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A Novel Method for Determining Kinetic Parameters of Dissociating Enzyme Systems
ANALYTICAL BIOCHEMISTRY ARTICLE NO.
264, 8 –21 (1998)
AB982818
A Novel Method for Determining Kinetic Parameters of Dissociating Enzyme Systems Zhi-Xin Wang National Laboratory of Biomacromolecules, Institute of Biophysics, Academia Sinica, Beijing 100101, People’s Republic of China
Received April 3, 1998
The theoretical analysis has been presented for the kinetics of dissociating–associating enzyme-catalyzed reactions. On the basis of the kinetic equation of substrate reaction, a general procedure is developed for determining the kinetic constants of dissociating–associating enzyme reactions. By analyzing the experimental data of initial velocity and steadystate velocity as functions of enzyme and substrate concentration, all unknown kinetic parameters can be determined from several simple, sequential calculations. This method is simple and rigorous, and the required experiments may also not be difficult for most dissociating enzyme systems. Therefore, the present method should be a useful addition to the available methods for studying subunit dissociation of enzymes. In comparison to other physical methods, the advantage of this method is not only its usefulness in the study of self-associating reactions at very low protein concentration but its convenience in the study of substrate effects on subunit–subunit interactions. © 1998 Academic Press
Key Words: dissociation–association; hysteretic enzyme; subunit–subunit interaction; rate constant.
Many systems are known in which ligands can induce changes in enzyme activity that occur much more slowly in comparison with the rate of the enzymatic reaction, resulting in a time lag in the response of the enzyme activity to changes in the ligand concentrations (1–3). The term hysteretic was introduced by Frieden to describe such slowly responding enzymes (4). Several mechanisms can account for slow responses including ligand-induced conformational changes of the enzyme, displacement of a tightly bound ligand by a different ligand, and enzyme dissociation–association (2, 4, 5). The kinetic behavior of the first two mechanisms has been systematically studied during the last two decades (4, 8
6 –17). However, relatively few experimental results for the dissociation–association kinetics have been reported in the literature. This probably is due to difficulties in two aspects. Experimentally, kinetic study of a dissociating–associating enzyme system needs to be performed over a wide range of both enzyme and substrate concentrations. Since the selfassociation process of enzyme will give rise to a lag or burst of product formation, the full-time progress curve of the reaction should be utilized for investigating this phenomenon. When such a lag or burst time is sufficiently long, it is readily observable in a progress curve. However, enzymes can have transition time in the millisecond range, so definitive results may require a continued assay system and rapid measurement techniques. Another related problem to deal with is that of substrate depletion where the initial substrate concentration is low. Velocity measurements may not be difficult at low enzyme concentrations, but are quite difficult at high enzyme concentrations where there will be a considerable depletion of substrate during the activity measurement. The difficulty here, aside from the fact that a continued assay method and stopped-flow equipment are almost essential for collecting the data, is the analysis of the full time course itself. It has been assumed in all the theoretical treatments of hysteretic behavior that the substrate level does not change over the time course of the measurement. When the problem of substrate depletion becomes of major importance, the mathematical description for this kinetic system can be quite complex and may be difficult to handle without utilizing some type of computer simulation analysis (18). There are a large number of enzymes which can reversibly dissociate and reassociate in response to an effector ligand (3). Such a change in subunit assembly usually is accompanied by a change in enzyme activity, providing a possible mechanism for 0003-2697/98 $25.00 Copyright © 1998 by Academic Press All rights of reproduction in any form reserved.
9
DETERMINING KINETIC PARAMETERS OF DISSOCIATING ENZYME SYSTEMS
regulation. The design and interpretation of proper kinetic experiments have received considerable attention (2, 4, 19 –27). In spite of the fact that kinetic studies should provide much useful information, some experimental difficulties may have restricted the application of these procedures. In the present paper, we describe a general method to determine the kinetic constants for dissociating– associating enzyme systems. By analyzing the data of initial and steady-state velocity, all kinetic parameters can be determined from several simple, sequential calculations. This method is simple and rigorous, and the required experiments also may not be difficult for most enzyme systems. Since low enzyme concentrations are usually used in enzyme activity assays, the present method has advantages when other physical methods may not be available.
where [E]0 is the total molar concentration of enzyme. Assuming that (i) the enzymatic reaction is irreversible and not inhibited by the product formed, (ii) the substrate concentration is constant throughout the activity measurement, (iii) the monomer contains a single active site and the dimer contains two equivalent and independent active sites, and (iv) the enzyme and substrate equilibrate rapidly with the enzyme–substrate complex and the dissociation–association step is a relatively slow processes, then the following relations hold at any time, @M# 5
K S@M T# , K S 1 @S#
[5]
@MS# 5
@S#@M T# , K S 1 @S#
[6]
K *S2@D T# K *S2@D T# 5 , 2 ~K *S 1 @S#! 2 K * 1 2K *S@S# 1 @S#
[7]
@DS# 5
2K *S@S#@D T# , ~K *S 1 @S#! 2
[8]
@DS 2# 5
@S# 2@D T# , ~K *S 1 @S#! 2
@D# 5
THEORY
Let us consider the simplest system of the enzyme undergoing a monomer– dimer reaction (Scheme 1),
2 S
and
[9]
where K S and K *S are the dissociation constants of substrate for the monomer and dimer, respectively. Substituting Eqs. [5]–[9] into Eq. [4] yields d@D T# 5 A@M T# 2 2 B@D T#, dt
[10]
where
SCHEME 1
where M, D, and S represent monomer, dimer, and substrate, respectively. On the basis of Scheme 1, we have @M T# 5 @M# 1 @MS#,
[1]
@D T# 5 @D# 1 @DS# 1 @DS 2#,
[2]
@E# 0 5 @M T# 1 2@D T#,
[3]
and
d@D T# 5 ~k 100@M# 2 1 k 110@M#@MS# 1 k 111@MS# 2! dt 2 ~k 200@D# 1 k 210@DS# 1 k 211@DS 2#!, [4]
A5
k 100K 2S 1 k 110K S@S# 1 k 111@S# 2 , ~K S 1 @S#! 2
B5
k 200K *S2 1 2k 210K *S@S# 1 k 211@S# 2 . ~K *S 1 @S#! 2
and
[11] @12#
According to the relationship d@D T# 1 d@M T# 52 , dt 2 dt
[13]
we have d@M T# 5 22A@M T# 2 1 2B@D T# dt 5 22A@M T# 2 1 B$@E# 0 2 @M T#% 5 22A@M T# 2 2 B@M T# 1 B@E# 0.
[14]
10
ZHI-XIN WANG
If the consumption of substrate is negligible during the course of reaction, A and B can be regarded as apparent rate constants of the following reaction:
of substrate shifts the equilibrium 2M T º D T toward formation of D T . In this case, we have
b5
A
2M T | 0 D T.
4A~@M T# 0 2 @M T# e!
4A@M T# 0 1 B 1 ÎB 2 1 8AB@E# 0
. 0.
B
Thus, B/A is the apparent dissociation constant for the dimerization reaction in the presence of substrate, B/A5[M T ] 2 /[D T ]. Integrating Eq. [14] with the boundary conditions t50, [M T ]5[M T ] 0 gives ~@M T# 2 l 1!~@M T# 0 2 l 2! 5 exp$2a t%, ~@M T# 2 l 2!~@M T# 0 2 l 1!
Therefore, b can be taken as a criterion for the direction of the equilibrium shift. The rate of the formation of product is given by d@P# k 2@S#@M T# 2k *2@S#@D T# 5 1 dt K S 1 @S# K *S 1 @S# 5
[15]
k 2@S#@M T# k *2@S#~@E# 0 2 @M T#! 1 K S 1 @S# K *S 1 @S#
5 a M@M T# 1 a D $@E# 0 2 @M T#%
where
5 a D@E# 0 1 ~a M 2 a D!@M T#
l1 5
2B 1 ÎB 2 1 8AB@E# 0 , 4A
l2 5
2B 2 ÎB 2 1 8AB@E# 0 , 4A
5 a D@E# 0 1
Let
n0 5 @M T# 0 2 l 1 b5 . @M T# 0 2 l 2
l 1 2 l 2b e 2at . 1 2 b e 2at
2B 1 ÎB 2 1 8AB@E# 0 @M T# e 5 . 4A
[17]
When [M T ] 0 ,[M T ] e , the influence of substrate shifts the equilibrium 2M T º D T toward formation of M T . According to the definition of b, we have 4A@M T# 0 1 B 1 ÎB 1 8AB@E# 0 2
@P# 5 n St 1 g ln
[16]
When t approaches infinity, [M T ]5[M T ] e 5 l 1 , and therefore the molar concentration of monomer at steady-state is given by
b5
u
d@P# 5 a M@M T# 0 1 a D $@E# 0 2 @M T# 0%. dt t50
[19]
Integrating Eq. [18] with the boundary conditions t50, [P]50 yields
Equation [15] can be rewritten as
4A~@M T# 0 2 @M T# e!
[18]
where aM5k2[S]/(KS1[S]) and aD5k*2[S]/(K *S1[S]) are the specific activities for monomer and dimer, respectively. The initial rate of the enzyme-catalyzed reaction is given by
and
a 5 ÎB 2 1 8AB@E# 0.
@M T# 5
~a M 2 a D!~ l 1 2 l 2b e 2at! , 1 2 b e 2 at
, 0.
On the other hand, when [M T ] 0 .[M T ] e , the influence
1 2 b e 2 at , 12b
[20]
where
a 5 ÎB 2 1 8AB@E# 0, b5 g5
4A@M T# 0 1 B 2 ÎB 2 1 8AB@E# 0 4A@M T# 0 1 B 1 ÎB 2 1 8AB@E# 0 aM 2 aD , 2A
,
and
n S 5 2a D@D T# e 1 a M@M T# e 5 a D@E# 0 1 ~a M 2 a D!
H
J
2B 1 ÎB 2 1 8AB@E# 0 . 4A
n S is the steady-state rate of enzyme-catalyzed reaction when t approaches infinity. The specific activity,
DETERMINING KINETIC PARAMETERS OF DISSOCIATING ENZYME SYSTEMS
11
a5 n S /[E] 0 , is the function of the total enzyme concentration and can be written as
a 5 a D 1 ~a M 2 a D!
H
J
2B 1 ÎB 2 1 8AB@E# 0 . 4A@E# 0
[21]
Case 1: b , 0, aM , aD The pattern of the progress curve of product formation is determined by both the specific activity of the monomer and dimer forms and the ratio of the initial and final monomer concentrations. When b ,0, a M ,a D , the equilibrium 2M T º D T shifts toward formation of less active monomer. Therefore, the rate of the enzyme-catalyzed reaction decreases during the process, and the progress curve of product formation has a initial burst phase. It can be seen from Eq. [21] that if one plots the specific activity, a, as a function of the total enzyme concentration, a is increasing on [E] 0 . Figure 1 illustrates theoretical progress curves for product formation at various concentrations of enzyme. Indicated are those portions of the curve from which initial and steady-state velocities may be obtained. The inset of this figure shows a schematic representation of a plot of the specific activity, a, versus the total concentration of enzyme, [E] 0 .
FIG. 2. Schematic plot of product formation, [P], versus reaction time, t, for a dissociating–associating enzyme reaction (Case 2). The two dashed lines represent the initial velocity and the steady-state velocity, respectively. The progress curves were generated by using Eq. [21] with the same kinetic parameters as Fig. 1 except that k 2 5 1.0 s21 and k *2 50.1 s21. Substrate concentration is 1 3 1023 M, and enzyme concentrations for curves 1–3 are 10 3 1027, 5 3 1027, and 2 3 1027 M, respectively. Inset: dependence of the specific enzyme activity on the total enzyme concentration. The steady-state velocities were obtained from the corresponding progress curves.
