J. Phys. Chem. 1994,98. 2072-2077
2072
Chaos in a Farey Sequence through Period Doubling in the Peroxidase-Oxidase Reaction T. Hauck and F. W.Schneider' Institute of Physical Chemistry, University of Wurzburg, Marcusstrasse 9-11, 0-8700Wiirrburg, Germany Received: September 2. 1993; In Final Form: Nouember 4, 199P
The dynamic properties of the peroxidaseoxidase reaction are investigated in a continuous flow stirred tank reactor (CSTR) in the free running mode. We report the first experimental period-doubling route to chaos i n a Fareysequencebyvaryingthe02flow. Froma l'startingpattern,perioddoubling,chaos,andsubsequently 12, 13, and 1 4 oscillations are observed. The chaotic state is clearly identified by the one-dimensional map and the fractal natureof the Renyi dimensions. It also shows an increased dissipation, compared with the dissipation of the periodic time series. A slight change in pH leads to the observation of another Farey sequence containing a concatenated state but no chaos. Model calculations are compared with the experimental results.
9 input
Introduction ThePOreactionisoneofa few knownexamp1esl"ofa nonlinear biochemicalreaction. Usinga fill reactor witha continuousinflow of oxygen and NADH, a number of nonlinear phenomena including bistability of two steady states,' bistability of a steady state and an oscillatory state,l periodic oscillations,6 and quasiperiodicity' have becn observed. In 1977, Olsen and found chaos in the peroxidase-oxidase (PO) reaction in a fill reactor. In recent work,lO." these chaotic oscillations were reproduced under similar conditionsand a period-doublingroute involving PI, PZ, P4, chaos, and P3 was found. Recently we used a CSTR in the PO reaction and found quasiperiodicityand a Farey sequence.12 Periodic perturbations oftheP01eactionl~intheCSTRlead toa richvarietyofresponse bchavior including the phenomenon of chaotic resonancel4in a periodically driven focus.' There are several examples of the existence of deterministic chaos in chemical reactions.ls The Belousov-Zhabotinskii (BZ) reaction is one of the best-known systems exhibiting chemical chaos.lG'8 In addition to the well-known period-doubling route,'9,2Ochemical chaos may be obtained from the wrinkling of a torus'l or from homoclinic orbits.2' Other routes to chaos are known from calculations but not yet found in chemical systems. For example, a 'period adding" route has been accidentallyfound in a coupled system of two reduced NFT models.z3 In circle maps, chaos between Farey-ordered states has been obtained through period d ~ u h l i n g ' ' ~or~ through ~ a quasiperiodic route.26 Chaos via period doubling has been observed between Fareyordered states in the four-variable Montanator model" and in the DOP model.2s We present experiments describing the route to chemical chaos through period doubling in a Farey sequence. The chaotic state and the neighboring periodic states of the peroxidase-oxidase (PO) reaction are observed in a CSTR (continuous flow stirred tank reactor). The chaotic time series isclearlyidentified by itsnext amplitudemap, itsstrangeattractor, the one-dimensional map, and the spectrum of generalized dimensions Horseradish peroxidase (HRP) catalyzes the aerobicoxidation of reduced nicotinamide adenine dinucleotide (NADH) in the peroxidaseoxidase (PO) reaction according to
-
[HRPI
ZNADH
+ 0,
2NAD'
+ 2H20
(1)
where NAD+, also known as coenzyme I, is 8-nicotinamide adenine dinucleotide. The use of a CSTR has the advantage of obtaining real steady states, whereas dynamicstates in a fill reactor *Abstract published in Advance ACS Absrroetr. February I . 1994.
0022-3654/94/2098-2072$04.50/0
I
Figure 1. CSTR with two inlets for the reactant solutions, gas inlet, &liquid outlet, and 0 2 Clark elcctrcde. Quartz glass walls are in the paper plane: Plexiglas walls and UV/vis light beam are perpendicular to the plane.
