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Enzymic parameters: Measurement of V and Km
BIOCHIMICAET BIOPHYSICAACTA
519
BBA 66392 ENZYMIC P A R A M E T E R S : MEASUREMENT OF V AND Km
H Y U N - J A E LEE AND I R W I N ]3. WILSON
Department of Chemistry, University of Colorado, Boulder, Colo. 80302 (U.S.A.) (Received March 22nd, 1971)
SUMMARY
A slight modification of the Lineweaver-Burk equation (and other variants) in which the form is retained but the instantaneous velocity is replaced by the negative change in substrate concentration divided by the time and the initial substrate concentration is replaced by the arithmetic average substrate concentration during the time interval, allows this linear reciprocal form to be used with little error even when as much as 50% of the substrate is consumed. This result is in sharp contrast to the prevailing belief that the initial velocities and initial substrate concentrations must be used.
The kinetic behavior of the majority of enzymes follows the Michaelis-Menten form : diS]
v-
dt
V I+--
Km
(i)
[s]
where v is the instantaneous velocity at a time t when the substrate concentration is IS]. This same form is predicted by many quite complicated enzyme mechanisms although the physical interpretation of the constants may, in different cases, be very different. The usual way of evaluating V and Km is to form the double reciprocal 1. I
I
v
v + v
~m
1
[S]
(2)
or a variant form2, 3 and measure the velocity at several different initial substrate concentrations. The comparative advantages of different modifications of the double reciprocal have been discussed at some length 4-6 but there has been little or no discussion of the practical experimental meaning of initial velocity. Clearly a velocity calculated on the basis of a change of I or 2% in substrate concentration during the time interval could be taken as an initial velocity, but it is not clear just how large a change in substrate concentration would be acceptable. In many cases it is not practical to limit the change in substrate concentration Biochim. Biophys. Acta, 242 (I97 I) 519-522
520
H . - J . L E E , I. B. W I L S O N
to I or 2% and procedures must be used which involve the integrated form of Eqn. I, namely [So]
Vt = [So] -- [St] + Km I n - -
(3)
[St]
where [So] is the substrate concentration at zero time and [St] the concentration after the time interval t (ref. 7)The gist of this paper is t h a t a simple modification of the double reciprocal form enables t h a t form to be retained even when as much as 50% of the substrate is utilized during the time interval. The best we can do in estimating v is to form the quotient, ([So]--[St])/t = ~. I f [So] -- [St] = A [ S ] is an appreciable part of [So], it is clear t h a t it would be unwise to use either [So] in Eqn. 2 or [St] at the end of the time interval. Some intermediate value is suggested. This intuitive procedure is given mathematical support by the Theorem of the Mean s which states t h a t for IS] = f (t) some value of [S] exists in the interval [So] -- [St] where d[S]/dt does equal (IS0] -- [St])/t. I f we could always select this value of IS], or a sufficient approximation to it, we should be able to always use Eqn. 2, in an exact manner. We shall show t h a t the arithmetic mean value of [S], IS] = ½ ([So] + [St]) is a good approximation to the correct value of IS] for Eqn. 2, and its use introduces negligible error even when one third of the substrate is utilized in the time interval, and an acceptable error when as much as one half of the substrate is used up. This result is in sharp contrast to the prevailing belief t h a t "initial velocities" must be measured in order to use the double reciprocal equation. The integrated Michaelis-Menten equation, Eqn. 3, can be put in the form I
I
= T
K m
+
V
I
[So]
[St] In ----[St]
[So] --
(4)
The proposed modification of the Lineweaver-Burk equation is i -
i = -
v
Km +
- -
v
w h e r e ~ = ([So] - -
I (5)
.[~]
[St])/t
a n d IS] = ([So] + [St])/2.
The question is "how closely does I / I S ] approximate to I
In -[SO] .?"
