Analysis of Anchor-Size Effects on Pinned Scroll Waves and Measurement of Filament Rigidity
Elias Nakouzi, Zulma A. Jiménez, Vadim N. Biktashev, Oliver Steinbock
aa r X i v : . [ n li n . PS ] J a n Analysis of Anchor-Size Effects on Pinned Scroll Waves andMeasurement of Filament Rigidity
Elias Nakouzi, Zulma A. Jim´enez, Vadim N. Biktashev † , and Oliver Steinbock ∗ Florida State University, Department of Chemistryand Biochemistry, Tallahassee, FL 32306-4390 and † University of Exeter, Department of Mathematical Sciences, Exeter, UK
Abstract
Inert, spherical heterogeneities can pin three-dimensional scroll waves in the excitable Belousov-Zhabotinsky reaction. Three pinning sites cause initially circular rotation backbones to approachequilateral triangles. The resulting stationary shapes show convex deviations that increase withdecreasing anchor radii. This dependence is interpreted as a transition between filament termina-tion at large surfaces and true, local pinning of a continuous curve. The shapes of the filamentsegments are described by a hyperbolic cosine function which is predicted by kinematic theory thatconsiders filament tension and rigidity. The latter value is measured as (1.0 ± × − cm /s. PACS numbers: 05.45.-a, 82.40.Ck, 82.40.Qt NTRODUCTION
In systems far-from-equilibrium, macroscopic patterns can emerge from processes at themolecular level. These selforganized structures show fundamental universalities across a widerange of disciplines and applications such as type II superconductors [1], neural networks[2], geochemical systems [3], and reaction-diffusion (RD) media [4]. Frequently studiedexamples are two-dimensional spiral waves in excitable and oscillatory RD systems. Thesewave patterns rotate around a zero-dimensional phase singularity [5]. Their tip describessystem-specific trajectories [6] which, in the simplest case, are circles with radii much smallerthan the wavelength of the spiral. These rotors exist in chemical and biological systems suchas catalytic surface reactions [7] and giant honey bees defending their nests against hornets[8]. Furthermore, spiral waves have been linked to medical phenomena such as contractions ofthe uterus during childbirth [9] and life-threatening cardiac arrhythmias [10]. Many of thesebiological processes occur in sufficiently thick tissue to require a spatially three-dimensionaldescription. Under such conditions, spirals extend to more complex rotors called scroll waves[11].A scroll wave can be viewed as a continuum of stacked spirals rotating around a one-dimensional phase singularity [12]. This curve is called the filament and organizes thesurrounding wave field. The motion of the filament is controlled by its own curvature,associated phase gradients (”twist”) of the vortex, as well as other factors. Examples includeself-shrinking, translating, and chaotic motion. In simple cases the motion of the filamentcan be described by the kinematic equation d s dt = ( α ˆN + β ˆB ) κ, (1)where s is a position vector (pointing at the filament) with the corresponding normal ( ˆN )and binormal ( ˆB ) unit vectors, t is time, and κ is the local filament curvature. The constants α and β are system-specific parameters [13, 14]. Several studies have modeled specific casesusing this equation e.g. [15, 16].For a filament tension of α >
0, the filament contracts and vanishes in finite time. Thiscase is prevalent among most experimental systems [17, 18]. Negative values of α causefilament expansion and lead to scroll wave turbulence [19–21]. The translational drift coef-ficient β controls the motion in the out-of-plane direction and equals zero in systems wherethe activator and inhibitor species have the same diffusion coefficient [13, 22].2he pinning of vortex waves to unexcitable heterogeneities [23, 24] is of importance notonly for fundamental reasons but also due to its potential relevance to cardiac arrythmia.For instance, tachycardia is caused by rotating waves of electrical activity. Recent experi-mental results suggest that these reentrant waves can become pinned to heterogeneities suchas remodeled myocardium [25]. Furthermore, tachycardia