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 calibrationreaction with no HRP present. Then J ~ ~ =D d(NADH)/dt; H this rate remains constant throughout the experiment. A typical value for our experimentsis 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-variablemodel. 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 experimentsdescribed 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 N A D H is a nonessential species while the calculations suggest that N A D H 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 ~ prior work on this of quasiperiodicity in the PO r e a ~ t i o n .Some 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
The Journal of Physical Chemistry, Vol. 96, No. 18, 1992 1343
The Horseradish Peroxidasdxidase Reaction
TABLE I: The Eight Step Mechanism and the Corresponding Four Nonlinear Differential Bate Equations of the DOP Model"
if A
A
kS
+ B + X *kl 2X
Y*Q
k2
2x
4 6
J .5
il
The numbers refer to the steps shown in Table I.
dynamic states of the PO reaction. The model used is a simple four-variable, eight step mechanism formulated by Degn, Olsen, and Perram? The perturbation is applied to the rate constant corresponding to the oxygen gas flow into the reaction system and is chosen to be a sinusoidal or a tweterm Fourier series waveform. The periodic perturbations are applied to a stable focus (sinusoidal perturbations), a periodic (limit cycle) region (two-term Fourier series perturbations), and a chaotic region (sinusoidal perturbations) of the PO system. In section 11, we discuss the model reaction mechanism used in the calculations and the methods of analysis. In section 111, we present results of two-term Fourier series perturbations on a model limit cycle of the PO reaction and compare them with our previous experimental results. In general, the dissipation of a perturbed state is lowered relative to the autonomous state for both experiments and calculations, although the DOP model is stiffer than the experiments. In section IV, we present results of sinusoidal perturbations on a stable focus of the DOP model of the PO reaction and compare them with our previous experimental results given in the preceding paper. The experiments suggest that NADH is a nonessential species while the calculations suggest that NADH is an essential species in the DOP model. Bistability between a stable focus and a limit cycle in the DOP model is shown. In section V, the results of sinusoidal perturbations on a chaotic theoretical state of the PO reaction are presented. Both chaotic and periodic respom to the sinusoidal perturbations are seen, depending on the amplitude of perturbation. The dissipation generally increases as the amplitude of perturbation increases, which is the opposite of the findings for perturbations on a limit cycle and stable focus.
II. Numerical Model and Analysis The calculations were performed on a simple four-variable, eight step model of the PO reaction proposed by Degn, Olsen, and Perram, commonly called the DOP model.* The DOP model resulted from the authors' search for a simple, general model which could reproduce the experimental sawtooth oscillations and the bistability between a high and a low oxygen steady state. Although it is not based on known mechanistic steps, but is instead based on autocatalysis and quadratic chain branching, the DOP model adequately reproduces these experimentally observed behaviors. The DOP model also shows chaos, mixed-mode oscillations, quasiperiodicity, and phase-locking on a t o r u ~ . ~The * ' ~network diagram for the DOP model is shown in Figure 1 and the eight kinetic step and the corresponding four differential equations are shown in Table I, where A corresponds to oxygen, B corresponds to NADH, X is an intermediate (almost definitely NAD'), and Y is another intermediate (probably an oxidation state of the HRP e n ~ y m e ) . ~ J lThe J ~ autocatalysis is readily seen in step 1 where a molecule of the intermediate X forms two molecules of X,while the branching is evident in the second step where the intermediate X forms the other intermediate Y. The third step, where the one Y forms two X's, is a second way to produce X;the branching step (step 2) links this step to the autocatalytic step (step 1). The fourth and fifth steps are the termination reactions for the in-
k6
xo-, x
2Y
k8
Bo* B
7 -7
Figure 1. Stoichiometric network diagram for the abstract 4-variable DOP model of the PO reaction where A = 02,B = NADH, X is probably NAD', and Y is probably an oxidation state of the HRP enzyme.
