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Kinetic implications in the analytical use of enzymes
ANALYTICAL
BIOCHEMISTRY
165, 13-19
( 1987)
REVIEW Kinetic Implications
in the Analytical Use of Enzymes A. B.
ROY
Protein Chemistry Group, John Curtin School of Medical Research, Australian National University, GPO Box 334, Canberra, ACT 2601 Australia
they should be generally applicable. It will be assumed that the enzymes to be used will be, if not homogeneous, at least highly purified, if they are not, then any attempt at a kinetic treatment, like the interpretation of the results obtained by the use of unpurified enzymes, could be valueless because of the likely presence of endogenous modifiers or substrates. The specificity of the enzymes will not be considered, the implicit assump tion is that the compounds of interest are indeed substrates. It must be stressed, however, that a completely general sulfatase or &$.tcuronidase does not exist (1); by definition, enzymes are specific catalysts.
Enzymes are frequently used to aid in the characterization and quantitation of those of their substrates which occur naturally or are formed by the metabolism of foreign compounds. Frequently used for this purpose are the hydrolytic enzymes, the J?-glucuronidases (B-D-glucuronoside glucuronosohydrolase, EC 3.2.1.3 1) and sulfatases (aryl-sulfate sulfohydrolase, EC 3.1.6.1; steryl-sulfate sulfohydrolase, EC 3.1.6.2) because of the almost ubiquitous occurrence of their respective substrates as metabolites of endogenous compounds and xenobiotics. While the value of this technique is considerable, little attention seems to have been paid to its kinetic basis although it should be obvious that in a situation where the essentially complete hydrolysis of a substrate, or frequently a mixture of substrates, is required, an emphasis on initial velocities is inappropriate. The conditions chosen are often far from the optimum. Unfortunately it will often be impossible to give an exact kinetic treatment of this type of experiment because the required kinetic parameters may well be unknown, but the following comments will emphasize the different views of the kineticist and the analyst. Although it may not always be possible to make full use of the advocated approach, the comments should at least point out some of the more obvious pitfalls of the usual empirical one, and may even stimulate further work in the area. These remarks will often be directed toward the sulfatases and &h.rcuronidases but
SUBSTRATE
CONCENTRATION
Perhaps the first, and most obvious, point is that the initial substrate concentration, so, will frequently be less than the K,,, for that substrate. Under such conditions, when so -4 K,, the usual Michaelis relationship (Eq. [ 11) simplifies to the linear relationship in Eq. [2]. This is trivial, but important consequences follow: ds J’s,, “O = - z = Km + so ds V u()= --&=K,‘so.
111 PI
Consider the situation where there are n substrates of initial concentrations s& sf, s& . . . , st. In the usual situation, where the 13
0003-2697187
$3.00
Copyright 8 1987 by Academic Press, Inc. All rights of reproduction in any form reserved.
14
A. B. ROY
concentration of each substrate is high compared to its K,,, (sh > Kh for all z), the rate equation for substrate 1 is given by Eq. [3]. It is clear that the inhibition caused by the effect of competing substrates could be very great. Similar expressions give the values of uo for each of the other substrates:
that calculated from Eq. [3]; the mean experimental value was 104 + 3% (range lOO109%) of the calculated value. It is clear, therefore, that the use, by choice or by necessity, of low substrate concentrations can be advantageous. This is especially so with the /3-glucuronidases which are commonly subject to substrate inhibition (1); that is, the initial velocity falls when the substrate concentration exceeds an optimum value. The velocity equation is given by Eq. [5] in which K2 is the dissociation constant, analogous to K,,,, for the enzyme-substrate complex ES, :
However, if for each of the substrates sk <
vs, “O= Km+ so+ sS/K**
With /3glucuronidase n is generally 2 (2). Unpublished work from this laboratory has shown that the arylsulfatase of H. pomatia becomes increasingly subject to substrate inhibition as the pH is lowered from the optimum of about 7.5 and it has also been noted in a few cases with the arylsulfatase of Takadiastase (3).
V’ v; = -.s& K!?l That is, the individual initial velocities are additive when & Q Kk. Some data for the hydrolysis of the isomeric nitrophenyl sulfates by the arylsulfatase of Helix pomatia ( 1) are given in Table 1; this clearly shows the dependence of the competing substrate effect upon the ratio so/K,. In every case in Table 1 the experimental value of the velocity obtained with mixed substrates agreed well with
PROGRESS
Substrate 2-NPS 3-NPS 4-NPS 2- + 3-NPS 2- + 4-NPS 3- + 4-NPS 2-+3-+4-NPS
%lL 3.8 3.9 4.0
” 28.3 37.4 50.4 39.1 48.1 50.9 48.1
rzIJ
65.7 78.7 87.7 116.1
CURVES
Although these relationships may be interesting, they are, as already pointed out, of little general interest in the present context.
