Kinetic characterization of a model for zymogen activation: An experimental design and kinetic data analysis

Catalysis, 79 (1993) 347-363 Elsevier Science Publishers B.V., Amsterdam

Journal of Molecular

347

MO28

Kinetic characterization of a model for zymogen activation: an experimental design and kinetic data analysis A. Vgzquez”, R. Varbn”, J. Tudelab and F. Garcia-CQnovas*Tb “Departamento de Quimica-Fisica, Universidad de Castilla- La Mancha (Spain) bDepartamento de Bioquimica y Biologia Molecular, Universidad de Murcia, Espinardo (Spain); fax. (+34-68)835418 or (+34-68)305101

(Received June 23,1992; accepted September 30,1992)

Abstract The kinetic equations of both the transient phase and the steady state of some mechanisms, considered as particular cases of a general model for enzyme activation through limited proteolysis, are obtained. Two alternative experimental approaches, an excess of zymogen, or of activating enzyme, are used. Procedures for the evaluation of the kinetic parameters and rate constants are developed and these methods are applied to the simulated data obtained by using a personal computer in order to verify the reliability of the method. Key words: enzyme kinetics; experimental design; kinetic data analysis; proteolysis; zymogen activation

Introduction Proteolytic enzymes are normally synthesized and secreted as inactive precursors, zymogens, to protect the cells which produce them. These zymogens must undergo an activation process, usually a limited proteolysis to attain their full catalytic activity [ 11. Many of the zymogen activating enzymes operate by Uni-Bi mechanisms [ 2,3,4]. Thus, a general mechanism describing the process of zymogen activation would be: E+G)

k:: k:,

G

EG+EZ’

k”

E+Z

Ua)

W

where E is the activating protease, G, is the inactive precursor of Z, Z is the activated enzyme and W is the peptide released from G during the formation of z. Transient phase kinetic studies of Uni-Bi mechanisms such as reaction (Ia) have been carried out by several authors with either the concentration of substrate greatly in excess of that of enzyme [ $61 or uice versa [ 7,8]. However, mechanism (IA) presents an experimental difficulty in that, generally, neither *Author to whom correspondence should be addressed. 0304-5102/93/$06.00

0 1993 - Elsevier Science Publishers B.V. All rights reserved.

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the concentration of W or Z can be followed directly with ease. The procedure normally used to measure the concentration of the activated enzyme, Z, is the periodic removal of aliquots of the reaction mixture followed by an assay of the activity of Z in the aliquots [9,10]. The activity of Z is determined by measuring the initial rate of a reaction with one of the substrate concentrations. This procedure is laborious and prone to many kinds of error, such as continuation of the reaction in the aliquot removed. Therefore, the measured activity does not exactly coincide with the activity at the time the aliquot was taken. Reaction (Ia) can be indirectly followed by coupling it with another enzyme reaction, where the activated enzyme Z acts on a chromogenic substrate (A) to give a chromophoric product (P, Q or both), the concentration of which can be experimentally followed. The action of E on G and that of Z on A follows the Uni-Bi mechanism. Thus, the overall scheme proposed is: E+Z

E+G

Z+A

1

k” k:,

ZA+

ZP--

k”

z+p

(Ia)

UIa)

Q

Scheme 1. Recently, a study of the transient phase kinetics of the reactions in Scheme 1 was carried out assuming that the concentration of zymogen, G, is greatly in excess of E [ 41 and vice versa [ 111. A number of activation mechanisms can be considered as particular cases of the mechanism indicated in Scheme 1, if one or more of the rate constants have much higher values than the others. The aim of this work is (a) to obtain the kinetic equation of some of those mechanisms considered as particular cases of the general mechanism indicated in Scheme 1. (b) To develop a procedure for the evaluation of the kinetic parameters and rate constants involved in these mechanisms. (c) To apply the method to the simulated data obtained by using a personal computer, in order to illustrate the reliability of the method proposed for the kinetic study of experimental systems with zymogen activation.

