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Chaos in Glycolysis
J. theor. Biol. (1997) 186, 303–306
Chaos in Glycolysis K N, P G S F H rsted Institute, University of Copenhagen, Department of Chemistry and CATS, H. C. O Universitetsparken 5, DK-2100 Copenhagen, Denmark (Received on 9 February 1996, Accepted in revised form on 6 December 1996)
Glycolysis occurs in almost every living cell as part of the energy metabolism. It forms a complex dynamical system, and might thus be capable of exhibiting complex phenomena. Simple oscillations have been observed frequently in suspensions of intact cells and in cell extracts, but only as transients. We have obtained sustained simple and complex oscillations in glycolysis of cell-free yeast extract in a flow-reactor. Sustained oscillations enable a powerful, proven method of dynamical system theory to unravel the kinetics and make it possible to observe chaos. Chaos was predicted from models long ago but has not previously been observed experimentally. We report the first experimental observation of unforced chaotic oscillations in glycolysis. 7 1997 Academic Press Limited
Organisms regulate their operating conditions (maintain homeostasis) through elaborate control systems involving feedback. Such controls may keep complex chemical reactions like glycolysis in a stationary state where all concentrations are independent of time. However, stationary states of complex systems may become unstable under certain conditions, and, in fact, oscillations have been observed in living cells as well as in cell-free extracts (Hess & Boiteux, 1971; Chance et al., 1973). Most oscillations observed so far have been simple (approximately periodic but transient), but in view of the complexity of glycolysis, much more complex behavior including chaos may be expected, as has been realized in recent years. Chaos is one of the most spectacular phenomena observed in homogeneous chemical reactions. Its existence is well-established for several open chemical systems (Field & Gyo¨rgyi, 1993; Marek & Schreiber, 1991; Scott, 1991), and it has been observed as transients in closed systems (Wang et al., 1994, 1995). Chaotic oscillations have also been observed in biochemical reactions and in biological systems. In fact, the first experimental result that was interpreted as showing chaotic oscillations was obtained with a peroxidase reaction (Olsen & Degn, 1977). Later 0022–5193/97/110303 + 04 $25.00/0/jt960366
observations of chaos and of a period doubling cascade leading to chaos in pseudo-open (Geest et al., 1992; Larter et al., 1993) and open (Hauck & Schneider, 1994) systems have confirmed the conclusion of the early experiment on the peroxidase reaction. In glycolysis, chaotic oscillations have been induced by time dependent forcing with a periodically varying flow of glucose (a possible substrate of glycolysis) (Markus et al., 1984, 1985a, b). This observation is important, but perhaps not very surprising (Scott, 1991). However, sustained complex oscillations or chaos have not previously been observed in unforced glycolysis, although abstract enzyme models suggest such possibilities (Decroly & Goldbeter, 1982; Goldbeter, 1993). To obtain sustained oscillations, we have run the reaction in a continuous-flow stirred tank-reactor (CSTR) with inflow of yeast extract as well as substrate and outflow of reaction mixture. (The classic procedure only flows substrate). A crucial step for CSTR experiments is therefore to produce and keep sufficient amounts of extract in stable form (Hess & Boiteux, 1968; Das & Busse, 1985; Ehlert-Oelkers, 1995). 7 1997 Academic Press Limited
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304
The reactor consists of a 1 cm by 1 cm quartz cuvette placed in a thermostatic jacket, by which the temperature of the reaction mixture is kept constant at 30.0°C. The volume of the mixture is kept constant at 1.7 ml by removing surplus liquid with a vacuum pump. The reaction mixture is stirred at a rate of 880 rpm. The thermostated reactor is placed in an HP8452A diode array spectrophotometer and the oscillations are observed continuously by monitoring the absorption of light by NADH (nicotinamide
adenine dinucleotide) at 340 nm using the absorption at 400 nm as a reference. The reactor is fed with constant flows using a peristaltic pump and by stepper motor driven piston burets from two stock solutions. One flow contains glucose in water, the other contains yeast extract dissolved in phosphate buffer (pH = 6.5) with added NADH, ATP (adenosine triphosphate) and Mg2 + , to give a mixed flow protein concentration of 4.9 mg ml−1. The solution is made from freeze-dried
