Aster swarming by symmetry breaking of cortical dynein transport and coupling kinesins

Neha Khetan and Chaitanya A. Athale *
Div. of Biology, IISER Pune, Dr. Homi Bhabha Road, Pashan, Pune 411008, India. E-mail: cathale@iiserpune.ac.in; Tel: +91-20-2590-8050

Received 12th June 2020 , Accepted 3rd August 2020

First published on 6th August 2020


Microtubule (MT) radial arrays or asters establish the internal topology of a cell by interacting with organelles and molecular motors. We proceed to understand the general pattern forming potential of aster–motor systems using a computational model of multiple MT asters interacting with motors in cellular confinement. In this model dynein motors are attached to the cell cortex and plus-ended motors resembling kinesin-5 diffuse in the cell interior. The introduction of ‘noise’ in the form of MT length fluctuations spontaneously results in the emergence of coordinated, achiral vortex-like rotation of asters. The coherence and persistence of rotation require a threshold density of both cortical dyneins and coupling kinesins, while the onset is diffusion-limited with relation to the cortical dynein mobility. The coordinated rotational motion emerges due to the resolution of a ‘tug-of-war’ of multiple cortical dynein motors bound to MTs of the same aster by ‘noise’ in the form of MT dynamic instability. This transient symmetry breaking is amplified by local coupling by kinesin-5 complexes. The lack of widespread aster rotation across cell types suggests that biophysical mechanisms that suppress such intrinsic dynamics may have evolved. This model is analogous to more general models of locally coupled self-propelled particles (SPP) that spontaneously undergo collective transport in the presence of ‘noise’ that have been invoked to explain swarming in birds and fish. However, the aster–motor system is distinct from SPP models with regard to the particle density and ‘noise’ dependence, providing a set of experimentally testable predictions for a novel sub-cellular pattern forming system.


1 Introduction

Self-organized pattern formation is observed almost universally in biological systems ranging in scale from large scale structures of swarming birds and fish,1 through cells undergoing collective migration patterns,2–7 to single-cell polarization by reaction–diffusion networks of proteins.8–10 The self-organized patterns of the mechanical elements of the cell, the cytoskeleton and molecular motors are particularly distinct, arising as they do from purely mechanical interactions. Of these the most distinct are the in vitro patterns reported from the reconstitution of ATP containing mixtures of microtubules (MT) and motors11–14 and actin and myosin.15,16 MT–motor activity in a circular boundary has been shown to break symmetry and result in vortex like motility.11,17 Evidence that such vortices are not just restricted to minimal in vitro reconstituted systems has come from in vivo studies demonstrating MT–motor driven cytoplasmic streaming in Caenorhabditis elegans embryos18 and Drosophila oocytes19 during development and the cells of the plant Chara.20 However, the range of motility patterns seen in these structures is specific to the geometry of the cell type and specific mix of motors. In order to understand the general principles of such pattern formation, theoretical models that take into consideration a wider range of cell and filament geometries are required.

A general model of self propelled particle (SPP) motion describing the emergence of collective motion or swarming based on local coupling interactions and ‘noise’ has been described by Vicsek et al.21 While this class of models minimally requires active particles with local interactions and ‘noise’, they do not capture the polarity of filaments and motors, since active particles are typically considered to be point particles. MT filaments have a kinetic polarity of plus- and minus-ends, which determines the direction of motor activity – kinesins walk towards the plus-ends and dyneins towards the minus-ends.22 Detailed models of self-organized patterns of linear filaments have shown good agreement with experiments as seen with MTs in the presence of kinesin11–13 or dynein14 as well as actin with myosin activity.15,16,23 However, inside most animal cells MTs have a characteristic orientation of minus-ends near the nucleus and plus-ends at the cell periphery, forming a mechanical positioning system for organelle transport, cell polarization and cell division. This characteristic organization of MTs is determined by microtubule organizing centers (MTOCs) that serve as nucleation points forming radial arrays or asters. At the same time, most studies of collective MT transport have used linear MTs. The effect of the MT geometry on mobility is seen when comparing the outcome of these two kinds of MTs encountering a sheet of immobilized motors of either plus- or minus-ended type. While linear MTs undergo collective transport and glide, radial asters undergo a tug-of-war due to the geometry as seen with kinesins for 1D doublets24 or dyneins transporting asters.25 The effect of diffusible motors on aster movement is variable and determined by both the motor type – tetrameric kinesins – and MT orientation. Pairs of asters coupled by kinesins bound to anti-parallel MTs will form bipolar spindle-like structures in simulations,26 confirming the importance of kinesin-5 in spindle assembly seen in experiments.27 Such anti-parallel MTs from asters of neighboring spindles in syncytial embryos of Drosophila result in even spacing of spindle asters.28 Parallel MTs, on the other hand, result in ‘zippering’ by movement of motors on parallel MTs as seen with MTOC asters during mouse oocyte spindle assembly.29,30 Dyneins coalesce asters independently of whether MTs are parallel or anti-parallel and the effect is observed of supernumerary centrosome clustering due to dynein.31 In many cells, however, dyneins are immobilized in the cell cortex. Asters contacting these membrane anchored motors result in pulling forces acting on asters, driving them to the cell center in cells with sizes comparable to the asters.32 In multi-nucleate cells such as the filamentous fungus Ashbya gossypii, these astral–MT interactions with cortical dynein are essential for maintaining a regular spacing between nuclei.33 Cortical dynein localization forms the mechanical basis of the asymmetric cell division of one-celled embryos of C. elegans34 and the cortical density of dynein has been shown to determine spindle oscillations.35 This suggests that mechanical interactions of MT asters with diffusible kinesin tetramers and cortical dyneins constitute conserved mechanical modules across a wide variety of cell types. While a general theory for the mechanics of a single aster in a confined cellular geometry has been developed previously,36 a model integrating motor localization seen in diverse cells with multiple asters is lacking.

