Scroll Waves Pinned to Moving Heterogeneities
aa r X i v : . [ n li n . PS ] D ec Scroll Waves Pinned to Moving Heterogeneities
Hua Ke, Zhihui Zhang, and Oliver Steinbock
Florida State University, Department of Chemistry and Biochemistry, Tallahassee, FL 32306-4390 (Dated: July 8, 2018)Three-dimensional excitable systems can selforganize vortex patterns that rotate around one-dimensional phase singularities called filaments. In experiments with the Belousov-Zhabotinskyreaction and numerical simulations, we pin these scroll waves to moving heterogeneities and demon-strate the controlled repositioning of their rotation centers. If the pinning site extends only alonga portion of the filament, the phase singularity is stretched out along the trajectory of the hetero-geneity which effectively writes the singularity into the system. Its trailing end point follows theheterogeneity with a lower velocity. This velocity, its dependence on the placement of the anchor,and the shape of the filament are explained by a curvature flow model.
PACS numbers: 05.45.-a, 82.40.Ck, 82.40.QtKeywords:
Processes far from equilibrium can create complex pat-terns that are difficult to predict from their atomistic orlocal dynamics. This emergence of spatial complexity of-ten results from comparably simple transport processes.A classic example are reaction-diffusion media which gen-erate dissipative structures such as stationary Turing pat-terns, traveling waves, and spatio-temporal chaos [1–3].These structures are universal in the sense that they areobserved across a wide range of physical, chemical, andbiological experiments. Specifically, rotating spiral wavesof excitation are observed in systems as diverse as activegalaxies [4], catalytic reactions [5], and bee colonies [6].In addition, they can orchestrate important biologicalfunctions such as the timing of contraction waves dur-ing child birth [7] or induce life-threatening conditionssuch as cardiac arrhythmias [8].While spiral waves have been studied intensively overthe past decades, their three-dimensional counter-partshave attracted less attention. These scroll waves rotatearound one-dimensional phase singularities called fila-ments. In general, these space curves are not static butmove according to their local curvature κ and differencein rotation phase (“twist”) [9–11]. In simple cases, thismotion obeys d s d t = ακ ˆ N , (1)where s , ˆ N and α denote the filament position, its unitnormal vector, and a system-specific line tension, respec-tively. Negative values of α can induce a turbulent mo-tion of the filament [9, 12], whereas positive values causecurve shrinking dynamics for which filament loops annihi-late and filaments connecting external surfaces convergeto straight lines.Recent studies show that filaments can attach to in-active heterogeneities [13, 14]. Most experiments on thistype of vortex pinning employ the Belousov-Zhabotinsky(BZ) reaction [15] which is an important model of ex-citable and oscillatory reaction-diffusion media. Scrollwaves, however, exist also in biological systems such as the human heart [8] for which pinning could occur atanatomical features (e.g. blood vessel and papillary mus-cle insertion points) as well as infarction-induced remod-eled myocardium. Regardless of the specific system, pin-ning of scroll waves implies wave rotation around theheterogeneity whereas simple filament termination is ob-served at heterogeneities much larger than the free ro-tation orbit [16]. Pinning is subject to topological con-straints, alters the rotation frequency, reshapes the globalwave field, and potentially induces twist [13, 16–18]. Re-cent studies have also shown that scroll waves self-wraparound thin cylindrical heterogeneities [19] and unpin dueto advective perturbations such as external electric fields[20].In this Letter, we report the pinning of scroll waves tomoving heterogeneities and show that a partially pinnedfilament stretches out along the trajectory of the anchor.The tail end of the filament does not remain stationarybut follows the heterogeneity at a speed that is indepen-dent of the anchor speed. Its velocity and shape dependon geometric aspects and the curvature flow dynamics ofthe homogeneous system. These experimental and nu-merical results open up interesting possibilities for thestudy of excitable systems with dynamic heterogeneities.Our experiments use a thick layer of BZ solution in acylindrical glass vessel (diameter 5.6 cm). The systemhas a free solution-air interface and its viscosity is in-creased by addition of xanthan gum (0.4 % w/v) and agar(0.05 % w/v). The initial concentrations of the reactantsare: [NaBrO ] = 62 mmol/L, [H SO ] = 175 mmol/L,[malonic acid] = 48 mmol/L, and [Fe(phen) SO ] =37.5 mmol/L. Details regarding the chemical prepara-tion and viscosity measurements have been published in[21]. All experiments are carried out at room tempera-ture. We use a monochrome video camera equipped witha dichroic blue filter to monitor the chemical wave pat-terns. The heterogeneity is a vertical glass rod (diameter1.1 mm) attached to a motor-driven linear actuator. Therod is submerged into the solution from the top down tocreate a constant gap of depth d between the bottom ofthe rod and the surface of the container base [Fig. 1(a)]. BZ solution pin to scroll waverod speed v glass rod glass vessel bottomcamera h d ab cd e r FIG. 1: (color online) (a) Schematic drawing of the exper-imental set-up. (b)-(e) Image sequence of two scroll waves.The right vortex is pinned to a rightward moving glass rod.Time between subsequent frames: 20, 48, and 87 min. Fieldof view: 2.3 cm × In our experiments, we vary the value of d between 0.2and 0.75 cm while keeping h constant at 1.1 ± ±
30 s and theirwavelength is about 0.5 cm. Initially both filaments arelinear and oriented parallel to the optical axis of our set-up. The associated wave fields are untwisted. Accord-ingly the three-dimensional vortices are detected as sim-ple spiral-shaped patterns (b). We then pin the right vor-tex to a glass rod which appears as a small disk-shapedregion in (c). We emphasize that the rod does not touchthe bottom of the reaction vessel but by choice, generatesa gap d of 0.45 mm. After five rotation periods, we beginto translate this anchor rightwards at a constant speedof v r = 0.1 mm/min (d). In response, the pinned scrollwave loses its initial, pseudo-two-dimensional characterand a diffuse, bright (excited) region is formed in thewake of the anchor. We continue to observe wave rota-tion around the moving rod (see movie in [22]) but alsodetect a trailing spiral-shaped feature (e). Notice thatthe unpinned vortex on the left is essentially unaffectedby these processes.We interpret the observed deformation of the pinnedscroll wave as the result of an increasingly deformed fil-ament. While its top portion is anchored to the movingglass rod, its unpinned connection to the base of the re-action vessel becomes stretched out along the trajectory x (cm) t ( m i n ) x (cm) t ( m i n ) ba FIG. 2: Space-time plots of scroll waves pinned to a movingglass rod. The intensity profiles are obtained along the tra-jectory of the rod. The experiments in (a) and (b) differ onlyin the gap height underneath the rod, which equals d ≈ of the rod. This stretching process is governed by i) thetopological requirement of a continuous filament connec-tion between the glass rod and the lower system bound-ary and ii) the flux-related requirement that filaments atNeumann boundaries must terminate in normal directionto the boundary. Accordingly, the pattern in Fig. 1(e)can be understood as a pinned (and probably twisted)scroll wave in the top portion of the system, a morehorizontally oriented filament left of the anchor, and adown-curving filament terminus near the lower systemboundary. The latter two regions account for the broadand diffuse feature behind the rod and its spiral-shapedtermination.The dynamics of scroll waves pinned to moving hetero-geneities are further analyzed in Fig. 2. Both space-timeplots are constructed from intensity profiles along thetrajectory of the rod but describe an experiment with anegligibly small gap underneath the rod in (a) and theexperiment shown in Fig. 1(b)-(e) for which d = 0.45 mm.The moving rod itself generates the bright, diagonal bandthat connects the lower left to the upper right corner ofthe plots. The thinner bright bands result from exci-tation waves of the pinned scroll wave. Notice the V-shaped features left of the rod in (b) that are absent in(a). These features are caused by the alternating emis-sion of left- and rightward moving pulses and are henceevidence for a rotating vortex. Accordingly, they corre-spond to the trailing end of the scroll wave filament andallow us to analyze its position and velocity.Figure 3 analyzes the elongation of partially pinnedfilaments in more detail. Figure 3(a) shows the tem-poral evolution of the distance L between the rod andthe trailing filament end for three representative experi-ments that differ only in the gap height d . Notice that L is the length of the filament’s projection into the imageplane. The data sets reveal