Simulation Studies of Self-Organization of Microtubules and Molecular Motors

Zhiyuan Jia,1, 2 Dmitry Karpeev,2 Igor S. Aranson,3 and Peter W. Bates11Department of Mathematics, Michigan State University, East Lansing, MI 48824

2Mathematics and Computer Science Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, IL 604393Materials Science Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, IL 60439

(Dated: January 17, 2008)

We perform Monte-Carlo-type simulation studies of self-organization of microtubules interact-ing with molecular motors. We model microtubules as stiff polar rods of equal length exhibitinganisotropic diffusion in the plane. The molecular motors are implicitly introduced by specifyingcertain probabilistic collision rules resulting in realignment of the rods. This approximation of thecomplicated microtubule-motor interaction by simple a instant collision allows us to by-pass the“computational bottlenecks” associated with the diffusion and dynamics of the motors and the re-orientation of the microtubules. Consequently, we are able to perform simulations of large ensemblesof microtubules and motors on a very large time scale. This simple model reproduces all importantphenomenology observed in in vitro experiments: formation of vortices for low motor density andray-like asters and bundles for higher motor density.

PACS numbers: 87.16.-b, 05.65.+b, 87.18.Hf

I. INTRODUCTION

Organization of complex networks of long biofilamentssuch as microtubules and actin filaments in the courseof cellular processes and division is one of the primaryfunctions of molecular motors [1]. A number of in vitroexperiments were performed [2–7] to study the inter-action of molecular motors and microtubules energizedby the hydrolysis of adenosine triphosphate in isolationfrom other biophysical processes simultaneously occur-ring in vivo. The experiments clearly demonstrated thatat large enough concentration of molecular motors andmicrotubules, the latter organize into ray-like asters androtating vortices depending on the type and concentra-tion of molecular motors. These experiments spurred nu-merous theoretical studied addressing various aspects ofself-organization of active filaments systems [8–16].

The experiments [4–7] suggested the following qualita-tive picture of motor-filament interaction. After a molec-ular motor has bound to a microtubule at a random po-sition, it marches along it in a definite direction untilit unbinds without appreciable displacement of micro-tubules (since the size of a molecular motor is small incomparison with that of the microtubule). However, if amolecular motor binds to two microtubules (some molec-ular motors (e.g. kinesin) form clusters with at least twobinding sites), it exerts significant torques and forces, andcan change the positions and orientations of the micro-tubules significantly, leading eventually to the onset oflarge-scale ordered patterns.

Small-scale molecular dynamics simulations were per-formed to elucidate the nature of self-organization [4, 5].In these simulations the microtubules were modelled bysemi-flexible rods diffusing in viscous fluids. Molecularmotors were correspondingly modelled by short stiff lin-ear springs with large diffusion coefficient. Once the mo-tor diffuses to within a certain small distance from the in-tersection point of two microtubules, it attaches to them

with a certain probability pon and marches along withvelocity v. The action of the motor is to exert forcesand torques on the microtubules, resulting in their mu-tual displacement and realignment. Then the motor de-taches with a probability poff . To model the dwellingeffects of the motors on the end-points of microtubules,observed for some types of molecular motors, an addi-tional probability pend to leave the end-point was as-signed. The corresponding typical dwelling time tend is ofthe order 1/pend. The simulations in [4, 5] indeed repro-duced certain features of the observed phenomenology,like stability and transition between vortices and asters.However, in this approach many fundamentally differenttime scales had to be simultaneously resolved computa-tionally (e.g. fast diffusion of the motors and very slowpattern formation). As a result, the method is very CPU-intensive, and only a small number of microtubules werestudied numerically, leaving many important questions,such as the nature of the transition and structure of thephase diagram, unanswered.

In Refs. [12, 13] a continuum probabilistic model ofalignment of microtubules mediated by molecular mo-tors was developed. The theory was formulated in termsof a stochastic master equation governing the evolutionof the probability density of microtubules with a givenorientation at a given location. The theory is based on anumber of simple assumptions on the interaction rules be-tween microtubules and molecular motors. In particular,only binary instant interactions of microtubules calledinelastic collisions are considered. These are mediatedby molecular motors in a two-dimensional microtubule-motor mixture of constant motor density. The motors areimplicitly introduced into the model by specifying theprobability of interaction of intersecting microtubules.Despite all the above simplifications of the biological pro-cess of self-organization of the cytoskeleton, the modelreproduced, on a qualitative level, key experimental ob-servations, such as the onset of an oriented (polar) phase

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above a critical density of motors, formation of asters forlarge density of motors and vortices for lower density, di-rect transition towards asters from the isotropic state forlarge dwelling times of the motors at the end of micro-tubules, and a density instability and the onset of bundleformation at very high motor density.