-------
TABLE 1: AL Model with Rate Constantsn (1.1) Perf3+H102 COI kl = 106 M-l r1 (1.2) CoI + NADH CoII + NAD' kt = I06 M-l SKI (1.3) CoII +NADH Per+3+ NAD' k3 = 106 M-' 6-I (1.4) CoIll+ NAD' Per+3+ 01'(1.6) NAD'+Oi (1.7) NADH + O,'+
(1.5)
(1.8) (1.9)
2NAD'
(1.10)
0 2
H+
[o2i0
(1.11) [NADH10 (1.12) Per+'+NAD' (1.13)
Per+'+Ol
TABLE 2
-----
+
k, = 106 M-'s-l ks = 6 X IO'M-l c1 NAD+ + 0 2 - ks = 106 M-' s-I NAD'+ HlOz k7 = 106 M-'sd (NADh ks = IO6 M-' s-I CoI NAD+ COIII
0 2
kp = bifurcation
parameter [OllO
klo = 0.1 s-' NADH kll = 1V' M-'s-' Per+' + NADf k n = 106 M-ls-' COIII k13= IO6 M-' d
Olsen Model with Rate con st ants^
(2.1) (2.2) (2.3) (2.4) (2.5) (2.6) (2.7)
B+X 2 x A+ B+ Y X Y [XI0 [Ah
(2.8)
[Blo
2X 2Y 3X P
Q X A
B
k, =0.347 kz=250 k3 = 0.035 k=20 ks = 5.35 k = 0.00001 k7 = bifurcation parameter: k7. = 0.1 k, = 0.825
areapproximationstosteady states. Thedisadvantageofa CSTR isthe larger expenditureofenzymeandsubstrates. Therecycling of NAD* produced in reaction 1 may be achieved by another 0 1994 American Chemical Society
The Journal of Physical Chemistry, Vol. 98. No. 8. 1994 2073
Chaos in a Farey Sequence
a
P1
2 1
11
P1
10
i3
i4
15
13 14 0 content (TO)
12
b
12
llf
11
l2
(11)2 chaos
12
11
16
13
13
14
15
14
O2 content (70)
P1
C
.
i.io
P2
P4 PS
1 .io
chaos
... .'..
P1"
ss
--
.
1.30
1.40
k9
.
1 .io 1.00 m =chaos
I
0.xo k7
0.60
T
0.40 HB
Figure 2. Bifurcation diagrams: (a) Experimental bifurcation diagram at pH = 5.40 as a function of the 0 2 content. The I ' state wexists with a state. (b) Experimental bifurcation diagram at pH = 5.80 as a function of the 0 2 content. (c) Bifurcation diagram calculated with the AL model as a function of the O2 inflow rate k9. (d) Bifurcation diagram calculated with the Olsen model as a function of the inflow rate of A. k i . HB = supercritical Hopf bifurcation point. 2'
enzyme reaction, which ultimately permits the use of lower flow rates in the CSTR
-
En"%
+
,I..,
NAD' G6P 6PGL + NADH + H* (2) where DH is glucose-6-phosphatedehydrogena~e.~~ We calculate the period-doubling transition from period 1 to chaos using the Aguda and Larter (AL) model3a of the PO reaction. Chaotic regions may be reached via period doubling between Farey-ordered states in the Olsen model," as observed in the present experiments.
Experimental Section Materials. Peroxidase from horseradish (HRP, RZ 3.05, activity 270 units per mg solid) was purchased from Sigma as a
salt-free powder. Glucose-6-phosphatedehydrogenase(DH) from leuconostoc mesenteroides (540 units &NAD+ per mg protein) was purchased from Sigma as a biuret suspension. p-NAD+ from yeast (99%)and D-glucose-6-phosphatedisodium salt (G6P. 99%) were purchased from Sigma. 2,4-Dichlorophenol (DCP) and methylene blue (MB) were purchased from Aldrich.
