J. Phys. Chem. 1992, 96, 7338-7342
7338
Further Experimental Studtes on the Horseradish Peroxldase-Oxldase Reaction Marjorie S. Samples, Yu-Fen Hung,and John Ross* Department of Chemistry, Stanford University, Stanford, California 94305 (Received: February 20, 1992; In Final Form: May 14, 1992)
We study experimentally the changes in the kinetics and the thermodynamics for the periodically forced oxidation of reduced nicotinamide adenine dinucleotide (NADH) by molecular oxygen (referred to as the PO reaction), which is catalyzed by the horseradish peroxidase enzyme. We choose different forms of external periodic perturbations on the inflow of molecular oxygen to observe the effect of such forms on the dissipation and the efficiency of the system. On forcing an experimental limit cycle with a two-term Fourier series waveshape, we observe that the dissipation (efficiency) of the system is lowered (raised) relative to the autonomous system and that this change is related to the frequency of perturbation. Sinusoidal and nonsinusoidal perturbations of a stable focus in the experimental system also show that the dissipation is lowered relative to the autonomous state and that as the perturbation amplitude increasesthe dissipation decreases. Left-sawtooth perturbations are out-of-phase with the autonomous waveshape and on the stable focus lead to the lowest dissipation. Furthermore, the perturbation experiments on a stable focus indicate that NADH is a nonessential species in the PO reaction. Experimental evidence for quasiperiodicity in the PO reaction is presented for the first time. In the succeeding paper, we compare these experimental results with numerical results.
I. Introduction
II. Experimental Section
The horseradish peroxidase (HRP) enzyme catalyzed oxidation of reduced nicotinamide adenine dinucleotide (NADH) by molecular oxygen is a widely studied nonlinear biochemical reaction known as the peroxidaseoxidase (PO) reaction. The stoichiometry of the overall reaction is
The PO experiments presented here were performed in a well-stirred reaction vessel which is continuously supplied with oxygen. The apparatus has been described elsewhere,sB6and we give a brief summary, Figure 1. The cylindrical reaction vessel (Pierce Chemical Co., Reacti-Vial) has a volume of 5 mL and is well-stirred. Oxygen is supplied to the vessel via two computer-gwemed gas mass flow controllers (Vacuum General, Inc., UltraFlo Model UC2-21), where one mass flow controller controls a nitrogen tank, while the other controls a compressed air tank. These mass flow controllers maintain the desired gas flow rate and the oxygen composition of the gas flowing into the reaction vessel. Before entering the reaction vessel, the outputs from the tanks are mixed in a two gas blender (Union Carbide, Linde Division, Model FM 4621-13). The periodic perturbation waveforms are created by a waveform generator board installed in a computer which regulates the perturbation waveform, perturbation frequency, and perturbation amplitwk t h e paramem may be changed at any time during the experiment without interfering with data collection. During a perturbation, the total gas flow rate into the reaction vessel is kept constant, and the perturbation is applied such that the average oxygen concentration during a perturbation cycle equals the oxygen concentrationduring a steady oxygen inflow. The oxygen concentration in the solution in the reaction vessel is continuously monitored by an oxygen microelectrode (Microelectrodes, Inc., Model MI-730), and the NADH concentration is continuously monitored by a UV-vis spectrophotometer (Bausch and Lomb, Spectronic 2000) set at 309 nm. As the reaction vessel is outside the spectrophotometer, 10% of the reaction solution is continuously pumped into a flow-through cell (Hellma, Model 176,051-QS) in the spectrophotometer. The residence time in this recirculating loop is 10-15
NADH
-
+ H' + f/zOz
NAD'
+ H20
(1)
