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Eur. Phys. J. E 18, 41–54 (2005)DOI 10.1140/epje/i2005-10029-3THE EUROPEANPHYSICAL JOURNAL ENonlinear competition between asters and stripes infilament-motor systemsF. Ziebert and W. Zimmermann aTheoretische Physik, Universität des Saarlandes, D-66041 Saarbrücken, Germany andTheoretische Physik, Universität Bayreuth, D-95440 Bayreuth, GermanyReceived 17 January 2005 and Received in final form 5 May 2005 /Published online: 7 October 2005 – c○ EDP Sciences / Società Italiana di Fisica / Springer-Verlag 2005Abstract. A model for polar filaments interacting via molecular motor complexes is investigated whichexhibits bifurcations to spatial patterns. It is shown that the homogeneous distribution of filaments, suchas actin or microtubules, may become either unstable with respect to an orientational instability of a finitewave number or with respect to modulations of the filament density, where long-wavelength modes areamplified as well. Above threshold nonlinear interactions select either stripe patterns or periodic asters.The existence and stability ranges of each pattern close to threshold are predicted in terms of a weaklynonlinear perturbation analysis, which is confirmed by numerical simulations of the basic model equations.The two relevant parameters determining the bifurcation scenario of the model can be related to theconcentrations of the active molecular motors and of the filaments, respectively, which both could be easilyregulated by the cell.PACS. 87.16.-b Subcellular structure and processes – 47.54.+r Pattern selection; pattern formation –89.75.-k Complex systems1 IntroductionIn eukaryotic cells the polar filaments actin and microtubulesinteracting with motor proteins play a crucial rolein intracellular organisation and transport as well as forthe static and dynamical structure of the cytoskeleton [1,2]. Most prominently, microtubules and kinesin are involvedin highly connected dynamical structures, such asthe mitotic spindle in cell division [3,4], while the motilityof the cell as a whole is governed by acto-myosin complexes[5,6]. The dynamical behavior of such assembliesdepends on the available biological fuel ATP (Adenosinetriphosphate),which is consumed during the motion ofmotor proteins along the filaments. Vesicles for instanceare transported across a cell by motors moving along thetracks defined by microtubules, or —which is the scope ofthis work— oligomeric motor proteins that attach to twoor more filaments induce relative motion between neighboringfilaments and cause dynamical networks. The latterprocess is vitally important in cells, since the cytoskeletonconstituted of the filaments has to be self-organized andeven actively reorganized not only during cell-locomotionbut also in order to react to outer stimuli (chemotaxis).During mitosis, microtubules attach to the chromosomes,which are then divided and the two halves finally areae-mail: walter.zimmermann@uni-bayreuth.detransported by the motor-induced filament sliding into thetwo evolving daughter cells.Since the situation is very complex in a living cell,well-designed in vitro experiments are the agent of choicefor controlled explorations of prominent aspects of cellularsystems. Recent experimental progress yielded indeedimportant insights into organization and dynamical propertiesof the cell, which in turn call for modeling activitiesto foster their deeper understanding. These experimentscomprise investigations of self-organization in filament-motormixtures of microtubules in the presence of asingle type of motor protein [7–9] and more recently also inactin-myosin networks [10,11] as well as assays where twotypes of motors interact with microtubules [12]. Even insuch model systems, simple compared to a living cell, therehas been found a great variety of different two-dimensionalpatterns, such as stripe patterns, asters, vortices and irregulararrangements.Works on modeling active filament-motor systemscomprise molecular-dynamics simulations [8,12,13] andmean-field approximations for spin-like models displayingstripe-like patterns [14]. More recently, a phenomenologicalmodel of a motor density interacting with a phenomenologicalvector field has been considered which isable to reproduce asters and vortex-like solutions [15,16]as well as models for bundle contraction in one spatialdimension [17–19].


42 The European Physical Journal EHere we follow a more microscopic approach startingfrom the Smoluchowski equation for the spatial and angulardistribution of rigid rods [20], which approximatethe rather stiff microtubules and with limitations also theactin filaments. It is well known since Onsager’s theorythat an ideal rigid-rod system exhibits a nematic liquidcrystalline order beyond some critical rod density [21,22]which is still present in generalizations for semiflexiblepolymers, albeit at a slightly higher filament density [23],and has been observed in vitro for both kinds of biopolymers[24,25]. To use Onsager’s equilibrium argument ofexcluded-volume–induced transitions in a nonequilibriumsystem like the cell, one can formulate phenomenologicalmodels as done recently to describe how nucleationand decay of the filaments in a cell at a density closeto the isotropic-nematic transition may give rise to spatiallyperiodic patterns [26]. Alternatively, one can use theaforementioned Smoluchowski equation for rigid rods [20],which is an approach widely used in polymer and colloidalscience and has recently been supplemented by active currentsto describe filament-motor systems [27] in a way inspiredby a macroscopic model for filament bundling in onespatial dimension [18]. In this approach one makes use ofthe fact that the small motor proteins diffuse much fasterthan the filaments. Hence the density of the motors can beassumed to be homogeneous and —as well as propertieslike the mean velocity and the duty ratio of the motors—enters into the model only via the coefficients. The phenomenologicaldescription of the active currents can be derivedby symmetry considerations and includes three majorcontributions, whereof in this work we focus on the effectson pattern formation of only two of them. Since mostof the experimental assays are quasi-two-dimensional, werestrict our analysis also to two spatial dimensions.The filament-filament interactions induced by the motorsas well as by