J . Phys. Chem. 1994,98, 7452-7454
1452
Ring-Shaped Waves of Inhibition in the Belousov-Zhabotinsky Reaction Milob Marek and Petr KaBtslnek Department of Chemical Engineering, Prague Institute of Chemical Technology, Technicka 5, 16628 Prague 6, Czech Republic
Stefan C. Miiller’ Max- Planck-Institut fur molekulare Physiologie, Rheinlanddamm 201, 0-441 39 Dortmund, Germany Received: May 5, 1994’
Both inhibiting (reducing) and activating (oxidizing) reaction4iffusion waves were observed in a shallow liquid layer of the Belousov-Zhabotinsky reaction under the exclusion of oxygen. The wave velocity of the rarely observed reducing waves is lower than the velocity of the oxidizing waves. It increases with increasing malonic acid concentration and depends only slightly on the concentration of the other reaction components, in contrast to the dependence of oxidizing waves. Growing and shrinking ring patterns resulting from the interaction of both types of waves were observed for the first time and are described by spectrophotometrically determined space-time portraits. The reducing wave is classified as a trigger wave of B r ions. An a priori estimate of its velocity is in agreement with the currently estimated rate of B r generation. The possible connection to spatiotemporal patterns of morphogenes is pointed out.
Chemical waves are a frequently investigated phenomenon of dynamic pattern formation under nonequilibrium conditions. They occur in diverse systems such as liquid reacting solutions, developmental processes in cells and living tissues, catalytic reactions on platinum surfaces, or the course of information transmission in neural and cardiac tissue.’ The model case for these traveling reaction4iffusion fronts is the Belousov-Zhabotinsky (BZ) reaction, preferably in its ferroin-catalyzed version. In a front a redox transition from the reduced red state of the catalyst takes place to its oxidized blue form, indicating a fast excitation of the solution via autocatalysis of an activator species (HBr02) followed by a slow relaxation back to the quiescent red state. The blue fronts thus formed propagate through the red solution layer as expanding circles or rotating spirals.* On rare occasions an inverse situation has become known, when an inhibitory reducing front moves into an activated oxidized territory of the r e a ~ t i o n . ~Red disks expanding into a blue oxidizing environment are observed at low levels of the ratio of the reducing substrate (malonic acid) to the oxidizing agent (bromate) and move with lower velocity than the blue oxidized fronts. In the rear portion of each reducing front an oxidized region appears that propagates with higher velocity. When it overtakes the reducing front, the oxidizing wave disappears (see Figure 1A). It has been suggested that this phenomenon is analogous to “incremental propagation” used in electrophysiology to describe propagation of pulses of diminishing a m p l i t ~ d e . ~ However, Tyson and Fife5 proposed earlier that the observation of wave cancellation resulted from the low optical contrast and that stationary wave propagation might actually be established. In this report we investigate the dependence of the velocity of oxidizing and reducing waves on the initial conditions (concentrations of the reactants). Interesting scenarios are studied that evolve when both types of fronts coexist in the same layer. In particular, growing and shrinking ring patterns are reported that result from the competitive dynamics of these complementary propagation processes. It is shown that both propagation of reducing waves and cancellation of oxidizing and reducing waves may occur, depending on the global malonic acid concentration and the actual concentration history in the studied pacemaker.
* To whom correspondence should be addressed. Abstract published in Aduunce ACS Absrracrs, July 15, 1994.
