J . Phys. Chem. 1993,97, 5649-9653
5649
Universality in the Peroxidase-Oxidase Reaction: Period Doublings, Chaos, Period Three, and Unstable Limit Cycles Curtis G. Steinmetz,t Torben Geest,?**and Raima Larter'vt Department of Chemistry, Indiana University Purdue University Indianapolis (IUPUI), 402 N.Blackford St., Indianapolis, Indiana 46202, and Institute of Biochemistry, Odense University, Campusvej 55, DK-5230 Odense M,Denmark Received: February 4, I993
Recently we reported on a period doubling route to chaos in the peroxidase-oxidase (PO) reaction. This was the first report of a universal route to chaos in the PO reaction. Here we present further evidence of universality in the transition to chaos in the PO reaction. We have observed period three oscillations under conditions near where chaotic oscillations are found, and we have found that chaotic oscillations in this reaction contain unstable periodic orbits. We also report numerical results from a simple model of the PO reaction. These theoretical calculations qualitatively reproduce the spectrum of behaviors observed in this reaction.
Introduction The peroxidase-oxidase (PO) reaction is an important bridge between chemical oscillators such as the Belousov-Zhabotinsky reaction' and biological oscillators such as intracellular CaZ+ oscillations.2 It is now clear that the PO reaction shows a wide spectrum of interesting behaviors including simple 0scillations,3~~ bistability,S chaos: and quasiperiodi~ity.~.~ The PO reaction was also recently shown to undergo a period doubling route to chaos.9 These behaviors have all been observed in vitro under well controlled laboratory conditions. The PO reaction also occurs in viuo,I0 and its detailed study therefore has significant implications for understanding the role of low dimensional complex dynamics in biological systems. The PO reaction is the peroxidase enzyme catalyzed oxidation of an electron donor in which molecular oxygen acts as the electron acceptor. This reaction occurs in plants as part of the process of lignification,Iz with nicotinamide adenine dinucleotide (NADH) as the electron donor. From now on when we refer to the PO reaction we will mean specifically the following overall reaction catalyzed by horseradish peroxidase: 2NADH
+ 0, + 2H'
-
2NAD'
+ 2H,O
The modifiers methylene blue (MB) and 2,4-dichlorophenol (DCP) are also present in the experiments that we will discuss. The role of DCP is still not completely understood, but it is especially important since the concentration of DCP was the bifurcation parameter for these experiments. Oscillationsin the PO reaction have been found only in the presence of DCP, whereas without DCP the PO reaction exhibits bistability but not oscillations.5 DCP, a synthetic (and carcinogenic) monophenol is not present in vivo, but naturally occurring monophenolsmight be able to replace DCP in the PO reaction.12J3 Damped periodic oscillations were first observed in the PO reaction in 1965.3 Sustained periodic oscillations were found in 1969: and chaos was first reported in 1977.6 Until recently there remained some doubt concerning the identification of the latter aperiodicoscillations in the PO reaction as chaotic, since no route to chaos had been observed. Now, however, a period doubling route to chaos has been found in the PO reaction, which lends strong support to the identification of chaos in the 1977 experiments. Experimentallimitations do not, at least at present, allow for the investigation of quantitative aspects of universality, i.e., the calculation of scaling exponents. More qualitativeaspects +
Indiana University F'urdue University Indianapolis.
* Odense University.
0022-3654/93/2097-5649%04.00/0
of universality, however, such as the existence and ordering of stable periodic windows and the existence of unstable limit cycles as part of a chaotic attractor are more accessible. Observation of these universal features provides important additional evidence of deterministic chaos in the PO reaction. In 1983, a model of the PO reaction, now commonly referred to as the Olsen model," was proposed. Simulations with the Olsen model qualitatively reproduce both the simple and chaotic oscillations of the PO reaction. The original studies with this model showed that decreasing the parameter kl caused thesystem to undergo a transition from simple oscillations to chaos, in a manner analogous to the transition to chaos observed experimentally when the enzyme concentration was decreased (this is also the case with the Degn-Olsen-Perram, or DOP model).5J4 In this paper we show that increasing the parameter k3 also induces a transition to chaos, and we believe that the parameter k3 is analogous to the concentrationof DCP. In both cases,decreasing kl and increasing k3, chaos arises in the Olsen model via a cascade of periodic doubling bifurcations.
