Aster Swarming by Collective Mechanics of Dyneins and Kinesins
AAster Swarming by Collective Mechanics ofDyneins and Kinesins
Neha Khetan and Chaitanya A. Athale,Div. of Biology, IISER Pune, Dr. Homi Bhabha Road,Pashan, Pune 411008, India.Email: [email protected] 11, 2020
Microtubule (MT) radial arrays or asters establish the internal topology ofa cell by interacting with organelles and molecular motors. We proceed tounderstand the general pattern forming potential of aster-motor systems us-ing a computational model of multiple MT asters interacting with motorsin a cellular confinement. In this model dynein motors are attached to thecell cortex and plus-ended motors resembling kinesin-5 diffuse in the cell in-terior. The introduction of ‘noise’ in the form of MT length fluctuationsspontaneously results in the emergence of coordinated, achiral vortex-like ro-tation of asters. The coherence and persistence of rotation requires a thresh-old density of both cortical dyneins and coupling kinesins, while the onset ofrotation if diffusion-limited with relation to cortical dynein mobility. The co-ordinated rotational motion arises due to the resolution of the ‘tug-of-war’ ofthe rotational component due to cortical motors by ‘noise’ in the form of MTdynamic instability, and such transient symmetry breaking is amplified bylocal coupling by kinesin-5 complexes. The lack of widespread aster rotationacross cell types suggests biophysical mechanisms that suppress such intrinsicdynamics may have evolved. This model is analogous to more general modelsof locally coupled self-propelled particles (SPP) that spontaneously undergocollective transport in presence of ‘noise’ that have been invoked to explain1 a r X i v : . [ q - b i o . S C ] J un warming in birds and fish. However, the aster-motor system is distinct fromSPP models with regard to particle density and ‘noise’ dependence, providinga set of experimentally testable predictions for a novel sub-cellular patternforming system. Radial microtubule (MT) arrays or asters determine the geometry of animalcells and determine their plane of division. The positioning and transportof asters in cells has been addressed in previous work based on multipleforce-generators. We develop a computational model of multiple asters ina cell that combines previously reported forces and examines the potentialfor collective motion. Forces are generated in our model by either the walk-ing activity of biomolecular motors or MT bending mechanics. Motors oftwo kinds are included - minus-end directed dyneins localized on the cellboundary and coupling kinesins in the cytoplasm. The model geometry isbased on experimental reports of animal cells ranging from epithelial cellsto one-celled embryos. Simulations result in spontaneous random motion ofthe asters when ‘noise’ due to MT length dynamics is added. The rotationalmotion is equally likely to result in clockwise and anti-clockwise rotations,due to the random nature of the initiation of motility. Additionally, we findthe collective motion requires the dyneins to be mobile on the cell boundary,with diffusive mixing essential for pattern formation. This model predictsa form of collective motion, similar to that observed during mitosis in em-bryonic development. Our model predicts such effects are intrinsic to cells,and mechanisms to suppress such swarming motility of asters might haveevolved that are yet to be tested in experiment. Aster swarming based onlocal interaction of components and ‘noise’ that cannot be predicted from thecomponents alone, is analogous to collective schooling of fish and swarmingof birds, which lack an explicit leader. This suggests it belongs to a gen-eral class of self-propelled particle (SPP) models, with differences in specificfeatures from such general models. 2
Introduction
Self-organized pattern formation is observed almost universally in biologicalsystems ranging in scales from large scale structures of swarming birds andfish [11], through cells undergoing collective migration patterns[17, 68, 57, 69,56, 64], to single-cell polarization by reaction-diffusion networks of proteins[2, 5, 71]. The self-organized patterns of the mechanical elements of thecell, the cytoskeleton and molecular motors are particularly distinct, arisingas they do from purely mechanical interactions. Of these the most distinctare the in vitro patterns reported from the reconstitution of ATP containingmixtures of microtubules (MT) and motors [49, 48, 66, 65] and actin andmyosin [8, 15]. MT-motor activity in a circular boundary has been shownto break symmetry and result in vortex like motility [49, 67]. Evidence thatsuch vortices are not just restricted to minimal in vitro reconstituted systems,has come from in vivo studies demonstrating MT-motor driven cytoplasmicstreaming in
Caenorhabditis elegans embryos [63] and
Drosophila ooocytes[42] during development and the cells of the plant
Chara [75]. However,the range of motility patterns seen in these structures are specific to thegeometry of the cell type and specific mix of motors. In order to understandthe general principles of such pattern formation, theoretical models that takeinto consideration a wider range of cell and filament geometries are required.A general model of self propelled particle (SPP) motion describing theemergence of collective motion or swarming based on local coupling interac-tions and ‘noise’ has been described by Vicsek et al. [77]. While this class ofmodels minimally requires active particles with local interactions and ‘noise’,they do not capture the polarity of filaments and motors, since active par-ticles are typically considered to be point particles. MT filaments have akinetic polarity of plus- and minus-ends, that determines the direction ofmotor activity- kinesins walk towards the plus-ends and dyneins towards theminus-ends [26]. Detailed models of self-organized patterns of linear fila-ments have shown good agreement with experiments as seen with MTs inpresence of kinesin [49, 48, 66] or dynein [65] as well as actin with myosinactivity [8, 59, 15]. However, inside most animal cells MTs have a charac-teristic orientation of minus-ends near the nucleus and plus-ends at the cellperiphery, forming a mechanical positioning system for organelle transport,cell polarization and cell division. This characteristic organization of MTs isdetermined by microtubule organizing centers (MTOCs) that serve as nucle-ation points forming radial arrays or asters. At the same time most studies3f collective MT transport have used linear MTs. The effect of MT geometryon mobility is seen when comparing the outcome of these two kinds of MTsencountering a sheet of immobilized motors of either plus- or minus-endedtype. While linear MTs undergo collective transport and glide, radial astersundergo a tug-of-war due to geometry as seen with kinesins for 1D doublets[38] or dyneins transporting asters [7]. The effect of diffusible motors on astermovement is variable and determined by both the motor type - tetramerickinesins and MT orientation. Pairs of asters coupled by kinesins bound toanti-parallel MTs will form bipolar spindle-like structures in simulations [47],confirming the importance of kinesin-5 in spindle assembly seen in experi-ments [80]. Such anti-parallel MTs from asters of neighboring spindles insyncytial embryos of
Drosophila result in even spacing of spindle asters [73].Parallel MTs on the other hand result in ‘zippering’ by movement of motorson parallel MTs as seen in with MTOC asters during in mouse oocyte spindleassembly [60, 31]. Dyneins coalesce asters independent of whether MTs areparallel or anti-parallel and the effect is observed supernumary centrosomeclustering due to dynein [55]. In many cells however, dyneins are immobi-lized in the cell cortex. Asters contacting these membrane anchored motorsresult in radial pulling forces acting on asters driving them to the cell cen-ter in cells with sizes comparable to the asters [36]. In multi-nucleate cellssuch as the filamentous fungus
Ashbya gossypii , these astral-MT interactionswith cortical dynein are essential for maintaining a regular spacing betweennuclei [18]. Cortical dynein localization forms the mechanical basis of theasymmetric cell division of one-celled embryos of
C. elegans [20] and the cor-tical density of dynein has been shown to determine spindle oscillations [53].This suggests mechanical interactions of MT asters with diffusible kinesintetramers and cortical dyneins constitute a conserved mechanical modulesacross a wide variety of cell types. While a general theory for the mechanicsof a single aster in a confined cellular geometry has been developed previ-ously [43], a model integrating motor localization seen in diverse cells withmultiple asters is lacking.Here, we have modeled the mechanics of a mixture of multiple MT astersacted on by dyneins at the cell cortex and kinesins in the cytoplasm, toexamine the potential of this system for collective transport. We test theeffect of the forces generated on asters arising from local coupling by cyto-plasmic kinesins, gliding of MTs on the circular boundary lined with corticaldyneins and inward pushing due to MT mechanics in confinement and therole of stochastic MT polymerization dynamics. Our model demonstrates4 ulti aster-motor systemCytoplasmic Forces (r a, d a ) D k Antiparallel MTs
Sliding apart
Parallel MTs
Zippering (r d , F d ) Kinesin-5 complex Dynein
Components
Aster
MTOCGrowing MT Shrinking MT -++ (b) + D d dx ~ dx (r d ,F d ) (r a, d a ) Cortical Forces (e) F M T Pushing forces Pulling force
Microtubule dynamics
Time
Shrinkage
MT lossGrowth CatastropheNucleationRescue (c)(a)(d)
Figure 1:
Model of aster-motor mechanics. (a) The model consists ofmultiple asters and motors in confinement. (b) Aster MTs are radially nu-cleated with minus-ends of MTs embedded in the Microtubule (MT) orga-nizing center (MTOC) and plus-ends out. MTs can switch between growing(black) and shrinking (gray) states. Motors are modeled as springs MT bind-ing sites as seen in case of anchored dynein (blue) and kinesin-5 (red) withtwo motor domains. (c) MTs dynamic instability is modeled with transitionsbetween growth and shrinkage states, while MTs per aster remain constant.(d) In the cell interior unbound kinesins diffuse and and can generate forceswhen bound a pair of MTs, serving to separate or aggregate asters, based onMT orientation. (e) At the cortex asters are pushed inwards by MT bend-ing forces ( F MT ), while anchored dyneins when bound drive aster transport.Free dyneins diffuse along the cortex. Orange arrows: force on the asters.Black arrows: direction of motor movement.that while coupled mechanics alone results in local and uncorrelated astermotility, the addition of ‘noise’ transforms it into coherent rotational mo-tion. This emergence of coherent rotation or swarming of asters depends oncortical dyneins, kinesins and the MT stochasticity. The spontaneous coher-ent streaming motion of asters predicted by the model is discussed in the5ontext of experimental evidence for the nature of dynein anchoring to thecell cortex. Components and interactions:
We have modelled a multi-aster system ina cellular compartment in the over damped regime with mechanics andstochastic binding kinetics of kinesin and dynein (Figure 1), based on a previ-ously developed computational agent-based model of MT-motor interactions[50]. MTs form a radial structures, asters, that interact with one anotherwhen they are cross-linked by molecular motors that bind to and walk onMTs. This mechanical coupling of asters at micrometer scales originates frommotor-driven forces spanning a few nanometers. The net force experiencedby a complex of coupled asters determines whether they are transported orstatic. Stochasticity in the model originates from three sources: MT lengthfluctuations, diffusion and the binding kinetics of motors. The mechanicalinteractions vary according to position either at (a) the cell cortex or (b) inthe cytoplasm and are described separately.(a)
Cell cortex:
Dynein-like motors are modeled as being bound to the cellcortex by their stalk domains and can bind to and walk on MT filamentsresulting in aster transport along the cell boundary. Such movement is com-parable to a ‘gliding assay’ described in previous work [7, 31, 27], with arotational component due to the circular geometry. MTs when bound todyneins or non-specifically encountering the cell-membrane at the cortex,bend and generate an inward restoring force on the aster (Figure 1). In addi-tion, MT-bound motors can also be dragged through the membrane. Whenmultiple motors bind to oppositely oriented MTs of the aster a tug-of-wararises in the MT-motor system. Such tug-of-wars emerge from the radialgeometry of asters, as previously described [7]. Dyneins that are not boundto MTs, can diffuse in the membrane.(b)
Cytoplasm:
In the cell interior, diffusible tetrameric kinesin motors aremodeled based on the mitotic kinesin-5 complexes. They can bind MTs andwalk towards the plus end and when bound to astral MTs from neighbor-ing asters simultaneously, they produce forces on asters. The resultant forcedepends on MT orientation- parallel or anti-parallel, leading to either ‘coa-lescence’ or ‘separation’ of asters respectively (Figure 1). Model parametersare taken from experimental reports where possible (Table 1).6
T asters:
A microtubule organizing center (MTOC) is modeled to nu-cleate a finite number of MTs in a radial manner forming an aster. MTsare modelled as semi-flexible rods with a bending modulus κ of 20 N/m ,based on previous reports [19]. When MTs at the cell boundary bend, theyexperience a restoring force based on beam-theory, F bend = π · κ/L , where Lis the filament length and κ is the flexural rigidity of microtubules. Bendingcan occur either because the aster is held at the rigid cell boundary, or MTsare bound to multiple dyneins. MT lengths ( L MT ) are modelled as eitheruniform and of fixed lengths (stabilized) or fluctuating (dynamic instability)with a mean length. We use the mean length of asters from measurementsmade on Xenopus oocytes [76] of (cid:104) L MT (cid:105) = 4.25 µm . The length dynamics aredescribed by the frequencies of catastrophe ( f c ) and rescue ( f r ) and velocitiesof growth ( v g ) and shrinkage ( v s ) with values taken from those reported for Xenopus laevis oocyte extracts [76, 6]: f c = 0 .
049 1/s and f r = 0 . v g = 0 . µm/s and v s = 0 . µm/s . The values of f c and f r are 0.049 1/sand 0.0048 1/s respectively in all calculations unless mentioned. Motor mechanics:
The motors modeled are of two kinds, cortical dyneinsand diffusible kinesin-5 complexes. Cortical dyneins are modelled as sin-gle walking motors with a spring-like stalk domain (Figure 1), the springconstant k d of which determines the stretch force F s = − k d ∆ h for a ∆ h change in stalk length [7, 31]. Bound motors are modeled as discrete step-pers with step size determined by the load. A load-free motor is modeled totake constant sized steps while increasing opposing forces reduces the stepsize in a piecewise manner based a previously described model [7, 31]. Ad-ditionally, the anchored dyneins attached to the MT are dragged along thecell-boundary based on the stretch on motor due to the opposing force orig-inating from the spring stretch force ( F s ). The detachment and stall forcesare identical ( f d = f s ) for dynein. Dyneins not attached to the MT diffuse inthe membrane with an effective diffusion coefficient D d , the value of which isvaried (Table 1). Kinesin-5 motor complexes are modeled as two motor headsjoined by a Hookean spring like connector representing the stalk (Figure 1),based on the stalk-linked dimer-of-dimers structure of this motor, that walkson two MTs simultaneously, thus coupling them [80]. The complexes are dif-fusible throughout the interior of the cell with an effective diffusion coefficient D k , the value of which we take to be 20 µ m /s based on typical cytosolicproteins (Table 1). Motors stochastically bind to MTs within a distance d a based on an attachment rate of r a . The second motor-domain can similarlybind another MT, independent of the MT orientation. A bound motor walks7 i ne s i n - C o r t i c a l F G s + K i ne s i n - C o r t i c a l F G s X ( µ m) -15 0 15-15015 Y ( µ m ) -15 0 15-15015 Y ( µ m ) -15 0 15-15015 Y ( µ m ) Stabilized MTs -15 0 15-15015 Y ( µ m ) -15 0 15-15015 Y ( µ m ) -15 0 15 X ( µ m) -15015 Y ( µ m ) Time (s)
Dynamic MTs (a)(b)(c)
Figure 2:
Spontaneous aster rotation.
Asters with MTs that were either( left ) of fixed length with MT flux J = 0 or ( right ) stochastically fluctuat-ing in length with J = -0.3 µ m/s. Simulations were run in presence of oneof the motor combinations: (a) diffusible cortical dyneins (blue circles) (b)cytoplasmic tetrameric kinesin-5 motors (red circles) or (c) dyneins and ki-nesins. The last 30 s of the trajectories of aster centers of the correspondingsimulations are plotted (the colorbar represents time). Kinesin density 10motors/ µm , cortical dynein density 10 motors/ µm , η = 0 . N · s/m , N a = 20, N MT per aster = 40, R cell = 15 µ m and total time 300 s. Video S1represents the time-series.on the MT at constant velocity v = v , if there is no opposing load acting onit. In the presence of an opposing load, the velocity becomes v = v · (1 − f (cid:107) f s ),where f s is the stall force and f (cid:107) is the projection of the extension force( f ex ) along the MT. In the case of the multi-aster system, a kinesin complexbound to two filaments simultaneously, will experience a load, resulting in arestoring force that drives aster movement. This is based on similar collec-tive motor mechanics models used to model kinesin-5 in previous work [41].Both dynein and kinesin-5 motors detach based on Kramers theory [35] with8 rate r d = r · e | f ex | /f d where r is the load-free basal detachment rate and f d is the detachment force. When a kinesin-5 motor walks to the end of theMT it is modelled to detach immediately [32, 41]. MT dynamics:
MT dynamics is modeled based on the two state modelof growth and shrinkage, associated with 4 parameters, the filament growthvelocity v g and shrinkage velocity is v s and two transition frequencies be-tween the two states: f cat the frequency of catastrophe (growth to shrinkagetransition) and f res the frequency of rescue (shrinkage to growth transition)[24, 10]. Work by Verde et al. [76] demonstrated how these parameters relateto mean MT length (L MT ) as: L MT = v g · v s v s · f c − v g · f r . (1)and. The related variable of flux in MT lengths J is then calculated by: J = v g · f r − v s · f c f c + f r (2)which determines whether the length of MTs on an average is in the ‘boundedstate’ ( J <
0) or unbounded state (
J >
0) or not dynamic ( J = 0).Thus the position of asters is determined by the net force that results fromall these sources (Figure 1) i.e. inward pushing force due to MT bending atthe cortex ( F MT ), a force that pulls the asters to the cell boundary due todyneins and separating or ‘zippering’ forces in the cytoplasm due to kinesin-5 motility when bound to pairs of parallel or anti-parallel MTs respectively,with MT dynamics as a major source of stochasticity (Figure 1). Aster motility is a result of symmetry breaking in forces that arises from acombination of multiple forces: (i) kinesin-5 complexes that either zipper orseparate asters based on the orientation of astral MTs, (ii) the bending forcesfrom polymerizing MTs at the cell boundary, (iii) dynein pulling forces atthe cell boundary (iv) stochasticity in astral MT lengths and (v) Brownianforces corresponding to the energy k B T . However, none of these individuallyhave any innate ability to drive directional motion due to the radial sym-metry of asters and Brownian motion, as illustrated in Figure 1. We find9able 1: Model parameters.
