1544 Notizen
Studies of the Chaotic Behaviour in the Peroxidase-Oxidase Reaction
Lars F. Olsen
I n s t i t u t e o f B i o c h e m i s t r y , O d e n s e U n i v e r s i t y , D e n m a r k
Z . N a t u r f o r s c h . 3 4 a , 1 5 4 4 - 1 5 4 6 ( 1 9 7 9 ) ; r e c e i v e d O c t o b e r 2 7 , 1 9 7 9
T h i s n o t e p r e s e n t s e x p e r i m e n t a l results w i t h an e n z y m e - c a t a l y s e d r e a c t i o n w i t h a c o n t i n u o u s s u p p l y o f t h e s u b - s t r a t e s in w h i c h w e o b s e r v e a t r a n s i t i o n f r o m periodic b e h a v i o u r t o a p p a r e n t l y n o n p e r i o d i c b e h a v i o u r b y decreas- ing t h e e n z y m e c o n c e n t r a t i o n . A m o d e l o f t h e r e a c t i o n is a l s o d e s c r i b e d . T h i s m o d e l , b a s e d o n l y o n e l e m e n t a r y r e a c t i o n s t e p s , c a n s i m u l a t e t h e p e r i o d i c b e h a v i o u r o b s e r v e d e x p e r i m e n t a l l y . B y c h a n g i n g o n l y o n e r a t e c o n s t a n t t h e b e h a v i o u r c h a n g e s f r o m periodic t o q u a s i - p e r i o d i c b e h a v i o u r . T h e w o r k o f earlier i n v e s t i g a t o r s s u g g e s t s t h a t s u c h q u a s i p e r i o d i c b e h a v i o u r (toroidal o s c i l l a t i o n s ) is a s t e p t o w a r d s c h a o t i c b e h a v i o u r .
Introduction
A chemical reaction in a homogeneous solution may exhibit a variety of types of dynamic behaviour such as bistability, oscillations or chaos. The possibility of the latter phenomenon in chemical systems was suggested only a few years ago by Rössler, who examined a few abstract reaction schemes involving at least 3 chemical species [1).
Since then some experimental observations have been made which may be interpreted as chaotic behaviour in chemical reaction systems [2—6].
A common feature of these systems is that they all involve a continuous supply of the reactants. Some of these experiments have been criticized by Showalter et al. [7] who suggested that the observed nonperiodic behaviour originates from uncontrolled fluctuations in the experimental system. At the same time Showalter et al. [7] expressed serious doubts about the possibility of finding truly chaotic behaviour in a chemical reaction scheme involving only elementary processes. This note describes an experimental biochemical system in which the transition from periodic behaviour to apparently nonperiodic behaviour occurs, and a model which can simulate the periodic behaviour and which also contains the conditions for the emergence of chaos
R e p r i n t r e q u e s t s t o D r . L . F . O l s e n , I n s t i t u t e o f B i o - c h e m i s t r y . O d e n s e U n i v e r s i t y , C a m p u s v e j 5 5 , D K - 5 2 3 0 O d e n s e M , D e n m a r k .
The reaction in question is the oxidation of reduced nicotinamide adenine dinucleotide (NADH) by molecular oxygen catalysed by peroxidase.
This reaction — named the peroxidase-oxidase reaction — is capable of demonstrating damped oscillations in a system open to oxygen [8, 9] and sustained oscillations [10, 11] as well as complex dynamic behaviour [2, 12] when both substrates are supplied continuously.
Methods and Materials
Experiments were performed in a 4.5 ml hexago- nal glass cuvette fitted with a stirrer for efficient mixing. Oxygen was supplied to the reaction mixture from the gas phase above the liquid, containing a mixture of O2 and N2. If the oxygen partial pressure in the gas is constant, the rate of diffusion of oxygen into the liquid follows the simple rate law [13]
d[0
2]/d* = fc([0
2]eq- [0
2]),
where [O2] denotes the oxj^gen concentration in the liquid, [02]eq is the oxygen concentration in the liquid when the gas and the liquid are in equilibrium with respect to oxygen, and k is a constant. A solu- tion of NADH was infused into the reaction mixture through a capillary, whose tip was below the surface of the liquid, using a high precision infusion pump (Harvard Apparatus Co., model 971).