TABLE THE EFFECT OF THE MAGNITUDE OF THE RATIO so/K, DROLYSIS OF THE THREE ISOMERIC MONONITROPHENYL
[51
1
ON COMPETING SUBSTRATE EFFECTS DURING THE HYSULFATES BY THE ARYL~ULFATASE OF Helix pomatia Activity
0.60 0.61 0.58 0.41
%lKll
”
zv
0.19 0.19 0.20
6.1 8.9 11.0 13.2 15.5 17.1 19.7
15.0 17.1 19.9 26.0
Activity
0.88 0.9 1 0.86 0.76
Note. The assays were carried out in a pH-stat using a partially purified enzyme (1). The column headed Zv is the activity expected in the absence of competing substrate effects. Substrates: potassium salts of 2-nitrophenyl sulfate (2-NPS; K,, 9.4 mM); 3-nitrophenyl sulfate (3-NPS; Km, 3.5 mM); 4-nitrophenyl sulfate (4-m; K,,,, 3.4 mM).
KINETIC
IMPLICATIONS
IN ANALYTICAL
ENZYME
1.5
USE
Here the integrated Michaelis equations must be used to supply the necessary information. Those corresponding to Eqs. [I], [2], and [3] are given in Eqs. [6], [7], and [8], respectively, in which s is the substrate concentration at time t so that the concentration of products is then (sO - s):
1
t = jp K,ln:
+ (,sO- 3)
161 Reaction
(mid
FIG. 2. Progress curve for an enzyme functioning under first-order conditions, when s, Q K,,,. The conditions are as for curve A in Fig. 1 except that so is 4 pm01 * liter-‘.
t=$!*n?!! s t=-
time
1 K,ln~+(~O-s) V[ s
+&6-s”) 2
.
1
VI are no extraneous interfering factors: for ex-
Neither Eq. [6] nor Eq. [8] can be rearranged to give s as an explicit function of t but Eq. [7], which holds when so 4 K, and is formally identical to the integrated rate equation for a first-order reaction, can be treated to give Eq. [9] which is often a more convenient form to use: s = Soe-‘Vl”“. PI It must be stressed, however, that all of these integrated equations apply only when there
ample, product inhibition or inactivation of the enzyme. Such a situation is admittedly unlikely in the present context. Computed plots of Eqs. [6] and [8] are given in Fig. 1 to show the effect of substrate inhibition on the shape of the progress curve while Fig. 2 shows the corresponding plot of Eq. [7], when so 4 K,,, . The different shapes of these curves, particularly the purely exponential curve corresponding to Eq. [7], should be noted. CHOICE OF ENZYME CONCENTRATION
Equations [6] and [7] can be normalized in terms of two parameters, CYand p; cy is any chosen degree of hydrolysis of the substrate and is, therefore, equal to 1 - s/s0 while fl is the relative substrate concentration, equal to so/K,,,. Substitution of cyand ,&3into Eqs. [6] and [7] and rearrangement give Eqs. [lo] and [ 111, respectively: 0
25
50 Reaction
time
75
I 100
(mid
FIG. I. Progress curves for a typical enzyme reaction showing Michaelis kinetics (Curve A) and one subject to substrate inhibition (Curve B) computed from Eqs. [6] and [S], respectively. The kinetic parameters in the latter are those for @-glucuronidase (2). K,,,, 0.29 mmoI.liter-‘; K2, 7.8 mmol -liter-‘; soI 10 mmol. liter-‘, V, 0.2 rrnol. liter-’ . min-‘.
V=T
a-bln(l
-CX)
[ V=$ln(l = ?.-jln(l
1
[lOI
-cu) - cu)
[**I
16
A. B. ROY
A similar expression can be obtained from Eq. [8] but is of little practical use because it contains K2 and 12which are rarely known. Figure 3 shows computed plots of [CY- (l//3) ln(1 -a)]and-(l/P)ln(l -a)forauseful value of (Y, 0.99. The two functions are very similar at low values of /3 but the former tends asymptotically to (Yat high values of /3, that is, at high relative substrate concentrations. In principle, Eqs. [lo] and [ 1 I] can be used to estimate the amount of enzyme required to give a 99% hydrolysis of a substrate, at initial concentration, so, in any chosen time, t. An example of such a calculation is given in Table 2. In practice, unfortunately, it will frequently be difficult to obtain such estimates because of the lack of knowledge of the appropriate values of K,,, and of the specific activity. Available values of these parameters will rarely have been obtained for conditions comparable to those in, for example, bile. Nevertheless, such estimates should be sought because they will allow the amount of enzyme to be kept to a minimum. Not only will this minimize both the expense and the effects of any contaminating enzyme which might inadvertently be present, but it will make less ambiguous the interpretation of experiments in which the activity of the enzyme is to be inhibited. Once again the problem is simplified by
TABLE 2 CALCIJLATIONOFTHEAMOUNTOFSULFATASE REQUIREDTOHYDROLYZEAGIVENAMOIJNT OFSUBSTRATEINACHOSENTIME Substrate, so Kn Reaction time a = 1 - s/so B = solKn
Estrone sulfate, 1 X IO-’ M 0.1 x lo-‘M 300 min 0.99 10
From Eq. [lo] v= 1.0 x 1o-3 X 1.45 mol. liter-’ *min-’ 300 = 5 X IO+ mol. liter-’ . min-’ Note. One unit of enzyme corresponds to a rate of 1 X 10m6mol. liter-’ * min-I; therefore the required concentration is 5 units * liter-‘.
