Destabilization of Three-Dimensional Rotating Chemical Waves in an

Feb 1, 1996 - Scroll waves in isotropic excitable media: Linear instabilities, bifurcations, and restabilized states. Hervé Henry , Vincent ... Forma...
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J. Phys. Chem. 1996, 100, 1975-1983

1975

Destabilization of Three-Dimensional Rotating Chemical Waves in an Inhomogeneous BZ Reaction Sergey Mironov,†,‡ Michael Vinson,§ Scott Mulvey,† and Arkady Pertsov*,† Department of Pharmacology, SUNY Health Sciences Center, Syracuse, New York 13210; Institute of Cellular Biophysics, Puschino, Moscow Region 142292, Russia; and Department of Physics, Shippensburg UniVersity, Shippensburg, PennsylVania 17257 ReceiVed: August 30, 1995; In Final Form: October 24, 1995X

The mechanism of destabilization of rotating vortices of chemical activity (scroll waves) induced by a parameter gradient is studied in a gel-immobilized Belousov-Zhabotinsky (BZ) reaction. Destabilization occurs when a temperature gradient above a certain critical value is applied parallel to the filament of an initially simple scroll wave. Direct observation of the vortex filament using video imaging provides evidence that the mechanism of destabilization involves the spontaneous creation of a helical perturbation of the filament which grows and eventually collides with the boundaries of the medium, fragmenting into multiple filaments. Evidence is presented that is phenomenon is the result of an instability caused by a gradient-induced filament drift.

1. Introduction Nonlinear waves in excitable media often organize themselves into vortex-like patterns of activity, known as “spiral waves” in two dimensions and “scroll waves” in three dimensions (for reviews, see refs 1-4). These vortices have been studied in autocatalytic chemical reactions such as the Belousov-Zhabotinsky (BZ) reaction5,6 and biological systems such as electrochemical propagation in myocardial tissue,7 aggregation of slime molds,8,9 spreading depression waves in the brain and retina,10 and calcium waves in oocytes.11 In the case of myocardial tissue, these vortices are thought to underlie pathological high-frequency rhythms of the heart.12,13 Experimental observations have demonstrated the significant effect of heterogeneities on the dynamics of these vortices (see ref 14 and references therein). One important effect of heterogeneities in three dimensions is the possibility of destabilization of otherwise stable vortices. Observations in threedimensional chemical excitable media have demonstrated that when a sufficiently strong parameter gradient is applied, the vortex pattern evolves into a complicated, turbulent-like state.15 The importance of this phenomenon is that it may serve as a model for the disorganized dynamics observed in the heart during fibrillation, a dangerous pathological arrhythmia.14 However, the mechanism of destabilization remains unclear. In this paper we study this phenomenon in a gel-immobilized BZ reaction in which an inhomogeneity has been introduced by a temperature gradient. We present the first direct observations of the dynamics of the filament (the phase singularity at the center of the scroll4,16,17) during the destabilization process. The rectilinear filament, stable to small perturbations in the absence of a gradient due to the stabilizing effect of filament tension,18 becomes unstable when a sufficiently strong gradient parallel to the filament is applied. We demonstrate that during destabilization the initially rectilinear filament develops a helical perturbation (similar to the effect called “sproing” in ref 19), * To whom correspondence should be addressed at Department of Pharmacology, SUNY Health Sciences Center, Syracuse, NY 13210. E-mail: [email protected]. † SUNY Health Sciences Center. ‡ Institute of Cellular Biophysics. § Shippensburg University. X Abstract published in AdVance ACS Abstracts, January 1, 1996.

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which grows until it interacts with the boundary of the medium and breaks up, leading to a complicated, turbulent-like state. In this paper, we focus mainly on the dynamics and mechanisms of growth of the perturbation, leaving aside its interactions with the boundaries. We find that the gradient-induced filament drift (occurring when the filament has a projection perpendicular to the gradient) is a destabilizing factor, competing with the stabilizing effect of filament tension. 2. Methods The Chemical Reaction. The gel-immobilized BZ reaction was produced with the following recipe (all chemicals from Aldrich): chemical

vol

final concn

1 mL 0.05 M 1 M NaBrO3 5 M H2SO4 0.8 mL 0.2 M 1 M CH2(COOH)2 1 mL 0.05 M 0.025 M ferroin 0.5 mL 6.25 × 10-4 M H2O 6.7 mL agarose (solid) 100 mg in 10 mL of H2O