However, due to significant complexity of the derivedstochastic master equation governing the evolution of theprobability density of microtubules, the analysis in Refs.[12, 13] was carried out in a relatively narrow range of pa-rameters, namely, in the vicinity of the orientational in-stability, which allowed rigorous reduction of the stochas-tic master equation to a set of much simpler amplitudeor Ginzburg-Landau type equations for the local coarse-grained density and orientation of microtubules. Thisapproach yields some insights into the self-organizationprocess, but, it obviously has its own limitations.

In this paper we perform Monte-Carlo-type simulationstudies of self-organization of microtubules interactingwith molecular motors. Instead of modelling the self-organization process in all details as was done in Refs.[4, 5], we use simplified interaction rules suggested byour previous works [12, 13]. This simplification allowsthe elimination of fast time scales associated with the dif-fusion and motion of the motors. Consequently, one mayfocus on relevant time and length scales associated withlarge-scale pattern formation and evolution. We studiedvery large ensembles of microtubules and addressed ques-tions related to the structure of the corresponding phasediagram and the transitions between various patterns.In agreement with early experiments, we were able to re-produce aster-like structures for high motor density andvortices for lower density, as well as transitions to bun-dles. Our approach provides direct access to the stochas-tic master equation and obtains insights far beyond theamplitude equations approach. Moreover, our methodprovides an efficient and fast tool for simulation of com-plex biological process of cytoskeleton self-organizationand can be possibly extended to rather different systems,such as anisotropic granular media and systems of self-propelled particles.

II. ESSENTIALS OF THE MODEL

We model microtubules as stiff polar rods of equallength l exhibiting anisotropic diffusion in the plane. Dif-fusion of the rod is characterized by three diffusion coef-ficients, diffusion parallel to the rod orientation D‖, per-pendicular to its orientation D⊥, and rotational diffusionDr. In the following we assume D‖ = 2D⊥ [17] for stiffrods diffusing in viscous fluid.

The key ingredient in the theory proposed in Refs.[12,13] was the approximation of the complicated process ofinteraction of one molecular motor with two microtubulesby a simple instant alignment process, see Fig. 1. We fo-cus on the two dimensional situation, and describe theorientation of microtubules by the planar angles ϕ1,2.

bs

a

L

++

- -

+ +

- -

ϕ ϕ ϕ ϕ1 2 1 2b b

a a

FIG. 1: Schematics of an alignment event (inelastic collision)between two microtubules interacting with one multi-headedmolecular motor. (a) A multi-headed molecular motor clusterattached at the intersection point of microtubules moves frompositive (+) towards negative (−) end of the microtubules.(b) After the interaction, the orientational angles ϕ1,2 andthe corresponding positions of the midpoints R1,2 becomealigned.

The microtubules before the collision posses initial an-gles ϕb

1,2. The action of the molecular motor binding si-multaneously to two microtubules results in their mutualalignment, and the angles after interaction become

ϕa1 = ϕa

2 =ϕb

1 + ϕb2

2. (1)

By analogy with the physics of inelastically collidinggrains, we call this kind of process fully inelastic collision(see e.g. [18]). Such an inelastic collision is a simple andreasonable approximation of the complicated interactionprocess [13], and is, in fact, an effect of simultaneous ac-tion of several motors or motors and static crosslinkingpolymers. An analysis of the interaction of two stiff rodswith one motor shows that the overall change in the anglebetween the rods is rather small: the angle decreases onlyby 25-30 % in average [13]. However, simultaneous ac-tion of a static crosslink, serving as a hinge, and a motormoving along both filaments results in a fast and com-plete alignment of the filaments [19]. This justifies theassumption of fully inelastic and instantaneous collisionsfor the rods interaction. Complete alignment also occursfor the case of simultaneous action of two motors movingin opposite direction, as in experiments on kinesin-NCD(gluththione-S-transferase-nonclaret disjunctional fusionprotein) mixtures [5]. The same is true for two motorsof the same type moving in the same direction but withdifferent speeds, where the variation in speed is due tovariability of motor properties and the stochastic charac-ter of the motion.