using conditions such that so 4 K,. Under these conditions Eq. [ 71 applies and the reaction is kinetically first order. It should be relatively simple to chose a concentration of enzyme and estimate, approximately, t,,* for the reaction: that is, the time taken for the substrate concentration to fall to half its initial value. This parameter is independent of so for a first-order reaction. It follows from Eq. [8] that K,,,/V is equal to t&n 2. If a degree of hydrolysis a! is now required, the appropriate time, t, , for the same concentration of enzyme will be given by Eq. [ 121, which can be derived from Eq. [7] as follows: I,=$.ln
looOl-
= -$.ln(l = -t,/2*
O.OJ.Jo
B
FIG. 3. Plots of [a - (l/@ln(l - a)] (curve A) and -( l/@ln( 1 - a) (curve B) against 8. Details are given in the text.
So so(l - 4 -(Y) ln(1 - (Y) In2 *
[Ql
For CYequal to 0.99, t, is approximately 7t1,2. It is obvious from the above relationships that t, is indirectly related to V so that the concentration of enzyme can readily be adjusted should the estimated value of t, be inappropriate. While only approximate, to
KINETIC
IMPLICATIONS
IN ANALYTICAL
the extent that the estimated value of tllZ is approximate, this method of calculating the required amount of enzyme is preferable to the empirical approach usually adopted and it must be stressed that the value of K,,,/V which it provides pertains to the actual system under investigation.
Once again in the present context ui (uO in the presence of an inhibitor) is of little interest but it is worth pointing out that at low relative substrate concentrations the usual rate equations for competitively and noncompetitively inhibited reactions both reduce to Eq. [ 131, V
“‘= K,(l
1131
+ ilK,jso’
17
USE
0.75 “0° y 1 lA y=10 \
$
y=100 \
0.50
0.25
-
0
INHIBITED REACTIONS
ENZYME
1
10
102 Reaction
time
103 (mid
y=1000 \
10
105
FIG. 4. Plots showing the variation with time of pi/p0 (p,,,pi: amount of product produced in uninhibited and inhibited reactions, respectively) at different relative inhibitor concentrations. First-order conditions, s,, * K,,,, therefore the values of poand pi are given by Eqs. [7] and [ 161, respectively. K,,,, 0.1 mmol *liter-‘; V, 1.5 pm01 * liter-’ * min-‘; y = i/K,, shown on the appropriate curves.
effects of competitive or noncompetitive inhibitors on an enzyme reaction subject to from which it follows that in either case the substrate inhibition can be complex (4) and, inhibition is kinetically noncompetitive be- although pertinent, cannot be considered cause the relative activity of the inhibited re- here. action, ui/uo, is equal to Ki/(Ki + i) and Equations [ 141 and [ 161 show what is aptherefore independent of the substrate con- parently often forgotten: that no matter what centration. the degree of inhibition of u. (provided it is Appropriate modifications of the inteless than 100%) the reaction will go to comgrated equations, Eqs. [6] and [7], must be pletion given sufficient time, as exemplified used to provide the required information. in Fig. 4. The curves therein were computed Equation [6] gives Eqs. [ 141 and [ 151 for from Eqs. [7] and [ 161 with different values competitive and noncompetitive inhibition, of i/Ki (y) in the latter. This comment also respectively, and both reduce to Eq. [ 161, applies to the competing-substrate effects alwhich can also be derived directly from Eq. ready mentioned: they will disappear as the [8] when so 4 K,,,: reaction time increases. Figure 5 illustrates the importance of set=b 1 + i K,ln: + (so - S) [I41 lecting the concentration of enzyme approI [( I1 priate to the chosen time of hydrolysis, taking as an example the hydrolysis of estrone K,,,ln~+(~o-s) 1151 sulfate, at an initial substrate concentration of 1 mM, by the sulfatase of H. pomatia. The appropriate concentration of enzyme was [161 calculated in Table 2 and the curves in Fig. 5 were computed using the values of the paEquation [ 161 can be rearranged to give s as rameters in the former. Curve A shows the an explicit function oft which is of the same course of the hydrolysis under these condiform as Eq. [lo], differing only in that Km in tions while curve B shows it in the presence the latter is replaced by ( 1 + i/Ki)K,. The of a competitive inhibitor at a concentration
