a cytoskeleton with a potential barrier between theoocyte centre and its posterior pole. It also shows that modulation of MT nucleation alongthe posterior-lateral cortex alone is sufficient to capture this. Because the potential barrieronly affects the bound state, diffusion and flows can aid posterior localisation in this contextby 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 alongthe cortex not only explain mRNA localisations in wild-type but can also account for themislocalisations of bicoid and oskar mRNAs in polarity mutants.The creation of stable and unstable fixed points in the transition from wild-type tostrong par-1 hypomorph topology constitutes a classical saddle-node bifurcation (Fig. 4K-11, Materials and methods M3.3) with the cortical MT seeding density as the bifurcationparameter. Interestingly, in both the polymer model and the rod model, the mean lengthof MTs acts as a second bifurcation parameter. For example, for sufficiently short MTfilaments (Fig. 4K), few MTs from the anterior can reach and contribute to orientationsin the posterior half of the oocyte. In the posterior half, MTs from the lateral side eitherpoint 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 theunstable tipping-point.To understand more generally how the MT lengths and the posterior-lateral distributionof 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 areabsent at the posterior pole, the wild-type topology (Fig. 5D i, vi) covers a large fractionof parameter space (Fig. 5A, red) for sufficiently long MTs (large (cid:15) ) and cortical MTnucleation gradient (large k P ). The creation of two fixed points splits the oocyte intodifferent 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 MTnucleation (Fig. 5A, horizontal dashed arrow; Figure 4 - figure supplement 1 I,H). For almostuniform nucleation along the entire posterior-lateral cortex, the unstable fixed point exitsthe 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 hypomorphtopology expands substantially (Fig. 5B, C, green). Interestingly, if MTs are sufficientlylong, MTs from the lateral and anterior cortex can reach and overpower those from theposterior pole, thus restoring wild-type-like cytoskeletal topology that allows oskar mRNAlocalisation. Therefore, a low level of posterior nucleation of MTs does not necessarily leadto mislocalisation of oskar mRNA. Instead, and in addition to the distribution of corticalMT nucleation, the length of MT filaments emerges as another key regulator for polarisationand function of the non-centrosomal network.12 ISCUSSION
Non-centrosomal MT networks represent a large, yet poorly understood class of MT ar-rangements that often fulfil specialised functions [44]. Non-centrosomal MTs are frequentlyaligned in parallel, thereby forming linear arrays as in epithelia or neurons [1, 3, 4]. Thenon-centrosomal MT cytoskeleton in
Drosophila oocytes was also first hypothesised to forma highly polarised linear array with MTs growing from the anterior surface towards theposterior pole [18, 45]. Instead, the MT cytoskeleton is a complex network without a vis-ibly discernible polarity along the anterior-posterior axis [15, 17, 46], and its organisationremained ambiguous [12–14, 16].We showed that in two different theoretical models, in which MTs grow from the oocytecortex into the volume, the cytoskeleton features an on average compartmental organisationand is therefore more ordered than previously thought. In combination with cytoplasmicflows and diffusion, this cytoskeletal organisation successfully reproduces localisations of os-kar and naive bicoid mRNAs in wild-type oocytes. Such a spatially varying, compartmentalorganisation suggests that a single statistical measure of polarity, derived by averaging datafrom the entire oocyte [14, 16], may not be a sufficient metric to characterise mRNA locali-sation.Cytoplasmic flows may generally contribute to mRNA transport. Flows at later stages ofoogenesis are fast and well-ordered, and occur concurrently with a late-phase enhancementof oskar mRNA accumulation [47]. This is consistent with a contribution of flows to mRNAlocalisation [24, 25]. At stage 9 of oocyte development, which is the focus of the workpresented here, flows are slow and chaotic. At this stage, mutants with reduced flowsshowed oskar mRNA localising normally [26], thus suggesting that slow and chaotic flowsmay not play the same role as in late oogenesis. However, flows in these mutants weremerely reduced rather than abolished completely, and the measurements underestimatedflow speeds [26], hence leaving the interpretation of flows unclear. We here find that theeffects of slow cytoplasmic flows on mRNA transport are negligible, and that cytoskeletaltransport 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-driventransport on the cytoskeleton rather than as an evolutionarily selected trait. This appearsto mirror findings in the
C. elegans zygote in which P-granules segregate by dissolution and13ondensation rather than via transport by cytoplasmic flows [48].Interestingly, neither MT-based transport alone nor combined with cytoplasmic flows anddiffusion is sufficient to reproduce the anterior localisation of nurse cell conditioned bicoid mRNA irrespective of the position of injection into the oocyte [17]. This Exuperantia-dependent mechanism may rely either on an unobserved population of MTs in the oocytethat can be specifically recognised by conditioned bicoid mRNA [12, 17], or on an unknownMT-independent mechanism.The central finding of our work is that gradients of cortical MT nucleation are sufficientfor the assembly of a functional compartmentalised MT cytoskeleton in wild-type oocytes.While many non-centrosomal MT arrays are linear and emphasise questions about estab-lishment and maintenance of parallel filament orientations [49, 50], our result stresses theneed to understand gradients in MT nucleation as an alternative strategy for the assemblyof functional non-centrosomal arrays. Whether MTs in
Drosophila oocytes are differentiallynucleated along the cortex itself or created elsewhere and then differentially anchored at thecortex remains an interesting open question, but both scenarios are compatible with ourmodel. Understanding how this gradient is established will therefore depend on discoveringhow PAR-1 regulates MT interactions with the cortex.Cortical MT gradients cannot only account for the wild-type cytoskeletal configurationbut also for the phenotypes observed in a hierarchy of par-1 hypomorphs. Modulation ofMT gradients along the posterior-lateral cortex alone are sufficient to explain the splittingof oskar mRNA between the centre and the posterior pole of the oocyte [22] via a saddle-node bifurcation, suggesting that the anterior and posterior-lateral surfaces of the oocyteare functionally decoupled. Generally, bifurcations in temporal behaviour govern importantqualitative transitions in many biological systems, such as the lactose network in E. coli[51], cell cycle in yeast [52], and collapses of bacterial populations [53]. In
Drosophila , oneimportant qualitative change is the temporal transition between stages 7/8 and 9. Duringthis transition the MT cytoskeleton reorganises from uniform nucleation around the cor-tex 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 ofPAR-1 mutants and the underlying bifurcation represent static snapshots of this dynamicdevelopmental transition in wild-type.In conclusion, the present work provides a model that describes the assembly, organisation14nd function of the non-centrosomal MT array in
Drosophila oocytes and directs futureattention to the molecular mechanisms that enable differential MT nucleation or anchoringat the cortex.