The parameters that determine the me-chanics and dynamics of motors are taken from experimental measurementsreported in literature, and where missing estimated.
Symbol Parameter Dynein Kinesin-5 Reference D Diffusion coefficient 0-100 µ m /s 20 µ m /s [81] v Motor velocity 2 0.04 µ m/s [7, 41] d a Attachment distance 0.02 µ m 0.05 µ m [7, 31, 41] r a Attachment rate 12 s − s − ” k Linker strength 100 pN/ µ m 100 pN/ µ m ” f s Stall force 1.75 5 pN ” r (cid:48) d Basal detachment rate 1 s − s − ” r (cid:48) d,end Basal end-detachment rate 1 s − immediate ” f d Detachment force 0.5 pN 1.6 pN [32], this studythe collective interactions result in three distinct form of aster patterns thatdepend on motors and MT dynamics: (I) centering, (II) hexagonal latticeand (III) spontaneous rotation (Figure 2). Cortical dyneins exert an out-ward pulling opposing the inward force generated by MT bending and whencombined with MT dynamics result in the inward force dominating resultingin steady state (I) centering of asters (Figure 2a). Even though MT lengthsare identical in dynamic and static cases, the rare long filament bendingagainst the cell membrane produces sufficient inward-forces to overcome theoutward pulling due to dynein. Kinesin-5 motors results in a steady state (II)hexagonal lattice resembling molecular crystal arrangements, as a result ofkinesin-5 pushing forces on asters due to anti-parallel MTs (Figure 2b). Dy-namic instability abolishes these structures due to fluctuations in the overlaplengths.Cortical dyneins and diffusible kinesin-5 combined result in slidingmotion along the circular cell boundary and coupling of asters respectively,which when combined with ‘noise’ in form of MT dynamic instability breaksymmetry and result in steady-state (III) spontaneous rotation , analogousto ‘swarming’ dynamics (Figure 2c). Based on the role of dynein motors inforce generation for the emergence of coherent aster rotational motion, weexpected the motor density to be an important parameter and proceeded totest the systematic effect of motor density.10 (a) X ( μ m) Y ( μ m ) (b)(c)(d) X ( μ m) X ( μ m) X ( μ m) -15 0 15 -15 0 15 -15 0 15 -15 0 1515 0 -15 -5 0 5
0 300 0 300
4 0-6
0 300 -10 0 10
0 300 d (Dynein/ m) f s w i t c h ( s - ) d (Dynein/ m) T i m e s pen t ( s ) d (Dynein/ m) < C > < || > CCWCW -4 0 4 -4 0 2 -5 0 5 10 -5 0 5 10 F r equen cy T i m e ( s ) (e) ω ( s - ) ω (s -1 ) ω (s -1 ) ω (s -1 ) ω (s -1 ) Time (s)
01 01 01 01
Time (s) Time (s) Time (s) -10 0 10 d (Dynein/ m) (f) Figure 3:
Onset of sustained aster rotation at threshold dynein den-sity. (a) Aster trajectories over the last 30 s are plotted for varying densitiesof cortical dynein ( ρ d = 10 to 10 motors/ µ m). Colorbar: time. (b) Theinstantaneous angular velocity ( ω ) for each aster (gray) and the mean an-gular velocity (black) in time is plotted for increasing ρ d . (c) Representsthe frequency distribution of ( ω ) for the corresponding simulations. (d) Theswitching frequency and (e) total time spent in either clockwise or counter-clockwise direction is plotted for increasing densities of dynein over the entiretrajectory. (f) The mean coherence in rotation ( φ C ) and the mean angularvelocity ( (cid:104)| ω |(cid:105) ) over the last 30 s is plotted as a function of increasing dyneindensity. Cell radius: 15 µ m, N a =20, surface density of kinesin is ( ρ k = 10 motors/ µ m ) and diffusion coefficient ( d d ) of dyneins is 10 µ m /s, J = -0.3 µ m/s. N runs = 5. Error bars: standard error.11 .2 Dynein density dependent onset of rotation pat-terns The localization of cortical dyneins on a circular boundary that walk on MTsis expected to result in a ‘gliding’ transport of filaments, with a strong ro-tational component. However, due to the radial geometry of asters, MTsfrom the same aster have been shown to encounter antangoistic forces re-sulting in a tug-of-war [7]. In our model, stochasticity in opposing forcesresulting from MT length fluctuations produce an element of randomnessthat is expected to transiently resolve the tug of war. We find increasingthe density of dynein ( ρ d ) results in increasingly sustained rotation (Figure3(a)), confirming the central role for dynein force generation when combinedwith stochastic MTs and kinesin-5 motors. The time traces of instantaneousangular velocity ( ω ) exhibit are smaller in magnitude and fluctuate more inpresence of few motors, while increasing numbers of dyneins result in higheramplitudes and fewer variations in angular velocity (Figure 3(b)). Increasingsynchronization between the individual traces is observed for increased motordensities. The pooled frequency distribution of ω at low dynein density isdistributed sharply around zero, increases in spread for ρ d of 10 motors/ µ m,is bi-modal for 100 motors/ µ m and is biased to one side at a high density of10 motors/ µ m, indicative of persistent motion (Figure 3(c)). The switch-ing frequency, f switch , is calculated as the total number of switch events perunit time and quantifies the persistence in the direction. Consistent with thetrend in ω we find increasing ρ d results in decreased f switch (Figure 3(d)).Rotating asters are not chiral since the time spent by individual asters mov-ing in clockwise (CW) and counter clockwise (CCW) orientations is equalbetween multiple iterations. Individual simulations result in a spontaneouschoice of an orientation and the collective movement is entrained, with noparticular preference for CW or CCW motion (Figure 3(e)). However, dueto the shorter sliding events at low densities, the proportion of time spentin either state is comparable which begins to diverge for increasing dyneindensities.In order to quantify the emergent collective order in a system of rotatingasters, we measure a rotational order parameter φ R comparable to previousreports for cells [39] and active particles [1]. φ R ( t ) is the mean rotational12
15 0 15
X ( -15 0 15 -15 0 15 -15 0 15 m) -15015 Y ( -15015 -15015 -15015 m ) X ( m) T i m e ( s ) X ( m)
X ( m) ρ k (motors/ μ m ) -2 k (motors/ μ m ) < C > < || > -2 k (motors/ μ m ) f s w i t c h ( s - )
10 10 10 k (motors/ μ m ) T i m e s pen t ( s ) -10-50510 -30-20-10010 -100102030 -4-202468 ω ( d θ / s ) Time (s) Time (s) Time (s) Time (s)
0 100 200 300 0 100 200 300 0 100 200 300 0 100 200 300 (a)(b)(c) (d) (e) -2 0
CCWCW
Figure 4:
Strength of inter-aster interactions dictate onset of sus-tained collective rotation. (a) Representative XY trajectories of asters forvarying densities of kinesin-5 motors in the cytoplasm is plotted over thelast 30 s. Color encodes for the trajectory time. (b) The instantaneous an-gular velocity, ω is plotted in time. Gray curves represent individual astertrack while the ensemble mean is represented in black. (c) The switchingfrequency over the entire trajectory is plotted for varying kinesin-5 densitieson (X-axis, log scale). (d) The total time spent in either clockwise (blue) orcounter-clockwise (red) direction is plotted for increasing kinesin-5 densityover the entire trajectory. (e) The measure of coherence in rotation, φ c aver-aged over the last 30 s is plotted over the varying values of kinesin-5 density( X axis- on the log scale) on the left y-axis and the ensemble mean overthe last 30 s for the angular velocity is plotted on the right y-axis. ρ d = 10 motors/ µ m, ρ a = 0.03 asters/ µ m , N runs =3. Error bar indicates SEM.order parameter at time t over N asters expressed as: φ R ( t ) = 1 N · N (cid:88) i =1 ˆ e θ i ( t ) · ˆ v i ( t ) (3)Here, ˆ v is the unit velocity vector and ˆ e θ i is the unit angular direction vector13f the aster. The dot product of these two terms is averaged over the totalnumber of asters (particles) N where i is the index of each aster (particle). φ R can take values ranging between -1 for counter clock wise motion (CCW)and +1 for clock-wise motion, while zero indicates the absence of rotation.Since there is no preference observed for CW or CCW motion, the modulus ofthe rotational order averaged over time and between multiple asters measuresthe coherence of motion ( φ C ) as described by the expression: φ C = T (cid:88) t = t s (cid:104)| φ R ( t ) |(cid:105) (4)The time average is taken at steady state between t s the time for onset ofrotation (typically 270 s) and total time T (300 s). We observe the measure ofrotational coherence increases with ρ d and saturates at high dynein densities,which is mirrored by the mean angular velocity over the last 30 s transitioningfrom low to high value at a ρ d of ∼
