Two Kinds of Wave in an Oscillating Chemical Solution BYA. T. WINFREE Department of Biological Sciences Purdue University West Lafayette Indiana 47907 USA. Received 6th September 1974 Wavelike phenomena in a chemical solution oscillating at period T fall into two classes (1) diffusion-independent but oscillation-dependent structures repeating at intervals T and (2) diffusion-dependent but oscillation-independent structures derived from threadlike filaments of “ scroll axis ” repeating at intervals TOmuch less than T. This paper experimentally examines a typical example of each. In unfiltered solution one usually observed a third class of wave source forming “ target patterns ” at diverse periods intermediate between T and To. KINEMATIC WAVES AND TRIGGER WAVES Distributed throughout a large enough space any unstirred chemical oscillation generally exhibits local parameter variations affecting the local period.Thus phase gradients develop and steepen even from initial synchrony of the bulk oscillation. Unless reactants and products have identical optical properties these changing phase gradients can be seen as moving bands of colour passing every point at regular intervals of time equal to the bulk oscillation period T. Called “ kinematic waves ” by Kopell and Howard,’ such waves have also been studied theoretically by Smoes and Dreitlein,2 by Th~enes,~ and by Ortoleva and Ross.~’ Kopell and Howard,l and Thoenes have additionally exhibited such waves in the oscillating reagent of Belousov and of Zhabotinsky and Zaikin.’ Provided that the oscillating reagent is initially well stirred and left in a homo- geneous environment no chemical gradients remain.The oscillation period being therefore everywhere the same any phase gradients subsequently introduced by a local disturbance will persist without change until molecular diffusion becomes sig- nificant on a scale embracing the affected area. In the diffusion-free uniform- period approximation the corresponding kinematic waves have been called “ pseudo-waves ”. With or without spatial variations of period such waves merely expose a shallow timing gradient in the spatially distributed but locally autonomous oscillation. Wave velocity is inversely proportional to the steepness of the phase gradient and so has no upper bound.Such waves do not involve diffusion are not conducted through the medium and are not impeded by impermeable barriers. However in sufficiently steep phase gradients diffusion cannot be ignored. In the chemical reagent of Belousov6 and of Zhabotinsky and Zaikin7-called 2 reagent henceforth-a new phenomenon emerges from steep phase gradients which assigns a lower bound to pseudowave velocity a pulse of chemical activity is trig- gered and propagates at a velocity v characteristic of the chemical medium. These have been called “trigger waves ”.9* They are arrested at impermeable barriers and have spatially uniform velocity rather than spatially uniform repeat time. Both 38 A. T. WINFREE solitary trigger waves and regularly spaced wave trains appear in both oscillating and non-oscillating versions of Z reagent.THE SLOWEST PSEUDOWAVES Though pseudowaves slower than the conduction velocity are not observable (because they trigger the faster wave) pseudowaves at all velocities down to this limit are readily demonstrable in oscillating Z reagent. In fact pseudowaves travelling at almost exactly the conduction velocity are the commonest sort. This is because every volume element turns blue spontaneously at intervals Tafter passage of a trigger wave. 35 END OF BULK OSCILLATION 30 25 c -3 2 8 20 M (d 2 cw 0 p, 2 15 10 5 0 0 10 20 40 60 period/s FIG.1.-The periods of three kinds of wave in filtered Z reagent are plotted against age of the reagent.Open circles represent high-frequency waves deriving from scroll rotation all have the same period To,which gradually increases. Solid circles represent bulk oscillation ; its period T approaches infinity after half an hour. Letters represent concentric rings filling the holes left by extinct scroll sources the period Tidentifies them as pseudowaves. Raw data underlying the dotted circle and the open circle connected to D by dashes are given in fig.2. For example when a source of trigger waves abruptly ceases to radiate (see Appendix) the last wave emitted (at t =0) sets up a radial phase gradient as it propa- gates away leaving an initially wave-free hole. But within that hole volume elements at distance r from the centre spontaneously turn blue at times nT+r/u.Thus behind the packet of trigger waves all spaced apart by distance Too,there appear concentric pseudowaves spaced apart by distance Tu > Tov all following the last trigger wave TWO KINDS OF WAVE IN AN OSCILLATING CHEMICAL SOLUTION outward at approximately* the conduction velocity u. This abruptchangeof period is shown in fig. 1 and 2. In fig. 1 are recorded the periods of various phenomena in a 2 mm deep layer of oscillating Z reagent at short intervals during a 40 min period after mixing of the ingredients (see Appendix for details). The open circles record a slow increase in the SCROLL RING @ # PERIOD 13 SECONDS PSEUDOWAVE D SCROLL WAVE PERIOD 29 SECONDS PERIOD II SECONDS event number FIG.2.