The sequence of preparation was as follows: The sodium bromate, sulfuric acid, malonic acid, and water were first mixed together. The agarose gel was heated to 70 °C so that the solid agarose would dissolve in 10 mL of water. It was then added to the other chemicals. Finally, the ferroin was added for a final volume of 20 mL. After careful mixing, the solution was poured into rectangular fluorometric 1 × 1 × 5 cm cuvettes and placed in a refrigerator to allow the agarose to solidify. This recipe produces a BZ reaction with convenient time and space scales, and moreover the reaction is visible to the unaided eye: excited regions appear blue due to oxidation of the ferroin. Data Recording. Images of the reaction were acquired using a Cohu-6500 CCD video camera via an EpixBio framegrabber and recorded in digital form on an optical disc. The frame-toframe interval was usually 15 s, providing approximately 2035 frames per rotation cycle. To increase the contrast of the image we used a band-pass filter centered at 490 nm. Image analysis was performed on a Pentium-100 PC computer using software developed by W. Baxter at the Department of Pharmacology, SUNY Health Science Center, Syracuse, NY. © 1996 American Chemical Society

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Figure 1. (A) Schematic of experimental setup for simultaneous recording of two orthogonal projections of scroll waves in BZ reaction. (B) Initiation of a simple scroll using the partition method. The left panel shows a wave front initiated by a silver wire in one compartment just prior to the removal of the partition. The right panel shows curling of the wave front and the formation of the scroll after the partition has been removed. The dashed line shows the location of the partition prior to removal. (C) The top and lateral projections of a simple scroll wave using the experimental setup shown in (A). Excited (oxidized) regions appeared brighter than unexcited regions. The slight curvature of the wave front (seen in the lateral projection) is due to small temperature variations between the center of the cuvette and the boundaries (which are maintained at a constant, and in this case equal, temperature). (D) Schematic visualization of a scroll wave in three dimensions, showing the filament as the phase singularity about which the wave rotates.

Figure 2. (A) A twisted scroll wave, formed from a simple scroll wave such as that of Figure 1C by application of the temperature gradient parallel to the filament. Twist is obtained by maintaining the bottom of the curvette at T1 ) 15 °C, and the top at T2 ) 25 °C, for a gradient of 10 °C/cm. At the moment shown in the figure, the filament is rectilinear. The upper panel is a projection parallel to the filament. Here the wave is rotating in a clockwise sense. The bottom panel shows a projection perpendicular to the filament (lateral projection); here the branching pattern characteristic of a twisted scroll can be seen. (B) Schematic visualization of a twisted scroll in three dimensions. The funnel-like shape is a consequence of the phase lag of the spiral as one travels down the filament. (C) Dependence of scroll wave period and wavelength on temperature (in the absence of gradient).

To study the vortex in three dimensions, we used the setup shown in Figure 1A. The cuvette was placed near an inclined

mirror which allowed simultaneous recording of two orthogonal projections of the reaction (as described in ref 20).

Chemical Waves in an Inhomogeneous BZ Reaction

Figure 3. Verification of the averaging method for determining filament location. (A) Lateral projection of the twisted scroll wave at one instant of time. The horizontal lines located at Z1 and Z2 indicate the z slices that are used in part B. (B) Time-space diagram for the two horizontal slices at Z1 (upper panel of part B) and Z2 (lower panel). This diagram is obtained by stacking slices vertically at sequential times, revealing a characteristic “fir tree” pattern. The filament location is determined by the central line between the branches, giving location X1 for the slice at Z1 and X2 for Z2. (C) The averaging method, described in the text, is shown after being applied to one full rotation of the scroll wave. The dark curve in the middle is the filament. The other dark areas are artifacts of the averaging method, resulting from spatial differences in the rotation period due to the gradient. They can be distinguished from the filament because they change their color and location rapidly when the averaging interval is changed slightly or shifted in time, whereas the filament remains unchanged. The coordinates X1 and X2 (white stars) obtained from the time-space method are superimposed on the filament. The averaging method yields identical coordinates for the filament, and moreover is much more convenient in that the entire filament is determined at once.