where NAD' is &nicotinamide adenine dinucleotide, also known as Coenzyme 1. Under acidic conditiom with a continuousoxygen supply, this reaction is known to exhibit bistability between two oxygen steady states,' bistability between a steady state and an oscillatory state,* periodi~ity,~ intermittency: and chaos.4 In previous papers, we presented the effects of external sinusoidal perturbations in the oxygen inflow on the kinetics and thermodynamics, including the dissipation and efficiency, of a limit cycle of the PO r e a ~ t i o n .In ~ ~these ~ studies?" there occur changes in the externally perturbed reaction compared to the autonomous oscillatory reaction with respect to the average concentrations of products and reactants, the average Gibbs free energy change, AG,the average rate of the PO reaction, J H R p I the phase shifting, 9,between the AG and the J H R p , the average dissipation, D, and hence also the efficiency. These changes are expected properties of "alternating current chemistry". The dissipation changes depending on the amplitude and frequency of the sinusoidal perturbation and is lowered the most relative to the autonomous state when the perturbation frequency is half the autonomous frequency. In this present work, we continue, by means of experiments, the study of the effects of external periodic perturbations of varying forms on the kinetics and thermodynamics of various dynamic states of the PO reaction. The perturbation is applied to the oxygen gas flow into the reaction system and is chosen to be sinusoidal, a square wave, a left or right sawtooth wave, or a two-term Fourier series wave. The periodic perturbations are applied to a stable focus and a periodic (limit cycle) region of the PO system. In the succeeding paper' we compare these experimental results to numerical predictions obtained from the Degn-Olsen-Perram (DOP) model.* In section I1 of this paper, we describe the experimental apparatus and in section I11 we present the methods of measurements and the data analysis. In section IV, we present results of twetenn Fourier series perturbations on an experimental limit cycle of the PO reaction; in section V, the same is done for periodic perturbations, either sinusoidal, sawtooth, or square waveforms, on a stable focus of the PO reaction. Finally, in section VI, we present the first experimental evidence for the existence of quasiperiodicity in this reaction, which has been predicted theoretically. 0022-3654/92/2096-7338S03.00/0
S.
The NAD+ produced by reaction 1 is recycled back to NADH by a second enzyme via the reaction NAD'
+ G6P
-
6PGL
+ NADH + H'
(2)
where reaction 2 is catalyzed by glucose-6-phosphate dehydrogenase (G6PDH), and the product 6-phosphogluconolactone (6PGL) does not interfere with the PO reaction. As long as the concentration of the substrate glume-6-phosphate (G6P) is much higher than the concentration of NAD', reaction 2 ensures a continuous supply of NADH. The reaction solution is maintained at atmospheric pressure at a temperature of 24 O C . The reaction solution has a volume of 4 mL, is maintained at pH 6.0 by a 0.5 M (2-(Nmorpho1ino)ethanesulfonic acid) buffer (MES from Sigma 0 1992 American Chemical Society
The Journal of Physical Chemistry, Vol. 96, No. 18, 1992 7339
The Horseradish Peroxidase-Oxidase Reaction
Q -
0,
CHART RECORDER
is the time average of the instantaneous AG values. The rate of the reaction, JHRp, is calculated from the slope of the NADH measurements since we have
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where &pDH is the known12 rate of the regeneration of NADH from NAD' via reaction 2. The average J H R p for a given time period is simply the time average of the corresponding instantaneous J H R p values. For a process occurring at a nonzero rate under isothermal conditions, the dissipation of the process is defined as
i >\\\\\\\
2
-
VIS SPECTROPHOTOMETER
D = dissipation = T(dS/dt)
(5)
that is, the temperature times the rate of entropy production. Furthermore, at constant pressure, the dissipation of an isothermal reaction is D = force X flux = A G ( J H R p )
The instantaneous dissipation is calculated from eq 6. For an oscillatory reaction, or one subject to a periodic external perturbation, the average dissipation is taken over one cycle of oscillation or over a long time period
GENE RATOR TANK TANK I AID
(6)
'I
D = ( WJHRP) )
(7)
WA
Figure 1. Schematic diagram of the experimental apparatus.
where the brackets denote an averaging over time.