(a)
Absorbance
1.0 0.9 0.8 0.7 0.6 0.5
(b)
Absorbance
1.0 0.9 0.8 0.7 0.6 0.5
(c)
Absorbance
0.9 0.8 0.7 0.6 0.5 0.4 1
2
3
4
5
6
7
8
9
t (hr) F. 1. Time series showing the absorbance at 340 nm relative to that at 400 nm. The absorption of light is mainly due to NADH. The three experiments differ in the total specific flow rate j and in the mixed flow concentrations of NADH and ATP (denoted [NADH]0 and [ATP]0). (a) Simple periodic relaxation oscillations at j = 2.5 × 10 − 4 s − 1 with [NADH]0 = 0.25 mM and [ATP]0 = 1.7 mM. (b) More complex, periodic oscillations at j = 2.0 × 10 − 4 s − 1. Otherwise the conditions are the same as in (a). (c) Chaotic oscillations at j = 3.3 × 10 − 4 s − 1 with [NADH]0 = 0.3 mM and [ATP]0 = 1.2 mM.
305
Absorbance
0.8
0.6
0.4 0
2
4
6
8
10
12
14
16
t (hr)
F. 2. Transient evolution from regular mixed-mode oscillations through period doubling to simple oscillations. Note that oscillatory patterns similar to the regular mixed-mode oscillations appear with varying number of small peaks per large peak in the aperiodic oscillations of Fig. 1(c). The experimental conditions are the same as in the experiment shown in Fig. 1(b) except for the specific flow rate which is j = 2.2 × 10 − 4 s − 1 during the time interval shown in the figure. It was adjusted to that value shortly before the start of the recording shown.
cell-free yeast extract, prepared from commercial baker’s yeast. (Saccharomyces cerevisiae by the method described by Hess & Boiteux (1968), Das & Busse (1985) and Ehlert-Oelkers (1995). The mixed flow concentration of glucose is 50 mM, and that of magnesium is 5 mM. The flow rates of the two flows are equal, but the total flow rate differs in the different experiments reported. These are described in Figs 1 and 2. The ability to sustain oscillations opens new ways of analysing the full glycolytic system using methods from the theory of dynamical systems. It enables glycolysis to be maintained in well defined dynamical states (such as stationary states, limit cycles, invariant tori, or chaotic attractors) and to characterize the system by the properties of such attractors and the bifurcations between them (normal forms, embedding of attractors, bifurcation diagrams, relaxation of perturbations). In particular, one may obtain quantitative data of direct kinetic significance for a complete reaction system by a set of special perturbation experiments at a Hopf bifurcation (quenching experiments) (So rensen & Hynne, 1989; Hynne et al., 1990). We intend to carry out such experiments, which may provide information about glycolysis which cannot be obtained by other methods (Hynne et al., 1993a, b). With the CSTR, we have obtained simple and complex periodic oscillations in glycolysis. Figure 1(a) shows simple sustained oscillations. Such oscillations have been run under constant conditions for 33 hr (until the supply of extract gave out). Figure 1(b)
shows a section of typical complex periodic oscillations lasting 20 hr. The conditions of Figs 1(a) and (b) are the same except for different flow rates. By also changing the mixed flow concentrations of NADH and ATP, we have found chaotic oscillations [Fig. 1(c)]. Note that, strictly speaking, chaos is defined only for sustained oscillations, which require an open system such as the one used here. For the aperiodic time series shown in Fig. 1(c), we have calculated the Lyapunov exponents from a three-dimensional Takens construction by the method of Sano & Sawada (1985) as 0.006 s − 1, −0.0005 s − 1, and −0.01 s − 1. The positive exponent confirms the visual impression of chaos. A positive Lyapunov exponent l expresses the exponential divergence of neighboring trajectories per unit time. It should be compared with a typical frequency of oscillation v. Thus l/v which expresses the exponential divergence per radian of oscillation is a dimensionless measure of chaos. It is close to unity in the experiment of Fig. 1(c) indicating manifest chaos. The existence of chaos is also supported by the appearance of complex but regular (non-chaotic) oscillations in other experiments. Figure 2 shows an evolution from regular mixed-mode oscillations through a long transient with period doubling to simple oscillations. Various irregular patterns similar to the regular mixed-mode oscillations of Fig. 2 appear in the chaotic experiment [Fig. 1(c)]. Transient mixed-mode oscillations similar to those of Fig. 2 have also been reported by Pye (Change et al., 1973). However, his transient is shorter because he used a closed system with trehalose as substrate.