4
+
dA/dt = -k,ABX - k,ABY k,Ao - k-,A dB/dt -klABX - kiABY + k,Bo d X / d t k,ABX - 2 k 8 + 2kjABY - kJ dY/dt -k,ABY + 2 k 8 - k5Y
+ kdo
" A is dissolved oxygen, A. is the gaseous oxygen flowing into the reaction, B is the NADH concentration in the reaction, Bo is the NADH cancentration which is flowing into the reaction, X is probably NAD', step 6 accounts for the spontaneous formation of X, and Y is an unknown intermediate which is most likely one of the HRP oxidation states.
termediates X and Y,the sixth step accounts for the spontaneous produdon of X,the seventh step corresponds to the equilibration of the oxygen gas phase with the aqueous phase, and the eighth step corresponds to the inflow of NADH into the system. The four nonlinear differential equations arc readily derived from these eight step. In this study, these four differential kinetic equations are solved by the B O D E integrating routine which uses a GEAR method.13 In these numerical studies, only the rate constants kl (which includes the HRP concentration) and k7 (corresponding to the gaseous oxygen diffusion rate constant) are varied as these are the variables which are generally varied in experiments. Depending on the values of these two constants, various autonomous dynamics are observed, including steady states, periodicity, quasiperiodicity, and cham. For a numerical calculation, the rate constants and initial conditions are chosen and the integration commences. After calculating the transients 80 that the desired dynamic state has stabilized, the periodic perturbation, either sinusoidal or a two-term Fourier series, is applied to the oxygen inflow kinetic equation, for a minimum of 20 cycles. The perturbation is then removed and the system is then allowed to relax-either to the original autonomous state or to a new state. The perturbation has the form k7,= k7(l A(cl sin (rut) c2 sin (mot))) (2)
+
+
where is the perturbed oxygen diffusion rate constant, k7 is the unperturbed rate constant, A is the amplitude of perturbation, c1 and c2 are the Fourier coefficients, w is the frequency of perturbation, and t is the time. For a Sinusoidal waveform, c2 k zero. Furthermore, for the two-term Fourier series perturbations we make the restri~tion'~ k71
+ c*2 = 1
(3) For sinusoidal perturbations, the amplitude of perturbation (A) was varied from 10%to 75% and the reduced frequency of perturbation (r = w/wo, where w is the driving frequency and oois the autonomous frequency) was varied from 0.94 to 1.0. For the two-term Fourier series perturbations, the amplitude of perturbation varied from 10% to 10096, the Fourier coefficient c2 varied from 0.0 to 0.894, and the reduced frequency of perturbation was varied from 0.5 to 1.0. From the numerical time series, the instantaneous and average Gibbs free energy change, AG, the instantaneousand average rate of the PO reaction, JHRP,and the instantaneous and average dissipation, D, are calculated. The instantaneous AG is calculated from the equation c12
AG = AGO
+ RT In ([NAD']
/ [NADH] [0211/2) (4)
where AGO is -106.127 kJ/m0l.'9~~-" The average AG over a time
7344 The Journal of Physical Chemistry, Vol. 96, No. 18, 1992
Samples and Ross
TABLE Ik Comparhm between Calcuhted (de)and Experinnet.l (exp) R e d d DWpatioa ( D / D o )for a Llmit Cycle Perlurbd with a Two-Term Fourier Series of 50% Amplitude, and with Varying Fourier Coefficients c2 and R e d d F r e p d e s ( r )
0.0 0.0 0.316 0.316 0.447 0.447 0.548 0.548 0.632 0.632 0.707 0.707 0.775 0.775 0.837 0.837 0.894 0.894
1.o 0.5 1.o 0.5 1.o 0.5 1.o 0.5 1.o 0.5 1.o 0.5 1.o 0.5 1.o 0.5 1.o 0.5
0.999 0.999 0.994 0.994 0.992 0.993 0.992 0.996 0.994 0.996 0.985 0.996 0.995 0.994 0.995 0.993 0.998 0.993
0.9925 0.9858 0.9382 0.9557 0.9377 0.9719 0.9504 0.9815 0.9785 0.9790 0.9666 0.9884 0.9907 0.9995 0.9840
, . . , l , . , . l , . ' , l , , , , I , , , , I
6300
F i i 2. Calculated response of a limit cycle to a two-term Fourier series perturbation; time series of the oxygen concentration in solution showing a transient periodic response followed by a stable chaotic response. External perturbation: A = 100%; r = 1.O;c2 = 0.837; see eqs 2and 3. 1.2 -+-L-
period 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 (5)
WJHRP)
1 1 . 4
0.8
+ 0.6
The instantaneous dissipation is calculated from
D
8800
Tim(8ec)
1.0260 0.9090
d[NADH] /dt = JHRP
6 4 0 0 6 6 0 0 8 8 0 0 6 7 0 0
6
(6)
The average dissipation is taken over one cycle of oscillation or over a long time period
D=
(WJHRP))
(7)
where the brackets denote an averaging over time. For the perturbations, the reduced dissipation (DIDo),where D is the average dissipation of the perturbed state and Do is the average dissipation of the autonomous state, is also calculated.