Chemical Co.), and initially contains 1.0 or 1.5 mM /3-NAD+ (crystalline /3-NAD+from Sigma, Product N1511, Grade 111-C), 25 mM G6P (Sigma), 1 pM methylene blue (Sigma), 50 pM 2,4-dichlorophenol (Sigma), and 3 w 5 0 activity units per mL of HRP (obtained from Sigma, Type X with a Reinheitszahl, A403/A275, of 3.0 and an activity of 250 units/mg protein (Biuret), as a crystalline suspension in a 3.2 M (NH4)2S04solution buffered at pH 6.0 with potassium phosphate), where one activity unit of HRP will form 1.O mg of purpurogallin from pyrogallol in 20 s at pH 6.0 at 20 "C. The amount of HRP used depended on the autonomous dynamics desired. For a limit cycle, 1.5 mM NAD' and 150 units/mL HRP were used; for a stable focus, 1.0 mM NAD+ and 100 units/mL HRP were used; and for quasiperiodicity, 1.O mM NAD' and 30-60 units/mL HRP were used. The solution is equilibrated with a 3 mol 96 02/N2gas mixture with a flow rate of 3 mL/s. Reaction 2 is started by the addition of 5 activity units of G6PDH, where one activity unit of G6PDH per minute in the oxidizes 1.O pM G6P to 6-phospho-~-gluconate presence of NADPH at pH 7.8 at 30 OC, whereupon reaction 1 commences. The G6PDH is produced from Leuconostoc Mesenteroids and is obtained from Sigma as a suspension in a 3.2 M (NH4)2S04solution buffered at pH 6.0 (Sigma, Type XXIII, with 300-600 NADP activity units per mg protein (Biuret)). After the autonomous dynamics have stabilized, generally 5-10 min after the reaction has started, the periodic perturbation is applied and is maintained for at least eight perturbation periods. As a reaction may last for 1-3 h, several perturbations may be applied during the course of a reaction.
IV. Effects of Two-Term Fourier Series Perturbations on an Experimental Limit Cycle of the PO Reaction Two-term Fourier series perturbations to the oxygen gas inflow were applied experimentally to a limit cycle of the PO reaction, where the oxygen flow is
III. Measurements and Data Analysis The time series for the NADH absorbance obtained from the spectrophotometer, the voltage reading from the oxygen electrode inside the reaction vessel, and the voltage readings from the two mass flow controllers are measured, digitized, and recorded every 0.5 s by a computer. The NADH and oxygen readings are then converted to concentration values and the resulting concentration time series are used to calculate the instantaneous and average AG, JHRp, and the dissipation. For both autonomous and driven oscillations, the phase shifting between the maxima of the AG and the J H R p oscillation periods are also obtained from the measurements. The instantaneous AG is calculated by AG = AGO
+ RTln ((NAD+)/(NADH)(02)1/2)
(3)
where AGO is known.6.9-11The average AG over a time period
O2 flow = steady O2flow(1
+ A(cl sin (ut) + c2 sin ( 2 4 1 ) (8)
where A is the amplitude of perturbation, u is the frequency of perturbation, and c1 and c2 are the Fourier coefficients. Furthermore, we make the restriction13
The values of c1and c2were varied from 0.0 to 1.O. Perturbations were applied at an amplitude of 50%, and with reduced frequencies of 0.5 and 1.0, where the reduced frequency is defined as the frequency of perturbation divided by the autonomous frequency
r = u/uo
(10)
The autonomous frequency was determined by allowing the attractor to evolve to the limit cycle and then measuring the frequency of the stable oscillations. The autonomous frequency could also be found from the Fourier spectrum of the time series. As in the sinusoidal perturbations on a limit cycle of the PO reaction, the two-term Fourier perturbations change the average concentrations of products and reactants, the average AG, the average JHRp, the phase shift d, the average dissipation, and the efficiency of the system, as compared to the respective quantities in the autonomous PO reaction. In general, the dissipation of the perturbed state is lowered up to 6% relative to the autonomous state; however, the extent of the change in the dissipation varies as c2 varies. The c2 values which are optimum for lowering the dissipation range from 0.3 to 0.5, as shown in Table I and Figure 2. Therefore, a two-term Fourier series is more efficient at lowering the dissipation that is a simple sinusoidal perturbation. This is in agreement with previous findings on a methane combustion rea~ti0n.l~ Table I and Figure 2 also illustrate the finding that when r = 1.0, the dissipation is generally lower than when r = 0.5. As the two-term Fourier series has a two-peaked shape, a two-term Fourier series perturbation of r = 1.0 roughly corresponds to a simple sinusoidal perturbation of r = 0.5, which has been shown in ATP-ase proton pump studies15 to be the frequency of (sinusoidal) perturbation which maximizes the efficiency of the proton pump.