. .
306
A living cell is an open system, and the constant inflow of cell extract and glucose in the CSTR of our experiments is a way of imitating realistic conditions of glycolysis in the cytosol. Such experiments may thus be used to determine conditions for various types of stationary or oscillatory behavior. In intact cells, species like NADH and ATP participate in many other biochemical reactions, however, so glycolysis is coupled to other biochemical subsystems. Therefore, the existence of complex oscillations and chaos in glycolysis is potentially very important to the range of behavior that is possible in living cells. It is interesting to speculate whether chaos in glycolysis may be linked to pathological conditions of living organisms. We are indebted to Professor H.-G. Busse and Dr R. Ehlert-Oelkers, Christian-Albrechts-Universita¨t zu Kiel, Germany, for introducing us to working with glycolysis and for teaching us their method of preparing the yeast extract. Dry extract can be obtained from the organization GfBB, Olshausenstrasse 40, D-24098 Kiel, Germany. We thank Merete Torpe for assistance with the experiments and the Danish Natural Science Research Council for support.
REFERENCES 1. C, B., P, E. G., G, A. K. & H, B. (1973). Biological and Biochemical Oscillators. New York: Academic Press. 2. D, J. & B, H.-G. (1985). Long term oscillation in glycolysis. J. Biochem. 97, 719–727. 3. D, O. & G, A. (1982). Biorhythmicity, chaos and other patterns of temporal self-organization in a multiply regulated biochemical system. Proc. Natl. Acad. Sci. U.S.A. 79, 6917–6921. 4. E-O, R. (1995). Simultane photometrische und massenpektrometrische Untersuchungen an glykolytischen Oszillationen zellfreier Hefeextrakte. Dissertation. Kiel: Christian-Albrechts-Universita¨t. 5. F, R. J. & G¨, L. (1993). eds. Chaos in Chemistry and Biochemistry. Singapore: World Scientific. 6. G, T., S, C. G., L, R. & O, L. F. (1992). Period-doubling bifurcations and chaos in an enzyme reaction. J. Phys. Chem. 96, 5678–5680.