MATERIAL AND METHODSM1 Coordinates and oocyte geometries
For calculation of the microtubule (MT) cytoskeleton, the anterior-posterior (AP) axisis aligned with the z -axis of a cartesian coordinate system, and results are later shifted androtated to align with the x -axis for visualization, for subsequent computations of cytoplas-mic flows and simulations of cargo transport. Dimensional coordinates cover the ranges x ∈ [ − L, L ], y ∈ [ − L, L ] and z ∈ [0 , z L ] with length scale L = 50 µ m. After nondimension-alization with scale L , coordinates are ranged as x (cid:48) ∈ [ − , y (cid:48) ∈ [ − ,
1] and z (cid:48) ∈ [0 , z ].Similarly, nondimensionalization of the shape parameter k P in the MT seeding density leadsto k (cid:48) P = k P /L , and all primes will be dropped subsequently. Methods figures are shownin nondimensional spatial coordinates, and reported parameter values are nondimensionalunless 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 capsare parameterized in cylindrical coordinates ρ = (cid:112) x + y ∈ [0 ,
1] and φ ∈ [0 , π ) as σ i ( ρ, φ ) = ρ ˆ e ρ + z i (cid:0) − ρ (cid:1) ˆ e z , (3)where ˆ e ρ = (cos( φ ) , sin( φ ) ,
0) and ˆ e z = (0 , ,
1) are the unit vectors in radial and z -direction,respectively. The line element along the parabola is calculated asd s i = (cid:13)(cid:13)(cid:13)(cid:13) ∂ σ i ∂ρ (cid:13)(cid:13)(cid:13)(cid:13) d ρ = (cid:113) z i ρ ) d ρ with arclength s i ( ρ ) = ρ (cid:113) z i ρ ) + 14 z i sinh − (2 z i ρ ) (4)defined such that s i ( ρ = 0) = 0 denotes the tip of the parabolic cap while s i ( ρ = 1) = s i denotes the distance from the tip to the anterior corners (Fig. 6 A,G), and hence 0 ≤ s i ( ρ ) ≤ i . The surface element for parabolic caps (in units of L ) is given bydΣ i = (cid:13)(cid:13)(cid:13)(cid:13) ∂ σ i ∂ρ × ∂ σ i ∂φ (cid:13)(cid:13)(cid:13)(cid:13) = ρ (cid:113) z i ρ ) d ρ d φ , (5)with total surface area Σ i = π z i ) (cid:16)(cid:2) z i ) (cid:3) / − (cid:17) . (6)The inward pointing normals for posterior and anterior caps are given by ˆ n P ( ρ, φ ) = − z P ρ ˆ e ρ + ˆ e z (cid:112) z P ρ ) (7) ˆ n A ( ρ, φ ) = 2 z A ρ ˆ e ρ + ˆ e z (cid:112) z A ρ ) . A special case arises for z A = 0 when the anterior parabolic cap becomes a flat disc. In thiscase, the arclength (eq. (4)) reduces to s A ( ρ ) = ρ , and the surface area (eq. (6)) simplifiesto Σ A = π . The inwards pointing normals for this case are given by ˆ n P (eq. (7)) for theposterior cap and ˆ n A = ˆ e z for the anterior disc, respectively.To investigate the robustness with respect to changes in oocyte shape, we test our modelfor the cytoskeleton, cytoplasmic flows and mRNA transport in two different geometries fora stage 9 Drosophila oocyte. Geometry-1 is used as standard geometry, and is comprisedof two parabolic caps (eq. (3)) with z P = 1 .
48 for the posterior cap and z A = 0 . z P − z A = 1 .
28 with an aspect ratio of 1 .
56 that qualitatively resembles a typical stage 9oocyte. Geometry-2 is tested as alternative geometry, and consists of a posterior paraboliccap with z P = 1, and an anterior flat disc with z A = 0 (Fig. 1 - figure supplement 1), givingan AP-axis length of 1 and aspect ratio of 2. We find that results are robust with respectto this change in geometry. M2 Polymer model for MT cytoskeleton
M2.1 MT nucleation probability
The first step in the computation of the MT cytoskeleton is the generation of MT seedingpoints that are randomly positioned along the oocyte membrane. We define the following16robability density along the arclength s i ( ρ ) of the anterior and posterior parabolic caps p Σ ( s i ( ρ ) | h i , k i ) = 1 A ˜ p Σ ( s i ( ρ ) | h i , k i ) , (8)with normalization over the oocyte surface A = (cid:90) Σ A dΣ A ˜ p Σ ( s A ( ρ ) | h A , k A )+ (cid:90) Σ P dΣ P ˜ p Σ ( s P ( ρ ) | h P , k P ) , (9)where the surface elements were specified in eq. (5) and the arclength was given in eq. (4).For the functional form of ˜ p Σ ( s ( ρ ) | h , k ) we use the expression˜ p Σ ( s ( ρ ) | h , k ) = h + (1 − h ) (cid:34) (cid:18) ks (cid:19) (cid:35) s ( ρ ) k + s ( ρ ) , (10)wherein the index i was dropped for simplicity. Note that for the maximum arclength s = s it holds that ˜ p Σ ( s = s | h , k ) = 1. h ∈ [0 ,
1] and k ∈ (0 , ∞ ) represent two parameters governing the shape of the probabilitydensity. For small values k (cid:28) s the nucleation density approaches a Hill function with Hillcoefficient n = 2 and depth h lim k (cid:28) s ˜ p Σ ( s ( ρ ) , | h , k ) = h + (1 − h ) s ( ρ ) k + s ( ρ ) . (11)In the opposite limit k (cid:29) s , the probability density approaches a parabola with depth h (Fig. 6B,H, red curve) lim k (cid:29) s ˜ p Σ ( s ( ρ ) , | h , k ) = h + (1 − h ) (cid:18) s ( ρ ) s (cid:19) , (12)thus creating a pronounced gradient of MT nucleation from the pole ( s = 0) to the corners( s = s ).MT nucleation along the arclength can increase from the gradient to a homogeneousnucleation along the arclength in two different ways: (i) either by increasing nucleation fromthe corners laterally towards the posterior pole, or (ii) by increasing nucleation uniformly at17he posterior pole. Each shape parameter h and k accounts for one of these possibilities.Increasing the parameter h leads to shallower gradients by increasing the probability fornucleation at the pole from 0 for h = 0 to the value for homogeneous nucleation for h = 1(Fig. 6H). Contrary, decreasing the parameter k leads to narrower regions of low nucleationprobability by decreasing its width laterally (Fig. 6B).To compare to experimental data, we stained α -tubulin in fixed stage 9 wild-type Drosophila oocytes and extracted fluorescence intensity profiles along the cortex from theposterior pole to the anterior corners (Fig. 7A,B). An ensemble of smoothed fluorescenceprofiles shows