might develop into a turbulentstate (ventricular fibrillation) which is a leading cause of sudden cardiac death [26]. Theinfluence of pinning sites on vortices in this turbulent state is entirely unknown and also ourunderstanding of scroll wave pinning in non-turbulent cases is poorly developed.Recently developed experimental procedures allow the deliberate pinning of 3D vorticesin chemical RD media and have opened up a wide range of opportunities for controlledstudies. Examples include investigations of scroll waves pinned to inert obstacles such as tori,double-tori, spheres, and cylinders [24, 27–29]. These experiments have revealed that pinningqualitatively alters the evolution of the filament and often results in life-time enhancementor complete stabilization of vortices that in the absence of pinning sites, would rapidly shrinkand annihilate. For instance, Jim´enez and Steinbock [28] reported that filament loops pinnedto three and four spherical heterogeneities converge to nearly polygonal filaments whiletwo pinning sites either fail to stabilize the vortex or create lens-shaped filaments that arestabilized by short-range filament repulsion. In addition, it was shown that filaments havethe tendency to self-wrap around thin heterogeneities [24] suggesting that stable, pinnedscroll waves should be a common feature in heterogeneous systems.Filament pinning is governed by two basic rules. Firstly, the total topological charge overany (external or internal) closed surface must equal zero [23, 30]. This condition assigns anindividual topological charge to the end point of each filament. For an n -armed vortex thischarge has an absolute value of n and a sign that reflects the sense of rotation. A simpleexample is the circular filament of a one-armed scroll ring pinned to spherical heterogeneities(see Fig. 1(a)). The filament loop touches each sphere twice and the corresponding chargesare +1 and -1 because (as viewed from the sphere’s interior) the rotation sense of the localspirals is different. A similar example is a straight filament spanning from one external wallof a box-like system to another. Secondly, a filament ending on a no-flux boundary mustbe oriented in normal direction to the surface [31]. This condition might be violated onlyin singular events such as a collision of a filament with a wall but will reestablish itself veryquickly. 3n this Article, we utilize the Belousov-Zhabotinsky (BZ) reaction as a convenient exper-imental model system for the study of scroll wave pinning. Traveling waves in this systemare driven by the autocatalytic production and diffusion of bromous acid. Our experimentsreveal a seemingly small effect that exposes an unexpected difference between filament pin-ning to small objects and filament termination at large objects. These two limiting cases areillustrated in Figs. 1(b,c), respectively and address the question in how far the surroundingwave fields enforce a smooth, ”kink-free” transition of the filament line through the inertand impermeable heterogeneity. In addition, we show that the observed effects cannot beexplained in the framework of Eq. (1) but require a higher-order term that we interpret asfilament rigidity [32]. We also report the first measurement of this quantity. EXPERIMENTAL METHODS
The BZ system consists of a bottom gel layer and a top aqueous layer, each of thickness0.48 cm. The reactant concentrations are identical in the two layers: [NaBrO ] = 0.04 mol/L,[malonic acid]= 0.04 mol/L, [H SO ] = 0.16 mol/L, and [Fe(phen) SO ] = 0.5 mmol/L. Atthe present gel composition (0.80% agar w/v), also all diffusion coefficients are expected tobe identical throughout the system. For these reaction conditions, the filament tension isfound to be α = (1 . ± . × − cm /s [33], which implies that free scroll rings collapse.Additionally, free filament motion in binormal direction is not observed and we can henceassume that β ≈ bubbles does not affect our experiments for at least the first four hours ofreaction. All experiments take place at a constant temperature of 21 . ◦ C. Subsequentanalyses are performed using in-house MATLAB scripts.