7. G, A. (1993). From Periodic Behaviour to Chaos in Biochemical Systems In. Chaos in Chemistry and Biochemistry. (Field, R. J.; Gyo¨rgyi, L., eds). Singapore: World Scientific. 8. H, T. & S, F. W. (1994). Chaos in a Farey sequence through period doubling in the peroxidase-oxidase reaction. J. Phys. Chem. 98, 2072–2077. 9. H, B. & B, A. (1968). Mechanism of glycolytic oscillation in yeast, I. Hoppe-Seyler’s Z. Physiol. Chem. 349, 1567–1574. 10. H, B. & B, A. (1971). Oscillatory phenomena in biochemistry. Ann. Rev. Biochem. 40, 237–258. 11. H, F., S , P. G. & M , T. (1993a). Current and Eigenvector analyses of chemical reaction networks at Hopf bifurcations. J. Chem. Phys 98, 211–218. 12. H, F., S , P. G. & M , T. (1993b). Complete optimization of models of the Belousov-Zhabotinsky reaction at a Hopf bifurcation. J. Chem. Phys 98, 219–230. 13. H, F., S , P. G. & N, K. (1990). Quenching of chemical oscillations: general theory. J. Chem. Phys 92, 1747–1757. 14. L, R., O, L. F., S, C. G. & G, T. (1993). Chaos in Biochemical Systems: The Peroxidase Reaction as a Case Study. In (Field, R. J. & Gyo¨rgyi, L., eds). Chaos in Chemistry and Biochemistry. Singapore: World Scientific. 15. M, M. & S, I. (1991). Chaotic Behaviour of Deterministic Dissipative Systems. Cambridge: Cambridge University Press. 16. M, M., K, D. & H, B. (1984). Chaotic dynamics in yeast glycolysis under periodic substrate input flux. FEBS Lett. 172, 235–238. 17. M, M., K, D. & H, B. (1985a). Properties of strange attractors in yeast glycolysis. Biophys. Chem. 22, 95–105. 18. M, M., M¨, S. C. & H, B. (1985b). Observation of entrainment, quasiperiodicity and chaos in glycolyzing yeast extracts under periodic glucose input. Ber. Bunsenges. Phys. Chem. 89, 651–654. 19. O, L. F. & D, H. (1977). Chaos in an enzyme reaction. Nature 267, 177–178. 20. S, M. & S, Y. (1985). Measurement of the Lyapunov spectrum from a chaotic time series. Phys. Rev. Lett. 55, 1082–1085. 21. S, S. K. (1991). Chemical Chaos. Oxford: Clarendon Press. 22. S , P. G. & H, F. (1989). Amplitudes and phases of small-amplitude Belousov-Zhabotinskii oscillations derived from quenching experiments. J. Phys. Chem. 93, 5467–5474. 23. W, J., S , P. G. & H, F. (1994). Transient period doublings, torus oscillations, and chaos in a closed chemical system. J. Phys. Chem. 98, 725–727. 24. W, J., S , P. G. & H, F. (1995). Transient complex oscillations in a closed Belousov-Zhabotinsky reaction: experimental and compulational studies. Z. Phys. Chem. 192, 63–76.
Chaos in Glycolysis K N, P G S F H rsted Institute, University of Copenhagen, Department of Chemistry and CATS, H. C. O Universitetsparken 5, DK-2100 Copenhagen, Denmark (Received on 9 February 1996, Accepted in revised form on 6 December 1996)
Glycolysis occurs in almost every living cell as part of the energy metabolism. It forms a complex dynamical system, and might thus be capable of exhibiting complex phenomena. Simple oscillations have been observed frequently in suspensions of intact cells and in cell extracts, but only as transients. We have obtained sustained simple and complex oscillations in glycolysis of cell-free yeast extract in a flow-reactor. Sustained oscillations enable a powerful, proven method of dynamical system theory to unravel the kinetics and make it possible to observe chaos. Chaos was predicted from models long ago but has not previously been observed experimentally. We report the first experimental observation of unforced chaotic oscillations in glycolysis. 7 1997 Academic Press Limited
Organisms regulate their operating conditions (maintain homeostasis) through elaborate control systems involving feedback. Such controls may keep complex chemical reactions like glycolysis in a stationary state where all concentrations are independent of time. However, stationary states of complex systems may become unstable under certain conditions, and, in fact, oscillations have been observed in living cells as well as in cell-free extracts (Hess & Boiteux, 1971; Chance et al., 1973). Most oscillations observed so far have been simple (approximately periodic but transient), but in view of the complexity of glycolysis, much more complex behavior