We set the molecular motor concentration m to be uni-form in space. This assumption is justified by a largevalue of the motor diffusion Dm compared to the micro-tubule diffusion, Dm ≈ 500D‖. Due to advection of themotors along the microtubules there is some accumula-

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tion of motors at the centers of asters and vortices, seee.g. [7]. However, the motor inhomogeneity has only aquantitative effect on the self-organization process anddoes not affect the qualitative features, such as the mor-phology or the phase diagram and the nature of the tran-sitions [13]. The effects of large-scale motor distribu-tion inhomogeneities can be easily incorporated into ourmodel by modification of the interaction rules accordingto the local density and orientation of microtubules.

The motor concentration m affects the probability ofinteraction between two microtubules, Pint. In the fol-lowing we set Pint ∼ σ2m, where σ2 is the effective in-teraction cross-section. The value of σ ≈ 30−50 nm, theorder of size of a kinesin-type molecular motor.

In addition, we assume that the interaction proba-bility Pint depends on the position of the intersectionpoint, see Figure 1. The intersection position is indi-cated by the signed distance s1,2, − l

2 ≤ s1,2 ≤ l2 , from

the midpoint of each rod. According to Refs. [12, 13],due to the polar nature of the microtubules, the depen-dence of the interaction probability on the intersection(and, correspondingly, attachment position) results in ananisotropic probability kernel in the collision integral inthe master equation. The anisotropy of the kernel, char-acterized by the parameter β, ranging between -1 and 1,depends on the motor dwelling time tend, which is smallfor kinesin-type motors and large for NCD-type motors.In Ref. [13] the relation between kernel anisotropy βand the motor dwell time at the end of the microtubuletend in the limit of tend À 1 was estimated as follows:β ∼ (v − const/tend)/poff , where v is the motor speed,poff is the motor unbinding rate. Thus, one sees thatβ increases with the increase in tend. In contrast, themotor attachment rate pon has a little effect on the ker-nel anisotropy, in agreement with experiments [4, 7]. Asit was shown in Refs. [12, 13], the anisotropy parame-ter controls the transition between asters and vortices; inthe continuum model no vortices were observed for largevalues of the kernel anisotropy.

In order to accommodate the anisotropy effect, i.e.dwelling of the motors at the end of microtubules, inour model we introduce the following dependence of in-teraction probability on the attachment positions:

Pint = P0

(1 + β

s1 + s2

l

). (2)

Here the constant P0 depends on the motor concentrationm, 0 < P0 ≤ 0.5 and the value (and the sign) of β de-pends on the type of motor. We believe that this genericlinear dependence captures on the qualitative level theeffects of kernel anisotropy. Our studies with differentdependencies of the probability Pint on s1,2 yielded qual-itatively similar results.

After the interaction we postulate that not only the an-gles, but also the midpoint positions of the microtubulesR1,2 coincide

Ra1 = Ra

2 =Rb

1 + Rb2

2. (3)

This approximation is reasonable in the case of largedwelling times tend of the motors, which guarantees thatafter the interaction the end points of the microtubuleswill coincide. Then together with the alignment interac-tion this effect will justify the assumption on the align-ment of midpoints as well. A large value of the dwellingtime tend is a reasonable approximation for NCD mo-tors, however tend is small for kinesin-type motors. Aswe will show later, the midpoint alignment assumptionmay produce under some conditions specific effects, suchas layering of the microtubules, or “smectic ordering”[20]. In future work we plan to introduce more realisticrules for the midpoint displacements.

III. ALGORITHM DESCRIPTION

We performed simulations in a two-dimensional squaredomain with periodic boundary conditions. Initially, mi-crotubules were randomly distributed over the domain.At each time step, i.e. from tn to tn+1, the update of po-sitions and orientations of the microtubules is comprisedof one substep processing anisotropic diffusion and onesubstep processing inelastic collision.