excluded-volume effects are nonlocal,thus they can be approximated by a gradient expansion.Since in vitro aster-like patterns evolve at much lower filamentdensity than the isotropic-nematic transition, a momentexpansion of the probability density function can betruncated to derive a closed set of equations for the physicalobservables, which are the density and the orientationfield of the filaments. After a detailed linear stability analysisof the homogeneous filament distribution, the majorgoal of this work is a characterization of the weakly nonlinearbehavior of patterns beyond their threshold. Forthis purpose we employ, on the one hand, numerical simulationsof the coupled equations for the filament densityand the orientational field. On the other hand, weuse the powerful method of amplitude expansion, whereequations of motion for the amplitudes of the spatiallyperiodic pattern are derived close to the pattern-forminginstability [28–30].The generic patterns in two-dimensional systems closeto threshold are stripes, squares or hexagonal patterns andfor the amplitudes of these patterns generic nonlinear amplitudeequations may be derived [28,31]. In our numericalsimulations we found in the range of small modulationsof both the homogeneous filament density and the orientationeither stripe or square patterns. The absence ofhexagonal structures in the simulations is confirmed bythe perturbational calculation and therefore we focus onthe amplitude equations for stripe and square patterns (asdescribed by Eq. (27) below), which are of the form [28,32]τ 0 ∂ t X = εX − g 1 |X| 2 X − g 2 |Y | 2 X ,τ 0 ∂ t Y = εY − g 1 |Y | 2 Y − g 2 |X| 2 Y .(1a)(1b)Here ε denotes the deviation of the control parameter fromits value at the threshold of pattern formation, while Xand Y describe the amplitudes of a spatially periodic patternin one (stripe) or two orthogonal directions (square),namely the x- and y-direction. These amplitude equationsare universal for all pattern-forming systems of a givensymmetry and the specific information of each system isonly encoded in the coefficients, namely τ 0 , g 1 , g 2 , whichare functions of the parameters of the underlying systemunder consideration. The nonlinear analysis of equations(1) is worked out analytically and by this advantagea whole phase diagram for the various patterns close tothreshold is presented. In contrast, such an investigationwould be very cumbersome and time consumptive usingnumerical simulations of the model equations. The solution|X| = |Y | corresponds to a square pattern of thesmall-amplitude modulation of the filament orientationalfield which resembles very much a structure of periodicallyarranged asters.The work is organized as follows. The underlying microscopicmodel for the filament distribution is presentedin Section 2 and in the same section we derive also thecoupled equations for the density and orientation of thefilaments, whereby the technical parts are moved to theappendices. The thresholds of the pattern-forming instabilitiesand its dependence of the parameters are determinedin Section 3, where we find an instability leadingto density modulations, an orientational instability as wellas a liquid-crystalline isotropic-nematic (I-N) transition.With regard to the formation of asters, the orientationalinstability is the most interesting one and its nonlinear behavior,namely the nonlinear stripe solutions and the periodiclattice of asters and inverse asters as well as their stabilityare investigated in more detail in Sections 4 and 5.We compare our analysis to available experimental resultsin Section 6 and the work is finished with a summary andoutlook.2 The modelThe probability of finding a rod (with fixed length L) atthe position r with the orientation u (with |u| = 1) attime t is described by the distribution function Ψ(r, u, t) 1and the governing so-called Smoluchowski equation [20] isjust the continuity equation for the probability∂ t Ψ + ∇ · J t + R · J r = 0 . (2)1 In the following we will write Ψ(r, u) for reasons ofbrevity and Ψ(r, u, t) only if we want to emphasize the timedependence.


F. Ziebert and W. Zimmermann: Nonlinear competition between asters and stripes 43Homogeneously distributed molecular motors are supposedto interact with the rods inducing active currentson them. The total translational and rotational currentsare given in Cartesian coordinates byJ t,i = −D ij [∂ j Ψ + Ψ∂ j V ex ] + J a t,i , (3a)J r,i = −D r [R i Ψ + ΨR i V ex ] + J a r,i . (3b)The terms with an upper index a are the motor-mediatedactive currents proposed in reference [27], the translationaldiffusion matrix readsD ij = D ‖ u i u j + D ⊥ (δ ij − u i u j ) ; (4)D r is the rotational diffusion constant and the operator ofrotational diffusion is given byR = u × ∂ u . (5)V ex describes the excluded-volume interaction∫ ∫V ex (r, u) = du ′ dr ′ W (r−r ′ , u, u ′ )Ψ(r ′ , u ′ ) (6)between rods, where the interaction kernel is expressed interms of the so-called Straley coordinates ζ and η in twodimensions [33]:∫L/2∫L/2W (r−r ′ , u, u ′ ) = |u×u ′ | dζ dη δ (r−r ′ +uζ+u ′ η) . (7)−L/2 −L/2This expression takes into account that there is only an interactionbetween rods with coordinates (r, u) and (r ′ , u ′ )in the case of a finite overlap, i.e. if the connection vectorr−r ′ can be constructed by a linear combination uζ +u ′ ηof the rod orientations with −L/2 < ζ, η < L/2. The activecurrents induced by the homogeneous motor densityare∫ ∫J a t = Ψ(r, u) du ′ dr ′ v(r−r , ′ u, u ′ )W (r−r , ′ u, u ′ )Ψ(r, ′ u ′ ) ,∫ ∫J a r= Ψ(r, u) du ′ dr ′ ω(u, u ′ )W (r−r , ′ u, u ′ )Ψ(r, ′ u ′ ), (8)where the translational and rotational velocity can bewritten following reference [27] asv(r−r , ′ u, u ′ ) = α r − r ′2 L1 + u · u ′|u × u ′ |ω(u, u ′ ) = γ(u · u ′ ) u × u′|u × u ′ |+ β 2u ′ − u|u × u ′ |, (9), (10)respectively. The active currents are normalized to the interactionvolume and they fulfill both, the conservation oftranslational and rotational momentum in the absence ofexternal forces and torques, as well as translational androtational invariance (cf. Ref. [27] and App. A).According to its functional dependence 1 + u · u ′ , thefirst term in equation (9) contributes at maximum to thefilament transport for a parallel orientation, which identifiesit as the leading pattern-forming mechanism in counteractionwith the diffusive motion. It arises from a differencein