steep increases in cortical MT density from posterior to anterior, albeit withhigh variability. Still, a quadratic fit of the ensemble shows that a parabolic function isa plausible representation of the increasing MT density (Fig. 7C). Several oocytes showfluorescent profiles that exhibit a local maximum at the posterior. This likely stems fromout of plane fluorescence due to the geometric shape of the oocyte pole, and we neglectthis minor effect here. At the anterior surface, MT density tends to be more homogeneouswith only shallow increases in cortical MT density towards the corners. In summary, for theposterior cap we choose a probability density with parabolic shape that reaches zero at theposterior pole h P = 0 , k P = 20 (Fig. 7D, bottom), whereas for the anterior cap we chose adensity with parabolic shape but only shallow depth h A = 0 . , k P = 20 (Fig. 7D, top). M2.2 Generation of seeding points
Discrete seeding points that are randomly positioned along the oocyte membrane accord-ing to the probability density p Σ ( s i ( ρ ) | h i , k i ) (eq. (8)) can be achieved by inverse transformsampling. To this end, the cumulative distribution function on the anterior ( i = A ) orposterior ( i = P ) cap can be computed as C ( ρ (cid:48) | h i , k i ) = (cid:90) π (cid:90) ρ (cid:48) dΣ i p Σ ( s i ( ρ ) | h i , k i ) . (13)Note that C ( ρ (cid:48) = 1 | h i , k i ) (cid:54) = 1 due to normalization across the total oocyte surface ratherthan across each individual cap. Therefore, random seeding points on each cap can bedrawn by applying the inverted cumulative distribution function to a random sample thatis uniformly distributed over the interval [0 , C ( ρ (cid:48) = 1 | h i , k i )]. In this formulation, the total18umber of seeding points is N = N A + N P (see Fig. 6). The ratio of anterior ( N A ) andposterior ( N P ) seeding points to be drawn must equal the ratio of total probability for apoint to fall on the anterior and posterior caps, respectively. For a fixed value N A , thisresults in N P = N A (cid:82) dΣ P p Σ ( s P ( ρ ) | h P , k P ) (cid:82) dΣ A p Σ ( s A ( ρ ) | h A , k A ) . (14)Here, we generated seeding points in an equivalent way by renormalizing the probabilitydensity (eq. (8)) on each cap individually as P Σ ( s i ( ρ ) | h i , k i ) = 1 A ˜ p Σ ( s i ( ρ ) | h i , k i ) , (15)wherein the new normalization factor A is obtained by integration over only one paraboliccap A ( h i , k i ) = (cid:90) dΣ i ˜ p Σ ( s i ( ρ ) | h i , k i ) . (16)Hence, the corresponding cumulative distribution function C ( ρ (cid:48) | h i , k i ) = (cid:90) π (cid:90) ρ (cid:48) dΣ i P Σ ( s i ( ρ ) | h i , k i ) (17)covers the range [0 ,
1] and seeding points are obtained as C − ( u ) for a random sample u withuniform distribution u ∈ [0 , i = A ) discin the alternative geometry-2, the normalization can be solved in closed form as A ( h A , k A ) = (cid:90) π dφ (cid:90) dρ ρ ˜ p Σ ( ρ, h A , k A ) = (18) π (cid:2) (1 − h A ) k A + 1 − (1 − h A ) k A ( k A + 1) ln(1 + 1 /k A ) (cid:3) . The corresponding cumulative distributions function evaluates to C ( ρ (cid:48) ) = (cid:90) π dφ (cid:90) ρ (cid:48) dρ ρ P Σ ( ρ, h A , k A ) = (19) ρ (cid:2) (1 − h A ) k A + 1 (cid:3) − (1 − h A ) k A ( k A + 1) ln[1 + ( ρ/k A ) ](1 − h A ) k A + 1 − (1 − h A ) k A ( k A + 1) ln[1 + (1 /k A ) ] , A i along the contact line ( s = s i ) on the anterior ( i = A ) and posterior ( i = P ) cap which contain d N i = N i P Σ ( s = s i | h i , k i ) d A i seeding points. Setting the numberof anterior points fixed, and enforcing the point densities to be equal in the corner ringsd N A / d A A = d N P / d A P for equal ring areas d A A = d A P allows to compute the number ofposterior seeding points as N P = N A P Σ ( s = s A | h A , k A ) P Σ ( s = s P | h P , k P )= N A A ( h P , k P ) A ( h A , k A ) , (20)where the second line follows because ˜ p Σ ( s i = s i | h i , k i ) = 1. Inserting the expression for A ( h i , k i ) (eq. (16)) shows that forcing seeding point densities on the anterior and posteriorcaps to be equal at the anterior corners (eq. (20)) is identical to chosing points according tothe ratio of total probabilities on the anterior and posterior caps (eq. (14)). Thus, chosingpoints from p Σ normalized over the entire surface (eq. 8) or from P Σ normalized over eachcap (eq. 15) is equivalent. Alternative seeding densities
Our model is not sensitive to the exact choice of the proba-bility density for MT seeding points, and here we define alternative seeding densities that aremathematically tractable. Specifically, as randomly positioned seeding points are generatedby operating the inverse CDF on a uniformly distributed sample u ∈ [0 ,
1] and φ ∈ [0 , π ), itis 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 probabilitydensity function p ( s i ( ρ )) along the arclength (eq. (13)) leads to C ( ρ (cid:48) ) = (cid:90) π d φ (cid:90) ρ (cid:48) d ρ p ( s i ( ρ )) ρ (cid:113) z i ρ ) . (21)For the case of a gradient of MT seeding points along the arclength of a parabolic cap we20an define the density to be p ( s i ( ρ )) ≡ p n ( ρ | n, z i ) = ( n + 1) ρ n − π (cid:112) z i ρ ) . (22)From eq. (21) this results in a CDF u ≡ C n ( ρ (cid:48) ) = ρ (cid:48) n +1 which can be analytically invertedas ρ (cid:48) = C − n ( u ) = u / ( n +1) .For the case of a uniform distribution of seeding points along the arclength, the probabilitydensity is constant and given by the inverse surface area of the parabolic caps (eq. 6) p ( s i ( ρ )) ≡ p u ( ρ | z i ) = 1Σ i . (23)For p u ( ρ | z i ) we find the CDF u ≡ C u ( ρ (cid:48) ) = (1 + (2 z i ρ (cid:48) ) ) / − z i ) ) / − , (24)which can again be inverted analytically to give C − u ( u ) = 12 z i (cid:114)(cid:16) u (cid:104) (1 + (2 z i ) ) / − (cid:105) + 1 (cid:17) / − . (25)For the