The diffusion of rigid rods in a viscous fluid is char-acterized by three diffusion coefficients, parallel D‖, per-pendicular D⊥ and rotational Dr. We used the follow-ing relations between the diffusion coefficients from Kirk-wood’s theory for polymer diffusion in three dimensions,D‖ = 2D⊥, Dr = c

l2 D‖. We used c = 1.5 and l = 0.5 inour simulation [21]. The “diffusion step” is introduced asan anisotropic random walk of the microtubules’ centerposition R = (x, y) and random rotation of its orienta-tion ϕ. The positions and orientations are updated ateach time step as follows:

Rn+1 = Rn + ζ1∆‖Un + ζ2∆⊥Nn

ϕn+1 = ϕn + ζ3∆r (4)

where ζi ∈ (−0.5, 0.5) are three random numbers gen-erated each time, and ∆i =

√2Di∆t, where Di is ei-

ther D⊥ ,D‖ or Dr, and vectors Un = (cos ϕn, sin ϕn),Nn = (− sin ϕn, cos ϕn) are directed along (U) and per-pendicular (N) to the orientation of the microtubule.The timestep was chosen as ∆t = 0.1.

At the “collision step”, after diffusion, we checkwhether any pairs of microtubules intersect, and if sowe locate the intersection points of the microtubules andassign an interaction probability to those pairs accordingto (2). In all of those intersections, some of them aresimple binary intersections, but others may be multipleintersections, i.e., a microtubule intersecting with morethan one other microtubule. Certainly, at a low densityof microtubules binary collisions are more typical. Re-gardless of whether intersections are binary or multiple,we calculate all interaction probabilities and sort themin descending order. Starting with the greatest Pint, we

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a b c

FIG. 2: (Color online) Snapshots illustrating configuration of 6, 000 rods for different motor densities, i. e. different values ofP0. Arrows represent microtubules, circles depict the cores of vortices or asters. (a) vortices, t = 620 , β = 1.0, P0 = 0.08 (lowmotor density); (b) asters, t = 602, β = 0.95, P0 = 0.10, (high motor density); (c) bundles, t = 400, β = 1.0, P0 = 0.15. Seealso [22] for movies # 1,2 illustrating self-organization process.

compare it with a randomly generated number ζ4 ∈ [0, 1].If Pint > ζ4, then we update that pair of microtubulesaccording to the collision rules Eqs.(1) and (3). If eitherof these two microtubules had other intersections, theyare ignored, i.e. these interaction probabilities are set tozero. We then proceed with the next largest interactionprobability, repeating until all have been acted upon.

A. Coarse-grained variables

In the simulations, the rods move freely within the do-main and with fluctuations in both position and orienta-tion of the rods, it is difficult to identify relatively stablepatterns. For this reason, and as an aid for computingdivergence and curl, we impose a square grid on the do-main with mesh length d and introduce a coarse-grainingprocedure to extract observable values, such as local ori-entation τ and local density ρ. Using W to denote thetwo-dimensional position vector of a grid point (Xi, Yj),

we calculate the number of rods N whose midpoint po-sitions are in the box [Xi − d,Xi + d] × [Yj − d, Yj + d].The following coarse-grained functions are employed tocompute τ and ρ at this grid point(Xi, Yj):

τ(W) =∑N

k=1 Φ(|W −Rk|)Uk∣∣∣∑Nk=1 Φ(|W −Rk|)Uk

∣∣∣ρ(W) = N (5)

Here |·| is Euclidean length and Φ is a weighting function.We take

Φ(s) = e−s2

l2 (6)

where l is chosen to be the length of each microtubule. Inthe simulation we also have to include contributions from“image particles” originating from the periodic boundaryconditions.

IV. SIMULATION RESULTS

We applied our model to 6,000 microtubules in a 20×20domain varying parameters P0 and β in a wide rangewith 7,000 time steps in the simulation for each choice of(P0, β). We impose a 40×40 grid on the domain to calcu-late the coarse-grained field. A snapshot was taken every10 iterations and so 700 snapshots were obtained for eachsimulation process. For most of the parameter valueschosen, it took about 300 snapshots (3000 time steps) torelax towards relatively stable large scale patterns, andmore than 500 snapshots (5000 steps) to become station-ary. The movies illustrating typical simulation resultscan be found in [22]. Some simulations clearly showed

a pattern of asters and/or vortices while others resultedin ambiguous patterns. Moreover, the clear-cut distinc-tion between asters and vortices appears to be difficultbecause of fluctuations. To examine the parameter space(P0, β) where there are transition regions between astersand vortices, we have devised a pattern characterizationscheme. The simulation results obtained from the firstfour thousand iterations were ignored as they representtransient states. The details of pattern characterizationprocedure are presented in the Appendix.