Reactor. A Plexiglas reactor is used with two quartz glass side wallsandanoxygenelectrodefittedononePlexiglassside(Figure I). The reactor has a square base of 18-mm side length and 18-mm height with an effective liquid volume of 4.3 mL. The gas volume above the liquid volume is 2.0 mL. We used an unsymmetric magnetic stirrer driven by a small motor with a stirring rate of 750 rpm. This was found to be a compromise
2074
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The Journal of Physical Chemistry, Vol. 98, No. 8,1994
0
1000
oxygen-transfer rate was determined as 1.11 X 10-3 s-l. The 02 content in the Oz/N2 mixture was variable as the bifurcation parameter. Temperature. The syringes containing the input chemicals, the tubing, and the reactor were thermostated at 25.0 OC for all experiments. Detection. The time series of NADH absorption (360-nm detection wavelength) and compound I11 absorption (418 nm) were obtained with a diode array spectrophotometer (Hewlett Packard 8452A). The time seriesoftheoxygen electrodepotential were measured with an oxygen selective Clark electrode and a microprocessor oximeter (WTW Oxi96). Thus the NADH concentration, the compound I11 concentration, and the oxygen concentration are detected simultaneously with a sampling rate of 2 Hz as shown in ref 12. All figures show the NADH absorption time series only. For the dissipation calculations the 02 potential and the NADH absorption are used. Calculation of the Dissipation. We follow the description in ref 14 which is based on the equations of Ross and co-~orkers:~J3
3000
Zoo0 time [SI
Figure 3. Period 1 oscillations: time series of NADH absorption (360 nm).
between efficient mixing of the liquid and a constant oxygen transfer through the liquid surface. Preparation of InflowComponents. All reactants were dissolved in aqueous 0.1 m phosphate buffer (pH 5.8) containing 1pmol/L MB and 50 pmol/L DCP. All solutions were prepared under saturated argon before each measurement. We used two gastight syringes (Hamilton) filled with NAD+/G6P solution and DH/HRP solution with the following reactor concentrations: syringe 1, [HRP] = 50 units/mL and [DH] = 2.5 units/mL; syringe 2, [NAD+] = 1.5 mmol/L and [G6P] = 25 mmol/L. Continuous Flow Conditions. A self-designed high-precision syringe pump driven by a computer-linked stepping motor was employed to control the flow rate of the reactants into the CSTR. The stepping frequency was 5 Hz. The flow rate of the inflow of the two solutions was constant at kf (liquid) = 5.78 X l e 3 min-1 (residence time T = 172 min). Two computer-linked mass flow controllers (MKStype 1259C) for 0 2 and N2, respectively, were employed to mix and regulate any chosen gas flow rate and 02/Nz ratio. The average flow rate of the gaseous mixture O,/ N2 was constant at kdgas) = 1.1 mL/s in all experiments. The
D = (force)(flux) = AGJm AG = AGO
(3)
+ RTln(NAD+/(NADH[02]'/2)] (4)
in which AG is the Gibbs free energy, J ~ iso the rate of the PO reaction (eq l), JDH= 1.0 X l e 5 mol/s from our own stopped flow measurements, and AGO = -106.1 kJ/m01.'~
Model Calculations The experimental results are compared with the calculations of two models, namely the Aguda-Larter (AL) model30 and the C
a
Y
0
1000
time [SI
Zoo0
3000 I
L
X
d
b 5e-05
1 1
'
1
3e-05 &-05
le-05 0
0
0.05
0. I 0.15 fkququency [ d s l
0.2
Figure 4. 1' Pattern: (a) time series of NADH absorption (360 nm); (b) Fourier spectrum; (c) attractor of the first three SVD dimensions (A', Y, 2 axes) with PoincarC sectional plane; (d) one-dimensional map.
Chaos in a Farey Sequence
The Journal of Physical Chemistry, Vol. 98, No. 8, 1994 2075
a
C
7 Y
0
1000
2000 time
3000
[SI
X
b
d
4e-05
.
3~-05 2a-05
.
le-05 -1.5
n
0
0.05
0.15
0.1
fiqucncy
[WSI
0.2
-1.5
-0.5
0.5
X(n)
Figure 5. Period-doubled (1')2 pattern: (a) time series of NADH absorption (360 nm); (b) Fourier spectrum; (c) attractor of the first three SVD dimensions (A', Y,2 axes) with Poincar6 sectional plane; (d) one-dimensional map. Olsen model.3' All integrations were done using the Gear algorithm.33~3~ The AL model (Table 1) represents a mechanistic model based on some elementary steps of the PO reaction; it consists of 10 variables and 13 steps. The experimental results are simulated by varying kg which represents the 0 2 inflow. The Olsen model (Table 2) is a dynamical four-variable model that consists of eight steps. The abstract variables are A = 0 2 and B = NADH, and X and Y correspond to radical intermediates NAD* and compound 111, respectively. The inflow rate of A, k7, is varied. Both the AL model and the Olsen model represent approximations to the CSTR boundary conditions. They are used to provide comparisons with the calculations of other workers.
