A Model Illustrating Amplification of Perturbations in an Excitable Medium BY RICHARD AND RICHARD J. FIELD M. NOYES Department of Chemistry University of Oregon Eugene Oregon 97403 U.S.A. Received 6th July 1974 The oscillatory Belousov-Zhabotinskii reaction can be modelled approximately by five irreversible steps A+Y-+X (M1) X+Y+P (Ma B+ X-+2X+ Z 043) 2X+Q (M4) Z+fY. (M5) These equations are based on the chemical equalities X = HBr02 Y = Br- 2 = 2Ce(IV) and A = B = BrO;. If the rate constants kMl to kMs are assigned by experimental estimates from oxybromine chemistry the kinetic behaviour of the model depends critically upon the remaining parameters kMs and$ Whenfdoes not differ too greatly from unity and when kMsis not too large the steady state is unstable to perturbation and the system oscillates by describing a limit cycle traject- ory.Whenfand kMs lie outside the range of instability the steady state is stable to very small perturb- ations. However the steady state may still be excitable so that perturbation of the control inter- mediate Y by a few percent will instigate a single excursion during which concentrations of X Y and Z change by factors of about lo5before the system returns to the original steady state. This ability of a small perturbation of the steady state to trigger a major response by the system is just the type of behaviour necessary to explain the initiation of a trigger-wave by a heterogeneous"pacemaker " as has been observed by Winfree. The same type of excitability of a steady state has important implica- tions for the understanding of biochemical control mechanisms.The so-called Belousov-Zhabotinskii reaction occurs in one of the most versatile of chemical systems. If a sulphuric acid solution of bromate ion and malonic acid contains a catalytic amount of a one-equivalent redox couple (such as Ce(II1)-Ce(IV) or Mn(1I)-Mn(II1)) with a reduction potential between about 1.0 and 1.5 V several remarkable types of behaviour are possible. In a stirred homogeneous solution the degree of oxidation of the redox couple may oscillate repeatedly with time. In an unstirred solution containing a gradient in temperature or in the concentration of some reactant variations in the phase of local oscillation can cause alternating bands of oxidation and reduction to propagate through the system ; these phase-related oscillations have been called pseudo-waves by Winfree.' In an unstirred but initially homogeneous solution local disturbances such as dust particles may generate regions of oxidation that propagate more rapidly than individual molecules could diffuse and leave refractory reduced regions behind them ; these disturbances are called trigger-waves by Winfree.' Finally if developing trigger-waves are disturbed in various ways they may develop very complicated spiral structures called scroll-waves by Winfree.2 References to these various types of behaviour may be found in a recent re vie^.^ 21 AMPLIFICATION OF PERTURBATIONS The detailed chemical mechanism of the temporal oscillations has now been elucidatedy4# and a simplified mathematical model developed to exhibit the same type of limit cycle behaviour.(j The same mechanism has been used both quali- tatively ’and quantitatively to describe the chemical processes taking place in the sharp leading edge of a trigger-wave.Although the previous discussion *demonstrated how a region of oxidation could propagate through the medium as a trigger-wave it did not concern itself with the initiation of such oxidation. In the present paper we show that the same model suggests how a modest perturbation in the concentration of bromide ion could initiate a transient region of oxidation that would then propagate as a trigger-wave. CHEMICAL MECHANISM The significant features of the chemical mechanism can be summarized by the following processes BrO +Br-+2H++HBr02 +HOBr HBrO +Br-+H++2HOBr BrO; +HBrO +H+-+2Br02 +H,O Ce3++BrOz+H+-,Ce4++HBrO 2HBrO,-+BrO +HOBr +H+ (C4) nCe4++BrCH(COOH),+nCe3+ +Br-+oxidized products.