18924
J. Phys. Chem. 1996, 100, 18924-18930
Further Refinements of the Peroxidase-Oxidase Oscillator Mechanism: Mixed-Mode Oscillations and Chaos Raima Larter* and Sheryl Hemkin Department of Chemistry, Indiana UniVersity Purdue UniVersity Indianapolis (IUPUI), 402 North Blackford Street, Indianapolis, Indiana 46202 ReceiVed: April 8, 1996; In Final Form: July 1, 1996X
A model of the peroxidase-oxidase (PO) reaction which explicitly includes the species 2,4-dichlorophenol (DCP) is proposed. Simulations with this model are run at different pH values, resulting in mixed-mode oscillations at pH ) 6.3, which go through period-doubling bifurcations into chaos, in agreement with recent experimental reports of studies at the higher pH. The dependence of features of these oscillations on NADH feedstream concentration and DCP initial concentration is also explored. Finally, comparison of detailed features of the simulated strange attractor to the recent experimental attractor reveals minor differences which may turn out to be important in elucidating the role of DCP in the mechanism which leads to chaos in the PO reaction.
I. Introduction Some years after the discovery of oscillatory behavior in the peroxidase-oxidase (PO) reaction,1 the first hints of chaotic behavior were detected in this reaction,2 making it, along with the Belousov-Zhabotinskii reaction, one of the earliest examples of chemical chaos. Since that time, a number of groups have investigated the PO reaction, carrying out experimental studies and theoretical analyses in attempts to elucidate the mechanism by which oscillatory and chaotic behavior arise in this reaction (see ref 3 for a recent review). Because the PO reaction involves a single enzyme and, yet, exhibits a whole range of complex dynamic behavior, including multistability, simple and complex oscillations, and chaotic behavior, it has become an important minimal case whose study has illuminated the possibilities for complex behavior in biology. The full understanding of complex dynamics in biological systems, such as oscillations in the glycolysis reaction, cellular calcium oscillations and waves, and other phenomena,4 depends on determining what is possible in the simplest of biochemical systems. The PO reaction has become a prototype for these kinds of studies. The PO reaction is the oxidation of organic electron donors by molecular oxygen catalyzed by horseradish peroxidase (HRP). When this reaction takes place in a flow system with reduced nicotinamide adenine dinucleotide (NADH) as the reductant, the concentrations of reactants (O2 and NADH) as well as some enzyme intermediates have been found to oscillate5 with periods ranging from several minutes to about an hour, depending on the experimental conditions. The PO reaction corresponds to the following overall reaction HRP
2NADH + O2 + 2H+ 98 2NAD+ + 2H2O and occurs as the first step in a sequence of reactions in plants which eventually culminates in the production of lignin;6,7 it also is involved in the important processes of the photosynthetic dark reactions.8 At this point, it is not known whether the oscillations observed in the flow system have any bearing on behavior in ViVo. However, oscillatory behavior is known to occur in other biochemical settings and seems to be a ubiquitous phenomenon at many levels in biological systems. The existX
Abstract published in AdVance ACS Abstracts, November 1, 1996.