RESULTS
Figure 2 shows still images from two experiments that differ only in the radius of theemployed beads and the resulting wave patterns. Image contrast stems from the varying ratiobetween the chemically reduced and oxidized form the catalyst (ferroin/ferriin). Accordinglybright and dark regions can be interpreted as excited and excitable areas, respectively.However, this situation is complicated by (unresolved) vertical variations along the depthof the three-dimensional reaction medium. Despite this limitation, the filament loop canbe readily detected from image sequences because, due to the wave rotation around thefilament, it sequentially emits waves in the outward and inward direction. As such, we canextract the filament coordinates by locating the set of pixels which undergo minimal contrastchange for the period of one full rotation. This method for filament detection was pioneeredby Vinson et al. [17].Figure 2 shows the result of this analysis as superposed bright (cyan) curves. At thegiven reaction time of 120 min (which corresponds to approximately 15 rotation periods)the initially circular filament loops have reached a stationary, polygon-like shape. We reem-phasize that in the absence of pinning sites, the filament would remain a circle, self-shrink,and vanish. This collapse clearly does not occur. In the following, we characterize the re-laxation dynamics into the stabilized, pinned state in terms of the distance d between thecenter of a given bead and the opposite filament segment. The distance is measured alongthe height of the equilateral triangle defined by the three pinning sites (see inset in Fig. 3).Two representative examples for the temporal evolution of d are shown in Fig. 3. The data5re well described by compressed exponentials of the form d ( t ) = d ss + ( d − d ss ) e ( − t/τ ) b . (2)The values d and d ss denote the initial loop diameter and the asymptotic distance, respec-tively. Analysis of various data sets yields an average b value of 2.35, which is 68% largerthan the earlier reported value ( b = 1.4) for filaments pinned to two beads [33]. The timeconstant τ varies between 45 to 85 minutes. The high values within this range are typicallyfound for larger filament loops. Note that the observed dynamics are qualitatively differentfrom the contraction of free circular filaments for which the diameter is well approximatedby the simple square root law d ( t ) = d − αt .As suggested by the examples in Fig. 2, the shape of the stationary filament as well asthe corresponding bead-to-filament distance d ss depend on the bead radius R . The results ofsystematic measurements are shown in Fig. 4 for a constant inter-bead distance of ∆ = 7 mm.The steady-state value d ss decreases with increasing values of R . Furthermore, the data fallwithin two simple geometric limits. Firstly, considering filament repulsion and the system’stendency to establish smooth wave patterns, it is unlikely that the tangential vectors ofthe terminating filament pair can form an angle above 180 degrees. Accordingly, the circledefined by the three bead centers constitutes the upper limit of d ss ≥ / √
3. Secondly, onlyfilament attraction could establish a filament angle below 60 degrees but would also causethe detachment of the filament from the pinning bead. Hence, the equilateral triangle withcorner points in the bead centers creates the lower limit of d ss ≤ √ /
2. In Fig. 4 both d ss limits are plotted as dashed lines. We find that our data are closer to the triangular than thecircular limit but nonetheless span about 40 % of the possible range. Unfortunately, we arenot able to pin vortex loops for bead radii below 0.75 mm, possibly because of limitationscreated by the size of the vortices’ natural rotation core. The continuous lines in Figs. 4and 5 relate to a theoretical description that is discussed later.The experimental data shown in Fig. 4 provide valuable information regarding our ini-tial question whether filament termination at large surfaces is qualitatively different fromfilament pinning to small heterogeneities. The former case corresponds to experiments withlarge beads and, because filament interaction is absent, to the limiting case of triangularfilaments. The latter case corresponds to very small beads and the circular limit. Given thenegative slope of the data in Fig. 4, we find that our data strongly support the hypothe-6is of qualitative differences between filament termination and pinning. One can speculatethat the transition from filament pinning to termination occurs if the circumference of theheterogeneity is comparable to the wavelength of the vortex structure. For our BZ system,the wavelength is λ = 4.85 mm which suggests a transition at R ∼ R = 1.0 mm. Figure 5(a) showsthe resulting data which reveal that d ss increases with increasing values of ∆. The dashedlines again represent the limiting cases of perfect circles and triangles. To obtain a betterunderstanding of these results, we remove the trivial linear scaling of d ss (∆) and find forincreasing values of the bead distance a smooth transition from the triangular limit towardthe circular limit (Fig. 5(b)). Experiments with even larger values of ∆ have not yieldedreliable