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The calculation of reaction-velocity constants for enzyme reactions from the kinetics of the disappearance of enzyme-substrate compounds
ARCHIVES
OF
BIOCHEMISTRY
AND
BIOPHYSICS
62, 284-292 (1956)
The Calculation of Reaction-Velocity Constants for Enzyme Reactions from the Kinetics of the Disappearance of Enzyme-Substrate Compounds’ David S. Greenstein From the Johnson
Research Foundation, University Philadelphia, Pennsylvania Received
October
of PennsyEvania,
10, 1955
INTRODUCTION Experimental studies have furnished numerous examples of enzymesubstrate compounds which disappear by reactions which are pseudofirst order with respect to the concentration of hydrogen donor. ES+AH”4E+SH+A
(1)
For example, Chance has shown that the disappearance of catalase hydrogen peroxide is pseudo-first order with respect to the concentration of ethanol (1). In such cases it seems reasonable to identify the pseudofirst-order constant with kca , where k4 is the rate constant for the combination of the intermediate complex with the donor and a is the concentration of donor. This is equivalent to assuming that the reaction of the complex with donor is the only pathway for the disappearance of the complex. This assumption is reasonable when experiments show a linear variation of the first-order constant with a over a wide range. However, in the case of catalase, experiments by Chance show that at high concentrations of alcohol, this is definitely not the case (2). One explanation which has been considered for this nonlinear behavior is the formation of an enzyme-substrate-donor complex, which has not yet been observed experimentally. Chance (2) postulated that the nonlinearity might be explained without any such modification of the reaction mechanism. His explanation (which is justified by our results) is that Hz02 not bound to catalase acts as a hydrogen donor and provides a second pathway for the disappearance of the complex. The type of mathematical analysis 1 This research
was supported
in part by the Office of Naval 2a4
Research.
REACTION-VELOCITY
CONSTANTS FOR ENZYME REACTIONS
285
used by Chance, however, leads only to a qualitative explanation of the nonlinear behavior. In the present paper, a mathematical approach is presented which quantitatively explains these deviations, shows when the first-order constant equals Ic(a, and shows how k1 , the rate constant for the combination of enzyme with substrate, can be measured from the kinetics of the disappearance of the enzyme-substrate compound. This approach is general enough to apply not only to enzyme-substrate compounds which follow Eq. (l), but also to the classical Michaelis-Menten enzyme-substrate compounds, which follow the relation. ES&E+P METHOD
(2)
FOR THE CATALASE SYSTEM
Let us first consider the case of catalase, for which the following reaction scheme and differential equations have been found to hold (3): catalase + HzOz 4 (e--P>
x
catalase HzOz ’
catalase HzOz + Hz02 kr’ catalase HzOz + RCHzOHo kr
(3)
P catalase + 2Hz0 + O2
(4)
catalase + RCOH
(5)
+ 2HZ0
(Unless otherwise noted, RCHzOH is added in sufficient excess to keep its concentration approximately constant.)
dp - = ka(e - p) - lclxp - &Z + k4a)p dt
dx -= dt
-klx(e da -= dt
- p) - k4’xp + hp -
k4ap
(6) (7)
(8)
Experimentally, it has been found that the bulk of the Hz02 has disappeared before the enzyme-substrate complex begins to disappear. Inspection of Eq. (6) suggests that the substrate concentration may then be so low .that the first two terms on the right become negligible and dp,ldt = - (k, + k4a)p, corresponding to the first-order decay of p. This
286
DAVID
S. GREENSTEIN
assumption was, in fact, used in interpreting experimental data, kz having been considered to be the value of the first-order constant obtained without added alcohol (a = 0). Some justification for this assumption comes from the analog computer data, which shows that for kz = 0 and low a, the first-order constant is k4a (2). The validity of the assumption that the Hz02 concentration is negligible in the latter stages of the reaction can be checked mathematically. The use of limits is suggested since dp/dt = - (kz + kda)p only if 2 = 0; that is when the reaction has gone to completion.2 Therefore, we define:
If this limit exists, it is the first-order constant for the disappearance of p. If the limit is ever zero, the order of decay is greater than one. Such a case will be considered later. From Eq. (6) we have
‘t!!!=k
-pdt
’
+ka
4
- kle E + kix + klx
Since lim-, dp/dt = lim,,, dx/dt = 0, it follows from Eqs. (6) and (7) that lim z = lim p = 0, and the terms of Eq. (10) involving 2 alone will vanish: = kz + k4a - kle lim E t-m 0p
01)
Equation (11) leads us to investigate limt+, (E) = (E).. Weshall first obtain bounds for 2 . In the last stages of the reaction both substrate 0 and complex are dis:piearing, i.e., dx/dt and dp/dt --j 0. Applying these conditions to Eqs. (6) and (7), we have kzp - k4’xp 6 ks(e - p) 6 (k, + h4 p + k&p
(1%
* In evaluating limits, it is assumed that all the time derivatives of concentrations tend to zero as time approaches infinity. This assumption is physically reasonable; it merely says that the chemical system comes to equilibrium. Another reasonable assumption is that if all the concentrations are initially non-negative, they will not attain negative values at any later time.