Experimental Section The reaction was carried out in a 20 mm (light path) X 18.5 mm X 38 mm quartz cuvette (Hellma Cells, custom made QS402) placed in an Aminco DW2000 dual-wavelength spectrophotometer, where the NADH concentration could be measured as the difference between the absorbances at 360 and 380 nm. The reaction solution was 7 mL of 0.1 M sodium acetate buffer at pH 5.1 with 0.1 pM methylene blue and 20-36 pM 2,4-dichlorophenol. The temperature was kept at 28 OC with a thermostatic jacket. DCP was added as a 10 mM solution in 96% ethanol.15 A solution of 0.20 M NADH in distilled water was pumped continuously into the sample at a rate of between 21.5 and 24.0 pL/h using a syringe pump (Harvard Apparatus, Model 22). The concentration of oxygen was measured with a Clark-type electrode (Radiometer, Copenhagen,Denmark) inserted into the sideof the cuvette (see Figure 1). Oxygen was added by diffusion from a gas headspace above the sample. The 02/N2 gas mixture in the head space was maintained at a constant compositionwith 1.42%(v/v) 0,at atmospheric pressure. With a constant oxygen partial pressure above the sample the rate of diffusion of oxygen into the liquid follows the rate law d[O,lldt = k([O,I, - [OZI) where [O,] is the actual oxygen concentration in the solution and [02]q is the concentration in solution when it is in equilibrium with the gas phase. The value of k was found to be 0.27 min-I 0 1993 American Chemical Society
5650 The Journal of Physical Chemistry, Vol. 97, No. 21, 195’3
Steinmetz et al.
Sllrrer
Hole for oxygen electrode
Figure 1. Experimental apparatus. The reaction is carried out in a quartz cuvette fitted with an oxygen electrode and placed inside a UV-visible spectrophotometer. Details are given in the experimental section.
when the sample was stirred with a high precision stirrer at 1420 rpm. Data from the oxygen electrode and the spectrophotometer were collected by a microcomputer through an interface board, at a rate of one data point every second, and stored on disk for further analysis. The data from the oxygen electrode were an average of 200 measurements.
The Olsen Model The Olsen model involves four variables: molecular orygen, NADH, and two intermediate species. One of the intermediates is very likely NAD’, while the other is possibly oxyferrous peroxidase, also known as compound I11(one of the five oxidation states of the peroxidase enzyme). Olsen originally designated these species as A (oxygen), B (NADH), X (NAD’), and Y (compound 111), and we willcontinuethispracticeforconsistency. Because of the idealized nature of the Olsen model, however, correlation of the model variables X and Y with actual chemical species remains tentative. The central feature of the Olsen model is the existence of two routes for the autocatalytic production of the intermediate X (NAD’). In one route two X’s are formed in a reaction of X with B (NADH) (for a net gain of one X via the first route). The other route involves two steps, First NAD’ reacts with the native enzyme to form compound 111, i.e., Y (compound 111) is formed from X. Then three X’s are formed by a termolecular reaction of A, B, and Y (for a net gain of two X s via the second route). The other steps in the mechanism are two (linear) radical termination steps, one radical initiation step, the equilibration of gaseous and dissolved oxygen, and the inflow of NADH. The complete eight-step mechanism and the corresponding system of four differential equations are given below: ks
ki
B+X-.2X kz
Y+Q k6
x,+x
2X+2Y
k5
k3
A+B+Y-.3X
&*A k-i
k4
X-P
k’s
Bo-B
A = k, - k-,A - k,ABY B = k, - k,BX - k,ABY X = k,BX-2k2X2+ 3 k 3 A B Y - k J +
k,
Y = 2k2X2 - k , Y - k,ABY
More details of the Olsen model are given in ref 11. Table I summarizes the important correspondences between the actual species and the Olsen model variables.