Initiation of Scroll Waves. To initiate scroll waves with a controlled position and orientation, we used the partition method.15 This method is illustrated in Figure 1B. In the middle of a cuvette containing the gelled BZ reaction, we placed a plastic screen (0.1 mm thick) along the 5 cm axis of the cuvette, dividing it into two equal compartments. In one of the compartments, a cylindrical wave was initiated by inserting a thin (0.3 mm diameter) silver wire along the short edge of the screen. When a rectilinear breaking edge of this wave, moving along the screen surface, reached the center of the

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Figure 4. Destabilization of a simple scroll wave. The bottom and top of the cuvette are maintained at T1 ) 15 °C and T2 ) 28 °C, respectively, for a gradient of 13 °C/cm. (A) Lateral projections at three different times. At t ) 0 (the moment when the gradient is first applied), the scroll wave is untwisted. At t ) 25 min (center panel), the wave is twisted, but not yet disorganized. At t ) 52 min (lower panel), the scroll wave no longer follows any discernible pattern. (B) Time series of the positions labeled a-c in the top panel of part A. The vertical dotted lines show the moments of time at which the snapshots of part A were taken.

cuvette, the screen was removed, and the breaking edge curled into a simple scroll wave with rectilinear filament. The location of the breaking edge of the wave before removal of the screen determined the location of the filament of the emerging scroll wave. Figure 1C,D shows a simple scroll obtained by the method described above. The Temperature Gradient. To study the effects of temperature gradients, the cuvette was placed between two rectangular stainless steel heat exchangers connected to two separate thermostat-controlled circulating baths. The temperature difference between the two, ∆T ) T2 - T1, varied from 0 to 17 °C. To slow down the reaction and to prevent formation of CO2 bubbles, the lower temperature was set below room temperature at T1 ) 15-17 °C. The gradient was applied perpendicular or parallel to the scroll wave filament. Figure

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Figure 5. Filament evolution during destabilization. The gradient is 5 °C/cm, smaller than in Figure 4 so that the evolution is slower and the earlier stages may be seen in more detail. (A) The top panels show snapshots of the lateral projection of the wave at different times. (B) The bottom panels show the filament as obtained from the averaging method. (The filament is the dark curve in each panel.)

2A,B shows a twisted scroll wave formed after application of a gradient of 10°/cm parallel to the filament. Twist was the result of differences in the natural rotation period along the filament due to the gradient (at the top is was 280 s; at the bottom, 730 s). The effect of gradients on local filament dynamics can be estimated from Figure 2C, which presents the experimental dependence of wavelength and rotation period on temperature for a simple scroll wave in the absence of a gradient. Filament Reconstruction. The filament was detected by time-averaging sequential frames over one rotation cycle of the scroll wave. The method is based on the established fact that the filament is surrounded by an area (the core) which is never excited by the circulating excitation wave.19,21 As a consequence, after time averaging this area appeared darker (i.e., less excited) than the rest of the medium, thus revealing the shape of the filament. To suppress variations in illumination that were comparable to the signal itself, background subtraction was required. The background was generated by applying a digital low-pass filter to one of the averaged images. We used a conic linear 55 × 55 pixel filter (the frames themselves were 100 × 300 pixels). The data obtained using this technique agree well with the time-space plot method of filament reconstruction.20 This is illustrated in Figure 3, which shows the reconstruction of a nonrectilinear filament, using both the averaging and the time-space methods. The averaging method allows determination of the whole filament at one, instead of requiring pointby-point reconstruction as in the time-space method. The time evolution of the filament was observed by shifting the averaging interval in time. The typical time resolution in our filament reconstructions was 2.5 min, corresponding to approximately one-third of a rotation period. The averaging method is limited to cases where the filament does not drift more than one core diameter in one rotation period. In our experiments, this method required a local filament speed of less than approximately 120 µm/min. In all of the experiments reported here, this condition was met. 3. Results To study the phenomenon of destabilization, a simple scroll was placed in a gradient parallel to the filament. Figure 4 shows evolution of such a scroll to a complicated, disorganized state, as shown from the lateral projection, under the influence of a constant-temperature gradient of 13 °C/cm. At the earliest times, the scroll wave was untwisted (Figure 4A, t ) 0 min). At a later time, the difference in rotation period due to the temperature gradient caused the scroll to develop twist, as can