REACTION-VELOCITY
CONSTANTS
FOR
ENZYME
REACTIONS
287
Taking out --k~xp and dividing by p, this gives h - k4’x 4- klx f he 5 < k2 -I- k4a + (k4’ + kl)x. and .letting t -j ~0, kze<
b
‘-pm’ 0
’
< k2 + ha
(13)
kle
Another equation for k can be obtained as follows. Adding Eqs. (6) and (7) gives -k4ap - 2ka)xp
(14) (15) 06) (17)
By the rule of L’Hospital:
=
k
Now, combining
(1%
Eqs. (11) and (18) we obtain: k4a
= k2 + k4a - kle : 0 Pm
(19)
which yields k2 -iG= Thi.s quadratic
o
(20)
equation has one positive and one negative root, and
288
DAVID S. GREENSTEIN
> 0. Hence, (;)*=;r*-l+m&] and, substituting k=
(21) into Eq. (18), 2k4a
1 + h +
k4a
+
1
_
he
h
+
ha
he
For low a (k4u << kz), Eq. (22) is well approximated k -
2 >
(22)
kz +4&i
by
k4a
kz
(23)
I+&? If e is large [kle > (kz + kra)], k m k4a , in agreement with the earlier assumption, mentioned above. For large a (k4a > k2) k M kle Equation
(24)
(22) assumes a special form as k2/kle tends to zero: k = k4a; k = kle;
a > kle/k4
(The special form of Eq. (25) depends on the mathematical d(i-zp
(25b) identity
= ) 1 - u I.>
A FURTHER APPLICATION
TO CATALASE
This method can also be used to shed light on another problem in catalase action. It has been pointed out that the catalase HzOz complex would be stable if there were no alcohol present and the combination of catalasewith Ha02 wereirreversible (i.e., if kz and k4a were simultaneously zero). Thus it becomes important to determine whether reversibility of reaction (3) or residual alcohol from the catalase preparation method is responsible for the instability which is observed when no alcohol is added.
REACTION-VELOCITY
CONSTANTS
FOR
ENZYME
REACTIONS
289
We now consider the case
In this case, as t -+ 00, p 3 0. Equations spectively:
(21) and (22) then become, re-
(26) and k=O
(27)
These equations also agree with reactions (5) and (9). Equation (27) implies that the order of the decay reaction is greater than one. We now investigate the second-order rate constant, 1 dP k’ = lim t--m ( -pzz ) From Equation
(2%
(14) we obtain
(29) (30)
(31)
and, substituting
from Equation
(26),
2k4/kz k’ = 2k4/kz/kle 1 + kz/k,e = kz ____ + be
(32)
This demonstrates that for limf,, (kdcz)/p = 0, and kz and k[ not zero, the decay of catalase peroxide complex is second order. This confirms analog computer data, which showed that log p vs. t is not linear in such a case (4). A physical explanation of this second-order decay is that the decay depends on the reaction of the complex with
290
DAVID S. GREENSTEIN
Hz02 , and kz serves to produce the Hz02 in the latter stages of the reaction. DISCUSSION OF THE RESULTS FOR CATALASE
The mathematical analysis gives two important results: a delineation of the range of concentrations for which the relation Ic = k4a is valid, and a formula for k which holds for all conditions. Equations (22) to (25) indicate that if kq is to be measured from the rate of disappearance of the catalase Hz02 complex, the conditions for doing so are low alcohol concentration and high enzyme concentration. It is interesting to note that this is Chance’s conclusion (2). However, he was unable to give a quantitative explanation of the effect of varying enzyme concentration. At high concentrations of alcohol, he found experimentally a linear variation of what he thought was k, with e, which he was unable to explain. A quantitative explanation of this phenomenon is given by Eq. (25b). What Chance called kd is equal to k/a which according to Eq. (25b) should be given by