motor activity between the filament center andthe filament endpoints, where motors stall for some finitetime τ f , which has been recognised as a crucial conditionfor aster formation [34]. It depends also on the relativefilament separation r − r ′ , hence no relative motion is obtainedif the motor connects two filaments having the samecenters of mass.The second term in equation (9), proportional to β,causes filament translations along u ′ −u and therefore becomesmaximal for antiparallel filament orientations. Thecontribution in equation (10) causes rotational currents.In this work we focus mainly on active currents caused bythe α-term and especially on its effects on pattern formationin two spatial dimensions.Equation (2) together with the currents as defined byequations (3) and (8) cannot be solved analytically forΨ(r, u, t) and even numerically, being a nonlinear integrodifferentialequation, it would be rather intricate. However,these equations may be simplified by expressing thedistribution function in terms of its moments. For a roddistribution, the zeroth, first and second moment with respectto the orientation correspond to the filament densityρ(r, t), the polar orientation t(r, t) and the nematic orderparameter S ij (r, t), which are defined by the following expressions:∫ρ(r, t) = du Ψ(r, u, t) ,∫t(r, t) = du u Ψ(r, u, t) ,∫S ij (r, t) = du u i u j Ψ(r, u, t) . (11)In contrast to a usual lyotropic liquid crystal [21], here thefilaments are polar with respect to the motor action whichbreaks the ±u-symmetry and therefore the first momentmay become finite.The critical filament density of the homogeneous isotropic-nematic(I-N) transition can be determined by thestability of the second moment of the rotational diffusioncontribution, i.e. where −D r∫uα u β R [RΨ + ΨRV ex ]changes its sign. This condition yields the critical densityρ IN = 3 2 π (12)in two spatial dimensions where the I-N transition is ofsecond order.In order to obtain a closed set of equations of motion,the distribution function Ψ(r, u, t) may be expanded withrespect to the leading moments as given by equations (11).In the parameter range well below the homogeneous I-Ntransition, where aster formation is observed in vitro andon which we concentrate in this work, the first two momentsare sufficient and the distribution function may berepresented in terms of ρ(r, t) and t(r, t) (for more details


44 The European Physical Journal Ewe refer to App. C)Ψ(r, u, t) = 1 ( )ρ(r, t) + 2u · t(r, t) . (13)2πTo get rid of the integral kernel in equation (7), a gradientexpansion can be performed as described in more detailin Appendix B. After a rescaling of variables viat ′ = D ‖L 2 t , x′ = 1 L x , ρ′ = L 2 ρ , t ′ = L 2 t ,α ′ = L D ‖α , D ′ r = L2D ‖D r, D = D ⊥D ‖, (14)where we have introduced the ratio D of the diffusion coefficients,we obtain in two spatial dimensions the followingcoupled equations for the density and the two componentsof the polar orientation field t i (i = x, y) of the filaments:∂ t ρ = 1 + D [ 1 + D∆ρ + − α ]∇ · (ρ∇ρ)2π 24− α ]48 ∂ i[t i ∂ j t j + t j ∂ i t j + t j ∂ j t i− α C 1{38∇ · (ρ∇∆ρ) + 11∂ i (t j ∂ i ∆t j )+ 16∂ i[t i ∆∂ j t j + 2t j ∂ j ∂ i ∂ l t l + t j ∂ j ∆t i]}, (15a)∂ t t i = −D r t i + 3D + 1 ∆t i + 1 − D ∂ i ∇ · t42+ 3D + 1 ∂ j (t i ∂ j ρ) + 1−D [](∂ j (t j ∂ i ρ) + ∂ i (t j ∂ j ρ))2π2π− α []96 ∂ j 3t i ∂ j ρ + t j ∂ i ρ + ρ(∂ i t j + ∂ j t i )− α ]96 ∂ i[t l ∂ l ρ + ρ∂ l t l − 16α []∂ i ρ∆∂ l t l + t l ∂ l ∆ρC 2− α )∂ j[ρ(11∂ j ∆t i + 16∂ i ∆t j + 32∂ j ∂ i ∂ l t lC 2]+ 16t j ∂ i ∆ρ + 32t l ∂ l ∂ i ∂ j ρ + 44t i ∂ j ∆ρ+ 1 [ γ48 4 − 4 {r]π D t j ∂ j ∂ i ρ − 1 }2 t i∆ρ(15b)with C 1 = 23040 and C 2 = 2C 1 .The contributions to equations (15) that include alsofourth-order derivatives are indispensible for the determinationof the instabilities evolving from the homogeneousfilament distribution and the nonlinear patterns in the followingsections. They were not taken into account in [27]and their importance was pointed out recently [35]. Itshould be noted that fourth-order derivatives proportionalto the translational diffusion have been neglected herebecause they are overcompensated by the fourth derivativesresulting from the active current and therefore influencethe presented results only quantitatively by a smallamount (but considerably complicate the equations).The instabilities of the homogeneous filament distributioncaused by the α-contribution to the active current,cf. (9, 10), and the nonlinear solutions are determined inthe following sections. The active current proportional toβ leads to additional effects, which are responsible, e.g.,for propagating modes in the system and for the breakingof the unexpected t → −t symmetry of equations (15). Toremind the reader, we have allowed t to become nonzero,so we allowed the breaking of the ±u-symmetry of equation(2), but our model equations (15) nevertheless have a±t-symmetry, since terms like t 2 appear in the equationfor t only due to the β-contribution of the current. Theseeffects are determined elsewhere [36] and the correspondingcontributions to equations (15) are not shown for thesake of brevity. The active rotational current proportionalto γ is nonlinear and therefore influences only the nonlinearbehavior of pattern selection beyond threshold asshown briefly in Section 4.3 Threshold for density and orientationalinstabilitiesHaving defined the model, the first issue in the field of patternformation is the determination of possible instabilities.As calculated in this section by a linear stability analysis,in a certain parameter range the homogeneous basicstate, which consists here in a constant filament density ρ 0and a vanishing polar orientation t = 0, becomes unstablewith respect to inhomogeneous perturbations ˜ρ(r, t) andt(r, t). For this purpose, by the ansatz ρ(r, t) = ρ 0 + ˜ρ(r, t)we separate the constant part ρ 0 of the filament densityfrom the spatially inhomogeneous one, ˜ρ(r, t), and linearizeequations (15) with respect to small inhomogeneouscontributions ˜ρ(r, t) and t(r, t). Accordingly a set of threecoupled linear equations is obtained:⎛⎞∂ t w(r, t) = L 0 w(r, t) =⎜⎝L (0)11 0 00 L (0)22 L(0) 230 L (0)32 L(0) 33for the three components of the vector⎛⎟⎠ w(r, t) , (16)w(r, t) = ⎝ ˜ρ t x⎠ (r, t) . (17)ty⎞The components of the linear operator L 0 are given bythe expressions[ (L (0) 1 + D11 = 1 + 2 )2 π ρ 0 − αρ ]0∆ − 19 αρ 024 11520 ∆2 ,L (0)22 = −D r + 3D + 1 ∆ + 1 − D ∂x 2 − αρ 04 2 96 (∆ + 2∂2 x)− αρ 046080(L (0) 1 − D23 = − αρ 02 48(11∆ 2 + 64∆∂x2 ),)∂ x ∂ y − αρ 0720 ∆∂ x∂ y (18)