special case of a flat disc, the uniform distribution of seeding points simplifies theCDF to u ≡ C u = (cid:90) π d φ (cid:90) ρ d ρ π ρ = ρ (26)with inverse ρ = C − u ( u ) = √ u .For a wild-type cytoskeleton, we impose a uniform MT density p u ( ρ | z i ) along the anteriorcap and a gradient of MTs p n ( ρ | n, z i ) along the posterior cap. At the contact line betweenanterior and posterior caps, the seeding densities need to be matched by adjusting the totalnumber of posterior seeding points N P . A ring on the anterior cap at the anterior cornerscontains d N A = ( N A / Σ A ) d A A seeding points, while a ring on the posterior cap at theanterior corners contains d N P = N P p n ( ρ = 1 | n, z P ) d A P points. Using eq. (22) and eq. (6)21nd forcing densities to be equal d N A / d A A = d N P / d A P results in N P = N A Σ A p n ( ρ = 1)= 3 (cid:0) z A (cid:1) N A n + 1 (cid:112) z P ) (1 + (2 z A ) ) / − . (27)For the alternative geometry-2 comprised of a posterior parabolic cap and flat anterior disc,matching of seeding densities results in N P = 2 N A n + 1 (cid:113) z P ) . (28)Using these alternative seeding densities does not qualitatively change the behavior of themodel. M2.3 MT growth
We describe MTs as persistent random walks, i.e. a chain of straight segments of constantlength λ that show some flexibility in their relative orientations. The orientation ˆ µ i of the i -th segment is drawn from a von Mises-Fisher probability distribution on a 2D sphere f ( ˆ µ i | ˆ µ i − , κ ) = κ π sinh( κ ) e κ ˆ µ i · ˆ µ i − , (29)where κ is the concentration parameter around the orientation ˆ µ i − of the previous segment(see sec. 2.5 for parameter values). The first orientation ˆ µ is drawn from the uniformangular distribution on a 2D sphere ( κ = 0) where orientations are rejected if pointinglocally outwards of the geometry ˆ n A,P ( ρ, φ ) · ˆ µ < l drawn from a probability distribution. The probability density is based onthe experimental finding that MTs in-vitro undergo a three-step aging process leading tocatastrophe, resulting in catastrophe lengths ( l c ) that follow a Gamma distribution [28].From this catastrophe length distribution, the length distribution observed in an ensemble22f growing MTs without shrinkage was calculated as φ Γ ( l | n, Λ) = Γ( n, l/ Λ) n Λ Γ( n ) , (30)where Γ( n, l ) is the incomplete Gamma function, Γ( n ) is the Gamma distribution, n =3 is the number of steps to reach catastrophe and Λ is the step length (equivalent to atransition time per step for constant growth velocity, see [28]). We set the expectation value E [ φ Γ ( l | n = 3 , Λ)] = 2 Λ ≡ N − as fraction (cid:15) of the anterior-posterior axis length, henceΛ = (cid:15) (cid:0) z P − z A (cid:1) / Γ ( l | n = 3 , Λ) = 1 − + 4 l Λ + l e − l/ Λ . (31)Similar to the MT seeding densities, the model is not sensitive to the exact choice of theMT length distribution. For example, using the one-parameter exponential distribution φ e ( l | Λ) = 1Λ e − l/ Λ (32)as an alternative length distribution and again regulating the expectation value E [ φ e ( l | Λ)] =Λ = (cid:15) (cid:0) z P − z A (cid:1) of MT lengths via a parameter (cid:15) does not qualitatively change our results. M2.4 MT persistence length
The stiffness of a polymer is commonly characterized by its persistence length P , whichis defined as the decay length of tangent-tangent correlations. To calculate the persistencelength of MT random walks, we first define the end-to-end vector for a polymer with N s segments of length λ and given initial orientation ˆ µ as R ( N s | ˆ µ ) = λ N s (cid:88) i =1 ˆ µ i . (33)23rientations are drawn from a von Mises-Fisher distribution (eq. (29)) with mean (cid:104) ˆ x | ˆ µ (cid:105) = (cid:90) ˆ x f ( ˆ x | ˆ µ , κ ) d ˆ x = (cid:18) κ ) − κ (cid:19) ˆ µ ≡ σ ˆ µ . (34)In order to calculate the persistence length of these polymers, we first consider the meanorientations of the second and third MT segments (cid:104) ˆ µ | ˆ µ (cid:105) = (cid:90) ˆ µ f ( ˆ µ | ˆ µ , κ ) d ˆ µ = σ ˆ µ , (cid:104) ˆ µ | ˆ µ (cid:105) = (cid:90) ˆ µ (cid:89) i =2 f ( ˆ µ i | ˆ µ i − , κ ) d ˆ µ i = σ ˆ µ , and we find in general the relation (cid:104) ˆ µ N s | ˆ µ (cid:105) = (cid:90) ˆ µ N s N s (cid:89) i =2 f ( ˆ µ i | ˆ µ i − , κ ) d ˆ µ i = σ N s − ˆ µ . (35)Hence, using eq. (35) and shifting indices, the expectation value of the end-to-end vector (cid:104) R ( N s | ˆ µ ) (cid:105) is given by λ N s (cid:88) i =1 (cid:104) ˆ µ i | ˆ µ (cid:105) = λ N s − (cid:88) n =0 σ n ˆ µ = λ − σ N s − σ ˆ µ . (36)For stiffness parameter κ → σ → κ → (cid:104) R ( N s | ˆ µ ) (cid:105) = lim σ → (cid:104) R ( N s | ˆ µ ) (cid:105) = λ ˆ µ . (37)In the opposite limit κ → ∞ ( σ → κ →∞ (cid:104) R ( N s | ˆ µ ) (cid:105) = lim σ → (cid:104) R ( N s | ˆ µ ) (cid:105) = λ N s ˆ µ , (38)thereby showing that the polymers interpolate between an undirected random walk andcompletely straight rods.The persistence length P is defined as the spatial decay length of the correlations of24olymer segment orientations e − L/P = (cid:104) cos( θ ) (cid:105) ≡ (cid:104) ˆ µ N s · ˆ µ (cid:105) . For large L = N s λ , we have1 P = − lim L →∞ L ln( (cid:104) ˆ µ N s · ˆ µ (cid:105) )= − lim N s →∞ λ N s ln( σ N s − ) ≈ − λ ln( σ ) . (39)For κ (cid:28)
1, it holds that σ ≈ κ/ O ( κ ) and hence P ≈ λ ln(3 /κ ) , (40)whereas for κ (cid:29) P ≈ λ κ . (41)Therefore, the persistence length is approximated as segment length λ multiplied by con-centration parameter κ . M2.5 Motor velocity field
From each realization of the MT cytoskeleton in the polymer model, we derive a nondi-mensional vector field termed motor-velocity field v (cid:48) m by computing the local vectorial sumof MT segments. To this end, we define a coarse-grained cubic grid with nondimensionalside length d G = 0 .