Here, we have modeled the mechanics of a mixture of multiple MT asters acted on by dyneins at the cell cortex and kinesins in the cytoplasm to examine the potential of this system for collective transport. We test the effect of the forces generated on asters arising from local coupling by cytoplasmic kinesins, gliding of MTs on the circular boundary lined with cortical dyneins and inward pushing due to MT mechanics in confinement and the role of stochastic MT polymerization dynamics. Our model demonstrates that while coupled mechanics alone results in local and uncorrelated aster motility, the addition of ‘noise’ transforms it into coherent rotational motion. This emergence of coherent rotation or swarming of asters depends on cortical dyneins, kinesins and the MT stochasticity. The spontaneous coherent streaming motion of asters predicted by the model is discussed in the context of experimental evidence for the nature of dynein anchoring to the cell cortex.

2 Model

2.1 Components and interactions

We have modeled a multi-aster system in a cellular compartment in the over damped regime with mechanics and stochastic binding kinetics of kinesin and dynein (Fig. 1), based on a previously developed computational agent-based model of MT–motor interactions.37 MTs form radial structures, asters, that interact with one another when they are cross-linked by molecular motors that bind to and walk on MTs. This mechanical coupling of asters at micrometer scales originates from motor-driven forces spanning a few nanometers. The net force experienced by a complex of coupled asters determines whether they are transported or static. Stochasticity in the model originates from three sources: MT length fluctuations, diffusion and the binding kinetics of motors. The mechanical interactions vary according to position, either at (a) the cell cortex or (b) in the cytoplasm, and are described separately.
image file: d0sm01086c-f1.tif
Fig. 1 Model of aster–motor mechanics. (a) The model consists of multiple asters and motors in confinement. (b) Aster MTs are radially nucleated with minus-ends of MTs embedded in the microtubule (MT) organizing center (MTOC) and plus-ends out. MTs can switch between growing (black) and shrinking (gray) states. Motors are modeled as springs with MT binding sites (circles) as seen in case of anchored dynein (blue) and kinesin-5 (red) with two motor domains. (c) MT dynamic instability is modeled with transitions between growth and shrinkage states, while the MTs per aster remain constant. (d) In the cell interior unbound kinesins diffuse and generate forces when bound to a pair of MTs, serving to separate or aggregate asters, based on the MT orientation. (e) At the cortex asters are pushed inwards by the forces of MT bending and growth (FMT), 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.
(a) Cell cortex. Dynein-like motors are modeled as being bound to the cell cortex by their stalk domains and can bind to and walk on MT filaments, resulting in aster transport along the cell boundary. Such movement is comparable to a ‘gliding assay’ described in previous work,25,30,38 with a rotational component due to the circular geometry. MTs, when bound to dyneins or non-specifically encountering the cell-membrane at the cortex, bend and buckle, generating a restoring force that pushes the aster inwards (Fig. 1). In addition, MT-bound motors can also be dragged through the membrane. When multiple motors bind to oppositely oriented MTs of the aster a tug-of-war arises in the MT–motor system. Such a tug-of-war emerges from the radial geometry of asters, as previously described.25 Dyneins that are not bound to MTs can diffuse in the membrane.
(b) Cytoplasm. In the cell interior, diffusible tetrameric kinesin motors are modeled based on the mitotic kinesin-5 complexes. They can bind MTs and walk towards the plus end and when bound to astral MTs from neighboring asters simultaneously they produce forces on asters. The resultant force depends on the MT orientation, parallel or anti-parallel, leading to either ‘coalescence’ or ‘separation’ of asters, respectively (Fig. 1). Model parameters are taken from experimental reports where possible (Table 1).
Table 1 Model parameters. The parameters that determine the mechanics and dynamics of motors are taken from experimental measurements reported in the literature, and where missing are estimated
Symbol Parameter Dynein Kinesin-5 Ref.
D Diffusion coefficient 0–100 μm2 s−1 20 μm2 s−1 48
v 0 Motor velocity 2 0.04 μm s−1 25 and 43
d a Attachment distance 0.02 μm 0.05 μm 25, 30 and 43
r a Attachment rate 12 s−1 2.5 s−1 25, 30 and 43
k Linker strength 100 pN μm−1 100 pN μm−1 25, 30 and 43
f s Stall force 1.75 pN 5 pN 25, 30 and 43
r d Basal detachment rate 1 s−1 0.05 s−1 25, 30 and 43
r d,end Basal end-detachment rate 1 s−1 Immediate 25, 30 and 43
f d Detachment force 0.5 pN 1.6 pN 45, this study


2.2 MT asters

A microtubule organizing center (MTOC) is modeled to nucleate a finite number of MTs in a radial manner forming an aster. MTs are modeled as semi-flexible rods with a bending modulus κ of 20 N m−2, based on previous reports.39 Small forces are exerted on MTs, resulting in bending at the cell boundary. The restoring force is calculated based on Euler beam-theory. If the forces exceed a threshold force the MTs buckle and the force is given by Fb = π2·κ/L2, where L is the filament length and κ is the flexural rigidity of the microtubules.37 The MTs bend either because the aster is held at the rigid cell boundary, or the MTs are bound to multiple dyneins. The MT lengths (LMT) are modeled as either uniform and of fixed length (stabilized) or fluctuating (dynamic instability) with a mean length. We use the mean length of asters from measurements made on Xenopus oocytes40 of 〈LMT〉 = 4.25 μm. The length dynamics are described by the frequencies of catastrophe (fc) and rescue (fr) and velocities of growth (vg) and shrinkage (vs) with values taken from those reported for Xenopus laevis oocyte extracts:40,41fc = 0.049 s−1 and fr = 0.0048 s−1, vg = 0.196 μm s−1 and vs = 0.325 μm s−1. The values of fc and fr are 0.049 s−1 and 0.0048 s−1, respectively, in all calculations unless mentioned.