4
6
6
10
1’2
1’4
16
Time (minutes)
1’8
k
0
Figure 2. Period three mixed mode oscillations in the PO reaction. The upper trace is NADH absorbance, and the lower trace is oxygen concentration. The concentration of DCP was 35.7 pM,the pumping rate of NADH was 21.5 pL/h (using a 0.20 M solution), and the concentration of enzyme was 0.60 pM. Other details are given in the experimental section.
TABLE I: Correspondencea between the PO Reaction and the Ohsea Model PO reaction
Olsen model
PO reaction
Olsen model
1021 [NADH] [NAD’]
A B
[compound 1111 [enzyme] [DCPI
ki
X
Y k3
All calculations in this paper were done on a VAX 8800 computer, but we have obtained identical results on a Silicon Graphics Personal Iris workstation. Numerical integration was carried out in double precision with a relative error tolerance of 10-10 using both LSODEI6 (an implementation of the Gear method) and the IMSL routine DVPRK” (an implementation of the Runge-Kutta method). Lyapunov exponents were calculated by directly applying the algorithm of Wolf et ala1*to the full set of equationsfor the Olsen model. We scaled the Lyapunov exponents to “bits per excursion” by calculating the number of “excursions”per time unit, i.e., the inverseof the first fundamental frequency. The divdrgence calculations were done by running two calculations with initial conditions differing by one part in lo* and plotting the logarithm of the distance between these trajectories as a function of time (as described in ref 19).
Resdtt4
Experiments. In the course of experiments intended to study chaotic oscillations we found, on two occasions, that period three oscillations occurred instead. In both cases we observed two distinct period doublings followed by a chaotic transient which lead to stable period three oscillations. This scenario is almost identical to that of a typical experimentin which chaos is observed in the PO system, except that in those experiments (1) the chaos is not transient and (2) no stable period three oscillations occur. The concentration of DCP and all other experimentalparameters in the two experiments in which period three was observed were very close to those in which stable chaos was observed. We interpret these results to mean that there is a stable period three window in the PO reaction, possibly slightly “beyond” the transition to chaos. Such windows are a universal feature of the period doubling route to chaos. Figures 2 and 3 show time series from one of the period three experiments. The period three oscillations are of the mixed mode type, that is, they consist of large and small amplitudes that are clearly separated in size. The repeating pattern is one large followed by two small peaks (see Figure 2). Figure 3 shows the evolution of the experiment from simple oscillations through a period doubling cascade and transient chaos into period three.
The Peroxidase-Oxidase Reaction
The Journal of Physical Chemistry, Vol. 97, No. 21, 1993 5651
-Ia
I
Unstable period 3 osc.
C
0 .c
9 4
c C
E c 8 2 C 0,
_x
o n! -0
I
10
30 Time (minutes)
20
4.
I
40
50 I
0
0
2
4
-
6
10
8
12
14
16
18
20
Time (minutes)
Unstable period 5 osc.
” 0501
I
80 90 100 Time (minutes) Figure 3. Transition from period one, through chaos, to period three. (a) Transient period-one oscillations undergo a period-doubling bifurcation to period two, and a subsequent addition of DCP (arrow) causes a second period-doubling to period four oscillations. Before the addition the concentration of DCP was 32.9 pM, while after the addition it was 35.7 pM (all other conditions are as given in Figure 2 and in the Experimental Section). (b) The period four oscillationsbecomechaotic and then finally settle into period three. There is no time lag between parts (a) and (b). Part (b) shows the beginning of the same period three oscillationsshown in Figure 2.