F. Ziebert and W. Zimmermann: Nonlinear competition between asters and stripes 45and the two further components L (0)32obtained by permuting ∂ x and ∂ y in L (0)and L(0) 22 , respectively.Naturally the mode ansatzand L(0) 3323may beλ a) b)00w(r, t) = E exp (σt + ik · r) (19)with the wave vector k = (q, p) and the eigenvector Esolves the linear homogeneous set of equations (16) andthe solubility condition provides a third-order polynomialfor the eigenvalue σ, which factorizes into a linear and aquadratic polynomial describing different types of instabilities.Considering moderate filament densities, for intermediatevalues of α, an orientational instability with afinite wavelength is preferred, whereas the density modedoes not couple on the level of the linear equations. Forlarge values of α a further mode becomes unstable whichresembles a spinodal decomposition of the filament densitydriven by the motor proteins.The latter instability with respect to density fluctuationsis governed by L (0)11 and the eigenvalue[ ( 1 + Dσ d = − 1 + 2 )2 π ρ 0 − αρ ]0k 2 − 19 αρ 024 11520 k4 (20)with k 2 = q 2 + p 2 . The term ∝ k 4 is always stabilizing,but the homogeneous basic state becomes unstable withrespect to density modulations for a positive coefficient ofk 2 leading to the corresponding critical filament densityρ d =1α12(1+D) − 2 πproviding that this value is positive, i.e. if α > 24πholds. The corresponding eigenvector is E d(21)(1 + D)= (1, 0, 0) Tand the dispersion is shown as the dotted lines in Figures1a) and b) for subcritical and supercritical values,respectively.Of the remaining two eigenvalues, σ I and σ S , only thefirst may become positive in a finite range of k as indicatedby the solid lines in Figure 1. It describes an instabilitywith respect to orientational fluctuations with the dispersionrelationσ I = −D r − 1 4 k2 (3 + D − αρ 08 + 5768 αρ 0k 2 ), (22)while σ S , shown as the dashed lines in Figure 1, is alwaysdampened and related to diffusion of orientational modes.The two corresponding eigenvectors are⎛ ⎞⎛ ⎞E I = ⎝ 0 q ⎠ , E S = ⎝ 0 q ⎠ . (23)p−pIt should be mentioned that the eigenvalues σ d (k),σ I (k), σ S (k) depend only on even powers of the wave numbermodulus reflecting the rotational symmetry of the basicstate (ρ 0 , t). The restabilizing k 4 -term in equation (22)was missing in reference [27], as pointed out recently [35].− 0.1− 0.20 0.4 0.8 1.2 1.6 2.00 0.4 0.8 1.2 1.6 2.0kFig. 1. The real parts λ(k) = Re[σ(k)] of the eigenvalue σ dcorresponding to the instability with respect to density fluctuations(dotted line) as well as the one with respect to orientationalfluctuations, σ I (solid line), are shown as a functionof the wave number k for D r = 0.15. The third eigenvalue σ Sis always dampened as depicted by the dashed line. In part a)we have chosen α = 21 and therefore ρ c < ρ 0 < ρ d leading toan orientational instability, whereas in part b) one has α = 26and ρ d < ρ c < ρ 0, hence both modes σ d and σ I have positivereal parts in a finite wave number range (see also Fig. 2).However, only the interplay between the k 2 and k 4 contributionas in equation (22) allows the identification ofa finite wave number instability. The extremal conditiontogether with the neutral stability conditiondσ Idk∣ = 0 , σ I (k = k c ) = 0 , (24)kcallow the determination of the critical filament density ρ cat the critical wave number k c , above which the orientationalinstability takes place:⎛⎛ √⎞ρ c = 8 ⎝(3+D) + 5α12 D r⎝1 + 1 + 24(3+D)⎠ , (25)5D r( ) 1/4 12Drk c = 4. (26)5αρ cBefore going further, one has to specify the value of thediffusion ratio D = D ⊥D ‖, which from the physical point ofview is not simply a parameter but distinguishes differentmodel classes or modelling regimes: If the rods wereisotropically diffusing, D equals one. In reality, a rod candiffuse much easier in the direction parallel to its orientationthan perpendicular to it. A hydrodynamic calculationaccounting for the rod-like shape [20] yields D = 1/2. Onecan also consider the case D = 0 meaning that D ⊥ = 0which applies for semi-dilute solutions, where the so-calledtube approximation holds and D ‖ dominates the diffusionbehaviour [20,37]. Very recently [38], a microscopic considerationof the molecular motors established that if D ≠ 1,i.e. D ⊥ ≠ D ‖ , there is not only relative filament motionas accounted for in (8), but also a net motion of the centerof mass of the considered filament pair. Nevertheless,performing the weakly nonlinear and the numerical calculationsof the following paragraphs, D mathematically


46 The European Physical Journal Eρ6Ninfluence on the nonlinear behavior of stationary periodicpatterns is one of the issues discussed in the next section.4O4 Nonlinear analysis of the orientationalinstability20S4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34αFig. 2. The critical densities ρ c (solid line) and ρ d (dashed line)for the orientational and density instability, respectively, areshown as a function of α and for D r = 0.15. The dash-dottedhorizontal line is the critical density ρ IN = 3 π, above which2the isotropic-nematic transition due to excluded-volume interactionstakes place. In the range referred to as N one has apure homogeneous transition to nematic order, in range O onehas a motor-driven spatially periodic orientational order andin range D one has modulations of the filament density. Theregion S denotes the parameter range where the homogeneoussolution is stable. In the following sections of the paper weinvestigate the nonlinear behavior in the O-region.is just a parameter which influences the behaviour onlyquantitatively, and even only to a small amount. Hencein the following we chose D = 1/2, which reflects theanisotropic diffusion behaviour of the rods. Center-of-massmotion may be however present, but for our analysis weexpect only minor changes.Figure 1 displays the dispersions, i.e. the wavenumber-dependent growth rates, σ d (k), σ I (k) and σ S (k)as dotted, solid and dashed lines, respectively. In Figure2, the critical density ρ d (dashed line) for an instabilitywith respect to inhomogeneous density fluctuations and ρ c(solid line) with respect to inhomogeneous orientationalfluctuations t(r, t) are shown as a function