a linear increase of L . The L ( c m ) t (min) −4 v t ( c m / s ) d (cm −1 ) ba FIG. 3: (color online) (a) Temporal evolution of the filamentlength L . Diamonds, triangles, and squares correspond togap sizes of d = 0.2, 0.45, and 0.75 cm, respectively. Thedashed lines are obtained by linear regression of these threedata sets. (b) Velocity of the trailing filament terminus as afunction of the inverse gap size (open circles). The straightlines are fits assuming v t ∝ /d (continuous, black), validity ofEq. 3 (dashed, red), and a circular termination of the filament(dotted, red). rate of filament elongation equals the difference v r − v t between the externally controlled rod speed v r and thereaction-diffusion-controlled velocity of the trailing fila-ment end v t . Figure 3(b) shows the latter speed as afunction of the inverse gap distance 1 /d . In these exper-iments, the rod speed was kept constant at 0.1 mm/min(i.e. 1.67 × − cm/s). The data are well described by v t = δ/d and yield an average of δ = 3.1 × − cm /s(solid black line). The red lines are discussed later. No-tice that v t cannot be larger than v r and 1 /d cannot besmaller 1 /h (here 0.9 cm − ).Our experimental results reveal only two-dimensionalprojections of the spatially three-dimensional wave pat-terns and filament shapes. To obtain a better under-standing of the unresolved vertical dimension, we per-formed numerical simulations using the Barkley model[23]: FIG. 4: (color online) Numerical simulation of a scroll wavepartially pinned to a moving anchor. (a) Snapshot of thethree-dimensional wave field v (orange) and the cylindricalheterogeneity (cyan). (b) Partial top view of the same pat-tern. (c) Time-space plot generated from a sequence of imagessimilar to the one in (b). (d) Superposition of seven filamentcurves (blueish) obtained during a single rotation period. Theanchor moves from the dotted, cyan position to the solid, cyanposition. Fitting of Eq. (2) to the filaments yields the redcurve. ∂u∂t = D ∇ u + 1 ǫ (cid:26) u (1 − u ) (cid:18) u − v + ba (cid:19)(cid:27) , (2a) ∂v∂t = D ∇ v + u − v. (2b)Although this dimensionless model is not derived from areaction mechanism, the variables u and v can be asso-ciated to the concentrations of the autocatalytic speciesHBrO and ferriin (Fe(phen) ), respectively. Our simu-lations use the parameter set ( D, ǫ, a, b ) = (1.0, 0.02, 1.1,0.18) which generates an excitable system in which sta-ble scroll waves exist [24]. Since the diffusion coefficients D in Eqs. (2a,b) are identical, the filament tension obeys α = D and filaments with small curvature and twist donot move in binormal direction [24, 25]. Accordingly,unperturbed, planar filaments perform curve-shrinkingdynamics within their initial plane of confinement. Allsimulations are based on forward Euler integration with atime step of 6 × − . The box-shaped system is resolvedby 600 × ×
150 grid points at a spacing of 0 . u, v ) =(0 , v r = 0.33). This cylinder extends onlythrough the top half of the system. Solid (orange) re-gions indicate that the local v values are high ( v > . v over theentire range of vertical z values for each ( x, y ) location.A representative example of the resulting image data isshown in Fig. 4b. The snapshot qualitatively agrees withthe experimental data shown in Fig. 1e. The small dif-ferences between our computational and experimental re-sults are likely due to a more pronounced twist of the sim-ulated vortex and/or local effects caused by the Stokesflow in our experiments. Figure 4c is a space-time plotgenerated from the temporal changes of the projectiondata. Its overall structure is very similar to the exper-imental results in Fig. 2b, thus supporting our earlierinterpretation.In the following, we discuss the physical origins of theobserved filament dynamics. Figure 4d combines sevensnapshots of the filament obtained during one rotationperiod of the vortex. The overall pattern resembles abundle of helices. This structure is the result of the lo-cal rotation around nearly circular trajectory and someweak twist caused by the partial pinning to the translat-ing anchor. The bundle clearly reveals the stretched outstructure of the filament and shows a sharp, nearly per-pendicular transition between a horizontal mid-sectionand the pinned top portion. At the lower terminus, thefilament is oriented perpendicular to the system bound-ary and highly curved. This curvature controls the mo-tion of the trailing end point according to Eq. (1). Wefind that the shape of the filament is well described byan analytical solution of Eq. (1) that had been previouslyconsidered in the context of freely moving filaments [26] and ideal grain boundary motion in two dimensions [27] x ( z ) = − αv t ln cos (cid:0) v t α ( z − z ) (cid:1) + x . (3)This curve has a constant hairpin-like shape and moveswith a constant speed v t that is related to the asymptotic,maximal height w of the curve according to v t = πα/ (2 w ) . (4)The solid (red) curve in Fig. 4d is the best fit of Eq. (3)to the helix bundle. Notice that we only evaluate datawith x <