[So] - [st]
[s,]
Let IS] [So]
-
I n - -[So] -= I +a
[st]
where a is the relative error involved in using I
[So] -
[Sd
(6)
[st]
in--
I/IS]
rather than
[So] [St]
Substitution from Eqn. 6 into Eqn. 4 yields I
~-v
I
Km
I
+~-(i+a)--[,q]
Biochim. Biophys. Acta, 242 (1971) 5 1 9 - 5 2 2
(7)
MEASUREMENT OF V AND K m
52I
Values of a were c a l c u l a t e d from Eqn. 6 for different e x t e n t s of r e a c t i o n in percent,
[~- [st]] x~oo [So]J a n d t h e results are p l o t t e d in Fig. I. T h e calculation indicates t h a t the r e l a t i v e error, a, is less t h a n I % even when t h e r e a c t i o n has gone to t h e e x t e n t of 30%. The rel a t i v e error is 2 % at 4 0 % e x t e n t of r e a c t i o n a n d < 4 % at 50% e x t e n t of reaction. Eqn. 7 which is e x a c t i n d i c a t e s t h a t if t h e a p p r o x i m a t e Eqn. 5 is used, the value o b t a i n e d for Km will be high b y t h e error a. T h u s if the e x t e n t of the reaction is 5 o % , Km will be high b y 4 % . T h e c o m m o n p r a c t i c e is to use Eqn. 2 w i t h v = ~[S]/t a n d [S] = [So]. W h e t h e r this is good practice d e p e n d s u p o n how closely i/IS0] a p p r o x i m a t e s I/[S0] - - [St] In [So]/[St], the a p p r o x i m a t i o n is poor! E v e n when the reaction has p r o c e e d e d to o n l y lO% t h e error in using [So] a l r e a d y exceeds 5 % (Fig. I). These results i n d i c a t e
7f5 ' ,/Cso]'
'
'
nr"
0
~
~0
20 PERCENT
i
i
so
40
i
50
i
eo
REACTION
Fig. I. Percent errors at different extents of reaction. Percent errors, Iooa, were calculated from Eqn. 6 for different extents of reaction,
[ I - [St]] X IOO [So] J
t h a t the use of Eqn. 2 w i t h initial velocities e s t i m a t e d b y A[S]/t a n d the initial subs t r a t e c o n c e n t r a t i o n s m u s t be r e s t r i c t e d to e x t e n t s of r e a c t i o n t h a t are less t h a n lO% a n d p r e f e r a b l y less t h a n 5%. The use of average s u b s t r a t e c o n c e n t r a t i o n s enables the i n v e s t i g a t o r to r e t a i n the L i n e w e a v e r - B u r k form even when q u i t e s u b s t a n t i a l e x t e n t s of reaction h a v e occurred. I n discussing this question we have a s s u m e d t h a t there is no p r o d u c t inhibition. This is a question t h a t is easily answered e x p e r i m e n t a l l y a n d is in a n y case, one t h a t m u s t be answered before a n y e q u a t i o n can be used. However, it is more l i k e l y to be significant when large e x t e n t s of r e a c t i o n are involved. Similarly,
Biochim. Biophys. Acta, 242 (1971) 519-522
522
H.-J. LEE, I. B. WILSON
the reaction m u s t n o t be significantly reversible. This is required also in the case of the Michaelis-Menten or L i n e w e a v e r - B u r k equation. However, reversibility is more likely to be a significant factor when large e x t e n t s of reaction are involved. I n e x p e r i m e n t a l work with angiotensin converting enzyme, it has not been feasible for us to restrict the e x t e n t of reaction to m u c h less t h a n 4o%. However, identical values for K m (4.2"IO 5 M) were o b t a i n e d whether Eqn. 4 or Eqn. 5 was usedg, 10. Our conclusion t h a t the L i n e w e a v e r - B u r k plot can be used for relatively large e x t e n t s of reaction, if the average value of ES], ES] is used, extends to all the v a r i a n t s of this plot including those for reactions occurring in the presence of inhibitors. ACKNOWLEDGMENT This work was supported b y G r a n t H E - I I 2 O I , N a t i o n a l I n s t i t u t e s of Health.
REFERENCES
i H. LINEWEAVERAND D. BURK, J. Am. Chem. Soc., 56 (I934) 658. 2 C. S. HANES, Biochem. J., 26 (1932) 14o6. 3 B. H. J. HOFSTEE, J. Biol. Chem., 179 (633) 1949. 4 B. H. J. HOFSTEE, Nature, 184 (1959) 1296. 5 M. DIXON AND E. C. WEBB, Nature, 184 (1959) 1298. 6 P. STUTTSAND I. FRIDOVICH, Science, 149 (1965) 447. 7 V. HENRI, C. R. Acad. Sci. Paris, 135 (19o2) 916. 8 I. S. SOKOLNIKOFF,Adv. Calculus, McGraw Hill, New York and London, 1939. 9 H-J. LEE, J. N. LARUE AND I. ]3. WILSON, Biochim. Biophys. Acta, 235 (1971) 521. IO H-J. LEE, J. N. LARuE AND I. ]3. WILSON, Arch. Biochem. Biophys., 142 (1971) 548. Biochim. Biophys. Acta, 242 (1971) 519-522
519
BBA 66392 ENZYMIC P A R A M E T E R S : MEASUREMENT OF V AND Km
H Y U N - J A E LEE AND I R W I N ]3. WILSON
Department of Chemistry, University of Colorado, Boulder, Colo. 80302 (U.S.A.) (Received March 22nd, 1971)
SUMMARY
A slight modification of the Lineweaver-Burk equation (and other variants) in which the form is retained but the instantaneous velocity is replaced by the negative change in substrate concentration divided by the time and the initial substrate concentration is replaced by the arithmetic average substrate concentration during the time interval, allows this linear reciprocal form to be used with little error even when as much as 50% of the substrate is consumed. This result is in sharp contrast to the prevailing belief that the initial velocities and initial substrate concentrations must be used.
