title: Swirling Instability of the Microtubule Cytoskeleton
authors: David B. Stein Gabriele De Canio Eric Lauga Michael J. Shelley Raymond E. Goldstein
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  These authors contributed equally
  Center for Computational Biology, Flatiron Institute, 162 5th Ave., New York, NY 10010
  
  These authors contributed equally
  Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical 
  Sciences, University of Cambridge, Cambridge CB3 0WA, United Kingdom
  
  e.lauga@damtp.cam.ac.uk
  Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical 
  Sciences, University of Cambridge, Cambridge CB3 0WA, United Kingdom
  
  mshelley@flatironinstitute.org
  Center for Computational Biology, Flatiron Institute, 162 5th Ave., New York, NY 10010
  Courant Institute, New York University, 251 Mercer St., New York, NY 10012
  
  R.E.Goldstein@damtp.cam.ac.uk
  Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical 
  Sciences, University of Cambridge, Cambridge CB3 0WA, United Kingdom
  
  
  
  
  
  In the cellular phenomena of cytoplasmic streaming, molecular motors carrying cargo 
  along a network of microtubules entrain the surrounding fluid.
  The piconewton forces produced by individual motors are sufficient to deform 
  long microtubules, as are the collective fluid flows generated by many moving motors.  
  Studies of streaming during oocyte development in the fruit
  fly D. melanogaster have shown a transition from a spatially-disordered 
  cytoskeleton, supporting flows with only 
  short-ranged correlations, to an ordered state with a cell-spanning vortical flow.
  To test the hypothesis that this transition is driven by fluid-structure interactions 
  we study a discrete-filament model and a coarse-grained continuum theory 
  for motors moving on a deformable cytoskeleton, both of which 
  are shown to exhibit a swirling instability to spontaneous large-scale rotational 
  motion, as observed.
  
  
  Swirling Instability of the Microtubule Cytoskeleton
      Raymond E. Goldstein
      December 30, 2023
  ====================================================
  
  
  
  
  A striking example of fluid-structure interactions within cells 
  <cit.> occurs in oocytes of the fruit fly 
  Drosophila melanogaster <cit.>.  These develop over 
  a week from a single cell through repeated rounds of cell division, 
  differentiation and growth, ultimately reaching hundreds of microns across.  
  This pathway has historically
  been divided into 14 stages, and it is in stages 9-11, at days 6.5-7 <cit.>, 
  that fluid motion is most noticeable.  In stage 9 (Fig. <ref>), microtubules (MTs) reach
  inward from the oocyte periphery, forming a dense assembly along which 
  molecular motors (kinesins) move at tens of nm/sec, carrying messenger 
  RNAs and other nanometric particles.  This motion 
  entrains the surrounding fluid, producing cytoplasmic streaming <cit.>
  that can be visualized several ways: in brightfield by the motion of endogenous 
  particles <cit.>, via their
  autofluorescence <cit.>, and through a combination of particle image velocimetry
  and fluorescently labelled microtubules 
  <cit.>.
  Previous work <cit.>
  revealed that these flows initially take the form of transient, recurring
  vortices and jets whose correlation length is a fraction of the cell scale, with 
  no long-range order.  But by stage 11, a dramatic reconfiguration of the cytoskeleton occurs, coincident with the appearance of a vortex 
  spanning the entire cell <cit.>. 
  
  Kinesin motors move from the minus ends of microtubules (attached to the 
  oocyte periphery) to the plus ends (free in the interior).
  Transport of cargo through the network 
  depends on motor-microtubule binding details <cit.> and 
  on the mesh architecture <cit.>.  As a motor pulls cargo toward the plus 
  end the filament experiences a localized minus-end-directed compressive force,
  as in Euler buckling.  For a filament of length L and 
  bending modulus A <cit.>, 
  the buckling force is ∼π A/L^2∼ 60  pN/L^2, 
  where L is measured in microns.  Thus, a 
  kinesin's force of several pN <cit.> can buckle 
  MTs 10-40 μm long.  
  
  
  
  The coupled filament-motor problem is richer than
  Euler buckling because a motor exerts a “follower force" <cit.>  
  that is aligned with the filament.  This feature breaks the 
  variational structure of the problem and allows a filament pinned at its minus end to 
  oscillate even at zero Reynolds number <cit.>.  
  By exerting a force on the fluid a motor induces long-range flows 
  which, if compressive, can further deform 
  filaments <cit.>.
  