Theory In Table 1 are given the mechanisms which can be considered as particular cases of the mechanism indicated in Scheme 1. We denote the activating route by Ai (i= 1,2,3,4); its monitoring route by Mi (j= 1,2,3,4), and the complete mechanism by A,M,. Since the step EG-+EZ+ W requires the cleavage of a peptide bond,

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TABLE 1 Mechanisms which can be considered to be particular cases of mechanism indicated in Scheme 1 Activating route

A,

E+G +EG+ZXE+Z -1 W kG E+G-‘E&E+Z+W k:,

-he

E+G W

Ada

ZC E+G&E&E+Z+W

Monitoring route

Ml

M2

MSe Mde

k” Z+A&ZA$Z&Z+P k!, kA A Z+AI-ZALZ+P+Q k”, Z+A&ZA$ZPxZ+P

K: * Z+AwZAk’Z+I=‘+Q

“In the steps As, A4, M3 and M4, rapid equilibrium conditions are given: KF = k%,/ky k!,/k:.

and K? =

whereas the step EZ+E+Z is a simple deacylation process, the relation lzg < ky is generally achieved [ 3,121. Direct measurement using the temperature jump technique shows that the binding of the protein substrates is much faster than the subsequent steps [ 131. In addition, the kf/k$ ratio shows a low value because the substrate of such an enzyme is usually an amide. In these conditions, the resulting mechanism is that indicated as AzMz in Table 1. The kinetic equation for this mechanism can be obtained from that corresponding to the mechanism indicated in Scheme 1, assuming that the constants k: and kf tend to 00. When the zymogen and substrate concentrations are much higher than the activating protease, the accumulation of the products is given by the following expression: [Q] = [P] =y+Bt+(yt2+S1e’1t+$e~2t

(11

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350

where: 1, = - (k:[G]o+kG_, A2 = -

(kf

[Alo

+k:)

+k!, +k$)

(2) (3) (4)

Equation (4) can be written as:

&ax [Alo

LX= [AlO +KA,

dim

[Glo

= [Gh, +Kg

(5)

where

Cl:,=

k;k:

[Alo [Elo

NAl,+~td

aA_= k:kt[Glo[Elo

2( [Glo +%d

(6) (7)

and

(8) (9) (10) (11) Under these conditions, the system does not attain a steady state, since the dependence of the concentrations of P and Q on the time for high values of t is not linear, but parabolic. If the rapid equilibrium conditions prevail in one route (activating or monitoring: mechanism A,M, or A,M, ) the accumulation equations of the products are uniexponential. The linear dependence of ;1 us. [G] ,, indicates that the monitoring route is in rapid equilibrium, and the linear dependence of a us. [AlO indicates that the activating route is in rapid equilibrium. When both routes are in rapid equilibrium the product accumulation equation is parabolic without exponential terms. For high concentrations of activating enzyme and making the constants k: and kf tend to 00 in the equations given in the literature [ 111, the results would be:

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351

=p+cmil&exp &J)

(12)

A, = [-P,

+ (P, -4Q#“]/2

(13)

I, = [-I’,

- (P; -4&,)“‘]/2

(14)

[Q]= [PI where

being - (2, +L2)=P1

=k:: [E],,+k?, [El0

A,& =Q1 =kf;kz

+kf

(15) (16)

and

A, = - (kf [Alo + k!, + k$) p=__a

[

$+$ 1

2

1

k$ [Glo[Alo t-x= [A],+KA, 6 _k%%‘kf [El,[Gl,[Al, lm~,--~,H~,--~,~ d _k%%‘kf [Elo[Gl,[Al, 2- %(~,-~,)(~,-A,) 6

(18) (19) (20)

(21)

Wlo[Glo[Alo

_k%%kt

3-

(17)

(22) %(~,-~,)(~,--A,)

When the monitoring route is in rapid equilibrium the product accumulation equation is biexponential and I, tends to GO.Moreover, when the activating route is also in rapid equilibrium, the equation is uniexponential and ;1 is given by

k? [El, ‘= - [El,, +I@

(23)

When t+oo the exponential term of eqn. (12) can be neglected and the equation for the concentration of products becomes linear, cxbeing the steady state accumulation velocity of the product.