data yet because, among other complications, rogue waves and bulk oscillations tendto interfere with the preparation of these large scroll rings.Additional information on the stationary vortex states can be obtained by analyzing theshape of the filament segments that extend from one bead to another. In most experiments,the deviations between the three individual segments are small and the results shown inthe following are the average shape of the stationary filament segment as measured fromdifferent experiments and different sides but for identical experimental conditions. Figure 6shows filament shapes for two different values of R . The s axis extends along the lineconnecting the two bead centers and is zero halfway between those centers. The ordinate isa generalized form of the earlier d variable and measures the distance of the filament fromthe line that is parallel to the s axis and passes through the third bead center. We find thatthe filament shape d ( s ) is in excellent agreement with the hyperbolic cosine function d ( t ) = d ss − k cosh qs (3)as demonstrated by the solid and dashed curves in Fig. 6. Notice that the values of d ss discussed above (e.g. in the context of Figs. 4, 5) are obtained from direct measurements ofthe filament position and not from fits to Eq. (3).7o gain a better understanding of these experimental results, we performed a semi-phenomenological analysis of the system. We suggest that the filament shape is determinedby the delicate interplay between the filament tension ( α ) and the filament rigidity ( ǫ ).Accordingly the equation ∂ t ~s = α∂ σ ~s − ǫ ( ∂ σ ~s ) ⊥ = 0 (4)describes the equilibrium of the planar filament ~s ( σ ) with σ denoting its arclength. Thisequation is a special case of the asymptotic theory presented in [32] for which several terms(not shown) vanish because (i) the diffusion coefficients of the chemical species in our reactiondiffer only slightly and (ii) the filament curvature is not too high. A more detailed descriptionand analysis of this kinematic model is presented as supplementary material [34]. Thisanalysis also considers the short-range, repulsive interaction between the filament segmentsin the vicinity of the beads, which we describe by a simple exponential decay with a decayconstant p . This assumption is in accord with experimental and numerical studies of spiraland scroll wave interaction [33, 35, 36].If we assume that the arclength σ of the filament segment differs only slightly from thelinear space coordinate s , Eq. (4) readily yields the experimentally observed hyperboliccosine function (Eq. (3)) for the shape of the stationary filament. Furthermore, it identifiesthe fitting parameter q as the square root of the ratio between the filament tension and thefilament rigidity d ( s ) = d ss − kq cosh qs, (5) q = p α/ǫ, (6)where k is an integration constant. Since the filament tension for this BZ system is known( α =(1.4 ± × − cm /s, [33]), Eqs. (5,6) allow us to measure the system-specificfilament rigidity ǫ from the shape of the stationary filament. The results are summarized inTable 1. The average value of ǫ is (1.0 ± × − cm /s. We believe this to be the firstmeasurement of filament rigidity in any excitable or oscillatory system.Notice that the hyperbolic cosine function can be difficult to distinguish from a quadraticpolynomial. Resolving fourth order terms, however, is necessary for the desired measure-ment of q and ǫ because Eq. (5) implies that the second-order term has no q -dependence8 d ( s ) = const − k s + O ( s )). Although both functions involve three fitting parameters,the lowest root mean square deviations are generally obtained for fits with the hyperboliccosine function. For example, the upper data set in Fig. 6 (closed, blue circles) yields rootmean square errors of 9.3 µ m and 10.9 µ m for the cosh and quadratic fits, respectively. Forthe lower data set (open, red circles), these numbers are 4.6 µ m and 5.2 µ m, respectively.Our model predicts that significantly larger values of ∆ (twice and more) cause a transi-tion from a rigidity-dominated to a tension-controlled filament with the latter showing morepronounced deviations from a parabola. However, the execution of such experiments is, asmentioned above, fraught with technical difficulties.Our theoretical analysis also describes the experimentally observed dependencies of sta-tionary filament distance. As detailed in the Supplemental Material [34], we obtain d ss (∆ , R ) = √