S0022-3654(96)01058-1 CCC: $12.00
ence of chaotic behavior in the PO reaction brings to mind many intriguing questions, e.g., if chaos can occur in a single enzyme reaction (such as the PO reaction), does the existence of multiple enzyme networks of reactions make chaos inevitable in ViVo? Is chaos a sign of health or disease? John Ross, to whom this issue is dedicated, has played an important role in elucidating the mechanism of the PO reaction and in placing its study center stage in the quest for understanding basic issues in nonlinear chemical kinetics. Some years ago, Ross and his associates carried out studies9,10 of the effect of periodic perturbations to the oxygen inlet flow on the PO reaction and found that these perturbations affect the overall reaction rate and the free energy dissipation. The frequency of the perturbation was varied, and it was found that while all frequencies lower the dissipation (and hence increase the efficiency), the smallest effect ocurred for frequencies near the autonomous frequency of the reaction. Ross and co-workers have also pioneered the use of stoichiometric network analysis (SNA),11,12 originally developed by Bruce Clarke13 in 1980, to categorize chemical oscillators into different classes in an attempt to determine the types of mechanisms which can give rise to oscillatory behavior. The idea is to find ways to use specific features of experimental observations in drawing conclusions about the type of mechanism which must be operative for that particular observation to have occurred. This helps to guide and simplify modeling efforts and, in the process, leads to further insights into the kinetic bases of nonlinear dynamic behavior. The application of the SNA method to a reaction network reveals the subnetworks which can support stationary states and determines the conditions under which these stationary states become unstable through a Hopf bifurcation. The roles which the chemical species play in these instabilities leads to the classification of “essential” species as autocatalytic, exit, feedback, and recovery species; others species are termed “inessential” if they play no role in rendering the network unstable. These investigations have led to a useful classification scheme for the contending mechanisms and have suggested experiments which can be used to distinguish between possible mechanisms for a given reaction. The PO reaction has, again, been central in the Ross group’s efforts to develop this technique. Another example of the contributions made by Ross et al. to the understanding of the PO reaction was the experimental observation in his lab of quasiperiodic behavior in this system.14 © 1996 American Chemical Society
Peroxidase-Oxidase Oscillator Mechanism Others had previously observed a period-doubling route to chaos in the PO reaction,15 but Ross’ observation of quasiperiodicity was the first to confirm suggestions from modeling studies16 that a torus attractor might be involved in the route to chaos in this system. A full understanding of the chaotic behavior of the PO reaction still eludes investigators of this reaction, but we seem to be narrowing the possibilities. This paper reports on steps toward including an as-yet little understood species, 2,4-dichlorophenol, in the latest contending mechanism for chaos in the PO reaction. A recent flury of papers reporting new models11,17,18 and new experimental observations19 have served to focus attention on a subset of the steps comprising the so-called “detailed models” of the PO reaction.3 Early work on this reaction by Yamazaki (the discoverer of damped oscillatory behavior in the PO reaction) and co-workers led to a long list of reactions which were thought to occur in this system. It has often been assumed that only a subset of these is essential for complex behavior (i.e., oscillations and chaos), but determining which subset to retain has taken many years and many experimental investigations to ascertain. It seems that a consensus is finally emerging regarding the essential subset of steps which lead to oscillatory behavior in the PO reaction; determining the source of chaotic behavior remains more elusive, however. The possibilities have begun to narrow considerably, but there are still some disagreements as to the roles which different steps play in the origin of chaotic behavior in the PO reaction network. In the following section we present a model that includes, as a core, the essential subset of reactions which most investigators agree lead to oscillatory behavior in the PO reaction. It is a slight modification of a recent model proposed by Olsen and co-workers18 which is, itself, a modification of a mechanism known as the Urbanalator, proposed by Scheeline and coworkers.17 The latter was created by considering only steps which had been directly observed experimentally during the course of oscillations and not inferred from other studies. It is similar to an earlier mechanism known as model A,20,21 differing only in one or two steps; the essential feature of model A of a double autocatalytic cycle is retained in the Urbanalator as well as its modification proposed here. Two species, DCP and methylene blue (MB), have not often been included in any of the detailed models because so little is known about their role in producing oscillations. Both were reported, early on, to be necessary for stabilizing oscillatory behavior, but oscillations have recently been observed when both DCP17 and MB22 are absent. However, chaotic behavior and associated complex periodic oscillations have not, as yet, been observed in the absence of these two species, so their role in the origin of chaos is still unclear. DCP has been known to be an important species since at least the 1992 experiments by Geest et al.15 which revealed a period-doubling route to chaos when [DCP] was varied. Bronnikova et al.18 were able to make an indirect comparison to these experiments by varying a rate constant they presumed to be proportional to [DCP]. The exact nature of the role which DCP plays has not been experimentally elucidated. However, Ross and co-workers have suggested12 two possible ways to include it in the reaction network and shown, through simulation and SNA techniques, that either of these are possible as they retain the essential network features of the skeleton mechanism that lead to oscillatory behavior. Here, we consider a third possibility that is consistent with the simulations of Bronnikova et al.18 in that important relationships between key species in the reaction network are retained.