  
  
  It has been hypothesized <cit.> that the transition from disordered 
  streaming flows to a single vortex in stage 11 is a consequence of the kinds of fluid-structure 
  interactions described above, facilitated by a decrease in cytoplasmic viscosity that accompanies
  the disappearance of a coexisting network of the biopolymer f-actin.
  Here, through a combination of direct computations on the coupled filament-flow problem 
  <cit.>
  and studies of a recent continuum theory for dense filament suspensions 
  <cit.>, we confirm this hypothesis 
  by showing the existence of a novel swirling instability of the cytoskeleton.
  
  The swirling instability can be understood in a 
  simplified model of the oocyte: a rigid sphere of radius R containing a fluid of viscosity
  μ, with N elastic filaments reaching inwards from clamped attachment points
  equally spaced around the equator.  A slice in the filament plane 
  (Fig. <ref>(a)) appears like a confocal slice of the oocyte (Fig. <ref>).  
  The filaments have a radius r, a constant length L, bending modulus A and a uniform line density f of 
  follower forces (Fig. <ref>(b)).  Although free microtubules have a complex 
  dynamics of growth and decay, recent evidence <cit.> for 
  `superstable' cortically-bound microtubules in stages displaying unidirectional 
  streaming justifies the constant-length approximation.
  
  Microtubules are the quintessential slender bodies <cit.> of biophysics, with aspect 
  ratios ε=r/L of O(10^-3).  As their self-interactions are weak, we use local slender-body theory <cit.> to obtain the dynamics.
  In an arclength parameterization s, 
  the j^ th filament r^j(s,t) evolves as
  
      η(_̊t^j - ^j) = (𝕀 + _̊s^j_̊s^j)(-A_̊4s^j + (Λ^j_̊s^j)_s + f^̊j_s),
  
  where ^̊j_s is the unit tangent, η=8πμ/c, with c=|ln (eε^2)|, and the
  Lagrange multiplier 
  Λ^j enforcing inextensibility obeys a second-order PDE <cit.>. 
  In the background flow U^j= u^j+ u^i→ j + 
   v^i→ j, u^j is that produced by the motors on
  j, u^i→ j is due to the motors on
  i j, and v^i→ j is due to motion of filaments i j.
  For example, the induced flow due to the jth fiber is u()=∫_0^L ds 
  f _̊s^j(s)· G(-^̊j(s)) (see Supplemental Material <cit.>), with G the Greens 
  function appropriate to the interior of a no-slip sphere <cit.>.
  Filament clamping at the sphere implies that ^̊j(0,t) remains fixed and that
  _̊s^j(0,t) is the inward sphere normal at the attachment point. The free end is torque- and force-free: ^̊j_ss(L,t)=^̊j_sss(L,t)=Λ(L,t)=0.
  
  A single fiber clamped at a flat wall displays
  a supercritical 
  Hopf bifurcation which, expressed in terms of the dimensionless motor force σ≡ fL^3/A, occurs 
  at σ^*≃ 124.2, beyond which the
  filament exhibits steady oscillations 
  whose amplitude grows as √(σ-σ^*) <cit.>.  
  When several filaments interact within the sphere (<ref>c) they 
  also oscillate, but with their motions synchronized
  in phase, very much like eukaryotic 
  flagella <cit.>.  The dynamical model (<ref>) contains
  two ingredients often found necessary for such synchronization <cit.>: hydrodynamic interactions and
  the ability of a filament to change shape and thereby 
  adjust its phase in response to those flows.
  
  As the filament density and motor strength are increased we find the swirling instability: a
  transition to a steady configuration of bent filaments whose distal parts are oriented almost parallel
  to the wall (Fig. <ref>(d)).  The bent configuration is maintained 
  by azimuthal flows,
  induced by the motors, that generate drag along the 
  distal part of the filaments, and thus a torque opposing the bending torques near the filament base.
  As with any such spontaneous breaking of symmetry,
  both left- and right-hand configurations are possible; the choice between the two is dictated by
  initial conditions.  This transition is reminiscent of the self-organized
  rotation of cytoplasmic droplets extracted from plants <cit.> and 
  the spiral vortex state of confined bacterial suspensions <cit.>, both
  modeled as suspensions of stresslets <cit.>.
  
  While direct computations on denser arrays of discrete filaments are possible <cit.>, cortically 
  bound oocyte microtubules are so tightly packed, with an inter-fiber spacing 
  δ≪ L <cit.>, that a continuum approach is
  justified.  The description we use <cit.>, in which microtubules form an 
  anisotropic porous medium, is based on the map =(̊),
  where the Lagrangian coordinate =(α,s) encodes the location α 
  of the minus ends of the microtubules and arclength s.  
  In a system of units made
  dimensionless by the length L and elastic relaxation time η L^4/A,
  we obtain a continuum version of (<ref>),
  
      _̊t - |̆_(̊)  = (𝕀+)·(-_̊ssss + (Λ)_s 
           - σ) .
  