32 ∆ + ( 1 pC − π qS )( RqS + C − , (7)where S = sinh q (∆ / − R ) and C = cosh q (∆ / − R ). Equation (7) can be readily com-pared to the measurement results in Figs. 4 and 5. For this purpose, we use the average q value of 3.74 cm − which corresponds to the measured filament tension and rigidity (Eq.(6)) and the appropriate constant bead distance (for Fig. 4) and constant bead radius (forFig. 5). Least square fitting then yields the continuous lines in the latter figures, which arein very good agreement with the experimental data. The fits in Figs. 4 and 5 are carriedout separately for each data set and yield p values of 3.41 cm − and 3.03 cm − , respectively.We also investigated the angle φ between the tangent to the filament at the bead and the s -axis. While the data are not fully conclusive (Table 1), we can tentatively identify sometrends. Firstly our measurements show that there is only a mild dependence of φ on thebead radius R . Secondly, the experimental data suggest that φ increases with increasingvalues of the distance ∆ to saturate at approximately 50 ◦ . This saturation behavior seemsreasonable if one considers that the central portion of long filaments (large ∆) is increasinglyflat. A theoretical description of the angle φ is presented in the Supplemental Material [34]and is in agreement with the experimentally observed trends.All of the above experiments and analyses are carried out for scroll filaments pinned tospherical objects located at the corners of equilateral triangles. Figure 7 shows an exampleof a pinning experiment in which the beads form an isosceles triangle in which the uniqueangle measures 40 ◦ . The still frames are taken at four different times and the corresponding9lament coordinates are superposed in bright cyan color. The filament loop in Figs. 7(a,b)is pinned to all three beads and its curve-shrinking dynamics are similar to the behaviorshown in Fig. 2. However, instead of establishing a stationary state, the filament unpinsfrom the lower bead to create a strongly curved segment that quickly withdraws towardthe upper bead pair [Fig. 7(c)]. The resulting filament loop is not stable and annihilatesas shown in Fig. 7(d). The unpinning from the bottom bead is a direct consequence ofthe small angle between the two lower filament segments. This process shares similaritieswith electric-field-induced unpinning events that were recently reported by Jim´enez andSteinbock [37]. In addition, it is reminiscent of filament loops detaching from planar no-fluxboundaries and hence also related to filament reconnections [38]. Lastly, we note that arecent study of scroll rings pinned to two spherical heterogeneities suggest that detachedfilament loops, like the one shown in Fig. 7(c), could generate stable, lens-shaped structuresif the inter-bead distance ∆ is sufficiently large. More experiments are needed to test thisprediction. CONCLUSIONS
Our experiments provide strong evidence that filaments ending at no-flux boundariesshow different behavior depending on the size of the impermeable heterogeneity. In thesimplest case, a filament terminates at a planar, external wall or other large structure. Inthis situation, wave rotation around the obstacle is irrelevant. However, if the circumferenceof the heterogeneity is comparable to or smaller than the pattern wavelength, rotation mustbe considered and the overall filament will feature shape variations that aim to reducegradients of the surrounding wave field. The latter force can be also interpreted in terms ofa repulsive interaction between the filament segments that end on the same surface.The global filament shapes resulting from this local pinning process cannot be explainedsolely by the contractive motion of the underlying curvature flow (Eq. (1)) but rather reveala higher-order phenomenon that we refer to as filament rigidity. For the specific situation ofthree pinning sites located at the corners of an equilateral triangle, the individual filamentsegments are described by hyperbolic cosine functions that deviate only slightly from simpleparabolas. We note that this outcome is reminiscent not only of the shape but also of thehistory of the catenary (”chain”) curve which Galileo described as an approximate parabola1039].We propose that filament rigidity is also crucial for explaining stationary and/or long-lived states of filaments pinned to thin, cylindrical heterogeneities. Such cases were recentlyreported for three-dimensional BZ systems [24] in which filament loops were attached to longglass rods. From the end points of these rods, the rotation backbone of the vortex extended asa free, C-shaped filament segment that despite its curvature remained essentially stationary.Similarly to our new results, this equilibrium state was likely the result of the antagonisticinterplay between filament tension and rigidity.We believe that future studies should aim to characterize the exact boundary conditionsfor pinned filaments as the current lack of this information severely hampers more detailedanalyses of the experimental observed dependencies. Other challenges include the study ofscroll wave pinning to non-inert heterogeneities such as regions with decreased excitabilityand/or diffusion coefficients. Filaments pinned to such ”soft” anchors can be expected tounpin more readily than filaments attached to inert and impermeable pinning sites. Weemphasize that both types of anchor regions are relevant to excitable and oscillatory RDsystems in biology where heterogeneities result from numerous sources including variationsin cell density, cell type, and gap junctions as well as anatomical features such as bloodvessels.
ACKNOWLEDGEMENT
This material is based upon work supported by the National Science Foundation underGrant No. 1213259. 11
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14) mm , respectively. The images are obtained120 min after the start of the reaction and show some CO bubbles (dark spots).