Samples et al.
7340 The Journal of Physical Chemistry, Vol. 96, No. 18, 1992 TABLE I: Experimental NADH (N)md Oxygen Responses (0)and the Reduced Dissipation, D/DWfor a Two-Term Fourier Series Waveform Perturbation with an Amplitude of 50%and with Varying Reduced Freuuencies ( r )and Fourier Coefficient c2 0 DIDO c2 r N 0.0 1.0 periodic periodic 0.9925 periodic 0.9858 0.0 0.5 periodic 0.9382 1.0 periodic periodic 0.316 0.9557 0.5 periodic periodic 0.316 periodic 0.9377 0.447 1.0 periodic periodic 0.9719 0.447 0.5 periodic periodic 0.9504 0.548 1.0 periodic periodic 0.9815 0.548 0.5 periodic periodic 0.9785 0.632 1.0 periodic periodic 0.9790 0.632 0.5 periodic 0.707 1.0 periodic periodic 0.9666 0.5 periodic periodic 0.9884 0.707 0.775 1.0 periodic periodic 0.9907 0.775 0.5 periodic periodic 0.9995 periodic 0.9840 0.837 1.0 periodic periodic 1.0260 0.894 1.0 periodic 0.9090 0.5 periodic periodic 0.894
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Fourier Coefficient C, Figure 2. Reduced dissipation, DIDo, vs Fourier coefficient c2, eq 8, for w = wo and w = 0 . 5 ~for 0 experimental perturbations on a limit cycle of the PO reaction.
The waveshapes for c2 = 0.3-0.5 look very similar to a left sawtooth waveshape, whereas for other c2 values, there is little or no resemblance to a sawtooth (either left or right) waveshape. The autonomous oscillations in the PO reaction are relaxation oscillations which most resemble a right sawtooth waveshape. So, the lowest dissipation is observed when the perturbation waveshape opposes the autonomous waveshape. It is clear that the dissipation of the perturbed state is not just sensitive to the amplitude and frequency of the external perturbation, but it is also sensitive to the perturbation waveshape. An interesting alternating-frequency response to the two-term Fourier series perturbations is observed when c2 is high and when r = 1. This is illustrated in Figure 3, which shows the oxygen time series and phase portrait for an experiment. In the time series, the frequency of the cycles alternates between two values whose average equals the perturbation frequency. The phase portrait shows this alternating response and also shows how the trajectories of the perturbed state have spread. We believe that this spread in the attractor is the result of its vicinity to a bifurcation point, and the perturbation pushes the autonomous state closer to a bifurcation. In earlier work, Ross and Vance found such an effect in experiments on the periodic forcing of the hydration of an epoxide. ti V. Effects of Sinusoidal and Nonsinusoidal Periodic Perturbations on an Experimental Stable Focus of the PO Reaction Sinusoidal and nonsinusoidal periodic perturbations to the oxygen gas inflow were applied experimentally to a stable focus of the PO reaction. The forcing waveshapes were sinusoidal,
0.5
1
1.5
2
2.5
3
Oxygen Conc. (t) Figure 3. Experimental response of a limit cycle to a two-term Fourier series perturbation;time series (a) and phase portrait (b) of the oxygen concentration in solution. Note the alternating-frequencyresponse in the time series and the spread of the trajectories in the phase portrait. External perturbation: A = 50%; r = 1.0; c2 = 0.775; see eq 8.