Kinetic data analysis We propose a new experimental design and kinetic data analysis based

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upon the analytical expressions obtained previously for the mechanism A2M2, using two alternative experimental approaches: an of excess of zymogen or of activating enzyme. High zymogen concentration conditions Under these conditions the following experimental design is proposed: Firstly, several progress curves with different values of [El,, [ Cl0 and [AlO are obtained. According to the kinetic analysis, the system can evolve through an equation with two (for mechanism AzM2), one (for mechanism A,M, and A,M2) or no (for mechanism A4M4) exponential terms. The fitting of the experimental recording to the corresponding equations is characterized by the x2 parameter, which can be compared by means of the F test to determine the best equation. Step 1: The monitoring reaction (Z + A) can be characterized kinetically by carrying out several assays at different values of [A] Owith the initial condition [A] ,, B [Z],, according to the method proposed by Gdlvez et al. [ 141. Step 2: A set of kinetic assays is carried out, in which [G],, and [AlO remain constant and [El0 has different values. The fitting of the experimental progress curves to the corresponding equation gives the y, J3,(IIand, when possible ;1, and 1, values. The linear dependence of cy us. [ El0 (eqn. (4) ) and the independence of ;11 and II, in respect of the activating enzyme (eqns. (2) and (3 ) ) can be proved. Step 3: A set of kinetic assays, in which [El0 and [ Cl0 remain constant and [A] Ohas different values. The fitting of the experimental progress curves to the corresponding equation gives the corresponding kinetic parameters. cr shows a hyperbolic dependence us. [AlO (eqn. (5) ), which permits the K& values to be obtained. 1, and A2, in the case of biexponential behaviour must fulfil eqns. (2 ) and (3 ). If the behaviour is uniexponential, the linear dependence of 1 us. [AlO indicates that the activating route is in rapid equilibrium; if 1 is independent of [A],, the monitoring route will be in rapid equilibrium. Step 4: A set of kinetic assays, in which [El0 and [AlO remain constant and [G] Ohas different values. The fitting of the experimental progress curves to the corresponding equation gives the corresponding kinetic parameters. The non-linear regression fitting of the cxparameter to eqn. (5) allows us to obtain Kg and a:,, . From this last parameter, according to eqn. (6) and bearing in mind the information on the monitoring reaction, one can obtain lzf. The linear regression analysis of the values of J.1 us. [G ] Ogives k? 1 and k y . High activating enzyme concentration conditions First, the experimental progress curves are obtained for different values of [El,, [ Cl0 and [A] ,,. The system can evolve according to an equation with three, two or one significant exponential terms. The fittings are characterized by the x2 value, which can be compared by the F test to determine the best equation. Fitting to an equation with three exponential terms indicates that