100 motors/ µ m (Figure 3(f)). Takentogether, it suggests that a minimal number of dyneins (10 dyneins/ µ m) issufficient for the onset of rotation while a sustained, coherent and persistentrotation emerge only at a higher dynein density (100 dyneins/ µ m). Whilecortical dynein is expected to play a role in the onset of rotation, kinesin-5motors are thought to entrain the collective transport. In order to examinethe importance of kinesin in the ordered rotational motion of asters, we testthe effect of their density. Kinesin-5 motors are local coupling factors between asters that produce asegregating force generated in the absence of dynein and MT dynamic insta-bility (Figure 2). Thus their role in entraining collective rotational motility isnot necessarily intuitive, other than as a mechanism of exerting local couplingforces. Locally, kinesin-5 complexes bound to a pair of MTs from neighbor-ing asters can generate pushing or pulling forces between the asters based onthe orientation of the astral MTs to which they are bound- anti-parallel MTsresult in asters being pushed apart or segregated, while parallel MTs resultin asters being pulled together or ‘zippered’ (Figure 1(d)). Interestingly alow density of kinesin-5 fails to produce any sign of rotational motion, andonly above a threshold motor density does collective rotation emerge (Figure14igure 5:
Effect of MT length fluctuations on persistence of rota-tion. (a) The aster trajectories over the last 30 s is plotted for increasingvalues of MT flux (noise). Color encodes for the trajectory time. (b) Theinstantaneous angular velocity, ω is plotted in time. Gray curves representindividual aster track while the ensemble mean is represented in black. (c)The measure of coherence in rotation, φ c averaged over the last 30 s is plottedover the varying flux values (log scale) on the left y-axis and the ensemblemean over the last 30 s for the angular velocity is plotted on the right y-axis. Inset:
Linear plot. (d) The switching frequency over the entire trajectoryaveraged for n=3 runs ( ± s.d.) is plotted against the MT flux rate J (x-axis,log scale). The inset represents the same data with a linear x-axis. J wasvaried by modifying the frequencies of catastrophe ( f c ) and rescue ( f r ). Here, ρ d = 10 motors/ µ m, ρ k = 10 motors/ µ m , N a =20.4(a)). This is also evidenced by the low mean angular velocity ∼ MT dynamics in vivo differ between interphase when the flux
J >
J < f c and rescue f r while keeping mean length < L > and veloc-ity of growth v g and shrinkage v s constant, we solved Equations 1 and 2 toobtain a range of values for J . MT dynamics is expected to influence thestability of overlaps of pairs of MTs, the instantaneous number of motorsthat can bind to MTs and fluctuations in the bending energy. To our sur-prise, increasing the magnitude of flux from 0 to − × − resulted in morepersistent rotation of asters (Figure 5(a)). This is confirmed by the reducedfluctuation in the angular velocity at steady state (Figure 5(b)) and rapidincrease and saturation of the mean rotational coherence φ c and angular ve-locity < | ω | > variables for increasing flux (Figure 5(c)). The decrease of16 a) Y ( μ m ) X ( μ m) D d ( μ m / s ) T i m e ( s ) -1 C D d ( μ m /s) (b) ω ( s - ) -1 (d) D d ( μ m / s ) -10 15 -15 -15 4 -4 -5 300 -4 6 -6 8 -2 10 -10 Time (s) -6 -10 -15 ω ( s - ) ρ d ( μ m -1 ) (c)(e) D d ( μ m /s)D d ( μ m /s) (f) ρ d ( μ m -1 ) (g) -1 f s w i t c h ( s - ) D d μ m /s = D d = 10 μ m /s ρ (motors/ μ m) Initial Final d ρ (motors/ μ m) d d ( ρ μ m -2 ) Figure 6:
Diffusion of dynein restores the uniform redistribution atthe cortex essential for the sustained rotation. (a) The aster trajec-tories over the last 30 s and (b) the instantaneous angular velocity ( ω ) intime is plotted for varying dynein diffusivity, D d (column) and dynein den-sity, ρ d (row) . (c) The switch frequency over the entire trajectory is plottedas a function of dynein diffusion coefficient. Colors indicate varying dyneindensities. The ensemble average over last 30 s for (d) mean angular velocity( (cid:104)| ω |(cid:105) ) and (e) mean coherence measure ( φ C ) is plotted for increasing valuesof dynein diffusion coefficient,( D d ). Distribution of dyneins at the cortexfrom a single run at the start (red) and end (gray)of the simulation for lowand high dynein density is represented for (f) low and (g) high dynein diffu-sivity. N a = 20, J = -0.3 µ m/s, ρ k = 10 motors/ µ m . N runs = 5. Error barsindicate SEM.switching frequency ( f switch ) with increasing magnitude of J further confirmsthe role that MT dynamics appears to play in reinforcing coherent motion17Figure 5(d)). This feature of our model of increase in rotational order as afunction of increasing noise diverges from the Viscek-type where increasing‘noise’ decreases the order of collective motion [77]. The difference couldrelate to the nature of ‘noise’, since in our model it only indirectly affectsmobility, while in SPP models ‘noise’ increases the randomness of particlemotion.A higher degree of MT length flux likely results in higher turnover ofbinding events, and a more equal distribution of cortical force generators,dyneins. This is the likely cause for the increased coherence in the system.Another means by which cortical forces could be redistributed is the mobilityof dynein. As a result we proceed to test whether the diffusive mobility ofcortical dyneins plays a role in the collective transport of asters. Inside cells, a fraction of dynein is bound to the cell cortex in dynamicclusters with a rapid turnover due to unbinding and diffusion [46]. We modelthis mobility of dyneins through diffusion of the motors along the corticalregion determined by an effective diffusion coefficient of dynein ( D d ). Inorder to test whether this diffusive mobility has any effect on the aster-mobility, we tested the effect of varying D d in a simulated cell with optimalkinesin-5 density and cell size. In continuation with the idea that a minimaldensity of dynein is required for the onset of rotations, we also varied ρ d ,the density of dynein. We find, decreasing dynein diffusivity from 10 to 0 µ m /s resulted in complete abolition of rotational motion even when ρ d wasgreater than the threshold density required for rotation (Figure 6(a)). Inother words, the presence of dynein alone is not sufficient to drive rotationalmotion, but also requires diffusive redistribution. Increasing D d above thethreshold resulted in a saturation of coordinated motility as quantified byangular velocity (Figure 6(b)), with the obvious absence of dynein abrogatingaster motility altogether. We observe a diffusion ( D d ) dependent phase-transition like behaviour in the switching frequency (Figure 6(c)), the steadystate angular velocity (Figure 6(d)) and the coherence in rotational orderparameter φ C (Figure 6(e)).This diffusion-limited behaviour of asters arises from the uniformity offree dyneins. When D d is below the threshold, dyneins do not adequatelyredistribute when unbound from MTs, resulting in formation of clusters ir-respective of dynein density (Figure 6(f)) while diffusive dynein results in18igure 7: Aster density dependent multi-aster motility patterns. (a)The aster trajectories over the last 30 s and (b) instantaneous angular velocity( ω ) in time is plotted for varying aster density, ρ a (column) and dyneindiffusion coefficient, D d (row) . (c) The switch frequency over the entiretrajectory is plotted as a function of aster density. Colors indicate dyneindiffusion coefficient. The ensemble average over last 30 s for (d) angularvelocity and (e) measure of coherence in rotation is plotted for increasingvalues of aster density (X-axis, log scale). The gray shaded area in (c-e)corresponds to data with 1 aster in the cell. ρ d = 10 motors/ µ m, ρ k = 10motors/ µ m , J = -0.3 µ m/s. N runs = 5.a steady state unform distribution (Figure 6(g)). Taken together, it sug-gests that homogeneity in dynein distribution at the cortex is essential foruniform pulling that can sustain sliding and drive collective motion. Sinceboth kinesin and dynein act to transport the MT asters, we proceeded toask whether aster density is likely to play a major role in collective motility,based on the predicted role for particle density in SPP models . In SPP models, collective ordered transport of particles emerges when thestrength of local coupling and particle density are both optimal. In our19 ( μ m / s ) ρ (asters/ μ m ) -2 -1 -1 Motors/aster d a Figure 8:
Motility patterns dependent on motor numbers anddynein distribution.