60
70
In addition to stable period three oscillations we have found unstable period three oscillations within a chaotic time series. Figure 4a shows a chaotic time series from one of the experiments reported in our previous paper:9 notice the three consecutive repetitionsof the period-three waveformin themidst of the chaotic time series. Figure 4b shows part of the chaotic transient from one of the period-three experiments in which two repetitions of an unstable period-five oscillation are indicated. These observations are evidence that unstable period-three and period-five orbits form part of the strange attractor in the chaotic PO system. It is another universal feature of the period doubling route to chaos that the resulting strange attractor contains unstable period three and period five orbits. Modeling. We were able to qualitatively reproduce the period doubling route to chaos in the PO reaction using the Olsen model. In these simulations the model parameter k3 causes the same types of changes which occur when [DCP] is changed; hence, we are identifying k3 with DCP much as kl has been identified with the enzyme concentration.6J’ Since DCP is known to accelerate the breakdown of compound 111,the identification we make here for k3is chemically reasonable as well.23 Figure 5 shows a phase diagram for the Olsen model. This phase diagram is actually a two dimensional projection onto the kl-k3 plane of the ninedimensional parameter space of the Olsen model in which the regions of simple oscillations and complex oscillations are demarcated. Figure 6 shows a sequence of time series for increasing values of k3 from simpleoscillationsthrough chaos to period three. Figure 6 should be compared with Figure 3, which shows a similar sequence in the transient leading to experimental observation of stable period three oscillations. Figure 7 is a bifurcation diagram for the period doubling route to chaos in the Olsen model with k3as the bifurcation parameter. In ref 11b a bifurcation diagram similar to Figure 7 is shown, but in which kl is the bifurcation parameter.
04
10 15 Time (minutes)
5
0
20
5
Figure 4. Unstable limit cycles associated with chaos. (a) The three repetitions of the “one large two small” pattern are part of a chaotic time series. The concentration of DCP was 27.9 pM, the pumping rate of NADH was 21.7 pL/h, and the concentration of enzyme was 0.65 pM. Other conditions are as given in Figure 2 and in the experimental section. (b) Here is shown a chaotic transient containing two repetitions of an unstable period five orbit. This is part of the chaotic transient between period four and period three in Figure 3b.
0.52
k,
I
0.371
/&/; complex oscillations
0.22
j
0.004
,
~
,.
1
0.024
,
1
I
0.044
k3 Figure 5. Phase diagram for oscillations in the Olsen model. The line labeled HB (Hopf bifurcation) marks the boundary between a stable steady state and stableoscillations. The line labeled PD (period-doubling bifurcation) marks the boundary between simpleoscillationsand complex oscillations (including period two, period four, chaos, and period three). Values of kl and k~ are indicated on the axes. The values of the other parameters are: k2 = 250, kq = 20, k5 = 5.35, k& = k7 = 0.1, A0 = 8, and k& = 0.825.
We have verified the existence of deterministic chaos in the Olsen model both by calculating Lyapunov exponents and by directly testing for divergence of initially nearby trajectories. For example, the largest positive Lyapunov exponent for the chaotic state pictured in Figure 6d is 0.45 bits/excursion. The divergence (or lack of it) of initially nearby trajectories was calculated for a number of different values of k l and k3,and the results were
Steinmetz et al.
5652 The Journal of Physical Chemistry, Vol. 97, No. 21, 1993
lo = 0.027
Discuapion
1
b = 0.033
lo = 0.0336
kj = 0.035
lo = 0.038
7.0
A
3.5 0.0
90
210
330
time Figure 6. Time series from the Olsen model. As kj increases the model goes from period one to period two to period four to chaos to period three. The value of kl is indicated for each time series, and the value of kl was 0.35. Other parameters are as given in Figure 5 .