04 ranging from − G/ − d G/ z A + d G/ z P − d G/ v (cid:48) m . v (cid:48) m is used subsequently to compute the cytoplasmic flow field,and to simulate active motor-driven cargo transport on the cytoskeleton. M2.6 Parameter values
For all computations of the MT cytoskeleton, we used N A = 25000 anterior seedingpoints, giving a total number of points between N = 55764 for wild-type cytoskeleton and25 = 84953 for par-1 null mutants.The length of an individual MT segment λ needs to be short compared to the systemlength L , yet long enough to allow simulations of even long MTs with feasible computationaleffort. Here, we chose λ = 0 . µ m. The maximum number of segments per MT is set to N max s = 200. Care was taken that few MTs reach the maximum length λ N max s .Experimental measurements of MT persistence lengths P are on the order of millimeters[54]. However, the effective persistence lengths of MTs in oocytes is likely shorter due toeffects of cytoplasmic flows as well as high density of yolk granules and other obstaclesin the cytoplasm that induce MT bending. MTs in the oocyte are neither seen to becompletely straight (Fig. 8, right), nor strongly curled up (Fig. 8, left). Therefore, we chosean intermediate value of κ = 18, corresponding to an effective persistence length (eq. (41))of P ≈ . µ m (Fig. 8, center).Due to their curved nature, the reach of a MT polymers is effectively shorter than thelength of a straight rod composed of the same number of segments. For example, for thechosen values of κ and λ , 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, tocompare the MT polymer model to a model in which MTs are represented as straight rodsnecessitates an adjustment in the mean MT length. M3 Rod model for the MT cytoskeleton
M3.1 Model setup
In addition to the polymer model for MTs, we test the net orientation of the cytoskeletonby comparing with a second model in which MTs are nucleated continuously around the cellmembrane and act as straight rods with a given length distribution. A single MT from apoint σ ( ρ, φ ) on the boundary that hits an observation point x in the volume contributesan orientation vector pointing from σ to x to the observation point. Heuristically, thiscontribution is expected to be weighted by three different factors: (i) the probability densityof MT nucleation p Σ ( σ ) at boundary point σ ( ρ, φ ), (ii) the probability that the nucleatedMT rod is oriented in a direction such that it hits the observation point x , and (iii) theprobability that the MT rod is at least long enough to reach from σ to x across a distance26 σ = (cid:107) x − σ (cid:107) . Summing up all the weighted orientational contributions from the entireboundary 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 dΣ around apoint σ reaches a volume element d V around x by p ( x , σ )dΣ d V . p ( x , σ ) is normalizedover the entire oocyte surface area Σ and volume V as (cid:90) Σ dΣ (cid:90) V d V p ( x , σ ) = 1 . (42)The marginal distribution p Σ ( σ ) given by p Σ ( σ ) = (cid:90) V d V p ( x , σ ) (43)defines the probability density of MT nucleation on the oocyte surface. Conversely, themarginal distribution p V ( x ) p V ( x ) = (cid:90) Σ dΣ p ( x , σ ) (44)is proportional to the total density of MTs that reach the observation point x from theentire boundary. Each individual MT rod contributes an orientation unit vector ˆ e xσ =( x − σ ) / (cid:107) x − σ (cid:107) . Thus, the net MT orientation at observation point x corrected for thedensity of contributing MTs can be calculated as o ( x ) = 1 p V ( x ) (cid:90) Σ dΣ ˆ e xσ p ( x , σ ) . (45)It is convenient to split the joint probability p ( x , σ ) into the conditional probability andthe probability for MT nucleation at the surface p ( x , σ ) = p ( x | σ ) p Σ ( σ ) . (46)For the nucleation of MTs on the oocyte surface p Σ ( σ ), we use the same probability densityas in the polymer model (eq. (8)). To specify the remaining conditional probability p ( x | σ ),first note that inserting eq. (46) into eq. (43) imposes the normalization condition1 = (cid:90) V d V p ( x | σ ) . (47)27onsider a local spherical polar coordinate system ( r σ , θ σ , φ σ ) centered at a given point σ on the oocyte surface with the z σ -axis identical to the locally inwards pointing normal ˆ n σ .MT rods are assumed to be oriented in any direction into the positive half space z σ > p ( x | σ ), we first assume MT rods of cross sectional area A and fixed length l corresponding to a length distribution φ δ ( r σ ) = δ ( l − r σ ) , (48)with cumulative distribution functionΦ δ ( r σ ) = (cid:90) r σ d r δ ( l − r ) = Θ( r σ − l )= 1 − Θ( l − r σ ) . (49)The volume V c = A l of such a cylindrical MT rod can be approximated as integral over thepositive half space by V c = A (cid:90) l d r σ = A (cid:90) ∞ d r σ Θ( l − r σ )= A (cid:90) π d φ σ (cid:90) π/ d θ σ sin( θ σ ) (cid:90) ∞ d r σ r σ Θ( l − r σ )2 πr σ ⇔ (cid:90) V d V Θ( l − r σ )2 πr σ l , (50)where the last step resulted from division by A l . Comparing this result with eq. (47) andusing eq. (49), we identify the conditional probability density function for randomly orientedMTs rods of fixed length as p ( x | σ ) = N δ πr σ [1 − Φ δ ( r σ )] . (51)Herein, the factor N δ = 1 /l corresponds to the mean MT length, the term (2 πr σ ) − rep-resents the uniform angular distribution on a half sphere, and the expression [1 − Φ( r σ )] isthe probability that MT rods are at least r σ long.To account for the experimentally measured MT length distribution, eq. (51) is finallygeneralized by replacing Φ δ ( r σ ) by the cumulative distribution used in the polymer model28 Γ ( r σ | n = 3 , Λ) (eq. (31)) with N Γ = 1 / (2Λ). Thus, the complete joint probability dis-tribution p ( x , σ ) (eq. (46)) for a MT emanating from σ ( ρ ) and contributing to the netorientation at observation point x in the oocyte volume is defined as p ( x , σ ) = p Σ ( s | h , k ) N Γ πr σ [1 − Φ Γ ( r σ | , Λ)] . (52)Note that MT straight rods always cover the distance between nucleation and observationpoint in a straight line. If the anterior surface is curved inwards, these straight lines cantemporarily pass outside of the oocyte geometry before reaching the observation point insidethe volume again. The rod model does not prevent such unphysical contributions (thoughthe polymer model does prevent this). However, this error does not occur in a geometry witha 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 inpractice this error is negligible. M3.2 Parameter space and comparison to polymer model
The continuum description of the rod model allows one to follow the saddle-node bifur-cation in the MT cytoskeleton precisely throughout parameter space. The creation of stableand unstable fixed points always occurs along the anterior-posterior (AP) axis because thesystem is rotationally symmetric. We therefore compute the uncorrected net orientation o ( x ) p V ( x ) (eq. (45)) for observation points x only along the symmetry axis as a functionof cortical nucleation parameter k P and mean MT length (cid:15) .For a cytoskeletal topology without any fixed point, the x -component o x ( x ) p V ( x ) willbe pointing towards the posterior pole everywhere along the AP axis. A positive value o x ( x ) p V ( x ) > o x ( x ) p V ( x )will be negative at at least one point along the AP axis, while at the same time it will bepositive at the grid point closest to the posterior pole x P where the unstable node createsa limited domain of attraction. Combination of both criteria therefore defines the region ofparameter space in which the oocyte is split into two different regions of attraction. Finally,a negative value at the posterior pole o x ( x P ) p V ( x P ) < k P if (cid:15) is large, but shifts to larger values of k P if (cid:15) becomes smaller.Furthermore, the bifurcation point can be crossed solely by varying (cid:15) while keeping k P fixed(Fig. 4 - figure supplement 1). In summary, this confirms that the polymer model exhibitsthe same parameter space structure as the rod model (Fig. 5), thus further supporting thesimilarity of the two models. Moreover, as in the rod model, this parameter space structureis again insensitive to the oocyte shape. M3.3 Bifurcation normal form