2.3 Motor mechanics

The motors modeled are of two kinds, cortical dyneins and diffusible kinesin-5 complexes. Cortical dyneins are modeled as single walking motors with a spring-like stalk domain (Fig. 1), the spring constant kd of which determines the stretch force Fs = −kdΔh for a Δh change in the stalk length.25,30 Dyneins are modeled as discrete steppers that step at a constant rate, but with a variable step-size determined by the load. A load-free dynein motor is modeled to take constant sized steps while increasing opposing forces reduce the step size in a piecewise manner based on a previously described model.25,30 Additionally, the anchored dyneins attached to the MT are dragged along the cell-boundary based on the stretch on the motor due to the opposing force originating from the spring stretch force (Fs). The detachment and stall forces are identical (fd = fs) for dynein. Dyneins not attached to the MT diffuse in the membrane with an effective diffusion coefficient Dd, the value of which is varied (Table 1). Kinesin-5 motor complexes are modeled as two motor heads joined by a Hookean spring like connector representing the stalk (Fig. 1), based on the stalk-linked dimer-of-dimers structure of this motor, which walks on two MTs simultaneously, thus coupling them.27 The complexes are diffusible throughout the interior of the cell with an effective diffusion coefficient Dk, the value of which we take to be 20 μm2 s−1 based on typical cytosolic proteins (Table 1). Motors stochastically bind to MTs within a distance da based on an attachment rate of ra. The second motor-domain can similarly bind another MT, independent of the MT orientation. The motor is modelled as a continuous stepper, with the velocity of translocation based on the experimentally measured force–velocity relation of Eg5.42 The bound motor walks on the MT at constant velocity v = v0 in the absence of an opposing load, while in the presence of an opposing load, the velocity reduces linearly with the opposing force image file: d0sm01086c-t1.tif. Here, fs is the stall force and f is the projection of the opposing force (fex) along the MT. In the case of the multi-aster system, a kinesin complex bound to two filaments simultaneously will experience a load, resulting in a restoring force that drives aster movement, as seen in a model with linear filaments in the spindle.43 Both dynein and kinesin-5 motors detach based on Kramers theory44 with a rate rd = r0·e|fex|/fd, where r0 is the load-free basal detachment rate and fd is the detachment force. When a kinesin-5 motor walks to the end of the MT it is modeled to detach immediately.43,45

2.4 MT dynamics

The MT dynamics is modeled based on the two state model of growth and shrinkage, associated with 4 parameters, the filament growth velocity vg and shrinkage velocity vs and two transition frequencies between the two states: fc the frequency of catastrophe (growth to shrinkage transition) and fr the frequency of rescue (shrinkage to growth transition).46,47 Work by Verde et al.40 demonstrated how these parameters relate to the mean MT length (〈LMT〉) as:
 
image file: d0sm01086c-t2.tif(1)
The related variable of flux in MT lengths J is then calculated by:
 
image file: d0sm01086c-t3.tif(2)
which determines whether the length of MTs on average is in the ‘bounded state’ (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 from all these sources (Fig. 1) i.e. a net inward pushing force due to MTs bending at the cortex producing a restoring bending force Fb and MT growth, resulting in a net MT force FMT, a force that pulls the asters to the cell boundary due to dyneins 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 (Fig. 1).

3 Results

3.1 Spontaneous emergence of collective aster rotation

Aster motility is a result of symmetry breaking in forces that arises from a combination of multiple forces: (i) kinesin-5 complexes that either zipper or separate asters based on the orientation of astral MTs, (ii) the bending forces from polymerizing MTs at the cell boundary, (iii) dynein pulling forces at the cell boundary, (iv) stochasticity in astral MT lengths and (v) Brownian forces corresponding to the energy kBT. However, none of these individually have any innate ability to drive directional motion due to the radial symmetry of asters and Brownian motion, as illustrated in Fig. 1. We find that the collective interactions result in three distinct forms of aster patterns that depend on the motors and MT dynamics: (I) centering, (II) hexagonal lattice and (III) spontaneous rotation (Fig. 2). Cortical dyneins exert an outward pulling opposing the inward force generated by MT bending and buckling and when combined with MT dynamics result in the inward force dominating resulting in steady state (I) centering of asters (Fig. 2a). Even though the MT lengths are identical in the dynamic and static cases, the rare long filament bending against the cell membrane produces sufficient inward-forces to overcome the outward pulling due to dynein. Kinesin-5 motors result in a steady state (II) hexagonal lattice resembling molecular crystal arrangements, as a result of kinesin-5 pushing forces on asters due to anti-parallel MTs (Fig. 2b). Dynamic instability abolishes these structures due to fluctuations in the overlap lengths. Cortical dyneins and diffusible kinesin-5 combined result in sliding motion along the circular cell boundary and coupling of asters, respectively, which when combined with ‘noise’ in the form of MT dynamic instability break symmetry and result in steady-state (III) spontaneous rotation, analogous to ‘swarming’ dynamics (Fig. 2c). Based on the role of dynein motors in force generation for the emergence of coherent aster rotational motion, we expected the motor density to be an important parameter and proceeded to test the systematic effect of motor density.
image file: d0sm01086c-f2.tif
Fig. 2 Spontaneous aster rotation. Asters with MTs that were either (left) of fixed length with MT flux J = 0 or (right) stochastically fluctuating in length with J = −0.3 μm s−1. Simulations were run in the presence of one of the motor combinations: (a) diffusible cortical dyneins (blue circles), (b) cytoplasmic tetrameric kinesin-5 motors (red circles) or (c) dyneins and kinesins. The last 30 s of the trajectories of aster centers of the corresponding simulations are plotted (the color bar represents time). Kinesin density 10 motors per μm2, cortical dynein density 102 motors per μm, η = 0.05 N s m−2, Na = 20, NMT per aster = 40, Rcell = 15 μm and total time 300 s. Video S1 (ESI) represents the time-series.