of motor activityα. The dash-dotted line describes the critical densityρ IN (12) above which the homogeneous isotropic-nematictransition takes place. On the left side of the dotted line,orientational fluctuations have lowest threshold, while onthe right side density fluctuations become unstable at first.For increasing values of the rotational diffusion coefficientD r , which determines how negative the dispersions σ I (k)and σ S (k) start from k = 0, the solid line is shifted upwardsin Figure 2 decreasing the α-range wherein the orientationalinstability has lowest threshold with ρ c < ρ IN .Beyond some critical value of β, the β-contribution tothe active current, equations (8, 9), induces an oscillatorybifurcation from the homogeneous filament distribution asdescribed in a forthcoming work, where also the nonlinearbehavior of the oscillatory patterns is discussed [36]. Thecontribution of the rotational current equations (8, 10) toequations (15) includes only nonlinear terms and thereforedoes not influence the thresholds of the instabilities. ItsDThe amplitude of the linear solution in equation (19)is limited by the terms in equations (15) that are nonlinearwith respect to ˜ρ and t. Here we investigate theweakly nonlinear behavior beyond the orientational instability,i.e. in the region referred to as O in Figure 2 whereρ c < ρ 0 < ρ d holds. This can be done numerically as exemplifiedin the next section and even analytically if ρ 0 isonly slightly beyond ρ c defined in (25): for a supercriticalbifurcation, the amplitude of the mode initially growingwith σ I (k) is small immediately above threshold and maybe determined in this range by a perturbative analysis,the so-called amplitude equation [28], whose derivation issketched in Appendix D.The generic form of amplitude equations dependson the preferred pattern beyond a stationary supercriticalbifurcation. In two spatial dimensions the patternis either spatially periodic in one direction (stripes), intwo (squares/rectangles) or in three directions (hexagonalpatterns) [28]. Numerical simulations of the basic equations(15) indicate, as described in Section 5, that stripeand square patterns are favored immediately above thethreshold of the orientational instability. Moreover, hexagonalstructures are not driven in this system due to theoverall up-down symmetry of equations (15) with respectto t, cf. Appendix D as well.Square patterns at threshold may be described analyticallyby a superposition of two linear modes (i.e. stripes)with orthogonal wave numbers which can be chosen withoutrestriction as k 1 = (k c , 0) and k 2 = (0, k c ) with k cgiven by equation (26) leading to the ansatz⎛w 1 = ⎝ ρ ⎞ ⎛1t 1x⎠ = ⎝ 0 ⎞ ⎛X⎠ e ikcx + ⎝ 0 ⎞0 ⎠ e ikcy + c.c. , (27)t 1y 0Ywhere c.c. means the complex conjugate. In a small neighborhoodof the threshold ρ c , measured by the dimensionlesscontrol parameterε = ρ 0 − ρ cρ c, (28)the time-dependent amplitudes X(t) and Y (t) of thesquare pattern are determined by two generic coupled nonlinearequations, the so-called amplitude equations [32,28]τ 0 ∂ t X = εX − g 1 |X| 2 X − g 2 |Y | 2 X ,τ 0 ∂ t Y = εY − g 1 |Y | 2 Y − g 2 |X| 2 Y .(29a)(29b)The coefficients τ 0 , g 1 and g 2 are derived in Appendix D.The specific system under consideration enters into thedescription only in these coefficients via the parameters


48 The European Physical Journal Ewith b the rod diameter and η the solvent viscosity. In ourscaled units this means D ′ r = L2D ‖D r = 6, lying far in therange of squares (asters).For semi-dilute solutions one can estimate [20,37]D r = 6D ‖ /(L 2 (1 + c r ρ ′ ) 2 ) ora) b)D r ′ = L26D r =D ‖ (1 + c r ρ ′ ) 2 (33)in scaled units, where c r ≃ 1 is a geometry factor from atube model calculation. Since stripes or bundle-like structuresare stable for D ′ r < 0.3–0.4, cf. Figures 3 and 4,one needs a rather high (but possible) filament density forsuch a one-dimensional ordering.According to these estimates, asters are the most likelypattern occurring above a stationary bifurcation in diluteor semi-dilute two-dimensional motor-filament systems.For bundle-like structures to emerge rather high filamentdensities are needed, which is physically intuitivefrom the overlap nature of all the interactions, namelyexcluded-volume and motor-induced filament-filament interaction.5 Results of numerical simulationsBesides the weakly nonlinear analysis described in the previoussection, the basic equations (15) have been solvednumerically in order to check the validity range of the perturbationanalysis and further explore the solution space.For this purpose a Fourier Galerkin pseudo-spectral methodhas been used imposing periodic boundary conditionson the system.Since the validity range of the amplitude expansionwith respect to the control parameter ε is not knownX 20.020.010 0.002 0.004 0.006 0.008 0.010εFig. 5. A comparison of the amplitude of a stripe solutionas obtained in simulations (crosses) with the analytical calculationX 0 = ±(ε/g 1) 1/2 as given by equation (30) is shown.The agreement remains well for higher values of ε, but in therange between ε ≃ 0.006 − 0.007 a secondary bifurcation takesplace rendering the high-ε solutions unstable. Parameters areD r = 0.5, α = 21 and γ = 0.X ,Y c)0.040.030.020.010 5 10 15 20 25 30tFig. 6. A simulation scenario is shown which numericallyconfirms the predictions of our weakly nonlinear analysis. Westarted with the aster (square pattern) displayed in a) with theanalytically calculated amplitude described by equation (31) ata point in parameter space where stripes should be preferredaccording to Figure 3, namely at α = 21 and D r = 0.15 (andε = 5 · 10 −4 ). Part c) shows now the temporal evolution of theamplitudes X and Y of the polar orientation components t xand t y as the dotted lines. One can clearly see that one of thetwo modes building up the square pattern is dampened andthat the other mode grows to the analytically predicted valuedescibed by equation (30) and displayed as the upper solid line.Part b) shows the final stripe pattern.a priori, in Figure 5 we compare the amplitude of a stripesolution as obtained by a numerical solution of the basicequations (15) with the analytical results given byequation (30). Close to threshold there is nearly perfectagreement between both approaches. However, the validityrange of the amplitude equations is actually restrictedto a range below ε ∼ 0.006 for the parameters used inFigure 5. Around this value a secondary instability takesplace, which is not taken into account in the perturbationexpansion. The numerical simulations show that beyondthis secondary instability a pronounced accumulation ofthe filaments to densities even higher than ρ IN appear accompaniedwith high alternating orientations. These solutionsare numerically stable but nevertheless in an invalidrange of our model since the nematic order parameter,cf. equations (11), has been neglected in the moment expansionwhich is not legitimate anymore —so we rejectshowing pictures here.In addition to the validity range with respect to ε, onemay also confirm numerically the stability of the nonlinearsolutions as predicted in Figure 3 by the weakly nonlinearanalysis. As an example, we start with a square solutionas shown in Figure 6a) at the point α = 21 and D r = 0.15in parameter space (and ε = 5·10 −4 ) belonging, according