The saddle-node bifurcation normal form in two dimensions is given by v x v y = x + λ − y , (53)wherein λ is the bifurcation parameter. For negative λ , stable and unstable fixed pointsexist and are located at x ± = ±√− λ . The critical bifurcation point occurs at λ = 0 (Fig. 4- figure supplement 1P-T). M4 Additional Methods
M4.1 Nondimensionalization and mathematical methods
In dimensional units, the system of transport equations for bound and unbound cargofractions c b and c u is defined as ∂ t c b + ∇ · ( v m c b ) = k b c u − k u c b (54) ∂ t c u + ∇ · ( u c u ) = − k b c u + k u c b + D ∇ c u , while the Navier-Stokes-Equations and volume forces are defined by ρ ( ∂ t u + ( u · ∇ ) u ) = − ∇ p + µ ∇ u + f (55) f = a v m , (56)30ith no-slip boundary conditions on the oocyte surface. Note that cytoplasmic streaming ispresent in oocytes even when oskar mRNA is not expressed [32], showing that some cargoother than oskar mRNA is transported by kinesin and responsible for driving flow. Thisjustifies that the parameter a in eq. (56) is independent of the concentration of the boundcargo c b .Computations of the fluid flow field and simulations of cargo transport are performedusing a nondimensionalized version of eqs. (54-56). For nondimensionalization, we select alength scale L = 50 µ m for scaling of space x = L x (cid:48) , and an advection scale derived fromthe active motor-driven transport V = 0 . µ m/s [14] for scaling of velocities v m = V v (cid:48) m ,implying an advection time scale τ = L/V = 100 s for scaling of time t = τ t (cid:48) . v (cid:48) m denotes thecoarse-grained motor-velocity field computed from each realization of the MT polymer model(Sec. S2.4). Defining the mean raction rate constant K = ( k b + k u ) / β = k b / (2 K ) we find the nondimensional version of eqs. (54) ∂ t (cid:48) c (cid:48) b + ∇ (cid:48) · ( v (cid:48) m c (cid:48) b ) = 2Da ( βc (cid:48) u − (1 − β ) c (cid:48) b ) (57) ∂ t (cid:48) c (cid:48) u + ∇ (cid:48) · ( u (cid:48) c (cid:48) u ) = 2Da ( − βc (cid:48) u + (1 − β ) c (cid:48) b )+Pe − ∇ (cid:48) c (cid:48) u , where Da = L K/V is the Damk¨ohler number and Pe =
L V /D is the Peclet number.For nondimensionalization of eqs. (55-56), we define scales for the pressure P = µ V /L ,force density F = µ V /L and force-velocity scaling A = F /V to obtainRe ( ∂ (cid:48) t u (cid:48) + ( u (cid:48) · ∇ (cid:48) ) u (cid:48) ) = − ∇ (cid:48) p (cid:48) + ∇ (cid:48) u (cid:48) + f (cid:48) (58) f (cid:48) = a (cid:48) v (cid:48) m (59)wherein p (cid:48) = p/ P , f (cid:48) = f / F , a (cid:48) = a/ A , and Re = ρ V L/µ is the Reynolds number. Notethat the Reynolds number is defined using the scale V of the active transport field becausethe flow velocity field u is derived from it and hence varies. To estimate an upper boundon the Reynolds number, we use the viscosity of water µ = 10 − Pa s and ten times thedensity of water ρ = 10 kg / m as lower and upper bounds for the viscosity and density ofthe cytoplasm in the oocyte. This results in an upper bound Re = 2 . × − , thus justifyingto neglect the inertia terms in eq. (58). 31 The unbinding constant k u can be estimated as ratio of the mean active transport velocityand the mean track length of oskar mRNA in stage 9 oocytes as k u = 0 . s − [14]. Thenondimensional parameter β 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 β = 0 .
13 [14].The unbinding constant k u and β can be used to calculate k b = 0 . s − . Therefore, themean reaction rate constant and the Damkoehler number are K = 0 . . oskar mRNA (2.9 kB) in mammalian cells are in the range of D = 0 . µ m/s or lower [55].With this value, the Peclet number evaluates to Pe = 1250.The nondimensional scaling parameter a (cid:48) (eq. (56)) which converts the active transportvelocities into forces acting on the cytoplasmic fluid absorbs several unknown quantitiesincluding the size of the cargo that drives fluid flow, drag forces on the fluid with localviscosity, the density of such cargo-motor complexes on MTs and any collective effects. Thevalue of a (cid:48) determines the mean nondimensional and dimensional fluid flow speeds (cid:104)| u (cid:48) |(cid:105) and V (cid:104)| u (cid:48) |(cid:105) . We regard a (cid:48) as a macroscopic, phenomenological parameter. The value a (cid:48) = 45( a (cid:48) = 55 for alternative geometry-2) is calibrated such that the resulting mean dimensionalfluid flow speeds V (cid:104)| u (cid:48) |(cid:105) match the average fluid flow speeds measured experimentally forstage 9 Drosophila oocytes (see Fig. 2) (compare to similar approach in [56]).To test the anchoring mechanism, the bound state is replaced by an anchored state towhich cargo can only bind at the extreme posterior pole. For the anchoring state we set k anch u = 0 s − and k anch b = 10 k b , resulting in β anch = 1 (no unbinding), K anch = 0 . anch = 12 . M4.3 Computational methods
For computation of the cytoplasmic flow field and subsequent transport simulations, themotor-velocity field v (cid:48) m and forces f (cid:48) from the MT polymer model are shifted and rotatedsuch that the AP axis aligns with the x -axis and the origin of the coordinate system islocated in the oocyte center. To compute the cytoplasmic flow field we use the open source32reeFem++ finite element solver [57]. Forces f defined on a 3D cubic grid are interpolatedto the unstructured 3D grid used by FreeFem++, and resulting flow velocity vectors areinterpolated back to the cubic grid and derived staggered grids.Cargo simulations of eqs. (57) are performed using custom written Matlab code in finitevolume formulation on staggered grids as described in [58]. As nondimensional time step weused ∆ t = 0 . v (cid:48) m and their corresponding cytoplasmic flow fields u (cid:48) to account for a dynamically remodellingcytoskeleton and temporally changing cytoplasmic flow fields. Each pair of v (cid:48) m - u (cid:48) -fields isactive for 432 time steps, corresponding to 3.6 minutes of simulated real time. M4.4 Fly stocks and experimental methods
Oocyte microtubules were stained as described by Theurkauf et al. [15] using a FITC-coupled anti-alpha-tubulin antibody at 1:200 (Sigma). The general shape and size of theoocyte was imaged by recording the fluorescence from a par-1 protein trap (PT) GFP line[59]. Homozygous or heterozygous par-1
PT flies were also used for measurements of cyto-plasmic flows by imaging movements of autofluorescent yolk granules. The allelic combina-tion par-1 W / par-1 was used for imaging cytoplasmic flows in strong par-1 hypomorphs[41]. In all cases of flow measurements, flies were yeast fed overnight, dissected under Voltalef10S oil and imaged at 40x magnification in a confocal microscope. Cytoplasmic flows wererecorded with a 405 nm laser, 4 µ s pixel dwell time, collecting 13 frames in 25 s intervals fora 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 [60]. For cold experiments,flies were kept at room temperature (22C), yeast fed overnight at 25C, and subsequently keptat 25C, 7C and 4C degrees for 6 hours before dissection and fixation. In situ hybridizationsand imaging of living tissue were performed according to standard methods [32]. The bicoid mRNA probe was labeled with Digoxigenin-UTP (Boehringer Mannheim).33 To compare topologies of in-vivo cytoplasmic flows with topologies of numerically com-puted cytoplasmic flow fields, we used particle image velocimetry (PIV) to generate vectorfields capturing the instantaneous direction and magnitude of particle movements. PIV wasperformed using the open source code PIVlab [61], and all PIV vector fields obtained frompairs of consecutive movie frames were averaged over the movie duration. For each movieof cytoplasmic streaming, the mean flow speeds were calculated as the average over all PIVvector magnitudes. Flow speeds obtained from PIV were confirmed by automatic particletracking using open source code by Blair and Dufresne [62]. Only particles that could betracked in at least three consecutive frames were accepted. Note that instantaneous speedswere 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 [26],thereby ensuring that our conclusions about the neglibible contribution of flows to transportremain sound.