including chaos may be expected, as has been realized in recent years. Chaos is one of the most spectacular phenomena observed in homogeneous chemical reactions. Its existence is well-established for several open chemical systems (Field & Gyo¨rgyi, 1993; Marek & Schreiber, 1991; Scott, 1991), and it has been observed as transients in closed systems (Wang et al., 1994, 1995). Chaotic oscillations have also been observed in biochemical reactions and in biological systems. In fact, the first experimental result that was interpreted as showing chaotic oscillations was obtained with a peroxidase reaction (Olsen & Degn, 1977). Later 0022–5193/97/110303 + 04 $25.00/0/jt960366
observations of chaos and of a period doubling cascade leading to chaos in pseudo-open (Geest et al., 1992; Larter et al., 1993) and open (Hauck & Schneider, 1994) systems have confirmed the conclusion of the early experiment on the peroxidase reaction. In glycolysis, chaotic oscillations have been induced by time dependent forcing with a periodically varying flow of glucose (a possible substrate of glycolysis) (Markus et al., 1984, 1985a, b). This observation is important, but perhaps not very surprising (Scott, 1991). However, sustained complex oscillations or chaos have not previously been observed in unforced glycolysis, although abstract enzyme models suggest such possibilities (Decroly & Goldbeter, 1982; Goldbeter, 1993). To obtain sustained oscillations, we have run the reaction in a continuous-flow stirred tank-reactor (CSTR) with inflow of yeast extract as well as substrate and outflow of reaction mixture. (The classic procedure only flows substrate). A crucial step for CSTR experiments is therefore to produce and keep sufficient amounts of extract in stable form (Hess & Boiteux, 1968; Das & Busse, 1985; Ehlert-Oelkers, 1995). 7 1997 Academic Press Limited
. .
304
The reactor consists of a 1 cm by 1 cm quartz cuvette placed in a thermostatic jacket, by which the temperature of the reaction mixture is kept constant at 30.0°C. The volume of the mixture is kept constant at 1.7 ml by removing surplus liquid with a vacuum pump. The reaction mixture is stirred at a rate of 880 rpm. The thermostated reactor is placed in an HP8452A diode array spectrophotometer and the oscillations are observed continuously by monitoring the absorption of light by NADH (nicotinamide
adenine dinucleotide) at 340 nm using the absorption at 400 nm as a reference. The reactor is fed with constant flows using a peristaltic pump and by stepper motor driven piston burets from two stock solutions. One flow contains glucose in water, the other contains yeast extract dissolved in phosphate buffer (pH = 6.5) with added NADH, ATP (adenosine triphosphate) and Mg2 + , to give a mixed flow protein concentration of 4.9 mg ml−1. The solution is made from freeze-dried
(a)
Absorbance
1.0 0.9 0.8 0.7 0.6 0.5
(b)
Absorbance
1.0 0.9 0.8 0.7 0.6 0.5
(c)
Absorbance
0.9 0.8 0.7 0.6 0.5 0.4 1
2
3
4
5
6
7
8
9
t (hr) F. 1. Time series showing the absorbance at 340 nm relative to that at 400 nm. The absorption of light is mainly due to NADH. The three experiments differ in the total specific flow rate j and in the mixed flow concentrations of NADH and ATP (denoted [NADH]0 and [ATP]0). (a) Simple periodic relaxation oscillations at j = 2.5 × 10 − 4 s − 1 with [NADH]0 = 0.25 mM and [ATP]0 = 1.7 mM. (b) More complex, periodic oscillations at j = 2.0 × 10 − 4 s − 1. Otherwise the conditions are the same as in (a). (c) Chaotic oscillations at j = 3.3 × 10 − 4 s − 1 with [NADH]0 = 0.3 mM and [ATP]0 = 1.2 mM.
305
Absorbance
0.8
0.6
0.4 0
2
4
6
8
10
12
14
16
t (hr)
F. 2. Transient evolution from regular mixed-mode oscillations through period doubling to simple oscillations. Note that oscillatory patterns similar to the regular mixed-mode oscillations appear with varying number of small peaks per large peak in the aperiodic oscillations of Fig. 1(c). The experimental conditions are the same as in the experiment shown in Fig. 1(b) except for the specific flow rate which is j = 2.2 × 10 − 4 s − 1 during the time interval shown in the figure. It was adjusted to that value shortly before the start of the recording shown.