Case 2: b , 0, aM . aD When b ,0, a M .a D , the equilibrium 2M T º D T shifts from less active dimer to more active monomer. Since the enzyme after starting the reaction converts from a state of low activity to an active one of high activity, the time-dependent curve of product formation has a lag period, and a plot of a versus [E] 0 shows a monotonically decreasing curve (Fig. 2). Case 3: b . 0, aM , aD When b .0, a M ,a D , the equilibrium 2M T º D T shifts from less active monomer to more active dimer, and the rate of enzymatic reaction is enhanced during the process. Therefore, the progress curve of product formation has a initial lag phase, and a monotonically increases with enzyme concentration, [E] 0 (Fig. 3). Case 4: b . 0, aM . aD
FIG. 1. Schematic plot of product formation, [P], versus reaction time, t, for a dissociating–associating enzyme reaction (Case 1). The two dashed lines represent the initial velocity and the steady-state velocity, respectively. The progress curves were generated by using Eq. [21] with the following kinetic parameters: k 2 50.1 s21, K S 5 131024 M, k*251.0 s21, K *S5531024 M, k100513105 M21 s21, k 11050.5310 5 M21 s21, k 11150.2310 5 M21 s21, k 20050.05 s21, h 50.1. Substrate concentration is 1 3 1023 M, and enzyme concentrations for curves 1–3 are 10 3 1027, 5 3 1027, and 2 3 1027 M, respectively. Inset: dependence of the specific enzyme activity on the total enzyme concentration. The steady-state velocities were obtained from the corresponding progress curves.
In this case, the equilibrium 2M T º D T shifts from more active monomer to less active dimer, and therefore the progress curve of product formation will show an initial burst phase, and a monotonically decreases with increasing enzyme concentration, [E] 0 (Fig. 4). In summary, a shift of equilibrium between monomeric and dimeric forms can be induced by substrates or effectors. If the enzyme converts from a less active form to a more active form during the reaction process, the progress curve of product formation will give rise to
12
ZHI-XIN WANG
an initial lag phase. Otherwise, the time-dependent progress curve will have an initial burst period. On the other hand, because the concentration of enzyme influences the equilibrium between monomer and dimer, and dilution will change the ratio of these two species, the specific activity will depend on enzyme concentration provided that the specific activity of monomer (a M ) and dimer (a D ) differ. These later might be equivalent to specific activities extrapolated to infinitely low and high enzyme concentrations, respectively. Therefore, in light of these two features, it is often possible to distinguish the different reaction mechanisms given above. DATA ANALYSIS
It can be seen from Eq. [20] that any particular progress curve contains five independent parameters, [M T ] 0 , a M , a D , A, and B. In theory, the fitting of data from a single progress curve to Eq. [20] will yield values for [M T ] 0 , a M , a D , A, and B. If the values for these parameters are obtained at different substrate concentrations, all the microscopic rate constants can then be determined from the expressions of these parameters. In practice, however, Eq. [20] is quite complex and, due to the fact that experimental data always contain experimental uncertainties, may be difficult to treat. With a set of unfortunate starting values, the fitting procedure will converge to the wrong values or perhaps
FIG. 3. Schematic plot of product formation, [P], versus reaction time, t, for a dissociating–associating enzyme reaction (Case 3). The two dashed lines represent the initial velocity and the steady-state velocity, respectively. The progress curves were generated by using Eq. [21] with the following kinetic parameters: k250.1 s21, KS5531024 M, s21, K *S5131024 M, k100513105 M21 s21, k*251.0 k110523105 M21 s21, k111553105 M21 s21, k20050.05 s21, h50.9. Substrate concentration is 1 3 1023 M, and enzyme concentrations for curves 1–3 are 2 3 1027, 1 3 1027, and 0.5 3 1027 M, respectively. Inset: dependence of the specific enzyme activity on the total enzyme concentration. The steady-state velocities were obtained from the corresponding progress curves.
FIG. 4. Schematic plot of product formation, [P], versus reaction time, t, for a dissociating–associating enzyme reaction (Case 4). The two dashed lines represent the initial velocity and the steady-state velocity, respectively. The progress curves were generated by using Eq. [21] with the same kinetic parameters as Fig. 3 except that k 2 51.0 s21 and k *2 50.1 s21. Substrate concentration is 1 3 1023 M, and enzyme concentrations for curves 1–3 are 2 3 1027, 1 3 1027, and 0.5 3 1027 M, respectively. Inset: dependence of the specific enzyme activity on the total enzyme concentration. The steady-state velocities were obtained from the corresponding progress curves.
enter a nonconvergent cycle. Therefore, it is necessary to develop some new approaches to overcome this difficulty. To determine values of the kinetic constants for the dimerization reaction, the first experiments would be to measure initial velocity over a wide range of enzyme concentrations. In general, it is best to perform the kinetic experiments at saturating levels of substrates. Under these conditions, a change in specific activity as a function of enzyme concentration reflects different intrinsic activities for different molecular weight species. Let us assume that the activity assay is performed in the following manner. An enzyme solution (of a certain concentration of enzyme) is preincubated until the equilibrium between the dimer and monomer is reached. The concentrations of monomer and dimer prior to the substrate addition are given by (27) @M#9 5
1 $2K 0 1 ÎK 20 1 8K 0@E#90%, 4
@D#9 5
1 $@E#90 2 @M#9% 2
1 1 5 @E#90 2 $2K0 1 ÎK20 1 8K0 @E#90 %, 2 8
[22]
[23]
where K 05k 200/k 100 is the dissociation constant for the monomer– dimer reaction and [E]90 the total molar
13
DETERMINING KINETIC PARAMETERS OF DISSOCIATING ENZYME SYSTEMS
concentration of enzyme in the absence of substrate. When the assay reaction is started by mixing substrate with the preincubated enzyme solution, at the beginning instant, the concentrations of each enzyme species in the assay system are @M T# 0 5
V9@M#9 5 h @M#9, V9 1 V c
@D T# 0 5
V9@D#9 5 h @D#9, V9 1 V c
@E# 0 5
V9@E#90 5 h @E#90, V9 1 V c
[24] and
[25] [26]
where h5V9/(V91Vc), and V9 and Vc are the volumes of preincubated enzyme solution and substrate solution, respectively. Substitution of Eqs. [22]–[26] into Eq. [19] yields
tration. In this case, the initial specific activity, a 0 , as a function of substrate concentration, could be represented as a0 5
where f 0 5(1/4[E] 0 ) {2hK 0 1 =( h K 0 ) 2 18 h K 0 [E] 0 }, and therefore is a constant at given enzyme concentration. This equation is analogous to the case of two enzymes acting on the same substrate, and would be expected to show kinetic behavior equivalent to negative cooperativity (32). It can be seen from Eq. [20] that when t approaches infinity, the asymptote of this equation is given by @P# 5 n St 1 g ln
5 ~aM 2 aD!h@M#9 1 aD@E#0 1 5 ~aM 2 aD !$2hK0 1 Î~hK0 !2 1 8hK0 @E#0 % 1 aD @E#0 . 4 [27]
n S 5 a D@E# 0 1
1 , 12b
[31]
H
@P#* 5 [28]
where a05n0/[E]0. Therefore, if one measures the initial velocities at various enzyme concentrations, and determine the values of a M and a D by extrapolating a 0 to infinitely low and high enzyme concentrations, the dissociation constant for the dimerization reaction in the absence of substrate, K 0 , can then be calculated from Eq. [28]. A rather detailed description for determining the value of K 0 has been formulated by Kurganov (27). Experimentally, it is often difficult to determine the initial velocities accurately from the linear parts of the progress curves. In these cases, the initial velocities can be estimated by fitting the initial parts of the reaction progress curves to second- or third-order polynomial equations in product concentration (28 –30). [29]
The initial velocity, (d[P]/dt) t50, was taken to be a measure of 1/c 1 (31). In order to study effects of substrate on the kinetic behavior of the dimerization reaction, the second group of experiment should be performed at various substrate concentrations and a fixed low enzyme concen-
~a M 2 a D! 4
B 3 2 1 A
Equation [27] can be rewritten as
t 5 c 0 1 c 1@P# 1 c 2@P# 2 1 c 3@P# 3 1 · · ·
[30]
and the slope and the intercept of the asymptote (steadystate velocity) are
n0 5 aM@MT#0 1 aD$@E#0 2 @MT#0%
2@E# 0~a 0 2 a D! 2 , K0 5 h ~a M 2 a D!~a M 2 a 0!
k 2@S#f 0 k *2@S#~1 2 f 0! 1 , K S 1 @S# K *S 1 @S#
ÎS D B A
2
18
J
B @E# 0 , A
and
[32]
aD 2 aM aD 2 aM ln~1 2 b ! 5 2A 2A 3 ln
2 Î~B/A! 2 1 8~B/A!@E# 0
4@M T# 0 1 ~B/A! 1 Î~B/A! 2 1 8~B/A!@E# 0
.
[33] Equation [32] can be rewritten as 2~ n S 2 a D@E# 0! 2 B 5 A ~a M@E# 0 2 n S!~a M 2 a D!
[34]
B 2@E# 0~a 2 a D! 2 5 . A ~a M 2 a!~a M 2 a D!
[35]
or
Substituting Eq. [35] into [33] yields A5
aD 2 aM 2@P#* 3 ln
~a 2 a D!@~a M 2 a D! 1 ~a M 2 a!# . 2@~a M 2 a!a 0 1 a M~a 2 a D! 1 a D~a D 2 a M!# [36]
14
ZHI-XIN WANG
Equations [35] and [36] allow experimental evaluation of B/A and A as a function of substrate concentration from the progress curves, providing the values of a M and a D at different concentrations of substrate are known. These equations are simplified, of course, if either a M or a D is equal to 0. As pointed out by Kurganov et al. (26), according to the conditions of detailed balance, we have k 210 K 0 K *S 5 , k 110 2 KS
S D
K *S k 211 5 K0 k 111 KS
2
.
[37]
Thus, for determining the values of all unknown kinetic parameters, the following calculations are performed: (i) Measure the time course of product formation as a function of the total enzyme concentration at a fixed high substrate level (so that it remains relatively constant during the activity measurement), and determine the values of n0 and n S from the progress curve of product–time at each particular enzyme concentration (Figs. 1– 4). (ii) Diagnose the reaction mechanism for the selfassociation process according to the pattern of the progress curves and the relationship between a and [E] 0 (insets of Figs. 1– 4). (iii) Determine the values of a M and a D by extrapolating a 0 5 n 0 /[E] 0 to infinitely low and high enzyme concentrations, and then calculate the K 0 according to Eq. [28]. (iv) Measure the progress curves of product formation at various substrate concentrations and a fixed low enzyme concentration (so that the substrate depletion can be neglected during the activity measurement), and determine the values of n0, n S , and [P]* from the progress curve of product–time at each particular concentration of substrate (Fig. 5). (v) Determine the values of k 2 , K S , k *2 , and K *S by fitting Eq. [30] to the experimental data obtained at different concentrations of substrate, {(n0)i , [S]i } (as h , [E] 0 and K 0 are known, inset of Fig. 5). (vi) Calculate the values of a M and a D at each particular concentration of substrate according to the following equations: aM 5
k 2@S# , K S 1 @S#
aD 5
k *2@S# . K *S 1 @S#
(vii) Calculate the values of B/A at different concentrations of substrate according to Eq. [35]. (viii) Calculate the values of A at different concentrations of substrate according to Eq. [36]. (ix) Determine the values of k 100, k 110, and k 111 by fitting Eq. [11] to the experimental data obtained at different substrate concentrations, {A i , [S] i }.
FIG. 5. Schematic plot of product formation, [P], versus reaction time, t, for a dissociating–associating enzyme reaction at different substrate concentrations. The two dashed lines represent the initial velocity and the steady-state velocity, respectively. The progress curves were generated by using Eq. [21] with the same kinetic parameters as Fig. 1. Enzyme concentration is 10 3 1027 M, and substrate concentrations for curves 1–3 are 8 3 1023, 6 3 1024, and 2 3 1024 M, respectively. Inset: dependence of the initial velocity on the substrate concentration. The initial velocities were obtained from the corresponding progress curves.