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353

the system operates according to mechanism A,M,. The fitting to a two exponential term equation indicates that the monitoring reaction is in rapid equilibrium. In this case, the kinetic parameters obtained are j?, cy, c&,S,, ;I 1 and ;12. Fitting to a uniexponential equation indicates that both reactions, activation and monitoring, are under rapid equilibrium conditions. In this case the kinetic parameters obtained are (x, p, 6 and 2. Once the proper equation has been determined, it is used to fit the progress curves obtained in all later assays. Step 1: [Equivalent to the first step of the previous alternative approach]. According to the method proposed by Galvez et al. [ 141 of the monitoring reaction under the initial condition that [Alo is much larger than [Z],, carrying out several assays at different values of [A] 0 in order to obtain lzf, k? 1 and &‘. Step 2: A set of kinetic assays is carried out, in which [El0 and [A],, remain constant and [G],, has different values. The linear dependence of a! vs. [G],, (eqn. (19 ) ) and the independence of the exponential argument to [C- _ (eqns. (13), (14) and (17)) canbeproved. Step 3: A set of kinetic assays, in which [El0 and [Cl0 remain constant and [A] 0has different values. The steady state velocity (a) shows a hyperbolic dependence of [A] 0 allowing us to calculate ke and K&. In the general case of mechanism A2M2, one of the three exponential arguments shows a linear dependence us. [A] 0 (eqn. (17) ). In the rapid equilibrium conditions of the monitoring route, both exponential arguments will be independent of [A] 0.Finally, if the behaviour of the system is uniexponential, its argument will be independent of [A],, too. Step 4: A set of kinetic assays, in which [G] 0 and [A] ,, remained constant and [El0 has different values was studied. The independence of (Y us. [E] 0 (eqn. (19) ) can be proved. In the general case of mechanism A2M2, A3 is independent of [El0 (eqn. (17)). The plot of -(A,+&) us. [El0 is a straight line of slope k? and intercept (kc, + k$ ) (eqn. (15) ). Likewise, the plot of ;1,1, us. [El0 is a straight line, which gives kg (eqn. (16)). In the more probable case, when the monitoring reaction is in rapid equilibrium, fitting the experimental progress curves to a biexponential equation gives the (x, ;1, and 2, parameters and the kinetic constants can be calculated similarly to the case indicated previously. Finally, uniexponential behaviour gives the j?, (Y and il parameters. 1 shows a hyperbolic dependence us. [ El0 (eqn. (23) ). Non-linear regression analysis enables k$ and Kz to be obtained.

Experimental In order to prove the reliability of the experimental design and the kinetic data analysis proposed previously, they have been applied to a set of simulated progress curves obtained by using a personal computer. The simulated progress curves are obtained by plotting the analytical solution of the differential equa-

354

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7

Start

Introduction of the kinetic constant values and initial reagent concentrations

(

FN(t)

= [PI

)

?I fo fa

t

31

0,

639,3

i FN(j)=foxFn(jxfa) E = ERROR FN(J) = FN(j) + E

-

t

Fig.1.Flow diagram of the computer

program used to obtain the simulated progress curves.

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355

TABLE 2 Kinetic constant values used to obtain the “simulated progress curves” Constant k:: (M-r

s-r)

k?, (s-r) k,” (s-l) k: (M-r

s-l)

k!, (s-l) k: (s-l)

Constants and parameters which can be calculated

Case a

Case b

Case c

Case d

10s 100 2 1O6 100 10

lo7 lo3 10 lo6 100 10

106 100 10 107 103 10

lo7 lo3 2 lo7 103 10

AZ

ki?,ki’ k!,,

k:

K$,, K&

TABLE 3 Constant kinetic values obtained from the analysis of the simulated progress curves with [G],, [A],, >> [El,, using the proposed method. The constant values used to obtain the simulated progress curves are indicated in the 3rd column Constant Case a k:: (M -’ s-l) k!, (s-l) k? (s-‘) k: (M-‘s-l) k!, (s-l) k$ (s-l)

Value calculated (1.5kO.l) x106 114+73 1.9kO.4 (8.8+0.4)x105 121?50 10.4t0.3

Value used 10s 100 2 106 100 10

Case b” k:: (s-l) Case c k:: (M-‘s-l) k!!, (s-l) kg (s-‘) kf (M-Is-l) k!, (s-l) k$ (s-l) Case d” k,G (s-l)

9.6kO.2

1.23? 0.06) x lo6 129*11 11+4 (1.5f0.1)X107 (1.1~0.4)x103 9.1* 0.4 2.0t0.2

10 lo6 100 2 lo7 103 10 2

“The constant kinetic values k:, kt and kt for cases (b) and (d) are the same as those of cases (a) and (c), respectively.