I
18
A. B. ROY
of 100 Kj, which would cause a 90% inhibition of vo. It is clear that at the chosen time, 300 min (Table 2), the extent of the inhibition is considerable but, as already pointed out, this decreases as the reaction time is increased. Curve C shows the progress in the presence of the same concentration of inhibitor but at 10 times the calculated concentration of enzyme. It is obvious that if the time of 300 min had been arbitrarily chosen and the appropriate amount of enzyme not used then it might well have been concluded that no inhibition had occurred, especially when it is remembered that these methods are frequently used only qualitatively, or at best semiquantitatively, to characterize compounds located by, for example, autoradiography of TLC plates. The curves in Fig. 5 should demonstrate quite clearly the interplay among the time of reaction, enzyme concentration, and inhibitor concentration and therefore the importance of checking experimentally that the correct conditions have indeed been chosen, Again there are advantages in working at low relative substrate concentrations, as illustrated by the computed curves in Fig. 6 which is directly comparable to Fig. 5 except
3
0.25
i 01
4 Reaction
time
hlid
FIG. 5. Progress curves for uninhibited (curve A) and inhibited (curves B and C) enzyme reactions computed using Eqs. [6] and [ 141. The vertical line shows 300 min, the time chosen in calculating the required enzyme concentration (Table 2). K,,,, 0.1 mmol.liter-‘; sa, 1 mmol. liter-‘; ilKi, (curves B and C) 100; V, 5 pm01 . liter-’ . min-’ (curves A and B), 50 rrmol . liter-’ - min-’ (curve C).
10.0
A
B
2 1 7.5 P .? 5.0 E 3 2.5 Lt '-I%,
0
1
10
103 Reaction
time
103 (mid
104
105
FIG.6. Progress curves for uninhibited (curve A) and inhibited (curve B) enzyme reactions under first-order conditions. Computed from Eqs. [7] and [ 161. K,,,, 0.1 mmol . liter-‘; so, 10 pmol . liter-‘; i/Ki (curve B), 100; V, 1.5 rrmol . liter-’ . min-‘.
that so was 0.01 mM (/3 = 0.1). The appropriate enzyme concentration, again for a reaction time of 300 min, was obtained from Eq. [ 111 in the same way as shown in Table 2. Curve A shows the control hydrolysis and curve B shows the hydrolysis in the presence of a competitive inhibitor at a concentration of 100 Ki, the same as in Fig. 5. It can be seen by comparing these two figures that the same concentration of inhibitor has a much greater effect under first-order conditions (Fig. 6) than under Michaelis conditions (Fig. 5), as is of course to be expected since the inhibition is competitive. In terms of initial velocities, the relative activity, ui/uo, is 0.1 in Fig. 5 (@ = 10) and 0.01 in Fig. 6 (j3 = 0.1). As already noted, the effect of an inhibitor on u. is not informative when essentially complete hydrolysis of the substrate is required and a more useful parameter under these conditions is AtliZ, defined as the difference between the values of tI12 for the inhibited and uninhibited reactions. It follows from Eqs. [7 ] and [ 141 and [7] and [ 151 that this parameter is given by Eqs. [ 171 and [ 181 for competitive and noncompetitive inhibition, respectively: At 112
K,i =
E
ln
2
[I71
KINETIC
K,i At ,,* = K
I
IMPLICATIONS
In 2 + 0.5 +$
IN ANALYTICAL
I
,
(K,,Jn 2 + 0.5.~~).
USE
19
firming the much greater extent of the inhibition in the latter, and showing it more informatively than by the effect on uo.
; = &
ENZYME
[ 181
When so 4 K,,,, Eq. [ 181 reduces to Eq. [ 171, which can also be derived directly from Eq. [ 161, again showing that competitive and noncompetitive inhibitors are kinetically indistinguishable when so 4 Km. The values of AtllZ for the pairs of curves in Figs. 5 and 6 are 1390 and 4620 min, respectively, con-
REFERENCES 1. Roy, A. B. (1987) Anal. Biochem. 165, 1-12. 2. Walker, P. G., and Levvy, G. A. (1953) Biochem. J. S&56-65.
3. Robinson, D., Smith, J. N., Spencer, B., and Williams. R. T. (1952) , B&hem. J. 51.202-208. 4. Krupka, R. M., and Laidler, K. J. (196 1) J. Amer. Chem. Sm. 83, 1448-1454, 1454-1458.
