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An alternative method for determining inhibition rate constants by following the substrate reaction
J. theor. Biol. (1990) 142, 531-549
An Alternative Method for Determining Inhibition Rate Constants by Following the Substrate Reaction Z H I - X I N W A N G t AND C H E N - L u
Tsout
Laboratory o f Molecular Enzymology, Institute of Biophysics, Academia Sinica, Beijing 100080, China (Received on 1 August 1989, Accepted on 13 October 1989) An alternative plotting method is described by which microscopic inhibition rate constants can be determined by following the substrate reaction in the presence of the irreversible inhibitor. Not only does this method keep the advantage of Tsou's original method (Tsou, 1965a. Acta Biochern. biophys. Sin. 5, 1028-1032; 1965b. Acta Biochem. biophys. Sin. 5, 409-417), but also is suitable for the cases where the consumption of substrate and accumulation of product must be taken into account.
Introduction
Some years ago, a systematic study on the kinetics of irreversible modification of enzyme activity was proposed (Tsou, 1965a, b; see Tsou, 1988). From the equations derived for the substrate reaction in the presence of inhibitor, the apparent rate constant for the irreversible inhibition of enzyme activity can be obtained in one single experiment. Since the republication of this approach in a Western Journal (Tian & Tsou, 1982) experimental studies based on this approach have produced useful results for the irreversible inhibition kinetics of enzyme activity in this and other laboratories (Bieth, 1984; H a r p e r & Powers, 1984; Harper et al., 1985; Mason et al., 1985; Liu & Tsou, 1986; Soulie et al., 1987; Wijnands et al., 1987, Kam et al., 1987; Crawford et al., 1988; Zhou et al., 1989). Moreover, although previous kinetic treatments have dealt almost entirely with enzymes involving a single substrate, this a p p r o a c h can be easily adopted to inhibition studies of enzyme reactions with two substrates (Wang & Tsou, 1987; Wang et al., 1988). In our previous publications it was assumed that substrate concentration remains constant during the course o f irreversible inhibition of enzyme activity. At the same time the possible inhibitory effect of product was not considered. However, in some cases where the inactivation process is slower or there is stronger product inhibition, effects of substrate depletion or product accumulation cannot be neglected. It is the propose of the present paper to examine the kinetics of substrate reaction during irreversible inhibition in such cases.
t t Current address: Box 445, Baker Laboratory, Department of Chemistry, Cornell University, Ithaca, NY 14853-1301, U.S.A. 531 0022-5193/90/040531 + 19 $03.00/0
(~) 1990 Academic Press Limited
532
z.-x.
WANG
C . - L . TSOU
AND
Theory 1. S I M P L E
ONE-SUBSTRATE
REACTION
(a) Noncomplexing inhibition Y
Y
+
k,
E+S.
k~
k~
+
' ES
k'~ EY + S ~k, '
~E+P
ESY
SCHEME 1
The irreversible inhibition of enzyme E, by the inhibitor Y, during reaction of the substrate S, can be represented by Scheme 1, where ES, E Y and ESY are the respective binary and ternary complexes. Let [Er] and [E*] represent the total concentration of those complexes of the enzyme without and with Y respectively, we have [ET]=[E]+[ES], [E*]=[EY]+[ESY] and [E]o=[Er]+[E*]. As the irreversible inhibitions are usually slow reactions compared to the set up of the steady state of the enzymatic reaction, applying the steady-state assumption to E and ES, we have the Michaelis relations;
[E] [ES] =
Km[ ET]
(1)
Km+[S]' [S][ET] Km+[S]
(2) '
where Km is the Michaelis constant. As is usually the case, both IS] and [Y] are >>[E]o, the rate of inhibition of the enzyme can be written as; d[Er] dt
k+o[ Y][E] + k'o[ Y][ES] =
k+oKm+ k'+o[S] [ Y][ET]. Km+[S]
(3)
If the change of substrate concentration cannot be neglected, (k+oKm+k'o[S])/ (KIn+IS]) is no longer a constant and eqn (3) cannot therefore be treated as before (Tsou, 1965a, b; 1988). The rate of disappearance of substrate is;
d[S] d[P] . . . . dt
dt
k2[ ES]
k2[S][ET] Km+[S] "
(4)
Combining eqns (3) and (4), d[Er] -
k+oK~ + k'o[S] [ Y] d[S], k2[S]
(5)
with the boundary condition [S] = [S]o and [ET] = [E]o, this integrates to;
[E]o_[ET]_k+oKm[Y] k2
[S]o+k'o[Y]
In [S]
k2
([S]o- [S]).