50 10066.577.588.5 t (min) d ( mm ) − d ss(1) = 6.6 − d ss(2) = 5.8 d FIG. 3 (Color online) Relaxation dynamics of the pinned filament into the stationary state. Thedistance d is measured from one bead center to the midpoint of the filament fragment opposite toit. This distance is shown schematically in the inset (dashed line). The experimental data sets areobtained for a bead distance of 6 mm (open, red circles) and 7 mm (closed, blue circles). The beadradius in both cases is 1 mm. Solid curves represent best-fit compressed exponentials. .0 1.5 2.06.06.26.46.66.88.08.2 d ss ( mm ) R (mm)
FIG. 4. Distance d ss of the stationary filament from the opposite bead as a function of theradius R of the pinning beads. The distance is measured along the central symmetry line of theequilateral bead triangle. The two dashed lines are the geometric limits of a circular (top) andtriangular (bottom) filament. With respect to the three bead centers, they equal the diameter ofthe circumcircle and the height of the bead triangle, respectively. The continuous curve is the bestfit of Eq. (7) to the experimental data. The bead distance is kept constant at 7 mm. .6 0.8 1 1.20.40.60.811.21.4 ∆ (cm) d ss ( c m ) (a) ∆ (cm) d ss / ∆ (b) FIG. 5. Distance d ss of the stationary filament from the opposite bead as a function of thebead distance ∆ (a). In (b) the same data are shown in terms of the ratio d ss / ∆. As in Fig. 4,the dashed lines correspond to the geometric limits of a circular (top) and triangular (bottom)filament. The continuous curves are the best fit of Eq. (7) to the experimental data. The beadradius is kept constant at 1.0 mm. s (mm) d ( mm ) FIG. 6 (Color online) Averaged shape of the stationary filament for R = 0.75 mm (open, redcircles) and 1.5 mm (closed, blue circles). The dashed and solid curves represent the correspondinghyperbolic cosine fits (Eq. (3)) which allow the measurement of the filament rigidity ǫ (see Table 1).For both experiments the bead distance is ∆ = 7 mm. a) (b)(c) (d) FIG. 7 (Color online) Still frames illustrating the evolution of a filament pinned to three beadslocated on an isosceles triangle. The snapshots are taken at t = 27 min (a), 67 min (b), 123 min(c), and 140 min (d). The filament unpins at the tight angle and collapses. r X i v : . [ n li n . PS ] J a n Supplemental Material:Analysis of Anchor-Size Effects on Pinned Scroll Waves andMeasurement of Filament Rigidity
Elias Nakouzi,
1, 2
Zulma A. Jim´enez,
1, 2
Vadim N. Biktashev † ,
1, 2 and Oliver Steinbock
1, 21
Florida State University, Department of Chemistryand Biochemistry, Tallahassee, FL 32306-4390 † University of Exeter, Department of Mathematical Sciences, Exeter, UK
1n the main paper, we have derived the shape of the stationary filament without any detailedassumptions regarding the boundary conditions at the pinning bead and the nature of therepulsive filament interaction. In the following, we present a more detailed analysis that,despite the simplifications made, aims to describe additional details including the depen-dencies of the central distance d ss between the filament and the opposing bead center on thebead radius R and the bead distance ∆. We reemphasize that the following analysis of ourexperiments is semi-phenomenological.We suggest that the filament shape is determined by the delicate interplay between thefilament tension, filament rigidity, and filament repulsion close to the beads, ∂ t ~s = α∂ σ ~s − ǫ (cid:0) ∂ σ ~s (cid:1) ⊥ + b | ∂ σ ~s | ∂ σ ~s + ( ~v ( ~s )) ⊥ = 0 , which is an equation of equilibrium of the filament, where the first three terms are as in theasymptotic theory presented in [1], with γ = e = b = 0 because the diffusion coefficientsof all the reagents are close to each other, γ = α and e = ǫ to agree with the notations withthe main paper, and the last term is added to describe phenomenologically the repulsion offilaments from each other in the vicinity of the bead. ∆ Rxy φ θ XY FIG. 1.
Sizes, coordinates, and angles.