That is, the slope of the straight line “kq” vs. e obtained by Chance should be equal to kl/a. Application of this assumption to Chance’s data (2) gives kl equal to 0.8 X lo’, which compares favorably with values of 0.5-0.6 X lo’, obtained by independent methods. Thus our analysis has led to an unexpected result; it has given us a new method of measuring kl , a rather striking substantiation of the idea (5) that transient-state studies are capable of yielding more information than steady-state studies. Equation (22) always gives the correct value of L and furnishes us with a better insight into the effects of the various rate constants. From Eq. (6) and the fact that kz represents a reaction which uses up the HzOz complex, it would seem entirely reasonable to expect k to have a larger value for ICZ> 0 than for kz = 0. Equation (22) indicates quite the opposite. Chemically this is explained by the fact that when complex disappears via kz , substrate is formed, which may then react with enzyme to form more complex; i.e., kz helps to make a larger catalase turnover possible during the final stages of the reaction [cf. Eq. (3)]. These arguments apply to the conditions when the reactions of Eqs. (4) and (5) proceed relatively slowly.
REACTION-VELOCITY
CONSTANTS FOR ENZYME REACTIONS
291
Our analysis enables us to form some conclusions about the instability of the catalase hydrogen peroxide complex. Since decay has always been observed to be first order, one must conclude that in the preparations studied so far there has been residual alcohol or other donor present. APPLICATION
OF THE METHOD
TO THE PEROXIDASE
SYSTEM
Our method can also be applied to enzymes which follow the scheme: E+S+ES (e - P) x ES + A P
P
E + products
a
dp - = hx(e - P> dt dx -= dt
--klx(e
(h + k‘dp
- P> + b2P
We may now derive equations identical with Eqs. (10) through (25), indicating a similarity between the terminal kinetics of the catalase hydrogen peroxide complex I and of the peroxidase complex II. There is one significant difference which our analysis does not indicate, namely, that the decay of the peroxidase intermediate is not first order until the latter stages of the reaction. Some analog computer data indicate that by the time the concentration has declined to one-fourth its maximum, the decay is first order. ACKNOWLEDGMENTS The author wishes to express his gratitude to Mr. Joseph Rutledge for completing this manuscript and to Dr. Chance for numerous discussions and criticisms .
SUMMARY
A mathematical approach has been developed which leads to a solution of the terminal kinetics of the catalase hydrogen peroxide complex and of the peroxidase complex. The first order (k) constant is given as a function of the rate constants and of the concentrations of enzyme (e) and donor (a). This functional relation shows not only that ka , the rate constant for the reaction of complex with donor, can be measured at
292
DAVID S. GREENSTEIN
high e and low a, but that with high e, ICI, the rate constant for the combination of enzyme with substrate, can be measured at high a. The value k1 = 0.8 X 10’ M-l X sec.-’ obtained in this manner compares favorably with values of 0.5-0.6 X 10’ obtained by other methods. The formula further indicates that k is a more complicated function of k2 than had previously been supposed, and suggests how kz may be properly measured. The analysis here presented justifies Chance’s explanation of the nonlinearity of the k vs. a relation, namely, turnover of enzyme during decay of the complex. Thus there is no need for postulating a ternary complex on the basis of this nonlinearity. REFERENCES 1. CHANCE,B., Actu Chem.Scmd.1,236
(1947).
2. CHANCE, B., J. Bid. Chem. 182,643 (1950). 3. CHANCE, B., GREENSTEIN, D. S., AND ROUOHTON, F. J. W., Arch. Biochem. and Biophys. 37, 301 (1952). 4. CHANCE, B., GREENSTEIN, D. S., HIGGINS, J., AND YANG, C. C., Awh. Biochem. and Biophys. 37, 322 (1952). 5. CHANCE, B., in “Modern Trends in Physiology and Biochemistry” (E. 8. G. Barron, ed.), p. 25. Academic Press, New York, 1952.