Select simulation results are shown in Figs 2, 3, 4 whereD‖ = 0.10 in all simulations. In agreement with the ex-periments [4, 5] and the theoretical models [12, 13], weobtained an isotropic phase for low motor densities (not

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a b

FIG. 3: (Color online) Coarse-grained images corresponding to parameters of Fig.2 for vortices (a) and asters (b). Arrowsrepresent the orientation field τ . The color (grey levels) shows the density ρ, red (bright) corresponds to the maximum of ρ,and blue (dark) to its minimum. See also [22] for movies # 3,4.

shown), and then vortices, transient aster-vortices (struc-tures which resemble vortices near the core and aster farfrom the core), asters, and bundles with gradual increaseof the motor density. Representative snapshots of therod configurations for three different values of the motordensity P0 are shown in Fig. 2 and the two correspond-ing coarse-grained snapshots of them superimposed withthe rod density field are shown in Fig. 3. As is evi-dent from our simulation results, a transition from anisotropic (disordered) phase to an oriented phase hap-pens with the increase in the motor density characterizedby the parameter P0. While due to the small size of thesystem (only 6,0000 particles) we have very strong fluc-tuations in the number of vortices, asters and anti-asters(structures similar to asters but with the opposite orien-tation of microtubules), see Fig. 4, general trend can beidentified: with the increase in the interaction probabil-ity P0 the averaged number of vortices decreases whilethe number of aster increases. For small values of theanisotropy parameter β asters and anti-asters appears tooccur with equal probability. However, with the increasein β the number of anti-asters rapidly decreases whilethe number of aster increases. For very high motor den-sity we observed an additional instability resulting in theformation of dense bundles of filaments with the same

orientation, see Fig. 2c. The bundles are also associatedby certain layering (smectic ordering) of the filaments.This ordering is due to the microscopic interaction lawwhich results in alignment of the rod midpoints as in Eq.(3). While this might be a case for the NCD motors withlarge dwelling time, for the kinesin motors the bundlesmay have a different structure which is not necessarilycaptured by these simulations. These results are in goodagreement with earlier theoretical predictions [12, 13].

The phase diagram delineating various regimes of self-organization is shown in Fig. 5. It bears a strong re-semblance to the experimental observations [4, 5] andthe theoretical model of Refs. [12, 13]. While the bound-aries are quite blurred due to strong fluctuations (see Fig.4), there is a transition from vortices to asters with theincrease of interaction rate P0 [23]. Moreover, the do-main of stability of vortices decreases with the increaseof anisotropy parameter β related to the dwell time ofthe motors, as observed experimentally and in agreementwith the continuum model of Refs. [12, 13]. However,we need to emphasize that all the boundaries shown inFig.5 are rather blurred; instead of sharp phase transi-tions we observed only smooth crossovers between dif-ferent regimes due to strong fluctuations and relativelysmall number of particles in the system [24].

The coarse-graining allows for easier identification ofaster and vortex structures, see Fig. 3. In the moviesmade using coarse-grained fields we are able to follow

the formation, interaction and evolution of asters andvortices. A typical scenario of the dynamical evolutionof the system is that small vortices and asters can coa-

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0 0.05 0.1P

0

0

2

4

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<N

>vorticesastersanti-asters

FIG. 4: (Color online) Averaged number of asters (squares),anti-asters (diamonds), and vortices (circles) as a functionof the interaction probability P0 for two different values ofparameter β. The data for β = 0.35 is shown in dashed lines,open symbols, and for β = 0.95 is shown in dotted lines,closed symbols.

0 0.05 0.1 0.150

0.1

0.2

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0.4

0.5

0.6

0.7

0.8

0.9

1

P0

β

FIG. 5: (Color online) Phase diagram of various regimes asthe function of the motor density P0 and the anisotropy pa-rameter β. The disordered region is blue (black) here; thevortex region is green (grey); the transition from vortex toaster happens at the yellow region (white) and red (dark grey)denotes aster regions. The dash line denotes the boundarywhere the rods become bundled.

lesce to form a larger vortex or aster (see the movies in[22] for parameters P0 = 0.12,β = 1.0). In accordancewith the experiments, vortices have suppression of themicrotubule density in the center (holes) and asters leadto an increase of the density of microtubules. We havealso observed the transformation of vortices into astersin the course of the simulations, likely due to fluctuationand fine size effects.