(6)
METHOD FOR DETERMINING INHIBITION RATE CONSTANTS
533
When t approaches infinity [ E r ] = 0 , if the substrate has not been completely consumed and its remaining concentration is I S l e , eqn (6) can be written as: [E]o
k÷oKm[ Y] In [S]o + k~-o[ Y] k2 [S]~ k_, ([S]o - [ S]~.) k+oKm[ Y] In [S]o ~-k'~o[ Y] [p]~. k2 [ S ] o - [P]~ k2
(7)
For noncompetitive, k+o= k'o, competitive, k +' o-- 0, and uncompetitive k+o = 0, inhibitions, eqn (7) becomes respectively; Noncompetitive
[E]o
Competitive
[E]o
Uncompetitive
[E]o-
k+oKm[ Y] k2
In
[S]o k+o[ Y] + [P]~, [S]o-[P]~ k2
k+oKm[ Y] In [S]o k2 [S]o-[P]~' k~-o[ r][P]oo k2
(8) (9) (10)
Laidler & Bunting (1973) also discussed the substrate reaction during enzyme inactivation where substrate provides complete protection (competitive) or has no effect (noncompetitive) on the inhibition of enzyme activity. The case of uncompetitive irreversible inhibition (sensitization by substrate was the terminology) i.e. only ES complex was susceptible for inhibition was not considered kinetically. However, it should be pointed out in this connection that although [ES] does decrease during the course this does not prevent a similar kinetic treatment as in the cases for competitive and noncompetitive inhibition in which as [E-c] decreases, [El and [ES] will likewise decrease. Substitution of eqn (6) into eqn (4) yields; -
d[ d ~S -] - Kin+ k2[S][S] ( [E]o k+oKm[ k 2 Y] In [[S]o S~ k----7([S]o-[S]) k'o[ Y] ].
(11)
From eqn (11) it can be easily shown that when t ~ 0, [S] ~ [S]o. The initial rate of enzyme catalyzed reaction is the same as the uninhibited rate. Equation (7) may be rearranged to give: [P]~ [E]o
k+oKm - - I n k~-o[E]o
[S]o k2 +- I S ] o - [P]~ k~-o[Y ]
(12)
By keeping [S]o constant and changing [E]o, a series of [P]~ will be obtained and from eqn (12), it can be seen that a plot of [P]oo/[E]o against (1/[E]o) ln{[S]o/([S]o-[P]~)} should give a straight line with a slope of (-k+oKm/k'o[E]o) and intercepts on the vertical and horizontal axes of k2/k'o[ Y] and k2/k+o[ Y]Km respectively, as shown in Fig. 1. As Kin, k2 can be obtained from experiment without the inhibitor, the microscopic rate constants, k÷o and k'0 can then be determined. For competitive inhibition, k + o - 0, the plot becomes a vertical straight line, and for uncompetitive inhibition, k+o=O, a horizontal line will be obtained. t
--
534
Z.-X. W A N G
AND
C.-L. T S O U
kz
k.~o [Y]
This point corresponds [El 0 = O, [P]oo=O
/ /
/
/
/
/
,\
I
k+oKm
k~o
/ / /
///''"~Slope = [5"1o / /
/
kz k+oK,.[Y]
I
--
I
[£]o FIG. 1. Schematic plot of the Scheme 1.
[n
[P]~o/[E]o against
IE]o
[S]o -[P]®
( l / [ E ] o ) In
{[S]o/([$]o-[P]~o)} for
a reaction obeying
As in the case of the Foster plot (Foster & Niemann, 1953) a straight line from the origin with a slope o f [S]o will strike the line at a point that corresponds to the limit condition of [E]o = 0 and [P]oo = 0. This may be seen from the fact that the slope of the line connecting the origin to the point in which [E]o = 0, [P]oo = 0 is equal to
[P]~/[E]o
lim - [S]o. [p]=-o (1/[E]o) In {[S]o/([S]o-[P]=)} Of course, this point is only a limit point approached with the decrease of [E]o and is unobtainable from experiments directly. However, it is important for choosing suitable values of [S]o. If the value of [S]o is too small, however change the value of [E]o, all of experimental points will be centered under the limit point. Hence, in order to obtain a more proper distribution for experiment points, the value of [S]o should be suitable increased. (b) Complexing inhibition For inhibitors complexing reversibly with the enzyme to form E Y before the irreversible modification step leading to EY' as shown in Scheme 2. We have now [ET] = [ E ] + [ES] + [EY] and [E]o = [ET] + [EY']. The complexing inhibitors are usually sufficiently similar to the substrate in structure, they most
METHOD
FOR
DETERMINING
INHIBITION
RATE
CONSTANTS
535
Y +
E+S.
kl
k2
"ES
k_l
=E+P
EY EY' SCHEME 2
probably occupy the substrate binding site so that ternary complex EYS does not form and for the same reason, it is also assumed in the following discussion that the complexing forming step is fast relatively to the subsequent modification step. Applying steady-state assumption to the reaction enclosed by dashed lines in Scheme 2, from the [ ES] and [EY] derived, the rate of inactivation and of product formation can be written as; d[Er]
dt
,
(13)
Km ( I + [ Y]~ + [S] Kt /
d[S]
- --
k,[ Y]Km[ET]/K,
k,[EY] -
kE[S][ ET]
= k2[ES] =
(14)
Km( I + [ Y]'~ +[ S] K~ ]
dt
From the above, we have
k,Km[ Y] diS]. KIk2[ S]
d i E t ] - -
(15)
When t = 0, IS] = [S]o and [ET] = [E]o, the above equation can be integrated to give; [ E ] o - [ET] =
k,Km[ Y] [S]o In K,k2 [ S ] o - [P]"
(16)
When t ~ co, [ E T ] = 0 and [ P ] = [ P ] ~ , eqn (16) becomes k,Km[ Y] [E]o
[S]o
K,k--------~In [S]o-[P]oo"
(17)
By plotting [E]o against In {[S]o/([S]o-[P]~)}, at constant [S]o, a straight line can be obtained with a slope of k;[ Y]Km/Ktk: from which the ratio of kffK~ can be calculated. Substitution of eqn (16) into eqn (14) gives;
-
d[S] d--~=v=
k2[S] {[E]o4 k,Km[ Y] [S] ( +[Y]~+[S] K , k - - - ~ l n [ - ~ o J.