  The fluid velocity 
  $̆ arises from the force distribution along the filaments and is evaluated at the
  Eulerian positionaccording to an inhomogeneous Stokes equation,
  
      -∇^2+̆ p = χ_mtρ[𝒥^-1(-_̊ssss + (Λ_̊s)_s)]|_^̊-1() ,
  
  subject to the incompressibility constraint·=̆ 0. The indicator functionχ_mtis 
  supported where the MT array is present (Fig. <ref>a).  Here,ρ=8πρ_0 L^2/cis the
  rescaled areal number density of microtubules, expressible asρ=ϕ(L/δ)^2, where the constantϕdepends only on the MT slenderness and packing geometry at the wall;ϕ≈4whenc≈10and the MTs are hexagonally packed.
  The quantity𝒥=[∂/̊∂]measures the change in 
  microtubule density due to deformations of the array;𝒥^-1increases as fibers move closer together.
  
  
  
  The simplest geometry is an infinite planar array of MTs, as shown in  
   Fig. <ref>(a).  As in the discrete model, the MTs are normally 
  clamped to a no-slip wall and are 
  force- and torque-free at their plus ends. At a distanceHabove the wall, 
  no-penetration and zero-tangential stress conditions are imposed on the fluid. For dynamics homogeneous 
  alongx, the fluid flow is unidirectional and constant above the MTs, soHplays no role.
  Nonlinear computations <cit.> reveal both oscillatory dynamics and the emergence of steady streaming. 
  Fig. <ref>(b) shows the dynamics whenρ=4.65andσ=70: self-sustaining oscillations of the 
  MT array are observed, similar to those in Fig. <ref>(c). Note that while Fig. <ref>(b) shows only a single filament, it represents the common dynamics of 
   all of the collectively beating filaments in the array. Whenσis decreased to≈39, the 
  MT array deforms and stabilizes into a steady 
  bent state (Fig. <ref>(c)). This 
  represents the continuum description of the swirling 
  transition, with similar dynamics to those observed in the discrete results.  
  
  An equilibrium of the system occurs when filaments are aligned straight alongz, with=̆0andΛ=-σ(1-z). Forσ>0, the motor-force is 
  compressive and buckling may occur.  
  A small transverse perturbation in fiber shape of the form=ẑ+ϵg(z)x̂(ϵ≪1) evolves as 
  
      g_t = -g_zzzz -σ[(1-z)g_z]_z + ρ[σ(1-z)g+g_zz].
  
  The first two terms are like those of an elastic filament under an aligned gravitational load, with an internal tension varying linearly
  from one end to the other <cit.>. The third is the fiber forces filtered through the 
  non-local Stokes operator, capturing hydrodynamic interactions within the fiber array (and hence theρprefactor). That this term is local is both fortunate and surprising, and follows from the simplicity of the 
  Stokes flow in this case. The termρg_zzcaptures the additional resistance to bending from flow: if a MT is to bend, it must move the 
  fluid around it, bending other MTs; the termρσ(1-z)gis destabilizing: if a 
  MT is to remain straight, it must resist the fluid motions generated by MTs around it.
  
  While the planar geometry reproduces all qualitative features of the streaming transition <cit.>, to capture
  the key feature of confined hydrodynamic interactions in the oocyte we extend the analysis to a cylindrical domain, 
  where the no-flow steady state is an array of straight MTs 
  pointing inwards. Fig. <ref> shows the results of a linear stability analysis for an experimentally 
  relevant ratio of cylinder diameter to MT length of10:1. 
  
  
  
  Forρ≪1, the continuum model behaves like isolated fibers with negligible collective fluid entrainment. 
  For smallσ, straight fiber arrays are stable (regions I&II, with region II having oscillatory decay to 
  equilibrium), but with increasingσthere is a Hopf bifurcation to a state that nonlinear simulations 
  show has oscillations (cf. Fig. <ref>(c)). Forρ≳2.8(δ≲1.2L), a new region of 
  instability (IV) appears, with real and positive eigenvalues; nonlinear simulations show this 
  leads to collective MT bending and swirling flows.  
  
  Figure <ref>(b) shows a nonlinear simulation of the transition to streaming
  in region IV.  The upper inset shows the development of the instability,
  with successive MTs bending over to form a dense canopy above their
  highly curved bases. At steady state, the concentrated motor forces within
  the canopy are azimuthally aligned, almost aδ-function a distance∼L/4above the
  wall, and drive the large-scale
  streaming flow. The ooplasmic flow beneath the MT canopy is nearly a linear shear flow, 
  transitioning above to solid body rotation, the solution to Stokes flow forced at a
  cylindrical boundary.
  