7.5
5'7 3.9
1
I
I
i
1 L
0.032
0.038
0.044
Figure 7. Bifurcation diagram for the Olsen model. As k, increases the model goes through a cascade of period doubling bifurcations leading to chaos, and then to period three. Maxima of the variable A ("Amar")are plotted versus the corresponding value of the parameter ks. Other parameters are as given in Figure 5.
always consistent with the Lyapunov exponents we calculated, Le., states with no positive Lyapunov exponents showed no divergence, while states with a positive exponent showed rapid divergence of initially nearby trajectories.
-
By varying the concentration of DCP in the PO reaction the sequence: simple oscillations period doubling cascade- chaos -period three can be induced. The same sequence occurs in the Olsen model when k3 is varied. Recall that k3 is the rate constant for the step in which the variable Yis consumed and the variable Xis produced, corresponding to the breakdown of compound I11 into native enzyme and the concomitant production of NAD'. The role of kl as a bifurcation parameter for the period doubling route to chaos in the Olsen model, and the relationship between kl and the total enzyme concentration has been discussed elsewhere at lengthell The bifurcation sequencedescribedaboveis a universal feature of unimodal iterated maps of a single variable, which are the simplest conceivable systems to undergo the transition to chaos. Chaos arising via the period doubling route has been observed in externally driven biological oscillators, such as Onchidium neurons20 (which also has a period-three window), but as far as we are aware the PO reaction provides the only example of an endogenous period doubling route to chaos in a biochemical or biological system (by endogenous we mean that there is no timedependent external forcing). An important part of the mathematical definition of deterministic chaos is the coexistenceof infinitely many unstable limit cycles.21 These unstable limit cycles are the key to the concept of "controlling chaos" in which small, carefully selected perturbations can stabilize one of the infinitely many unstable limit cycles on a strange attractor.22 We have shown direct evidence for the existence of two such unstable limit cycles as part of the strange attractor in the PO reaction. The role of DCP in the PO reaction is becoming more clear. There is much evidence that DCP and other monophenols, both synthetic and naturally occurring, accelerate the breakdown of compound I11 to the native enzyme.23 This is a crucial step since compound I11 is inactive and breaks down very slowly in the absence of monophenols. In terrestrial plant lignification monophenols accelerate peroxidase catalyzed NADH oxidation, and, therefore, H202 generation, and then are polymerized in an HzO2-dependent reaction also catalyzed by peroxidase.12J3 This raises the possibility that DCP is degraded during the PO reaction, and more work is necessary to determine the fate of DCP in the course of experiments that take several hours. Models of the PO reaction have not generally included the concentration of D C P indeed, if DCP's role is catalytic its concentration should not appear in the rate equations. In the Olsen model we might think of DCP concentration as being included implicitly through the rate constant for the step in which compound I11 is broken down (k3). The qualitative agreement between experiment and theory in this regard lends support to the view that k3 is proportional to the concentration of DCP.
conclusions Period-three mixed-mode oscillations have been observed in the peroxidase+xidase reaction, under conditions close to where chaotic oscillations,arising out of a period-doublingcascade, were recently found. Because such periodic windows are a universal feature of the period-doubling route to chaos, their observation contributes significantlyto the study of chaos in the peroxidase+ oxidase reaction, and to the study of biological chaos in general. Unstable periodic orbits were also found in a chaotic time series and in a chaotic transient. These unstable limit cycles are a further qualitative indication of universality of the peroxidase+ oxidase reaction. Further experimental work is needed to better elucidate the role of DCP, the bifurcation parameter in these experiments, in the origin of oscillations and chaos in the PO reaction. This will necessarily entail a better understanding of the intermediate species compound 111. It should also be possible to find more
The Peroxidase-Oxidase Reaction periodic windows and to more quantitativelydetermine the ranges of parameter values for both complex periodic and chaotic states. Although quasiperiodicity in thePO reaction has been reported,7J it remains to be seen whether a quasiperiodic route to chaos, in addition to the period doubling route, exists in this reaction. We found that the Olsen model, an elementary four-variable model of the PO reaction (which, though highly idealized, obeys mass action kinetics) is quite successful in qualitatively reproducing both of the two transitions to chaos that have now been observed in the PO reaction (that involving enzyme concentration as the bifurcation parameter and that involving DCP concentration). As noted above, quasiperiodicityhas now been observed in the PO reaction, and in a future paper we will report on the interaction of period doublings and quasiperiodicity in the Olsen model.24 It is important to look for reductions of more detailed models25 that can approach the Olsen model in its ability to simulate chaotic behavior in this reaction.