Results and Discussion In our experimentswe exclusively find P1 relaxationoscillations at pH = 4.80 for all 0 2 contents between two supercritical Hopf bifurcations. At pH = 5.40, a Farey sequence (Figure 2a) is observed between large-amplitude relaxation oscillations (period 1) and small-amplitude sinusoidal oscillations (period 1*) in the neighborhood of a supercritical Hopf bifurcation. This Farey sequencecontainstheconcatenated state 1' lzwhose firing number is F(1'12) = 2/5 (F = L/(L+ S) = p/q, where L and S are the number of small and large peaks in one oscillation period, respectively). The firing number F(1'l2) = 2/5 represents the Farey sum of the firing numbers F(1') = l / z and F(12) = I/3, where the Farey sum is defined aspl/ql,$ p2/q2 = @I + p2)/(q1 42). Further concatenated states could not be resolved experimentally. At pH = 5.80, a Farey sequence (Figure 2b) is found where period doubling and chaos are located between a 1' and a l 2 mixed-mode pattern. Individual states of the Farey sequence at pH = 5.80 are shown in Figures 3-8. The time series of NADH absorption (360 nm) is shown for 10.8% 0 2 content for period 1 relaxation oscillations (Figure 3).
+
The NADH time series of the 11 pattern for 11.7% 02 (Figure 4a) and the period-doubled (11)2pattern for 12.3% 0 2 (Figure 5a) are presented. Period doubling is characterized by the corresponding Fourier spectra (Figures 4 and 5b), attractors (Figures 4 and 5c), and one-dimensional maps (Figures 4 and 5d), respectively. A chaotic time series of NADH absorption is detected for a 02 content of 13.1% (Figure 6a). We characterize the deterministic chaos in the PO reaction in a CSTR by its Fourier spectrum (Figure 6b), its next amplitude map (Figure 6c), and its attractor reconstructed by the SVD method35 (Figure 6d). The analysis of the attractor leads to the identification of deterministic chaos by an extremum in the one-dimensionalmap (Figure 6e) and the D,spectrum of generalized Renyi dimensions36 (Figure 6f) calculated with (nearest neighbor analysis). From Figure 6f, the Hausdorff dimension is DH = D,-o = 2.90, the information dimension DI = Dq51= 2.65, and the correlation dimension DC = Dq12 = 2.40. They follow the conjecture of Kaplan and Y ~ r k e that ' ~ all dimensions must be fractal and greater than 2.0 for chaos. The analysis of the time series of compound I11 absorption and of the 0 2 potential leads to similar conclusions (not shown). The SVD attractor reconstruction method is used since it is able to filter nonamplified noise by an orthonormalization algorithm.'* The strange attractor (Figure 6d) is plotted in the first three normalized dimensions (X, Y, Z axes). The PoincarC section (not shown) may be obtained from the intersectionsof the trajectories in the attractor with an arbitrary plane (normalized X , Yaxes). The one-dimensionalmap (Figure 6e) is constructed by plotting successive trajectory intersections of the PoincarC section (X,+I versus X,,). Futhermore, Farey-ordered states 12 for 13.6% 02 (Figure 7), 13 for 14.2%0 2 (Figure 8), and 14 for 14.7% 0 2 (not shown) are detected. The bifurcation diagram of all experimentallyobserved
2076
Hauck and Schneider
The Journal of Physical Chemistry, Vol. 98, No. 8, 1994 a
d I
I
I
I
1.6 3
1.4 $2
9 1.2 Y 1.o
I
b
e 1.5
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/.so -
4
-
2
0
2
4
6
8
4
Figure 6. Chaotic state: (a) time series of NADH absorption (360 nm); (b) Fourier spectrum; (c) next amplitude map; (d) attractor of the first three SVD dimensions (X,Y,Z axes) with Poincar6 sectional plane; (e) one-dimensional map; ( f ) Dq spectrum of generalized Renyi dimensions.