  We now estimate ranges of density and force that are consistent with 
  observed streaming speedsu≈100-400 nm/s (Fig. <ref> and <cit.>). TakingL=20 μm,μ=1 Pa s <cit.> andA=20 pNμm^2, we obtain a velocity scaleA/ηL^3 ≈1 nm/s and a force-density scaleA/L^3≈2.5 fN/μm. 
  Figure <ref>c shows the streaming speeds calculated from nonlinear 
  simulations in region IV. Those with maximum speeds falling in the experimental range  
  lie in the hatched area. Increasingρonly marginally increases streaming speeds, and so to increase flow speed while remaining in region IV requires increasing bothρandσ. The minimum value ofρ≈20that is consistent with observed streaming velocities corresponds toδ≲0.4 L, a more stringent constraint than that required for the streaming transition. The force densities consistent with streaming speeds aref∼ 0.1-0.6 pN/μm. Speeds on the higher end of the physical range approach the≈ 700 nm/s of kinesin-1 under negligible load <cit.>, while cargo speeds on oocyte MTs are200-500 nm/s <cit.>. Assuming a linear force-velocity relation and a stall force of6 pN <cit.> gives a single motor force of≈ 2 pN; approximately1-6kinesins are required per20  μm MT to generate these force densities.
  
  It may be surprising that the streaming speed only weakly depends onρ. A heuristic argument for the flow speeds views the cytoskeleton as a porous medium of permeabilityk∼δ^2, in which speedu∼(k/μ)∇p, where the pressure gradient (force/volume) from motors isf/δ^2, yieldingu∼f/μ∼(A/ηL^3)(8π/c)σ, independent ofρ.  This relationship is surprisingly accurate <cit.>.
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  When the densityρis sufficiently high, the swirling instability first appears for force densitiesσsubstantially smaller than those that induce buckling instabilities in a single filament. Thus this transition 
  must be driven by the additional hydrodynamic destabilization that neighboring fibers impart (in the simplest geometry, 
  given by the termρσ(1-z)gin Eq. <ref>). This observation motivates a simple heuristic argument 
  for the instability, in which a filament is bent by the flow produced primarily 
  by its upstream neighbor, whose distal half is nearly parallel to the wall.  
  Seen from a distance, that bent portion acts on the fluid like a point force <cit.>F∼(fL/2)_̊s(L)oriented along its distal tangent vector 
  (Fig. <ref>), displaced a distanceh∼L/2from the surface.  Near a no-slip wall, the far-field flow 
  alongxdue to a forceF∥x̂a distanceδupstream is simple shear 
  <cit.>,
  
      U(x,z)=γ̇z ê_x ,
  
  whereγ̇=3h F/2πμδ^3. Self-consistency requires the magnitude of the force driving the shear be 
  given by the projection ofFalongx, soγ̇→γ̇sin(θ(L)).
  
  
  
  The very simplest model to illustrate the self-consistency condition 
  is a rigid MT with a torsional spring at its base  that provides a restoring
  torque-kθ(Fig. <ref>(i)).  
  Withz(s)=scosθandηn̂n̂·Uthe local normal force on a segment, the local torque about the points=0isηγ̇s^2 cos^2θwhich, when integrated along the filament and balanced
  against the spring torque, yields the 
  self-consistency condition 
  
      θ=Bsinθcos^2θ ,
  
  whereB=ηγ̇L^3/k.
  ForB<1(slow flow or a stiff spring)θ=0is the only fixed point, while forB≳1two mirror-image swirling solutions appear through a pitchfork bifurcation,θ_±≃(6(B-1)/7)^1/2.
  
  To study the interplay between filament oscillations and swirling we use (<ref>)
  in the filament dynamics (<ref>), where the control parameter for the shear flow is <cit.>
      M=ηγ̇L^4/A∼3σ/c(ρ/ϕ)^3/2,
  
  where the second relation uses the above estimates forFandh.  
  Since a clamped elastic filament
  behaves like a torsional spring with a spring constantk=A/L, we see consistency with the
  parameterBdefined above. 
  A numerical self-consistent calculation confirms the existence of a swirling instability <cit.>.
  
  Through simplified discrete and continuum models we have demonstrated the existence of
  a novel swirling instability of arrays of elastic filaments, thus lending support to the hypothesis 
  <cit.>
  that cytoplasmic streaming flows in Drosophila 
  oocytes are tied to self-organization of the microtubule cytoskeleton.  Future studies could shed light on the detailed mechanism involved in the untangling of the 
  Drosophila oocyte cytoskeleton when it transitions to the vortical state,
  and the possibility of reproducing this transition in vitro. Lastly, this study highlights the role of active force dipoles in the self-organization of fluid-biopolymer systems <cit.>.
  