We assume that the repulsion distance is muchsmaller than the distance between the beads and theshape of a filament is symmetric so we consider onehalf of it, from one bead to its middle point betweentwo beads. To define the repulsion ”force”, we as-sume that all the filaments lie in one plane, that twofilaments touching the same bead are mirror reflec-tions of each other, introduce Cartesian coordinatesin the plane with the origin at the center of the bead, x axis along the mirror symmetry axis, and focus onthe filament with y >
0. In these terms, we assume,following numerical and experimental results in [2, 3], that the repulsion velocity decreasesexponentially with the distance to the mirror, | ~v | = a exp( − py ), where a > p > φ of the filament with respect to the mirrorline, so e.g. a filament perpendicular to the mirror line is not affected by its reflection. Let σ
2e the arclength coordinate along the filament, starting from the point it touches the bead,and let φ ( σ ) be the angle of the tangent vector ∂ σ ~s with respect to the x axis, so the localfilament curvature is κ ( σ ) = ∂ σ φ . Then, rewriting the above equation, we obtain κ ′′ = Γ κ + Bκ + A e − py cos( φ ) , (1) φ ′ = κ, (2) y ′ = sin( φ ) , (3)where Γ = α/ǫ , B = 1 + b /ǫ , and A = a/ǫ . The dash designates differentiation with respectto σ and the last two equations are definitions of φ and y added to complete the system.The first simplification we make is to assume that the curvature is not too high throughoutthe length of the filament, so we can neglect the Bκ term. Further, we assume thatthe filament does not change its orientation significantly within the range of the repulsivevelocity. Then the filament curvature profile is described by the equation κ ′′ ≈ Γ κ + A e − p sin φ ( R + σ ) cos( φ ) , (4)the general solution of which is κ = C e qσ + C e − qσ + Lp sin φ − q e − p sin( φ ) σ , where q = Γ / , L = A cos( φ ) e − p sin( φ ) R , and we assume that q = p sin( φ ) (an equalityhere would be a rare event in any case). The constants C and C depend on the boundaryconditions. Ideally, further boundary conditions for this equation should be obtained fromthe same asymptotics that were used to derive the filament equations of motions; these arenot available at present. In our simplified approach, we suppose that both κ (0) and κ ′ (0)are zero or negligibly small, then C = L q ( q + p sin φ ) ,C = L q ( q − p sin φ ) . (5)The scale of the inner region is p − as beyond a few of those lengths the interactionterm is negligible. The consistency condition of our approximation is that the change of φ throughout the inner region, which assuming sin φ ∝
1, cos φ ∝
1, is ∆ φ ∝ A/p , is small.3or σ beyond the inner region, we can neglect the term proportional to exp( − p sin φ σ ), sothe solution is simply κ ≈ C e qσ + C e − qσ = − κ m cosh ( q ( σ − σ m ))where κ m > σ m is the arclength coordinate of the middle point. With account of (5), thisgives sin( φ ) = qp tanh( qσ m ) . (6)In particular, this predicts a plateau for φ at large values of σ m , satisfyingsin( ¯ φ ) = qp . (7)Notice that the latter relation is in good agreement with the experimental data presentedin Table 1 of the main paper because the measured angles φ are essentially constant. Eventhe small decrease of φ for the largest bead radius ( R ∼ R -dependent decrease of σ m , consistent with the more accurate formula (6). For φ ≈ ◦ , q = ( α/ǫ ) / , α = 1 . × − cm / s, and ǫ = 10 − cm / s, the characteristic length scale offilament repulsion is then estimated as p − ≈ . X ( s ) = s Z cos( κ m sinh( qs ) /q ) d s,Y ( s ) = − s Z sin( κ m sinh( qs ) /q ) d s, where ( X, Y ) are Cartesian coordinates centered at the midpoint with the X along thetangent to the filament at that point, and s = σ − σ m is the arclength coordinate measuredfrom the midpoint. These integrals cannot be evaluated in elementary functions, but usingthe assumption of smallness of curvature already made, in terms of the filament angle θ inthe ( X, Y ) coordinate system, we can approximate sin( θ ) ≈ θ , cos( θ ) ≈
1, leading simply to Y ( X ) = κ m q (1 − cosh( qX )) .
4n the same approximation we have φ = π/ θ ≈ π − κ m sinh( qX ) /q where the π/ x, y ) and ( X, Y ) frames(see figure 1). Combining these results with the previously obtained (6), we obtain, afterelementary transformations, the following expression for the midpoint curvature κ m = q p cosh( qσ m ) − πq qσ m ) . We estimate the the distance d ss from the center of a bead to the middle of the contraposedfilament as d ss = ∆ cos( π/
6) + R sin( θ ) + | Y ( − σ m ) | (see figure 1), and the estimate for σ m ,in the same limit of small θ as already used above, is σ m = ∆ / − R . This results in thefollowing: d ss (∆ , R ) = √
32 ∆ + (cid:18) p C − π q S (cid:19) ( RqS + C − , where S = sinh (cid:18) q (∆ / − R ) (cid:19) and C = cosh (cid:18) q (∆ / − R ) (cid:19) , which is the equation (7) ofthe main paper. [1] H. Dierckx, H. Verschelde, ¨O. Selsil, and V. Biktashev, Phys. Rev. Lett. , 174102 (2012).[2] M.-A. Bray and J. P. Wikswo, Phys. Rev. Lett. , 238303 (2003).[3] Z. Jim´enez and O. Steinbock, Phys. Rev. Lett. , 098301 (2012)., 098301 (2012).