We followed the trajectory of individual rods in the

vicinity of the vortex core in the steady-state. We havefound that the particles generally do not rotate aroundthe vortex core. This stems from the fact that in ourbinary collision algorithm the center of mass of two in-teracting rods is not displaced in the course of collision,Eq. (3). This restriction suppresses directed motionof the rods, and, consequently, global rotation. Thus,the rotation of microtubules seen in experiment [4] islikely related to the interaction with the substrate or theboundary of the container [13, 16], or, possibly is relatedto multi-particle interactions and anisotropic interactionwith the fluid [11] neglected in our model.

In our simulations we also observed that the centersof the asters typically exhibit a drift, reminiscent to theacceleration instability of aster cores predicted in Ref.[13]. This phenomenon especially appears at the stage offormation of asters. However, the precise nature of thedrift is still an open question since it could be also dueto fluctuation effects.

V. CONCLUSION

In this paper we developed a Monte Carlo type stochas-tic approach to study molecular motors mediating self-organization of microtubules. The approach allows us tobypass the fast time scale associated with the diffusionand the motion of individual molecular motors and con-centrates on the relevant features of long-time and large-scale behaviors associated with the self-organization phe-nomena.

While a direct comparison with earlier algorithms byRef. [7] is not always possible due to different nature ofapproximations, some rough estimates are useful. Totalsimulation time in Ref [7] was 1500 sec. The characteris-tic time scale of the simulations of the order of 1 sec canbe inferred from the density of microtubules (about 0.05µm−2, or about 500 microtubules in the box 100 × 100microns) and motor diffusion ( D = 20µm2/s) whichroughly corresponds to 103 dimensionless units of time.Our simulations, performed with much higher number ofmicrotubules (6000) and in the bigger boxes were per-formed for about 1000 dimensionless units, i.e. aboutthe same order of magnitude as in Refs. [4, 7].

Our method can be easily adapted to new experimentalsettings, such as a motor and microtubule system witha fraction of motors permanently bound to the substrate[16]. Our results are complimentary to analytical stud-ies of self-organization in the framework of amplitudeequations derived from the stochastic master equations,and provide valuable tests for a variety of phenomeno-logical continuum theories of cytoskeleton formation [8–11, 15]. Moreover, our simulations shed new light on mi-croscopic details of self-organization not available in thecontinuum formulation. We anticipate that somewhatsimilar approaches can be applied to a broad range ofsystems, such as networks of actin filaments interactingwith myosin motors [6], patterns emerging in granular

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systems with anisotropic particles [25–29], and systemsof self-propelled objects [30, 31].

We thank Lev Tsimring, Jacques Prost, FrancoisNedelec, Falko Ziebert, and Walter Zimmermann for use-ful discussions. This work was supported by the U.S.Department of Energy, grant DE-AC02-06CH11357 andin part by NSF DMS-0401708 and DARPA FunBio HR0011-05-1-0057.

APPENDIX A: PATTERN CHARACTERIZATION

We computed the discrete divergence, div = ∇· τ , andthe curl, rot = ∇ × τ , of the coarse-grained field of thepattern from the last 3000 iterations. Here div and rotdepend on the mesh size of the coarse-grained field. Byusing the central difference scheme, the extrema of divand rot can be −4 or 4 for an ideal aster or vortex underthe 40× 40 grid on the 20× 20 domain.

The basic idea for pattern characterization is that anaster would have its local divergence greater or less thana threshold value at the center. The similar observa-tions are the same to a vortex and its curl. To realizethe pattern characterization, we generated the followingprocedure:

• Firstly, using the snapshot at t = 700, determinethe local extrema of div and rot with values suffi-ciently far from zero. Specifically:

1. Compute the minimal value, min(div), of thedivergence, div. Suppose that it occurs at(i, j);

2. Eliminate the surrounding square area con-sisting of (2q + 1) × (2q + 1) mesh points.We chose q = 4 in our computation, i.e.,temporarily set div(k, l) = 0, i − 4 ≤ k ≤i + 4, j − 4 ≤ l ≤ j + 4. Locate the nextminimal value of div from the remaining re-gion;

3. Repeat step 2 on the remaining region untildiv > −2.5;

4. Use the above three steps to find the maximaof div with div > 2.5;

5. Go through step 1 to step 3 to locate the localminima of rot with rot < −2.5;

6. Apply a similar procedure to find local max-ima of rot with rot > 2.5;

7. If two of the selected extrema of |div| and |rot|occur in one of the selected squares, then wediscard the square that is not centered at apoint where the greater of |div| and |rot| oc-curs.