3.2 Dynein density dependent onset of rotation patterns

The localization of cortical dyneins that walk on MTs on a circular boundary is expected to result in a ‘gliding’ transport of filaments, with a strong rotational component. However, due to the radial geometry of asters, MTs from the same aster have been shown to encounter antagoistic forces resulting in a tug-of-war.25 In our model, stochasticity in opposing forces resulting from MT length fluctuations produces an element of randomness that is expected to transiently resolve the tug of war. We find that increasing the density of dynein (ρd) results in increasingly sustained rotation (Fig. 3(a)), confirming the central role of dynein force generation when combined with stochastic MTs and kinesin-5 motors. The time traces of the instantaneous angular velocity (ω) are smaller in magnitude and fluctuate more in the presence of few motors, while increasing numbers of dyneins result in higher amplitudes and fewer variations in angular velocity (Fig. 3(b)). Increasing synchronization between the individual traces is observed for increased motor densities. The pooled frequency distribution of ω at low dynein density is distributed sharply around zero, increases in spread for ρd of 10 motors per μm, is bi-modal for 100 motors per μm and is biased to one side at a high density of 103 motors per μm, indicative of persistent motion (Fig. 3(c)). The switching frequency, fswitch, is calculated as the total number of switch events per unit time and quantifies the persistence in the direction. Consistent with the trend in ω we find that increasing ρd results in decreased fswitch (Fig. 3(d)). Rotating asters are not chiral since the time spent by individual asters moving in clockwise (CW) and counter clockwise (CCW) orientations is equal between multiple iterations. Individual simulations result in a spontaneous choice of orientation and the collective movement is entrained, with no particular preference for CW or CCW motion (Fig. 3(e)). However, due to the shorter sliding events at low densities, the proportion of time spent in either state is comparable, which begins to diverge for increasing dynein densities.
image file: d0sm01086c-f3.tif
Fig. 3 Onset of sustained aster rotation at the threshold dynein density. (a) Aster trajectories over the last 30 s are plotted for varying densities of cortical dynein (ρd = 100 to 103 motors per μm). Color bar: time. (b) The time dependence of the instantaneous angular velocity (ω) for each aster (gray) and the ensemble average (black) are used to plot (c) the frequency distribution of ω for increasing ρd. (d) The switching frequency and (e) total time spent in either the clockwise or counter-clockwise direction is plotted for increasing densities of dynein over the entire trajectory. (f) The mean coherence in rotation (ϕC) and the mean angular velocity (〈|ω|〉) over the last 30 s are plotted as a function of increasing dynein density. Cell radius: 15 μm, Na = 20, the surface density of kinesin is ρk = 101 motors per μm2 and the diffusion coefficient of dynein (Dd) is 10 μm2 s−1, J = −0.3 μm s−1. Nruns = 5. Error bars: standard error.

In order to quantify the emergent collective order in a system of rotating asters, we measure a rotational order parameter ϕR, comparable to previous reports for cells49 and active particles.50ϕR(t) is the mean rotational order parameter at time t over N asters expressed as:

 
image file: d0sm01086c-t4.tif(3)
Here, [v with combining circumflex] is the unit velocity vector and êθi is the unit angular direction vector of the aster. The dot product of these two terms is averaged over the total number of asters (particles) N where i is the index of each aster (particle). ϕR can take values ranging between −1 for counter clockwise motion (CCW) and +1 for clockwise motion, while zero indicates the absence of rotation. Since there is no preference observed for CW or CCW motion, the modulus of the rotational order averaged over time and between multiple asters measures the coherence of motion (ϕC) as described by the expression:
 
image file: d0sm01086c-t5.tif(4)
The time average is taken at steady state between ts the time for onset of rotation (typically 270 s) and total time T (300 s). We observe that the measure of rotational coherence increases with ρd and saturates at high dynein densities, which is mirrored by the mean angular velocity over the last 30 s transitioning from a low to high value at a ρd of ∼100 motors per μm (Fig. 3(f)). Taken together, it suggests that a minimal number of dyneins (10 dyneins per μm) is sufficient for the onset of rotation, while sustained, coherent and persistent rotation emerges only at a higher dynein density (100 dyneins per μm). While cortical dynein is expected to play a role in the onset of rotation, kinesin-5 motors are thought to entrain the collective transport. In order to examine the importance of kinesins in the ordered rotational motion of asters, we test the effect of their density.