F. Ziebert and W. Zimmermann: Nonlinear competition between asters and stripes 49Fig. 7. A simulation of equations (15) in the range of stablestationary asters is shown as a superposed plot of the orientationfield (arrows) and the filament density (dark color codinglow density, light color high density). Parameters are α = 21,D r = 0.5, ε = 5 · 10 −5 .to Figure 3, to the region of stable stripe patterns. Aftera slight perturbation, the simulated temporal evolution inFigure 6c) shows that only one of the initially equal amplitudesremains finite in the long-time limit leading tothe predicted stationary stripe pattern displayed in Figure6b). By several numerical runs we confirmed the analyticallypredicted stability diagrams in Figures 3 and 4.In the parameter range of stable asters, a vector plotof the orientation field superposed on the color-coded filamentdensity is presented in Figure 7. The filament orientationis indicated by arrows, the length being a measureof the degree of orientation. The density is high in thebright regions and low in the dark ones. At the right-handside in Figure 7, one can spot an aster with arrows pointingradially from a center with lowered filament densityopposing to an inverse aster top left with arrows pointingradially into the center (we remind the reader thatperiodic boundary conditions are imposed). The centersof the asters have lowered filament densities which can beexplained by the nonlinear analysis: in the derivation ofthe amplitude equations in Appendix D one can clearlysee that the growing amplitudes of the orientation modulationsexcite higher density modes —which then limit theorientation amplitudes to render the system stable— andtherefore in the center of an aster, where the orientationvanishes, there is no need for a high density. Both densityand the degree of orientation reach their maximum in betweenthe asters and two saddle-like structures buildingup a square with the two opposing aster centers completethe repetitive structure of the pattern at threshold in themotor-filament system.6 Comparison to experimentsIn this section we compare the results of our model analysisto the in vitro experiments. Taking intermediate parametervalues, namely α ≃ 20 and D r ≃ 0.3, the criticalwave number of the periodic modulation in scaled units isk c ≃ 1.5 leading for filaments with a mean length of 5 µmto a wavelength of 20 µm, which lies within the experimentalrange. The motor contribution for a microtubulekinesinmixture can be estimated [38] and lies aboutαL ≃ 1.2 · 10 −12 m 2 s −1 , while filament diffusion is ofthe order 10 −13 –10 −15 m 2 s −1 , depending on the filamentlength. Hence values of the scaled parameter α ′ = αLD ‖≈ 20as used by us are sensible. Since even higher values of αare possible, also the density instability, which occurs forall D r if α is large enough, may be relevant in experiments.For any comparison between the nonlinear behaviorof the filament density and orientational patterns as describedin this work and the experimentally observed inhomogeneousfilament distributions, one has to bear inmind the following fundamental difference. In our work themodulation amplitudes around the homogeneous filamentdensity as well as the emerging modulation of orientationare small and periodic and the generic patterns in this situationcan only be stripes, squares or hexagons [28], leadingto a discrete rotational symmetry of the aster pattern.In the experiments however, the modulation amplitudesare rather strong, the pattern is irregular and most structureshave a local continuous rotational symmetry. Thereason for this is simply that our analysis corresponds tothe so-called weakly nonlinear regime immediately abovethe threshold of the pattern-forming instability, while theexperiments correspond to the strongly nonlinear regimefar beyond threshold. For the strong modulations as inthe experiments, where both motors and filaments accumulatein the aster centers and the regions in between theasters are nearly depleted, it is not surprising that thepattern is not regular anymore and that the restrictionto the simple patterns possible at threshold is no morevalid. In spite of this however, from models and experimentson pattern formation in driven fluid systems [28],there are well-known examples that the small amplitudeexpansion captures the qualitative behavior of patterns ina considerable range beyond threshold.With regard to this experience with other systems it isvery reasonable to compare the stability trends suggestedby the phase diagrams presented in Figure 3 and Figure 4to experimental results as described for instance in references[7–9]. The patterns occurring there in unconfined geometriesfor increasing motor concentration, are vortices,mixtures of asters and vortices, asters and finally bundlesof microtubules for very high motor densities. The laststep, the transition from asters to bundles, is in agreementwith our calculations visualized in Figure 3, wherefor increasing values of α, which is proportional to the(homogeneous) density of motors, a transition from astersto stripes takes place. We did not find vortices. This maybe either due to the omission of the β-contributions tothe active current or even more likely due to the fact thatat lower motor concentration the spatial variations of themotor density may become a relevant dynamical degree offreedom that has to be taken into account in the modeling.