BIOCHEMISTRY
165, 13-19
( 1987)
REVIEW Kinetic Implications
in the Analytical Use of Enzymes A. B.
ROY
Protein Chemistry Group, John Curtin School of Medical Research, Australian National University, GPO Box 334, Canberra, ACT 2601 Australia
they should be generally applicable. It will be assumed that the enzymes to be used will be, if not homogeneous, at least highly purified, if they are not, then any attempt at a kinetic treatment, like the interpretation of the results obtained by the use of unpurified enzymes, could be valueless because of the likely presence of endogenous modifiers or substrates. The specificity of the enzymes will not be considered, the implicit assump tion is that the compounds of interest are indeed substrates. It must be stressed, however, that a completely general sulfatase or &$.tcuronidase does not exist (1); by definition, enzymes are specific catalysts.
Enzymes are frequently used to aid in the characterization and quantitation of those of their substrates which occur naturally or are formed by the metabolism of foreign compounds. Frequently used for this purpose are the hydrolytic enzymes, the J?-glucuronidases (B-D-glucuronoside glucuronosohydrolase, EC 3.2.1.3 1) and sulfatases (aryl-sulfate sulfohydrolase, EC 3.1.6.1; steryl-sulfate sulfohydrolase, EC 3.1.6.2) because of the almost ubiquitous occurrence of their respective substrates as metabolites of endogenous compounds and xenobiotics. While the value of this technique is considerable, little attention seems to have been paid to its kinetic basis although it should be obvious that in a situation where the essentially complete hydrolysis of a substrate, or frequently a mixture of substrates, is required, an emphasis on initial velocities is inappropriate. The conditions chosen are often far from the optimum. Unfortunately it will often be impossible to give an exact kinetic treatment of this type of experiment because the required kinetic parameters may well be unknown, but the following comments will emphasize the different views of the kineticist and the analyst. Although it may not always be possible to make full use of the advocated approach, the comments should at least point out some of the more obvious pitfalls of the usual empirical one, and may even stimulate further work in the area. These remarks will often be directed toward the sulfatases and &h.rcuronidases but
SUBSTRATE
CONCENTRATION
Perhaps the first, and most obvious, point is that the initial substrate concentration, so, will frequently be less than the K,,, for that substrate. Under such conditions, when so -4 K,, the usual Michaelis relationship (Eq. [ 11) simplifies to the linear relationship in Eq. [2]. This is trivial, but important consequences follow: ds J’s,, “O = - z = Km + so ds V u()= --&=K,‘so.
111 PI
Consider the situation where there are n substrates of initial concentrations s& sf, s& . . . , st. In the usual situation, where the 13
0003-2697187
$3.00
Copyright 8 1987 by Academic Press, Inc. All rights of reproduction in any form reserved.
14
A. B. ROY
concentration of each substrate is high compared to its K,,, (sh > Kh for all z), the rate equation for substrate 1 is given by Eq. [3]. It is clear that the inhibition caused by the effect of competing substrates could be very great. Similar expressions give the values of uo for each of the other substrates:
that calculated from Eq. [3]; the mean experimental value was 104 + 3% (range lOO109%) of the calculated value. It is clear, therefore, that the use, by choice or by necessity, of low substrate concentrations can be advantageous. This is especially so with the /3-glucuronidases which are commonly subject to substrate inhibition (1); that is, the initial velocity falls when the substrate concentration exceeds an optimum value. The velocity equation is given by Eq. [5] in which K2 is the dissociation constant, analogous to K,,,, for the enzyme-substrate complex ES, :
However, if for each of the substrates sk <
vs, “O= Km+ so+ sS/K**
With /3glucuronidase n is generally 2 (2). Unpublished work from this laboratory has shown that the arylsulfatase of H. pomatia becomes increasingly subject to substrate inhibition as the pH is lowered from the optimum of about 7.5 and it has also been noted in a few cases with the arylsulfatase of Takadiastase (3).
V’ v; = -.s& K!?l That is, the individual initial velocities are additive when & Q Kk. Some data for the hydrolysis of the isomeric nitrophenyl sulfates by the arylsulfatase of Helix pomatia ( 1) are given in Table 1; this clearly shows the dependence of the competing substrate effect upon the ratio so/K,. In every case in Table 1 the experimental value of the velocity obtained with mixed substrates agreed well with
PROGRESS
Substrate 2-NPS 3-NPS 4-NPS 2- + 3-NPS 2- + 4-NPS 3- + 4-NPS 2-+3-+4-NPS
%lL 3.8 3.9 4.0
” 28.3 37.4 50.4 39.1 48.1 50.9 48.1
rzIJ
65.7 78.7 87.7 116.1
CURVES
Although these relationships may be interesting, they are, as already pointed out, of little general interest in the present context.