-Successive closures of the inward wave from a tiny scroll ring are recorded by plotting event time against event number in the upper trail of data points.The lower trail shows a scroll source discontinuing its periodic wave emission and the same space being filled with pseudowaves ; their times of appearance at the centre are recorded. period of diverse trigger-wave sources this is the period of scroll rotation. The solid circles record an increase in the period of bulk oscillations in wave-free regions and their eventual termination after half an hour (note that scroll sources continue). The letters A-F record the intervals between eruption of circular waves at the centre of the “hole ” left by a receding packet of nearly circular trigger waves after its source vanishes each letter represents the mean period of all the waves within one expanding hole.* Since u does increase somewhat with wave spacing it is possible that the conduction slightly outraces the pseudowave. If so then if an impermeable barrier could be implanted without deform- ing the liquid (I have not found a way!) the wave striking one side would continue from the other side after a slight delay proportional to the barrier’s distance from the centre or from the next wave approaching from the centre whichever is less. A. T. WINFREE The observations underlying three of these data points are shown in fig. 2 where the time of each event (appearance of a wave wave passing a marker or collision of waves) is plotted in serial order.The series of 12 observations plotted in the lower right depicts a scroll source emitting progressively less eccentric oval waves at an 11 s period until the last one is nearly circular. Thereafter circular waves appear at 29 s intervals close to the simultaneously-measured period of bulk oscillations. This transition is indicated by the dashed line in fig. 1. The other 5 lettered cases show that the period of these hole-filling target patterns approximates to the bulk oscillation period. After the end of bulk oscillations no such waves appear in new holes. The upper series of observations in fig. 2 shows the 13 s period of a horizontal scroll ring approximately ;t mm in diameter shedding waves to the inside and outside all of which appear circular in projection.This observation is entered in fig. 1 as the dotted open circle. In unfiltered reagent the space between curves T and Toin fig. 1 is typically filled with many trails of data points depicting the diverse and increasing periods of indi-vidual pacemaker centres each generating a “ target pattern ” of concentric trigger- wave circles at its own period intermediate between Tand To. ROTATING PSEUDOWAVES? The phase gradients constituting pseudowaves A-F were radial spanning many cycles of bulk oscillation from centre to receding periphery. But unlike most thermodynamic variables the phase of an oscillation is defined mathematically on the unit circle rather than on the real line. Thus it seems possible in principle for phase to continually increase through any number of complete cycles around a closed path in space.The pseudowave seen on such a circular gradient would rotate about a fixed centre. Such pseudowaves have been invoked to explain the rotating spiral waves seen in 2 reagent but these turn out to be trigger waves by the criteria that their period To is much less than T that they are totally blocked by impermeable barriers and that they appear in non-oscillating reagent as well as in oscillating reagent. Rotating pseudowaves remain to be observed in chemically oscillating media though there may be biological examples.lo ROTATING TRIGGER WAVE SOURCES However the principal mode of diffusion-dependent organization in Z reagent (at the ubiquitous period To)seems to be rotation of appropriately crossed concen- tration gradients.A two-dimensional region (a thin film of liquid) organized in this way emits a rotating trigger wave shaped like an Archimedes’ spiral. Its colour pattern rotates about a slightly wobbly pivot. In three dimensions the pivot becomes a 1-dimensional filament threading through the liquid like a vortex line in classical hydrodynamics. However here there is no fluid motion. From this filament there emerges a scroll-shaped wave of excitation. Its perpendicular cross-section resembles an Archimedes’ spiral. The geometry of this pivot-filament the scroll axis deter- mines the spatial and temporal organization of the oscillating reaction. The scroll axis tends to close in rings except where a nearby interface prevents