M5 Flow fields and oocyte nucleus
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 ofindividual flow fields we calculate the unbiased, discrete vector autocorrelation over time forthe sequence of PIV flow fields in each movie according to the expression C ( k ) = (cid:42) N − k (cid:88) i = x,y N − k (cid:88) n =0 ( u i ( n + k ) u i ( n )) (cid:43) (60)where angular brackets denote the average across all points of the PIV grid, and the peakvalue of C ( k ) is normalized to one. Autocorrelation functions from N = 48 movies exhibita large spread across generally low correlation values, and an exponential fit gives a decaytime constant of 4.4 minutes. 34 In 2D confocal images, the oocyte nucleus occasionally appears to cover a significantfraction of oocytes, particularly in young oocytes up to stage 8. To quantify the size ofthe nucleus compared to the size of the oocyte, we therefore estimate the fraction of theoocyte volume that is covered by the nucleus. We first extract the entire boundary of theoocyte using a custom written macro in Fiji [63]. Averaging the boundary shape from belowand 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 ante-rior ( x A ( t ) , y A ( t )) and posterior ( x P ( t ) , y P ( t )) curve, the volume of the respective solids ofrevolutions can be computed as V i = π (cid:90) d t ( y i ( t )) d x i d t . (61)While the anterior surface is typically curved inwards for stage 9 oocytes, the anterior surfacein young oocytes up to stage 8 can be outwards or inwards curved. Hence, depending onthe shape of the anterior the volumes of the anterior V A and V P have to be suitably addedor 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 V σi = π (cid:90) L z i d z (cid:18) L (cid:113) − z/ ( L z i ) (cid:19) = πL z i / . (62)Given that the anterior surface in the standard geometry-1 is curved inwards, the totaloocyte volume is given by the difference between between the posterior and the anterior capvolumes V σP − V σA = πL (cid:0) z P − z A (cid:1) / / π r .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 youngstage 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,35he nucleus is unlikely to influence oskar mRNP transport towards the posterior, and wetherefore neglect it in simulations of cargo transport.
M5.3 Impact on flow field
The nucleus may be expected to disturb the cytoplasmic flow field by means of adding anadditional no-slip boundary condition inside the volume. We tested this effect by includinga spherical excluded volume at the oocyte anterior surface in the solution of the 3D Stokesequations (Fig. 2 - figure supplement 1A,F,H). The nucleus leads to a clear local disturbanceof 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 theflow fields in which the nucleus is not visible shows that the nucleus rarely changes the flowfield (Fig. 2 - figure supplement 1G,H). This low sensitivity with respect to excluded internalvolumes 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 impactof the nucleus because it generally is not spherical and often seen to squeeze deeply into theanterior corners, therefore reaching far less into the oocyte volume than the idealized sphereused here.
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 ResearchFellowship [080007] (D. StJ.), the Leverhulme Trust, and the European Research CouncilAdvanced Investigator Grant [247333] (R. E. G.).36
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Figure. 1. Models for the MT meshwork show local order in the cytoskeleton. A) 3D geometryof a stage 9
Drosophila oocyte (grey, anterior to the left, posterior to the right) containing morethan 55000 MT seeding points. From the corners to the centre, MT seeding density decreasesweakly along the anterior ( k A = 1000 µ m, h A = 0 .
8) and strongly along the posterior-lateral cortex( k P = 150 µ m, h P = 0, see Materials and methods M2.1). Nucleated MT polymers are stiff randomwalks, 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 distributionthat accounts for the MT aging process. The mean target length is set to a fraction (cid:15) of the AP-axislength, here (cid:15) =0.5 (Material and methods M2.3). The inset shows the 3D angular distribution of0.5% of all MT segments with 3D statistical bias. B) Cross section through the MT cytoskeletonshown 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 sectionin panel B on a coarse-grained grid shown as streamlines that visualize local directionality. D)Staining of α -tubulin (green) shows MT density distribution in a fixed stage 9 oocyte. Nuclei inblue (DAPI), scale bar is 30 µ m. E) Schematic detailing the work flow in the model and comparisonsto experiments. F) Local directionality of MT cross section as in panel C for an average over 50independent realizations of the cytoskeleton. G) Local directionality computed from the rod modelwith the same parameters as in panels A-B but shortened MT lengths (cid:15) = 0 .
25 (Material andmethods M2.5). Orange arrows show the separatrices between subcompartments. H) MT densitydistribution computed from 50 realizations of the polymer model. igure. 1 - figure supplement 1. Compartmentalization of the MT cytoskeleton is robust to changesin oocyte geometry. A) Alternative geometry for a stage 9 oocyte comprised of a posterior paraboliccap and an anterior disc. MT segments intersecting a cross section in one realization of the polymermodel with nondimensional seeding density parameters k P = 150 µ m, h P = 0 , k A = 1000 µ m, h A = 0 . (cid:15) = 0 . α -tubulin staining of an early stage 9 oocyte. Scale bar is 25 µ m. E) Schematicof the work flow in our model. F) Streamplot of the local vectorial sum of 50 realizations of thepolymer MT cytoskeleton with parameters as in A,B. G) Local net orientation of straight rod MTsin the rod model with parameters as in A,B but reduced mean MT target length (cid:15) = 0 .