OF
BIOCHEMISTRY
AND
BIOPHYSICS
62, 284-292 (1956)
The Calculation of Reaction-Velocity Constants for Enzyme Reactions from the Kinetics of the Disappearance of Enzyme-Substrate Compounds’ David S. Greenstein From the Johnson
Research Foundation, University Philadelphia, Pennsylvania Received
October
of PennsyEvania,
10, 1955
INTRODUCTION Experimental studies have furnished numerous examples of enzymesubstrate compounds which disappear by reactions which are pseudofirst order with respect to the concentration of hydrogen donor. ES+AH”4E+SH+A
(1)
For example, Chance has shown that the disappearance of catalase hydrogen peroxide is pseudo-first order with respect to the concentration of ethanol (1). In such cases it seems reasonable to identify the pseudofirst-order constant with kca , where k4 is the rate constant for the combination of the intermediate complex with the donor and a is the concentration of donor. This is equivalent to assuming that the reaction of the complex with donor is the only pathway for the disappearance of the complex. This assumption is reasonable when experiments show a linear variation of the first-order constant with a over a wide range. However, in the case of catalase, experiments by Chance show that at high concentrations of alcohol, this is definitely not the case (2). One explanation which has been considered for this nonlinear behavior is the formation of an enzyme-substrate-donor complex, which has not yet been observed experimentally. Chance (2) postulated that the nonlinearity might be explained without any such modification of the reaction mechanism. His explanation (which is justified by our results) is that Hz02 not bound to catalase acts as a hydrogen donor and provides a second pathway for the disappearance of the complex. The type of mathematical analysis 1 This research
was supported
in part by the Office of Naval 2a4
Research.
REACTION-VELOCITY
CONSTANTS FOR ENZYME REACTIONS
285
used by Chance, however, leads only to a qualitative explanation of the nonlinear behavior. In the present paper, a mathematical approach is presented which quantitatively explains these deviations, shows when the first-order constant equals Ic(a, and shows how k1 , the rate constant for the combination of enzyme with substrate, can be measured from the kinetics of the disappearance of the enzyme-substrate compound. This approach is general enough to apply not only to enzyme-substrate compounds which follow Eq. (l), but also to the classical Michaelis-Menten enzyme-substrate compounds, which follow the relation. ES&E+P METHOD
(2)
FOR THE CATALASE SYSTEM
Let us first consider the case of catalase, for which the following reaction scheme and differential equations have been found to hold (3): catalase + HzOz 4 (e--P>
x
catalase HzOz ’
catalase HzOz + Hz02 kr’ catalase HzOz + RCHzOHo kr
(3)
P catalase + 2Hz0 + O2
(4)
catalase + RCOH
(5)
+ 2HZ0
(Unless otherwise noted, RCHzOH is added in sufficient excess to keep its concentration approximately constant.)
dp - = ka(e - p) - lclxp - &Z + k4a)p dt
dx -= dt
-klx(e da -= dt
- p) - k4’xp + hp -
k4ap
(6) (7)
(8)
Experimentally, it has been found that the bulk of the Hz02 has disappeared before the enzyme-substrate complex begins to disappear. Inspection of Eq. (6) suggests that the substrate concentration may then be so low .that the first two terms on the right become negligible and dp,ldt = - (k, + k4a)p, corresponding to the first-order decay of p. This
286
DAVID
S. GREENSTEIN
assumption was, in fact, used in interpreting experimental data, kz having been considered to be the value of the first-order constant obtained without added alcohol (a = 0). Some justification for this assumption comes from the analog computer data, which shows that for kz = 0 and low a, the first-order constant is k4a (2). The validity of the assumption that the Hz02 concentration is negligible in the latter stages of the reaction can be checked mathematically. The use of limits is suggested since dp/dt = - (kz + kda)p only if 2 = 0; that is when the reaction has gone to completion.2 Therefore, we define:
If this limit exists, it is the first-order constant for the disappearance of p. If the limit is ever zero, the order of decay is greater than one. Such a case will be considered later. From Eq. (6) we have
‘t!!!=k
-pdt
’
+ka
4
- kle E + kix + klx
Since lim-, dp/dt = lim,,, dx/dt = 0, it follows from Eqs. (6) and (7) that lim z = lim p = 0, and the terms of Eq. (10) involving 2 alone will vanish: = kz + k4a - kle lim E t-m 0p
01)
Equation (11) leads us to investigate limt+, (E) = (E).. Weshall first obtain bounds for 2 . In the last stages of the reaction both substrate 0 and complex are dis:piearing, i.e., dx/dt and dp/dt --j 0. Applying these conditions to Eqs. (6) and (7), we have kzp - k4’xp 6 ks(e - p) 6 (k, + h4 p + k&p
(1%
* In evaluating limits, it is assumed that all the time derivatives of concentrations tend to zero as time approaches infinity. This assumption is physically reasonable; it merely says that the chemical system comes to equilibrium. Another reasonable assumption is that if all the concentrations are initially non-negative, they will not attain negative values at any later time.