3.3 Kinesin-5 numbers dictate the strength of inter-aster interactions

Kinesin-5 motors are local coupling factors between asters that produce a segregating force generated in the absence of dynein and MT dynamic instability (Fig. 2). Thus their role in entraining collective rotational motility is not necessarily intuitive, other than as a mechanism of exerting local coupling forces. Locally, kinesin-5 complexes bound to a pair of MTs from neighboring asters can generate pushing or pulling forces between the asters based on the orientation of the astral MTs to which they are bound – anti-parallel MTs result in asters being pushed apart or segregated, while parallel MTs result in asters being pulled together or ‘zippered’ (Fig. 1(d)). Interestingly a low density of kinesin-5 fails to produce any sign of rotational motion, and only above a threshold motor density does collective rotation emerge (Fig. 4(a)). This is also evidenced by the low mean angular velocity ∼0 below the threshold density (Fig. 4(b)) and the frequent switching due to only local fluctuations (Fig. 4(c)). Only once the threshold density has been crossed do we observe a choice of either CW or CCW rotation (Fig. 4(d)) with a sudden jump in the mean angular velocity and increase and saturation of the measure of rotational coherence (Fig. 4(e)). This vital role of coupling kinesin motors is comparable to the local coupling introduced in SPP models. Here, astral MT overlaps are maintained only when sufficient numbers of kinesin-5 motors continue to remain bound at steady state. This appears to be essential for the onset of coherent multi-aster rotation.
image file: d0sm01086c-f4.tif
Fig. 4 The strength of inter-aster interactions dictates the onset of sustained collective rotation. (a) Representative XY trajectories of asters for varying densities of kinesin-5 motors in the cytoplasm are plotted over the last 30 s. The color encodes the trajectory time. (b) The instantaneous angular velocity, ω, is plotted in time. Gray curves represent individual aster tracks while the ensemble mean is represented in black. (c) The switching frequency over the entire trajectory is plotted as a function of the log of kinesin-5 density. (d) The total time spent in either the clockwise (blue) or counter-clockwise (red) direction is plotted for increasing kinesin-5 density over the entire trajectory. (e) The measure of coherence in rotation, υc averaged over the last 30 s (left y-axis) and the ensemble mean of the magnitude of angular velocity, 〈|ω|〉, over the last 30 s (right y-axis) are plotted as a function of increasing kinesin-5 density. ρd = 102 motors per μm, ρa = 0.03 asters per μm2, Nruns = 3. Error bar indicates the SEM.

The maintenance of local coupling through kinesin depends heavily on a minimal overlap of filaments between pairs of asters. While the aster density can ensure the proximity of MTs, the filaments will need to be stable for sufficient time for kinesin-5 motors to bind and walk. We therefore proceeded to test the sensitivity of the model to the dynamicity of MT lengths.

3.4 Coherence of rotation and the effect of the MT flux

The MT dynamics in vivo differ between interphase when the flux J > 0 corresponding to a state of unbounded growth and slow dynamics and mitosis when MTs are in a bounded state with J < 0 and fluctuations in MT lengths.40 The transition between these states has been attributed to the presence of microtubule associated proteins (MAPs) that bind to and modulate the dynamics of MTs, primarily by regulating the transition rates between growing and shrinking stages. The flux also varies across organisms, which could arise from intrinsic differences in tubulin polymerization or evolutionary differences in MAPs.

We capture this difference in our model by modulating the flux rates J while maintaining constant MT lengths 〈LMT〉 and examine their effect on the patterns. To this end we vary J while keeping 〈LMT〉 and the velocity of growth vg and shrinkage vs constant. Solving eqn (1) and (2) simultaneously for a range of J values, we obtain a range of values of fr and fc, which we use to examine the effect of MT dynamics. We expect the increased magnitude of the flux to affect the stability of overlaps of pairs of MTs, the instantaneous number of motors that can bind to MTs and the bending energy, due to their dependence on instantaneous MT lengths. To our surprise, increasing the magnitude of the flux from 0 to −3 × 10−1 resulted in more persistent rotation of asters (Fig. 5(a)). This is confirmed by the reduced variation in the angular velocity at steady state (Fig. 5(b)) and rapid increase and saturation of the mean rotational coherence ϕc and angular velocity 〈|ω|〉 variables for increasing flux (Fig. 5(c)). The decrease of the switching frequency (fswitch) with increasing magnitude of J further confirms the role that the MT dynamics appears to play in reinforcing coherent motion (Fig. 5(d)). This feature of our model of an increase in rotational order as a function of increasing noise diverges from the Vicsek-type where increasing ‘noise’ decreases the order of collective motion.21 The difference could relate to the nature of the ‘noise’, since in our model it only indirectly affects the mobility, while in SPP models ‘noise’ increases the randomness of particle motion.


image file: d0sm01086c-f5.tif
Fig. 5 Effect of MT flux on persistence of rotation. (a) The aster trajectories over the last 30 s are plotted for increasing values of MT flux (noise). The color encodes the trajectory time. (b) The instantaneous angular velocity, ω, is plotted in time. Gray curves represent individual aster tracks while the ensemble mean is represented in black. (c) The measure of coherence in rotation, ϕc, averaged over the last 30 s is plotted over varying flux values (log scale) on the left y-axis and the ensemble mean 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 trajectory averaged for n = 3 runs (±SEM) is plotted against the MT flux rate J (x-axis, log scale). The inset represents the same data with a linear x-axis. J was varied by modifying the frequencies of catastrophe (fc) and rescue (fr). Here, ρd = 102 motors per μm, ρk = 10 motors per μm2, Na = 20.