50 The European Physical Journal E7 Conclusions and outlookFor a model of interacting biopolymers and motor proteins,a nonlinear competition between the formation ofstripes and square patterns has been described for thefirst time. The model is based on a nonlinear (and nonlocal)Smoluchowski equation for the distribution functionof rigid rods extended by active currents [27] to accountfor the motor-mediated relative motion of the filaments.Under the assumption that the spatial variations are smallon the length scale of the filaments, this equation can beapproximated by a local equation of motion by performinga fourth-order gradient expansion [35]. The linear stabilityof the homogeneous basic state as well as the thresholdand the wave number of the evolving pattern are describedcomprehensively. The homogeneous state may become unstablewith respect to density, orientational and nematicinstabilities, but in different parameter ranges as shown inFigure 2. The two crucial model parameters are the rateof active translational transport and the rotational diffusioncoefficient which both can be regulated by the cellby creating filaments, motor proteins and the fuel ATP asdiscussed and estimated at the end of Section 4.Our subsequent nonlinear analysis focuses on the orientationalinstability, which may describe the early stateof the biological relevant aster-like structures as in the mitoticspindle which have also be seen in in vitro systemsrecently [8,11]. This analysis is to a large extent analyticallyin terms of an amplitude expansion technique yieldingtwo coupled equations, namely equations (29), for the(small) amplitudes of the spatial modulation of the orientationalfield in x- and y-direction. These two coupledequations determine the amplitudes of square (asters) andstripe patterns and they provide also the stability boundarybetween both solutions near threshold. Simulationswith a spectral code and periodic boundary conditionsconfirmed these analytical results.Estimates for the two model parameters α andD r of the basic equations (15), cf. end of Section 4,suggest that asters are the most likely pattern occurringabove a stationary bifurcation in dilute or semi-dilutetwo-dimensional motor-filament systems. Stripes may beexpected only for rather high filament densities whichcan be traced back to the overlap nature of both presentinteractions, namely excluded-volume and motor-inducedfilament-filament interaction. Once again both the type ofthe pattern, i.e. asters or stripes, as well as its wavelengthcould be governed by the cell by creating filaments, motorproteins and ATP.The model analyzed here is a first step towards a morecomplete description of the complex nonlinear structureslikely to occur in an active filament-motor system. Theeffects of the contribution describing the active antiparallelfilament transport within this model, cf. the β-termin equation (9), on the onset and the nonlinear behaviorof the respective patterns will be worked out in reference[36]. With β beyond some threshold, the homogeneousfilament distribution becomes unstable against afinite-wavelength oscillatory instability [27,35]. This hasto be contrasted to the solitary propagating modes foundin the one-dimensional version of the model presented inreference [15] and to the soliton solutions investigatedin more detail in the model of references [18,39], whichwas the starting point of the two-dimensional generalization[27]. Propagating modes may be relevant for vitalprocesses like cell spreading on a substrate as measured,for instance, in reference [40].There are several directions of an extension of the presentedmodel to which forthcoming work may be devoted.One direction is to extend the moment expansion schemeone step further yielding an equation for the nematic orderparameter tensor as defined in equation (11). The interplayof the nematic order with the polar orientation in thepresence of motors, may give rise to an interesting modelsystem of a new symmetry class.Experiments indicate that the spatially inhomogeneousdistribution of the motor density may become afurther relevant degree of freedom [41]. So an unstudiedand worthwhile extension of the present model is the inclusionof a field for the motor density in forthcoming works.Another interesting question was raised by [38], which accountedfor the motor proteins upon coarse-graining andgained new equations with some terms not present thatare however allowed by symmetry. A comparison of thismodel and the present one in the nonlinear regime wouldbe desirable.To tackle the problem of fully developed asters, especiallythe accumulation of filaments and motors in theaster center, it could be fruitful not to start from the instabilityand the nonlinear evolution of asters as in ourmodel but to use a complementary description as a defectstructure. A model describing the filament-motor systemas an active gel [42] could give an answer to this problemwhen analyzed in combination with an equation for thefilament density.The possibility of switching oligomeric motor clustersfrom an active state into an inactive crosslinking stateopens further interesting scenarios of pattern formation,as recently observed for actomyosin [11] and will be discussedelsewhere from a theoretical point of view in moredetail [36].Appendix A. Symmetries of themotor-mediated velocitiesThe explicit form of the motor-mediated translational andangular velocities, namely equations (9) and (10), can beobtained by writing down the simplest terms fulfilling theconservation laws and symmetries of the system.If one considers an interacting filament pair in theabsence of external forces and torques, both momentumand angular momentum of the pair have to be conserved.Hence the motor-mediated translational and angular velocities,v = ṙ − ṙ ′ and ω = ˙u − ˙u ′ , have to be odd underthe transformation (r, u; r ′ , u ′ ) → (r ′ , u ′ ; r, u). Translationalinvariance leads to f(r, r ′ ) = f(r − r ′ ) for f = v, ω.Together with the rotational invariance, v has to be oddand ω even under the transformation (r − r ′ ; u, u ′ ) →