III. Effects of Two-Term Fourier Series Perturbations on a Limit Cycle of tbe PO Reaction
A stable limit cycle (k,= 0.12, A d 7 = 0.890)of the numerical model was perturbed with a series of two-term Fourier series waveforms: the amplitude of perturbation was varied from 10% to 10096, the reduced frequency, r, was 0.5 or 1.O, and the Fourier coefficient c2 was varied from 0.0 to 0.894. From the calculated time series, the reduced dissipation, DIDo, is calculated and the results are compared to our experimental results of the previous paper,5 Table 11. The numerical NADH and oxygen responses to the two-term perturbation are of much smaller amplitude than the experimental responses. The change in the dissipation for the calculated responses is also smaller than the experimental dissipation changes, Table 11. In general, the calculated reduced dissipation, DIDo, is less than 1.O which is in agreement with the experiments, but the experiments show larger variations from 1.0. Although in the experiment$ the perturbation frequency which is optimum for lowering the reduced dissipation is clearly r = 1.0 (perturbations at the autonomous frequency), the calculations show no distinction between the reduced dissipation for perturbations at the autonomous frequency or at half the autonomous frequency. Furthermore, in the experiments, the magnitude of the Fourier coefficient c2 is found to be important, with c2 values from 0.3 to 0.5 lowering the r e d u d dissipation the most; however, for the calculations, there is not a strong relationship between the magnitude of c2and the reduced dissipation. Clearly, for the two-term Fourier series perturbations, the model is stiffer than the experiments. As the DOP model is not based on mechanistic steps and so is not a skeletal model, these discrepancies are undoubtedly a result of flaws in this widely studied model.
26
2.8
3
3.2
3.4 X
3.6
3.0
4
4.2
Figure 3. Poincarg scction of the oxygen response shown in Figure 2 (excluding the transient). Note the broken, fractal structure which indicates chaos.
Other interesting numerical responses to perturbation were observed. For A = 100% (Note that although the amplitude is 10096, k7,does not go to zero due to the interaction between the two terms of the Fourier series, eq 2), r = 1.0,and c2 = 0.837, there is a transient 1:l two-frequency periodic response followed by a stable chaotic response (as judged by the Poincard section which has a fractal, broken loop), Figures 2 and 3. For A = lo!%, r = 0.5, and c2 = 0.837,a stable chaotic response is observed immediately. These chaotic responses may be the result of the overlapping of neighboring entrainment bands.
IV. Effects of Shueoidal Periodic Perturbations w a Stable Focus of the PO Reaction Sinusoidal perturbations were applied to a stable focus (k,= 0.270,Aok7 = 0.890)of the DOP model and the results, along with a comparison to our previous experimental results, are summarized in Table 111. The perturbation amplitude varied from 10% to 75%. and reduced frequencies of r = 1.0 and r = 0.94 were applied. When the driving frequency equals the autonomous frequency of oscillation, the O2responses are periodic for all amplitudes of perturbation. This conflicts with the experimental fmdings outlined in our preceding paper,5where aperiodic O2responses were seen for low amplitudes of perturbation. However, when the driving frequency is changed slightly so that r = 0.94, then the calculated O2responses are quasiperiodic for an amplitude of perturbation of 10%and are periodic for all other amplitudes of perturbation. Therefore, it is probable that the experimentally chosen frequency of perturbation was not exactly equal to the autonomousfrequency although it was very close. (Due to the noise in the experimental
The Journal of Physical Chemistry, Vol. 96, No. 18, 1992 1345
The Horseradish Peroxidase-Oxidase Reaction TABLE Itk Comparison between Calculated ( d c ) and Experiw~tal (exp) NADH (N) and Oxygen (0)Responses and the Reduced D&dprtiOs( D / D , ) for a Stable Focus Siausoiddy Perturbed with Varying Amplitudes (A ) and Reduced Freqwacy ( r ) A expor calc 0.10 ex 0.10 calc r = 1.0 0.10 calc r = 0.94 0.25 exp 0.25 calc r = 1.0 0.25 calc r = 0.94