A higher degree of MT length flux likely results in a higher turnover of binding events, and a more equal distribution of cortical force generators, dyneins. This is the likely cause of the increased coherence in the system. Another means by which cortical forces could be redistributed is the mobility of dynein. As a result we proceed to test whether the diffusive mobility of cortical dyneins plays a role in the collective transport of asters.

3.5 Dynein diffusion limits symmetry breaking

Inside cells, a fraction of dynein is bound to the cell cortex in dynamic clusters with a rapid turnover due to unbinding and diffusion.51 We model this mobility of dyneins through diffusion of the motors along the cortical region determined by an effective diffusion coefficient of dynein (Dd). In order to test whether this diffusive mobility has any effect on the aster-mobility, we tested the effect of varying Dd in a simulated cell with optimal kinesin-5 density and cell size. In continuation of the idea that a minimal density of dynein is required for the onset of rotations, we also varied ρd, the density of dynein. We find that decreasing the dynein diffusivity from 10 to 0 μm2 s−1 resulted in complete abolition of rotational motion even when ρd was greater than the threshold density required for rotation (Fig. 6(a)). In other words, the presence of dynein alone is not sufficient to drive rotational motion, but it also requires diffusive redistribution. Increasing Dd above the threshold resulted in saturation of coordinated motility as quantified by the angular velocity (Fig. 6(b)), with the obvious absence of dynein abrogating aster motility altogether. We observe a diffusion (Dd) dependent phase-transition like behaviour in the switching frequency (Fig. 6(c)), the steady state angular velocity (Fig. 6(d)) and the coherence in rotational order parameter ϕC (Fig. 6(e)).
image file: d0sm01086c-f6.tif
Fig. 6 Diffusion of dynein restores the uniform redistribution at the cortex essential for sustained rotation. (a) The aster trajectories over the last 30 s and (b) the instantaneous angular velocity (ω) in time are plotted for varying dynein diffusivity Dd (column) and dynein density ρd (row). (c) The switching frequency over the entire trajectory is plotted as a function of the dynein diffusion coefficient. Colors indicate varying dynein densities. The ensemble average over the last 30 s for the (d) mean angular velocity (〈|ω|〉) and (e) mean coherence measure (ϕC) is plotted for increasing values of the dynein diffusion coefficient (Dd). The distribution of dyneins at the cortex from a single run at the start (red) and end (gray) of the simulation for low and high dynein density is presented for (f) low and (g) high dynein diffusivity. Na = 20, J = −0.3 μm s−1, ρk = 10 motors per μm2. Nruns = 5. Error bars indicate the SEM.

This diffusion-limited behaviour of asters arises from the uniformity of free dyneins. When Dd is below the threshold, dyneins do not adequately redistribute when unbound from MTs, resulting in formation of clusters irrespective of the dynein density (Fig. 6(f)), while diffusive dynein results in a steady state uniform distribution (Fig. 6(g)). Taken together, it suggests that homogeneity in the dynein distribution at the cortex is essential for uniform pulling that can sustain sliding and drive collective motion. Since both kinesin and dynein act to transport the MT asters, we proceeded to ask whether the aster density is likely to play a major role in collective motility, based on the predicted role of the particle density in SPP models.

3.6 Critical density of asters required for rotation

In SPP models, collective ordered transport of particles emerges when the strength of local coupling and the particle density are both optimal. In our multi-aster systems, while asters mechanically interact with the boundary, for coupling we require kinesin-5 activity while dynein acts at the boundary. The rotation onset was observed only at an optimal value of the aster density when NA was varied from 1 to 114, keeping all other conditions optimal for coherent aster rotations (Fig. 7(a and b), middle row). Both high and low aster densities ρa failed to produce rotation. On the other hand, due to the diffusion limitation of dynein when Dd was varied we found that the patterns did not change above a threshold value of dynein diffusion. Indeed for Dd > 1 μm2 s−1, the system undergoes collective rotation for an optimal range of aster density, attaining minimal switching transitions (Fig. 7(c)) and maximal velocity (Fig. 7(d)) and rotational coherence (Fig. 7(e)). The observed density dependence deviates from the kinetic phase transitions in the SPP models, which we understand results from an insufficient number of force generators required to drive large-scale rotation at high densities, and too few particle interactions at low densities. This is consistent with restoration of rotation when the dynein density is increased by ten fold (data not shown). Thus, we observe that collective rotation is limited by the aster density, where the low density effects arise from a lack of coupling, while at high densities the relative number of motors per asters plays a role.
image file: d0sm01086c-f7.tif
Fig. 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 are plotted for varying aster density ρa (column) and dynein diffusion coefficient Dd (row). (c) The switch frequency over the entire trajectory is plotted as a function of aster density. Colors indicate the dynein diffusion coefficient. The ensemble average over the last 30 s for the (d) angular velocity and (e) measure of coherence in rotation is plotted for increasing values of aster density (x-axis, log scale). The gray shaded area in (c–e) corresponds to data with 1 aster in the cell. ρd = 102 motors per μm, ρk = 10 motors per μm2, J = −0.3 μm s−1. Nruns = 5.

Our model therefore predicts a complex set of components and behaviour that is predicted to result in coherent, collective rotational transport of asters involving the density of motors and asters and the stochasticity of MTs and that is diffusion-limited in terms of cortical dynein.