• Secondly, take the local square area consisting of(2q + 1) × (2q + 1) mesh points for each of theextracted locations, and compute four quantities,

min(div), max(div), min(rot) and max(rot) in thispatch for each snapshot from t = 401 to t = 700.

• Thirdly, compute the time averages of those fourquantities for each patch from these 300 snap-shot values, denoted as mid = |min(div)|, mad =max(div), mic = |min(rot)|, and mac = max(rot).To distinguish between a vortex and an aster, weintroduced an additional parameter ξ = 0.6, whoseuse is explained below.

• Finally, to determine the type of pattern in eachlocal square area, we decide according to the fol-lowing criteria.

– If mid ≥ 3.0, mac < 3.0, mic < 3.0 and mad ≤ξmid, it is an aster.

– If mid ≥ 3.0, mac ≥ 3.0 (mic ≥ 3.0), mic ≤ξmid (mac ≤ ξmid) and mad ≤ ξmid, it is anintermediate form between an aster and a vor-tex and we assign it an aster-vortex pattern.

– If mad ≥ 3.0, mac < 3.0, mic < 3.0 andmid ≤ ξmad, the directions of the rods pointoutwards and it is an anti-aster pattern.

– If mad ≥ 3.0, mac ≥ 3.0 (mic ≥ 3.0), mic ≤ξmad (mac ≤ ξmad) and mid ≤ ξmad, it isan antiaster-vortex pattern.

– If mid < 3.0 and mad < 3.0 and mic ≥ 3.0(mac ≤ ξmic) or mac ≥ 3.0 (mic ≤ ξmac),then it is a vortex pattern.

– in any other case, it is isotropic.

The parameter space (P0, β) is in the range 0.01 ≤P0 ≤ 0.15 and 0.0 ≤ β ≤ 1.0. We made a grid withstepsizes ∆P0 = 0.01 and ∆β = 0.05 so that we had15 × 21 = 315 mesh points. For each pair of values,we used three different initial conditions for the simula-tions, using the characterization of the final states de-scribed above. We obtained the numbers of asters (A),aster-vortices (AV ), antiasters (AA), antiaster-vortices(AAV ) and vortices (V ) for each (P0, β) and we foundthat AV = 0 and AAV = 0. At each parameter gridpoint we computed two values according to the followingformula:

MA(i, j) =1

N1

i+1∑

i−1

j+1∑

j−1

(A + AA) (A1)

MV (i, j) =1

N1

i+1∑

i−1

j+1∑

j−1

(V ) (A2)

For the boundary points, the summations in (A1) and(A2) are taken only over the neighboring points around(i, j) within the parameter domain. N1 in (A1) and (A2)is the number of points in the summation. From MA(i, j)and MV (i, j), we calculated ra(i, j) = MA(i,j)

MA(i,j)+MV (i,j)

8

and rv(i, j) = MV (i,j)MA(i,j)+MV (i,j) . Finally we generated a

matrix, MP , whose entries give the pattern informationat that parameter point.

• If 13 (MA(i, j)+MV (i, j)) ≤ 1.5, then it belongs to

disordered region and MP (i, j) = −1.0.

• If ra(i, j) ≥ 0.6 and rv(i, j) ≤ 0.4, then it belongsto aster region and MP (i, j) = 1.0.

• If ra(i, j) ≤ 0.4 and rv(i, j) ≥ 0.6, then it belongsto vortex region and MP (i, j) = 0.0.

• Otherwise, it belongs to a transition region andMP (i, j) = 0.5.

We used MP martix to produce a pseudo-color phasediagram. The pixels with MP (i, j) = 1 are assigned red,

the pixels with MP (i, j) = 0.5 yellow, the pixels withMP (i, j) = 0.0 green and the pixels with MP (i, j) =−1.0 blue.

To identify the bundled region, we calculated the den-sity of the rods at each grid point, which is defined asthe number of rods whose positions are in the square boxwith the grid point as the center. Next we computedthe global minimal and maximal density in the domainat each time slice. Those minimal and maximal densi-ties were averaged over 300 slices and then over threesamples, i.e., minden = 1

3

∑31(

1300

∑300j=1 minden(j)),

maxden = 13

∑31(

1300

∑300j=1 maxden(j)). If minden < 0.2

and maxden > 60, then this point is marked bundled. Inthe bundled region the rods formed several, with thesestripes sometimes forming concentric circles. Moreoverasters appear to dominate vortex structures.

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