(x) Calculate the values of k 200, k 210, and k 211 according to Eq. [37]. Figure 6 shows a schematic representation of plots of A and B versus substrate concentration. It should be indicated that if k 2 (or k *2 ) is equal to 0, K S (or K *S ) cannot be determined by step (v) since the expression of n0 will contain only the K *S (or K S ) term. For example, when k 2 50 (a M 50), Eqs. [30], [35], and [36] can be simplified to a0 5
k *2@S#~1 2 f 0! 5 a D~1 2 f 0!, K *S 1 @S#
B 2@E# 0~a 2 a D! 2 , 5 A aa D A5
[38]
and
[39]
a 2D 2 a 2 aD . ln 2@P#* 2~a 2D 2 aa 0!
[40]
Substituting Eq. [37] into the expressions of A and B (Eqs. [11] and [12]) yields
H
B K *S~K S 1 @S#! 5 K0 A K S~K *S 1 @S#!
J
2
.
[41]
Therefore, instead of using steps (v)–(viii), the values of K S , K *S , B/A, and A can be determined as follows. (i) Calculate the values of aD at different substrate
15
DETERMINING KINETIC PARAMETERS OF DISSOCIATING ENZYME SYSTEMS
FIG. 6. Dependence of the apparent rate constants A and B on the substrate concentration. The curves were generated by using Eqs. [11] and [12] with the following kinetic parameters: KS5131024 M, K *S5531024 M, k100513105 M21 s21, k11050.53105 M21 s21, k11150.23105 M21 s21, k20050.05 s21.
concentrations, {(aD)i, [S]i}, according to aD5a0/(12f0) (as h, [E]0, and K0 are known). (ii) Determine the values of k *2 and K *S by fitting the expression of a D to the experimental data obtained at different concentrations of substrate. (iii) Calculate the values of B/A at different concentrations of substrate according to Eq. [39]. (iv) Determine the value of K S by fitting Eq. [41] to the experimental data obtained at different substrate concentrations, {(B/A) i , [S] i }. (v) Calculate the values of A at different concentrations of substrate according to Eq. [40]. The procedures given above for determining the kinetic parameters can be easily extended to situations of enzyme reactions involving two substrates. In these cases, however, the concentrations of both substrates have to be varied to obtain all kinetic parameters. It is important to emphasize the following consideration. In slowly dissociating enzyme systems (in which the dimer–tetramer interconversion is very slow in comparison with the rate of enzymatic reaction), the dissociation constant K0, deter-
mined from plots of a0 against [E]0, is appropriate to conditions under which the enzyme is preincubated before its activity is measured, and the dissociation constant B/A, determined from the steady-state velocity, nS, according to Eq. [35], is appropriate to the conditions under which the reaction takes place. When K0,B/A, the presence of substrate shifts monomer–dimer equilibrium toward formation of monomer, and when K0.B/A, the effect of substrate is such that the monomer– dimer equilibrium is shifted toward the formation of dimer. An example of the dissociating enzyme system is human immunodeficiency virus 1 protease (HIV-PR).1 HIV-PR is an aspartyl protease composed of two identical protomers. The time course of product formation for the enzyme-catalyzed reaction shows an initial burst phase (33). Following the initial loss of enzyme activity, a steady-state or equilibrium level of activity was obtained. No further loss of activity was observed over a period of 30 min. The effect of initial enzyme concentration on properties of the time-dependent loss of activity was determined. At enzyme levels of 6.2–50 nM, the specific activity, nS/[E]0, monotonically increased with increasing enzyme concentration. This result indicates that the progress curves for the loss of enzymatic activity are consistent with dissociation of the active HIV-PR dimer into inactive monomeric subunits. When the monomer contains several subunits and therefore several active sites, the association process between a monomer with i substrates and another with j substrates can be expressed as k1ij
MS i 1 MS j | 0 DS i1j. k2ij
If substrate binding is rapid relative to the catalytic reaction, and the binding sites are equivalent and independent in the monomer and in the dimer, we then have (26)
H
B k 200 K *S~K S 1 @S#! 5 A k 100 K S~K *S 1 @S#!
S D
k 2ij K *S 5 r ij K 0 k 1ij KS
J
2n
,
and
[42]
i1j
,
[43]
where r ij 51 at i5j, r ij 5 21 at iÞj, and n is the number of the total binding sites in the monomer. Thus, the apparent rate constants, A and B, can be written as
O O r k S niDS njDS @S# K D n
n
ij
A5
i50 j50
S
1ij
11
@S# KS
D
i1j
S
2n
,
and
[44]
1 Abbreviation used: HIV-PR, human immunodeficiency virus 1 protease.
16
ZHI-XIN WANG
phorylase a into active dimer. When phosphorylase a was preincubated with 0.5 mg/ml glycogen, the lag phase in the progress curve was completely abolished, and linear progress curves were obtained. Such linearity indicates that (i) the coupling enzyme is in sufficient quantity to ensure actual measurement of the primary enzyme’s behavior, and (ii) the substrate consumption can be neglected and the initial rate conditions are satisfied during activity measurement. At the enzyme concentrations of 0.29 to 1.74 nM, both the relaxation time, t, and specific activity, a5 n S /[E] 0 , are independent on enzyme levels, suggesting that the enzyme concentration in the assay system is far less than the corresponding apparent dissociation constant. It can be seen from Eq. [20] that when [E] 0 ! B/A, we have FIG. 7. Time course of the phosphorylase a-catalyzed reaction. The final enzyme concentrations (dimer) for curves 1, 2, 3, 4, 5, and 6 were 1.74, 1.45, 1.16, 0.87, 0.58, and 0.29 nM, respectively. The assay system contains 50 mM Bis-Tris, 50 mM KCl, 1 mM DTT, 0.1 mM EDTA, 2 mM MgAc, 2% glycerol, 97 mM 7-methyl-6-thioguanosine, 148 mg/ml purine nucleoside phosphorylase, 1.2% dimethyl sulfoxide, 0.5 mg/ml glycogen, and 4 mM glucose-1-P at pH 7.0, 25°C. The reactions were initiated by the addition of the preincubated phosphorylase a (4.69 mM dimer) into the assay system. In the control experiment (curve 0), 1.74 nM phosphorylase a was preincubated in 1.604 ml of assay solution except that glucose-1-P was absent for 20 min, and then 16 ml of glucose-1-P solution (0.4 M) was added to initiate the reaction.
O O k S niDS njDS @S# K *D n
n
2ij
B5
i50 j50
S
11
@S# K *S
D
i1j
S
2n
,
[45]
and k 2ij 5k 2ji , k 1ij 5k 1ji . According to the procedure mentioned above, once the values of A and B at different substrate concentrations are obtained, the microscopic rate constants k 1ij and k 2ij can then be determined by fitting Eqs. [44] and [45] to the experimental data. As an example, the experimental data for the substrate reaction kinetics of glycogen phosphorylase a is analyzed using the present method. A detailed kinetic study of this enzyme will be published elsewhere. Glycogen phosphorylase a from rabbit muscle is a tetrameric molecule consisting of four identical subunits. Enzyme activity was determined in the direction of glycogen synthesis by the method of Sergienko and Srivastava (34). Figure 7 shows the time courses of enzyme-catalyzed reactions at different enzyme concentrations. Before attaining the steady state the reaction showed a lag phase in the product formation. The increase of enzyme activity is connected with the reversible dissociation of inactive tetramer of phos-
@P# 5 n St 1 g ln 5 a D@E# 0t 1 5 n St 1
1 2 b e 2at 12b aD ~@D T# 0 2 @E# 0!~1 2 e 2Bt! B
n0 2 nS ~1 2 e 2Bt!, B
[46]
where n S 5a D [E] 0 and n 0 5a D [D T ] 0 . Therefore, at low enzyme concentrations, the rate of tetramer formation for the process 2D T 3T T is negligible, and the reaction can then be regarded as a simple irreversible dissociation process. In further studies, the effect of substrate on properties of the time-dependent increase of activity was determined. From the progress curves of product formation, n0, n S , and B can be determined by fitting Eq. [46] to the experimental data. A plot of B against glycogen concentration shows that the apparent dissociation rate constant is not affected by the glycogen concentration (Fig. 8A). The average value of the apparent dissociation rate constant is B57.963102361.4431024 s21. Figure 8B shows the dependence of the steadystate rate on the glycogen concentration. From these data, the values of K m and V max for glycogen were determined to be K m 5 0.360 6 0.029 mg/ml, V max 5 2.0 6 0.062 mM/min. Similar experiments at another fixed concentration of glucose-1-phosphate (6.17 mM) were also performed to illustrate the influence of the glucose-1-phosphate concentration on the apparent Michaelis constants (data not shown). The values of K m and V max so determined were K m 5 0.314 6 0.042 mg/ml and V max 5 2.85 6 0.143 mM/min. These results suggest that the binding of glycogen at one active site does not influence either the binding of glucose-1phosphate on the same subunit or the binding of either glycogen or glucose-1-phosphate on a different subunit. Since the present kinetic study can give rise only to information on effects of the substrates on the appar-
17
DETERMINING KINETIC PARAMETERS OF DISSOCIATING ENZYME SYSTEMS
where B/A is the apparent dissociation constant for the dimer–tetramer reaction of phosphorylase a in the presence of one substrate, and [E]90 is the total enzyme concentration in the preincubating system. Given that only the dimeric form of phosphorylase a is catalytically active, the initial velocity and the final velocity will be proportional to the initial and final concentrations of the dimeric form in the assay system, respectively. Therefore, when the reaction is started by diluting the preincubated mixture of enzyme–substrate into the assay system, the ratio of initial velocity to final velocity is given by f5
n 0 @D T# 0 @D T#9 5 5 nS @E# 0 @E#90
H
1 B 5 2 1 4@E#90 A
ÎS D B A
2
1 8@E#90
J
B , A
[48]
where [D T ] 0 and [E] 0 are the concentrations of dimer and total enzyme concentration in the assay system at t50, respectively. Equation [48] can be rearranged as B 2@E#90 f 2 5 . A ~1 2 f !
FIG. 8. (A) Dependence of the apparent dissociation rate constant on the concentration of glycogen. (B) Dependence of the steady-state rate on the concentration of glycogen. The solid line is the theoretical curve calculated according to the Michaelis–Menten equation with V max 5 2.0 mM min21, K m 5 0.360 mg/ml. The assay system contains 50 mM Bis-Tris, 50 mM KCl, 1 mM DTT, 0.1 mM EDTA, 2 mM MgAc, 2% glycerol, 95 mM 7-methyl-6-thioguanosine, 148 mg/ml purine nucleoside phosphorylase, 1.2% dimethyl sulfoxide, 3.08 mM glucose-1-P, and 0.58 nM phosphorylase a (dimer) at pH 7.0, 25°C. The reactions were initiated by the addition of the preincubated phosphorylase a (4.69 mM dimer) into the assay system.
ent dissociation rate constant, experiments other than kinetic studies are required to investigate the influence of substrates on the association process. As mentioned earlier, binding of glucose-1-phosphate to the enzyme has no influence on the binding affinity for glycogen, and therefore a study of the influence of a single substrate on the dimer–tetramer equilibrium will provide useful information. If phosphorylase a is preincubated with one substrate until the dimer–tetramer equilibrium is reached, the concentration of the dimeric phosphorylase a in the preincubating system is given by
@D T#9 5
H
1 B 2 1 4 A
ÎS D B A
2
18
J
B @E#90 , A
[47]
[49]
Preincubation of the enzyme with different concentrations of glucose-1-phosphate have no effect on the initial lag phase in the progress curve. This result indicates that the presence of glucose-1-phosphate does not shift the equilibrium between the dimeric form and the tetrameric form of phosphorylase a. When the enzyme was preincubated with different concentrations of glycogen, the lag phase in the progress curves was reduced. Similarly, the values of the initial and steady-state velocities of the phosphorylase a-catalyzed reaction, n0 and nS, were determined from the progress curves of the phosphorylase a reaction, and then the apparent dissociation constant, B/A, can be calculated according to Eq. [49]. Figure 9 shows the dependence of B/A on the concentration of glycogen. The fact that B/A increases with glycogen concentration indicates that glycogen binds preferentially to the dimer form of phosphorylase a. As mentioned above, because neither the dimer–tetramer equilibrium nor the binding of glycogen to enzyme is affected by glucose-1phosphate, it is reasonable to assume that the apparent dissociation constant, B/A, obtained in the absence of glucose-1-phosphate is equivalent to that in the assay system which contains a constant amount of glucose-1phosphate. Since the values of KS (50.36 mg/ml) and B5k200 (57.96 3 1023 s21) have been determined previously, and the concentration of enzyme is known [E]90 5 5.33 3 1026 M in the present case), k100 and K *S can then be determined by fitting Eq. [42] to the experimental data. The solid line in Fig. 9 represents the best fit of the experimental results with parameters k100 5 1.26 3 104
18
ZHI-XIN WANG
FIG. 9. Dependence of the apparent dissociation constant, B/A on the concentration of glycogen in the preincubation mixture. The values of B/A were calculated according to Eq. [49]. The solid line is the bestfitting result according to Eq. [42] with KS 5 0.36 mg/ml, k20057.9631023 s21, K *s51.67 mg/ml, and k10051.263104 M21 s21. Phosphorylase a (5.53 mM dimer) was preincubated at 25°C for at least 2 h in Bis-Tris buffer containing various concentrations of glycogen; 0.4 ml enzyme– glycogen mixture was then diluted into assay system (1.62 ml), and the full-time reaction curve were recorded. The assay system contains 50 mM Bis-Tris, 50 mM KCl, 1 mM DTT, 0.1 mM EDTA, 2 mM MgAc, 2% glycerol, 110 mM 7methyl-6-thioguanosine, 148 mg/ml purine nucleoside phosphorylase, 1.2% dimethyl sulfoxide, 0.5 mg/ml glycogen, 15 mM glucose1-P, and 1.36 nM phosphorylase a (dimer) at pH 7.0, 25°C.