4 Discussion

The emergence of self-organized multi-aster swarming in confinement is comparable to a wide range of biological systems that span several scales – from molecules, through bacterial populations, to large animal swarms. The specific properties of molecular motors and their critical role in cell physiology, growth and division make their study particularly important. The role of radial MT arrays seen here in the spontaneous emergence of patterns depends critically on four factors: (a) a rigid boundary resulting in bending of MTs, (b) diffusively redistributed cortical dynein forces that generate a tug-of-war of MTs which when resolved produces circumferential movement, (c) forces of local coalescence and separation of asters driven by kinesin-5 like motors producing coupling and (d) stochasticity due to filament polymerization kinetics that break the symmetry of the system. Dynein at the cortex is critical for sustained rotation with kinesin-5 resulting in local coupling that enhances the coherence of rotation. Counter-intuitively an increase in the magnitude of the MT flux (J) increases the rotational persistence measured by ϕc and decreases the switching frequency fswitch despite the increased ‘noise’. A critical density of asters 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 limitation of cortical dynein in determining the rotational onset, we cluster the angular velocity ω, rotational coherence ϕc and frequency of switching fswitch into three clusters (Fig. S1, ESI). We find that this clustering produces three regions in the phase plane of dynein diffusion Dd and aster density ρa – a region of sustained and persistent rotation, one completely lacking rotation and a transition zone between the two (Fig. 8). This picture of the onset of collective motion, dependent on density and noise, resembles the SPP reported by Vicsek et al.52 However, we observe that coherent rotation requires an optimal aster density, in contrast to the onset of ordered collective motility or swarming above a certain particle density. This reversal in trends compared to SPP models could arise from the discrete nature of coupling in our model, inspired by the need for mechanistic detail compared to the simplifying implicit coupling in SPP models. In particular, at high particle (aster) density the kinesin-5 motors per aster become limiting, as depicted in Fig. 8. An additional difference to general SPP models is the diffusion-limitation of cortical dynein mobility, again arising from the discrete and physically realistic model of boundary forces. Thus our model we believe predicts that physically inspired details of collective transport by local coupling can also result in qualitatively different behaviour than that predicted by the general SPP models. The physical detail in our model has the advantage over a more abstract model of the potential to test our predictions in experiment.


image file: d0sm01086c-f8.tif
Fig. 8 Motility patterns dependent on motor numbers and the dynein distribution. The effect of aster density ρa and dynein diffusion coefficient Dd was classified into three clusters (Fig. S1, ESI†) based on the calculated average angular velocity (ω), coherence in rotation (ϕc) and frequency of switching (fs). The three clusters correspond to qualitatively distinct forms of mobility: sustained, coherent and persistent rotation (green), coherent rotation with variable persistence (red) and a lack of rotation (blue), as seen in the representative XY trajectories. The dashed-line is a guide to the eye separating the clusters. Cell size R = 11 μm, the cortical dynein density is 100 motors per μm and the cytoplasmic kinesin-5 density is 10 motors per μm2.

The role of cortical dynein is critical in generating the rotational component of motion due to MT binding and dynein motility. Therefore, we believe that it is important to consider the behavior of cortical dyneins in the context of reported in vivo interactions in cells. For simplicity we consider the dynein to be bound to the cortical membrane throughout the simulation. The mobility of dynein is dependent on whether the motor is bound to MTs. When bound to MTs, the motors are dragged based on the motor stretch, independent of the motor stepping, while free motors undergo 1D diffusion in the cell boundary corresponding to the cortex. The model of cortical dynein mobility is based on previous reports based on dynamic microscopy of MT associated dynein speckles consisting of multiple molecules found at the cell cortex with a high turnover over second time-scales.51 In cells of the fission yeast Schizosaccharomyces pombe, cortical dyneins bound to MTs detach from the cortex and MTs diffuse in the cytosol and eventually bind at another cortical location.53,54 Such mobility is distinct from in vitro gliding assays with motors immobilized on a glass cover-slip at an anchoring point (Fig. 9(a)). Additionally, MTs have been seen to themselves mediate dynein localization and re-distribution at the cortex,51,55,56 which has motivated models of redistribution of the motor at the cortex (Fig. 9(b)). Additionally, in vitro MT–motor gliding assays with motors anchored in supported membrane bilayers suggest that the collective transport properties change due to motor ‘slippage’ and diffusion.57,58 We use the reported motor diffusion coefficients in lipids for our model of dynein diffusion (Fig. 9(c)), based on reports for kinesins and myosins.57–59 Additionally, the lipid mobility itself has been demonstrated to affect the motor distribution as seen in the case of kinesin motors that cluster along MTs by lateral diffusion of membrane lipids in a gliding assay, in the absence of motor activity.60 However, evidence from biochemical interactions and localization studies suggests that dynein is bound to adaptor proteins which in turn are membrane bound such as Num1 and dynactin,61 mcp562 or ternary complexes such as NuMA/LGN/Gαi.63–65 Therefore a future improvement to models of cortical dynein mechanics would also need to take into account adaptor mechanics (Fig. 9(d)). Actin networks that line the cell cortex, on the cytoplasmic side of the inner membrane of most animal cells, result in hindered diffusion of membrane proteins anchored in the plasma membrane, like for example GPCRs.66,67 The resulting ‘corralling’ of receptors by an actin meshwork modifies the hopping probability, resulting in local clustering and aggregation of receptors68 and hindering the recruitment of dynein at the cortex.69 Therefore the effect of spatial heterogeneity in dynein mobility could further improve our ability to predict localization patterns observed in specific cell types (Fig. 9(e)). Actin-associated proteins regulate the distribution of dynein anchors at the cortex during spindle positioning in cells64,69 and actin flows result in direct displacement of dyneins along the cortex in C. elegans embryos,70 suggesting that further details of cellular mechanics could allow for a direct comparison to in vivo experimental dynamics (Fig. 9(f)). Integrating the insights obtained from in vivo single-molecule imaging and in vitro reconstitution of lipid–motor60 and lipid–actin71 systems will go a long way to improve the quantitative precision and qualitative match between the model predictions and experiments in order to provide insights into the role of cortical dynein MT aster positioning in cells.