  
  We are indebted to Maik Drechsler and Isabel Palacios for sharing the data in Fig. <ref> and to them, Daniel St Johnston and Vladimir Gelfand for discussions on Drosophila streaming. 
  This work was supported in part by ERC Consolidator grant 682754 (EL), 
  Wellcome Trust Investigator Award 207510/Z/17/Z, 
  Established Career Fellowship EP/M017982/1 from the
  Engineering and Physical Sciences Research Council, and the Schlumberger Chair Fund (REG). MJS acknowledges the support of NSF Grant DMS-1620331.
  
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  SM  See Supplemental Material at <http://link.aps.org/supplemental/xxx> for
  further details and results.
  
  thesis Further examples and details are in: 
  G. De Canio, Motion of filaments induced by molecular motors:
  from individual to collective dynamics, PhD thesis, University of Cambridge (2018).
  
  MaulKim C. Maul and S. Kim, Image systems for a Stokeslet inside a rigid 
  spherical container,
  https://doi.org/10.1063/1.868223Phys. Fluids  6, 2221 (1994).
  
  synchro R.E. Goldstein, Green Algae as Model Organisms for Biological Fluid Dynamics,
  https://doi.org/10.1146/annurev-fluid-010313-141426Annu. Rev. Fluid Mech.  47, 343 (2015).
  
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  https://doi.org/10.1063/1.2956984Chaos  18, 037128 (2008).
  
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  protoplasme isolés. I. Mouvement rotatoire et le processus de son apparition,
  https://doi.org/10.1508/cytologia.18.146Cytologia  18, 146 (1953);
  Recherches sur les phéénomenès moteurs dans les fragments de 
  protoplasme isolés. II. Mouvements divers d'etermin'es par la condition de milieu,
  https://doi.org/10.1508/cytologia.18.202Cytologia  18, 202 (1953).
  
  confinement_expt H. Wioland, F.G. Woodhouse, J. Dunkel, J.O. Kessler, and R.E. Goldstein,
  Confinement Stabilizes a Bacterial Suspensions into a Spiral Vortex,
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  Saintillan D. Saintillan and M.J. Shelley, Instabilities and Pattern Formation in Active Particle Suspensions:
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  Confined Active Suspensions, 
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  NRZS E. Nazockdast, A. Rahimian, D. Zorin, and M.J. Shelley, A fast platform for simulating semi-flexible fiber suspensions applied to cell mechanics, https://doi.org/10.1016/j.jcp.2016.10.026J. Comput. Phys.  329 (2017).
  
  methods Differentiation of (<ref>) yields an equivalent equation 
  for the tangent-vector field <cit.>, which we have found to be 
  numerically more stable.
  
  Landau L.D. Landau and E.M. Lifshitz, Theory of Elasticity, 
  2nd ed. (Pergamon Press, Oxford, 1970), p. 99, Problem 7.
  
  ponytails R.E. Goldstein, P.B. Warren and R.C. Ball, Shape of a Ponytail and the
  Statistical Physics of Hair Fiber Bundles,
  https://doi.org/10.1103/PhysRevLett.108.078101Phys. Rev. Lett.  108, 078101 (2012).
  
  Loiseau P. Loiseau,, R. Davies, L.S. Williams, M. Mishima and I.M. Palacios, 
  Drosophila PAT1 is required for Kinesin-1 to transport cargo and to maximize its
  motility, https://doi.org/10.1242/dev.048108Development  137, 2763 (2010).
  
  eLife D.R. Brumley, K.Y. Wan, M. Polin and R.E. Goldstein, Flagellar synchronization through 
  direct hydrodynamic interactions,
  https://doi.org/10.7554/eLife.02750eLife  3, e02750 (2014).
  
  Blake J.R. Blake, A note on the image system for a stokeslet in a no-slip boundary, 
  https://doi.org/10.1017/S0305004100049902Math. Proc. Camb. Phil. Soc.  70, 303 (1971).
  
  lauga_pillars J.-B. Thomazo, E. Lauga, B. Le Révérend, E. Wandersman,
  and A.M. Prevost, Collective stiffening of soft hair assembles, 
  https://www.arxiv.org/abs/2002.02834arXiv:2002.02834.
  
  
  
  § SUPPLEMENTAL MATERIAL
  
  
  This file contains calculational details for computations of discrete filaments
  and the continuum model, further results from linear stability analyses in the
  latter, and quantitative analysis of streaming speeds in the swirling state.
  