cell-free yeast extract, prepared from commercial baker’s yeast. (Saccharomyces cerevisiae by the method described by Hess & Boiteux (1968), Das & Busse (1985) and Ehlert-Oelkers (1995). The mixed flow concentration of glucose is 50 mM, and that of magnesium is 5 mM. The flow rates of the two flows are equal, but the total flow rate differs in the different experiments reported. These are described in Figs 1 and 2. The ability to sustain oscillations opens new ways of analysing the full glycolytic system using methods from the theory of dynamical systems. It enables glycolysis to be maintained in well defined dynamical states (such as stationary states, limit cycles, invariant tori, or chaotic attractors) and to characterize the system by the properties of such attractors and the bifurcations between them (normal forms, embedding of attractors, bifurcation diagrams, relaxation of perturbations). In particular, one may obtain quantitative data of direct kinetic significance for a complete reaction system by a set of special perturbation experiments at a Hopf bifurcation (quenching experiments) (So rensen & Hynne, 1989; Hynne et al., 1990). We intend to carry out such experiments, which may provide information about glycolysis which cannot be obtained by other methods (Hynne et al., 1993a, b). With the CSTR, we have obtained simple and complex periodic oscillations in glycolysis. Figure 1(a) shows simple sustained oscillations. Such oscillations have been run under constant conditions for 33 hr (until the supply of extract gave out). Figure 1(b)
shows a section of typical complex periodic oscillations lasting 20 hr. The conditions of Figs 1(a) and (b) are the same except for different flow rates. By also changing the mixed flow concentrations of NADH and ATP, we have found chaotic oscillations [Fig. 1(c)]. Note that, strictly speaking, chaos is defined only for sustained oscillations, which require an open system such as the one used here. For the aperiodic time series shown in Fig. 1(c), we have calculated the Lyapunov exponents from a three-dimensional Takens construction by the method of Sano & Sawada (1985) as 0.006 s − 1, −0.0005 s − 1, and −0.01 s − 1. The positive exponent confirms the visual impression of chaos. A positive Lyapunov exponent l expresses the exponential divergence of neighboring trajectories per unit time. It should be compared with a typical frequency of oscillation v. Thus l/v which expresses the exponential divergence per radian of oscillation is a dimensionless measure of chaos. It is close to unity in the experiment of Fig. 1(c) indicating manifest chaos. The existence of chaos is also supported by the appearance of complex but regular (non-chaotic) oscillations in other experiments. Figure 2 shows an evolution from regular mixed-mode oscillations through a long transient with period doubling to simple oscillations. Various irregular patterns similar to the regular mixed-mode oscillations of Fig. 2 appear in the chaotic experiment [Fig. 1(c)]. Transient mixed-mode oscillations similar to those of Fig. 2 have also been reported by Pye (Change et al., 1973). However, his transient is shorter because he used a closed system with trehalose as substrate.
. .
306
A living cell is an open system, and the constant inflow of cell extract and glucose in the CSTR of our experiments is a way of imitating realistic conditions of glycolysis in the cytosol. Such experiments may thus be used to determine conditions for various types of stationary or oscillatory behavior. In intact cells, species like NADH and ATP participate in many other biochemical reactions, however, so glycolysis is coupled to other biochemical subsystems. Therefore, the existence of complex oscillations and chaos in glycolysis is potentially very important to the range of behavior that is possible in living cells. It is interesting to speculate whether chaos in glycolysis may be linked to pathological conditions of living organisms. We are indebted to Professor H.-G. Busse and Dr R. Ehlert-Oelkers, Christian-Albrechts-Universita¨t zu Kiel, Germany, for introducing us to working with glycolysis and for teaching us their method of preparing the yeast extract. Dry extract can be obtained from the organization GfBB, Olshausenstrasse 40, D-24098 Kiel, Germany. We thank Merete Torpe for assistance with the experiments and the Danish Natural Science Research Council for support.