image file: d0sm01086c-f9.tif
Fig. 9 Determinants of the cortical dynein mobility. The dynein (green) mobility at the cortex can be the result of multiple forms of anchorage: (a) immobilized on a rigid cortex (gray), (b) unbinding (arrow), diffusing in the cytoplasm and binding at another location (arrow), (c) diffusing in a lipid membrane (bidirectional arrow), (d) coupled to a stretchable linker (spring) that is embedded in the membrane, (e) attached via an adapter protein (box) that cross-links with the actin cortex or (f) the linker or actin cross-linker being actively transported by actin flows (red arrow).

The absence of more widespread spontaneous rotation of asters in animal cells suggests that biological systems may have evolved mechanisms to suppress rotations. In our model dynein must be diffusively redistributed at the cortex for sustained aster rotation to occur, in order to produce spatially homogeneous pulling forces throughout the cortex. Once motors are bound to MTs, they walk towards the minus end, resulting in local clustering at multiple locations where MTs contact the cortex. Upon unbinding from the MT, dynein diffusion results in dissipation of any clustering. On the other hand, experimental studies from cells indicate that cortical dyneins are localized in multi-protein clusters at the cortex.51,54,62,65,72 Additional sources of spatial heterogeneity in the dynein distribution could be the switching between states of activity by regulators as seen in budding yeast dynein activation and cortical localization by Num1.73 Our model also requires lateral sliding of MTs along the cortex for coherent rotational motility to emerge. However, in experiments the nature of MT–motor interactions depends on the cell shape, and may be end-on or lateral, in spherical or cylindrical cell geometries, respectively.74 In addition, dynein capture based shrinkage at the cortex is reported in yeast75 and T-cells,76 which may further act as a brake to the sustained lateral interactions, also discussed in ref. 77. Thus, the diversity of mechanisms responsible for precise positioning of spindles in evolutionarily distinct cell types, could indicate the need to suppress the sustained lateral interactions of MTs at the cortex, thereby preventing spontaneous aster rotation in cells.

Our study is distinct from the multiple reports of cytoskeletal filament vortices and sustained flows due to the MT geometry and multiplicity of asters. In contrast, cytoplasmic streaming in Drosophila oocytes driven by kinesin-dependent MT transport78 due to MTs sliding on other cortically immobilized MTs19 requires linear MTs. More recently, a combination of experiment and theory has demonstrated large motor driven MT vortices in space on either 2D surfaces14 or in 3D confined compartments with Xenopus extracts,17 both of which were performed using linear MTs. There is a clear lack of studies that examine collective properties of asters. In vivo MT asters during the first embryonic division of Caenorhabditis elegans are seen to oscillate due to pulling by cortical dyneins and a force asymmetry arising from MT bundling and motor density. While this has been studied in detail in experiment and simulation, the studies typically invoke only a pair of asters, coupled at the metaphase plate,79 but do not address the properties of multiple asters. In contrast, the centering ability of asters based on pushing at the cell membrane and pulling by cytosolic dynein motors seen in sea urchin embryos80 suggests that such cellular processes are robust to positioning errors arising from thermal noise. Our predictions of spatial patterns are based on similar mechanistic components – filaments, diffusible and cortically anchored motors and a rigid boundary – arising from the presence of many asters and multiple, diffusible motors. The choice of parameters of motor and MT mechanics is based on experimentally measured values, with the aim of predicting the outcome of potential experiments. Indeed, recent developments with linear MTs encapsulated in lipid droplets together with motors demonstrate a transition from random to astral geometries dependent on collective mechanics.81,82 In future, the encapsulation of multiple asters could be used to test some of our model predictions.

5 Conclusions

Our computational model predicts a mechanistic basis for the emergence of ordered motility of multiple asters resulting from a combination of local mechanical coupling due to kinesins, tug-of-war at the cortex due to dynein and stochastic fluctuations of MTs. The onset of collective motion limited by dynein diffusion and a critical density of asters are features that distinguish it from general swarming models, due to the discrete nature of the force generators. The asters transition from local rotational motion to coherent rotation, but the rotation itself is achiral, consistent with the stochasticity that drives coordinated motility. We find that increased ‘noise’ in terms of MT dynamics enhances the coherence of motility. These results point to a novel, biologically testable, self-organized mechanical pattern forming system.

Conflicts of interest

There are no conflicts to declare.

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. Fellowships from IISER Pune for integrated PhD, the Council for Scientific and Industrial Research (CSIR) India 09/936(0128)/2015-EMR-1, a project assistantship from a DBT grant BT/PR16591/BID/7/673/2016 and travel support from SERB ITS/2018/005639 supported Neha Khetan. We acknowledge feedback about the biological motivation of the model from Thomas Lecuit.

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Footnote

Electronic supplementary information (ESI) available. See DOI: 10.1039/d0sm01086c

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