0022-3654/94/2098-7452~04.50/0
Finally, we speculate on the mechanism of generation of reducing trigger waves. Solutions were prepared in a suitable range of concentrations ofNaBrO3 (0.20-0.45 M),NaBr(0.001-0.006 M), HzS04 (0.150.30 M), malonic acid (0.008-0.045 M), and ferroin (0.005 M). A thin layer of the solution (depth 0.5 mm) was sandwiched between two glass plates of 62-mm diameter to exclude the influence of oxygen from the air.6 Recently, it was found that the presence of oxygen can profoundly affect the number of B r ions generated in the oxidation of bromomalonic acid by the Ce4+ catalyst,’ and thus it is advisable to control the possible oxygen transfer. Temperature was kept constant at 25 f 0.1 OC. Pattern formation was observed visually and recorded a t a specific wavelength (490 nm, bandwidth 10 nm) by a charge-coupled device camera (Hamamatsu C 3077) and stored on a video recorder. The patterns in the sandwiched layer were investigated as a function of the initial malonic acid (MA) concentration. At the upper concentration values (0.024-0.045 M), several generations of reducing waves (RW) traveling between oxidizing waves (OW) can be observed, whereas at the lower values (0.008-0.00I5 M) reducing waves prevail. The lifetime of the reaction under the latter conditions is about 10 min. During this period only one or two generations of RW’s appear. In the entire investigated range of MA concentrations, the lifetime of visible pattern formation decreases from 40 min to almost zero with decreasing substrate concentration. The remarkable interaction and competition of blue and red fronts that will be focused on in the following can be observed for intermediate MA concentrations (0.015-0.024 M). The patterns in Figure 1A illustrate typical features of the observed scenario: A reducing wave begins to grow as a red spot (dark in the picture) with an initial velocity of 35 ”1s; after approximately 40 s a new blue oxidizing wave (bright in the picture) appears and propagates toward the RW with a velocity of about 167 pm/s. Within this period, the O W decelerates, until it catches up with the RW, leaving a thin red (dark) circle. This thin red circle then may decay, propagate in the outward direction, or grow to the outside and the inside with almost equal velocities on both sides. The latter behavior leads to a ring-shaped wave of inhibition, as shown the contrastenhanced image of Figure 1B. 0 1994 American Chemical Society
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Figure 2. Velocities of the oxidizing (open squares) and reducing (full squares) waves as a function of initial concentrations. Standard concentrations: [ N a B Q ] = 0.358 M, [NaBr] = 0.002 M, [H2SO4] = 0.2 M, [MA] = 0.005 M, [ferroin] = 0.005 M. In each graph one concentration is varied and the others remain constant.
Figure 1. (A) Overall view of the generated concentration patterns of oxidizing (OW) and reducing waves (RW) in a shallow layer of the Belousov-Zhabotinsky reaction in a Petri dish (diameter 62 mm). The red R W s appear as dark rings or spots on a blue (bright) background. (B) Snapshot of a ring-shaped reducing wave, shown as a contrastenhanced digital image. The black line indicates the spatial cut along which the space-time portrait of Figure 3 was extracted. Initial concentrations: 0.358 M NaBrO3,0.003 M NaBr, 0.30 M H2SO4.0.0198 M MA, 0.005 M ferroin. Image area approximately 1 1 X 1 1 mm2.
Obviously, the different propagation velocitiesof the coexisting waves provide a major key to the obsefyed pattern dynamics. In fact, as shown in Figure 2, the oxidizing fronts are always faster than the reducing ones. Furthermore, one finds that within the investigated concentrationranges the RW velocity depends mainly on the malonic acid concentration and very little on sulfuric acid and bromate concentrations, respectively. This behavior is just the opposite of the well-known dependence of the OW velocity on both H 2 S O 4 and NaBrO3 concentrations and the almost negligible influence of MA on OW propagation. Generation and evolution of the ring-shaped (doughnut) wave in Figure 1B are recorded in the space-time portrait of Figure 3A. At the bottom of the portrait (at time t l ) a RW disk is in the process of expanding into the surrounding medium. At time 12 a new wave is excited at the center of the now reduced and excitabledisk. Due to its higher front velocity is catches up with the RW and movesalong with it, until at time 23 the inner boundary