TABLE Jk Experimental NADH (N)and Oxygen (0)Responses md the Reduced Dissipation ( D / D o )for a Stable Focus Perturbed with Varying Amplitudes (A ) A waveform DIDn N 0 0.10 sinusoidal 1.006 negligible aperiodic to periodic 0.10 right sawtooth 0.986 negligible aperiodic 0.986 negligible aperiodic 0.10 left sawtooth periodic 0.99 1 periodic 0.10 square periodic 0.920 periodic 0.25 sinusoidal periodic 0.25 right sawtooth 0.998 periodic periodic 1.005 periodic 0.25 left sawtooth periodic 0.986 periodic 0.25 square periodic 0.98 1 periodic 0.50 sinusoidal periodic 0.50 right sawtooth 0.976 periodic periodic 0.940 periodic 0.50 left sawtooth periodic 0.987 periodic 0.50 square periodic 0.932 periodic 0.75 sinusoidal periodic 0.75 right sawtooth 0.990 periodic 0.904 periodic 0.75 left sawtooth periodic 0.974 periodic 0.75 square
square, right sawtooth, and left sawtooth. The amplitudes of perturbation ranged from 10% to 7596, and the driving frequency equaled the autonomous frequency. The autonomous frequency was determined by letting the stable focus evolve to the steady state and measuring the frequency of the oscillations as they approached the steady state. The experimental results for the sinusoidal and nonsinusoidal perturbatioins applied to a stable focus are summarized in Table 11. Aperiodic O2and negligible NADH responses were seen for low amplitudes of perturbation, and periodic NADH and O2 responses were observed for higher amplitudes of perturbation.
The Horseradish Peroxidase-Oxidase Reaction '
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The Journal of Physical Chemistry, Vol. 96, No. 18, 1992 7341
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A typical example of an aperiodic O2and a negligible NADH response is shown in Figure 4 for a 10% right sawtooth perturbation. The exact type of dynamics displayed in the perturbed state could not be determined due to the high level of noise present in the stable focus. The NADH response shows no oscillations, within experimental error, but does show a slight change in average concentration. This nonoscillatory NADH response, compared to the oscillations of NADH in the autonomous PO system, indicates that NADH is a nonessential species, Le., NADH need not oscillate in order for other species in the system to oscillate. For a discussion of the role of essential and nonessential species in oscillatory reactions, see refs 17-19. The dissipation for the perturbed states is, in general, lower than that for the autonomous state, by up to 10%. The tabulation of the reduced dissipation DIDo, Table 11, leads to two points. First, as the amplitude of perturbation increases, the dissipation de". Second, the left sawtooth and the sinusoidal waveforms are, in general, most efficient in lowering the dissipation, where the left sawtooth is the waveform least like the autonomous waveform.
VI. Evidence for the Existence of Quasiperiodicity in the PO Reaction Until now, quasiperiodicityhas not been presented in the PO reaction, although it has been predicted by theoretical models.8*20 Here, we present experimental evidence of quasiperiodicity in the PO reaction. The experimental apparatus described above was modified such that there as a continuous infusion of NADH-the method employed by Degn and Olsen4-into the reaction solution instead of recycling the NADH via reaction 2. The cylindrical reaction vessel was replaced with a 3.0 mL quartz, rectangular cuvette. The flow loop through the spectrophotometer was removed, so only the oxygen concentration was monitored continuously with time. The experimental apparatus is shown in Figure 5 . The typical parameters used were as follows: 24 OC;pH 5.1 in a sodium acetate buffer; 60-80 HRP activity units/mL; 1.5% O2 at a gas flow rate of 1 mL/s; O2 diffusion rate constant of O.OO984.015 s-l; 0.2 p M methylene blue; 15 pM 2,4-dichlorophenol; and 0.16 M NADH (obtained as a disodium salt from Sigma, product N8129) solution infused at 13 pL/h or a 0.020 M NADH solution infused at 80 pL/h. A typical quasiperiodic run and its characteristic looplike next-phase plot (equivalent to the next-amplitude plot) are shown in Figures 6 and 7, respectively. W. Conclusion We have shown with experiments that a tw+term Fourier series perturbation on a limit cycle of the nonlinear biochemical PO reaction is more efficient at lowering the dissipation than is a simple sinusoidal waveform. This is in agreement with previous experimental findings on a methane combustion system.I4 The