  
  
  § CALCULATIONAL DETAILS FOR DISCRETE FILAMENTS
  
  
   §.§ Image system inside a sphere
  
  
  By modelling the motor-cargo ensemble with fluid-entraining
  follower forces and
  approximating the system geometry by a spherical container,
  we take advantage of the analytical solution <cit.> for the velocity field
  of a point force inside a sphere with a no-slip wall.  
  These simplifications allow us to compute the
  background flow directly,
  instead of numerically solving the
  Stokes equations for the fluid flow.
  For a point force of magnitudeFlocated at_0inside a sphere of unit radius,
  the m^thcomponent of the velocity is
  
      u_m = F_j/8 πμ[ G_jm( - _0 )  +
                                  G̅_jm() ]   ,
  
  whereμis the dynamic viscosity
  of the fluid,G_jmis the Green's function
  
      G_jm( -_0 )  =
      δ_jm/r + ( x_j - x_0_j )
                              (  x_m - x_0_m)/r^3  ,
  
  and
  
      G̅_jm()   =      ê_mê_k{1-3R^2_0/2R^3_0  G_jk( - _0 )  
                                  -1-R^2_0/R^4_0 ê_l   G_jk,l( - _0 ) 
                               -(1-R^2_0)^2/4R^5_0 ∇^2  G_jk( - _0 ) }
          + ( δ_km -ê_kê_m)
                              {3R^2_0-5/2R^3_0  G_jk( - _0 ) 
                              +(1-R^2_0)^2/4R^5_0 ∇^2  G_jk(  - _0 ) }
          + 1-R^2_0/R^4_0 ê_k( δ_lm -ê_lê_m)
                                    G_jk,l( - _0 ) 
                               -3(R^2_0-1)/R^3_0( δ_jm -ê_jê_m)/r̅
                              +(R^2 - 1 ) ( δ_km -ê_kê_m)
                              ∂φ_k/∂ x_j  ,
  
  with
  
      φ_k = - 3(R^2_0 - 1)/2R^3_0 x_k/r̅ R-R̅_0 cosα +
      r̅cosα/R R̅^2_0 sin^2α  .
  
  Here,R_0andR̅_0are the norms of_0and its image_0, respectively, = _0/R_0is the unit vector in the axial direction,r = |-_0|,r̅ = |-_0|,Ris the norm of, andαis the
  angle betweenand.
  
  
  
   §.§ Geometrical formulation
  
  
  For both the discrete-filament computations and the self-consistent model, we use the tangent angle representation of the 
  filament.  For a filament of lengthL, parameterized by arclengths, 
  anchored at wall in thex-yplane, withzorthogonal to the plane, into the fluid, the position vector isr(s)=(x(s),z(s)), 
  and the unit tangent and normal vectorst̂(s)=r_sandn̂(s)are
  
      t̂(s)   =   cosθ(s)ê_x + sinθ(s)ê_z
      n̂(s)   =   sinθ(s)ê_x -cosθ(s)ê_z ,
  
  whereθ(s)is the angle the tangent vector makes with respect to thex-axis.
  The curve is obtained by integration oft̂, and assuming(x(0),z(0))=(0,0), we have
  
      x(s)=∫_0^s  ds' cosθ(s') ,    z(s)=∫_0^s ds' sinθ(s') .
  
  It follows from the above that 
  
      ∂t̂/∂ t=-θ_tn̂ ,
  
  from which we can obtain the equation of motion forθgiven that forr.
  
  The equation of motion is that of elastohydrodynamics within resistive force theory,
  
      (ζ_∥t̂t̂+ζ_⊥n̂n̂)
          ·( r_t- U)=-A r_ssss-(Λ r_s)_s ,
  
  whereAis the filament bending modulus,Λis the Lagrange multiplier, andU=Uê_x+Vê_zis the background flow experienced by the filament.  
  We express the derivatives in (<ref>) in terms of the curvatureκ=θ_s, using the
  Frenet-Serret equationst̂_s=-κn̂andn̂_s=κt̂, 
  and rescale the equation of motion via
  
      r   =L r' ,    z=Lz' ,    s=Ls' ,    κ = 1/Lκ'  , 
      Λ   =A/L^2Λ'  ,     t= ζ_⊥ L^4/A t'  ,    U= ζ_⊥ L^3/A U'.
  
  If we letζ_⊥/ζ_∥≃2, and useê_x=cosθt̂ + sinθn̂andê_z=sinθt̂ - cosθn̂, and then drop the primes, we obtain
  
      r_t=(Ucosθ +Vsinθ)t̂+(Usinθ-Vcosθ) n̂
          +(κ_ss-κ^3+κΛ)n̂
          +η(3κκ_s-Λ_s)t̂ .
  
