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detailed tests using harmonic and Morse oscillators which found that, while there was only a mild dependence on frequencies and force constants, the spectra showed large systematic changes for small changes in bond lengths. It was not obvious from the work on di- and triatomics that such a generalization would hold for larger polyatomic ions. In particular, testing of the iterative fitting of photoelectron intensities, with the Duchinsky rotation, has been done for only one larger species, f~rmaldehyde,'~ and found to give a good structure. We find a marked dependence of our simulation on bond lengths for c-C3H2'+which, upon iterative fitting, are close to ab initio predictions. Similar success has been found for the vibrational structure of the photoelectron detachment spectra of negative ions.22 More than a test of the general protocol, which may now find application for a variety of ions accessible by photoionization, this exercise pins down the geometry of an important ionic species whose structure would have been difficult to access by other methods.
Acknowledgment. We acknowledge the suggestion by Prof. Dr. Peter Botschwina (GBttingen) that we explicitly model the Franck-Condon structure in our photoelectron spectra. We also acknowledge Dr. A. D. McLean (IBM Almaden) and Prof. L. Radom (Australian National University) for details of their calculations on C3H2and C3H2'+. Funding from the Department of Energy, the National Science Foundation, and the Camille and Henry Dreyfus Foundation is gratefully acknowledged.
References and Notes (1) Clauberg, H.; Chen, P. J . Am. Chem. SOC.1991, 113, 1445. (2) Clauberg, H.; Minsek, D. W.; Chen, P. J . Am. Chem. SOC.1992,114, 99. (3) Smith, D.; Adams, N. G. Int. J . Mass Specrrom. Ion Processes 1984, 61, 15. Smith, D.; Adam, N. G.Inr. J. Mass Specfrom. Ion Processes 1987, 76, 307. Prodnuk, S. D.; DePuy, C. H.; Bierbaum, V. M. Int. J . Muss Specrrom. Ion Processes 1990, 100, 693. (4) Adams, N. G.;Smith, D. Asrrophys. J. 1987, 317, L25. (5) Wong, M. W.; Radom, L. Org. Mass Spectrom. 1989, 24, 539.
(6) Frisch, M. J.; Head-Gordon, M.; Trucks, G. W.; Foresman, J. B.; Schlegel, H. B.; Raghavachari, K.; Robb, M. A.; Binkley, J. S.;Gonzalez. C.; DeFrees, D. J.; Fox, D. J.; Whiteside, R. A.; Seeger, R.; Melius, C. F.; Baker, J.; Martin, R. L.; Kahn, L. R.; Stewart, J. J. P.; Topiol, S.;Pople, J. A. Gaussian 90; Gaussian Inc.: Pittsburgh, PA, 1990. (7) Thaddeus, P.; Vrtilek, J. M.;Gottlieb, C. A. Asrrophys. J . 1985, 299, L63. Bogey, M.; Demuynck, C.; Destombes, J. L. Chem. Phys. krr.1986, 125, 383. Kanata, H.; Yamamoto, S.;Saito, S . Chem. Phys. k i f . 1987, 140, 221. Bogey, M.;Demuynck, C.; Destombes, J. L.; Dubus, H. J . Mol. Specrrosc. 1987, 122, 313. (8) Reisenauer, H. P.; Maier, G.;Riemann, A.; Hoffmann, R. W. Angew. Chem., Inr. Ed. Engl. 1984, 23,641. Maier, G.;Reisenauer, H. P.; Schwab, W.; Carsky, P.; Hess, B. A,, Jr.; Schaad, L. J. J . Am. Chem. SOC.1987,109, 5183. Hirahara, Y.; Masuda, A,; Kawaguchi, K. J . Chem. Phys. 1991,95, 3975. (9) Wilson Jr., E. B.; Decius, J. C.; Cross, P. C. Molecular Vibrations; McGraw-Hill: New York, 1955. (10) Program No. QCMP 067, available from the Quantum Chemistry Program Exchange, Indiana University, Bloomington, IN, (11) Sharp, T. E.;Rosenstock, H. M. J . Chem. Phys. 1964. 41, 3453. Botter, R.; Dibeler, V. H.; Walker, J. A,; Rosenstock, H. M. J . Chem. Phys. 1966, 44, 1271. Rosenstock, H . M. Inr. J. Mass Specrrom. Ion Phys. 1971, 7, 33. (12) Duschinsky, F. Acra Physicochim. URSS 1937, 7, 551. (1 3) Chau, F.T.; McDowell, C. A. Spectrochim. Acra., Part A 1990,46A, 723. (14) We consider the relative intensities in this spectrum to be more reliable to those in the similar spectra we published in refs 1 and 2. The baseline is flatter, and signal-to-noise ratio is better. (15) Minsek, D. W.; Chen, P. J. Phys. Chem. 1990, 94, 8399. (16) Miller, T. A. Annu. Rev. Phys. Chem. 1982, 33, 257. (17) Saykally, R. J. Science 1988, 239, 157. Oka, T. Philos. Trans. R. SOC.London, A 1988, 324, 81. (18) Dunning, F. B., Stebbings, R. F., Eds.RydbergSrares ofAtoms and Molecules; Cambridge University Press: London, 1983. (19) Kanter, E.P.; Vager, Z.; Both, G.;Zajfman, D. J. Chem. Phys. 1986, 85, 7487. (20) Jonathan, N.; Morris, A.; Okuda, M.; Ross, K. J.; Smith, D. J. J . Chem. Soc., Faruday Trans. 2 1974, 70, 1810. Frost, D. C.; Lee, S. T.; McDowell, C. A. J . Chem. Phys. 1973,59, 5484. (21) Chau, F.T. THEOCHEM 1987,36,157. Chau, F. T. THEOCHEM 1987, 36, 173. (22) Ervin, K. M.; Ho, J.; Lineberger, W. C. J. Phys. Chem. 1988, 92, 5405. Ervin,K. M.; Lineberger, W. C. Adu. Gas Phase Ion Chem., in press.