6 7.87 3 102 M21 s21 and K *S 5 1.67 6 0.08 mg/ml. From these values, all the microscopic rate constants k1ij and k2ij can be obtained according to Eq. [43].
0.3 to 3 s. Thus, at an enzyme level of 1025 M or higher, the transition time may be too short to conveniently measure, whereas if both B/A and the enzyme concentration were as low as 1027 M, the self-association reaction could be a rather slow process. As mentioned previously, there are three basic mechanisms for describing hysteretic enzyme systems: the slow conformational change model, the ligand displacement model, and the enzyme dissociation–association model. With these different formulations and viewpoints of similar systems, it is necessary that the similarities and true difference between various models be pointed out. We will attempt to present here a general, unified picture of hysteretic enzymes. The following three common features are generally agreed to refer to hysteretic enzyme systems: (i) enzyme may exist in two interconvertible states with different kinetic properties, (ii) the interconversion between these two enzyme states is slow in comparison with establishment of the steady-state of enzymatic reaction, and (iii) the substrate can shift the equilibrium between the two states. In the presence of substrate, the schemes for describing hysteretic enzyme systems and the corresponding expressions of product formation can be written as shown in Schemes 2– 4. A. Slow Conformational Change Model (4) This mechanism (Scheme 2) assumes that the enzyme may exist in two conformational states that differ in their catalytic or binding properties, and that the rate of interconversion of conformational states is slow in comparison with the rate of enzymic reaction. The product formation at time t is given by
DISCUSSION
With the analytical expression of product formation as a function of time, an estimation of reasonable transition time can be made for the case where association is diffusion-controlled. It can be seen from Eq. [16] that the transition time can be approximately written as 1 1 t< 5 . a A Î~B/A! 2 1 8~B/A!@E# 0
[50]
As pointed out by Frieden (18), values for association constant, A, are normally in the range of 105 to 106 M21 s21. Thus, values of t will depend on the magnitudes of the apparent dissociation constant and the total enzyme concentration. A decrease in either B/A or [E]0 will cause an increase in the transition time. Since the greatest change of kinetic behaviors for a self-association system occurs in the region of the B/A, it is frequently satisfactory to vary the concentration from about 0.2 to 5.0 times this constant. For example, if B/A and [E]0 are assumed each to equal 1026 M, then the transition time is equal to
@P# 5 n St 1
n0 2 nS $1 2 e 2~A1B!t%, A1B
where
A5
k 10K S 1 k910@S# , K S 1 @S#
B5
SCHEME 2
k 20K *S 1 k920@S# . K *S 1 @S#
[51]
19
DETERMINING KINETIC PARAMETERS OF DISSOCIATING ENZYME SYSTEMS
If the enzyme exists in the E form only before substrate addition (that is, [E9]50 at t50), the initial velocity, n0, and the steady-state velocity, n S , are given by
n0 5
k 2@E# 0@S# , K S 1 @S#
nS 5
k 2@E# 0@S#B . ~K S 1 @S#!~A 1 B!
and
B. Ligand Displacement Model (4, 17) If an inhibitor, I, is bound to an enzyme surface quite tightly, and the dissociation rate of the inhibitor from the enzyme is relatively slow, then when the tightly bound inhibitor is displaced by an activator, L, the rate of increase in enzymatic activity could be measurable as a function of time. This mechanism would be represented by Scheme 3. Assuming that the binding steps of substrate and activator are equilibrated, and [S] 0 , [L] 0 @[E] 0 , the expression of product formation is then given by
n0 1 2 b e 2 at @P# 5 n St 1 ln , A@E# 0 12b
SCHEME 4
nS 5
H
3 @E# 0 2 @I# 0 2 1
[52]
a 5 Î~A@I# 0 1 A@E# 0 1 B! 2 4A @I# 0@E# 0, 2
@I# 0 1 @E# 0 1 ~B/A! 2 Î$@I# 0 1 @E# 0 1 ~B/A!% 2 2 4@E# 0@I# 0 b5 , @I# 0 1 @E# 0 1 ~B/A! 1 Î$@I# 0 1 @E# 0 1 ~B/A!% 2 2 4@E# 0@I# 0
n0 5
$k 2K9L 1 k92@L#%@S#@E# 0 , K9SK L 1 K9S@L# 1 K9L@S# 1 @S#@L#
ÎF
B A
@E# 0 1 @I# 0 1
S DG B A
2
J
2 4@E# 0@I# 0 ,
k 10K9SK L 1 k910K9L@S# , K9SK L 1 K9S@L# 1 K9L@S# 1 @S#@L#
B5
k 20K *S 1 k920@S# , K *S 1 @S#
and
and where the condition [EI]5[E] 0 at t50 has been assumed. C. Enzyme–Protein Interaction Model (Scheme 4) There are a number of examples of interaction of enzymes with other proteins. For these types of interaction systems, the protein effector may be considered simply as a direct analog of a small ligand effector. Since the rate of association of two proteins will be slower than that of an enzyme with a small ligand, one might expect to see hysteretic behavior. During the last two decades, the kinetic behaviors of slow binding and slow-tight binding modifier have been studied systematically (7, 10, 17). Thus, if the kinetic data of the enzyme–protein interaction can be obtained, treatment of the data would be similar to that of enzyme–small ligand interactions. When [M] 0 @ [E] 0 , @P# 5 n St 1
SCHEME 3
SD
A5
where 2
~k 2K9L 1 k92@L#!@S# 2~K9SK L 1 K9S@L# 1 K9L@S# 1 @S#@L#!
n0 2 nS $1 2 e 2~A@M#01B!t%, A@M# 0 1 B
A5
k 10K S 1 k910@S# , K S 1 @S#
B5
k 20K *S 1 k 20@S# , K *S 1 @S#
20
ZHI-XIN WANG
n0 5
k 2@E# 0@S# , K S 1 @S#
nS 5
k 2@E# 0@S#B . ~K S 1 @S#!~A@M# 0 1 B!
and [53]
When [M] 0 '[E] 0 , @P# 5 n St 1
n0 1 2 b e 2 at ln , A@E# 0 12b
[54]
where
a 5 Î~A@M# 0 1 A@E# 0 1 B! 2 2 4A 2@M# 0@E# 0, @M# 0 1 @E# 0 1 ~B/A! 2 Î$@M# 0 1 @E# 0 1 ~B/A!% 2 2 4@E# 0@M# 0 b5 , @M# 0 1 @E# 0 1 ~B/A! 1 Î$@M# 0 1 @E# 0 1 ~B/A!% 2 2 4@E# 0@M# 0
n0 5
k 2@E# 0@S# , K S 1 @S#
nS 5
B k 2@S# @E# 0 2 @M# 0 2 2~K S 1 @S#! A 1
ÎF
and
H
@E# 0 1 @M# 0 1
S DG B A
SD
2
J
2 4@E# 0@M# 0 .
1 2 b e 2 at 12b
[55]
where
a 5 ÎB 2 1 8AB@E# 0, 4A@M T# 0 1 B 1 ÎB 2 1 8AB@E# 0
The present investigation was supported in part by Grant 39421003 from the China Natural Science Foundation and the Pandeng Project of the National Commission for Science and Technology.
1. 2. 3. 4. 5. 6. 7. 8. 9.
12. 13. 14.
,
15.
aM 2 aD g5 , 2A
n S 5 a D@E# 0 1 ~a M 2 a D!
H
J
2B 1 ÎB 2 1 8AB@E# 0 , 4A
A5
k 100K 1 k 110K S@S# 1 k 111@S# , ~K S 1 @S#! 2
B5
k 200K *S2 1 2k 210K *S@S# 1 k 211@S# 2 , ~K *S 1 @S#! 2
2 S
2
k *2@S# . K *S 1 @S#
ACKNOWLEDGMENTS
11.
4A@M T# 0 1 B 2 ÎB 2 1 8AB@E# 0
aD 5
and
Finally, it should be indicated that in theory, the kinetic behavior of a self-association process should be dependent on the total enzyme concentration. However, at low enzyme concentration, this process may appear to be independent of enzyme concentration and could be misinterpreted to be a consequence of an isomerization process. Therefore, the activity measurements should be made in conjunction with other methods which measure different parameters. Especially, molecular weight measurements as a function of time or time-dependent conformational changes are important in determination of the mechanism.
10.
b5
k 2@S# , K S 1 @S#
REFERENCES
D. Enzyme Dissociation–Association Model @P# 5 n St 1 g ln
aM 5
16. 17. 18. 19. 20. 21. 22. 23.
Frieden, C. (1971) Annu. Rev. Biochem. 40, 653– 696. Frieden, C. (1979) Annu. Rev. Biochem. 48, 471– 489. Traut, T. W. (1994) Crit. Rev. Biochem. Mol. Biol. 29, 125–163. Frieden, C. (1970) J. Biol. Chem. 245, 5788 –5799. Ainslie, G. R., Jr., Shill, J. P., and Neet, K. E. (1972) J. Biol. Chem. 247, 7088 –7096. Tian, W. X., and Tsou, C. L. (1982) Biochemistry 21, 1028 –1032. Tsou, C. L. (1988) Adv. Enzymol. 61, 381– 436. Wang, Z. X., and Tsou, C. L. (1987) J. Theor. Biol. 127, 253–270. Wang, Z. X., Preiss, B., and Tsou, C. L. (1988) Biochemistry 27, 5095–5100. Wang, Z. X., Wang, H. R., and Zhou, M. H. (1995) Biochemistry 34, 6863– 6868. Morrison, J. F., and Walsh, C. T. (1988) Adv. Enzymol. 61, 201–301. Cha, S. (1975) Biochem. Pharmacol. 24, 2177–2185. Cha, S. (1976) Biochem. Pharmacol. 25, 2695–2702. William, J. W., Morrison, J. F., and Duggleby, R. G. (1979) Biochemistry 18, 2567–2573. Sculley, M. J., and Morrison, J. F. (1986) Biochim. Biophys. Acta 874, 44 –53. Zhao, K. Y., and Wang, Z. X. (1996) J. Theor. Biol. 181, 319 –327. Wang, Z. X. (1993) Anal. Biochem. 230, 370 –377. Frieden, C. (1981) in Protein–Protein Interactions (Frieden, C., and Nichol, L. W., Eds.), pp. 289 –314. Wiley, New York. Frieden, C. (1967) J. Biol. Chem. 242, 4045– 4052. Nichol, L. W., Jackson, W. J. H., and Winzor, D. J. (1967) Biochemistry 6, 2449 –2456. Nichol, L. W., and Winzor, D. J. (1976) Biochemistry 15, 3015–3019. Steiner, R. F. (1980) Mol. Cell. Biochem. 31, 5–24. Neet, K. E., and Ainslie, G. R. (1976) Trends Biochem. Sci. 1, 145–147.