REACTION-VELOCITY
CONSTANTS
FOR
ENZYME
REACTIONS
287
Taking out --k~xp and dividing by p, this gives h - k4’x 4- klx f he 5 < k2 -I- k4a + (k4’ + kl)x. and .letting t -j ~0, kze<
b
‘-pm’ 0
’
< k2 + ha
(13)
kle
Another equation for k can be obtained as follows. Adding Eqs. (6) and (7) gives -k4ap - 2ka)xp
(14) (15) 06) (17)
By the rule of L’Hospital:
=
k
Now, combining
(1%
Eqs. (11) and (18) we obtain: k4a
= k2 + k4a - kle : 0 Pm
(19)
which yields k2 -iG= Thi.s quadratic
o
(20)
equation has one positive and one negative root, and
288
DAVID S. GREENSTEIN
> 0. Hence, (;)*=;r*-l+m&] and, substituting k=
(21) into Eq. (18), 2k4a
1 + h +
k4a
+
1
_
he
h
+
ha
he
For low a (k4u << kz), Eq. (22) is well approximated k -
2 >
(22)
kz +4&i
by
k4a
kz
(23)
I+&? If e is large [kle > (kz + kra)], k m k4a , in agreement with the earlier assumption, mentioned above. For large a (k4a > k2) k M kle Equation
(24)
(22) assumes a special form as k2/kle tends to zero: k = k4a; k = kle;
a > kle/k4
(The special form of Eq. (25) depends on the mathematical d(i-zp
(25b) identity
= ) 1 - u I.>
A FURTHER APPLICATION
TO CATALASE
This method can also be used to shed light on another problem in catalase action. It has been pointed out that the catalase HzOz complex would be stable if there were no alcohol present and the combination of catalasewith Ha02 wereirreversible (i.e., if kz and k4a were simultaneously zero). Thus it becomes important to determine whether reversibility of reaction (3) or residual alcohol from the catalase preparation method is responsible for the instability which is observed when no alcohol is added.
REACTION-VELOCITY
CONSTANTS
FOR
ENZYME
REACTIONS
289
We now consider the case
In this case, as t -+ 00, p 3 0. Equations spectively:
(21) and (22) then become, re-
(26) and k=O
(27)
These equations also agree with reactions (5) and (9). Equation (27) implies that the order of the decay reaction is greater than one. We now investigate the second-order rate constant, 1 dP k’ = lim t--m ( -pzz ) From Equation
(2%
(14) we obtain
(29) (30)
(31)
and, substituting
from Equation
(26),
2k4/kz k’ = 2k4/kz/kle 1 + kz/k,e = kz ____ + be
(32)
This demonstrates that for limf,, (kdcz)/p = 0, and kz and k[ not zero, the decay of catalase peroxide complex is second order. This confirms analog computer data, which showed that log p vs. t is not linear in such a case (4). A physical explanation of this second-order decay is that the decay depends on the reaction of the complex with
290
DAVID S. GREENSTEIN
Hz02 , and kz serves to produce the Hz02 in the latter stages of the reaction. DISCUSSION OF THE RESULTS FOR CATALASE
The mathematical analysis gives two important results: a delineation of the range of concentrations for which the relation Ic = k4a is valid, and a formula for k which holds for all conditions. Equations (22) to (25) indicate that if kq is to be measured from the rate of disappearance of the catalase Hz02 complex, the conditions for doing so are low alcohol concentration and high enzyme concentration. It is interesting to note that this is Chance’s conclusion (2). However, he was unable to give a quantitative explanation of the effect of varying enzyme concentration. At high concentrations of alcohol, he found experimentally a linear variation of what he thought was k, with e, which he was unable to explain. A quantitative explanation of this phenomenon is given by Eq. (25b). What Chance called kd is equal to k/a which according to Eq. (25b) should be given by