The system was mounted in a dual wavelength spectrophotometer (Hitachi-Perkin Elmer 356).
NADH was measured spectrophotometrically at 380 nm minus 400 nm and oxygen was measured with a Clark-type electrode (Radiometer, Copen- hagen). A detailed description of the system has appeared previously [11, 12].
Horseradish perioxidase (RZ 1.0) and NADH were obtained from Boehringer. Methylene blue and 2,4-dichlorophenol were purchased from Merck.
Computer simulations were performed on a minicomputer with peripherals using a standard Runge-Kutta-Merson integration procedure.
Results and Discussion
The waveform of the oscillations in the peroxidase- oxidase reaction with continuous supply of NADH and O2 are very much dependent on the concentra-
0340-4811 / 79 / 1200-1544 $ 01.00/0. — P l e a s e order a reprint rather t h a n m a k i n g y o u r o w n c o p y .
Notizen 1 5 4 5
" J " 3 8 0 - 4 0 0 nm ( N A D H )
15 r
10
0
La a a a a a a
F i g . 1. N o n p e r i o d i c a n d periodic oscillations in t h e p e r o x i d a s e - o x i d a s e r e a c t i o n . E x p e r i m e n t a l c o n d i - t i o n s : a) 0 . 7 RZM p e r o x i d a s e , 0 . 1 [I.M m e t h y l e n e b l u e a n d 11 [JLM 2 , 4 - d i c h l o r o p h e n o l in 0 . 1 M a c e t a t e b u f f e r , p H 5 . 1 . O 2 c o n t e n t in t h e g a s p h a s e w a s 1 . 6 5 % b y v o l u m e . 0 . 2 5 M N A D H w a s i n f u s e d at a r a t e o f 1 2 . 5 [i\/h. T e m p e r a t u r e 2 8 ° C . b ) as in a ) e x c e p t t h a t t h e e n z v m e c o n c e n t r a t i o n w a s increased t o 1 . 2 Ü.M.
I- 20 min -M
tion of the enzyme as illustrated in Figure 1. It has previously been shown [2] that the oscillations shown in Fig. 1 a may be of the chaotic type since plots of a given amplitude of the oscillation against the preceding amplitude (next amplitude map)
defines a single valued bell-shaped curve satisfying Li and Yorke's [14] conditions for chaos.
The peroxidase-oxidase reaction can be simulated by a class of models involving linear and quadratic branching steps and linear termination [11, 12] as
115-
1 1 0 -
F i g . 2 . a ) N o n p e r i o d i c b e h a v i o u r in t h e m o d e l d e s c r i b e d in the t e x t . R a t e c o n s t a n t s : ki = 0 . 1 6 5 , k-2 = 3 . 7 5 X 1 02, fa = 3 . 5 X 1 0 ~2, k4 = 2 0 . 0 , fa = 5 . 3 5 , fa = 1 0 "3, fa = 7 = 0 . 1 , fa[B0] = 0 . 8 2 5 , [ A o ] = 8 . 0 . Initial c o n d i t i o n s : [ A ] = 8 . 0 , [ B ] = 1 0 0 . 0 , [ X ] = [ Y ] = 0 . 0 . T h e traces s h o w n b e g i n at a p p r o x i m a t e l y 4 0 0 t i m e u n i t s a f t e r t h e s i m u l a t i o n w a s s t a r t e d , b ) P l o t o f [ X ] a g a i n s t [ B ] for t h e oscillations s h o w n in F i g u r e 2 a .
Time
1546 Notizen
shown below
— 2 X , 2 X - - 2 Y.
—^ 2 X, X
k\ p
— Q, Xo
f
A - 7 A' Bo —^ B.
Here A denotes oxygen, B denotes NADH and X and Y are intermediate free radicals. The sixth reaction accounts for the spontaneous formation of free radicals in the reaction mixture. This model is a modification of a model proposed by Degn to explain the oscillations in the Bray reaction [15, 16].
The model can simulate the periodic behaviour observed in the experimental system. By changing only one parameter (ki) apparently nonperiodic
oscillations are observed as shown in Figure 2 a.