DETERMINING KINETIC PARAMETERS OF DISSOCIATING ENZYME SYSTEMS 24. Neet, K. E., and Ainslie, G. R. (1980) Methods Enzymol. 64, 192–226. 25. Kurganov, B. I. (1977) J. Theor. Biol. 68, 521–543. 26. Kurganov, B. I., Dorozhko, A. I., Kagan, Z. S., and Yakovlev, V. A. (1976) J. Theor. Biol. 60, 287–299. 27. Kurganov, B. I. (1982) Allosteric Enzymes. pp. 151–313. Wiley, New York. 28. Orsi, B. A., and Tipton, K. F. (1979) Methods Enzymol. 63, 159 –183. 29. Wang, M. H., Wang, Z. X., and Zhao, K. Y. (1996) Biochem. J. 320, 187–192.
21
30. Wang, Z. X., Killilea, S. D., and Srivastava, D. K. (1993) Biochemistry 32, 1500 –1509. 31. Philo, R. D., and Selwyn, M. J. (1973) Biochem. J. 135, 525– 530. 32. Dixon, M., and Webb, E. C. (1979) Enzymes, 3rd ed., pp. 75–78. Longman, London. 33. Cheng, Y.-S. E., Yin, F. H., Foundling, S., Blomstrom, D., and Kettner, C. A. (1990) Proc. Natl. Acad. Sci. USA 87, 9660 – 9664. 34. Sergienko, E. A., and Srivastava, D. K. (1994) Anal. Biochem. 221, 348 –355.
264, 8 –21 (1998)
AB982818
A Novel Method for Determining Kinetic Parameters of Dissociating Enzyme Systems Zhi-Xin Wang National Laboratory of Biomacromolecules, Institute of Biophysics, Academia Sinica, Beijing 100101, People’s Republic of China
Received April 3, 1998
The theoretical analysis has been presented for the kinetics of dissociating–associating enzyme-catalyzed reactions. On the basis of the kinetic equation of substrate reaction, a general procedure is developed for determining the kinetic constants of dissociating–associating enzyme reactions. By analyzing the experimental data of initial velocity and steadystate velocity as functions of enzyme and substrate concentration, all unknown kinetic parameters can be determined from several simple, sequential calculations. This method is simple and rigorous, and the required experiments may also not be difficult for most dissociating enzyme systems. Therefore, the present method should be a useful addition to the available methods for studying subunit dissociation of enzymes. In comparison to other physical methods, the advantage of this method is not only its usefulness in the study of self-associating reactions at very low protein concentration but its convenience in the study of substrate effects on subunit–subunit interactions. © 1998 Academic Press
Key Words: dissociation–association; hysteretic enzyme; subunit–subunit interaction; rate constant.
Many systems are known in which ligands can induce changes in enzyme activity that occur much more slowly in comparison with the rate of the enzymatic reaction, resulting in a time lag in the response of the enzyme activity to changes in the ligand concentrations (1–3). The term hysteretic was introduced by Frieden to describe such slowly responding enzymes (4). Several mechanisms can account for slow responses including ligand-induced conformational changes of the enzyme, displacement of a tightly bound ligand by a different ligand, and enzyme dissociation–association (2, 4, 5). The kinetic behavior of the first two mechanisms has been systematically studied during the last two decades (4, 8
6 –17). However, relatively few experimental results for the dissociation–association kinetics have been reported in the literature. This probably is due to difficulties in two aspects. Experimentally, kinetic study of a dissociating–associating enzyme system needs to be performed over a wide range of both enzyme and substrate concentrations. Since the selfassociation process of enzyme will give rise to a lag or burst of product formation, the full-time progress curve of the reaction should be utilized for investigating this phenomenon. When such a lag or burst time is sufficiently long, it is readily observable in a progress curve. However, enzymes can have transition time in the millisecond range, so definitive results may require a continued assay system and rapid measurement techniques. Another related problem to deal with is that of substrate depletion where the initial substrate concentration is low. Velocity measurements may not be difficult at low enzyme concentrations, but are quite difficult at high enzyme concentrations where there will be a considerable depletion of substrate during the activity measurement. The difficulty here, aside from the fact that a continued assay method and stopped-flow equipment are almost essential for collecting the data, is the analysis of the full time course itself. It has been assumed in all the theoretical treatments of hysteretic behavior that the substrate level does not change over the time course of the measurement. When the problem of substrate depletion becomes of major importance, the mathematical description for this kinetic system can be quite complex and may be difficult to handle without utilizing some type of computer simulation analysis (18). There are a large number of enzymes which can reversibly dissociate and reassociate in response to an effector ligand (3). Such a change in subunit assembly usually is accompanied by a change in enzyme activity, providing a possible mechanism for 0003-2697/98 $25.00 Copyright © 1998 by Academic Press All rights of reproduction in any form reserved.
9
DETERMINING KINETIC PARAMETERS OF DISSOCIATING ENZYME SYSTEMS
regulation. The design and interpretation of proper kinetic experiments have received considerable attention (2, 4, 19 –27). In spite of the fact that kinetic studies should provide much useful information, some experimental difficulties may have restricted the application of these procedures. In the present paper, we describe a general method to determine the kinetic constants for dissociating– associating enzyme systems. By analyzing the data of initial and steady-state velocity, all kinetic parameters can be determined from several simple, sequential calculations. This method is simple and rigorous, and the required experiments also may not be difficult for most enzyme systems. Since low enzyme concentrations are usually used in enzyme activity assays, the present method has advantages when other physical methods may not be available.
where [E]0 is the total molar concentration of enzyme. Assuming that (i) the enzymatic reaction is irreversible and not inhibited by the product formed, (ii) the substrate concentration is constant throughout the activity measurement, (iii) the monomer contains a single active site and the dimer contains two equivalent and independent active sites, and (iv) the enzyme and substrate equilibrate rapidly with the enzyme–substrate complex and the dissociation–association step is a relatively slow processes, then the following relations hold at any time, @M# 5
K S@M T# , K S 1 @S#
[5]
@MS# 5
@S#@M T# , K S 1 @S#
[6]
K *S2@D T# K *S2@D T# 5 , 2 ~K *S 1 @S#! 2 K * 1 2K *S@S# 1 @S#
[7]
@DS# 5
2K *S@S#@D T# , ~K *S 1 @S#! 2
[8]
@DS 2# 5
@S# 2@D T# , ~K *S 1 @S#! 2
@D# 5
THEORY
Let us consider the simplest system of the enzyme undergoing a monomer– dimer reaction (Scheme 1),
2 S
and
[9]
where K S and K *S are the dissociation constants of substrate for the monomer and dimer, respectively. Substituting Eqs. [5]–[9] into Eq. [4] yields d@D T# 5 A@M T# 2 2 B@D T#, dt
[10]
where
SCHEME 1
where M, D, and S represent monomer, dimer, and substrate, respectively. On the basis of Scheme 1, we have @M T# 5 @M# 1 @MS#,
[1]
@D T# 5 @D# 1 @DS# 1 @DS 2#,
[2]
@E# 0 5 @M T# 1 2@D T#,
[3]
and
d@D T# 5 ~k 100@M# 2 1 k 110@M#@MS# 1 k 111@MS# 2! dt 2 ~k 200@D# 1 k 210@DS# 1 k 211@DS 2#!, [4]
A5
k 100K 2S 1 k 110K S@S# 1 k 111@S# 2 , ~K S 1 @S#! 2
B5
k 200K *S2 1 2k 210K *S@S# 1 k 211@S# 2 . ~K *S 1 @S#! 2
and
[11] @12#
According to the relationship d@D T# 1 d@M T# 52 , dt 2 dt
[13]
we have d@M T# 5 22A@M T# 2 1 2B@D T# dt 5 22A@M T# 2 1 B$@E# 0 2 @M T#% 5 22A@M T# 2 2 B@M T# 1 B@E# 0.
[14]
10
ZHI-XIN WANG
If the consumption of substrate is negligible during the course of reaction, A and B can be regarded as apparent rate constants of the following reaction:
of substrate shifts the equilibrium 2M T º D T toward formation of D T . In this case, we have
b5
A
2M T | 0 D T.
4A~@M T# 0 2 @M T# e!
4A@M T# 0 1 B 1 ÎB 2 1 8AB@E# 0
. 0.
B
Thus, B/A is the apparent dissociation constant for the dimerization reaction in the presence of substrate, B/A5[M T ] 2 /[D T ]. Integrating Eq. [14] with the boundary conditions t50, [M T ]5[M T ] 0 gives ~@M T# 2 l 1!~@M T# 0 2 l 2! 5 exp$2a t%, ~@M T# 2 l 2!~@M T# 0 2 l 1!
Therefore, b can be taken as a criterion for the direction of the equilibrium shift. The rate of the formation of product is given by d@P# k 2@S#@M T# 2k *2@S#@D T# 5 1 dt K S 1 @S# K *S 1 @S# 5
[15]
k 2@S#@M T# k *2@S#~@E# 0 2 @M T#! 1 K S 1 @S# K *S 1 @S#
5 a M@M T# 1 a D $@E# 0 2 @M T#%
where
5 a D@E# 0 1 ~a M 2 a D!@M T#
l1 5
2B 1 ÎB 2 1 8AB@E# 0 , 4A
l2 5
2B 2 ÎB 2 1 8AB@E# 0 , 4A
5 a D@E# 0 1
Let
n0 5 @M T# 0 2 l 1 b5 . @M T# 0 2 l 2
l 1 2 l 2b e 2at . 1 2 b e 2at
2B 1 ÎB 2 1 8AB@E# 0 @M T# e 5 . 4A
[17]
When [M T ] 0 ,[M T ] e , the influence of substrate shifts the equilibrium 2M T º D T toward formation of M T . According to the definition of b, we have 4A@M T# 0 1 B 1 ÎB 1 8AB@E# 0 2
@P# 5 n St 1 g ln
[16]
When t approaches infinity, [M T ]5[M T ] e 5 l 1 , and therefore the molar concentration of monomer at steady-state is given by
b5
u
d@P# 5 a M@M T# 0 1 a D $@E# 0 2 @M T# 0%. dt t50
[19]
Integrating Eq. [18] with the boundary conditions t50, [P]50 yields
Equation [15] can be rewritten as
4A~@M T# 0 2 @M T# e!
[18]
where aM5k2[S]/(KS1[S]) and aD5k*2[S]/(K *S1[S]) are the specific activities for monomer and dimer, respectively. The initial rate of the enzyme-catalyzed reaction is given by
and
a 5 ÎB 2 1 8AB@E# 0.
@M T# 5
~a M 2 a D!~ l 1 2 l 2b e 2at! , 1 2 b e 2 at
, 0.
On the other hand, when [M T ] 0 .[M T ] e , the influence
1 2 b e 2 at , 12b
[20]
where
a 5 ÎB 2 1 8AB@E# 0, b5 g5
4A@M T# 0 1 B 2 ÎB 2 1 8AB@E# 0 4A@M T# 0 1 B 1 ÎB 2 1 8AB@E# 0 aM 2 aD , 2A
,
and
n S 5 2a D@D T# e 1 a M@M T# e 5 a D@E# 0 1 ~a M 2 a D!
H
J
2B 1 ÎB 2 1 8AB@E# 0 . 4A
n S is the steady-state rate of enzyme-catalyzed reaction when t approaches infinity. The specific activity,
DETERMINING KINETIC PARAMETERS OF DISSOCIATING ENZYME SYSTEMS
11
a5 n S /[E] 0 , is the function of the total enzyme concentration and can be written as
a 5 a D 1 ~a M 2 a D!