Period-Doubling Bifurcatlons and Chaos in an Enzyme Reaction Torben Geest,t*t Curtis G. SteinmetzJvt Raima Larter,* and Lars F. 019en**t Physical Biochemistry Group, Institute of Biochemistry, Odense University, Campusvej 55, DK-5230 Odense M,Denmark, and Department of Chemistry, Indiana University-Purdue University at Indianapolis (IUPUI), Indianapolis, Indiana 46205 (Received: March 19, 1992; In Final Form: May 22, 1992)
The peroxidaseoxidase (PO) reaction is the peroxidasecatalyzed oxidation of organic electron donors with molecular oxygen as the oxidant. With NADH as the electron donor, the PO reaction is one of the simplest biochemical reactions showing nonlinear behavior such as bistability and oscillations. A few experimental observations have also indicated chaotic dynamics in this reaction. However, until now the evidence for chaos has been inconclusive, partially because no route to chaos has been established experimentally. Here we present the first experimental observation of period doubling bifurcations leading to chaos in the PO reaction.
Introduction Within the past decade it has become apparent that aperiodically fluctuating time series in biology and medicine often have deterministic origins.'-3 It was recently suggested that in living organisms such deterministic chaos is often a normal mode of behavior, whereas periodic oscillations may be abnormal or even path~logical.~ To enhance our understanding of chaotic fluctuations in complex biological systems,it is useful fmt to study chaos in simple experimental systems such as homogeneous chemical + Odense
* IUPUI.
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0022-365419212096-5678$03.00/0Q
or biochemical reaction systems. There are now several observations of chaos in chemical reaction^,^ the most well-known example being the Belousov-Zhabotinskii (BZ) However, there are also indications of chaotic behavior in biochemical reactions such as the PO reaction9 and the glycolytic system.Io Although the existence of autonomous chaos in giycoIysis has been predicted theoretically," chaos has been observed only in this reaction network, involving 12 enzyme-catalyzed reactions, when glucose is supplied periodically. The PO reaction therefore remains the only biochemical candidate for the study of chaos in the absence of external forcing. The PO reaction has a special role as a laboratory model for biological chaos. As a homogeneous, 1992
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Figure 1. Diagram of the experimental setup. The reaction vessel is a 20 X 20 mm quartz cuvette inserted into an aluminum holder containing channels for thermostating water and support for the oxygen electrode which is fitted into the side of the cuvette. The vessel is mounted in a dual-wavelength spectrophotometer for simultaneous measurements of oxygen and NADH. In A we show schematically the position of the cuvette in the spectrophotometer together with the inlets for NADH and the 0 2 / N 2gas mixture. In B we show a side view of the cell holder with the oxygen electrode mounted. The tube displayed to the left represents the inlet for thermostating water.