J. Phys. Chem., Vol. 100, No. 49, 1996 18925
Figure 1. Network diagram corresponding to the proposed model. Explicit forms of the mechanistic steps are given in Table 1 along with assigned rate constant values. Steps 14 and 15 in this model (shown as dashed lines) are used in place of reaction 8 in the Urbanalator17 and its modification.18
An additional reason that it may be important to understand the role of DCP in this reaction is that the PO reaction in ViVo occurs in the presence of phenols. These phenolic monomers are polymerized through an HRP-catalyzed, H2O2-dependent set of reactions which occurs subsequent to the PO reaction. While it is not known whether the phenols play a part in the PO reaction per se, phenols of various kinds (including DCP) are known to accelerate several peroxidase-catalyzed reactions. In fact, this is the reason that DCP was originally added to the reaction mixture in which oscillatory behavior was detected. Recent experiments in Olsen’s laboratory23 have, furthermore, revealed that a variety of phenolic compounds strongly influence the oscillatory behavior of the PO reaction. Understanding the role of DCP might very well, therefore, lead to a better understanding of the role of phenols in the PO reaction in ViVo and may shed some light on the question of whether oscillations occur in ViVo. Recent experiments by Hauser and Olsen19 have shown that mixed-mode oscillations occur when experiments are carried out at a slightly higher pH than is normally the case in PO experiments. They showed that the model suggested by Bronnikova et al.18 could be adjusted to higher pH by taking into account the [H+] dependence of certain rate constants. We have carried out the same sort of analysis here using our model which includes DCP explicitly and have found that the range of parameter values over which mixed-mode oscillations occur becomes wider at higher pH, in agreement with Hauser and Olsen’s experimental observation.19 Nevertheless, certain disagreements in oscillatory waveform shape, features of the chaotic attractor, and other details remain, indicating that something is still missing or incorrect in this model. Therefore, we end with a discussion of possible directions for future work. II. Models The model we propose is shown in Figure 1; the explicit reaction steps for this model and rate constant values used are given in Table 1. The proposed model is based on modifications of the Urbanalator, a model that is similar, but not identical, to an earlier model known as model A. The Urbanalator and model A differ in three important ways: (1) the Urbanalator includes the direct conversion of superoxide radical into hydrogen peroxide via H+, i.e., 2H+ + 2O2•- f O2 + H2O2;(2) the Urbanalator also includes the direct oxidation of NADH by molecular oxygen, NADH + O2 + H+ f NAD+ + H2O2, a step which was not included in any of the previously suggested models (including model A) but which may be very important, particularly in the presence of MB which catalyzes this reaction;24 and (3) a reaction in model A, NADH + O2•- f
18926 J. Phys. Chem., Vol. 100, No. 49, 1996
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TABLE 1: Detailed Model of the Peroxidase-Oxidase Reaction reacn no.
reacns
rate const (pH 5.1)
ref
R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12 R13 R-13 R14 R15
NADH + O2 + H+ f NAD+ + H2O2 H2O2 + Per3+ f CoI CoI + NADH f CoII + NAD• CoII + NADH f Per3+ + NAD• NAD• + O2 f NAD+ + O2•O2•- + Per3+ f CoIII 2O2•- + 2H+ f H2O2 + O2 CoIII + NAD• f CoI + NAD+ 2NAD• f (NAD)2 Per3+ + NAD• f Per2+ + NAD+ Per2+ + O2 f CoIII NADH(stock) f NADH O2(gas) f O2(liq) O2(gas) f O2(liq) DCP + CoIII f CoI + DCP• DCP + NAD• f DCP + NAD+
k1[H+]a ) 3.0 × 100 M-1 s-1 k2 ) 1.8 × 107 M-1 s-1 k3 ) 4.0 × 104 M-1 s-1 k4 ) 2.6 × 104 M-1 s-1 k5 ) 2.0 × 107 M-1 s-1 k6 ) 1.7 × 107 M-1 s-1 k7[H+]2 b ) 2.0 × 107 M-1 s-1 k8c ) 1.35 × 108 M-1 s-1 k9 ) 5.6 × 107 M-1 s-1 k10 ) 1.8 × 106 M-1 s-1 k11 ) 1.0 × 105 M-1 s-1 k12[NADH](stock) ) variedd k13[O2](gas) ) 6.2415 × 10-8 M s-1 k-13 ) k13 ) 3.73 × 10-3 s-1 k14 ) 1.0 × 108 M-1 s-1 k15 ) 9.0 × 105 M-1 s-1
18 37, 32 33 33 18 34 34 25 35 18 36, 18 18 18 18
a Rate constant left unchanged at higher pH. b At pH 6.3, k [H+]2 ) 5.0 × 106 M-1 s-1. c Reaction 8 was replaced by reactions 14 and 15 in this 7 model; therefore, k8 was set equal to 0. d See figures.