be seen by the formation of conic patterns in the lateral projection (part A, t ) 25 min). By t ) 52 min the vortex had broken up into a complicated turbulent-like state near the cold end of the cuvette, as shown in the bottom panel of part A. Optical density recordings (part B) at the three points labeled a-c show that initially the phase was constant along a vertical line, as expected for an untwisted vertical scroll. During the following 30 min, the twist increased as seen from the increased phase difference between the three time series. After 30 min the patterns at points b and c become complex and nonperiodic, indicating the development of the turbulent state. To understand the mechanism of destabilization, we directly reconstruct the dynamics of the filament using the timeaveraging technique described above. We show that the instability starts as a spontaneous perturbation to the initially rectilinear filament. The perturbation then grows until the filament encounters a boundary of the medium and breaks, producing daughter filaments. Figure 5 shows evolution of the filament in a gradient of 5 °C/cm; this gradient is smaller than that of Figure 4. (The lower gradient significantly slows down the evolution of the perturbation, allowing more detailed observations.) At early times (0-67 min, about seven rotation periods), the filament was approximately straight. At 67 min, significant twist had developed (as can be seen from the upper panel), but the filament was still straight. After 92 min, however, a perturbation developed in the lower (colder) end, appearing as a small bump in the filament. The delay from the application of the gradient to the development of the initial perturbation comprised about 10 rotation cycles. As soon as the perturbation appeared, it grew rapidly (note the times indicated in the figure) and moved up the filament, until the entire filament had deviated from the initial rectilinear shape. In the case shown, the growth continued until the end of the filament touched the frontal surface of the cuvette and broke into two fragments (t ) 142 min). The smaller fragment (near the bottom, shown by an arrow) disappeared quickly, while the larger survived and continued to evolve, moving toward the left side of the cuvette. Three-dimensional reconstruction shows that during destabilization the filament acquires a helical shape. This can be seen by combining two orthogonal projections of the filament, as shown in Figure 6. The top projection reveals a horseshoe shape (Figure 6A), whereas the lateral projection shows an “S” shape (Figure 6B); from these two projections the helical shape of the filament can be deduced. By identifying points on the filament as they appear in the two projections, we obtained a

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Figure 6. Three-dimensional reconstruction of the filament, from two orthogonal projections recorded simultaneously. (A) The top projection. (B) The lateral projection. The left-hand panels show pictures of the wave front; the right-hand panels are projections of the filament obtained from the averaging method. The points marked f in (A) and (B) represent the same point on the filament, as do the points marked f′. (C) Threedimensional visualization of the filament, constructed from the data in parts A and B. Recording of the projection shown in (A), which is normally obscured by the heat exchangers, required removing the cuvette from the temperature gradient. Due to the slow filament evolution, this does not affect the reconstruction. (During the one rotation cycle required for averaging, the filament did not change appreciably.)

set of (x,y,z) values for points along the filament. These data allowed the filament to be reconstructed in three dimensions (Figure 6C), revealing its helical shape. The distance over which the helix made one full rotation was 0.6 cm, approximately the same as the twist rate at the time of the initial perturbation. The radius of the helix was approximately 0.2 cm at the moment shown. Near the top and the bottom of the cuvette the filament deviates from the helical shape due to the no-flux boundary conditions, which require the filament to be perpendicular to the wall. Our observations provide evidence that as its radius grew, the helix also rotated. A rotating helix projected laterally onto a two-dimensional surface should appear as a sinusoidal curve

that propagates along the axis of the helix (the direction of propagation being determined by the sense of rotation). Figure 7A shows a sequence of projections of a filament in a gradient of 4 °C/cm. In addition to the growth of the perturbation, the apparent propagation of the sinusoidal curve in the upward direction can also be seen, supporting the interpretation of the filament’s motion as a rotating helix with an expanding radius. The filament clearly made oscillatory excursions from its original location (compare filament positions 5 and 11). The oscillations occurred at a rate of about one cycle in 2 h (about 20 times slower than the rotation rate of the scroll wave itself). Figure 7B further illustrates the growing, oscillatory nature of the evolution of the perturbation by showing the maximal

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Figure 7. (A) Sequential filament configuration during development of the instability. The gradient is 4 °C/cm. The time interval between successively numbered filaments is 12 min. Filament number 16 has also collided with the boundary of the cuvette. (B) Maximum and minimum deviations of the horizontal (x) location of the filament as functions of time.