TABLE THE EFFECT OF THE MAGNITUDE OF THE RATIO so/K, DROLYSIS OF THE THREE ISOMERIC MONONITROPHENYL
[51
1
ON COMPETING SUBSTRATE EFFECTS DURING THE HYSULFATES BY THE ARYL~ULFATASE OF Helix pomatia Activity
0.60 0.61 0.58 0.41
%lKll
”
zv
0.19 0.19 0.20
6.1 8.9 11.0 13.2 15.5 17.1 19.7
15.0 17.1 19.9 26.0
Activity
0.88 0.9 1 0.86 0.76
Note. The assays were carried out in a pH-stat using a partially purified enzyme (1). The column headed Zv is the activity expected in the absence of competing substrate effects. Substrates: potassium salts of 2-nitrophenyl sulfate (2-NPS; K,, 9.4 mM); 3-nitrophenyl sulfate (3-NPS; Km, 3.5 mM); 4-nitrophenyl sulfate (4-m; K,,,, 3.4 mM).
KINETIC
IMPLICATIONS
IN ANALYTICAL
ENZYME
1.5
USE
Here the integrated Michaelis equations must be used to supply the necessary information. Those corresponding to Eqs. [I], [2], and [3] are given in Eqs. [6], [7], and [8], respectively, in which s is the substrate concentration at time t so that the concentration of products is then (sO - s):
1
t = jp K,ln:
+ (,sO- 3)
161 Reaction
(mid
FIG. 2. Progress curve for an enzyme functioning under first-order conditions, when s, Q K,,,. The conditions are as for curve A in Fig. 1 except that so is 4 pm01 * liter-‘.
t=$!*n?!! s t=-
time
1 K,ln~+(~O-s) V[ s
+&6-s”) 2
.
1
VI are no extraneous interfering factors: for ex-
Neither Eq. [6] nor Eq. [8] can be rearranged to give s as an explicit function of t but Eq. [7], which holds when so 4 K, and is formally identical to the integrated rate equation for a first-order reaction, can be treated to give Eq. [9] which is often a more convenient form to use: s = Soe-‘Vl”“. PI It must be stressed, however, that all of these integrated equations apply only when there
ample, product inhibition or inactivation of the enzyme. Such a situation is admittedly unlikely in the present context. Computed plots of Eqs. [6] and [8] are given in Fig. 1 to show the effect of substrate inhibition on the shape of the progress curve while Fig. 2 shows the corresponding plot of Eq. [7], when so 4 K,,, . The different shapes of these curves, particularly the purely exponential curve corresponding to Eq. [7], should be noted. CHOICE OF ENZYME CONCENTRATION
Equations [6] and [7] can be normalized in terms of two parameters, CYand p; cy is any chosen degree of hydrolysis of the substrate and is, therefore, equal to 1 - s/s0 while fl is the relative substrate concentration, equal to so/K,,,. Substitution of cyand ,&3into Eqs. [6] and [7] and rearrangement give Eqs. [lo] and [ 111, respectively: 0
25
50 Reaction
time
75
I 100
(mid
FIG. I. Progress curves for a typical enzyme reaction showing Michaelis kinetics (Curve A) and one subject to substrate inhibition (Curve B) computed from Eqs. [6] and [S], respectively. The kinetic parameters in the latter are those for @-glucuronidase (2). K,,,, 0.29 mmoI.liter-‘; K2, 7.8 mmol -liter-‘; soI 10 mmol. liter-‘, V, 0.2 rrnol. liter-’ . min-‘.
V=T
a-bln(l
-CX)
[ V=$ln(l = ?.-jln(l
1
[lOI
-cu) - cu)
[**I
16
A. B. ROY
A similar expression can be obtained from Eq. [8] but is of little practical use because it contains K2 and 12which are rarely known. Figure 3 shows computed plots of [CY- (l//3) ln(1 -a)]and-(l/P)ln(l -a)forauseful value of (Y, 0.99. The two functions are very similar at low values of /3 but the former tends asymptotically to (Yat high values of /3, that is, at high relative substrate concentrations. In principle, Eqs. [lo] and [ 1 I] can be used to estimate the amount of enzyme required to give a 99% hydrolysis of a substrate, at initial concentration, so, in any chosen time, t. An example of such a calculation is given in Table 2. In practice, unfortunately, it will frequently be difficult to obtain such estimates because of the lack of knowledge of the appropriate values of K,,, and of the specific activity. Available values of these parameters will rarely have been obtained for conditions comparable to those in, for example, bile. Nevertheless, such estimates should be sought because they will allow the amount of enzyme to be kept to a minimum. Not only will this minimize both the expense and the effects of any contaminating enzyme which might inadvertently be present, but it will make less ambiguous the interpretation of experiments in which the activity of the enzyme is to be inhibited. Once again the problem is simplified by
TABLE 2 CALCIJLATIONOFTHEAMOUNTOFSULFATASE REQUIREDTOHYDROLYZEAGIVENAMOIJNT OFSUBSTRATEINACHOSENTIME Substrate, so Kn Reaction time a = 1 - s/so B = solKn
Estrone sulfate, 1 X IO-’ M 0.1 x lo-‘M 300 min 0.99 10
From Eq. [lo] v= 1.0 x 1o-3 X 1.45 mol. liter-’ *min-’ 300 = 5 X IO+ mol. liter-’ . min-’ Note. One unit of enzyme corresponds to a rate of 1 X 10m6mol. liter-’ * min-I; therefore the required concentration is 5 units * liter-‘.