(C5) Steps (Cl) (C2) and (C4) are assumed to be bimolecular elementary processes involving oxygen atom transfer and accompanied by rapid proton transfers ; the HOBr so produced is rapidly consumed directly or indirectly with bromination of malonic acid. Step (C3a) is rate-determining for the overall process of (C3a) +2(C3b). The Ce4+ produced in step (C3b) is consumed in step ((25) by oxidation of bromo- malonic acid and other organic species with production of bromide ion. The complete chemical mechanism is considerably more compli~ated,~ but the simplified version presented here is sufficient to explain the oscillatory behaviour of the system.COMPUTATIONAL MODEL The significant kinetic features of the chemical mechanism can be simulated by the model we have called the Oregonator? A+Y+X X+Y+P B+X+2X+Z 2x-Q z+.. This computational model can be related to the chemical mechanism by the identities A =B =BrO; X =HBrO, Y =Br- and Z =2Ce(lV). The Oregonator functions because the switched intermediate X is generated by steps (Ml) and (M3) that are zero and first order in X and is destroyed by steps (M2) and (M4)that are first and second order in X. When the concentration of the control R. J. FIELD AND R. M. NOYES intermediate Y is sufficiently large the concentration of X attains a steady state approximated by Xsmall.When the concentration of Y is sufficiently small X attains a different steady state approximated by X,,,,,. kM3B-kM2Y Xlnrgc = 2kM4 ' The concentration of X is switched between the two steady states whenever the concentration of Y attains the critical value Ycrit. ycrit = kM3B/kM2. (3) The regeneration intermediate is produced in significant amount only when X is in the Xlargc steady state and it regenerates Y until the critical concentration is attained and X is switched to Xsma,,.The Concentration of Zthen decreases and Yis destroyed until it again reaches Ycritand permits the concentration of X to be switched again. Thedifferential equations describing temporal behaviour can be cast in terms of the dimensionless variables u 9 p and z and the dimensionless parameters q s and w.= JkM3B/kMlA kM5 w= Jk,,khf3AB' These differential equations then become da/dz = s(q -qu +a -qa2) dvld7 = (1 Is)( -9 -9a+.fP) (12) dp/dr = w(u-/I). (13) We have examined the Oregonator under conditions such that the rate constants for the first four steps correspond to our best estimates for the situation in a solution with 0.06M KBrO and 0.8 M H2S04. Then q = 8.375 x s = 77.27 and w = 0.1610 kM5. When these values are selected the kinetic behaviour depends upon the two parametersf and kM5. APPLICATIONS TO UNSTABLE AND EXCITABLE SYSTEMS It is possible to determine steady state values for the concentrations of the three intermediates such that daldz = dq/dz = dp/dr = 0.Then a secular equation analysis AMPLIFICATION OF PERTURBATIONS of the linearized kinetic equations will reveal whether small perturbations from the steady state will decay or whether they will grow and lead to a limit cycle trajectory around the steady state. I I 1 (a9935,4so.os) 0.0 1.0 2.0 F FIG.1.-Regions of stable and unstable Oregonator steady states for q = 8.375~ s = 77.27 and IV = 0.1610kM,. The steady state is stable to oscillation except for sufficiently small values of kM5 with fin the neighbourhood of unity. The results of such an analysis are shown in fig. 1. This figure differs slightly from one presented previously which was based only on a sufficient condition for instability of the steady state.It is impossible to have a steady state if k, = 0 and fig. 1 demonstrates that the steady state is always stable for sufficiently large values of kM5. Unstable steady states are observed for some finite positive values of kM5provided the stoichiometric parameter fdoes not differ too much from unity. Stable steady states with fappreciably less than unity correspond to Y < Ycritand X = Xlarge. Stable steady states with f appreciably greater than unity correspond to Y > Ycrit and x = L3ll. Trigger-waves of the type studied by Winfree involve regions of oxidation that advance into regions of excess bromide ion (Y > Ycrit)that are marginally stable to oscillation. Such a region can be modelled by a stable point near to the right bound- ary of the region of instability defined by fig.1. As Winfree has pointed out trigger-waves initiate at a " pacemaker " such as a speck of dust that locally perturbs the marginally stable solution and generates a region of oxidation that then propagates as a trigger-wave. If our model is appro6 priate a small perturbation of the steady state should generate a single pulse of oxidation followed by return to the steady state. Moreover the system should be particularly sensitive to perturbation of the control intermediate Y. Exactly these predictions are demonstrated by fig. 2 to 5 illustrating the behaviour of a system initially at the stable but excitable steady state for f= 1.5 and kM5= 2 s-I. Fig. 2 to 4 are logarithmic plots showing the behaviour of the various inter- mediates following a discontinuous reduction of Y (or q) by 6.5 % at time z = 153.R. 3. FIELD AND R. M. NOYES I -__ 1----r--I-si 1 I I 50 150.0 300.0 450.0 7 FIG. 2.