Kml-kS,/
(18)
536
Z.-X. WANG
AND
C.-L. TSOU
When t -~ O, [S] -~ [S]o, eqn (18) becomes, after rearrangement, 1 Km+[S]o Km[Y] ÷ Vo k2[S]o[E]o k2[S]o[E]oK,"
(19)
As [S]o, Km and k 2 a r e known, Kz can be obtained from the plot of 1/Vo against [ Y]. The expressions for the initial rates of enzyme reaction, Vo, are different for complexing and noncomplexing inhibitors in that eqn (19) contain the term [ Y], whereas the expression for the noncomplexing inhibition does not. Hence, not only can K~ be obtained from the plot of 1/Vo against [ Y], but also noncomplexing and complexing types of irreversible inhibition can be differentiated.
2. I N H I B I T I O N
BY P R O D U C T
(a) Noncomplexing inhibition First, let us consider the case in which the product of the reaction is a competitive inhibitor. The modification of enzymes in the presence of the irreversible inhibitor can be represented as in Scheme 3. Y +
E + S.
kl '
Y
Y
Y
+
+
+
ES
k, "
~
Kp E+P
E P .
"
r: EY
+ S .
• ESY
EPY
_
• EY
+ P
SCHEME 3
Assuming that the irreversible modification reactions are relatively slow reactions compared to the rapid establishment of the steady state of enzymatic reaction and the binding of product to the enzyme. Let [ E r ] = [ E ] + [ E S ] + [ E P ] , [E*]= [ E Y ] + [ E S Y ] + [ E P Y ] and [E]o=[ET]+[E*r], it can be shown that the rates of modification of the enzyme and the appearance of product are, respectively; d[Er] dt
(k+o[E] + k'o[ ES] + k'~o[ EP ])[ Y] k+oKm + k~-o[S] + k'~o[ P]Kml Kp [ Y][ Er],
Km(I+[P]~÷[S] Kp] d[S] - --= dt
k2[ES] =
k2[ S][ Er] K m ( l + ~ p ]) + I S ]
,
(20)
(21)
M E T H O D FOR D E T E R M I N I N G
INHIBITION
537
RATE C O N S T A N T S
and, d[ET]-
k+oKm+ k~o[S] + k~oKm([S]o- [S])/Kp [ y ] d[S]. k2[S] (22)
With the boundary conditions [S] = [S]o, [Er] = [E]o, this integrates to; [ E ] o - [Er] - k+o[ Y]Km In IS]o+ k~-o[ Y] ( I S ] o - IS]) k2 [S] k2
+k~°[Y]Km{[S]oln[S]°-([S]o-[S])}. Kpk2
(23)
--~
Substitution of eqn (23) into eqn (21) gives; I)=--
k2[S] { k+oKp+ k~o[S]o [ Y]Km In [S]o dt =Km{I+([P]/Kp)I+[S] [E]o Kpk2 [S---]
d[S]
-
) k '+oKp k~oKm [ Y]([S]o- [S])[. ) Kpk2
(24)
When t ~ 0, [S] ~ [S]o, eqn (24) becomes; Vo= lim
d[S]
k2[S]o[E]o
dt
Km+ [S]o"
,~o
(25)
The enzyme is completely inactive when t ~ co, we therefore have [ E r ] = 0, [S] = [S]~o, and eqn (23) can be written as; [E]o = k+oKp+ k~o[S]o [ Y]Km In
Kpk2
[S]o + k'+oKp- kgoKm [ y ] [ p ] ~ . [S]o - [P]~ Kpkz
(26)
Equation (26) can be rearranged to give; r ku [E]o_k+oKp+k'o[S]o[ n y]Kml [S]o ~_k.oKp+oKra[y], [P]~ Kpk2[P]oo [ S ] o - [P]~ Kpk2
(27)
or
[P]oo (k+oKp+k~o[S]o)Km [S]° 1 -I Kpk2 [E]o = - (k'+oKp- k'~oKm)-----~[E]---o In [ S ] o - [ P / ~ (k'+oKp- k~-ogm)[ Y]"
(28)
It can be seen from eqn (27) that [P]~ changes with the values of [E]o employed, and plots of [E]o/[P]oo vs. (1/[P]oo)In {[S]o/([S]o-[P]~)} at a series fixed [S]o values give a group of straight lines (Fig. 2) with slopes of; s-
k÷o[ Y]Km
k,
÷
k~-o[Y]Km
k2Kp
IS]o,
(29)
538
Z.-X. W A N G
AND
C.-L. T S O U
\
_ _ k + o I5'Io"~ KIn[Y] (A.I.O+ "~-~p J
k~.o[rl
k.~'OKm[ Y]
k2
k2%
I --In IP}=
FIG. 2. Plots of [E]o/[P]~ against (1/[P]~o) In tively by the product.