Experimental Section A stirred 7-mL sample in a 20 X 20 mm quartz cuvette mounted in a thermostatic jacket and containing 0.7 pM horseradish peroxidase (Boehringer Mannheim, immunoassay purity), 0.1 pM methylene blue, and 20-35 pM 2,4-dichlorophenol in 0.1 M sodium acetate buffer at pH 5.1 was in contact with an 02/N2gas mixture at atmospheric pressure containing 1.42% (v/v) 02.A solution of 0.08 M NADH in distilled water was pumped into the sample at a rate of 35-40 pL/h, using a Harvard Apparatus, Model 22 syringe pump. The infusion rate was chosen such that the average concentration of NADH was constant. The reactor was placed in an Aminco DW2000 dual-wavelength spectrophotometer, and the NADH concentration was measured as the difference between the absorbance at 360 and 380 nm. The oxygen concentration in the liquid was measured with a Radiometer oxygen electrode inserted into the side of the cuvette. Figure 1 shows diagrams of the experimental setup. The NADH and oxygen signals were sampled and digitized (sampling rate 1 Hz) by a microcomputer fitted with an analog-to-digitalconverter and stored on disk for further analysis. The temperature was 28
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Figure 2. Oscillations in oxygea concentration during the peroxidasecatalyzed oxidation of NADH in a system with continuous supplies of NADH and 02.Dichlorophenol was added to the reaction mixture from a 20 mM stock solution in ethanol to the final concentrations of (A) 20 pM, (B) 25 pM, (C) 30.8 pM,and (D) 32.2 pM. A 0.08 M solution of NADH was pumped into the sample at an initial rate of 39 pL/h, which .was later decreased to 36 pL/h to keep the average NADH concentration constant. Other conditions are described in the Experimental Section. 2.72
isothermal system it is of the simplest class of chemical oscillators. But as an enzyme-catalyzed reaction whose substrate, reduced nicotinamide adenine dinucleotide (NADH), is one of the fundamental molecules in metabolism, the PO reaction is far more "biological" than reactions with transition-metal catalysts and simple organic, or even inorganic, substrates (such as the BZ reaction). Although studied primarily as a model system the PO reaction also occurs in vivo in plants.'* In addition to chaos the PO reaction has previously been shown to produce periodic oscillation~,'~ bi~tability,'~ and coexistence of a limit cycle oscillation with a stable steady state.I5 Since the initial observation of chaos in the PO rea~tion,~ much progress has been made in the understanding of the origins of chaotic behavior, and the analysis of chaotic data from experiments has been refined. The evidence presented previously for chaos in the PO reaction has consisted of aperiodic time series, next-amplitude maps, and qualitative reproduction of chaotic time series in computer simulations.l69I7 It is now generally accepted that, whenever possible, a route to chaos should also be demonstrated.
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Results and Discussion In the PO reaction peroxidase catalyzes the oxidation of reduced substrates such as NADH with molecular oxygen as the electron acceptor. In a well-stirred homogeneous solution with continuous supplies of oxygen and NADH and in the presence of the modifiers 2,4-dichlorophenol and methylene blue, the concentrations of NADH and oxygen will oscillate. Figure 2 shows oscillations of oxygen at four different concentrations of dichlorophenol. Dichlorophenol has been shown to increase the rate of breakdown of oxyferrous peroxidase1*(also known as compound 111), which
5680 The Journal of Physical Chemistry, Vol. 96, No. 14, 1992 ~
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Figure 4. Three-dimensional phase plot with T = 6s (A), Poincart section (B) and return map (C) of the data from Figure 2D. In (B) we show the reconstructed flow projected in two dimensions and the Poincart plane used for slicing the flow (normal to the page and indicated by the line). The return map in (C) is constructed by plotting each point on the Poincart section against the preceding point.
is one of the enzyme intermediates of this reaction. With increasing dichlorophenol concentrations the waveform of the oscillations changes from a period 1 (Figure 2a) to a period 2 (Figure 2b) to a period 4 (Figure 2c) to an aperiodic oscillation (Figure 2d). In the type of experiment shown here, dichlorophenol was added incrementally to produce the various types of periodic and aperiodic oscillations. However, by adding the necessary amount of dichlorophenol at the beginning of an experiment, we were able to get all the different types of periodic oscillations shown in Figure 2 in separate experiments. Figure 3 shows 2-dimensional projections of the reconstructed phase plots of the oxygen oscillations from Figure 2. Reconstructions were made according to the method of time delay^'^**^ as ([O,](t),[O,](t + T), [O&t 2T), ...) where Tis a fixed time delay, here chosen as 6 s. The time series and the phase plots clearly demonstrate a transition from periodic oscillations to aperiodic oscillations through a perioddoubling cascade.2’s22This type of transition has not previously been observed in a biochemical system, but it is a well-known route to chaos occurring in a wide variety of systems. Similar experiments to those shown in Figure 2 were performed by infusing a more concentrated (0.2 M) solution of NADH at a corresponding slower rate. Here we also observed period doubling bifurcations to chaos when increasing the concentration of dichlorophenol in the reaction mixture. To analyze the aperiodic oscillations further, we constructed Poincar6 sections and return maps of the 3-dimensional phase portraits as shown in Figure 4. The return map in Figure 4c appears to have a single maximum and bears a striking resemblance to the next-return map computed recently23for a detailed model of the PO reaction. The next-return map published in 1977 as the original evidence for chaos in the PO reaction9has a slightly different shape but was obtained under slightly different experimental conditions. This map was also shown to agree very well with a map computed from a simple model of the PO reaction.24 It is not yet known whether the detailed model proposed in ref 23, or a modification of this model, will be able to reproduce both types of experimentally observed maps or, more generally, whether arbitrary variations of the experimental conditions will yield chaotic states which agree quantitatively with the model. We also computed the maximum Lyapunov exponent, A,, of the data from Figure 2d. Using the method of Wolf et al.25we obtained a value of XI = 0.0175 bit/s. Using the nonlinear forecastingmethod proposed by we obtained XI = 0.0184 bit/s. The maximum Lyapunov exponent is a measure of a dynamical system’s dependence on initial condition^,^' and a positive exponent is characteristic of chaotic motion. In biochemical textbooks it is generally assumed that cell metabolism sits still in a stable steady state. However, biochemical pathways are highly nonlinear through the regulation of key enzymes that often are activated by their products or inhibited
+
by their substrates. These properties are also found in the PO r e a ~ t i o n ’ ~and - ~ *are responsible for its chaotic dynamics. Since our work has demonstrated that chaos is possible even in a single enzyme reaction and metabolic pathways consist of complex networks of such reactions, we conjecture that chaos in metabolism should be quite common. However, a verification of this conjecture requires long-term continuous measurements of metabolites in individual cells or in cultures of synchronous cells and such measurements have yet to be made.