The effect of aster density ρ a and dynein diffusioncoefficient D d was based on the calculated average angular velocity ( ω ), co-herence in rotation ( φ c ) and frequency of switching ( f s ) into three clusters(Figure S1). The three clusters correspond to qualitatively distinct forms ofmobility: sustained, coherent and persistent rotation (green) , coherent ro-tation with variable persistence (red) and a lack of rotation (blue) , as seenin the representative XY trajectories. The dashed-line is a guide to the eyeseparating the clusters. Cell size R = 11 µ m, cortical dynein density is 100motors/ µ m and cytoplasmic kinesin-5 density is 10 motors/ µ m .multi-aster systems, while asters mechanically interact with the boundary,for coupling we require kinesin-5 activity while dynein acts at the bound-ary. Rotation onset was observed only at an optimal value of aster densitywhen N A were varied from 1 to 114, keeping all other conditions optimal forcoherent aster rotations (Figure 7 (a,b), middle row). Both high and lowaster asters ρ a failed to produce rotation. On the other hand, due to thediffusion limitation of dynein when D d was varied we found the patterns didnot change above a threshold value of dynein diffusion. Indeed for D d > µ m /s, the system undergoes collective rotation for an optimal range of asterdensity, attaining minimal switching transitions (Figure 7(c)) and maximalvelocity (Figure 7(d)) and rotational coherence (Figure 7(e)). The observeddensity dependence deviates from the kinetic phase transitions in the SPPmodels, which we understand as resulting from insufficient number of forcegenerators required to drive large-scale rotation at high densities, and too20ew particle interactions at low densities. This is consistent with restorationof rotation when the dynein density is increased by ten fold (data not shown).Thus, we observe that collective rotation is limited by aster density, wherethe low density effects arise from lack of coupling, while at high densities therelative number of motors per asters play a role.Our model therefore predicts a complex set of components and behaviourthat are predicted to result in coherent, collective rotational transport ofasters involving density of motors and asters, the stochasticity of MTs andis diffusion-limited in terms of cortical dynein.Figure 9: Determinants of cortical dynein mobility.
Dynein (green)mobility at the cortex can be the result of multiple forms of anchorage: (a)immobilized on a rigid cortex (grey), (b) unbind (arrow), diffuse in the cyto-plasm and bind at another location (arrow), (c) diffuse in a lipid membrane(bidirectional arrow), (d) coupled to a stretchable linker (spring) that is em-bedded in the membrane, (e) attached via an adapter protein (box) thatcross-link with the actin cortex or (f) linker or actin cross-linker are activelytransported by actin flows (red arrow).21
Discussion and Conclusions
The emergence of self-organized multi-aster swarming in confinement is com-parable to a wide range of biological systems that span several scales- frommolecules, through bacterial populations to large animal swarms. The spe-cific properties of molecular motors and their critical role in cell physiology,growth and division makes their study particularly important The role of ra-dial MT arrays seen here in the spontaneous emergence of patterns dependscritically on four factors: (a) rigid boundary resulting in bending of MTs, (b)diffusively redistributed cortical dynein forces that generate a tug-of-war ofMTs which when resolved produces circumferential movement, (c) forces oflocal coalescence and separation of asters driven by kinesin-5 like motors pro-ducing coupling and (d) stochasticity due to filament polymerization kineticsthat break the symmetry of the system. Dynein at the cortex is critical forsustained rotation with kinesin-5 resulting in local coupling that enhancescoherence of rotation. Counter-intuitively increased MT lengths increasesrotational persistence, in effect due to increased ‘noise’. A critical density ofasters is required for the persistence of steady-state rotation, with both low-and high-density limits resulting in a loss of rotation.In order to summarize the range of aster density and diffusion limitationof cortical dynein in determining rotational onset, we cluster the angularvelocity ω , rotational coherence φ c and frequency of switching f s into threeclusters (Figure S1). We find this clustering produces three regions in thephase plane of dynein diffusion D d and aster density ρ a – a region of sustainedand persistent rotation, one completely lacking rotation and a transition zonebetween the two (Figure 8). This picture of the onset of collective motion,dependent on density and noise resembles the SPP reported by Vicsek et al.[78]. However, we observe coherent rotation requires an optimal aster density,in contrast to the onset of ordered collective motility or swarming above acertain particle density. This reversal in trends compared to SPP modelscould arise from the discrete nature of coupling in our model, inspired bythe need for mechanistic detail compared to the simplifying implicit couplingin SPP models. In particular at high particle (aster) density, the kinesin-5motors per aster become limiting, as depicted in Figure 8. An additionaldifference to general SPP models is the diffusion-limitation of cortical dyneinmobility, again arising from the discrete and physically realistic model ofboundary forces. Thus our model we believe predicts that physically inspireddetails of collective transport by local coupling can also result in qualitatively22ifferent behaviour than that predicted by the general SPP models. Thephysical detail in our model has the advantage over a more abstract modelof the potential to test our predictions in experiment.The role of cortical dynein is critical in generating the rotational compo-nent of motion due to MT binding and dynein motility. Therefore, we believeit is important to consider the behavior of cortical dyneins in the context ofreported in-vivo interactions in cells. For simplicity we consider the dynein tobe bound to the cortical membrane throughout the simulation. Mobility ofdynein is dependent on whether the motor is bound to MTs. When bound toMTs, the motors are dragged based on the motor stretch, independent of themotor stepping, while free motors undergo 1D diffusion in the cell boundarycorresponding to the cortex. The model of cortical dynein mobility is basedon previous reports based on dynamic microscopy of MT associated dyneinspeckles consisting of multiple molecules were found at the cell cortex witha high turnover over seconds time-scales [46]. In cells of the fission yeast Schizosaccharomyces pombe , cortical dyneins bound to MTs detach from thecortex and MTs, diffuse in cyotosol and eventually bind at an another cor-tical location [79, 4]. Such mobility is distinct from in vitro gliding assayswith motors immobilized on the glass cover-slip at anchored point (Figure9(a)). Additionally, MTs have been seen to themselves mediate dynein lo-calization and re-distribution at the cortex [46, 44, 70], that has motivatedmodels of redistribution of the motor at the cortex (Figure 9(b)). Addition-ally, in vitro
MT-motor gliding assays with motors anchored in supportedmembrane bilayers suggest the collective transport properties change dueto motor ‘slippage’ and diffusion [51, 21]. We the reported motor diffusioncoefficients in lipids for our model of dynein diffusion (Figure 9(c)), basedon reports for kinesins and myosins [37, 21, 51]. Additionally, lipids mobil-ity itself has been demonstrated to affect motor distribution as seen in caseof kinesin motors that cluster along MTs by lateral diffusion of membranelipids in a gliding assay, in the absence of motor activity [40]. However, evi-dence from biochemical interactions and localization studies suggest dynein isbound to adaptor proteins which in turn are membrane bound such as Num1and dynactin [3], mcp5 [74] or ternary complexes such as NuMA/LGN/G α i[12, 33, 52]. Therefore a future improvement to models of cortical dyneinmechanics would also need to take into account adaptor mechanics (Figure9(d)). Actin networks that line the cell cortex, on the cytoplasmic side of theinner membrane of most animal cells, result in hindered diffusion of mem-brane proteins anchored in the plasma membrane, like for example GPCRs2330, 29]. The resulting ‘corralling’ of receptors by an actin meshwork mod-ifies the hopping probability resulting in local clustering and aggregation ofreceptors [14] and hindering the recruitment of dynein at the cortex [58].Therefore the effect of spatial heterogeneity in dynein mobility could furtherimprove our ability to predict localization patterns observed in specific celltypes (Figure 9(e)). Actin-associated proteins regulate the distribution ofdynein anchors at cortex during spindle positioning in cells [33, 58] and actinflows result in direct displacement of dyneins along the cortex in C. elegans embryos [13], suggesting further details of cellular mechanics could allow fora direct comparison to in vivo experimental dynamics (Figure 9(f)). Inte-grating the insights obtained from in vivo single-molecule imaging, in vitro reconstitution of lipid-motor [40] and lipid-actin [25] systems, will go a longway to improve the quantitative precision and qualitative match between themodel predictions and experiments in order to provide insights into the roleof cortical dynein MT aster positioning in cells.The absence of more widespread spontaneous rotation of asters in ani-mal cells suggests that biological systems may have evolved mechanisms tosuppress rotations. In our model dynein must be diffusively redistributed atthe cortex for sustained aster rotation to occur, in order to produce spatiallyhomogeneous pulling forces throughout the cortex. Once motors are boundto MTs, they walk towards the minus end, resulting in a local clustering atmultiple locations where MTs contact the cortex. Upon unbinding from theMT dynein diffusion results in dissipation of any clustering. On the otherhand experimental studies from cells indicate cortical dyneins are localizedmulti-protein clusters at the cortex [46, 45, 74, 4, 52]. Additional sources ofspatial heterogeneity in dynein distribution could be the switching betweenstates of activity by regulators as seen in budding yeast dynein activation andcortical localization by Num1 [62]. Our model also requires lateral sliding ofMTs along the cortex for coherent rotational motility to emerge. However,in experiments the nature of MT-motor interactions depend on cell shape,and may be end-on or lateral, in spherical or cylindrical cell geometries re-spectively [22]. In addition, dynein capture based shrinkage at the cortexis reported in yeast [16] and T-cells [82] which may further act as braketo the sustained lateral interactions also discussed in [23]. Thus, multiplemechanisms responsible for precise positioning of spindles could have beenan optimization to suppress the sustained lateral interaction of MTs at thecortex thereby preventing the rotations in the cell.Our study is distinct from the multiple reports of cytoskeletal filament24ortices and sustained flows due to the MT geometry and multiplicity ofasters. In contrast, cytoplasmic streaming in
Drosophila oocytes driven bykinesin-dependent MT transport [61] due to MTs sliding on other, corticallyimmobilized MTs [42], requires linear MTs. More recently, a combination ofexperiment and theory have demonstrated large motor driven MT vortices inspace on either 2D surfaces [65] and in 3D confined compartments with
Xeno-pus extracts [67], both of which were performed using linear MTs. There is aclear lack of studies that examine collective properties of asters.