using conditions such that so 4 K,. Under these conditions Eq. [ 71 applies and the reaction is kinetically first order. It should be relatively simple to chose a concentration of enzyme and estimate, approximately, t,,* for the reaction: that is, the time taken for the substrate concentration to fall to half its initial value. This parameter is independent of so for a first-order reaction. It follows from Eq. [8] that K,,,/V is equal to t&n 2. If a degree of hydrolysis a! is now required, the appropriate time, t, , for the same concentration of enzyme will be given by Eq. [ 121, which can be derived from Eq. [7] as follows: I,=$.ln
looOl-
= -$.ln(l = -t,/2*
O.OJ.Jo
B
FIG. 3. Plots of [a - (l/@ln(l - a)] (curve A) and -( l/@ln( 1 - a) (curve B) against 8. Details are given in the text.
So so(l - 4 -(Y) ln(1 - (Y) In2 *
[Ql
For CYequal to 0.99, t, is approximately 7t1,2. It is obvious from the above relationships that t, is indirectly related to V so that the concentration of enzyme can readily be adjusted should the estimated value of t, be inappropriate. While only approximate, to
KINETIC
IMPLICATIONS
IN ANALYTICAL
the extent that the estimated value of tllZ is approximate, this method of calculating the required amount of enzyme is preferable to the empirical approach usually adopted and it must be stressed that the value of K,,,/V which it provides pertains to the actual system under investigation.
Once again in the present context ui (uO in the presence of an inhibitor) is of little interest but it is worth pointing out that at low relative substrate concentrations the usual rate equations for competitively and noncompetitively inhibited reactions both reduce to Eq. [ 131, V
“‘= K,(l
1131
+ ilK,jso’
17
USE
0.75 “0° y 1 lA y=10 \
$
y=100 \
0.50
0.25
-
0
INHIBITED REACTIONS
ENZYME
1
10
102 Reaction
time
103 (mid
y=1000 \
10
105
FIG. 4. Plots showing the variation with time of pi/p0 (p,,,pi: amount of product produced in uninhibited and inhibited reactions, respectively) at different relative inhibitor concentrations. First-order conditions, s,, * K,,,, therefore the values of poand pi are given by Eqs. [7] and [ 161, respectively. K,,,, 0.1 mmol *liter-‘; V, 1.5 pm01 * liter-’ * min-‘; y = i/K,, shown on the appropriate curves.
effects of competitive or noncompetitive inhibitors on an enzyme reaction subject to from which it follows that in either case the substrate inhibition can be complex (4) and, inhibition is kinetically noncompetitive be- although pertinent, cannot be considered cause the relative activity of the inhibited re- here. action, ui/uo, is equal to Ki/(Ki + i) and Equations [ 141 and [ 161 show what is aptherefore independent of the substrate con- parently often forgotten: that no matter what centration. the degree of inhibition of u. (provided it is Appropriate modifications of the inteless than 100%) the reaction will go to comgrated equations, Eqs. [6] and [7], must be pletion given sufficient time, as exemplified used to provide the required information. in Fig. 4. The curves therein were computed Equation [6] gives Eqs. [ 141 and [ 151 for from Eqs. [7] and [ 161 with different values competitive and noncompetitive inhibition, of i/Ki (y) in the latter. This comment also respectively, and both reduce to Eq. [ 161, applies to the competing-substrate effects alwhich can also be derived directly from Eq. ready mentioned: they will disappear as the [8] when so 4 K,,,: reaction time increases. Figure 5 illustrates the importance of set=b 1 + i K,ln: + (so - S) [I41 lecting the concentration of enzyme approI [( I1 priate to the chosen time of hydrolysis, taking as an example the hydrolysis of estrone K,,,ln~+(~o-s) 1151 sulfate, at an initial substrate concentration of 1 mM, by the sulfatase of H. pomatia. The appropriate concentration of enzyme was [161 calculated in Table 2 and the curves in Fig. 5 were computed using the values of the paEquation [ 161 can be rearranged to give s as rameters in the former. Curve A shows the an explicit function oft which is of the same course of the hydrolysis under these condiform as Eq. [lo], differing only in that Km in tions while curve B shows it in the presence the latter is replaced by ( 1 + i/Ki)K,. The of a competitive inhibitor at a concentration