Increasing [S] 0
IS] o - [S]o - [ P ] =
{[S]o/([S]o-[P]~)} for
a reaction inhibited competi-
and ordinate intercepts of, k~-o[Y] i- - -
k2
k~o[ Y]Km
k2Kp
(30)
The slope, as can be seen from eqn (29) is linear function of [S]o. Secondary plot of s vs. [S]o gives the ordinate intercept and the slope as (Fig. 3): intercept - k~o[ Y] Km,
k2Kp
(31)
k+o[ Y]Km (32) k2 As Kin, Kp and k 2 c a n be obtained in experiments without the inhibitor, the microscopic rate constants, k+a, k+o and k+o can then be calculated from eqns (30-32). Alternative method of plotting, which is similar to that proposed by Foster et ai. (1953) may also be used to get k+o, k'o and k-~o. According to eqn (28), a plot of [P]~/[E]o against (1/[E]o)ln{[S]o/([S]o-[P]~)} at constant [S]o will give a straight line with the slope and intercept indicated in Fig. 4. As in the case without slope =
M E T H O D FOR D E T E R M I N I N G
INHIBITION
RATE C O N S T A N T S
539
k+'oK,.tYI J o
k*oKm[ kz Yl [S]o FIG. 3. Replot of the slopes against [S]o.
/i/ ~
k;oK~k+oKm
/ Line of slope [S]o ~
I (5"1o t n - IE] o IS]o - [P]=
FIG. 4. Schematic plot o f
[P]=/[E]o
against ( 1 / [ E ] o ) In
2Kp
{[S]e/([S]o-[P]~o)}
by the product. The experimental points shown (---0---) are hypothetical.
for a reaction inhibited
540
Z.-X. W A N G A N D C.-L. T S O U
product inhibition, a line drawn through the origin with a slope equal to the initial substrate concentration, IS]o, will again strike the line at a point corresponding to the condition of [E]o = 0, [ P ] ~ = 0. At various initial substrate concentrations the results will therefore be as represented schematically in Fig. 5. The points of intersection will fall on a straight line of slope -k+oKm/k'+o and of intercept k2/k'o[ Y] and k2/k+o[Y]Km, as in the uninhibited case, since the initial points are uninfluenced by the products.
k2 / Ko[×l
/E% I t, ]o3
!/[s]~ /
1//
/
,
IS] o
~-'oo~n IS]o- [p]= FIG. 5. The determination of the microscopic inhibition rate constants during enzyme modification reaction subject to competitive product inhibition. [S]o ,-[S]o4 represent different initial substrate concentrations. The experimental points shown (----43--) are hypothetical. For each value of [S]o, the full circle is obtained by extrapolating the line through the experimental points back to the limit point of [ P ] ~ = 0, [ E ] o = 0 . These extrapolated points lie on a straight line with slope of -k+oKm/k'+ o and intercepts of k2/k+o[ Y]Km and k2/k'+o[Y] on the abscissa and ordinate, respectively.
From the line through the limit points k+o and k~-o can be obtained when the values of k2, Km and Kp are known, and from the intercepts through the line for the individual runs (see Fig. 4) k2Kp/(k'oKp-k~oKm)[Y] can be found, it is therefore possible to separate k+o, k'o and k~-o. If the inhibition by a product is of the mixed type the reaction scheme is shown in Scheme 4.
METHOD
FOR
DETERMINING
INHIBITION K,
E . k*O[KY]
Ke EP ~
"
RATE
CONSTANTS
541
k2
ES
, E + P
,.
ESP
k.-,,[ r}
~:~{ r
EPY
ESPY SCHEME 4
The corresponding equation is;
k'o
[E]o =
k'oKs] [ Y] rp~ + k~'o[ Y] rp.2 K v I k 2 [ Joo 2k2K'p I Jo~ •
.
+Ks[Y]k2 k+o+
k" rs1JO/\
[S]o
+OL
Kp ]ln[s]o_[p]~.
(33)
In this case the plot of[P]oo/[E]o against (1/[E]o) In {[S]o/([S]o- [P]~)} at constant [S]o does not give a straight line. However, as in the case of the competitive product inhibition, the values of k÷o and k'o can still be obtained by the extrapolating method given above (see also the discussion of Comish-Bowden on Foster plot, Cornish-Bowden, 1976). Because the slope of the graph is no longer a constant but changes with [P]~, it is difficult to get the values of k~-o and k~o from the plot of [P]oo/[E]o against (1/[E]o){ln [S]o/([S]o-[P]oo)}. If the inhibitor cannot add on to ESP to form ESPY during the reaction, that is k~o = 0, eqn (33) reduces to; [E]o =
k'+o
k : o K ~ [ Y ] r p ~ + K , [ Y I [ k +k~-o[S]o'~l [S]o Kp / k : " jo~ k2 \ +o Kp ] n [ s ] o - [ P ] ~ '
(34)
which is identical with eqn (26). Hence, the same method can be used to obtain the values of k+o, k~-o and k~o. (b) Complexing inhibition Consider the following scheme in which the product is a competitive inhibitor with respect to substrate. Y + E+S.
k~
k_j
k, • ES
Kp ~ EP~
EY
EY' SCHEME 5
'
E+P
542
Z.-X. W A N G
AND
C.-L. T S O U
As complexing inhibitors are usually similar in structure to substrate, they very probably occupy the substrate binding site so that the ternary complexes E S Y and E P Y do not form. Let lET] = [ E ] + [ E S ] + [ E P ] + [ E Y ] and [E]o=[ET]+[EY']. Again applying steady-state assumption it can be shown that when t -~ oo, [Er] --- 0 and [S] = [S]~,
k,[Y]Km [S]o In Ktk2 [S]o-[P]~"
[E]o
(35)
This equation is identical with eqn (17) hence, the same method can be used for determination of kJ K~. Similarly, when t --> 0, IS] ~ [S]0, [P] --> 0, we have; d[S] -
v°=! i m - dt
k2[S]o[E]o Km(I+[Y]~+[S]o
(36)
K, / It can be seen that Kt can again be obtained by the plot of 1/Vo against [ Y].