The kinetic behavior of the majority of enzymes follows the Michaelis-Menten form : diS]
v-
dt
V I+--
Km
(i)
[s]
where v is the instantaneous velocity at a time t when the substrate concentration is IS]. This same form is predicted by many quite complicated enzyme mechanisms although the physical interpretation of the constants may, in different cases, be very different. The usual way of evaluating V and Km is to form the double reciprocal 1. I
I
v
v + v
~m
1
[S]
(2)
or a variant form2, 3 and measure the velocity at several different initial substrate concentrations. The comparative advantages of different modifications of the double reciprocal have been discussed at some length 4-6 but there has been little or no discussion of the practical experimental meaning of initial velocity. Clearly a velocity calculated on the basis of a change of I or 2% in substrate concentration during the time interval could be taken as an initial velocity, but it is not clear just how large a change in substrate concentration would be acceptable. In many cases it is not practical to limit the change in substrate concentration Biochim. Biophys. Acta, 242 (I97 I) 519-522
520
H . - J . L E E , I. B. W I L S O N
to I or 2% and procedures must be used which involve the integrated form of Eqn. I, namely [So]
Vt = [So] -- [St] + Km I n - -
(3)
[St]
where [So] is the substrate concentration at zero time and [St] the concentration after the time interval t (ref. 7)The gist of this paper is t h a t a simple modification of the double reciprocal form enables t h a t form to be retained even when as much as 50% of the substrate is utilized during the time interval. The best we can do in estimating v is to form the quotient, ([So]--[St])/t = ~. I f [So] -- [St] = A [ S ] is an appreciable part of [So], it is clear t h a t it would be unwise to use either [So] in Eqn. 2 or [St] at the end of the time interval. Some intermediate value is suggested. This intuitive procedure is given mathematical support by the Theorem of the Mean s which states t h a t for IS] = f (t) some value of [S] exists in the interval [So] -- [St] where d[S]/dt does equal (IS0] -- [St])/t. I f we could always select this value of IS], or a sufficient approximation to it, we should be able to always use Eqn. 2, in an exact manner. We shall show t h a t the arithmetic mean value of [S], IS] = ½ ([So] + [St]) is a good approximation to the correct value of IS] for Eqn. 2, and its use introduces negligible error even when one third of the substrate is utilized in the time interval, and an acceptable error when as much as one half of the substrate is used up. This result is in sharp contrast to the prevailing belief t h a t "initial velocities" must be measured in order to use the double reciprocal equation. The integrated Michaelis-Menten equation, Eqn. 3, can be put in the form I
I
= T
K m
+
V
I
[So]
[St] In ----[St]
[So] --
(4)
The proposed modification of the Lineweaver-Burk equation is i -
i = -
v
Km +
- -
v
w h e r e ~ = ([So] - -
I (5)
.[~]
[St])/t
a n d IS] = ([So] + [St])/2.
The question is "how closely does I / I S ] approximate to I
In -[SO] .?"