I
18
A. B. ROY
of 100 Kj, which would cause a 90% inhibition of vo. It is clear that at the chosen time, 300 min (Table 2), the extent of the inhibition is considerable but, as already pointed out, this decreases as the reaction time is increased. Curve C shows the progress in the presence of the same concentration of inhibitor but at 10 times the calculated concentration of enzyme. It is obvious that if the time of 300 min had been arbitrarily chosen and the appropriate amount of enzyme not used then it might well have been concluded that no inhibition had occurred, especially when it is remembered that these methods are frequently used only qualitatively, or at best semiquantitatively, to characterize compounds located by, for example, autoradiography of TLC plates. The curves in Fig. 5 should demonstrate quite clearly the interplay among the time of reaction, enzyme concentration, and inhibitor concentration and therefore the importance of checking experimentally that the correct conditions have indeed been chosen, Again there are advantages in working at low relative substrate concentrations, as illustrated by the computed curves in Fig. 6 which is directly comparable to Fig. 5 except
3
0.25
i 01
4 Reaction
time
hlid
FIG. 5. Progress curves for uninhibited (curve A) and inhibited (curves B and C) enzyme reactions computed using Eqs. [6] and [ 141. The vertical line shows 300 min, the time chosen in calculating the required enzyme concentration (Table 2). K,,,, 0.1 mmol.liter-‘; sa, 1 mmol. liter-‘; ilKi, (curves B and C) 100; V, 5 pm01 . liter-’ . min-’ (curves A and B), 50 rrmol . liter-’ - min-’ (curve C).
10.0
A
B
2 1 7.5 P .? 5.0 E 3 2.5 Lt '-I%,
0
1
10
103 Reaction
time
103 (mid
104
105
FIG.6. Progress curves for uninhibited (curve A) and inhibited (curve B) enzyme reactions under first-order conditions. Computed from Eqs. [7] and [ 161. K,,,, 0.1 mmol . liter-‘; so, 10 pmol . liter-‘; i/Ki (curve B), 100; V, 1.5 rrmol . liter-’ . min-‘.
that so was 0.01 mM (/3 = 0.1). The appropriate enzyme concentration, again for a reaction time of 300 min, was obtained from Eq. [ 111 in the same way as shown in Table 2. Curve A shows the control hydrolysis and curve B shows the hydrolysis in the presence of a competitive inhibitor at a concentration of 100 Ki, the same as in Fig. 5. It can be seen by comparing these two figures that the same concentration of inhibitor has a much greater effect under first-order conditions (Fig. 6) than under Michaelis conditions (Fig. 5), as is of course to be expected since the inhibition is competitive. In terms of initial velocities, the relative activity, ui/uo, is 0.1 in Fig. 5 (@ = 10) and 0.01 in Fig. 6 (j3 = 0.1). As already noted, the effect of an inhibitor on u. is not informative when essentially complete hydrolysis of the substrate is required and a more useful parameter under these conditions is AtliZ, defined as the difference between the values of tI12 for the inhibited and uninhibited reactions. It follows from Eqs. [7 ] and [ 141 and [7] and [ 151 that this parameter is given by Eqs. [ 171 and [ 181 for competitive and noncompetitive inhibition, respectively: At 112
K,i =
E
ln
2
[I71
KINETIC
K,i At ,,* = K
I
IMPLICATIONS
In 2 + 0.5 +$
IN ANALYTICAL
I
,
(K,,Jn 2 + 0.5.~~).
USE
19
firming the much greater extent of the inhibition in the latter, and showing it more informatively than by the effect on uo.
; = &
ENZYME
[ 181
When so 4 K,,,, Eq. [ 181 reduces to Eq. [ 171, which can also be derived directly from Eq. [ 161, again showing that competitive and noncompetitive inhibitors are kinetically indistinguishable when so 4 Km. The values of AtllZ for the pairs of curves in Figs. 5 and 6 are 1390 and 4620 min, respectively, con-
REFERENCES 1. Roy, A. B. (1987) Anal. Biochem. 165, 1-12. 2. Walker, P. G., and Levvy, G. A. (1953) Biochem. J. S&56-65.
3. Robinson, D., Smith, J. N., Spencer, B., and Williams. R. T. (1952) , B&hem. J. 51.202-208. 4. Krupka, R. M., and Laidler, K. J. (196 1) J. Amer. Chem. Sm. 83, 1448-1454, 1454-1458.