1.6
1.4
1.2 1.o
Figure 7. l 2 Pattern: time series of NADH absorption (360 nm).
Figure 8. l 3 Pattern: time series of NADH absorption (360 nm).
states at pH = 5.80 is given in Figure 2b. It is compared with the calculated bifurcation diagrams of the Aguda-Larter model (Figure 2c) and of the Olsen model (Figure 2d). The Aguda-Larter model exhibits a period-doubling route to chaos starting from period 1 where the experimentally observed Farey sequences are not found with the rate constants of Table 1.30 The simple Olsen model shows a similar scenario as the experiments. A Farey sequence including chaos through period doubling (Figure 2d) is obtained for changing values of k7 where
kl = 0.347 and all other rate constants are given in Table 2. If kl is increased (= 0.370),a periodic Farey sequence is obtained. A comparison of the dissipation D of the PO reaction is shown for period 1, 11, (11)*,and chaos (Figure 9). Since the values of AG and of J ~ are o similar for periodic and chaotic motions (not shown), the increased dissipation D of the chaotic regime mainly originates from a loss of phase between AG and Jpo. The phase difference was reported by Lazar and Rosd3to be the main reason for changes in dissipation of periodically perturbed oscillations.
Chaos in a Farey Sequence 1.m
1.01
The Journal of Physical Chemistry, Vol. 98, No. 8, 1994 2077
, I
rI - Il
I-
1
D12
1'
I
l
C
Figure 9. Reduced dissipation D/&l of the chaotic states and the neighboring periodic states.
Conclusion A chaotic region has been found within a Farey sequence (Figure 2b) in the PO reaction. The novel feature of this work is that the transition from a 1' pattern to deterministic chaos occurs via period doubling 1 (11)z. The sequence 11-4 is a Farey-ordered sequence whose concatenated states could not be resolved experimentally. However, the concatenated state 1112 could be experimentallydetected in a neighboring Farey sequence (Figure 2a). The chaotic state is identified by extrema in the next amplitude map and in the one-dimensional map, respectively, and by the generalizeddimensions &,DI, and DCshowing fractal values above 2.0. Deterministic chaos shows a higher dissipation than the neighboring periodic states. The experimental route to chaos through period doubling between Farey-ordered states is compared with model calculations. The Aguda-Larter model has not been found to reproduce any Farey sequences. The abstract Olsen model exhibits qualitative agreement with the experiments, although the model is chemically unrealistic. Further conclusions on the nonlinear mechanism of the PO reaction may be drawn from calculations39 and experiments concerned with the control of chaos.
-
Acknowledgment. This work was supported by the Stiftung Volkswagenwerk and the Fonds der Chemischen Industrie. We thank T. M. Kruel for the implementation of computer programs for the SVD attractor reconstruction method and the NNA dimension analysis method. References and Notes (1) Rapp, P. E. J. Exp. Biol. 1979, 81, 281. (2) Yamazaki, I.; Yokota, K.; Nakajama, R. Biochem. Biophys. Res. Commun. 1965, 21, 582. (3) Chance, B.; Hess, B.; Betz, A. Biochem. Biophys. Res. Commun. 1964, 16, 182.