3. R E V E R S I B L E
ONE-SUBSTRATE
REACTION
For a simple single-substrate, single-product reversible reaction, modification of the enzyme can be represented by Scheme 6. Y +
k~ E + S ~ k ,
Y + E
S
k, ~ k-z
Y + E
k? EY + S .
" ESY
EPY
k*l
P
k~ ~ k3
Y + E+P
" k* . ' EY + P k*~
SCHEME 6
Let [ E T ] = [ E ] + [ E S ] + [ E P ] , [ E * ] = [ E Y ] + [ E Y S ] + [ E Y P ] [E*]. According to steady-state assumption; [E]
-
[ES] [EP] -
(k-~k-2+ k-lk3+ k2k3)[ET]
E {kin(k_2 + k 3 ) [ S ] + E
and [ E ] o = [ E T ] +
,
(37)
k_2k-3[P]}[Er]
{k,k2[S] + k_3( k_, + k2)[ P ]}[ ET ]
,
,
where ~ = k_~ k_2 + k_l k3 + k2k3 + k, (k2 + k-2 + k3)[ S] + k-3(k-~ + k2 + k-2)[P].
(38) (39)
METHOD
FOR
DETERMINING
INHIBITION
RATE
CONSTANTS
543
The rates of enzyme modification and product appearance are respectively; d[Er] dt
(k+o[E] + k'o[ ES] + k'~o[EP])[ Y] =
{k+o(k_,k_2+ k-lk3+ k2k3)-~ k'o{k,(k_2+ k3)[S] + k_:k_3[P]} + k+o{k,k2[S] + k_3(k_,~ + k2)[P]}} [Y][ET]
diS] dt
diP] --= dt
k2EES]
-
(40)
k_2[EP]
( k,k2k3[ S] - k_,k_2k_3[ P ])[ Er]
E
V,Kp[ S] - VpK,[ P ] [G], K~Kv + Kp[ S] + K~[ P ]
(41)
where;
KKs--
k2 k3 k2 + k-2 + k3'
k-lk-2 + k-lk3 + k2k3 kl( k2 + k-2+ k3)
k_lk_ 2
Vo - k_l + k2 + k_2' k_lk-2 + k-lk3 + k~k3 Kp= k-3(k-t+k2+k-2)
From eqns (40) and (41), we have, d[Er] ..[k+o(k_,k_2+k_,k3+k2k3) k'o{k,(k_2+k3)[S]+k_2k_3[P]} d[S] - [ r j / ~ [ - ~ - )c--~k--2;_3-~ ÷ ktk2k3tS] - k-,k-2k-3[P] 4
k+o{k,k2[ S] + k-3( k-i + k2)[ P ]} ~ ~ - - ~ k-3--~ J'
(42)
If initially [ P ] = 0 , we have [ P ] = [ S ] o - [ S ] at any time. Equation (42) can be integrated with [ET] ----[ E ] o , [ S ] = [ S ] o to give: [E]o-[ET] -
k~-o(kl k_2 + kl k3 + k_2k_3) + k~-o(kl k 2 - k_ i k-3 - k2k_3) [ Y][P] klk2k3 + k-lk-2k-3
,, + k_,k3+k2k3)[S]o[Y] { (k'+ok_2+k+ok2)k,k_a(k_,k_,_ [P] ] (klk2k3+ k_ik_2k-3) 2 In 1 - [P]eqJ~ k+°(k-'k-2+k-'k3+k2k3)[Y] { ~ } klk2k3+k_lk_2k_3 In 1 -
,
(43)
where [Pica is the concentration of product present at equilibrium as defined by the Haldane relationship
[ P]eq kz k2k3 V,Ke _ [ S ] o - [P],q - k-~-~-2k-3- VpK, - K,q.