Ackwwledgmeat. The authors gratefullyacknowledge support of this work by the donors of the Petroleum Research Fund, administered by the American Chemical Society,and the National Science Foundation under Grant CHE-8913895.
References and Notes ( I ) Field, R. J., GyBrgyi, L., Eds. Chaos in Chemistry and Biochemistry; World Scientific: Singapore, 1993. (2) Cuthbertson, K.S. R., Cobbold, P. H., Eds. Oscillationr in Cell Calcium; special issue of Cell Calcium 1991, 12,61. (3) Yamazaki, I.; Yokota, K.; Nakajima, R. Biochem. Biophys. Res. Commun. 1965,21, 582.
The Journal of Physical Chemistry, Vol. 97, No. 21. 1993 5653 (4) Nakamura, S.;Yokota, K.;Yamazaki, I. Nature 1969,222, 794. (5) Degn, H.; Olsen, L. F.; Perram, J. W. Ann. N.Y. Acad. Sci. 1979, 316,623. (6) Olsen, L. F.; Degn, H. Nature 1977,267, 177. J.J. Phys. Chem. 1992,96,7338. (7) Samples,M.S.;Hung,Y.-F.;Ross, ( 8 ) Samples, M. S.;Ross, J. J . Phys. Chem. 1992, 96,7342. (9) Geest, T.;Steinmetz, C. G.; Larter, R.; Olsen, L. F. J . Phys. Chem. 1992,96,5618. Willmer, C. M. Phnta 1988, 174,44 and references (10) Pantoja, 0.; therein. (11) (a) Olsen, L. F. Phys. Lett. 1983. 9 4 4 454. (b) Olsen, L. F. In Stochastic Phenomena and Chaotic Behavior in Complex Systems; Schuster, P., Ed.; Springer-Verlag; Berlin; 1984;p 116. (c) Olsen, L. F. Biochem. Biophys. Acta 1978,527,212. (12) Gross, G.G.; Janse, C.; Elstner, E.F. Planta. 1977,136, 271. (13) Mider, M.; Fiissl, R. Plant Physiol. 1982,70, 1132. (14) Larter, R.; Bush, C. L.; Lonis, T. R.; Aguda, B. D. J . Chem. Phys. 1987,87,5765. (15).Unpublished results by Alex Scheeline and Dean Olsen at the Universityof Illinois indicate that the ethanol affects the oxygen transfer rate, but this effect has not been quantified. (16) Hindmarsh, A. C. ACM Signum Newsletr. 1980,15, 10. (17) International Mathematical and Statistical Library, MATH/LIBRARY, FORTRAN Subroutines for Mathematical Applications, Version 1.0,1987. (18) Wolf,A.;Swift,J.B.;Swinney,H.L.;Vastano,J.A.PhysicaD1985, 16,285. (19) Steinmetz, C. G.;Larter, R. J . Chem. Phys. 1991,94, 1388. (20) Hayashi, H.; Ishizuka, S.J . Theor. Biol. 1992,156,269. (21) Devaney, R. L. An Introduction to Chaotic DynamicalSystems, 2nd ed.;Addison-Wesley: Redwood City, CA, 1989;Chapter 1. (22) Ott, G.; Grebogi, C.; Yorke, J. A. Phys. Reu. Lett. 1990,64,1196. (23) Halliwell, B. Phnta 1978,140,81. (24) Steinmetz, C. G.; Larter, R. Manuscript in preparation. (25) Aguda, B. D.; Larter, R. J . Am. Chem. SOC.1990, 112,2167.