REFERENCES 1. C, B., P, E. G., G, A. K. & H, B. (1973). Biological and Biochemical Oscillators. New York: Academic Press. 2. D, J. & B, H.-G. (1985). Long term oscillation in glycolysis. J. Biochem. 97, 719–727. 3. D, O. & G, A. (1982). Biorhythmicity, chaos and other patterns of temporal self-organization in a multiply regulated biochemical system. Proc. Natl. Acad. Sci. U.S.A. 79, 6917–6921. 4. E-O, R. (1995). Simultane photometrische und massenpektrometrische Untersuchungen an glykolytischen Oszillationen zellfreier Hefeextrakte. Dissertation. Kiel: Christian-Albrechts-Universita¨t. 5. F, R. J. & G¨, L. (1993). eds. Chaos in Chemistry and Biochemistry. Singapore: World Scientific. 6. G, T., S, C. G., L, R. & O, L. F. (1992). Period-doubling bifurcations and chaos in an enzyme reaction. J. Phys. Chem. 96, 5678–5680.
7. G, A. (1993). From Periodic Behaviour to Chaos in Biochemical Systems In. Chaos in Chemistry and Biochemistry. (Field, R. J.; Gyo¨rgyi, L., eds). Singapore: World Scientific. 8. H, T. & S, F. W. (1994). Chaos in a Farey sequence through period doubling in the peroxidase-oxidase reaction. J. Phys. Chem. 98, 2072–2077. 9. H, B. & B, A. (1968). Mechanism of glycolytic oscillation in yeast, I. Hoppe-Seyler’s Z. Physiol. Chem. 349, 1567–1574. 10. H, B. & B, A. (1971). Oscillatory phenomena in biochemistry. Ann. Rev. Biochem. 40, 237–258. 11. H, F., S , P. G. & M , T. (1993a). Current and Eigenvector analyses of chemical reaction networks at Hopf bifurcations. J. Chem. Phys 98, 211–218. 12. H, F., S , P. G. & M , T. (1993b). Complete optimization of models of the Belousov-Zhabotinsky reaction at a Hopf bifurcation. J. Chem. Phys 98, 219–230. 13. H, F., S , P. G. & N, K. (1990). Quenching of chemical oscillations: general theory. J. Chem. Phys 92, 1747–1757. 14. L, R., O, L. F., S, C. G. & G, T. (1993). Chaos in Biochemical Systems: The Peroxidase Reaction as a Case Study. In (Field, R. J. & Gyo¨rgyi, L., eds). Chaos in Chemistry and Biochemistry. Singapore: World Scientific. 15. M, M. & S, I. (1991). Chaotic Behaviour of Deterministic Dissipative Systems. Cambridge: Cambridge University Press. 16. M, M., K, D. & H, B. (1984). Chaotic dynamics in yeast glycolysis under periodic substrate input flux. FEBS Lett. 172, 235–238. 17. M, M., K, D. & H, B. (1985a). Properties of strange attractors in yeast glycolysis. Biophys. Chem. 22, 95–105. 18. M, M., M¨, S. C. & H, B. (1985b). Observation of entrainment, quasiperiodicity and chaos in glycolyzing yeast extracts under periodic glucose input. Ber. Bunsenges. Phys. Chem. 89, 651–654. 19. O, L. F. & D, H. (1977). Chaos in an enzyme reaction. Nature 267, 177–178. 20. S, M. & S, Y. (1985). Measurement of the Lyapunov spectrum from a chaotic time series. Phys. Rev. Lett. 55, 1082–1085. 21. S, S. K. (1991). Chemical Chaos. Oxford: Clarendon Press. 22. S , P. G. & H, F. (1989). Amplitudes and phases of small-amplitude Belousov-Zhabotinskii oscillations derived from quenching experiments. J. Phys. Chem. 93, 5467–5474. 23. W, J., S , P. G. & H, F. (1994). Transient period doublings, torus oscillations, and chaos in a closed chemical system. J. Phys. Chem. 98, 725–727. 24. W, J., S , P. G. & H, F. (1995). Transient complex oscillations in a closed Belousov-Zhabotinsky reaction: experimental and compulational studies. Z. Phys. Chem. 192, 63–76.