and minimal horizontal (x) deviations of the filament as functions of time. The fact that these curves diverge is further evidence that the perturbation grew in time (until colliding with the boundary of the medium). There is a critical gradient below which the destabilization scenario does not occur. At smaller gradients, twist develops but the scroll reaches an equilibrium configuration with a rectilinear filament. An example of equilibration is shown in Figure 8, with a gradient of 2 °C/cm. At early times, the scroll was untwisted. During the first hour of the experiment, the scroll wave developed twist, finally (after approximately 60 min) reaching an equilibrium configuration which remained stationary for up to 6 h. Equilibration can be seen from the optical density recordings from the two points labeled a and b (part B). During the first hour (left-hand part of B), the phase difference between the two points grew. The right plot shows the same two points 6 h later. One can see significant phase shift (approximately π radians) which is stable and appears to be constant in time. We find that the critical gradient below which equilibration occurs is about 2.8 °C/cm. For gradients below this value, the rectilinear filament remained stable for the duration of the experiment (as long as 6 h); for larger gradients, the perturbation appeared and the filament ultimately destabilized. We never saw an equilibrium, nonrectilinear filament. The time for appearance of the initial perturbation depends on the temperature gradient, as shown in Figure 9. The delay between application of the gradient and development of the first perturbation was longer for gradients closer to the critical value. Our data did not allow us to determine whether the delay grows asymptotically; our accuracy in applying temperature was about 0.25 °C. If there is an asymptote, then, it is very sharp, growing toward infinity within only 0.25 °C; however, the reproducibility of the delay and lack of scatter close to the critical gradient suggest that in fact there is a discontinuity without an asymptote. The delay times were reproducible within one rotation period (approximately 10 min), as can be seen from the scatter in the data of Figure 9. (Note for example there are three measure-

Mironov et al. ments at the gradient of 5 °C/cm.) The delay times depended only on the gradient and not on the average temperature. The destabilization process always followed the same sequence, with the perturbation beginning in the colder part of the cuvette. The evolution of the shape, as well as the time for the instability to appear, followed the same reproducible sequence. When several scrolls were present in the same cuvette, the instability developed almost simultaneously for all of them. The twist of the scroll at the moment of the first appearance of the perturbation depended on the gradient. The larger the gradient, the larger the accumulated twist. At a gradient of 3 °C/cm, the twist rate was 3.9 ( 0.3 rad/cm, whereas at 17 °C/ cm is was 14 ( 1.2 rad/cm. The instability did not occur when the filament was perpendicular to the gradient. Instead, the filament remained rectilinear and drifted almost perpendicular to the gradient without significant deformation. (The average value of the component of the drift velocity parallel to the temperature gradient was about 1.2 ( 0.9% of the total velocity.) Figure 10 shows the drift of two scrolls in our chemical experiments, in a gradient of 8 °C/cm. The scrolls can be seen to drift in opposite directions (Figure 10, parts A and B). The direction of the drift was determined by the sense of the scroll wave rotation. The velocity of drift was approximately constant. Although the gradient was much larger than the gradient required to induce destabilization in the parallel case, the filament remained straight, as can be seen from the projection shown in Figure 10B, where the filament appears as a dark dot. The drift velocity was found to be proportional to the magnitude of the temperature gradient, as shown in Figure 10C. The constant of proportionality between the gradient and the drift velocity was found to be β ) 0.048 ( 0.005 mm2/(deg min). Although the drift in this case does not lead to destabilization, it may be a crucial ingredient in the mechanism of destabilization, as discussed below. 4. Discussion Historically, the hypothesis explaining the destabilization of a simple scroll wave in a gradient was that a large gradient produces a rotation frequency at the hotter end higher than the colder part of the medium can support, leading to multiple wave breaks on small heterogeneities in the medium.15 If this hypothesis were correct, filament visualization would reveal the spontaneous creation of several filaments near the cold end due to these wave breaks. Moreover, since these wave breaks would depend on the presence of random inhomogeneities, the particular pattern of spontaneous filament creation would also be random, varying from one experiment to the next. However, our data provide strong evidence that the destabilization is not due to random effects of the medium; instead, our results indicate that destabilization occurs through a reproducible change in the shape of the filament. Specifically, a small deviation from its initial straight-line shape begins at the cold end of the filament and evolves into a helical shape. The helix grows until it interacts with the boundaries of the medium, leading to the disorganized state. The instability we observe is reminiscent of the “sproing” instability, reported in computational experiments by Henze et al.19 In these experiments (which included the Oregonator model, often used to simulate the BZ reaction), a scroll wave in a homogeneous medium is given an initial twist and allowed to evolve. If the twist is large enough, the initially rectilinear filament rapidly expands (or “sproings”) into a helical shape and may then interact with the boundaries. The mechanism of this phenomenon is unclear. (Keener and Tyson4 have at-