0.50 0.50 0.50 0.75 0.75 0.75
N
negligible periodic quasiperiodic periodic periodic periodic periodic exp calc r = 1.0 periodic calc r = 0.94 periodic periodic exp calc r = 1.0 periodic calc r = 0.94 periodic
0 aperiodic to periodic periodic quasiperiodic periodic periodic periodic periodic periodic periodic periodic periodic periodic
DIDO 1.006 1.W29 1.0027 0.920 1.0013 1.0016 0.981 0.9983 0.9979 0.932 0.9952 0.9944
1 2900
stable focus, it is Wicult to pick the exact autonomous frequency.) Another difference between the experiments and the calculations is evident in the NADH responses to low amplitudes of perturbation. Experimentally, for an amplitude of lo%, no oscillations were ob~erved.~In the calculations, for an amplitude of lo%, periodic oscillations are obtained when r = 1.O,and quasiperiodic oscillations are obtained when r = 0.94. The experimental results suggest that NADH is a nonessential species-it is not necessary for the NADH concentration to oscillate in order for the system to oscillatein a stable focus of the PO rea~tion;~ in contrast, the calculated NADH responses suggest that, in the DOP model, NADH is an essential species in a stable focus. The reduced dissipation, DIDo, was also determined for the numerical calculations, and the results and comparison with previous experiments are shown in Table 111. The reduced dissipations for r = 1.0 and r = 0.94 are almost identical for all amplitudes of perturbation. Also,as in the previous experiments on a stable focus and on a limit as the amplitude of perturbation increases, the dissipation of the perturbed response generally decreases. However, for the calculations, the reduced dissipation is above 1.O for amplitudesof perturbation below 50% and is below 1.O for amplitudes of perturbation greater than or equal to 50%, while in the experiment^,^ the reduced dissipation is above 1.0 only when A = 10%. Bistability between a stable state and an oscillatory state, referred to as 0 - S bistability, was also observed. When the rate constant which includes the HRP concentration was lowered slightly from k, = 0.270to 0.269 (thus moving the system closer to an Oscillatory regime, although for both values of k l , kl = 0.269 or 0.270,the autonomous state is a stable focus) a bistable response to perturbations in the oxygen diffusion rate constant k7,is seen, Figure 4. For the bistability calculations, the stable focus corresponding to kl = 0.269 was perturbed sinusoidally at the autonomous frequency for more than ten periods. After the perturbation was removed, the perturbed state (which was periodic) relaxed, not to the original stable focus but to a stable periodic state, as seen in Figure 4. Thus, we have shown that the DOP model displays 0 - S bistability, as seen experimentally by Aguda, Hofmann Frisch, and Olsen2 and as predicted by Aguda and Larter in a more complex model (model A) of the PO r e a c t i ~ n . ~ Furthermore, it is clear from Figure 4 that the average oxygen concentration for the autonomous stable focus and the unperturbed limit cycle are the same. This is in good agreement with the experiments,2whereas model A predicts that the average oxygen concentration on the oscillatory state is higher than the oxygen concentration of the steady state.9
V. Effects of Sinusoidal Perturbations on a Chaotic State of the DOP Model of tbe PO Reaction
A chaotic state (k,= 0.1 178,Aok7= 0.775)of the DOP model is perturbed sinusoidally to determine the effects on the kinetics and thermodynamics. The autonomous chaotic state chosen to be perturbed is a mixed-mode chaotic state composed of two frequencies, one dominant and one minor, referred to as wM and 0 ., The amplitudes of perturbation are 10%and SO%, and the
3000
3100
3200 3300
3400
3500
3600
Time("