F i p e 5. Schematic diagram of the experimental apparatus used for finding quasiperiodicity. ' ' ' '
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optimum value for c2, eq 8, is found to be 0 . 3 4 3 , where the waveshape most closely resembles a left-sawtooth wave. When the driving waveform is out-of-phase or of a dissimilar shape to the autonomous oscillations, the dissipation is lowered. For the two-term Fourier series perturbations on a limit cycle, we also found that a reduced frequency of 1.O lowered the dissipation more
J. Phys. Chem. 1992, 96, 7342-7346
7342
relative to the autonomous state than did a reduced frequency of 0.5. As a two-term Fourier series has a two-peaked shape, a two-term Fourier series at r = 1.O corresponds to a simple sinusoidal perturbation at r = O S . Thus, this finding is in agrement with previous sinusoidal perturbation experiments on a limit cycle of the PO r e a ~ t i o n . ~ . ~ We found above that when the driving waveform is out-of-phase or of a dissimilar shape to the autonomous oscillations, the dissipation is lowered. This is confirmed experimentally by the sinusoidal and nonsinusoidal perturbations on a stable focus of the Po reaction. Here, we ohserved that a left-sawtooth waveform is most efficient at lowering the dissipation of the system; in contrast, the autonomous waveshape is most similar to a rightsawtooth waveshape. We also showed that as the amplitude of perturbation increases, the dissipation decreases, which confirms previous experimental findings?s6 For the perturbation experiments (sinusoidal and nonsinusoidal) on a stable focus, the dissipation is, in general, lowered relative to the autonomous state. For the experiments on a stable focus, a negligible NADH response is often observed for low amplitudes of perturbation ( A = 10%). This negligible NADH response suggests that NADH is a nonessential species in a stable focus: it is not necessary for the concentration of NADH to oscillate with time in order for other species in the system to oscillate. As several papers have recently outlined, important mechanistic information may be obtained upon the categorization of species as essential or Finally, we have presented the first experimental evidence for quasiperiodicity in the PO reaction as judged by the time series and the next-phase plots.
Acknowledgment. This work was supported in part by the National Institutes of Health.
Re@&y No. NADH, 58-68-4; 02,7782-44-7; peroxidase, 9003-99-0; oxidase, 9035-73-8.
References and Notes (1) Degn, H. Nature 1968, 217, 1047. (2) Aguda, B. D.; Hofmann Frisch, L.-L.;Olsen, L. F. J. Am. Chem. Soc. 1990, 112,6652. (3) Nakamura, S.; Yokota, K.; Yamazaki, I. Nature 1969, 222, 794. (4) Olsen, L. F.; Degn, H. Nature 1977, 267, 177. (5) Lazar, J. G.; Ross, J. Science 1990, 247, 189. (6) Lazar, J. G.; Ross, J. J . Chem. Phys. 1990,92, 3579. (7) Samples, M. S.; Ross, J. J . Phys. Chem. Following paper in this issue. (8) Degn, H.; O b ,L. F.; Perram, J. Ann. N.Y.Acad. Sei. 1979,316,623. (9) The AGO for reaction 1 is found from the following two half-cell reactions and their Eo values: 1/202 + 2H+ + 2e- H20, Eo (pH = 0) = 0.6145 V (ref IO); and NADH + H+ 4 NAD+ + 2H+ + 2 6 , Eo (pH 7) = 0.32 V (ref 11). These values are then converted to AG by AG = -nFE, adjusted to pH = 6.0, and then added together to obtain the overall AGO for reaction 1 . (10) CRC Handbook of Chemistry and Physics, 65th ed.; Weast, R. C., Astle, M. J., Beyer, W. H., Eds.; CRC Press: Boca Raton, FL, 1984-1985; p D-157. (1 1) Styrer, L. Biochemistry, 3rd ed.; W. H. Freeman and Co.:New York, 1988; p 401. (12) J G ~ ~isDdetermined H by a calibration reaction with no HRP present. H this rate remains constant throughout the Then J ~ ~ =D d(NADH)/dt; experiment. A typical value for our experiments is 0.1 absorbance unit/min. (13) Hjelmfelt, A.; Harding, R. H.; Tsujimoto, K. K.; Ross, J. J. Chem. Phys. 1990, 92, 3559. (14) Hjelmfelt, A.; Harding, R. H.; Ross, J. J . Chem. Phys. 1989, 91, 3677. (15) Schell, M.; Kundu, K.; Ross, J. Proc. Narl. Acad. Sci. U S . A . 