of the thin reduced ring that is left induces the R W propagation in the opposite direction, that is, back toward the center of the original disk (time t4). At time 25 the medium is again oxidized at the former location of the thin reduced ring (time t3), giving rise to a blue ring inside the growing red one (time fa). This sequence of events may occur repeatedly, creating a concentric set of expandingand shrinking rings. The prerequisite is that the original center keeps emitting activating and inhibiting pulses. The spatial intensity profiles of Figure 3B were extracted at the specified times 11 to t6 along the same cut through the ring pattern (compare Figure 1B). They reveal that the steepnessof the RW front is much smaller than that of the OW front. Figure 4 shows in quantitative detail that the velocity of the outer wave front remains constant during the disk-to-ring transition, whereas the emerging inner activating wave front starts to expand rapidly, slows down in the wake of the inhibitory wave, and reverses its direction to attain a velocity (now as a reducing wave front) about equal in magnitude to that of the outer reducing wave. Hence, we can conclude that velocities of both inner and outer reducing waves are stationary, and this confirms the speculation of Tyson and Fife.5 The authors believe that the propagating red reducing wave is a trigger wave. One can speculate that the mechanism of the generationof the target pattern is as follows: The reducing wave is connected with the front of increased B r concentration which is generated by the sequence of reactions in the BZ system responsiblefor B r production. At the center of the reduced spot the concentration of H B r O 2 is decreased to its stationary value (or the concentrationof B r is increased). The stimulationexcites the neighboring points and induces the trigger wave front of catalyst reduction (red spot) which starts to propagate into the oxidized medium. Then a wave front of catalyst oxidiation(OW)
7454 The Journal of Physical Chemistry, Vol. 98, No. 31, 1994
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propagation), or a sufficient level of B r concentration together with the excitability of the reduced state is preserved. Then reduced waves propagate again in both directions (cf. Figure 4). For a numerical estimatewe recall that thevelocityof the oxidized wave is dependent on the square root of the rate of generation of HBr02, which is approximately equal to 40 M2 s.I0 On the other hand, the velocity of the reduced wave is determined by the rate of B r generation which is given as 1 M2 s by Farsterling et al. in ref 6. Thus, the ratio of the velocities of oxidizing and reducing waves should be = 6.3, which qualitatively agrees with the results of our experimental observations. Developmental and control processes in living cells and tissues are frequently connected with the evolution of spatial and spatiotemporal patterns of morphogens and other controlling species, commonly classified as activators and inhibitors. We can view the observationsof ring patterns arising due to dynamic interaction of RW and OW as an experimental confirmation of such prepatternsarising in a controllableway in a simple chemical reaction-diffusion system.
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Acknowledgment. Fruitfuldiscussionswith A. M. Zhabotinsky and Zs. Nagy-Ungvaraiare acknowledged. We thank M. Dahlem for help with the graphical displays.
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Figure 3. (A, top) Space-time portrait showing the evolution of the ring-shaped wave along the coordinate x indicated in Figure 1B as a black line. Dark areas represent the reduced state of the reaction. Concentrations as specified in Figure 1B. (B,bottom) Spatial profiles of transmitted light intensity Z (wavelength 490 nm) along the coordinate x at times 11 to 16, as indicated in the space-time portrait of (A).
arises at the center and travels into the back of the reduced wave having high Br: concentration. It slows down when approaching the RW according to the actual high local value of B r concentration,obeyingthedispersionrelation. When theoxidized wave (HBr02 wave front) approaches the reduced wave ( B r peak), a competition between the processes consuming HBrO2 and producing B r takes place (see, for example, the original Field-KiirbNoyes mechanism8.9). The HBr02 front of the oxidized wave is consumed in the reaction with B r . Either the excitabilityis lost and both types of waves annihilate(decremental
References and Notes (1) Holden, A., Markus, M., Othmer, M. G., Eds. Nonlinear Wave Processes in Excitable Media; Plenum: New York,1990. Swinney, H. L., Krinsky, V. I., Eds. Waves and Patterns in Chemical and Biological Media. Physica D 1991, 49. (2) Field, R. J., Burger, M., Eds. Oscillations and Traveling Waves in Chemical Systems; Wiley: New York, 1985. (3) Smoes, M. L. In Dynamics ofsynergetic Systems; Haken, H., Ed.; Springer: Berlin, 1980; p 80. (4) Zhabotinsky, A. M.; Buchholtz, F.; Kyatkin, A. B.; Epstein, I. R.J. Phys. Chem. 1993.97.7578. ( 5 ) Tyson, J. J.; Fife, P. C. J. Chem. Phys. 1980, 73, 2224. (6) The inverted top of a Petri dish was used as the container bottom. The inverted bottom of the Petri dish was inserted as the top plate, positioned on a spacer ring made of Teflon. (7) Fhterling, H. D.;Stuk, L.; Barrand, A.; McCormick, W.D.J. Phys. Chem. 1993.97.2623. (8) Field, R. J.; K(lr(is, E.; Noyes, R. M. J. Am. Chem. Soc. 1970,94,
8649. (9) Field, R. J.; F(lrsterling, H. D.J. Phys. Chem. 1986, 90, 5400. (10) Nagy-Ungvarai, Zs.;Tyson, J. J.; H a , B. J. Phys. Chem. 1989.93, 707.