28, showingthe same compartmentalization of the oocyte as the average in the polymer model. H) MT densitydistribution in the ensemble of 50 realization of the polymer MT cytoskeleton. igure. 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 thecytoskeleton shown in Fig. 1A. The horizontal plane shows a 2D cross-section through the 3Dfield. Anterior to the left, posterior to the right. B) Cross-section through the 3D field shown inpanel A with arrows indicating flow directions and colouring indicating flow speeds. C) Confocalimage of a live stage 9 oocyte. Arrows show the flow field computed from PIV of streaming yolkgranules and averaged over ≈ µ m. D) Same as A, but showing themean flow organization for an average of 100 individual 3D flow fields, analogous to the meanorganization of the MT cytoskeleton in Fig. 1F. E) Same as B for the average in panel D. F) Meanfluid flow speeds were obtained by PIV (red, 13 . ± . ± sem) and automatic particletracking (orange, 15 . ± . ± sem) from 48 oocytes. Experimentally measured flowspeeds were used to calibrate the forces f in the Stokes equations Eq. (2) such that the computedmean speeds in 3D (blue, mean: 14 . . igure. 2 - figure supplement 1. The oocyte nucleus covers a negligible fraction of the oocytevolume and disturbs the flow field only locally. A) Streamlines visualizing the flow field computedin the 3D oocyte geometry with additional no-slip boundary condition on a sphere representingthe nucleus. Same forces as in Fig. 2A-B, main text. The nucleus occupies 2.2% of the oocytevolume. Horizontal plane shows a cut through the flow field. B) Young stage 9 oocyte stainedwith DAPI (blue), phalloidin (red) and showing oskar MS2 GFP (green) in the process of reachingthe posterior pole. Scale bar is 25 µ 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 computedfor N e = 9 early (red) and N m = 7 mid stage 9 oocytes (blue). Arrowhead marks the oocyteshown in panel B. E-F) Cross sections through the 3D flow fields for the same forces as in panelA with (F) and without (E) nucleus. G-H) Same as E and F, but vertical cross section that doesnot contain the nucleus. igure. 3. The model recapitulates oskar and bicoid mRNA transport, implying dominance ofcytoskeletal transport. A-C) Shown are fixed oocytes with oskar MS2 GFP (green) and stainedwith DAPI (blue) and Phalloidin (red). oskar mRNA forms a central cloud at late stage 8 (A), acollimated channel while moving to the posterior (B), and a posterior crescent at stage 9 (C). Scalebars are 25 µ m. D) 3D oocyte showing the initial oskar mRNA cargo distribution. E-F) Crosssections through a simulation of oskar mRNA transport with diffusion, motor-transport and flowsshowing the distribution of total cargo c b + c u at the indicated time points. No posterior anchor ispresent. Simulations of 3 (6) hours cycle through 50 v m - u -pairs in random order once (twice, seeFig. 4). For simulations of 1.5 hours, 25 v m - u -pairs were chosen at random. Compare to exper-imental observations in panels B and C. G) Same simulation as in D-F, but without cytoplasmicflows, showing largely identical localization as in F. H) Simulation of bicoid mRNA transport showsthe 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 injectionat 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 ante-rior 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 withoutactive motor-driven transport. Compare to panels F and G. igure. 4. The cortical MT seeding density determines sites of mRNA localisations and givesrise to a bifurcation in the cytoskeleton. A-E) Simulations of oskar (A-C) and bicoid (D) mRNAtransport with diffusion, cytoskeletal transport and flow for the cytoskeletal architectures shown inpanels E-H, and their corresponding flow fields. For oskar mRNA, simulations reproduce wild-typelocalization (A, same as Fig. 3F), and partial (B) or complete mislocalisation (C). Initial conditionas in Fig. 3D. Simulation times were occasionally increased to 6h (B) to rule out transient con-centration patterns. For bicoid mRNA, simulations capture mislocalisation to both anterior andposterior as in gurken/torpedo/cornichon mutants and in strong par-1 hypomorphs (D, arrow-heads). Time points as indicated. The inset in D shows initial condition. E-H) Average local MTorientations in ensembles of 50 realizations of the polymer model for varying MT seeding densitiesalong arclength s (see panel E) of the posterior-lateral cortex (insets). A MT seeding density thatincreases from a wild-type gradient (E, h P = 0 , k P = 150 µ m) laterally towards the posterior (F, h P = 0 , k P = 17 . µ m) to a near-uniform distribution (G, h P = 0 , k P = 1 µ m) shows a saddle-nodebifurcation by creating a pair of stable (green) and unstable (red) fixed points. H) A MT seedingdensity that is slightly lower at the posterior pole ( h P = 0 . , k P = 40 µ m) produces mean MTorientations virtually indistinguishable from uniform seeding density (compare to G). I-K) Vectorfield of the mathematical normal form of the 2D saddle-node bifurcation (Materials and methodsM3.3). Values of the bifurcation parameter λ as indicated. Fixed points are located at positions x = ±√ λ . 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 andat the posterior pole (compare to panel D, arrowheads). Scale bar is 25 µ m. igure. 4 - figure supplement 1. Either the MT seeding density or the MT length can act asbifurcation parameter. A-J) Each panel shows the mean local MT orientations for an ensemble of50 realizations of the polymer MT cytoskeleton under variations of the seeding density parameter k P and MT length parameter (cid:15) as indicated. Approximate location of the stable fixed point isshown in green, with region of attraction in blue, and domain of attraction of the posterior polein red. The point on the AP-axis between red and blue arrow region marks the unstable fixedpoint. Percentages indicate the the ensemble-averaged 3D directional bias. Experimentally, thedirectional bias was measured as 57 .
97% : 42 .
03% [16]. Cytoskeleton in panel I was used aswild-type cytoskeleton (Fig. 1F, main text). Anterior seeding density is unchanged in all panels k A = 1000 µ m, h A = 0 .
8. K-O) Same as panels A-J but for ensembles with 100 realizations ofthe cytoskeleton to ensure reliable visualization of vector field even when fewer MTs reaching theoocyte center. P-T) Vector fields computed from a saddle-node bifurcation normal form. Thecritical bifurcation point is λ = 0, and for λ <
0, fixed points are located at x ± = ±√− λ . igure. 4 - figure supplement 2. Lateral MT growth produces the clone-adjacent-mislocalizationphenotype (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 cy-toskeleton in the polymer model ( h A = 0 . , k A = 1000 µ m, h P = 0 , k P = 150 µ m). Additional MTshave been added dorsal to the posterior pole (black arrow) to mimick posterior follicle cell clones( RAS ∆ C b MARCM) that overexpress dystroglycan [40]. B) Density of MTs for 50 realizations ofthe cytoskeleton shows enrichment on one side of the posterior pole (white arrow). C) Streamplotshowing the local vectorial orientation averaged over 50 realizations of the cytoskeleton. Note theupwards tilt away from the central posterior pole. D) Simulations of oskar mRNA transport bydiffusion, cytoskeletal transport and corresponding cytoplasmic flows show that cargo is repelledfrom the site of additional MT nucleation on one side of the posterior pole, thereby capturing theCAM phenotype [40]. igure. 5. The parameter space of the rod model shows the relation and interconversion betweenall 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 (cid:15) and extent of the posterior seeding density k P . The bifurcation line between wild-type and weak par-1 hypomorph topologies can be crossedby either changing the seeding density laterally (horizontal dashed arrow), or by shortening theMTs (vertical dashed arrow). B-C) Parameter spaces as in A for increasing MT nucleation at theposterior pole (B: h P = 0 . , C : h P = 0 . h A = 0 . , k A = 1000 µ m). igure. 6. Seeding point density in the polymer model. Shown are N randomly drawn seedingpoints on the posterior cap of the standard oocyte geometry, according to eq. (15) distributedeither (near-)uniformly on the cap (A, G) or in a parabolic gradient from the posterior pole to theanterior corners (F, L). Between uniform and parabolic distribution, the seeding density can varyeither by laterally reducing the density (B, C-E, parameter k P ), or by reducing the density at theposterior pole (H, I-K, parameter h P ). Total number of points is calculated with a fixed numberof anterior points N A = 4000 (eq. (20)). For actual computations of wild-type cytoskeletons( k P = 3 , h P = 0 , k A = 20 , h A = 0 .
8, see main text for dimensional parameters), more than 55000seeding points are used. igure. 7. The cortical MT density increases approximately parabolically from the posterior poletowards the anterior corners. A) α -tubulin staining of a stage 9 oocyte (green) with DAPI stainingof DNA (blue). Arrows indicate arclengths s P from the posterior pole to the anterior corners andanalogously 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. Theminimum intensity value across all profiles was subtracted from each. Blue line shows a parabolicfit. 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). Blackdashed line in bottom panel shows exact parabola. Arclengths are nondimensional lengths withscale L = 50 µ m. igure. 8. von Mises-Fisher parameter κ determines MT stiffness. Shown are three cross sectionsthrough MT cytoskeletons for indicated values of κ . MTs become stiffer with increasing values of κ . Insets show angular distribution for 3 × points drawn from von Mises-Fisher distributionaround mean direction ˆ µ = ˆ e z ..