it.Rings several centimetres in circumference are common. The smallest rings yet seen are about one wavelength in circumference leaving just room enough for organization of the scroll core (itself of circumference equal to one wavelength) around the circular scroll axis. TWO KINDS OF WAVE IN AN OSCILLATING CHEMICAL SOLUTION DlSSECTlON OF A SCROLL RING In order to examine the anatomy of a scroll ring in more detail it is convenient to use a version of the oscillating reagent which conducts waves well while absorbed in the pores of a Millipore filter (see Appendix for details). Impregnated filters can be stacked to make a horizontally-laminated three-dimensional medium with roughly isotropic conduction properties. By assembling such stacks from filters already bearing waves it is possible to induce three-dimensional concentration patterns of peculiar geometry.The waveforms which eventually develop from such contrived initial conditions can be examined in detail by tossing the Millipore stack into a preservative bath then reassembling an image of the wave from its fixed horizontal sections. From computer siniulations of two-dimensional media resembling a thin layer of Z reagent one learns that crossed concentration gradients of the sort needed to create a scroll axis are formed when a solitary wave is suddenly brought into contact with inert medium for example as in fig. 3. The pivot of a spiral wave appears near the FIG.3.-A spiral wave can be created by abutting a block of inert reagent against the endpoint of a plane wave in another block.The spiral's centre appears near the initial point of contact. initial point of contact. Imagine fig. 3 spun about its right edge to form a pair of cylinders the bottom one now containing an outward propagating cylindrical wave. This suggests a trick suitable for implementation in a Millipore stack. Two stacks each about 0.7 mm deep are prepared by stacking 5 filters permeated with Z reagent as described in the Appendix. A hemispherical wave is started in one stack by touching the centre of its upper surface with an electrically heated filament. After a minute the wave becomes a cylinder several mm in diameter extending vertically through the stack and propagating outward. When the second stack still inert is set on top the situation of fig.3 is created in every vertical plane through the axis of the cylinder. The locus of pivots for the spiral wave emerging in each such plane is a horizontal circle roughly coincident with the upper edge of the cylindrical wave at the moment of contact. From this ring-shaped filament a scroll wave emerges propagating in all directions throughout the Millipore stack (fig 4.) Every horizontal cross-section of this wave consists of concentric circles. Every FIG.5.-A scroll ring in a stack of 10 Millipore filters is shown in serial section parallel to the circular scroll axis perpendicular to the axis of circular symmetry. The top side of each filter is shown. They are about 0.15 mm thick. Number 10 was on top.Each panel is a 1cm square. The black slash across each panel shows the section line along which the view in fig. 6 was taken. To face page 431 A. T. WINFREE vertical cross-section through the symmetry axis shows segments of two spirals one on the left and a mirror image on the right . Fig. 4 attempts a schematic of 6 stages during one rotation of the spiral showing only the left side spiral; the right edge of the box is the vertical symmetry axis of the Millipore stack. It will be noted that the wave erupts alternately at equal intervals of one half period through the top and bottom Millipores above and below the initial ring of contact. The inside half of this wave continues inward to annihilation on the symmetry axis while the outer half continues outward.rj 4-r2 FIG.4.-A spiral wave in a rectangular piece of medium is shown at 6 equally spaced times during the cycle. The successive panels were drawn by rotating an involute spiral 60" each time. In the actual experiment circular waves emerged through the opaque upper surface (filter No. lo) approximately above the initial cylinder after 25 s then after 50 s more then after another 50 s then after 40 s. The 40-50 s period of the 3-dimensional wave is close to the 45 s rotation period of a well-developed spiral in a parallel 2-dimensional control experiment in a single thickness of Millipore. (The bulk oscillation period is several minutes). 