H2O2 + NAD•, is eliminated from the Urbanalator since no direct evidence for it exists experimentally. The inclusion of this latter reaction in models of the PO reaction can be traced to an early suggestion of Yokota and Yamazaki25 (1965) who, while not reporting direct experimental evidence of its existence, reiterated its importance26 in 1977 by stating that it “...appears to be an essential reaction in propagation of the chain reaction....” The success of the Urbanalator and its derivatives in reproducing experimental observations of oscillation makes this reasoning less compelling. The second of the new steps introduced in the Urbanalator, the oxidation of NADH by molecular oxygen, is catalyzed by methylene blue (MB), although this has only been studied at pH 8-9, much higher than the normal pH 5.1 used in experimental studies of the PO reaction. Olson et al. suggested17 a catalytic role for the species MB+ involving the intermediate MBH but did not consider this cycle explicitly in simulations with the Urbanalator. Ross and co-workers12 have studied the role of the additional steps involving MB using SNA and concluded that no essential change in the stability properties occur whether MB is explicitly included or not. Bronnikova et al.18 also did not include MB explicitly in their simulations, so we continue with the assumption that this species can be taken to be proportional to the rate constant of step 1. Simulations with the Urbanalator revealed the existence of simple oscillations when experimentally realistic rate constants were used. No evidence of complex oscillations or chaos was reported for this model. To study the origin of chaotic behavior, Bronnikova et al. published simulations using a two-step modification of the Urbanalator; the modification involved the addition of the species Per2+, formed through a reaction of Per3+ with NAD•. The addition of these two steps to the Urbanalator is identical to the change made by Aguda and Larter in transforming model A into another model27 capable of sustaining complex oscillations and chaos (model C, for “chaos”). Bronnikova et al. found that this same modification of the Urbanalator produced a model which exhibited complex oscillations and chaos, arising via a period-doubling route and described by a chaotic attractor that bears a remarkable resemblance to that observed experimentally by Geest et al.15 The existence of Per2+ in the oscillatory reaction mixture has been inferred by many investigators, but direct detection of it during oscillations has not been reported, since its absorption spectrum is obscured by maxima associated with other oxidation states of the enzyme. The validity of the two important steps involving Per2+ depends on the existence of this species during
oscillations. Hence, Scheeline and co-workers did not include it in the Urbanalator. Recently, however,23 unpublished analysis involving deconvolution of the absorption spectra in the range 400-450 nm taken during oscillations confirms that Per2+ is, indeed, formed. These theoretical and experimental results are very gratifying, therefore, in confirming that we are beginning to understand the key features of the PO mechanism which, first, produce oscillation and, second, lead to chaotic behavior. At this point, it is important to point out that, while the SNA technique is very useful for elucidating the mechanistic origin of oscillatory behavior in a reaction network, it cannot be used, in its present form, to analyze a network for its ability to produce chaos. No other analytical technique is known that can be used to eliminate possible mechanisms for the origin of chaos, hence, the only way, currently, to choose between different possibilities is to compare the observed routes to chaos, the nature of attractors which govern the chaotic behavior, and other details which might distinguish between different mechanisms that lead to chaos. While the model proposed by Bronnikova et al. does not include DCP explicitly, a comparison to experiments in which [DCP] acted as a bifurcation parameter was made by these investigators by varying the rate constant for reaction R8, CoIII + NAD• f CoI + NAD+. Although the role of DCP is not certain, it has been suggested by Halliwell28 and others23 that DCP stimulates the degradation of CoIII, regenerating the “active” form of the enzyme. [Active means those forms which have the ability to catalyze the oxidation of substrates such as NADH.] No experimental evidence exists that would give us any clues as to the precise mechanism by which this is done, however, and several possibilities exist. Bronnikova et al. suggested18 two possibilities: a direct chemical effect of DCP or an indirect one in which DCP induces a conformational change in the enzyme, stabilizing a particular form. Ross and co-workers have considered a couple of possibilities of the direct chemical type,11,12 but neither of these would result in an overall reaction that retains the kinetic relationships between NADH, NAD+, and NAD•, which exist in all of the detailed models studied to date. One possibility studied by Ross et al. was the following:
DCP + CoIII f DCP• + CoI DCP• + NADH f DCP + NAD• For which the net reaction is:
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J. Phys. Chem., Vol. 100, No. 49, 1996 18927
CoIII + NADH f CoI + NAD• Another was similar but produced Per2+ instead of CoI:
DCP + CoIII f DCP• + Per2+ + O2•DCP• + NADH f DCP + NAD• For which the net reaction is:
CoIII + NADH f Per2+ + O2•- + NAD• Both of these suggestions reverse the role which NAD• plays in step 8; these suggestions produce an additional NAD•, while (R8) consumes one, forming the final product NAD+. Although simulations11 with models which contain the second possibility above show period doubling to chaos and an attractor that is similar to that observed experimentally, the model which was actually studied in these simulations retained step 8 and included many other steps as well, which might be the actual source of chaotic behavior. Thus, it is difficult to draw conclusions as to the role which the additional DCP loop plays in the origin of chaos in this model. Here we propose a different possibility, also of the direct chemical type and also involving DCP in a catalytic cycle. The overall reaction R8 in the Bronnikova model is replaced by two steps, R14 and R15:
DCP + CoIII f DCP• + CoI
(R14)
DCP• + NAD• f DCP + NAD+
(R15)
For which the net reaction is: •
+
CoIII + NAD f CoI + NAD
(R8)
This suggestion retains the kinetic relationships between NADH, NAD+, and NAD• in the Urbanalator, model A, and their derivatives. Since these models have been successful in reproducing most experimental observations to date, we felt it best to perturb the structure of the reaction network as little as possible in suggesting a modification which would include DCP explicitly. Since no experimental studies of the kinetics of DCP consumption in the PO reaction have been carried out, no rate constant values for reactions R14 and R15 are available. Indeed, the rate constant for reaction R8 (which is the overall reaction R14 + R15), has never been directly measured, either. The reported value29 of 1.3 × 108 M-1 s-1 is one of four rate constants determined in a single fitting of experimental data and, hence, should be considered as highly suspect. Therefore, very little is known even about order of magnitude values that should be considered in modeling this important part of the reaction network. We thus undertook a search of parameter space to determine the change in model dynamics which occur as k14 and k15 are varied over wide ranges and determined that a relatively narrow range of values resulted in oscillatory behavior. We also carried out simulated variations in parameters which are easily adjusted experimentally, such as [DCP], [NADH] in the feed stream, and the pH. Recent experiments involving variations of the latter two (pH and [NADH]) have revealed19 for the first time mixed-mode oscillations in the PO reaction which exhibit a period-adding sequence and associated chaotic behavior. In the following section we report results of simulations with this model which show agreement with these
Figure 2. Range of parameter values ([DCP]0 and k12, the flow rate of NADH into the reactor) over which oscillatory behavior is observed for (a) pH ) 5.1 and (b) pH ) 6.3. The region within the dashed lines includes mixed-mode oscillations arranged in a period-adding sequence; period-doubling of a mixed-mode state is observed within the dotted lines; highly complex oscillations, including chaos, are found within the region bounded by solid lines (pH 6.3 only).
experiments, providing further evidence for the validity of this most recent family of PO reaction models. Results The simulations were carried out using the LSODE implementation of Gear’s algorithm. In order to gain insight into the modified model, we carried out calculations which simulated, to the best of our abilities, the experimental studies of Hauser and Olsen. Rate constant values were the same as those used in Olsen’s modification18 of the Urbanalator.17 We studied the effect of varying the rate constants k14 and k15 and found that k15 exhibited a greater influence upon the complexity of the dynamics. The most complex bifurcation sequence was found when k14 was approximately 1.0 × 108 M-1 s-1 and k15 ) 9.0 × 105 M-1 s-1. Since Hauser and Olsen experimentally varied the [NADH] concentration in the feedstream, we probed the behavior of the model by varying the rate constant k12, which is associated with the NADH feed. Furthermore, their experiments were carried out at pH ) 6.3, while the usual experimental configuration involves pH ) 5.1. The rate constant k7 was allowed to vary with pH since it involves a second-order dependence on [H+]; the other potentially pH-dependent rate constant, k1, was kept at the same value as that used for pH 5.1 since it involves only a linear dependence on [H+]. All other rate constants are taken to be unchanged from those assumed to hold for the pH ) 5.1 experiment and are summarized in Table 1. Figure 2 shows the region in parameter space in which oscillatory behavior was found in the simulations. The two parameters considered are k12, the flow rate constant for NADH, and [DCP]0, the initial concentration of DCP. The region enclosed by the dashed line shows the range of parameter values
18928 J. Phys. Chem., Vol. 100, No. 49, 1996
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Figure 3. Bifurcation diagrams showing mixed-mode oscillations and period-doubling bifurcations for (a) pH )5.1 and (b) pH ) 6.3. All rate constants are given in Table 1. Initial conditions for the first value of k12 are [DCP]0 ) 51 µM, [Per3+]0 ) 0.9 µM, [O2]0 ) 16.7 µM, all other concentrations zero. 100 000 points are displayed with the first 250 000 points discarded as a transient. Subsequent values of k12 used the last point of the preceding time series as the initial value.