62 because the abrupt transition to the cylindri-cal anchor is not captured by this description. We findthat the fit captures the shape of the filament bundlewell and the asymptotic height ( w = 17.3) of the curveis only slightly larger than the gap ( d = 15) between theanchor and the lower system boundary. Furthermore,the fit yields α = 1.02 which is very close to the sys-tem’s known filament tension of 1.0. We conclude thatEq. (3) provides a very good description of the shape ofthe elongating filaments.Equations (3) and (4) can also be used to interpretour experimental measurements of v t if we assume that w = d . We first establish the filament tension α fromindependent experiments in which we follow the free col-lapse of scroll rings. In accordance with Eq. (1), theradius R of their circular filament obeys d R/ d t = − α/R and yields α = 2.05 × − cm /s. On the basis of Eq. (4),this value is used to plot the dashed, red curves inFig. 3(b). The graph is nearly identical with the pro-portionality fit (black curve) and hence a good descrip-tion of the experimental data. For comparison, we alsographed the dependence expected for a trailing filamentthat terminates as a circular segment of radius d . Thespeed of such as termination point is given by v t = α/d .The slope of the corresponding curve (red dotted line inFig. 3[b]) is π/ w = d fails.In conclusion, we have shown that scroll waves can bepinned to moving heterogeneities. Partial pinning of ascroll wave stretches the filament along the trajectoryof the anchor. In this process the terminus of the fila-ment is not stationary but follows the anchor at a lowerspeed that is determined by the filament’s local curvatureat the system boundary. We expect that filaments canbe also stretched out along nonlinear trajectories, whichprovides a powerful tool for preparing arbitrary shapesincluding examples that reveal filament interaction andreconnection events [28].This material is based upon work supported by theNational Science Foundation under Grant No. 1213259. [1] N. Tompkins, N. Li, C. Girabawe, M. Heymann, G. B.Ermentrout, I. R. Epstein, and S. Fraden, Proc. Natl.Acad. Sci. USA , 4397 (2014).[2] A. Padirac, T. Fujii, A. Est´evez-Torres, and Y. Rondelez,J. Am. Chem. Soc. , 14586 (2013).[3] M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys.
851 (1993).[4] L. S. Schulman and P. E. Seiden, Science , 425 (1986).[5] G. Ertl, Angew. Chem. Int. Ed , 3524 (2008).[6] G. Kastberger, E. Schmelzer, and I. Kranner, PLoS ONE , e3141 (2008).[7] E. Pervolaraki and A. V. Holden, Biosystems , 63(2013).[8] E. M. Cherry and F. H. Fenton, Am. J. Physiol. HeartCirc. Physiol. , H2451 (2012).[9] V. N. Biktashev, A. V. Holden, and H. Zhang, Phil.Trans. Roy. Soc. London A , 611 (1994).[10] J. P. Keener and J. J. Tyson, SIAM Rev. , 1 (1992).[11] M. Vinson, S. Mironov, S. Mulvey, and A. Pertsov, Na-ture , 477 (1997).[12] S. Alonso, F. Sagu´es, and A. S. Mikhailov, Science ,1722 (2003).[13] Z. A. Jim´enez, B. Marts, and O. Steinbock, Phys. Rev.Lett. , 244101 (2009).[14] A. M. Pertsov, M. Wellner, M. Vinson, and J. Jalife,Phys. Rev. Lett. , 2738 (2000).[15] A. S. Mikhailov and K. Showalter, Phys. Rep. , 79 (2006).[16] E. Nakouzi, Z. A. Jim´enez, V. N. Biktashev, and O.Steinbock, Phys. Rev. E , 042902 (2014).[17] M. Vinson, A. Pertsov, and J. Jalife, Physica D , 119(1993).[18] S. Dutta and O. Steinbock, J. Phys. Chem. Lett. , 945(2011).[19] Z. A. Jim´enez and O. Steinbock, Phys. Rev. E , 036205(2012).[20] Z. A. Jim´enez, Z. Zhang, and O. Steinbock, Phys. Rev.E , 052918 (2013).[21] H. Ke, Z. Zhang, and O. Steinbock, J. Phys. Chem. A , 6819 (2014).[22] See Supplemental Material at URL for movies of typicalexperiments.[23] D. Margerit and D. Barkley, Chaos , 636 (2002).[24] S. Alonso, R. K¨ahler, A. S. Mikhailov, and F. Sagu´es,Phys. Rev. E , 056201 (2004).[25] M. Gabbay, E. Ott, and P. N. Guzdar, Phys. Rev. Lett.78, 2012 (1997).[26] S. Dutta and O. Steinbock, Phys. Rev. E , 055202(R)(2010).[27] W. W. Mullins, J. Appl. Phys. , 900 (1956).[28] D. Kupitz and M. J. B. Hauser, J. Phys. Chem. A117