25 s later the stack was dispersed into cold fixative while the trailing edge of the fourth wave was still passing vertically through filter No.1. Fixed in this way wave thickness is about 3mm. Successive waves follow one another by 13 mm at a velocity of 2 mm/min. Fig. 5 shows the upper surfaces of corresponding 1 cm squares cut out of the ten filters in order from 1 (the bottom) to 10 (the top). Waves propagate with the sharp edge foremost the inner circle is moving inward the outer circles outward. In filters 1 and in 4-7 we also see vertically propagating waves only the trailing edge of one in filter 1 but the full thickness of 3 mm in filters 4-7. At the left edge a wave is TWO KINDS OF WAVE IN AN OSCILLATING CHEMICAL SOLUTION entering from another source. A gas bubble seems to have inhibited propagation between filters 3 and 2.Filters 5 and 8 show air bubbles caught under the glass during photography. Each of the 10 prints is marked with a section line fig. 6 assembles a vertical cross-section through the stack along this line shown as though seen from the top the upper half of each print discarded. The position of each wavefront in each layer is marked by a wedge pointing in the direction of propagation. The imaginary spirals connecting these data include arcs about 3wave spacing above the top filter J 1 CM. x FIG.6.-Wave fronts along the section lines in fig. 5 ale marked by wedges pointing in the direction of propagation the apex at the wave front. The successive sections are stacked 0.15 mm apart in this scale reconstruction of a vertical section through the Millipore stack.An imaginary spiral and scroll core boundary are superimposed. (fixed 25 s = 3cycle after passing a vertical wave) and less than 3 mm below the bottom filter (which shows that wave’s trailing edge as a broad swath). The assumed position of the scroll axis is indicated by two circles bounding the scroll core. Core circumference being 14 mm,9 its diameter is not distinguishable from the + mm thickness of waves fixed in this way. Fig. 6 may be compared with stage 2 of fig. 4. CONCLUSION Wave phenomena of two kinds distinguished by their periods arise in Z reagent Those with period T derive from phase gradients in the spontaneous bulk oscilla- tion. They have been analyzed extensively in the theoretical literature. They have been exhibited in Z reagent in the one-dimensional long-wavelength case.l* Their independence of diffusion has been shown experimentally by their passage through impermeable barriers1 Fig.1 and 2 describe such waves in the two- dimensional case in the short-wavelength limit. Those with period To < T occur only in two- and three-dimensional situations as spirals and scroll waves respectively. Some theoretical discussion of such waves has appeared. * By obstructing their passage with impermeable barriers they have been shown to be “ trigger waves ” critically dependent on diffusion. Fig. 5 and 6 show a ring-shaped scroll wave in Z reagent. Wave phenomena at intermediate periods are much more commonly observed in unfiltered Z reagent.7 Since most of these trigger wave ‘‘ target patterns ” are eliminated by careful filtration and can be restored by deliberate contamination with dust they are viewed as consequemxs of local shorter-period oscillation near a hetero-geneous nucleus.A. T. WINFREE These experiments were made possible by NSF Grant 37947 and an NIH Research Career Development Award. APPENDIX The pseudowave experiment uses an oscillating cerium+ malonate reagent similar to that of Kopell and Howard,l but with excess ferroin to enhance visibility 2.15 g cerous nitrate hexahydrate in 250 ml water 1 volume 36.0 g malonic acid in 250 ml water 2 volumes 13.5 g potassium bromate in 250 ml water 2 volumes 55.0 ml sulphuric acid plus 200 ml water 4 volumes 1.0 ml Triton X-100 surfactant in 15 1.water 1 volume 25 millimolar ferrous 1,lO-phenanthroline sulphate 2 volumes It is important to avoid the E. Merck ferroin widely available in Europe. This is made with a chloride salt ; the chloride is not removed and poisons the reaction. The solution is filtered through a Millipore GSWP 0.2 micron " Millex "filter into a fresh Falcolnware petri dish. Lining the dish with Sylgard silicone resin helps in eliminating pacemakers and the "target patterns " of trigger waves they emit. The dish is placed over a blue-green fluorescent light-box baffled to avoid heating. The lid is coated with a 61mof 0.1 % Triton X-100 to prevent fogging. A magnifying glass is helpful. The few remaining pacemakers produce circular waves (or a hot needle is used if there are none) which must be sheared by gently tilting the dish before the bulk oscillation annihilates them.From their wreckage a diversity of scroll sources emerge generating diversely con- voluted patterns (" intestines ") all characterized by the ubiquitous period and wavelength of the involute spiral wave. Some form tiny scroll rings or fragments of rings interrupted by a single airlliquid interface. Some of these contract until they abruptly vanish leaving a series of closely-packed waves to propagate away. The empty hole expanding around the annihilated source then fills from centre outward with concentric pseudowaves. The times of their appearances at the centre are noted to the nearest second with a stopwatch. Some-times instead their