In vivo
MTasters during the first embryonic division of
Caenorhabditis elegans are seento oscillate due to pulling by cortical dyneins and a force asymmetry arisingfrom MT bundling and motor density. While this has been studied in de-tail in experiment and simulation, the studies typically invoke only a pair ofasters, coupled at the metaphase plate [34], but do not address the propertiesof multiple asters. In contrast, the centering ability of asters based on push-ing at the cell membrane and pulling by cytosolic dynein motors seen in seaurchin embryos [72], suggests such cellular processes are robust to positioningerrors arising from thermal noise. Since our predictions of spatial patternsare based on similar mechanistic components- filaments, diffusible and cor-tically anchored motors and a rigid boundary-arising from the presence ofmany asters and multiple, diffusible motors. The choice of parameters ofmotor and MT mechanics is based on experimentally measured values, withthe aim of predicting the outcome of potential experiments. Indeed recentdevelopments with linear MTs encapsulated in lipid droplets together withmotors, demonstrate a transition from random to astral geometries depen-dent on collective mechanics [9, 28]. In future, the encapsulations of multipleasters could be used to test some of our model predictions.The system described here is based on experimentally reported mod-els and parameters of MTs, asters and the motor mechanics and kineticsof dynein and kinesin-5. Our computational model demonstrates how lo-cal mechanical coupling combined with stochastic fluctuations and circularboundary effects can result in the emergence of coherent swarming motilityof MT asters. The model demonstrates a diffusion limitation of motors, dueto the discrete nature of the force generators. This system transitions from‘flocking’ to ‘swirling’ behaviour, depending on the mobility of the dyneins onthe cell boundary. Thus we believe these results point to a novel biologicallytestable self-organized pattern forming system.25
Acknowledgements
This work was supported by a grant from the Dept. of Biotechnology (DBT),Govt. of India BT/PR16591/BID/7/673/2016 to Chaitanya Athale. Fellow-ships from IISER Pune for integrated PhD, council for scientific research(CSIR) India 09/936(0128)/2015-EMR-1 and project assistantship from aDBT grant BT/PR16591/BID/7/673/2016 supported Neha Khetan. Weacknowledge feedback about the biological motivation of the model fromThomas Lecuit.
Simulation code was based on C++ code of Cytosim, an OpenSource agentbased simulation engine for cytoskeletal mechanics [50]. A typical simulationrun with 20 asters and 7070 kinesin and 9500 dynein motors in a cell of radius15 µ m for 300 s required ≈
200 minutes on a 12 core Intel machine (XeonE5 2630) running Linux (Ubuntu 14.04) with 15.6 GB memory. All dataanalysis and plotting were performed in MATLAB R2017a (Mathworks Inc.,USA). For clustering analysis, custom made Python scripts was written usingscikit-learn package [54]. Heirarchial clustering (k-means) was performedusing angular velocity, coherence order parameter and switching frequencyas features. The silhouette score for 3 clusters of 0.758 was higher than for2 or 4 clusters and used to determine the clusters in variable space (FigureS1). The labels of each cluster were then used to determine colour classes ofaster motility (Figure 8).
10 Supporting Information arameter Description Value Reference f c Frequency of catastrophe 0.049 (s − ),varied [6], this studyf r Frequency of rescue 0.0048 (s − ),varied [6], this studyv g Growth velocity 0.196 ( µ m/s) [6]v s Shrinkage velocity 0.325 ( µ m/s) [6] κ MT bending modulus 20 (N/m ) [19] Aster: N a Number of asters 40, varied This studyN MT Number of MTs/aster 20 This studyTable SI MT parameters.
The microtubule and aster parameters usedin simulations were taken from experimental measurements reported in lit-erature, and where missing estimated.Table SI Transition frequencies and MT flux rates.
The parametersof MT dynamic instability: frequency of catastrophe (f c , s − ) and frequencyof rescue (f r , s − ) was varied to obtain different MT flux rates (J) at constantvalue of growth velocity (v g = 0.196 µ m/s), shrinkage velocity (v s = 0.325 µ m/s) and mean MT length (L MT = 4.25 µ m). The values were reportedpreviously in [76, 6]. f c (s − ) f r (s − ) J ( µ m/s) Stabilized:
None None 0
Dynamic: ( d e g r ee / s ) f s ( / s ) Figure S1:
Clusters of motility patterns for varying number of mo-tors per aster and degree of uniformity at the cortex.
The parameterspace for varying aster density and dynein diffusion coefficient was segregatedinto clusters using k-means clustering based on the measures of average an-gular velocity ( ω ), coherence in rotation ( φ c ) and frequency of switching ( f s ).The circles correspond to points in the parameter space and color representsthe clusters: (green) sustained, (red) coherent and persistent rotation and (blue) for coherent rotation with variable persistence and a lack of sustainedrotation . The ’stars’ indicate the centroid of the clusters. Cell size R =11 µ m, cortical dynein density = 100 motors/ µ and cytoplasmic kinesin-5density = 10 motors/ µ m .
12 Supplemental VideosReferences [1] F Alaimo, S Praetorius, and A Voigt. A microscopic field theoreticalapproach for active systems.
New Journal of Physics , 18(8):083008,28ideo SV1: The effect of (top row) cortical force generators, (mid-row) ki-nesin complexes and (bottom row) both kinds of motors on MT asters withthe two columns simulating MTs lengths as (left) fixed (right) dynamicallyunstable. The video corresponds to results depicted in Figure 1. Time is 300s, N a = 20, ρ d = 100 motors/ µ m, ρ k = 10 motors/ µ m , J = 0 and - 0.3 µ m/sfor stabilized and dynamic MTs.2016.[2] Steven J Altschuler, Sigurd B Angenent, Yanqin Wang, and Lani F Wu.On the spontaneous emergence of cell polarity. Nature , 454(7206):886–9,aug 2008.[3] V. Ananthanarayanan. Activation of the motor protein upon attach-ment: Anchors weigh in on cytoplasmic dynein regulation.
Bioessays ,38(6):514–525, 06 2016.[4] V. Ananthanarayanan, M. Schattat, S. K. Vogel, A. Krull, N. Pavin,and I. M. Toli?-N?rrelykke. Dynein motion switches from diffusive todirected upon cortical anchoring.
Cell , 153(7):1526–1536, Jun 2013.295] Y. Asano, A. Nagasaki, and T. Q. Uyeda. Correlated waves of actinfilaments and PIP3 in Dictyostelium cells.
Cell Motil. Cytoskeleton ,65(12):923–934, Dec 2008.[6] C A Athale, A Dinarina, M Mora-Coral, C Pugieux, F Nedelec, andE Karsenti. Regulation of microtubule dynamics by reaction cascadesaround chromosomes.
Science , 322:1243–1247, 2008.[7] C. A. Athale, A. Dinarina, F. Nedelec, and E. Karsenti. Collectivebehavior of minus-ended motors in mitotic microtubule asters glidingtoward DNA.
Phys Biol , 11(1):016008, Feb 2014.[8] F Backouche, L Haviv, D Groswasser, and A Bernheim-Groswasser. Ac-tive gels: dynamics of patterning and self-organization.
Physical Biology ,3(4):264–273, 2006.[9] H. Baumann and T. Surrey. Motor-mediated cortical versus astral mi-crotubule organization in lipid-monolayered droplets.
J. Biol. Chem. ,289(32):22524–22535, Aug 2014.[10] P M Bayley, M J Schilstra, and S R Martin. A simple formulation ofmicrotubule dynamics: quantitative implications of the dynamic insta-bility of microtubule populations in vivo and in vitro.
J Cell Sci , 93 (Pt 2):241–54, Jun 1989.[11] Scott Camazine, James Sneyd, Michael J. Jenkins, and J.D. Murray. Amathematical model of self-organized pattern formation on the combsof honeybee colonies.