Acknowledgment. We thank Lars Teil Nielsen, Tommy Narnberg, and Thomas Graf for technical assistance and The Danish Natural Science Research Council and Odense University Natural Science Faculty Council for financial support. Construction of phase plots and return maps and computation of Lyapunov exponents were done using Dynamical Software (Dynamical Systems Inc., Tucson, AZ).
References and Notes (1) Glass, L.; Mackey, M. C. From Clocks to Chaos, The Rhythms of Life; Princeton University Press: Princeton, NJ, 1988. (2) Olsen, L. F.; Degn, H. Q.Rev. Biophys. 1985, 18, 165. (3) Schaffer, W. M. IMA J. Math. Appl. Med. Biol. 1985, 2, 221. (4) Pool, R. Science 1989, 243, 604. ( 5 ) Scott, S.K. Chemical Chaos. International Series of Monographs on Chemistry, Clarendon Press: Oxford, 1991; Vol. 24. (6) Schmitz, R. A.; Graziani, K. R.; Hudson, J. L. J . Chem. Phys. 1977, 67, 3040. (7) Hudson, J. L.; Mankin, J. C. J . Chem. Phys. 1981, 74, 6171. (8) Roux, J. C.; Turner, J. S.;McCormick, W. D.; Swinney, H. L. In Non-linear Problems: Present and Future; Bishop, A. R., et al., Eds.; North-Holland: Amsterdam, 1982; p 409. (9) Olsen, L. F.; Degn, H. Nature 1977, 267, 177. (10) Markus, M.; Kuschmitz, D.; Hess, B. FEBS Lett. 1984, 172, 235. (11) Decroly, 0.;Goldbeter, A. Proc. Natl. Acad. Sci. U.S.A. 1982, 79, 6917. (12) MPder, M.; Amberg-Fischer, V. Plant Physiol. 1982, 70, 1128. (13) Nakamura, S.;Yokota, K.; Yamazaki, I. Nature 1969, 222, 794. (14) Degn, H. Nature 1968, 217, 1047. (15) Aguda, B. D.; Frisch, L.-L. H.; Olsen, L. F. J. Am. Chem. Soc. 1990, 11 2, 6652. (16) Steinmetz, C. G.; Larter, R. J . Chem. Phys. 1991, 94, 1388. (17) Olsen, L. F. Phys. Lett. 1983, 94A, 454. (18) Halliwell, B. Planta 1978, 140, 81. (19) Takens, F. Lect. Notes Math. 1981, 898, 366. (20) Packard, N.; Crutchfield, J. P.; Farmer, J. D.; Shaw, R. S.Phys. Rev. Lett. 1980, 45, 712. (21) May, R. M. Nature 1976, 261, 459. (22) Feigenbaum, M. J. J . Statisf. Phys. 1978, 19, 25. (23) Aguda, B. D.; Larter, R. J . Am. Chem. SOC.1991,113, 7913. (24) Olsen, L. F. In Stochastic Phenomena and Chaotic Behavior in Complex Systems; Schuster, P., Ed.; Springer-Verlag: Berlin, 1984; p 116. (25) Wolf, A.; Swift, J. B.; Swinney, H. L.; Vastano, J. A. Physica 1985, 160, 285. (26) Wales, D. J. Nature 1991, 350,485. (27) Ruelle, D. Ann. N.Y. Acad. Sci. 1979, 316, 408. (28) Degn, H.; Olsen, L. F.; Perram, J. W. Ann. N.Y. Acad. Sci. 1979,316, 623.