passage by a fixed scratch or bubble on the floor of the dish is noted.The results are similar and are plotted in fig. 1(A-F). The bulk oscillation period is measured by recording the times at which the reagent suddenly turns more transparent (in blue-green light) at a fixed wave-free place in the dish as well as in a separate wave-free dish of the same reagent. The interval between this event and the next appearance of a blue spot in the centre of one of the above holes remains the same within a few seconds attesting the synchrony of both bulk-oscillations. Scroll rotation period is measured by recording the times when the inward circular wave in a scroll ring contracts to a point and vanishes ; or when waves from two or more separate scroll sources collide and annihilate each other the intervals are the same within several percent.Usually several of these diverse sources were watched simultaneously noting event times in as many columns of a table. The times were later plotted and periods were read from the slopes of the resulting line segments as in fig. 2. The results from two independent experi- ments were indistinguishable ; fig. I combines them. Herman Gordon pointed out that these curves become much smoother when observations are restricted to one place in the dish. This is because the oscillation period varies several percent from place to place in the dish and because intervals between wave collisions or wave passages reflect the (generally shorter) period of wave appearances at their source some time earlier.The Millipore stack experiments use a published recipe * but with 1 ml instead of 2 nil of sulphuric acid. Consistent wave propagation depends on the purity of the sodium brom- ate ; even reagent grade samples sometimes need to be recrystallized. This reagent does not oscillate when poured in thin layers exposed to the air presumably due to oxygen interfering with free-radical processes. However it does oscillate in bulk with a period of several minutes when confined in a sealed tube or in a thin layer under oil or in Millipores under a glass coverslip. (Sometimes the reagent poured in a petri dish will oscillate in the deeper meniscus though not in the shallower centre ; then a blue wave starts around the rim of the dish and TWO KINDS OF WAVE IN AN OSCILLATING CHEMICAL SOLUTION propagates into the centre once every several minutes).The more acid reagent * behaves similarly in air but must be spread in a much thinner layer or divided into fine droplets to inhi bit oscillation completely. The filters are 23 mm diameter Millipore GSWP numbered with tiny Magic Marker dots and filled with 2 reagent by floating glossy-side-up on the liquid surface. With nylon forceps they are lifted free stacked on Plexiglas (Perspex) covered with a microscope cover- slip and drained of excess liquid into the corner of a Kleenex tissue. The wave is started with a 3 V penlight its bulb removed and the tungsten filament carefully stretched out to a point which becomes hot but does not glow.The two stacks are joined before a bulk oscillation erases this single cylindrical wave. Because carbon dioxide placques quickly accumulate between the filters the whole process must be terminated within several minutes (at 20°C) the stack is impaled with a needle to assist in orienting the numbered filters later then dispersed in cold 3 % perchloric acid. After several minutes each filter is sealed under a coverslip and photographed on high-contrast film in blue light. About 20 such experiments were run. Scroll rings were caught at several of the stages of rotation depicted in fig. 4. Several stacks produced very complicated structures possibly due to invisible gas bubbles separating the filters. In some of the very regular ones like fig. 5-6 the period of wave eruption on the visible surface of the stack was as long as 70 s.This sometimes happens in a single-Millipore spiraI wave too. The reason is not known but chemical contamination perhaps from sodium chloride or perchloric acid fingerprints is suspected. N. Kopell and L. Howard Science 1973 180 1171. M. Smoes and J. Dreitlein J. Chem. Phys. 1973 59 6277. D. Thoenes Nature (Phys. Sci.) 1973 243 18. P. Ortoleva and J. Ross J. Chem. Phys. 1973,58 5673. P. Ortoleva and J. Ross J. Chem. Phys. 1974 60 5090. B. Belousov Sb. Ref.Rod. Med. 1958 145. A. Zhabotinsky and A. Zaikin Nature 1970 225 535. A. Winfree Science 1972 175 634. E. Zeeinan Towards n Theoretical Biology ed. C. Waddington (Aldine New York 1972) vol. 4 pp. 8-67. A. Winfree Mathematical Probleiiis in Biology Lecture Notes in Biomatlzematics (Victoria Conference 1973) Vol.2. ed. P. van den Driessche (Springer-Verlag Berlin 1974) p. 241. A. Winfree Sci. Amer. 1974 230 82. A. Winfree Muihenmtical Aspects of Chemical aiid Biological Problems and Quaitturn Chemistry ed. D. Cohen (Amer. Math. SOC. Providence 1974) vol. 8 in press.