Journal of Theoretical Biology , 147(4):553 – 571,1990.[12] E. S. Collins, S. K. Balchand, J. L. Faraci, P. Wadsworth, and W. L.Lee. Cell cycle-regulated cortical dynein/dynactin promotes symmetriccell division by differential pole motion in anaphase.
Mol. Biol. Cell ,23(17):3380–3390, Sep 2012.[13] A De Simone, A Spahr, C Busso, and P G¨onczy. Uncovering the balanceof forces driving microtubule aster migration in c. elegans zygotes.
NatCommun , 9(1):938, 03 2018.[14] Sneha A Deshpande, Aiswarya B Pawar, Anish Dighe, Chaitanya AAthale, and Durba Sengupta. Role of spatial inhomogenity in gpcr30imerisation predicted by receptor association-diffusion models.
PhysBiol , 14(3):036002, 05 2017.[15] Hajer Ennomani, Gaelle Letort, Christophe Guerin, Jean-Louis Mar-tiel, Wenxiang Cao, Francois Nedelec, Enrique M. De La Cruz, ManuelThery, and Laurent Blanchoin. Architecture and Connectivity GovernActin Network Contractility.
Current Biology , 26(5):616–626, 2016.[16] Cassi Estrem, Colby P Fees, and Jeffrey K Moore. Dynein is regulatedby the stability of its microtubule track.
J Cell Biol , 216(7):2047–2058,07 2017.[17] P. Friedl, P. B. Noble, P. A. Walton, D. W. Laird, P. J. Chauvin, R. J.Tabah, M. Black, and K. S. Zanker. Migration of coordinated cell clus-ters in mesenchymal and epithelial cancer explants in vitro.
Cancer Res. ,55(20):4557–4560, Oct 1995.[18] R. Gibeaux, A. Z. Politi, P. Philippsen, and F. Nedelec. Mechanism ofnuclear movements in a multinucleated cell.
Mol. Biol. Cell , 28(5):645–660, Mar 2017.[19] Frederick Gittes, Brian Mickey, Jilda Nettleton, and Johnathon Howard.Flexural Rigidity of Microtubules and Actin Filaments Measured fromThermal Fluctuations in Shape.
J. Cell Biol. , 120(4):923–934, 1993.[20] Stephan W Grill, Jonathon Howard, Erik Sch¨affer, Ernst H K Stelzer,and Anthony a Hyman. The distribution of active force generators con-trols mitotic spindle position.
Science (New York, N.Y.) , 301(5632):518–521, 2003.[21] R. Grover, J. Fischer, F. W. Schwarz, W. J. Walter, P. Schwille, andS. Diez. Transport efficiency of membrane-anchored kinesin-1 motorsdepends on motor density and diffusivity.
Proc. Natl. Acad. Sci. U.S.A. ,113(46):E7185–E7193, 11 2016.[22] J. Guild, M. B. Ginzberg, C. L. Hueschen, T. J. Mitchison, and S. Du-mont. Increased lateral microtubule contact at the cell cortex is sufficientto drive mammalian spindle elongation.
Mol. Biol. Cell , 28(14):1975–1983, Jul 2017. 3123] G. G. Gundersen. Evolutionary conservation of microtubule-capturemechanisms.
Nat. Rev. Mol. Cell Biol. , 3(4):296–304, Apr 2002.[24] Terrell L. Hill.
Linear Aggregation Theory in Cell Biology . Springer,1987.[25] A. Honigmann, S. Sadeghi, J. Keller, S. W. Hell, C. Eggeling, andR. Vink. A lipid bound actin meshwork organizes liquid phase sepa-ration in model membranes.
Elife , 3:e01671, Mar 2014.[26] Johnathon Howard.
Mechanics of Motor Proteins and the Cytoskeleton .Sinauer Associates, Sunderland, 2001.[27] K. Jain, N. Khetan, and C. A. Athale. Collective effects of yeast cyto-plasmic dynein based microtubule transport.
Soft Matter , 15(7):1571–1581, Feb 2019.[28] M. P. N. Juniper, M. Weiss, I. Platzman, J. P. Spatz, and T. Surrey.Spherical network contraction forms microtubule asters in confinement.
Soft Matter , 14(6):901–909, Feb 2018.[29] Rinshi S Kasai and Akihiro Kusumi. Single-molecule imaging revealeddynamic gpcr dimerization.
Curr Opin Cell Biol , 27:78–86, Apr 2014.[30] Rinshi S Kasai, Kenichi G N Suzuki, Eric R Prossnitz, Ikuko Koyama-Honda, Chieko Nakada, Takahiro K Fujiwara, and Akihiro Kusumi. Fullcharacterization of gpcr monomer-dimer dynamic equilibrium by singlemolecule imaging.
J Cell Biol , 192(3):463–80, Feb 2011.[31] N. Khetan and C. A. Athale. A Motor-Gradient and Clustering Modelof the Centripetal Motility of MTOCs in Meiosis I of Mouse Oocytes.
PLoS Comput. Biol. , 12(10):e1005102, Oct 2016.[32] M. J. Korneev, S. Lakamper, and C. F. Schmidt. Load-dependent releaselimits the processive stepping of the tetrameric Eg5 motor.
Eur. Biophys.J. , 36(6):675–681, Jul 2007.[33] S. Kotak. Mechanisms of Spindle Positioning: Lessons from Worms andMammalian Cells.
Biomolecules , 9(2), 02 2019.3234] Cleopatra Kozlowski, Martin Srayko, and Francois Nedelec. CorticalMicrotubule Contacts Position the Spindle in C. elegans Embryos.
Cell ,129(3):499–510, 2007.[35] H Kramers. Brownian motion in a field of force and the diffusion modelof chemical reactions.
Physica , 7(4):284–304, 1940.[36] Liedewij Laan, Nenad Pavin, Julien Husson, Guillaume Romet-Lemonne, Martijn van Duijn, Magdalena Preciado L´opez, Ronald DVale, Frank J¨ulicher, Samara L Reck-Peterson, and Marileen Dogterom.Cortical Dynein Controls Microtubule Dynamics to Generate PullingForces that Position Microtubule Asters.
Cell , 148(3):502–14, 2012.[37] C. Leduc, O. Campas, K. B. Zeldovich, A. Roux, P. Jolimaitre,L. Bourel-Bonnet, B. Goud, J. F. Joanny, P. Bassereau, and J. Prost.Cooperative extraction of membrane nanotubes by molecular motors.
Proc. Natl. Acad. Sci. U.S.A. , 101(49):17096–17101, Dec 2004.[38] Cecile Leduc, Nenad Pavin, Frank Julicher, and Stefan Diez. CollectiveBehavior of Antagonistically Acting Kinesin-1 Motors.
Phys. Rev. Lett. ,105(128103):1–4, 2010.[39] J. L¨ober, F. Ziebert, and I. S. Aranson. Collisions of deformable cellslead to collective migration.
Sci Rep , 5:9172, Mar 2015.[40] J. Lopes, D. A. Quint, D. E. Chapman, M. Xu, A. Gopinathan, andL. S. Hirst. Membrane mediated motor kinetics in microtubule glidingassays.
Sci Rep , 9(1):9584, Jul 2019.[41] R. Loughlin, R. Heald, and F. Nedelec. A computational model predictsXenopus meiotic spindle organization.
J. Cell Biol. , 191(7):1239–1249,Dec 2010.[42] W. Lu, M. Winding, M. Lakonishok, J. Wildonger, and V. I. Gelfand.Microtubule-microtubule sliding by kinesin-1 is essential for normalcytoplasmic streaming in Drosophila oocytes.
Proc. Natl. Acad. Sci.U.S.A. , 113(34):4995–5004, 08 2016.[43] Rui Ma, Liedewij Laan, Marileen Dogterom, Nenad Pavin, and FrankJ¨ulicher. General theory for the mechanics of confined microtubuleasters.
New Journal of Physics , 16(1):013018, 2014.3344] S. M. Markus and W. L. Lee. Regulated offloading of cytoplasmic dyneinfrom microtubule plus ends to the cortex.
Dev. Cell , 20(5):639–651, May2011.[45] S. M. Markus, K. M. Plevock, B. J. St Germain, J. J. Punch, C. W.Meaden, and W. L. Lee. Quantitative analysis of Pac1/LIS1-mediateddynein targeting: Implications for regulation of dynein activity in bud-ding yeast.
Cytoskeleton (Hoboken) , 68(3):157–174, Mar 2011.[46] T. Mazel, A. Biesemann, M. Krejczy, J. Nowald, O. Muller, andL. Dehmelt. Direct observation of microtubule pushing by corticaldynein in living cells.
Mol. Biol. Cell , 25(1):95–106, Jan 2014.[47] F. Nedelec. Computer simulations reveal motor properties generatingstable antiparallel microtubule interactions.
J. Cell Biol. , 158(6):1005–1015, 2002.[48] F Nedelec, T Surrey, and A C Maggs. Dynamic concentration of motorsin microtubule arrays.