over which oscillatory behavior was found. The simulations yield a stable steady state for parameter values outside these lines; for parameter values inside the lines, both simple and complex oscillations are found in various regions, as indicated by the dotted and solid lines. Oscillatory behavior is found over a wider range of parameter space at the higher pH, a result which carries over to our study, to be described below, of the range of parameter space in which mixed-mode oscillations are observed. In both the pH ) 5.1 and 6.3 simulations, both simple and mixed-mode oscillations were found. The range of values over which the mixed-mode states could be found was much wider at the higher pH. This may suggest a reason why mixed-mode oscillations occurring in sequence and undergoing perioddoublings had never been observed in the PO reaction prior to Hauser and Olsen’s recent experiments. Since these experiments were carried out at a higher pH than normal, our simulations suggest that mixed-mode oscillations would have been easier to observe. While the pH ) 5.1 and 6.3 results were qualitatively similar at low [DCP]0 values, the disparity between these two pH values increased at higher [DCP]0. At low [DCP]0, mixed-mode oscillations arose via a period-adding sequence as the parameter k12 (the flow rate of NADH) was increased. Figure 3a shows a bifurcation diagram for this sequence. At higher [DCP]0, a similar sequence is followed (see Figure 3b) but results in more complex oscillations, including apparent chaos. For pH ) 5.1, the mixed-mode states arise via a period-adding sequence as described above; at a sufficient state of complexity, one of these mixed-mode states undergoes a period-doubling transition resulting, eventually, in a concatenated state involving the old mixed-mode state and the next one in the period-adding sequence. Figure 4 shows some typical examples for pH ) 5.1. At the higher pH, this same general sequence is found, but the highly complex, concatenated states which arise from a period-doubling sequence out of a mixed-mode state, eventually
Figure 4. Selected time series, calculated as described in Figure 3, for pH ) 5.1 and various k12 values: (a) k12 ) 0.1048; (b) blow-up of part a showing detail of small amplitude peaks; (c) k12 ) 0.1056. The state shown in part a is, thus, a 13 mixed-mode oscillation, while that in part c is a period-doubled version of this, i.e., (13)2.
give rise to what appears to be a chaotic state. Figure 5 shows behavior typical of this higher pH. Finally, Figure 6 shows the attractors ([NADH] Vs [O2]) associated with the two pH values. In the lower pH case we do not see chaos; the most complex oscillation observed at the lower pH is a period-doubled mixed-mode limit cycle. At the higher pH, however, the final attractor appears to be a chaotic one; it is similar to the attractor reported experimentally by Hauser and Olsen19 in having an apparent unstable focus with an associated inset and outset. Some differences exist between the calculated attractor and the experimental one, however, such as the fact that the trajectories spiral toward the unstable focus in the calculation but away from it in the experiment. Further study of this and other discrepancies is under way in our lab. Discussion Hauser and Olsen’s recent report of mixed-mode oscillations in the PO reaction which are arranged in a period-adding sequence can be partially understood by the simulations reported here in which we find a wider range of parameter values associated with mixed-mode behavior at a higher pH. Since
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J. Phys. Chem., Vol. 100, No. 49, 1996 18929
Figure 6. Attractors seen for the cases of (a) pH ) 5.1, k12 ) 0.1056 and (b) pH ) 6.3, k12 ) 0.1059. All other parameter values are given in Table 1. 60 000 time points from the time series data shown in Figure 4c are used in constructing part a; 100 000 points from Figure 5d correspond to part b.