  Differentiating (<ref>) with respect to s to obtain the equation of motion of
  the tangent angle θ and that for Λ:
  
      θ_t=-θ_ssss+3θ_s^2θ_ss-(θ_sΛ)_s  
           +2(3θ_s^2θ_ss-θ_sΛ_s)-U_ssinθ+V_scosθ ,
  
       while enforcing inextensibility leads to the equation for Λ,
       
      (∂_ss-1/2θ_s^2)Λ=
          -1/2θ_s^4+3θ_ss^2+7/2θ_sθ_sss
          +M/ηsin(θ)cos(θ) .
  
  The boundary conditions at the attachment point are those of a clamped filament,
  
      θ(0)=π/2 ,    θ_sss(0)-θ_s(0)^3+
          θ_s(0)Λ(0)=0 , 
         Λ_s(0)-3θ_s(0)θ_ss(0)=0 .
  
  whereas at the free end we have
  
      θ_s(1)=0 ,    θ_ss(1)=0 ,   Λ(1)=0 .
  
  
   §.§ Self-consistent model
  
  
  If we take the typical bent filament shape in the streaming regime (Fig. 2(d) of main text) and place
  it near a flat no-slip wall, we can understand in the simplest situation the flow it produces.  As shown 
  in the streamlines and velocity colormap of Figure <ref> there is a boundary-layer 
  phenomenon near the wall that arises from the combination of tangential forcing of the flow by the
  motors and the no-slip condition at the wall.  The most prominent part of the flow is a lobe of high speeds 
  emanating from the bent portion of the filament, directed downstream.  This feature forms the basis
  of the self-consistent model for the swirling transition.
  
  
  
  As Fig. <ref> shows, the flow downstream from a bent filament is approximately
  simple shear.  This can be seen directly in Blake's analysis 
  <cit.> of the flow due to a point force near a no-slip wall.  For the purposes of a
  simple self-consistent model, we consider only the asymptotic form of that flow evaluated atx=(x,y,z)due to a point forceFat(0,0,h):
  
      u_i≃   F_k/8πμ[12hx_ix_αx_3δ_kα/| x|^5
          +h^2δ_k3(-(12+6δ_i3)x_ix_3/| x|^5+
          30x_ix_3^3/| x|^7)] .
  
  Taking the leading-order term, andFalongx, we obtain
  
      u(x,z)=γ̇(x) z ê_x
  
  where the shear rate is
  
      γ̇(x)=3Fh/2πμ x^3 .
  
  In the notation used in the tangent angle dynamics (<ref>), we haveU=Mz(s)andV=0, where 
  
      M=ηγ̇L^4/A∼3σ/c(ρ/ϕ)^3/2 .
  
  a control parameter that appears also in the dynamics elastic filaments in extensional flows 
  <cit.>.  Here, the second relation uses the definitions of the dimensionless
  control parametersσandρfrom the main text.
  To complete the self-consistency calculation, we set thex-component of the forceFto be
  proportional to the projection of the tangent vector at the distal tip, soM →Msin(θ(L))The tangent angle dynamics (<ref>) and (<ref>) 
  were solved by a Crank-Nicholson method
  via a pentadiagonal matrix for the fourth derivative, 
  treating all the nonlinearities
  explicitly, and discretizing theΛproblem as a tridiagonal matrix in which
  the diagonal term involvingθ_s^2was updated after each time step.  The initial
  condition wasθ(0,s)=π/2-asin(πs/2)^2, witha=2.5and the time stepdt=0.1 ds^4, withds=1/(N-1), whereNis the total number of
  grid points (N=41is sufficient).
  
  Figure <ref> shows a typical result of the self-consistent 
  calculation in the regime of stable streaming, illustrating how a highly bent initial condition relaxes
  to a conformation like that seen in the calculations described in the main text (compare Figs. 3(c) from 
  discrete filament calculations and
  4(b) from continuum model). 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  § FURTHER RESULTS FROM CONTINUUM MODEL
  
  
   §.§ Dynamics of the transition to swirling
  
  
  In Supplemental Video 1, we show the dynamics of the transition from the unstable equilibrium of radially oriented filaments to stable streaming flow, corresponding to Fig. 4b (main text). In this parameter regime, the filament array initially buckles, and large azimuthal flows are generated by rapid deformation of the array. As the filament motion slows, the filaments become oriented parallel to the cylinder walls. Azimuthal flows are continuously generated by the streaming mechanism, and the drag from these flows moving past the stationary filaments is sufficient to pin them into the observed conformation.
  