F. Ziebert and W. Zimmermann: Nonlinear competition between asters and stripes 51(r ′ − r; −u, −u ′ ). The simplest terms fulfilling the aboveconditions for v are proportional to r − r ′ or u − u ′ andfor ω proportional to u×u ′ . One can thus write down [27]v(r−r , ′ u, u ′ ) = α r − r ′2 L1 + u · u ′|u × u ′ |ω(u, u ′ ) = γ(u · u ′ ) u × u′|u × u ′ |+ β 2u ′ − u|u × u ′ |, (A.1). (A.2)The term proportional to α is made orientation-dependent(it could be generalized by allowing two different prefactors)and the plus sign favours parallel alignment. In theterm proportional to γ, spatial dependence has been neglectedand the prefactor u·u ′ models the tendency of motorsto bind on two filaments which share an angle smallerthan π 2 . The common factor |u × u′ | −1 is just a normalizationto the interaction volume.Appendix B. The gradient expansion of theinteraction integralsThe excluded-volume interactions, cf. equation (6), aswell as the motor induced rod-rod interaction, cf. equations(8), are defined by overlap integrals. Hence the equationof motion (2) for the probability distribution functionΨ(r, u) is nonlocal and its solution is exceedingly difficult.Assuming that spatial variations are small on the lengthscale of a filament, in order to deal with a local equationone can perform a systematic expansion of the integralswith respect to gradients of the probability distributionfunction, as described in this appendix.The interaction kernel given by equation (7) can beexpressed in terms of Straley coordinates [33], which intwo dimensions are defined byr − r ′ = uζ + u ′ η ,(B.1)with the parameter constraint −L/2 < ζ, η < L/2 (Lthe filament length) and the Jacobian |u × u ′ |. Thus theexcluded-volume interaction, namely equation (6), is determinedby the four-dimensional integral∫ ∫V ex (r, u) = du ′∫L/2dr ′−L/2dζ∫L/2−L/2dη |u × u ′ |· δ (r − r ′ + uζ + u ′ η) Ψ(r ′ , u ′ ) , (B.2)which may be approximated by a Taylor series expansionof the δ-function with respect to η and ζ 2 . Performingthe η and ζ integration one finally obtains∫V ex (r, u) = L 2 du ′ |u × u ′ |{·[1 + L2(u · ∂ r ) 2 + (u ′ · ∂ r ) 2}] Ψ(r, u ′ ) , (B.3)242 Equivalently, after performing the r ′ -integration the shifteddistribution function Ψ (r + uζ + u ′ η) may be expanded.where terms up to second order in the spatial derivativeshave been taken into account. The prefactor L 2 reflectsthe two-dimensional excluded volume.By a similar procedure, the contribution to the activecurrent proportional to α can be evaluated as∫Jt,i a = − αL3 du ′ Ψ(r, u) (1 + u · u ′ )24{(· u i (u · ∂ r )[1 − L2 216 5 (u · ∂ r) 2 + 2 )]3 (u′· ∂ r ) 2+u ′ i(u ′· (∂ r )[1 − L2 216 5 (u′· ∂ r ) 2 + 2 )]}3 (u · ∂ r) 2 Ψ(r, u ′ ).(B.4)One should note that due to the ζ, η-integrations, oddpowers of ∂ r vanish in both expressions, which is themathematical reason underlying the rotational symmetryof equations (15).Appendix C. The moment expansion methodThe method for deriving the coupled equations (15) for thedensity and the orientation field of the filaments from theunderlying Smoluchowski equation, equation (2), is similarto the calculations presented in reference [43], whereit has been used to obtain equations describing the initialstage of spinodal decomposition during the isotropicnematictransition of a three-dimensional hard-rod fluid.In both cases, the starting point is an approximate descriptionof the probability distribution function Ψ(r, u, t)by its moments, but in contrast to a usual liquid crystal asconsidered in reference [43], now the rods are polar withrespect to the motor action. Thus the ±u-symmetry isbroken and the first moment, namely the filament orientationfield t, does not vanish here and cannot be omitted.Hence one has to approximate Ψ(r, u, t) by its first twomoments,Ψ(r, u, t) ≃ 1 ( )ρ(r, t) + 2u · t(r, t) , (C.1)2πwith the filament density ρ(r, t) and the orientation fieldt(r, t), cf. equations (11).The validity of the representation in equation (C.1)can be seen immediately by using it to evaluate the firstmoments which correctly yields ∫ du Ψ(r, u) = ρ(r) and∫du u Ψ(r, u) = t(r). Inserting the currents as definedby equations (3) and (8) as well as applying the gradientexpansion given in Appendix B, then an integrationof equation (2) by ∫ du and ∫ duu yields two evolutionequations for ρ(r, t) and t(r, t), respectively, as given byequations (15).The remaining integrals involved are merely orientationalaverages. If one defines〈A(u)〉 u =∫ ∫ du2π2π A(u) = dθ2π A(θ) ,0(C.2)


52 The European Physical Journal Ewhere θ parameterizes the unit vector u in two dimensions,the following formulas are useful and easily proven: Allmean values depending on odd powers of u vanish as wellas the mean values of any product of |u × u ′ | and oddpowers of u or u ′ , since |u × u ′ | = √ 1 − (u · u ′ ) 2 containsonly even powers of u, u ′ . For the even powers one getsand〈u α u β 〉 u = 1 2 δ αβ , (C.3)〈u α u β u µ u ν 〉 u = 1 8 (δ αβδ µν + δ αµ δ βν + δ αν δ βµ ) (C.4)〈u α u β u µ u ν u σ u τ 〉 u = 1 48 {δ αβδ µν δ στ + perm.} , (C.5)perm. meaning that all permutations of {α, β, µ, ν, σ, τ}generating different index combinations of the Kroneckerdelta products are present. Furthermore one needs∫ dθ〈|u × u ′ ′|〉 u ′ =2π | sin (θ − θ′ ) | = 2 (C.6)πand〈|u × u ′ |u ′ αu ′ β〉 u ′ = − 23π (u αu β − 2δ αβ ) .(C.7)The operator of rotational diffusion in two dimensions asgiven by equation (5) can be expressed as[R] i = [u × ∂ u ] i = δ i3 (u 1 u ′ 2 − u 2 u ′ 1)(C.8)and for the rotational terms, the following partial integrationformula:〈A(u)R [B(u)]〉 u = −〈R [A(u)] B(u)〉 uis crucial to simplify calculations.Appendix D. Derivation of the amplitudeequations(C.9)Here we describe the scheme for the derivation of the twocoupled amplitude equations (29) from the three underlyingnonlinear equations (15). First of all one assumessmall values for the amplitudes X and Y of the spatiallyperiodic deviations from the homogeneous basic state ρ 0and t = 0. At threshold, i.e. at ρ 0 = ρ c and for k c as calculatedin equations (25) and (26), these deviations areeither periodic in the x- or in y-direction as described bythe ansatz in equation (27).Similar to Section 3, the nonlinear equations (15) maybe rewritten in terms of the deviations w = (˜ρ, t x , t y ) fromthe basic state w 0 = (ρ 0 , 0, 0) as follows:∂ t w = L 0 w + N (ρ, t)⎛with N (ρ, t) = ⎝ N ⎞ρ (ρ, t)N x (ρ, t) ⎠ .N y (ρ, t)(D.1)(D.2)The linear operator is defined by equation (18) and thenonlinear operator N includes all the nonlinear termswith respect to ˜ρ and t on the right-hand sides of equations(15).Naturally, as the small expansion parameter the relativedistance from the threshold,ε = ρ 0 − ρ cρ c, (D.3)is chosen. Close to threshold the dynamics of the linearsolution in equation (19) is slow and accordingly a slowtime scaleT = εt(D.4)is introduced allowing the time derivatives in equations(15) to be replaced by∂ t → ε∂ T . (D.5)The solution w is expanded with respect to powers of ε 1/2w = ε 1/2 w 1 + εw 2 + ε 3/2 w 3 + . . . ,with w 1 as in equation (27), as are the nonlinearitiesN = εN 1 + ε 3/2 N 2 + . . . .(D.6)(D.7)Sorting the contributions to equation (D.1) with respectto powers of ε, one ends up with the following hierarchyof equations:ε 1/2 : L 0 w 1 = 0 ,(D.8a)ε : L 0 w 2 = −N ρ (t 1 )e ρ ,(D.8b)ε 3/2 : L 0 w 3 = ∂ T w 1 − L 2 w 1 − ∑N i (ρ 2 , t 1 )e i , (D.8c)i=x,ythat has to be solved successively. We have introducede ρ = (1, 0, 0), e x = (0, 1, 0), e y = (0, 0, 1) and⎛⎞L (2)11 0 0⎜L 2 = ⎝ 0 L (2) ⎟22 L(2) 23⎠(D.9)0 L (2)32 L(2) 33with(1L (2) 1 + D11ρ = − α )∆ − 19 α0 π 24 11520 ∆2 ,1L (2)22ρ = − α 0 96 (∆ + 2∂2 x) − α (11∆ 2 + 64∆∂ 2 )x ,460801L (2)23ρ = − α 0 48 ∂ x∂ y − α720 ∆∂ x∂ y . (D.10)The remaining two matrix elements L (2)32from L (2)23and L(2) 22 by permuting ∂ x and ∂ y .and L(2) 33 followThe equation in O(ε 1/2 ) is just the linear eigenvalueproblem already discussed as equation (16) in Section 3,i.e. it is solved by ˜ρ 1 = 0 andt 1x = X(T )e ikcx + c.c. ,t 1y = Y (T )e ikcy + c.c.(D.11)


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