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356

tion system previously obtained, using concrete kinetic constant values. The noise of the experimental instruments is introduced in the plot. In this way the corresponding equation is plotted and the noise is simulated. We denote by “simulated progress curves” the data points generated by using the equation and the simulated noise. The kinetic data analysis is applied to the simulated progress curves, and the final values of kinetic parameters and rate constants obtained are compared to their corresponding initial values used in the curve generation. A simple computer program allows us to simulate the progress curves. In

t (msl

0

0

2

L

6

0

[El,.106(M)

Fig. 2. (A) Progress curves of product accumulation corresponding to the mechanism of Scheme 1. The kinetic constants used are those indicated in Table 2 for case (a). [A],,= [Glo=5X lo-* M. The values of [E],x lo6 (M) used were: 1.5; 2.5; 3.5; 4.5 and 5.5. (B) Plot of CYus. [El,. The kinetic constants used in the simulation are those indicated in Table 2. ( q ) case (a); ( A ) case (b); (0)case (c); (0)case (d). [G],,and [A],were5x10-4M; [E],wasvariedfrom0.5x10-6 to 5.5 x 1O-6 M.

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351

Fig. 1, the flow diagram of the program is given. The program was installed on a Tandon personal computer PC/AT286 with an Epson LX-850 printer. The values of the kinetic constants used are given in Table 2.

Results and discussion High zymogen concentration conditions We first obtained a set of progress curves at different values of [E ] ,,, [G ] ,, A

,,“,&

,x1.,

:

,

10

20

30

t (ms)

Fig. 3. (A) Progress curves of product accumulation corresponding to the mechanism of Scheme 1. The kinetic constants used are those indicated in Table 2 for case (b ) . [G ] 0 = 5 x lop4 M and [E]0=5x10-sM. The values of [A],xlO* (M) used were: 0.2; 0.5; 0.8; 1.4 and 2.0. (B) Plot of CYUS.[A],. [G],=5xlOWM; [E],=5~10-~M; [A], varied from 0.2 x lop4 M to 5.5 X 10e4 M. The kinetic constants used in the simulation are those given in Table 2. ( q ) case (a); ( A ) case (b); (0) case (c); (0) case (d).

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358

and [A],, in which the initial conditions [G],>> [El0 and [A],>> [Cl0 are fulfilled. It has been proved that the accumulation of the product is given by eqn. (1) (case a; Table 2) or for particular cases, an equation with only one exponential term (cases b and c; Table 2) or by no exponential term (case d; Table 2). In the more complex situation (case a; Table 2) fitting by non-linear regression analysis of the progress curves to eqn. (1) gives the parameters y, p, (Y,c&,A1, S, and 1,. The initial estimation of these parameters is obtained as follows: For high time values the exponential terms of eqn. (1) are negligible and the accumulation of the product is given by the equation: [P] =y+pt+at2

(24)

the accumulation velocity of product P is: (25) The plot of d [ P] /dt us. time gives estimates of 20 and p from the slope and the intercept, respectively of the obtained straight line. The estimation of y can be obtained from a point on the progress curves at t>> 1, in which eqn. (24) is fulfilled. ;1, can be obtained from previous kinetic studies in which the activated enzyme (Z) and the auxiliary substrate (A) are the only species present at the start of the reaction. From eqn. (8)) A I can be calculated. The initial estimations of 6, and & are calculated from eqns. (10) and (11)) respectively. Step 1: The values of kf, KA_,and Iz; obtained are indicated in Table 3. Step 2: We have obtained a set of progress curves in which [ Cl0 and [A],, remain constant and [E] 0 has different values (Fig. 2A). The linear plot of cy us. [El0 (Fig. 2B) follows eqn. (4). The A1 and AZ values are independent of [El0 (eqns. (2) and (3)). In case (a), the A, and A2 values calculated were -643 2 32 s-l and -578235 s-l, respectively. For case (b), the II value obtainedwas -513223 s-l and for case (c) -629531 s-i. Step 3: In Fig. 3A is shown a set of progress curves obtained for the same values of [Cl, and [El0 and different values of [A],. cx shows a hyperbolic dependence us. [Alo according to eqn. (5) (Fig. 3B). Fitting by non-linear regression to eqn. (5) allows us to obtain K & for every case. The calculated TABLE 4 Values of K& and Kg determined for each case indicated in Table 2 Case