[So] - [st]
[s,]
Let IS] [So]
-
I n - -[So] -= I +a
[st]
where a is the relative error involved in using I
[So] -
[Sd
(6)
[st]
in--
I/IS]
rather than
[So] [St]
Substitution from Eqn. 6 into Eqn. 4 yields I
~-v
I
Km
I
+~-(i+a)--[,q]
Biochim. Biophys. Acta, 242 (1971) 5 1 9 - 5 2 2
(7)
MEASUREMENT OF V AND K m
52I
Values of a were c a l c u l a t e d from Eqn. 6 for different e x t e n t s of r e a c t i o n in percent,
[~- [st]] x~oo [So]J a n d t h e results are p l o t t e d in Fig. I. T h e calculation indicates t h a t the r e l a t i v e error, a, is less t h a n I % even when t h e r e a c t i o n has gone to t h e e x t e n t of 30%. The rel a t i v e error is 2 % at 4 0 % e x t e n t of r e a c t i o n a n d < 4 % at 50% e x t e n t of reaction. Eqn. 7 which is e x a c t i n d i c a t e s t h a t if t h e a p p r o x i m a t e Eqn. 5 is used, the value o b t a i n e d for Km will be high b y t h e error a. T h u s if the e x t e n t of the reaction is 5 o % , Km will be high b y 4 % . T h e c o m m o n p r a c t i c e is to use Eqn. 2 w i t h v = ~[S]/t a n d [S] = [So]. W h e t h e r this is good practice d e p e n d s u p o n how closely i/IS0] a p p r o x i m a t e s I/[S0] - - [St] In [So]/[St], the a p p r o x i m a t i o n is poor! E v e n when the reaction has p r o c e e d e d to o n l y lO% t h e error in using [So] a l r e a d y exceeds 5 % (Fig. I). These results i n d i c a t e
7f5 ' ,/Cso]'
'
'
nr"
0
~
~0
20 PERCENT
i
i
so
40
i
50
i
eo
REACTION
Fig. I. Percent errors at different extents of reaction. Percent errors, Iooa, were calculated from Eqn. 6 for different extents of reaction,
[ I - [St]] X IOO [So] J
t h a t the use of Eqn. 2 w i t h initial velocities e s t i m a t e d b y A[S]/t a n d the initial subs t r a t e c o n c e n t r a t i o n s m u s t be r e s t r i c t e d to e x t e n t s of r e a c t i o n t h a t are less t h a n lO% a n d p r e f e r a b l y less t h a n 5%. The use of average s u b s t r a t e c o n c e n t r a t i o n s enables the i n v e s t i g a t o r to r e t a i n the L i n e w e a v e r - B u r k form even when q u i t e s u b s t a n t i a l e x t e n t s of reaction h a v e occurred. I n discussing this question we have a s s u m e d t h a t there is no p r o d u c t inhibition. This is a question t h a t is easily answered e x p e r i m e n t a l l y a n d is in a n y case, one t h a t m u s t be answered before a n y e q u a t i o n can be used. However, it is more l i k e l y to be significant when large e x t e n t s of r e a c t i o n are involved. Similarly,
Biochim. Biophys. Acta, 242 (1971) 519-522
522
H.-J. LEE, I. B. WILSON
the reaction m u s t n o t be significantly reversible. This is required also in the case of the Michaelis-Menten or L i n e w e a v e r - B u r k equation. However, reversibility is more likely to be a significant factor when large e x t e n t s of reaction are involved. I n e x p e r i m e n t a l work with angiotensin converting enzyme, it has not been feasible for us to restrict the e x t e n t of reaction to m u c h less t h a n 4o%. However, identical values for K m (4.2"IO 5 M) were o b t a i n e d whether Eqn. 4 or Eqn. 5 was usedg, 10. Our conclusion t h a t the L i n e w e a v e r - B u r k plot can be used for relatively large e x t e n t s of reaction, if the average value of ES], ES] is used, extends to all the v a r i a n t s of this plot including those for reactions occurring in the presence of inhibitors. ACKNOWLEDGMENT This work was supported b y G r a n t H E - I I 2 O I , N a t i o n a l I n s t i t u t e s of Health.
REFERENCES
i H. LINEWEAVERAND D. BURK, J. Am. Chem. Soc., 56 (I934) 658. 2 C. S. HANES, Biochem. J., 26 (1932) 14o6. 3 B. H. J. HOFSTEE, J. Biol. Chem., 179 (633) 1949. 4 B. H. J. HOFSTEE, Nature, 184 (1959) 1296. 5 M. DIXON AND E. C. WEBB, Nature, 184 (1959) 1298. 6 P. STUTTSAND I. FRIDOVICH, Science, 149 (1965) 447. 7 V. HENRI, C. R. Acad. Sci. Paris, 135 (19o2) 916. 8 I. S. SOKOLNIKOFF,Adv. Calculus, McGraw Hill, New York and London, 1939. 9 H-J. LEE, J. N. LARUE AND I. ]3. WILSON, Biochim. Biophys. Acta, 235 (1971) 521. IO H-J. LEE, J. N. LARuE AND I. ]3. WILSON, Arch. Biochem. Biophys., 142 (1971) 548. Biochim. Biophys. Acta, 242 (1971) 519-522