Figure 4. Calculated autonomous and perturbed oxygen concentrations as a function of time showing the presence of bistability between a steady state and an oscillatory state. The autonomous system starts in a stable focus, but after a perturbation is applied and removed, the system relaxes to a limit cycle. Perturbation: A = 0.25, w = wo, Aok, = 0.269. TABLE I V Calculated NADH (N) and Oxygen (0)Responses and the Reduced Dissipation @ / D o ) for a Sinusoidal Perturbation with Varying Amplitudes (A) and Frequencies ( w ) on a Chaotic State of
the DOP Madel N chaotic chaotic chaotic chaotic periodic periodic periodic periodic
0 chaotic chaotic chaotic chaotic periodic periodic periodic periodic
DIDO 1.Ooo 1.Ooo 1.001 0.999 0.999 1.003 1.005 1.001
frequencies of perturbation are o.5wM, wM, 2 w M , and 0 ., For an amplitude of perturbation of lo%, all the NADH and O2responses are chaotic, as shown in Table IV. However, for an amplitude of perturbation of So%, the NADH and O2r e s p o w are periodic. Previously, a chaotic state of the Lorenz model has been observed to have a periodic (or quasiperiodic or chaotic) response upon application of a periodic (a cosine wave) perturbation. Table IV highlights another result observed. The dissipation of the chaotic responses (where the amplitude of perturbation is 10%) is unchanged from the dissipation of the autonomous chaotic state, except for the perturbation at w, which shows a lower dissipation than the autonomous chaotic state. The dissipation of the periodic responses (where the amplitude of perturbation is 50%) is slightly higher than the dissipation of the autonomous chaotic state, except for the perturbation at 0 . 5 with ~ ~A = 50%. As the amplitude of perturbation increases, the dissipation also increases. This is opposite to the result found for periodic perturbations on a stable focus and found previously for sinusoidal perturbations on a stable limit ~ y c l e . ~ - ~
'*
VI. Conclusion We have shown with calculations on the DOP model that a two-term Fourier series perturbation on a limit cycle of the PO reaction lowers the dissipation of the perturbed state relative to the autonomous state, in agreement with previous experiments. However, the calculations showed no relationship between the reduced frequency r or the Fourier coefficient c2 and the reduced dissipation, although these were found to be important experimental factors in determining the reduced dissipation. Calculations on a stable focus of the DOP model showed, for low amplitudes of perturbation with a reduced frequency of perturbation of r = 0.94,quasiperiodic NADH and oxygen responses,but for higher amplitudes of perturbation, only periodic responses were seen. For r = 1.O, periodic responses were seen
J. Phys. Chem. 1992, 96,7346-7351
7346
for all amplitudes of perturbation. So care must be taken when determining the autonomous frequency, as slight differences may result in varying dynamics. In our experiments on a stable focus, described in the preceding article,5 a negligible NADH response to perturbation suggests that NADH is a nonessential species in a stable focus, while the DOP calculations on a stable focus suggest that NADH is an essential species in a stable focus. This is a major disagreement, probably resulting from the simplicity of the DOP model. The numerical perturbation studies on a stable focus also showed that the DOP model exhibits 0 - S bistability, and the average oxygen concentration of the oscillatory state is the same as the oxygen concentration of the steady state. This is in good agreement with experiments performed by Aguda, Hofmann Frisch, and Olsen.2 Sinusoidalperturbations on a chaotic state of the DOP model show chaotic responses for low amplitudes of perturbation and periodic responses for high amplitudes of perturbation. The dissipation increases as the amplitude of perturbation increases, whereas the opposite is found for sinusoidal perturbations on an experimental limit cycle and for sinusoidal and nonsinusoidal perturbations on a calculated and experimental stable focus. Both the form of the external perturbation and the dynamic state of the system are of importance in determining the dissipation.