Phys. Rev. Lett. , 86:3192–3195, 2001.[49] F. J. Nedelec, T. Surrey, A. C. Maggs, and S. Leibler. Self-organizationof microtubules and motors.
Nature , 389(6648):305–308, Sep 1997.[50] Francois Nedelec and Dietrich Foethke. Collective langevin dynamics offlexible cytoskeletal fibers.
New Journal of Physics , 9(11):427, 2007.[51] S. R. Nelson, K. M. Trybus, and D. M. Warshaw. Motor couplingthrough lipid membranes enhances transport velocities for ensemblesof myosin Va.
Proc. Natl. Acad. Sci. U.S.A. , 111(38):E3986–3995, Sep2014.[52] M. Okumura, T. Natsume, M. T. Kanemaki, and T. Kiyomitsu. Dynein-Dynactin-NuMA clusters generate cortical spindle-pulling forces as amulti-arm ensemble.
Elife , 7, 05 2018.[53] Jacques Pecreaux, Jens-Christian R¨oper, Karsten Kruse, Frank J¨ulicher,Anthony A Hyman, Stephan W Grill, and Jonathon Howard. Spindleoscillations during asymmetric cell division require a threshold numberof active cortical force generators.
Current biology : CB , 16(21):2111–22,nov 2006. 3454] F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion,O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vander-plas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duch-esnay. Scikit-learn: Machine learning in Python.
Journal of MachineLearning Research , 12:2825–2830, 2011.[55] N. J. Quintyne, J. E. Reing, D. R. Hoffelder, S. M. Gollin, and W. S.Saunders. Spindle multipolarity is prevented by centrosomal clustering.
Science , 307(5706):127–129, Jan 2005.[56] S. Rausch, T. Das, J. R. Soine, T. W. Hofmann, C. H. Boehm, U. S.Schwarz, H. Boehm, and J. P. Spatz. Polarizing cytoskeletal tension toinduce leader cell formation during collective cell migration.
Biointer-phases , 8(1):32, Dec 2013.[57] M. Reffay, L. Petitjean, S. Coscoy, E. Grasland-Mongrain, F. Amblard,A. Buguin, and P. Silberzan. Orientation and polarity in collectively mi-grating cell structures: statics and dynamics.
Biophys. J. , 100(11):2566–2575, Jun 2011.[58] E. Sanchez, X. Liu, and M. Huse. Actin clearance promotes polar-ized dynein accumulation at the immunological synapse.
PLoS ONE ,14(7):e0210377, 2019.[59] V. Schaller, C. Weber, C. Semmrich, E. Frey, and A. R. Bausch. Polarpatterns of driven filaments.
Nature , 467(7311):73–77, Sep 2010.[60] M Schuh and J Ellenberg. Self-organization of MTOCs replaces cen-trosome function during acentrosomal spindle assembly in live mouseoocytes.
Cell , 130:484–498, 2007.[61] L. R. Serbus, B. J. Cha, W. E. Theurkauf, and W. M. Saxton. Dyneinand the actin cytoskeleton control kinesin-driven cytoplasmic streamingin Drosophila oocytes.
Development , 132(16):3743–3752, Aug 2005.[62] Brina Sheeman, Pedro Carvalho, Isabelle Sagot, John Geiser, DavidKho, M Andrew Hoyt, and David Pellman. Determinants of s. cerevisiaedynein localization and activation: implications for the mechanism ofspindle positioning.
Curr Biol , 13(5):364–72, Mar 2003.3563] Tamar Shinar, Miyeko Mana, Fabio Piano, and Michael J Shelley. Amodel of cytoplasmically driven microtubule- based motion in the single-celled Caenorhabditis elegans embryo.
Proc. Nat. Acad. Sci. USA ,108(26):10508–10513, 2011.[64] S. S. Soumya, A. Gupta, A. Cugno, L. Deseri, K. Dayal, D. Das, S. Sen,and M. M. Inamdar. Coherent Motion of Monolayer Sheets underConfinement and Its Pathological Implications.
PLoS Comput. Biol. ,11(12):e1004670, Dec 2015.[65] Yutaka Sumino, Ken H. Nagai, Yuji Shitaka, Dan Tanaka, KenichiYoshikawa, Hugues Chat´e, and Kazuhiro Oiwa. Large-scale vor-tex lattice emerging from collectively moving microtubules.
Nature ,483(7390):448–452, 2012.[66] T. Surrey, F. Nedelec, S. Leibler, and E. Karsenti. Physical proper-ties determining self-organization of motors and microtubules.
Science ,292(5519):1167–1171, May 2001.[67] Kazuya Suzuki, Makito Miyazaki, Jun Takagi, Takeshi Itabashi, andShin’ichi Ishiwata. Spatial confinement of active microtubule networksinduces large-scale rotational cytoplasmic flow.
Proc. Nat. Acad. Sci.USA , 114(11):2922–2927, 2017.[68] B. Szabo, G. J. Szollosi, B. Gonci, Z. Juranyi, D. Selmeczi, and T. Vic-sek. Phase transition in the collective migration of tissue cells: exper-iment and model.
Phys Rev E Stat Nonlin Soft Matter Phys , 74(6 Pt1):061908, Dec 2006.[69] D. T. Tambe, C. C. Hardin, T. E. Angelini, K. Rajendran, C. Y. Park,X. Serra-Picamal, E. H. Zhou, M. H. Zaman, J. P. Butler, D. A. Weitz,J. J. Fredberg, and X. Trepat. Collective cell guidance by cooperativeintercellular forces.
Nat Mater , 10(6):469–475, Jun 2011.[70] M. A. Tame, J. A. Raaijmakers, B. van den Broek, A. Lindqvist,K. Jalink, and R. H. Medema. Astral microtubules control redistri-bution of dynein at the cell cortex to facilitate spindle positioning.
CellCycle , 13(7):1162–1170, 2014. 3671] D. Taniguchi, S. Ishihara, T. Oonuki, M. Honda-Kitahara, K. Kaneko,and S. Sawai. Phase geometries of two-dimensional excitable waves gov-ern self-organized morphodynamics of amoeboid cells.
Proc. Natl. Acad.Sci. U.S.A. , 110(13):5016–5021, Mar 2013.[72] Hirokazu Tanimoto, Jeremy Sall´e, Louise Dodin, and Nicolas Minc.Physical forces determining the persistency and centring precision ofmicrotubule asters.
Nature Physics , 2018.[73] I. A. Telley, I. Gaspar, A. Ephrussi, and T. Surrey. Aster migrationdetermines the length scale of nuclear separation in the Drosophila syn-cytial embryo.
J. Cell Biol. , 197(7):887–895, Jun 2012.[74] J. M. Thankachan, S. S. Nuthalapati, N. Addanki Tirumala, andV. Ananthanarayanan. Fission yeast myosin I facilitates PI(4,5)P2-mediated anchoring of cytoplasmic dynein to the cortex.
Proc. Natl.Acad. Sci. U.S.A. , 114(13):E2672–E2681, 03 2017.[75] Jan-Willem van de Meent, Idan Tuval, and Raymond E. Goldstein.Nature’s microfluidic transporter: Rotational cytoplasmic streaming athigh p´eclet numbers.
Phys. Rev. Lett. , 101:178102, Oct 2008.[76] F. Verde, M. Dogterom, E. Stelzer, E. Karsenti, and S. Leibler. Controlof microtubule dynamics and length by cyclin A- and cyclin B-dependentkinases in Xenopus egg extracts.
J. Cell Biol. , 118(5):1097–1108, 1992.[77] Tam´as Vicsek, Andr´as Czir´ok, Eshel Ben-Jacob, Inon Cohen, and OferShochet. Novel type of phase transition in a system of self-driven parti-cles.
Phys. Rev. Lett. , 75:1226–1229, Aug 1995.[78] Tam´as Vicsek and Anna Zafeiris. Collective motion.
Physics Reports ,517(3):71 – 140, 2012. Collective motion.[79] Sven K Vogel, Nenad Pavin, Nicola Maghelli, Frank Juelicher, Iva MTolic-Norrelykke, Frank J¨ulicher, and Iva M Toli´c-Nørrelykke. Self-Organization of Dynein Motors Generates Meiotic Nuclear Oscillations.
PLoS Biology , 7(4):918–928, apr 2009.[80] C E Walczak, I Vernos, T J Mitchison, E Karsenti, and R Heald. Amodel for the proposed roles of different microtubule-based motor pro-teins in establishing spindle bipolarity.
Curr. Biol. , 8(16):903–913, 1998.3781] Z. Wu, M. Su, C. Tong, M. Wu, and J. Liu. Membrane shape-mediatedwave propagation of cortical protein dynamics.
Nat Commun , 9(1):136,01 2018.[82] Jason Yi, Xufeng Wu, Andrew H. Chung, James K. Chen, Tarun M.Kapoor, and John A. Hammer. Centrosome repositioning in T cells isbiphasic and driven by microtubule end-on capture-shrinkage.