  
  
   §.§ Effect of cylindrical confinement
  
  
  In the main text, we present the evolution equation for a small perturbation to the equilibrium solution in a planar geometry. This equation:
  
      g_t = -g_zzzz -σ[(1-z)g_z]_z + ρ[σ(1-z)g+g_zz],
  
  remains local due to the simplicity of the Stokes flow, allowing the individual terms to be easily identified and interpreted. In order to better capture the real geometry of the oocyte, in the main text we present the results of the linear stability analysis in a cylindrical geometry (Fig 4a, main text). The same analysis, over the same range of effective densities (ρ) and effective motor force-densities (σ) is shown in Fig. <ref>, computed for the planar geometry. These results are qualitatively the same as those shown in the main text in Fig 4a, indicating that the essential structure of the swirling transition is due to the interactions between nearby filaments in the fiber bed, and does not depend in a critical way on the specific geometry of the oocyte.
  
  
  In, Fig. <ref> we present the same analysis for different confinement ratios — 2.2:1 (Fig <ref>a), 4:1 (Fig <ref>b), and 20:1, (Fig <ref>c). Note that the ratio 2.2:1 indicates a cylinder diameter of2.2L, which is extremely confined, with filaments reaching≈91%of the way to the center of the cylinder. Even under such strong confinement, the qualitative structure remains, although the quantitative value at which the bifurcations occur changes significantly. The effect of strong confinement is to suppress the appearance of streaming at larger forcing values. This makes sense: when forcing is large, the tendency of lone fibers is to oscillate. In the planar system, when fibers are sufficiently dense, the strong flows caused by motors walking on neighboring fibers is sufficient to pin the fiber into a stable deformed state, suppressing oscillations (in fact, the drag from these flows produces extensile forces on the fibers, reducing the net contractile forcing generated by the motors). Under high confinement, the strength of such flows near the fiber tips is reduced, and the flow can no longer suppress the tendency of the fiber to oscillate. When the confinement is not so extreme, even the quantitative values change little, from 10:1 (Fig 4a, main text), to 20:1 (Fig <ref>c), to the flat geometry (Fig <ref>).
  
  It may surprise the reader that whenρ→0, the behavior (as a function ofσ) at all confinement ratios converges to exactly the same thing. This is because the continuum model imposes boundary conditions on the coarse-grained flow only; asρ→0this flow goes to0with it and the dynamics are those of a fiber moving in a quiescent background.
  
  
  
   §.§ Streaming Speeds in Dense Arrays
  
  
  By solving for-_̊ssss + (Λ_̊s)_sin Eq. 2 (main text), substituting into Eq. 3 (main text), definingξ≡χ_MTρ𝒥^-1, and ignoring frame transformations for the sake of simplicity, we find that:
  
      -^2+̆ p + ξ(𝕀-_̊s_̊s/2)(-̆_̊t) = ξσ_̊s.
  
  This is a forced Brinkman equation with an anisotropic permeability. Whenξis large, the skeletal drag term dominates the left-hand side, and at steady state_̊t=0. We may thus approximate the steady streaming velocity as simply≈̆σ(𝕀+_̊s_̊s)_̊s = 2σ_̊s. For homogeneous flows in the planar geometry, and axisymmetric flows in the cylindrical geometry, the flow is purely in thex̂orθ̂(azimuthal) direction, respectively. Since_̊sis not purely coincident withx̂orθ̂, some of the forcing generated by the molecular motors must be absorbed into the pressure gradient, and so we take this estimate to be an upper bound:v_streaming≤2σ. The maximum steady-state streaming speed as a function ofσ, as computed by the continuum model in 10:1 confinement, is shown in Fig. <ref>, forρ=10, 20, …, 50. For smaller values ofρ, the fiber is less deformed and drag contributes less to the balance in Eq. <ref>, leading to a relationship closer tov_streaming=σ. Whenρis larger, drag dominates Eq. <ref>, the fiber is nearly azimuthally aligned, and the streaming speed approaches the boundv_streaming=2σ.
  
  
  In dimensional units, this bound isv_streaming≤2σA/ηL^3. In the main text, we used a simple argument based on Darcy's law to estimatev_streaming=(σA/ηL^3)(8π/c). The leading factor of two in the more refined estimate presented in this SI comes from taking into account fiber anisotropy and the nearly azimuthal alignment of the fibers, while the disappearance of the factor of8π/carises from properly accounting for the fibers aspect ratio. In either case, the disappearance of the MT density from the estimate of streaming velocity arises due to the fact that both the forcing (ξσ_̊s) and the skeletal drag (ξ(𝕀-_̊s_̊s/2))scale with the densityρ.
  
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