K&x10”

rl

1.12 kO.11 1.21 k 0.05

1.16 + 0.06 1.18+0.15

:

0.81 0.92 + 0.08 0.07

0.96 1.14 k 0.13 0.06

(M)

K$x104

(M)

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359

14

52 00

0

c

![Gl,

2

O0

I

10

20

30

t hs)

Fig. 4. (A) Progress curves of product accumulation corresponding to the mechanism of Scheme 1. The kinetic constants used are those indicated in Table 2 for case (c ) . [A], = 5 X 10d4 M and [E10=5x 10m6M. The values of [G],x lo4 (M) usedwere: 0.2; 0.4; 1; 2; 3. (B) Plotof (Yus. [G],. [Ah,=5~10-~M; [E],=5~10-~M; [G], varied from 0.5 x 10e4 M to 5.5 X 10e4 M. The kinetic constants used in the simulation are those given in Table 2. (0 ) case (a); ( A ) case (b ); (0 ) case (c); (0) case (d).

values are indicated in Table 4. In case (a) 1, is independent of [AlO being - 554 5 33 s- ’ and A2 shows a linear dependence on [A] O,similarly to L in case (b ) . In case (c ) , the A value obtained was -539 2 27 s-l, independent of [A],,. Step 4: Figure 4A shows a set of progress curves obtained for the same values of [El,, and [AlO but different values of [G],. a shows a hyperbolic dependence on [ Cl0 according to eqn. (5) (Fig. 4B). The non-linear regression analysis allows us to calculate K& and (x$,, . The K$, values are listed in Table 2. From a$,,, according to eqn. (6)) and knowing the kinetic constant of the monitoring reaction, k f can be calculated. In case (a) & is independent of

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360

2

4 [G&JO5 (M)

Fig. 5. (A) Progress curves of product accumulation corresponding to the mechanism of Scheme 1. The kinetic constants used are those indicated in Table 2 for case (a). [A ] ,,= [E ] ,,= 5 X 10W4 M. The values of [G],X lo5 (M) used were: 0.5; 1.0; 1.5; 2.0; 2.5. (B) Plot of the product accumulation velocity in the steady state us. [Cl,. The kinetic constants used and the values of [A],, and [El0 were as indicated in Fig. 5A. [Cl0 was varied from 0.5X 10m5to 5.5X 1O-5 M.

[G] ,,, similarly to the A value obtained in case (b ). The plot of ;11 (case a) or A (case c) us. [Cl0 is a straight line. From the slope and intercept of these plots kC 1 and k’;: (eqn. (2) ) can be calculated. The values of the constants obtained for each case are listed in Table 3. The good agreement between the calculated values and those used to simulate the progress curves confirms the validity of the experimental design and the kinetic data analysis proposed.

Vbzquez et al./J. Mol. Catal. 79 (1993) 347-363

0

6

4

2 [AlJO

361

(Ml

Fig. 6. (A) Progress curves of product accumulation corresponding to the mechanism of Scheme 1. The kinetic constants used are those indicated in Table 2 for case (a). [E ],, = 5 x 10e4 M and [G],=5~10-a M. The values of [A],X104 (M) used were: 0.2; 0.4; 0.6; 0.8; 1.0. (B) Plot of product accumulation velocity in the steady state us. [A],. The kinetic constants used and the valuesof [G],,and [E],wereasindicatedinFig.6A. [A],wasvariedfrom0.2x10-4t05.5x10-4 M.