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Figure 8. Equilibration at low gradient. The gradient here is 2 °C/cm. (A) Snapshots at three different times. The top panels show lateral projections of the wave itself; the bottom panels are the filaments (dark curves) obtained from the averaging method. (B) Time series of the pixel intensity at the points marked a and b in the upper left panel of part A. The left-hand graph shows the formation of twist (increasing phase shift) during the first hour. The right-hand graph shows equilibrated phase shift.

tempted to explain this phenomenon by adding higher-order twist terms to the local equations of filament dynamics.22) Some features of our experiments are similar to the “sproing” phenomenon. First, the instability develops in a helical shape. Second, it occurs after the scroll has developed significant twist (see Figure 5). Finally, the observed twist rate for destabilization is about 0.7 rotation/wavelength, roughly consistent with the threshold for “sproing” reported for the Oregonator model in Henze et al. The major difference between our observations and “sproing” is the presence of the temperature gradient. Indeed, “sproing” occurs in homogeneous media in which the twist rate is spatially uniform and is locked in by the periodic boundary conditions.19 In our experiments, due to the gradient, twist is continually being

Figure 9. Dependence of instability time on the gradient. The “delay” is defined as the time between the application of the temperature gradient and the appearance of the first perturbation (as in the t ) 92 min panel of part B of Figure 5). For gradients below 3 °C/cm, no instability was found.

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Figure 10. Drift of scroll waves in a gradient perpendicular to the filament. (A) Snapshots of the wave fronts (left) and filaments (right) of two counterrotating scroll waves S1 and S2 in a gradient of 8 °C/cm. (B) As part A, but at a later time, t ) 90 min. The scroll on the left has drifted to the right; the one on the right has drifted to the left. At still later times, interaction between the waves coming from the faster scroll (S2) and the filament of S1 cause a sudden change in the motion of S1 (not shown). (C) Drift speed as a function of gradient.

created in the interior and being dissipated at the (no-flux) boundaries, and the twist rate is not uniform. The presence of the gradient may play a direct and crucial role in the destabilization process. Our observations of scroll waves perpendicular to the gradient indicate that the gradient introduces a velocity perpendicular to both the gradient and the filament. We propose that this velocity may account for the growth of the perturbation during the destabilization process. This drift velocity is present when there is a component of the filament perpendicular to the gradient (Figure 10). Its direction is perpendicular to both the filament and the gradient, and its magnitude is proportional to the value of the gradient (Figure 10C). These properties of the drift velocity can be summarized by

G ) βT ˆ ×g

(1)

where G is the drift velocity, T ˆ is the unit vector in the direction of the filament, and g is a vector whose magnitude is equal to the temperature gradient and whose direction is along the gradient (pointing cold to hot). This formula assumes that the magnitude of the drift velocity is proportional to the component of the filament that is perpendicular to the gradient. The parameter β is the constant of proportionality between the drift speed and the gradient. Our measurements indicate that β ) 0.048 ( 0.005 mm2/(deg min). The destabilizing effect of the drift velocity can be seen from Figure 11. This figure shows that an initially planar bump will evolve toward a helical shape under the influence of the drift velocity given in eq 1. The gradient velocity is oriented out of

Figure 11. Schematic of the gradient-induced velocities on a filament with a planar bump. Only those portions of the filament with projections perpendicular to the gradient can drift. The upper and lower parts of the bump drift out of the plane of the bump in opposite direction.

the plane of the bump, with the upper and lower portions moving in opposite directions, so as to deform the initially planar bump into a helical shape. When the filament becomes locally helical, the gradient velocity becomes oriented so as to increase the radius of the helix, thus causing a tendency toward destabilization. The destabilizing effect of the gradient velocity is opposed by the stabilizing effect of filament tension.18 At small gradients the stabilizing effect of filament tension prevails, and destabilization does not occur. When the gradient is large enough, the gradient-induced velocity dominates, leading to an increase