Figure 5. Selected time series found for pH ) 6.3 and various k12 values: (a) k12 ) 0.1048; (b) k12 ) 0.1050; (c) k12 ) 0.1056; (d) k12 ) 0.1059. We assign the following notation to these states: (a) 13; (b) (13)2; (c) (13)4; (d) chaos, i.e., the 13 state goes through a period-doubling cascade into chaos.
Hauser and Olsen were carrying out experiments at a pH value that is higher than almost all the previous PO oscillation experiments, our results suggest that the higher pH would have made it easier to find conditions under which mixed-mode behavior occurs. Several years ago, Steinmetz and Larter30 reported on a study of simulations with one of the earliest, but more abstract, models of the PO reaction, the Degn-Olsen-Perram31 or DOP model. This model, while reproducing qualitative features of the PO
oscillations, also exhibited a plethora of complex oscillatory states, including mixed-mode oscillations arranged in a Farey sequence and chaos. These studies with the DOP model revealed that chaotic states arose from mixed-mode oscillations via a period-doubling cascade. Both our earlier results and the current simulations indicate that chaos arising via a perioddoubling cascade from mixed-mode states would not be unexpected in the PO reaction. Even though we are very pleased at the excellent agreement between these simulations and recent experiments, some slight disagreement still remains, however. The time series observed experimentally is associated with a trajectory which is rapidly injected toward an apparent unstable saddle, from which it spirals outward until the small oscillations reach a critical amplitude before exploding out into phase space in a large excursion. The time series observed in our simulations, on the other hand, is associated with a trajectory which, essentially, runs in reverse: a large excursion is attracted toward a saddle point in phase space, and the trajectory undergoes damped oscillations toward this saddle. Once the amplitude becomes small enough, the trajectory comes under the influence of the unstable manifold and is blown away from the saddle point. The difference between experiment and simulation is most likely due to the additional steps involving DCP, since Hauser and Olsen did not find this particular disagreement between experiment and simulations with a model that is identical in every way to the one suggested here except that the DCP steps are not included.
18930 J. Phys. Chem., Vol. 100, No. 49, 1996 Conclusions We have carried out simulations with a modification of a recently proposed model for the PO reaction. These simulations show that a period-adding sequence of mixed-mode oscillations becomes more likely at higher pH values, a conclusion consistent with conditions required for the first experimental observations of mixed-mode oscillations in the PO reaction. While agreement is excellent, both qualitatively and quantitatively, a difference in some detailed features of the associated attractor was noted between experiment and simulation. This difference can be traced to our inclusion of explicit steps involving DCP and provide, therefore, an opportunity for exploring the variety of behaviors which might be observed when DCP is involved in different ways. We continue our investigation of this important species in the PO reaction. Acknowledgment. We gratefully acknowledge support of this work by the National Science Foundation through Grant NSF-CHE-9307549. References and Notes (1) Yamazaki, I.; Yokota, K.; Nakajima, R. Biochem. Biophys. Res. Commun. 1965, 21, 582. (2) Olsen, L. F.; Degn, H. Nature 1977, 267, 177. (3) Larter, R.; Olsen, L. F.; Steinmetz, C. G.; Geest, T. Chaos in Biochemical Systems: The Peroxidase Reaction as a Case Study. In Chaos in Chemical and Biochemical Systems; Field, R., Gyo¨rgyi, L., Eds.; World Scientific Press: Singapore, 1993; pp 175-224. (4) Goldbeter, A. Biochemical oscillations and cellular rhythms: The molecular bases of periodic and chaotic behaViour; Cambridge University Press: Cambridge, U.K., 1996. (5) Olsen, L. F.; Degn, H. Biochim. Biophys. Acta 1978, 523, 321. (6) Ma¨der, M.; Amberg-Fisher, V. Plant Physiol. 1982, 70, 1128. (7) Ma¨der, M.; Fu¨ssl, R. Plant Physiol. 1982, 70, 1132. (8) Pantoja, O.; Willmer, C. M. Planta 1988, 174, 44. (9) Lazar, J. G.; Ross, J. Science 1990, 247, 189.
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