544
Z.-X. W A N G
AND
C.-L. T S O U
When the reaction time is sufficiently long, [ E r ] equal to zero. At this moment, if the substrate reaction does not reach equilibrium, we have [P] = [P]~ and eqn (43) can then be written as:
[E]o -
k~-o(k~k_ 2 + k~k 3 +
k_2k_3) +
k~o(k, k 2 - k_ ~k-3 - k2k-3)
k I k 2 k 3 + k_ 1k_2k_3
[ Y][e]o~
', (k'+ok_2+k+ok2)k,k_3(k ~k_2+ k_~k3+ k2k3)[S]o[Y] f _FPlo~I'_ - ~ (k~k2k3+k_,k_2k_3) 2 ln~.l [p]~qj
k+o(k-~k-2 + k-,k3 + k2k3)[ Y] In ~ 1 - [P]~'~ k l k 2 k 3 + k_l k-2k-3 [ [ P]eqJ"
(44)
Similarly, a plot of [E]o/[P]~ against (1/[P]~)ln{1-([P]oo/[P]eq)} will give a straight line. The slope of the straight line is slope = -
(k'+ok_2 + k+ok2)k, k_3(k_~ k_2 + k_l k3 + k2k3)[ S]o[ Y] ( k Ik2k 3+ k_ i k_2k_3)2 k+o(k_, k-a+ k - , k 3 + k2k3)[ Y] klk2k3+k ik_2k_3
(45)
As (k_,k_2+k_,k3+k2k3)/(k, k2k3+k_,k_2k_3)= V,Kd(VsG+ V~Ks) and the values of V,, Vp, Ks and Kp can be obtained from experiments of steady-state kinetics in the absence of the inhibitor, k+o can be determined from a plot of the slope against [S]o. In general, the individual constants k2, k3 etc can not be evaluated by steady-state investigations. Consequently, in order to obtain the values of k~-o and k"+o, it is necessary to use the special methods for determining k2, k3 etc. Equations for complexing irreversible inhibitors can be obtained by similar method. 4. REACTIONS WITH MORE T H A N O N E SUBSTRATE
Although similar equations can be derived for enzyme reactions involving two different substrates and two different products, the equations generally are too complicated for practical use, particularly where reversibility or product inhibition is taken into account. Hence, only the simplest case will be considered, in order to illustrate the general principles. If irreversibility and the absence of product inhibition is assumed, for a two substrate reaction following a rapid equilibrium random mechanism can be represented by Scheme 7. E R - K~ " E R S
'~
E ,' ESK, ,L k-.[ Y] I E R Y - I ~ . , E R S Y
l :lY
EY-
k3 , E + P
I~'°t~!/' I~:ot,'J
' ESY
SCHEME 7
METHOD
FOR
DETERMINING
INHIBITION
RATE
CONSTANTS
545
According to fast equilibrium assumption and [Er] = [E] + lEg] + [ ES] + lEgS], [E*] = [ EY] + l e g Y] + [ ESY] + [ ERSY], and [E]o = [ET] + [E~], we have, [El
-
[ER] -
K 'RKs[ Er ] K'R Ks + Ks[R] + KR IS] + [R][S]' Ks[ R ][ ET ]
(46) (47)
K'R Ks + Ks[R] + KR[S] + [R][S]'
[~s] -
KR[S][ET] K'RKs + Ks[R] + KR [S] + [R][S]'
(48)
[ERS] -
[S][R][ET] K'RKs + Ks[R] + KR[S] + [R][S]'
(49)
then, d[Er] - ~ = {k+o[E] + k'+o[ER ] + k+o[ES] + k~.o[ERS]}[ Y] tit {k+oK'RKs + k'oKs[ R ] + k%KR[ S] + k+o[R ][ S]}[ Y][ ET ] K ~Ks + Ks[R] + K . [ S ] + [R][S] diP]
dt
= k3[ERS] =
ka[R][S][Er] K'RKs + Ks[R] + KR[S] + JR]IS].
(50) (51)
From eqns (50) and (51), d[Er] k+oK'RKs[Y] k%oKs[ Y] d[P] - k3([R]o-[P])([S]o-[P]) ~ k3([S]o-[P]) -~ k+oge[ Y]
+ k+o[ - - Y]
k3([R]o-[P]) k3 With the boundary conditions [ET]----[E]o and [P] = 0, this integrates to, [E]o-[Er]
KR[Y]( k3
k+oK's +
\[S]o-[R]o
+ Ks[Y]( k3
(52)
) [R]o k+o l n [ R ] o _ [ p ]
k+oK'R +k'o)
\[R]o-[S]o
r q- r[S]° ~_k~o[Y][p]. In t S j o - t P J
k3
(53)
When t --> co, [ E T ] = 0 and [ P ] = [ P ] ~ , eq (53) can be written as, k ' [ E ] o - .Ku[r] [R]o . . ( . + o K s +k.+o"~In k3 \ [ S ] o - [ R ] o } [R]o-[P]~ +
+oK. [R]o-[S]o
, +k+o
)
In
[ S]o ~-k.o[ Y] [S]o-[P]~ k3 [P]oo.
(54)
546
z.-x.
WANG
AND
C.-L. T S O U
This is a nonlinear equation; however, it can be linearized if one o f the substrate is saturating at all stages; in this case, an equation identical to eqn (7) is obtained in the non-saturating substrate. This would enable k ' o , k+o and k~o to be obtained from experiments where each substrate in turn was saturating, but not k.o. When k~-'o= 0, eqn (54) simplifies to, [ E ] o - KR[Y] [ k+oK's k---'~ [,[ S]-~- [-R]o +
.,, \ [R]o I¢+o) In [R ] o - [ P]~
Ks[ Y] (. k+og'R ~-k+o~ In +
k3
\ [ g ] o - [S]o
/
[S]o [ S ] o - [P]oo"
(55)
Rearranging, we obtain, In
[S]o IS]o- [P]~
{k+oK~+ k~-o([R]o- [S]o)}KR In [R]o {k+oK'n+k'o([R]o-[S]o)}Ks [ R ] o - [P]oo +
A plot of In
[ E ]o([R ]o - [ S]o) k3 {k+oK ~ + k~-o([R ]o - [S]o)}Ks[ Y]"
{[S]o/([S]o-[P]o~)} against
In
{[R]o/([R]o-[P]~)}
/[P]'~
(56) will give a straight
I
t FIG. 6. Progress curve of enzyme-catalyzedreaction in the presence of an irreversibleinhibitor. Curve 1: actual progress curve. Curve 2: Theoretical progress curve described by equation of t)
[P] = .':7~... (1 - e-At~q'). At r j
where v is the initial velocity of the enzyme catalyzed reaction without the inhibitor and A the apparent rate constant of the irreversible inhibition, and it is assumed that the substrate concentration remains a constant during the reaction.