Chemical Waves in an Inhomogeneous BZ Reaction in the helical radius and destabilization. As the helical radius increases, the instability develops at an accelerated rate due to the decrease in the filament tension, which is proportional to the local curvature. A question that remains in the gradient picture is what determines the initial perturbation. Since the time to destabilization is quite reproducible (Figure 9), it would appear that it is not merely a random fluctuation. One possibility is that the initial perturbation is provided by the “instantaneous filament”. The instantaneous filament is the line connecting the tips of the spirals as one traverses the filament. For a twisted vortex, the instantaneous filament forms a helix whose pitch is determined by the twist rate and whose radius is the core radius. If this is the case, then a larger core should result in a greater instability, since the stabilizing curvature-induced velocity would be smaller. The instantaneous filament assumption is consistent with the observation that the instability always begins at the cold end of the filament, where the core radius is larger. In addition to the effects of local filament dynamics (both the intrinsic dynamics and the local velocity produced by the gradient), the gradient may also contribute through nonlocal effects. Specifically, the higher-frequency waves emanating from the hot end of the filament may affect the dynamics of the filament near the cold end. Computational studies in two dimensions have indicated that external waves of higher frequency interacting with a spiral wave can cause the spiral to drift.23 In a three-dimensional medium with a gradient, the “external” waves are those emanating from the hotter end of the filament, which may then cause additional drift in the colder end. This is a nonlocal effect which is not accounted for in the local dynamics of refs 4 and 18. It remains to exploit our phenomenological explanation of destabilization into a full theory of filament instability in the presence of a gradient. Specifically, we still need to understand the relationship between our observations and the phenomenon of “sproing”, as well as nonlocal effects and the source of the initial perturbation.

J. Phys. Chem., Vol. 100, No. 5, 1996 1983 Acknowledgment. We thank Drs. Richard Gray, Omer Berenfeld, and Marcel Wellner for carefully reading the manuscript and Mr. William Baxter for his help with the computer visualizations. This work was supported in part by National Heart and Blood Institute Grant HL39707 and American Heart Association Grant-in-Aid 94016950. References and Notes (1) Self-organization: AutowaVes and Structures far from Equilibrium; Krinsky, V., Ed.; Springer-Verlag: Berlin, 1984. (2) Winfree, A. SIAM ReV. 1990, 32, 1. (3) WaVes and Patterns in Chemical and Biological Media, Vol. 49 of Physica D; Swinney, H., Krinsky, V., Eds.; North-Holland: Amsterdam, 1991. (4) Keener, J.; Tyson, J. SIAM ReV. 1992, 34, 1. (5) Zhabotinsky, A.; Zaikin, A. Oscillatory Processes in Biological and Chemical Systems; Pushchino: Russia, on Oka, 1971; Vol. 2, pp 279283. (6) Winfree, A. Science 1972, 175, 634. (7) Davidenko, J.; Pertsov, A.; Salomonsz, R.; Baxter, W.; Jalife, J. Nature 1992, 335, 349. (8) Gerisch, G. Wilhelm Roux ArchiV. Entwick Org. 1965, 156, 127. (9) Siegert, F.; Weijer, C. Physica D 1991, 49, 224. (10) Gorelova, N.; Bures, J. Neurobiology 1983, 14, 353. (11) Lechleiter, J.; Girard, S.; Peralta, E.; Clapham, D. Science 1991, 252, 123. (12) Pertsov, A.; Davidenko, J.; Salomonsz, R.; Baxter, W.; Jalife, J. Circulation Res. 1993, 72, 631. (13) Gray, R.; Jalife, J.; Panfilov, A.; Baxter, W.; Cabo, C.; Davidenko, J.; Pertsov, A. Circulation 1995, 91, 2454. (14) Pertsov, A.; Vinson, M. Philos. Trans. R. Soc. London. A 1994, 347, 687. (15) Pertsov, A.; Aliev, R.; Krinsky, V. Nature 1990, 345, 419. (16) Winfree, A.; Strogatz, S. Nature 1984, 311, 611. (17) Panfilov, A.; Pertsov, A. Dokl. Biophys. 1984, 274, 58. (18) Biktahsev, V.; Holden, A.; Zhang, H. Philos. Trans. R. Soc. London, A 1994, 347, 611. (19) Henze, C.; Lugosi, E.; Winfree, A. Can. J. Phys. 1990, 68, 683. (20) Pertsov, A.; Vinson, M.; Mu¨ller, S. Physica D 1993, 63, 233. (21) Mu¨ller, S.; Plesser, T.; Hess, B. Science 1985, 230, 661. (22) Keener, J. Physica D 1988, 31, 269. (23) Krinsky, V.; Agladze, K. Physica D 1983, 8, 50.

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