These oscillations are not of the same type as those shown in Figure 1 a. The next amplitude map is a closed curve and therefore cannot be used for diagnosing chaos. The flow appears to be confined to a distorted torus (or its 4-dimensional analog) a illustrated in Figure 2 b. These oscillations are therefore quasiperiodic and not truly chaotic.
However, Rössler has recently presented some simple differential equations in which the transition toroidal behaviour —> chaotic behaviour occurs [17, 18], Hence a more detailed analysis may reveal that this transition also occurs in the model described here because as shown by Rössler toroidal oscillation is a possible (though not necessary) step towards chaos. Such an analysis is now in progress.
[ 1 ] 0 . E . R ö s s l e r , Z . N a t u r f o r s c h . 3 1 a . 2 5 9 ( 1 9 7 6 ) . [ 2 ] L . F . O l s e n a n d H . D e g n , N a t u r e L o n d o n 2 6 7 , 1 7 7
( 1 9 7 7 ) .
[ 3 ] R . A . S c h m i t z , K . R . G r a z i a n i , a n d J . L . H u d s o n , J . C h e m . P h y s . 6 7 , 3 0 4 0 ( 1 9 7 7 ) .
[ 4 ] 0 . E . R ö s s l e r a n d K . W e g m a n n , N a t u r e L o n d o n 2 7 1 , 8 9 ( 1 9 7 8 ) .
[ 5 ] K . W e g m a n n a n d 0 . E . R ö s s l e r , Z . N a t u r f o r s c h . 3 3 a, 1 1 7 9 ( 1 9 7 8 ) .
[6] J . L . H u d s o n , M . H a r t , a n d D . M a r i n k o , J . C h e m . P h v s . 7 1 . 1 6 0 1 ( 1 9 7 9 ) .
[ 7 ] K . S h o w a l t e r , R , M . N o y e s , a n d K . B a r - E l i , J . C h e m . P h y s . 6 9 , 2 5 1 4 ( 1 9 7 8 ) .
[ 8 ] I . Y a m a z a k i , K . Y o k o t a , a n d R . N a k a j i m a , B i o c h e m . B i o p h v s . R e s . C o m m u n . 2 1 , 5 8 2 ( 1 9 6 5 ) .
[ 9 ] H . D e g n . B i o c h i m . B i o p h y s . A c t a 1 8 0 , 2 7 1 ( 1 9 6 9 ) .
[ 1 0 ] S. N a k a m u r a , K . Y o k o t a , a n d I . Y a m a z a k i , N a t u r e L o n d o n 2 2 2 , 7 9 4 ( 1 9 6 9 ) .
[ 1 1 ] L . F . O l s e n a n d H . D e g n , B i o c h i m . B i o p h y s . A c t a 5 2 3 , 3 2 1 ( 1 9 7 8 ) .
[ 1 2 ] H . D e g n , L . F . Olsen, a n d J . W . P e r r a m , A n n . N . Y . A c a d . Sei. 3 1 6 , 6 2 3 ( 1 9 7 9 ) .
[ 1 3 ] H . D e g n , J . L u n d s g a a r d , L . C. P e t e r s e n , a n d A . O r m i c k i , i n : M e t h o d s o f B i o c h e m i c a l A n a l y s i s ( D . Glick, e d . ) , V o l . 2 6 , J o h n W i l e y & S o n s , N e w Y o r k f 9 7 9 , p p . 4 7 - 7 7 .
[ 1 4 ] T . - Y . L i a n d J . A . Y o r k e , A m e r . M a t h . M o n . 8 2 , 9 8 5 ( 1 9 7 5 ) .
[ 1 5 ] H . D e g n , A c t a C h e m . S c a n d . 2 1 , 1 0 5 7 ( 1 9 6 7 ) . [ 1 6 ] P . L i n d b l a d a n d H . D e g n , A c t a C h e m . S c a n d . 2 1 ,
7 9 1 ( 1 9 6 7 ) .
[ 1 7 ] 0 . E . R ö s s l e r , Z . N a t u r f o r s c h . 3 2 a , 2 9 9 ( 1 9 7 7 ) . [ 1 8 ] 0 . E . R ö s s l e r , A n n . N . Y . A c a d . Sei. 3 1 6 , 3 7 6 ( 1 9 7 9 ) .