H
J
2B 1 ÎB 2 1 8AB@E# 0 . 4A@E# 0
[21]
Case 1: b , 0, aM , aD The pattern of the progress curve of product formation is determined by both the specific activity of the monomer and dimer forms and the ratio of the initial and final monomer concentrations. When b ,0, a M ,a D , the equilibrium 2M T º D T shifts toward formation of less active monomer. Therefore, the rate of the enzyme-catalyzed reaction decreases during the process, and the progress curve of product formation has a initial burst phase. It can be seen from Eq. [21] that if one plots the specific activity, a, as a function of the total enzyme concentration, a is increasing on [E] 0 . Figure 1 illustrates theoretical progress curves for product formation at various concentrations of enzyme. Indicated are those portions of the curve from which initial and steady-state velocities may be obtained. The inset of this figure shows a schematic representation of a plot of the specific activity, a, versus the total concentration of enzyme, [E] 0 .
FIG. 2. Schematic plot of product formation, [P], versus reaction time, t, for a dissociating–associating enzyme reaction (Case 2). The two dashed lines represent the initial velocity and the steady-state velocity, respectively. The progress curves were generated by using Eq. [21] with the same kinetic parameters as Fig. 1 except that k 2 5 1.0 s21 and k *2 50.1 s21. Substrate concentration is 1 3 1023 M, and enzyme concentrations for curves 1–3 are 10 3 1027, 5 3 1027, and 2 3 1027 M, respectively. Inset: dependence of the specific enzyme activity on the total enzyme concentration. The steady-state velocities were obtained from the corresponding progress curves.
Case 2: b , 0, aM . aD When b ,0, a M .a D , the equilibrium 2M T º D T shifts from less active dimer to more active monomer. Since the enzyme after starting the reaction converts from a state of low activity to an active one of high activity, the time-dependent curve of product formation has a lag period, and a plot of a versus [E] 0 shows a monotonically decreasing curve (Fig. 2). Case 3: b . 0, aM , aD When b .0, a M ,a D , the equilibrium 2M T º D T shifts from less active monomer to more active dimer, and the rate of enzymatic reaction is enhanced during the process. Therefore, the progress curve of product formation has a initial lag phase, and a monotonically increases with enzyme concentration, [E] 0 (Fig. 3). Case 4: b . 0, aM . aD
FIG. 1. Schematic plot of product formation, [P], versus reaction time, t, for a dissociating–associating enzyme reaction (Case 1). The two dashed lines represent the initial velocity and the steady-state velocity, respectively. The progress curves were generated by using Eq. [21] with the following kinetic parameters: k 2 50.1 s21, K S 5 131024 M, k*251.0 s21, K *S5531024 M, k100513105 M21 s21, k 11050.5310 5 M21 s21, k 11150.2310 5 M21 s21, k 20050.05 s21, h 50.1. Substrate concentration is 1 3 1023 M, and enzyme concentrations for curves 1–3 are 10 3 1027, 5 3 1027, and 2 3 1027 M, respectively. Inset: dependence of the specific enzyme activity on the total enzyme concentration. The steady-state velocities were obtained from the corresponding progress curves.
In this case, the equilibrium 2M T º D T shifts from more active monomer to less active dimer, and therefore the progress curve of product formation will show an initial burst phase, and a monotonically decreases with increasing enzyme concentration, [E] 0 (Fig. 4). In summary, a shift of equilibrium between monomeric and dimeric forms can be induced by substrates or effectors. If the enzyme converts from a less active form to a more active form during the reaction process, the progress curve of product formation will give rise to
12
ZHI-XIN WANG
an initial lag phase. Otherwise, the time-dependent progress curve will have an initial burst period. On the other hand, because the concentration of enzyme influences the equilibrium between monomer and dimer, and dilution will change the ratio of these two species, the specific activity will depend on enzyme concentration provided that the specific activity of monomer (a M ) and dimer (a D ) differ. These later might be equivalent to specific activities extrapolated to infinitely low and high enzyme concentrations, respectively. Therefore, in light of these two features, it is often possible to distinguish the different reaction mechanisms given above. DATA ANALYSIS
It can be seen from Eq. [20] that any particular progress curve contains five independent parameters, [M T ] 0 , a M , a D , A, and B. In theory, the fitting of data from a single progress curve to Eq. [20] will yield values for [M T ] 0 , a M , a D , A, and B. If the values for these parameters are obtained at different substrate concentrations, all the microscopic rate constants can then be determined from the expressions of these parameters. In practice, however, Eq. [20] is quite complex and, due to the fact that experimental data always contain experimental uncertainties, may be difficult to treat. With a set of unfortunate starting values, the fitting procedure will converge to the wrong values or perhaps
FIG. 3. Schematic plot of product formation, [P], versus reaction time, t, for a dissociating–associating enzyme reaction (Case 3). The two dashed lines represent the initial velocity and the steady-state velocity, respectively. The progress curves were generated by using Eq. [21] with the following kinetic parameters: k250.1 s21, KS5531024 M, s21, K *S5131024 M, k100513105 M21 s21, k*251.0 k110523105 M21 s21, k111553105 M21 s21, k20050.05 s21, h50.9. Substrate concentration is 1 3 1023 M, and enzyme concentrations for curves 1–3 are 2 3 1027, 1 3 1027, and 0.5 3 1027 M, respectively. Inset: dependence of the specific enzyme activity on the total enzyme concentration. The steady-state velocities were obtained from the corresponding progress curves.
FIG. 4. Schematic plot of product formation, [P], versus reaction time, t, for a dissociating–associating enzyme reaction (Case 4). The two dashed lines represent the initial velocity and the steady-state velocity, respectively. The progress curves were generated by using Eq. [21] with the same kinetic parameters as Fig. 3 except that k 2 51.0 s21 and k *2 50.1 s21. Substrate concentration is 1 3 1023 M, and enzyme concentrations for curves 1–3 are 2 3 1027, 1 3 1027, and 0.5 3 1027 M, respectively. Inset: dependence of the specific enzyme activity on the total enzyme concentration. The steady-state velocities were obtained from the corresponding progress curves.
enter a nonconvergent cycle. Therefore, it is necessary to develop some new approaches to overcome this difficulty. To determine values of the kinetic constants for the dimerization reaction, the first experiments would be to measure initial velocity over a wide range of enzyme concentrations. In general, it is best to perform the kinetic experiments at saturating levels of substrates. Under these conditions, a change in specific activity as a function of enzyme concentration reflects different intrinsic activities for different molecular weight species. Let us assume that the activity assay is performed in the following manner. An enzyme solution (of a certain concentration of enzyme) is preincubated until the equilibrium between the dimer and monomer is reached. The concentrations of monomer and dimer prior to the substrate addition are given by (27) @M#9 5
1 $2K 0 1 ÎK 20 1 8K 0@E#90%, 4
@D#9 5
1 $@E#90 2 @M#9% 2
1 1 5 @E#90 2 $2K0 1 ÎK20 1 8K0 @E#90 %, 2 8
[22]
[23]
where K 05k 200/k 100 is the dissociation constant for the monomer– dimer reaction and [E]90 the total molar
13
DETERMINING KINETIC PARAMETERS OF DISSOCIATING ENZYME SYSTEMS
concentration of enzyme in the absence of substrate. When the assay reaction is started by mixing substrate with the preincubated enzyme solution, at the beginning instant, the concentrations of each enzyme species in the assay system are @M T# 0 5
V9@M#9 5 h @M#9, V9 1 V c
@D T# 0 5
V9@D#9 5 h @D#9, V9 1 V c
@E# 0 5
V9@E#90 5 h @E#90, V9 1 V c
[24] and
[25] [26]
where h5V9/(V91Vc), and V9 and Vc are the volumes of preincubated enzyme solution and substrate solution, respectively. Substitution of Eqs. [22]–[26] into Eq. [19] yields
tration. In this case, the initial specific activity, a 0 , as a function of substrate concentration, could be represented as a0 5
where f 0 5(1/4[E] 0 ) {2hK 0 1 =( h K 0 ) 2 18 h K 0 [E] 0 }, and therefore is a constant at given enzyme concentration. This equation is analogous to the case of two enzymes acting on the same substrate, and would be expected to show kinetic behavior equivalent to negative cooperativity (32). It can be seen from Eq. [20] that when t approaches infinity, the asymptote of this equation is given by @P# 5 n St 1 g ln
5 ~aM 2 aD!h@M#9 1 aD@E#0 1 5 ~aM 2 aD !$2hK0 1 Î~hK0 !2 1 8hK0 @E#0 % 1 aD @E#0 . 4 [27]
n S 5 a D@E# 0 1
1 , 12b
[31]
H
@P#* 5 [28]
where a05n0/[E]0. Therefore, if one measures the initial velocities at various enzyme concentrations, and determine the values of a M and a D by extrapolating a 0 to infinitely low and high enzyme concentrations, the dissociation constant for the dimerization reaction in the absence of substrate, K 0 , can then be calculated from Eq. [28]. A rather detailed description for determining the value of K 0 has been formulated by Kurganov (27). Experimentally, it is often difficult to determine the initial velocities accurately from the linear parts of the progress curves. In these cases, the initial velocities can be estimated by fitting the initial parts of the reaction progress curves to second- or third-order polynomial equations in product concentration (28 –30). [29]
The initial velocity, (d[P]/dt) t50, was taken to be a measure of 1/c 1 (31). In order to study effects of substrate on the kinetic behavior of the dimerization reaction, the second group of experiment should be performed at various substrate concentrations and a fixed low enzyme concen-
~a M 2 a D! 4
B 3 2 1 A
Equation [27] can be rewritten as
t 5 c 0 1 c 1@P# 1 c 2@P# 2 1 c 3@P# 3 1 · · ·
[30]
and the slope and the intercept of the asymptote (steadystate velocity) are
n0 5 aM@MT#0 1 aD$@E#0 2 @MT#0%
2@E# 0~a 0 2 a D! 2 , K0 5 h ~a M 2 a D!~a M 2 a 0!
k 2@S#f 0 k *2@S#~1 2 f 0! 1 , K S 1 @S# K *S 1 @S#
ÎS D B A
2
18
J
B @E# 0 , A
and
[32]
aD 2 aM aD 2 aM ln~1 2 b ! 5 2A 2A 3 ln
2 Î~B/A! 2 1 8~B/A!@E# 0
4@M T# 0 1 ~B/A! 1 Î~B/A! 2 1 8~B/A!@E# 0
.
[33] Equation [32] can be rewritten as 2~ n S 2 a D@E# 0! 2 B 5 A ~a M@E# 0 2 n S!~a M 2 a D!
[34]
B 2@E# 0~a 2 a D! 2 5 . A ~a M 2 a!~a M 2 a D!
[35]
or
Substituting Eq. [35] into [33] yields A5
aD 2 aM 2@P#* 3 ln
~a 2 a D!@~a M 2 a D! 1 ~a M 2 a!# . 2@~a M 2 a!a 0 1 a M~a 2 a D! 1 a D~a D 2 a M!# [36]
14
ZHI-XIN WANG
Equations [35] and [36] allow experimental evaluation of B/A and A as a function of substrate concentration from the progress curves, providing the values of a M and a D at different concentrations of substrate are known. These equations are simplified, of course, if either a M or a D is equal to 0. As pointed out by Kurganov et al. (26), according to the conditions of detailed balance, we have k 210 K 0 K *S 5 , k 110 2 KS
S D
K *S k 211 5 K0 k 111 KS
2
.