(4) Degn, H. Nature 1968, 217, 1047. (5) Aguda, B. D.; Hofmann-Frisch, L.-L.; Olsen, L. F. J . Am. Chem. SOC.1990, 112,6652. (6) Nakamura, S.;Yokota, K.; Yamazaki, I. Nature 1969, 222, 794. (7) Samples, M.S.; Hung,Y.-F.;Ross, J.J.Phys. Chem. 1992,96,7338. ( 8 ) Olsen, L. F.; Degn, H. Nature 1977, 217, 1047. (9) Olsen, L. F.; Degn, H. Biochim. Biophys. Acta 1978, 527, 212. (10) Geest, T.; Steinmetz, C. G.; Larter, R.; Olsen, L. F. J . Phys. Chem. 1992, 96, 5678. (1 1) Larter, R.; Olsen, L. F.; Steinmetz, C. G.; Geest, T. In Chaos in Chemistry and Biochemistry; Field, R. J., Gybgyi, L., Eds.; World Scientific: New Jersey, 1993; p 175. (12) Hauck, T.; Schneider, F. W. J. Phys. Chem. 1993.97, 391. (13) Lazar, J. G.; Ross, J. J. Chem. Phys. 1990, 92, 3579. (14) Fbster, A.; Hauck, T.; Schneider, F. W. J . Phys. Chem., in press. (a) Scott, (15) Foradiscussionofreactionsexhibitingchemicalchaossee: S. K. Chemical Chaos; International Series of Monographs on Chemistry; Clarendon Press: Oxford, 1991;Vol. 24. (b) Marek, M.; Schreiber, 1. Chaotic Behavior of Deterministic Dissipative Systems; Academia: Praha, 1991. (16) Turner, J.; Roux, J . C.; McCormick, W. D.; Swinney, H. L. Phys. Lett. 1981, A85, 9. (17) Roux, J. C.; Simoyi, R. H.; Swinney, H. L. Physica 1981,8D, 257. (18) Schneider, F. W.; Miinster, A. F. J . Phys. Chem. 1991, 95, 2130. (19) Roesky, P.; Schneider, F. W. J . Phys. Chem. 1993, 97, 398. (20) Doumbouya, I. S.;Miinster, A. F.; Doona, C. J.; Schneider, F. W. J. Phys. Chem. 1993,97, 1025. (21) Doumbouya, I. S.; Schneider, F. W. J . Phys. Chem. 1993,97,6945. (22) Doona, C. J.; Blittersdorf, R.; Schneider, F. W. J . Phys. Chem. 1993, 97, 7258. (23) Holz, R.; Schneider, F. W. To be published. (24) Schell, M.; Fraser, S.;Kapral, R. Phys. Rev. 1983, 28A, 373. (25) Belair, J.; Glass, L. Phys. Lett. 1983, A96, 113. (26) MacKay, R. S.;Tresser, C. Physica 1986, 190, 206. (27) GyBrgyi, L.; Field,R. J.;Nosticzius, 2.;McCormick, W. D.;Swinney, H. L. J. Phys. Chem. 1992, 96, 1228. (28) Larter, R.; Steinmetz, C. G. Philos. Trans. R . Soc. l991,337A, 291. (29) Yamazaki, I.; Yokota, K. In BiologicalandBiochemicalOscillationF; Chance, B., Pye, E. K., Gosh, A. K., Hess,B., Eds.; Academic Press: New York, 1973; p 109. (30) Aguda, B. D.; Larter, R. J. Am. Chem. SOC.1991, 113, 7913. (31) Olsen, L. F. Phys. Lett. 1983, A94 454. (32) AGO is calculated from the following two half-cell reactions and their Eo values: l/2 0 2 + 2Ht -.H20, EO(pH = 0 ) = 0.6145 V (ref a): NADH + Ht -.NADt + 2Ht + 2e-, EO (pH = 7) = 0.32 V (ref b). These values are converted to AG by AG = -nFE, adjusted to pH = 6, and added together to obtain AGO. (a) CRC Handbook of Chemistry and Physics, 65th ed.; Weast, R. C., Astle, M. J., Beyer, W. H., Eds.; CRC Press: Boca Raton, FL, 19841985; p D-157. (b) Stryer, L. Biochemistry, 3rd ed.;W. H. Freeman and Co.: New York, 1988; p 401. (33) Gear, C. W. Numerical Initial Value Problems in Ordinary Differential Equations; Prentice-Hall: Englewood Cliffs, NJ, 1971; p 209. (34) Hindmarsh, A. C. Gear: Ordinary Differential Equations Systems Solver; VCID 2001, rev 3, Dec 1974. (35) Broomhead, D. S.; King, G. P. Physica 1986, 20D, 217. (36) Renyi, A. Probability Theory; North-Holland: Amsterdam, 1970. (37) Grassberger, P. Phys. Lett. 1985, A107, 101. (38) Kaplan, J . L.;Yorke, J. A. InLectureNotesinMathematics;Peitgen, H. O., Walther, H. O., Eds.; Springer: Berlin, 1979; p 204, Vol. 730. (39) Schneider, F. W.; Blittersdorf, R.; FBster, A.; Hauck, T.; Lebender, D.; Miiller, J. J . Phys. Chem. 1993, 97, 12244.