METHOD
FOR
DETERMINING
INHIBITION
RATE
CONSTANTS
547
line with a slope s, and an ordinate intercept, i, of {k+oK ~ + k'~o([ g ]o - [ S]o)} KR s - {k+oK ~ + k~-o([R]o- [S]o)}Ks' i-
(57)
[ E ] o ( [ g ] o - [S]o)k3 {k+oK ~ + k'o([R]o - [S]o)}Ks[ Y]"
(58)
It can be seen from eqn (58) that a replot of 1/i against 1/([R]o-[S]o) should give a straight line with a slope of k+oKsK'R[Y]/k3[E]o and an intercept of k'oKs[Y]/k3[E]o. As KR, K'R, Ks, k3 and [Y] are known, k+o and k'o can be calculated, and k+o can be obtained further from eqn (57). The microscopic inhibition rate constants for enzyme with other mechanisms can be obtained in a similar way. Discussion
It can be seen from the above that for the case where the effects of substrate consumption or product accumulation cannot be neglected, we can still determine the reaction rate constant of enzyme modification by following the substrate reaction. The present method not only keeps the advantages of Tsou's method, but also extends its range of application. However, although, in theory, the present method is more strict than that proposed by Tsou, in practice, it is less convenient, especially for enzyme catalyzed two-substrate reactions. If the consumption of substrate cannot be controlled within an appropriate range, we can use the initial part of the reaction curve and make the Guggenheim plot (Shoemaker & Carland, 1962) to determine the [ P ] * which is different from the actually measured value of [P]o~, and then obtain the inhibition kinetic parameters by the double reciprocal plot of 1 / [ P ] * against 1/[S]. However, as both the consumption of substrate and other complicated reaction mechanism can result in the curvature of Guggenheim plot (Zhou et al., 1988), in order to distinguish these two possibility, it is still necessary to plot [P]~/[E]o against (1/[E]o)In {[S]o/([S]o-[P]oo)} and see whether a straight line can be obtained. We will now discuss the conditions for which Tsou's method applies in the case of uncomplicated one-substrate reaction. Expanding the right hand side of eqn (11) to power series, we have,
[E]o_k~o[Y]
k+o[Y]Km ( -
k---~ [P]~
k2
In 1
_ 'k÷o[ Y] [ p ] ~ + k÷o[ V]Km ['[P]~
[P]o~'~ [S]o] 1 /[p]~\2
1 ([p]~s
+ When [P]oo/[S]o<<1, the term containing eqn (59) can then be written as,
1
(59)
[Sloj + .....
([P]~/[S]o)'- and higher can be neglected,
k'o[ Y] k+o[ Y]Km [E]o- - [P],+ [P]~, k2 k2[S]o
(60)
548
Z.-X. W A N G A N D C.-L. T S O U TABLE 1
Relative error caused by Tsou's method. For different values of [P]oo/[E]o [P]J[S]o
Relative error
0.01 0.03 0.05 0.10 0.15 0.20 0.25 0.30
<0.005 <0.015 <0.026 <0.056 <0-088 <0-125 <0.167 <0.214
[P]o~-
k2[ E]o[ S]o (k+oKm + k'o[S]o)[ Y]'
(61)
which is identical with eqn (12) in Tsou's paper. (Tsou, 1965b, 1988). By comparing eqns (59) and (60) it is obvious that the error caused by Tsou's method is,
oo 1 ([P]oo~" .~--2nt [S]oJ
I ( [ P ] = ~ 2 ~ ([P]oo~" 1 ([P]ool[S]o) 2 [P]~ 2 t [S]oJ .=o \ [-'~o/ =7 1-([P],~l[S]o) -2([S]o-[P]<~)[S]o"
When [P]= = 0.05[S]o,
[P]~/2([S]o- [ P]oo)[S]o
([P]J[S]o)
= 0.026.