[37]
Thus, for determining the values of all unknown kinetic parameters, the following calculations are performed: (i) Measure the time course of product formation as a function of the total enzyme concentration at a fixed high substrate level (so that it remains relatively constant during the activity measurement), and determine the values of n0 and n S from the progress curve of product–time at each particular enzyme concentration (Figs. 1– 4). (ii) Diagnose the reaction mechanism for the selfassociation process according to the pattern of the progress curves and the relationship between a and [E] 0 (insets of Figs. 1– 4). (iii) Determine the values of a M and a D by extrapolating a 0 5 n 0 /[E] 0 to infinitely low and high enzyme concentrations, and then calculate the K 0 according to Eq. [28]. (iv) Measure the progress curves of product formation at various substrate concentrations and a fixed low enzyme concentration (so that the substrate depletion can be neglected during the activity measurement), and determine the values of n0, n S , and [P]* from the progress curve of product–time at each particular concentration of substrate (Fig. 5). (v) Determine the values of k 2 , K S , k *2 , and K *S by fitting Eq. [30] to the experimental data obtained at different concentrations of substrate, {(n0)i , [S]i } (as h , [E] 0 and K 0 are known, inset of Fig. 5). (vi) Calculate the values of a M and a D at each particular concentration of substrate according to the following equations: aM 5
k 2@S# , K S 1 @S#
aD 5
k *2@S# . K *S 1 @S#
(vii) Calculate the values of B/A at different concentrations of substrate according to Eq. [35]. (viii) Calculate the values of A at different concentrations of substrate according to Eq. [36]. (ix) Determine the values of k 100, k 110, and k 111 by fitting Eq. [11] to the experimental data obtained at different substrate concentrations, {A i , [S] i }.
FIG. 5. Schematic plot of product formation, [P], versus reaction time, t, for a dissociating–associating enzyme reaction at different substrate concentrations. The two dashed lines represent the initial velocity and the steady-state velocity, respectively. The progress curves were generated by using Eq. [21] with the same kinetic parameters as Fig. 1. Enzyme concentration is 10 3 1027 M, and substrate concentrations for curves 1–3 are 8 3 1023, 6 3 1024, and 2 3 1024 M, respectively. Inset: dependence of the initial velocity on the substrate concentration. The initial velocities were obtained from the corresponding progress curves.
(x) Calculate the values of k 200, k 210, and k 211 according to Eq. [37]. Figure 6 shows a schematic representation of plots of A and B versus substrate concentration. It should be indicated that if k 2 (or k *2 ) is equal to 0, K S (or K *S ) cannot be determined by step (v) since the expression of n0 will contain only the K *S (or K S ) term. For example, when k 2 50 (a M 50), Eqs. [30], [35], and [36] can be simplified to a0 5
k *2@S#~1 2 f 0! 5 a D~1 2 f 0!, K *S 1 @S#
B 2@E# 0~a 2 a D! 2 , 5 A aa D A5
[38]
and
[39]
a 2D 2 a 2 aD . ln 2@P#* 2~a 2D 2 aa 0!
[40]
Substituting Eq. [37] into the expressions of A and B (Eqs. [11] and [12]) yields
H
B K *S~K S 1 @S#! 5 K0 A K S~K *S 1 @S#!
J
2
.
[41]
Therefore, instead of using steps (v)–(viii), the values of K S , K *S , B/A, and A can be determined as follows. (i) Calculate the values of aD at different substrate
15
DETERMINING KINETIC PARAMETERS OF DISSOCIATING ENZYME SYSTEMS
FIG. 6. Dependence of the apparent rate constants A and B on the substrate concentration. The curves were generated by using Eqs. [11] and [12] with the following kinetic parameters: KS5131024 M, K *S5531024 M, k100513105 M21 s21, k11050.53105 M21 s21, k11150.23105 M21 s21, k20050.05 s21.
concentrations, {(aD)i, [S]i}, according to aD5a0/(12f0) (as h, [E]0, and K0 are known). (ii) Determine the values of k *2 and K *S by fitting the expression of a D to the experimental data obtained at different concentrations of substrate. (iii) Calculate the values of B/A at different concentrations of substrate according to Eq. [39]. (iv) Determine the value of K S by fitting Eq. [41] to the experimental data obtained at different substrate concentrations, {(B/A) i , [S] i }. (v) Calculate the values of A at different concentrations of substrate according to Eq. [40]. The procedures given above for determining the kinetic parameters can be easily extended to situations of enzyme reactions involving two substrates. In these cases, however, the concentrations of both substrates have to be varied to obtain all kinetic parameters. It is important to emphasize the following consideration. In slowly dissociating enzyme systems (in which the dimer–tetramer interconversion is very slow in comparison with the rate of enzymatic reaction), the dissociation constant K0, deter-
mined from plots of a0 against [E]0, is appropriate to conditions under which the enzyme is preincubated before its activity is measured, and the dissociation constant B/A, determined from the steady-state velocity, nS, according to Eq. [35], is appropriate to the conditions under which the reaction takes place. When K0,B/A, the presence of substrate shifts monomer–dimer equilibrium toward formation of monomer, and when K0.B/A, the effect of substrate is such that the monomer– dimer equilibrium is shifted toward the formation of dimer. An example of the dissociating enzyme system is human immunodeficiency virus 1 protease (HIV-PR).1 HIV-PR is an aspartyl protease composed of two identical protomers. The time course of product formation for the enzyme-catalyzed reaction shows an initial burst phase (33). Following the initial loss of enzyme activity, a steady-state or equilibrium level of activity was obtained. No further loss of activity was observed over a period of 30 min. The effect of initial enzyme concentration on properties of the time-dependent loss of activity was determined. At enzyme levels of 6.2–50 nM, the specific activity, nS/[E]0, monotonically increased with increasing enzyme concentration. This result indicates that the progress curves for the loss of enzymatic activity are consistent with dissociation of the active HIV-PR dimer into inactive monomeric subunits. When the monomer contains several subunits and therefore several active sites, the association process between a monomer with i substrates and another with j substrates can be expressed as k1ij
MS i 1 MS j | 0 DS i1j. k2ij
If substrate binding is rapid relative to the catalytic reaction, and the binding sites are equivalent and independent in the monomer and in the dimer, we then have (26)
H
B k 200 K *S~K S 1 @S#! 5 A k 100 K S~K *S 1 @S#!
S D
k 2ij K *S 5 r ij K 0 k 1ij KS
J
2n
,
and
[42]
i1j
,
[43]
where r ij 51 at i5j, r ij 5 21 at iÞj, and n is the number of the total binding sites in the monomer. Thus, the apparent rate constants, A and B, can be written as
O O r k S niDS njDS @S# K D n
n
ij
A5
i50 j50
S
1ij
11
@S# KS
D
i1j
S
2n
,
and
[44]
1 Abbreviation used: HIV-PR, human immunodeficiency virus 1 protease.
16
ZHI-XIN WANG
phorylase a into active dimer. When phosphorylase a was preincubated with 0.5 mg/ml glycogen, the lag phase in the progress curve was completely abolished, and linear progress curves were obtained. Such linearity indicates that (i) the coupling enzyme is in sufficient quantity to ensure actual measurement of the primary enzyme’s behavior, and (ii) the substrate consumption can be neglected and the initial rate conditions are satisfied during activity measurement. At the enzyme concentrations of 0.29 to 1.74 nM, both the relaxation time, t, and specific activity, a5 n S /[E] 0 , are independent on enzyme levels, suggesting that the enzyme concentration in the assay system is far less than the corresponding apparent dissociation constant. It can be seen from Eq. [20] that when [E] 0 ! B/A, we have FIG. 7. Time course of the phosphorylase a-catalyzed reaction. The final enzyme concentrations (dimer) for curves 1, 2, 3, 4, 5, and 6 were 1.74, 1.45, 1.16, 0.87, 0.58, and 0.29 nM, respectively. The assay system contains 50 mM Bis-Tris, 50 mM KCl, 1 mM DTT, 0.1 mM EDTA, 2 mM MgAc, 2% glycerol, 97 mM 7-methyl-6-thioguanosine, 148 mg/ml purine nucleoside phosphorylase, 1.2% dimethyl sulfoxide, 0.5 mg/ml glycogen, and 4 mM glucose-1-P at pH 7.0, 25°C. The reactions were initiated by the addition of the preincubated phosphorylase a (4.69 mM dimer) into the assay system. In the control experiment (curve 0), 1.74 nM phosphorylase a was preincubated in 1.604 ml of assay solution except that glucose-1-P was absent for 20 min, and then 16 ml of glucose-1-P solution (0.4 M) was added to initiate the reaction.
O O k S niDS njDS @S# K *D n
n
2ij
B5
i50 j50
S
11
@S# K *S
D
i1j
S
2n
,
[45]
and k 2ij 5k 2ji , k 1ij 5k 1ji . According to the procedure mentioned above, once the values of A and B at different substrate concentrations are obtained, the microscopic rate constants k 1ij and k 2ij can then be determined by fitting Eqs. [44] and [45] to the experimental data. As an example, the experimental data for the substrate reaction kinetics of glycogen phosphorylase a is analyzed using the present method. A detailed kinetic study of this enzyme will be published elsewhere. Glycogen phosphorylase a from rabbit muscle is a tetrameric molecule consisting of four identical subunits. Enzyme activity was determined in the direction of glycogen synthesis by the method of Sergienko and Srivastava (34). Figure 7 shows the time courses of enzyme-catalyzed reactions at different enzyme concentrations. Before attaining the steady state the reaction showed a lag phase in the product formation. The increase of enzyme activity is connected with the reversible dissociation of inactive tetramer of phos-
@P# 5 n St 1 g ln 5 a D@E# 0t 1 5 n St 1
1 2 b e 2at 12b aD ~@D T# 0 2 @E# 0!~1 2 e 2Bt! B
n0 2 nS ~1 2 e 2Bt!, B
[46]
where n S 5a D [E] 0 and n 0 5a D [D T ] 0 . Therefore, at low enzyme concentrations, the rate of tetramer formation for the process 2D T 3T T is negligible, and the reaction can then be regarded as a simple irreversible dissociation process. In further studies, the effect of substrate on properties of the time-dependent increase of activity was determined. From the progress curves of product formation, n0, n S , and B can be determined by fitting Eq. [46] to the experimental data. A plot of B against glycogen concentration shows that the apparent dissociation rate constant is not affected by the glycogen concentration (Fig. 8A). The average value of the apparent dissociation rate constant is B57.963102361.4431024 s21. Figure 8B shows the dependence of the steadystate rate on the glycogen concentration. From these data, the values of K m and V max for glycogen were determined to be K m 5 0.360 6 0.029 mg/ml, V max 5 2.0 6 0.062 mM/min. Similar experiments at another fixed concentration of glucose-1-phosphate (6.17 mM) were also performed to illustrate the influence of the glucose-1-phosphate concentration on the apparent Michaelis constants (data not shown). The values of K m and V max so determined were K m 5 0.314 6 0.042 mg/ml and V max 5 2.85 6 0.143 mM/min. These results suggest that the binding of glycogen at one active site does not influence either the binding of glucose-1phosphate on the same subunit or the binding of either glycogen or glucose-1-phosphate on a different subunit. Since the present kinetic study can give rise only to information on effects of the substrates on the appar-
17
DETERMINING KINETIC PARAMETERS OF DISSOCIATING ENZYME SYSTEMS
where B/A is the apparent dissociation constant for the dimer–tetramer reaction of phosphorylase a in the presence of one substrate, and [E]90 is the total enzyme concentration in the preincubating system. Given that only the dimeric form of phosphorylase a is catalytically active, the initial velocity and the final velocity will be proportional to the initial and final concentrations of the dimeric form in the assay system, respectively. Therefore, when the reaction is started by diluting the preincubated mixture of enzyme–substrate into the assay system, the ratio of initial velocity to final velocity is given by f5
n 0 @D T# 0 @D T#9 5 5 nS @E# 0 @E#90
H
1 B 5 2 1 4@E#90 A
ÎS D B A
2
1 8@E#90
J
B , A
[48]
where [D T ] 0 and [E] 0 are the concentrations of dimer and total enzyme concentration in the assay system at t50, respectively. Equation [48] can be rearranged as B 2@E#90 f 2 5 . A ~1 2 f !