1987, 84, 424. (16) Vance, W.; Ross,J. J . Chem. Phys. 1988,88, 5536. (17) Eiswirth, M.; Freund, A.; Ross,J. J . Phys. Chem. 1991,95, 1294. (18) Eiswirth, M.; Freund, A.; Ross, J. Inr. Ser. Mathematical Numbers 1991, 97, 105. (19) Eiswirth, M.; Freund, A.; Ross, J. Adu. Chem. Phys. 1991,80, 128. (20) Steinmetz, C. G.; Larter, R. J . Chem. Phys. 1991, 94, 1388.
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Theoretical Studies and Comparison with Experiments on the Horseradish Peroxidase-Oxidase Reaction Marjorie S. Samples and John ROSS* Department of Chemistry, Stanford University, Stanford, California 94305 (Received: May 14, 1992) We study numerically the changes in the kinetics and the thermodynamics for the periodically forced oxidation of reduced nicotinamide adenine dinucleotide (NADH) by molecular oxygen. This reaction, which is catalyzed by the horseradish p x i d a s e enzyme, is referred to as the PO reaction. The model used for the calculations is the Degn-lsen-Perram (DOP) model, a simple four-variable model. We choose different forms of external periodic perturbations in the inflow of molecular oxygen to observe the effect of such forms on the dissipation of the system. On forcing a numerical limit cycle with a two-term Fourier series waveshape, we observe that the dhipation of the system is lowered relative to the autonomous system. Sinusoidal perturbatioins of a stable focus in the model system also show that the dissipation may be lowered relative to the autonomous state and that, as the perturbation amplitude increases, the dissipation decreases. The calculations indicate that NADH is an essential species in the PO reaction. By application of a perturbation to a stable focus, bistability between a steady state and an oscillatory state was observed. Numerical calculations on a chaotic state of the PO reaction show both periodic and chaotic responses, with the chaotic responses leading to lower dissipation than the periodic responses. Comparison of these calculations with the experiments described in the preceding paper leads to the conclusion that the DOP model is stiffer than the experiments; perturbation factors, such as the frequency of perturbation and the Fourier coefficient c2 for two-term series perturbations, lead to larger changes in the dissipation in the experiments than in the calculations. The experiments on a stable focus suggest that NADH is a nonessential species while the calculations suggest that NADH is an essential species.
I. Introduction A widely studied nonlinear biochemical reaction is the horseradish peroxidase (HRP) enzyme catalyzed oxidation of reduced nicotinamide adenine dinucleotide (NADH) by molecular oxygen. This reaction is also referred to as the peroxidaseoxidase (PO) reaction. The stoichiometry of the overall reaction is NADH + H'
+ f/202
-
NAD'
+ HzO
(1)
where NAD' is &nicotinamide adenine dinucleotide. Under acidic 0022-3654/92/2096-7342303.00/0
conditions and with a continuous oxygen supply, this reaction displays a variety of dynamics including bistability between two oxygen steady states,I bistability between a steady state and an oscillatory state: periodicity: intermittency," and chaos.4 In the prccading paper, we present experimental evidence for the existence Some prior work on this of quasiperiodicity in the PO rea~tion.~ reaction is discussed in ref 5. In the present work, we continue to study, by means of numerical calculations, the effwts of external periodic perturbations of varying fonns on the kinetics and thermodynamics of the diverse 0 1992 American Chemical Society