That is, the slope of the straight line “kq” vs. e obtained by Chance should be equal to kl/a. Application of this assumption to Chance’s data (2) gives kl equal to 0.8 X lo’, which compares favorably with values of 0.5-0.6 X lo’, obtained by independent methods. Thus our analysis has led to an unexpected result; it has given us a new method of measuring kl , a rather striking substantiation of the idea (5) that transient-state studies are capable of yielding more information than steady-state studies. Equation (22) always gives the correct value of L and furnishes us with a better insight into the effects of the various rate constants. From Eq. (6) and the fact that kz represents a reaction which uses up the HzOz complex, it would seem entirely reasonable to expect k to have a larger value for ICZ> 0 than for kz = 0. Equation (22) indicates quite the opposite. Chemically this is explained by the fact that when complex disappears via kz , substrate is formed, which may then react with enzyme to form more complex; i.e., kz helps to make a larger catalase turnover possible during the final stages of the reaction [cf. Eq. (3)]. These arguments apply to the conditions when the reactions of Eqs. (4) and (5) proceed relatively slowly.
REACTION-VELOCITY
CONSTANTS FOR ENZYME REACTIONS
291
Our analysis enables us to form some conclusions about the instability of the catalase hydrogen peroxide complex. Since decay has always been observed to be first order, one must conclude that in the preparations studied so far there has been residual alcohol or other donor present. APPLICATION
OF THE METHOD
TO THE PEROXIDASE
SYSTEM
Our method can also be applied to enzymes which follow the scheme: E+S+ES (e - P) x ES + A P
P
E + products
a
dp - = hx(e - P> dt dx -= dt
--klx(e
(h + k‘dp
- P> + b2P
We may now derive equations identical with Eqs. (10) through (25), indicating a similarity between the terminal kinetics of the catalase hydrogen peroxide complex I and of the peroxidase complex II. There is one significant difference which our analysis does not indicate, namely, that the decay of the peroxidase intermediate is not first order until the latter stages of the reaction. Some analog computer data indicate that by the time the concentration has declined to one-fourth its maximum, the decay is first order. ACKNOWLEDGMENTS The author wishes to express his gratitude to Mr. Joseph Rutledge for completing this manuscript and to Dr. Chance for numerous discussions and criticisms .
SUMMARY
A mathematical approach has been developed which leads to a solution of the terminal kinetics of the catalase hydrogen peroxide complex and of the peroxidase complex. The first order (k) constant is given as a function of the rate constants and of the concentrations of enzyme (e) and donor (a). This functional relation shows not only that ka , the rate constant for the reaction of complex with donor, can be measured at
292
DAVID S. GREENSTEIN
high e and low a, but that with high e, ICI, the rate constant for the combination of enzyme with substrate, can be measured at high a. The value k1 = 0.8 X 10’ M-l X sec.-’ obtained in this manner compares favorably with values of 0.5-0.6 X 10’ obtained by other methods. The formula further indicates that k is a more complicated function of k2 than had previously been supposed, and suggests how kz may be properly measured. The analysis here presented justifies Chance’s explanation of the nonlinearity of the k vs. a relation, namely, turnover of enzyme during decay of the complex. Thus there is no need for postulating a ternary complex on the basis of this nonlinearity. REFERENCES 1. CHANCE,B., Actu Chem.Scmd.1,236
(1947).
2. CHANCE, B., J. Bid. Chem. 182,643 (1950). 3. CHANCE, B., GREENSTEIN, D. S., AND ROUOHTON, F. J. W., Arch. Biochem. and Biophys. 37, 301 (1952). 4. CHANCE, B., GREENSTEIN, D. S., HIGGINS, J., AND YANG, C. C., Awh. Biochem. and Biophys. 37, 322 (1952). 5. CHANCE, B., in “Modern Trends in Physiology and Biochemistry” (E. 8. G. Barron, ed.), p. 25. Academic Press, New York, 1952.