References rad 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.; Yamazalti, I. Nature 1969, 222, 794. (4) Olsen, L. F.; Degn, H. Nature 1977, 267, 177. (5) Samples, M.S.; Hung, Y.-F.; Ross, J. J. Phys. Chem. P r d n g paper in this issue. (6) Lazar, J. G.; Ross, J. Science 1990, 247, 189. (7) Lazar, J. G.; Ross, J. J. Chem. Phys. 1990, 92, 3579. (8) Degn, H.; Olsen, L. F.; Perram, J. W. Ann. N.Y. Acad. Sci. 1979,316, 623. (9) Aguda, B. D.; Larter, R.J . Am. Chem. Soc. 1990, 112, 2167. (IO) Steinmetz, C. G.; Larter, R. J . Chem. Phys. 1991, 94, 1388. (11) Olson, D. L.; Schecline, A. Anal. Chim. Acta 1990, 237, 381. (12) Olsen, L. F.; Hofmann Frisch, L.-L.; Schaffer, W. M.The Peroxidase0xida.w Reaction: A Case for Chaos in the Biochemistry of the Cell. In A Chaotic Hierarchy; Baier, G., Klein, M.,Eds.; World Scientific Publishers: Singapore, 1991. (13) Hindmarsh, A. C. ACMSignum News. 1980, 15, 10. (14) Hjelmfelt, A.; Harding, R. H.; Tsujimoto, K. K.; Ross,J. J. Chem. Phys. 1990, 92,3559. (15) The AGO for reaction 1 is found from the following two halfell reactions and their Eo values: '/zOz + 2H+ + 2e- HzO, Eo (pH = 0) = 0.6145 V (ref 16); NADH + H+ NADt + 2Ht + 2e; Eo (pH = 7) = 0.32 V (ref 17). These values are then converted to AG by AG = -rrFE, adjusted to pH 6.0, and then added together to obtain the overall AGO for reaction 1. (16) CRC Handbook of Chemistry and Physics, 65th ed.; Weast, R. C., Astle, M.J., Beyer, W. H., Us.CRC ; Press: Boca Raton, FL. 1984-1985; Acknowledgment. This work was supported in part by the p D-157. National Institutes of Health. (17) Stryer, L. Biochemistry, 3rd ed.; W. H. Freeman and Co.:New York, Registry No. NADH, 56-68-8;O2,7782-44-7;peroXidase,9003-73-O; 1988; p 401. oxidase, 9035-73-8. (18) Aizawa, Y.; Ueza, T. Prog. Theor. Phys. 1982,68, 1864.
-
-
Fast Reactions between Diatomic and Poiyatomic Molecules T. Stoecklin Laboratoire de Physico- Chimie Thgorique, 351 Cows de la Libgration, 33405 Talence Cedex, France
and D. C. Clary* Department of Chemistry, University of Cambridge, Lenrfield Road, Cambridge CB2 I E W,U.K, (Received: March 3, 1992;In Final Form: May 20, 1992)
Calculations of rate constants are presented for fast neutral reactions between polar diatomic and polyatomic molecules. The method applies the infiniteorder sudden approximation to treat the rotations of the diatomic molecule and a rotationally adiabatic approximation with asymmetrictop wavefunctions to describe the rotations of the polyatomic. A captureapproximation is used to calculate reaction cross sections and rate constants. The interaction potential is expressed as a sum of dipoldpole and dispersion contributions. The calculated rate constants compare well with analytical formulas derived for the limits of high and very low temperatures. The NaO + H20, NaO + 03,CH + CH20, and SH + NO2 reactions are considered as examples and calculated rate constants are compared with experiment for these four fast reactions. Quite good agreement is obtained for the NaO O3and CH + CH20 reactions.
+
1. Introduction
A rotationally adiabatic capture theory (AC) combined with various sudden approximations has been applied previously to calculate cross sections and rate constants for a wide range of fast reactions involving neutral or charged particles.'-'O For such reactions, a large amount of experimental rate constant data is now available and is needed in areas such as atmospheric,"J2 comb~stion,'~ and interstellar chemistty.l4 Most previous theoretical studies on neutral fast reactions were concerned with r e actions involving only atoms or diatomic molecules, although some calculations have been reported on the reactions of ions with symmetric and asymmetric top m ~ l e c u l e s . ~ ~ ~ J ~ A particularly powerful version of the rotationally adiabatic theory for fast moleculemolecule reactions is the adiabatic capture partial centrifugal sudden approximation (ACPCSA).I This method uscs the infiniteorder-sudden approximation (IOSA) for one molecular partner with a small rotor constant and the rota-
tionally adiabatic theory for the other molecular partner. The technique has, so far, been applied to several reactions between diatomic molecules.' It is reviewed and compared with other methods in ref 7. We present here an extension of this method to the fast reaction between a polar diatomic molecule and a polar polyatomic molecule described as an asymmetric top. We use an analytical expansion of the long range part of the interaction potentialI5 including electrostatic and dispersion terms. Also, for each angular configuration of the diatomic molecule we apply a classical capture theoryI6 to calculate cross sections and rate constants which are state-selected in the rotational states of the asymmetric top molecule. All molecules are treated as having closed electronic shells. The theory gives rate constants that refer to a summation over all product states and is most appropriate for very exothermic reactions with strongly attractive long-range potentials in the reactant channel. We apply the ACPCSA to the fast reactions 0 1992 American Chemical Society