FIG. 8. (A) Dependence of the apparent dissociation rate constant on the concentration of glycogen. (B) Dependence of the steady-state rate on the concentration of glycogen. The solid line is the theoretical curve calculated according to the Michaelis–Menten equation with V max 5 2.0 mM min21, K m 5 0.360 mg/ml. The assay system contains 50 mM Bis-Tris, 50 mM KCl, 1 mM DTT, 0.1 mM EDTA, 2 mM MgAc, 2% glycerol, 95 mM 7-methyl-6-thioguanosine, 148 mg/ml purine nucleoside phosphorylase, 1.2% dimethyl sulfoxide, 3.08 mM glucose-1-P, and 0.58 nM phosphorylase a (dimer) at pH 7.0, 25°C. The reactions were initiated by the addition of the preincubated phosphorylase a (4.69 mM dimer) into the assay system.
ent dissociation rate constant, experiments other than kinetic studies are required to investigate the influence of substrates on the association process. As mentioned earlier, binding of glucose-1-phosphate to the enzyme has no influence on the binding affinity for glycogen, and therefore a study of the influence of a single substrate on the dimer–tetramer equilibrium will provide useful information. If phosphorylase a is preincubated with one substrate until the dimer–tetramer equilibrium is reached, the concentration of the dimeric phosphorylase a in the preincubating system is given by
@D T#9 5
H
1 B 2 1 4 A
ÎS D B A
2
18
J
B @E#90 , A
[47]
[49]
Preincubation of the enzyme with different concentrations of glucose-1-phosphate have no effect on the initial lag phase in the progress curve. This result indicates that the presence of glucose-1-phosphate does not shift the equilibrium between the dimeric form and the tetrameric form of phosphorylase a. When the enzyme was preincubated with different concentrations of glycogen, the lag phase in the progress curves was reduced. Similarly, the values of the initial and steady-state velocities of the phosphorylase a-catalyzed reaction, n0 and nS, were determined from the progress curves of the phosphorylase a reaction, and then the apparent dissociation constant, B/A, can be calculated according to Eq. [49]. Figure 9 shows the dependence of B/A on the concentration of glycogen. The fact that B/A increases with glycogen concentration indicates that glycogen binds preferentially to the dimer form of phosphorylase a. As mentioned above, because neither the dimer–tetramer equilibrium nor the binding of glycogen to enzyme is affected by glucose-1phosphate, it is reasonable to assume that the apparent dissociation constant, B/A, obtained in the absence of glucose-1-phosphate is equivalent to that in the assay system which contains a constant amount of glucose-1phosphate. Since the values of KS (50.36 mg/ml) and B5k200 (57.96 3 1023 s21) have been determined previously, and the concentration of enzyme is known [E]90 5 5.33 3 1026 M in the present case), k100 and K *S can then be determined by fitting Eq. [42] to the experimental data. The solid line in Fig. 9 represents the best fit of the experimental results with parameters k100 5 1.26 3 104
18
ZHI-XIN WANG
FIG. 9. Dependence of the apparent dissociation constant, B/A on the concentration of glycogen in the preincubation mixture. The values of B/A were calculated according to Eq. [49]. The solid line is the bestfitting result according to Eq. [42] with KS 5 0.36 mg/ml, k20057.9631023 s21, K *s51.67 mg/ml, and k10051.263104 M21 s21. Phosphorylase a (5.53 mM dimer) was preincubated at 25°C for at least 2 h in Bis-Tris buffer containing various concentrations of glycogen; 0.4 ml enzyme– glycogen mixture was then diluted into assay system (1.62 ml), and the full-time reaction curve were recorded. The assay system contains 50 mM Bis-Tris, 50 mM KCl, 1 mM DTT, 0.1 mM EDTA, 2 mM MgAc, 2% glycerol, 110 mM 7methyl-6-thioguanosine, 148 mg/ml purine nucleoside phosphorylase, 1.2% dimethyl sulfoxide, 0.5 mg/ml glycogen, 15 mM glucose1-P, and 1.36 nM phosphorylase a (dimer) at pH 7.0, 25°C.
6 7.87 3 102 M21 s21 and K *S 5 1.67 6 0.08 mg/ml. From these values, all the microscopic rate constants k1ij and k2ij can be obtained according to Eq. [43].
0.3 to 3 s. Thus, at an enzyme level of 1025 M or higher, the transition time may be too short to conveniently measure, whereas if both B/A and the enzyme concentration were as low as 1027 M, the self-association reaction could be a rather slow process. As mentioned previously, there are three basic mechanisms for describing hysteretic enzyme systems: the slow conformational change model, the ligand displacement model, and the enzyme dissociation–association model. With these different formulations and viewpoints of similar systems, it is necessary that the similarities and true difference between various models be pointed out. We will attempt to present here a general, unified picture of hysteretic enzymes. The following three common features are generally agreed to refer to hysteretic enzyme systems: (i) enzyme may exist in two interconvertible states with different kinetic properties, (ii) the interconversion between these two enzyme states is slow in comparison with establishment of the steady-state of enzymatic reaction, and (iii) the substrate can shift the equilibrium between the two states. In the presence of substrate, the schemes for describing hysteretic enzyme systems and the corresponding expressions of product formation can be written as shown in Schemes 2– 4. A. Slow Conformational Change Model (4) This mechanism (Scheme 2) assumes that the enzyme may exist in two conformational states that differ in their catalytic or binding properties, and that the rate of interconversion of conformational states is slow in comparison with the rate of enzymic reaction. The product formation at time t is given by
DISCUSSION
With the analytical expression of product formation as a function of time, an estimation of reasonable transition time can be made for the case where association is diffusion-controlled. It can be seen from Eq. [16] that the transition time can be approximately written as 1 1 t< 5 . a A Î~B/A! 2 1 8~B/A!@E# 0
[50]
As pointed out by Frieden (18), values for association constant, A, are normally in the range of 105 to 106 M21 s21. Thus, values of t will depend on the magnitudes of the apparent dissociation constant and the total enzyme concentration. A decrease in either B/A or [E]0 will cause an increase in the transition time. Since the greatest change of kinetic behaviors for a self-association system occurs in the region of the B/A, it is frequently satisfactory to vary the concentration from about 0.2 to 5.0 times this constant. For example, if B/A and [E]0 are assumed each to equal 1026 M, then the transition time is equal to
@P# 5 n St 1
n0 2 nS $1 2 e 2~A1B!t%, A1B
where
A5
k 10K S 1 k910@S# , K S 1 @S#
B5
SCHEME 2
k 20K *S 1 k920@S# . K *S 1 @S#
[51]
19
DETERMINING KINETIC PARAMETERS OF DISSOCIATING ENZYME SYSTEMS
If the enzyme exists in the E form only before substrate addition (that is, [E9]50 at t50), the initial velocity, n0, and the steady-state velocity, n S , are given by
n0 5
k 2@E# 0@S# , K S 1 @S#
nS 5
k 2@E# 0@S#B . ~K S 1 @S#!~A 1 B!
and
B. Ligand Displacement Model (4, 17) If an inhibitor, I, is bound to an enzyme surface quite tightly, and the dissociation rate of the inhibitor from the enzyme is relatively slow, then when the tightly bound inhibitor is displaced by an activator, L, the rate of increase in enzymatic activity could be measurable as a function of time. This mechanism would be represented by Scheme 3. Assuming that the binding steps of substrate and activator are equilibrated, and [S] 0 , [L] 0 @[E] 0 , the expression of product formation is then given by
n0 1 2 b e 2 at @P# 5 n St 1 ln , A@E# 0 12b
SCHEME 4
nS 5
H
3 @E# 0 2 @I# 0 2 1
[52]
a 5 Î~A@I# 0 1 A@E# 0 1 B! 2 4A @I# 0@E# 0, 2
@I# 0 1 @E# 0 1 ~B/A! 2 Î$@I# 0 1 @E# 0 1 ~B/A!% 2 2 4@E# 0@I# 0 b5 , @I# 0 1 @E# 0 1 ~B/A! 1 Î$@I# 0 1 @E# 0 1 ~B/A!% 2 2 4@E# 0@I# 0
n0 5
$k 2K9L 1 k92@L#%@S#@E# 0 , K9SK L 1 K9S@L# 1 K9L@S# 1 @S#@L#
ÎF
B A
@E# 0 1 @I# 0 1
S DG B A
2
J
2 4@E# 0@I# 0 ,
k 10K9SK L 1 k910K9L@S# , K9SK L 1 K9S@L# 1 K9L@S# 1 @S#@L#
B5
k 20K *S 1 k920@S# , K *S 1 @S#
and
and where the condition [EI]5[E] 0 at t50 has been assumed. C. Enzyme–Protein Interaction Model (Scheme 4) There are a number of examples of interaction of enzymes with other proteins. For these types of interaction systems, the protein effector may be considered simply as a direct analog of a small ligand effector. Since the rate of association of two proteins will be slower than that of an enzyme with a small ligand, one might expect to see hysteretic behavior. During the last two decades, the kinetic behaviors of slow binding and slow-tight binding modifier have been studied systematically (7, 10, 17). Thus, if the kinetic data of the enzyme–protein interaction can be obtained, treatment of the data would be similar to that of enzyme–small ligand interactions. When [M] 0 @ [E] 0 , @P# 5 n St 1
SCHEME 3
SD
A5
where 2
~k 2K9L 1 k92@L#!@S# 2~K9SK L 1 K9S@L# 1 K9L@S# 1 @S#@L#!
n0 2 nS $1 2 e 2~A@M#01B!t%, A@M# 0 1 B
A5
k 10K S 1 k910@S# , K S 1 @S#
B5
k 20K *S 1 k 20@S# , K *S 1 @S#
20
ZHI-XIN WANG
n0 5
k 2@E# 0@S# , K S 1 @S#
nS 5
k 2@E# 0@S#B . ~K S 1 @S#!~A@M# 0 1 B!
and [53]
When [M] 0 '[E] 0 , @P# 5 n St 1
n0 1 2 b e 2 at ln , A@E# 0 12b
[54]
where
a 5 Î~A@M# 0 1 A@E# 0 1 B! 2 2 4A 2@M# 0@E# 0, @M# 0 1 @E# 0 1 ~B/A! 2 Î$@M# 0 1 @E# 0 1 ~B/A!% 2 2 4@E# 0@M# 0 b5 , @M# 0 1 @E# 0 1 ~B/A! 1 Î$@M# 0 1 @E# 0 1 ~B/A!% 2 2 4@E# 0@M# 0
n0 5
k 2@E# 0@S# , K S 1 @S#
nS 5
B k 2@S# @E# 0 2 @M# 0 2 2~K S 1 @S#! A 1
ÎF
and
H
@E# 0 1 @M# 0 1
S DG B A
SD
2
J
2 4@E# 0@M# 0 .
1 2 b e 2 at 12b
[55]
where
a 5 ÎB 2 1 8AB@E# 0, 4A@M T# 0 1 B 1 ÎB 2 1 8AB@E# 0
The present investigation was supported in part by Grant 39421003 from the China Natural Science Foundation and the Pandeng Project of the National Commission for Science and Technology.
1. 2. 3. 4. 5. 6. 7. 8. 9.
12. 13. 14.
,
15.
aM 2 aD g5 , 2A
n S 5 a D@E# 0 1 ~a M 2 a D!
H
J
2B 1 ÎB 2 1 8AB@E# 0 , 4A
A5
k 100K 1 k 110K S@S# 1 k 111@S# , ~K S 1 @S#! 2
B5
k 200K *S2 1 2k 210K *S@S# 1 k 211@S# 2 , ~K *S 1 @S#! 2
2 S
2
k *2@S# . K *S 1 @S#
ACKNOWLEDGMENTS
11.
4A@M T# 0 1 B 2 ÎB 2 1 8AB@E# 0
aD 5
and
Finally, it should be indicated that in theory, the kinetic behavior of a self-association process should be dependent on the total enzyme concentration. However, at low enzyme concentration, this process may appear to be independent of enzyme concentration and could be misinterpreted to be a consequence of an isomerization process. Therefore, the activity measurements should be made in conjunction with other methods which measure different parameters. Especially, molecular weight measurements as a function of time or time-dependent conformational changes are important in determination of the mechanism.
10.
b5
k 2@S# , K S 1 @S#
REFERENCES
D. Enzyme Dissociation–Association Model @P# 5 n St 1 g ln
aM 5
16. 17. 18. 19. 20. 21. 22. 23.
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