-Logarithmic plot of v(Y or Br-) following a discontinuous perturbation of -6.5 :< from the steady state. Perturbation occurred at T = 153. 7 Fro. 3.-Logarithmic plot of a(X or HBrOJ following a 6.5 % decrease of 7from the steady state situation. Perturbation occurred at 7 = 153. 7 FIQ,4.-Logarithmic plot of p(Z or Ce(1V)) following a 6.5 % decrease of 7 from the steady state situation. Perturbation occutced at 7 = 153. AMPLIFICATION OF PERTURBATIONS -lr----- 7 II I 1 I500 I600 1700 180.0 7 FIG.5.-Linear plot of q(Y or Br-) showing subsequent effects of 6.0 and of 6.5 % discontinuous decreases from the steady state at T = 153.The sensitivity of the system clearly is very dependent upon the magnitude of the perturbation. A similar reduction of 6 % generated a ripple barely observable on this logarithmic scale. Fig. 5is a linear plot of the region near Ycritshowing that the 6.5 % perturbation is sufficient to cause the system to attain the critical switching condition while the 6 % perturbation is not sufficient to do so. Even much larger perturbations of the switched (Xor a) and regeneration (2or p) intermediates generated only very minor excursions from the steady state situation. DISCUSSION The above calculations offer a convenient model for the very large (non-linear!) amplification of a finite but minor chemical perturbation of a steady state system.Just such amplification must be involved in whatever process initiates a trigger-wave in a marginally stable Belousov-Zhabotinskii solution. Similar amplification is presumably involved whenever a minor change in conditions triggers a complicated sequence of events in a living organism or biological system. The transmission of a nerve impulse is an example of just such a triggered response. Although these calculations were initiated to model the excitable medium into which the trigger-waves of Winfree would advance the analogy may not be exact. Trigger-waves are observed in a thin sheet of solution very few millimetres thick.The solution in such a sheet is indeed marginally stable but excitable. However we find that the same solution in bulk is unstable to temporal oscillation although with a very long time period. We suspect that the thin sheet is maintained almost saturated with atmospheric oxygen and that the reaction is thereby influenced. We do not know enough detailed chemistry to model this suspected oxygen effect with confidence. Presumably the free radical processes of M3 or M5 are affected. We find that even a factor of two change in kM3changes by only 0.2 % the critical kM5 at which the steady state becomes unstable. It therefore appears that oxygen exerts its influence on the stoichiometry of the system. If oxygen inhibits the attack of BrO radicals on cerium(III) less than two X species would be produced in step M3.If oxygen promotes radical attack on bromomalonic acid with resulting bromide ion production theffactor in step M5 would increase; fig. 1 shows that such increase could stabilize a previously unstable steady state. Therefore the fact that Winfree solution is unstable to bulk oscillations does not affect the basic validity of the model we develop here for chemical amplification of a marginally stable system. R. J. FIELD AND R. M. NOYES ' A. T. Winfree Lecture Notcs 011Biomatlrenratics ed. P. van den Driessche (Springer-Verlag Berlin) in press. A. T. Winfree Science 1973,181 937. R. M. Noyes and R. J. Field Ann. Reu. Phys. Chem. 1974,25,95. R. M. Noyes R. J. Field and E. Koros J. Arner. Chern. Soc. 1972,94 1394. R. J. Field E. Korljs and R. M. Noyes J. Amer. Chem. SOC.,1972 94 8649. R. J. Field and R. M. Noyes J. Chem. Phys. 1974,60,1877. 'R. J. Field and R. M. Noyes Nature 1972 237 390. R. J. Field and R. M. Noyes J. Arner. Chem. Soc. 1974 96 2001. A. T. Winfree Science 1972 175,634.