High activating enzyme concentration conditions A set of simulated progress curves has been obtained using the kinetic constants listed in Table 2 with [El,, [AlO > [G],. Fitting to an equation similar to eqn. (12) but with only one exponential term, gives the parameters

362

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TABLE 5 Constant kinetic values obtained from the analysis of the simulated [A],>> [G],, using the proposed method Constant Kg (M) k:: (s-1) K$, (M) k;4 (s-1)

Value calculated

progress curves with [El,,

Value used

1.03+0.06)x10-4 1.96? 0.05 (1.220.1) x10-4 lO.O? 0.2

IO-* 2 1o-4 10

6

0

2

4 [El,.1’J4 (M)

Fig. 7. (A) Progress curves of product accumulation corresponding to the mechanism of Scheme 1. The kinetic constants used are those indicated in Table 2 for case (a). [A],,=5 x 10W4M and [G],=~x lo-‘M. The values of [E],X lo4 (M) used were: 0.2; 0.4; 0.7; 1.8. (B) Plot of -,I vs. [El,. The kinetic constants used and the values of [Cl0 and [Alo were as indicated in Fig. 7A. [El,, was varied from 0.2 x 10W4to 5 x 10v4 M.

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et al/J. Mol. Catal. 79 (1993) 347-363

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j?, CY,6 and 2 and the value of x2. Further fitting to a biexponential equation does not introduce any significant improvement. Step 1: Figure 5A shows a set of simulated progress curves obtained at the same values of [Alo and [El, but different values of [G],. All the tracings show the same induction period, r. The ;2.value obtained is independent of [G],, being -1.66kO.03 s-l. The plot of cy vs. [Cl0 is linear (Fig. 5B) according to eqn. (19). Step 2: The progress curves obtained at the same values of [E ] ,, and [ Cl0 but different values of [A] 0 show the same induction period, r (Fig. 6A), similar to the case above. The 1 value obtained is independent of [A],, being -1.61+0.02 s-l. cy shows a hyperbolic dependence US. [A],, (Fig. 6B). The values of K& and lze obtained by non-linear regression (eqn. (19) ) are indicated in Table 5. Step 3: Figure 7A shows a set of simulated progress curves obtained at the same [G],and [Alo values but different values of [E ] 0. It can be observed that the product is accumulated at the same velocity in the steady state for all the tracings. The plot of -13.us. [El,, is hyperbolic (eqn. (23); Fig. 7B). Non-linear regression analysis allows us to obtain the values of .lzg and K& . These values are indicated in Table 5. The good agreement between the calculated values and those used to simulate the progress curves confirms the validity of the experimental design and the kinetic data analysis proposed.

References 1 2

P. Cohen, Control of Enzyme Actiuity, Chapman & Hall, London, 1976, pp. 20-31. M. Delaage, P. Denuelle and M. Lazdunski, Biochem. Biophys. Res. Commun., 29 (1967) 235.

3

4 5 6

7 8 9

10 11 12 13 14

M. Dixon, E.C. Webb, C.J.R. Thorne and K.F. Tipton, Enzymes, 3rd edn., Longman, London, 1979, pp. 300-307. R. Varon, F. Garcia Ca’novas, F. Garcia Carmona, J. Tudela, M. Garcia, A. VQzquez and E. Valero, Math. Biosci., 87 (1987) 31. R.J. Maguire, N.H. Hijazi and K.J. Laidler, Biochim. Biophys. Acta, 341 (1974) 1. J. Galvez, R. Varon, F. Garcia C&novas and F. Garcia Carmona, J. Theor. Biol., 94 (1982) 413. H.P. Kasserra and K.J. LaidIer, Can. J. Chem., 48 (1970) 1793. N.H. Hijazi and K.J. LaidIer, Can. J. Chem., 50 (1972) 1440. E. Colomb and C. Figarella, Biochim. Biophys. Acta, 571 (1979) 343. M. Garcia Moreno, B.H. Havsteen, R. Varon and H. Rix-Matzen, Biochim. Biophys. Acta, 1080 (1991) 143. R. Varon, F. Garcia Cannovas,F. Garcia Carmona, J. Tudela, A. Roman and A. Vazquez, J. Theor. Biol., 132 (1988) 51. A. Cornish-Bowden and C.W. Wharton, Enzyme Kinetics, IRL Press, Oxford, 1988. B.H. Havsteen, Studia Biophys., 57 (1976) 117. J. Galvez, R. Varon, F. Garcia Canovas and F. Garcia Carmona, An. Quim., 79 (1983) 521.