That is to say, when the formation of product, [P]=, is 5% of [S]o, the relative error caused by Tsou's method is less than three percent, in general, it is negligible. The relative errors caused by Tsou's method for different values of [P]oo/[E]o is listed in Table 1. REFERENCES BIETH, J. G. (1984). Biochem. Med. 32, 387-397. CORNISH-BOWDEN, A. (1976). Principles o f Enzyme Kinetics. pp. 144-149. London: Butterworth. CRAWFORD, C., MASON, R. W., WIKSTROM, P. & SHAW, E. (1988). Biochem. J. 253, 751-758. FOSTER, R. J. & NIEMANN, C. (1953). Proc. hath. Acad. Sci. U.S.A. 39, 999-1003. HARPER, J. W. & POWERS, J. C. (1984). J. Am. Chem. Soc. 106, 7618-7619. HARPER, J. W., HEMMI, K. & POWERS, J. C. (1985). Biochemistry 24, 1831-1843. KAM, C. M., COPHER, J. C. & POWERS, J. C. (1987). J. Am. Chem. Soc. 109, 5044-5045. LAIDLER, K. J. & BUNTING, P. S. (1973). The Chemical Kinetics of Enzyme Action. pp. 163-180. London: Oxford University Press. LIu, W. & T s o u , C. L. (1986). Biochim. biophys. Acta 870, 185-190. MASON, R. W., GREEN, G. D. J. 8(. BARRETT, A. J. (1985). Biochem. J. 226, 233-241. SHOEMAKER,O. P. & CARLAND,C. W. (1962). Experiments in Physical Chemistry. p. 222. New York: McGraw-Hill. SOULIE, J. M., RIVIERE, M., BUC, J., GONTERO, B. & RICHARD, J. (1987). Eur. J. Biochem. 162, 271-284.
M E T H O D FOR D E T E R M I N I N G
INHIBITION
RATE C O N S T A N T S
TIAN, W. X. & TSOU, C. L. (1982). Biochemistry 21, 1028-1032. Tsou, C. L. (1965a). Acta. Biochim. biophys. Sin. 5, 398-408. TSOU, C. L. (1965b). Acta. Biochim. biophys. Sin. 5, 409-417. Tsou, C. L. (1988). Ado. Enzymol. Relat. Areas molec. Biol. 61, 381-436. WANG, Z. X. & TSOU, C. L. (1987). J. theor. Biol. 127, 253-270. WANG, Z. X., PREISS, B. & T s o u , C. L. (1988) Biochemistry. 27, 5095-5100. WIJNANDS, R. A., MULLER, F. & VISSER, A. J. W. G. (1987). Eur. J. Biochem. 163, 535-544. ZHOU, J. M., LIU, C. & T s o u , C. L. (1989). Biochemistry 28, 1070-1076.
549
An Alternative Method for Determining Inhibition Rate Constants by Following the Substrate Reaction Z H I - X I N W A N G t AND C H E N - L u
Tsout
Laboratory o f Molecular Enzymology, Institute of Biophysics, Academia Sinica, Beijing 100080, China (Received on 1 August 1989, Accepted on 13 October 1989) An alternative plotting method is described by which microscopic inhibition rate constants can be determined by following the substrate reaction in the presence of the irreversible inhibitor. Not only does this method keep the advantage of Tsou's original method (Tsou, 1965a. Acta Biochern. biophys. Sin. 5, 1028-1032; 1965b. Acta Biochem. biophys. Sin. 5, 409-417), but also is suitable for the cases where the consumption of substrate and accumulation of product must be taken into account.
Introduction
Some years ago, a systematic study on the kinetics of irreversible modification of enzyme activity was proposed (Tsou, 1965a, b; see Tsou, 1988). From the equations derived for the substrate reaction in the presence of inhibitor, the apparent rate constant for the irreversible inhibition of enzyme activity can be obtained in one single experiment. Since the republication of this approach in a Western Journal (Tian & Tsou, 1982) experimental studies based on this approach have produced useful results for the irreversible inhibition kinetics of enzyme activity in this and other laboratories (Bieth, 1984; H a r p e r & Powers, 1984; Harper et al., 1985; Mason et al., 1985; Liu & Tsou, 1986; Soulie et al., 1987; Wijnands et al., 1987, Kam et al., 1987; Crawford et al., 1988; Zhou et al., 1989). Moreover, although previous kinetic treatments have dealt almost entirely with enzymes involving a single substrate, this a p p r o a c h can be easily adopted to inhibition studies of enzyme reactions with two substrates (Wang & Tsou, 1987; Wang et al., 1988). In our previous publications it was assumed that substrate concentration remains constant during the course o f irreversible inhibition of enzyme activity. At the same time the possible inhibitory effect of product was not considered. However, in some cases where the inactivation process is slower or there is stronger product inhibition, effects of substrate depletion or product accumulation cannot be neglected. It is the propose of the present paper to examine the kinetics of substrate reaction during irreversible inhibition in such cases.
t t Current address: Box 445, Baker Laboratory, Department of Chemistry, Cornell University, Ithaca, NY 14853-1301, U.S.A. 531 0022-5193/90/040531 + 19 $03.00/0
(~) 1990 Academic Press Limited
532
z.-x.
WANG
C . - L . TSOU
AND
Theory 1. S I M P L E
ONE-SUBSTRATE
REACTION
(a) Noncomplexing inhibition Y
Y
+
k,
E+S.
k~
k~
+
' ES
k'~ EY + S ~k, '
~E+P
ESY
SCHEME 1
The irreversible inhibition of enzyme E, by the inhibitor Y, during reaction of the substrate S, can be represented by Scheme 1, where ES, E Y and ESY are the respective binary and ternary complexes. Let [Er] and [E*] represent the total concentration of those complexes of the enzyme without and with Y respectively, we have [ET]=[E]+[ES], [E*]=[EY]+[ESY] and [E]o=[Er]+[E*]. As the irreversible inhibitions are usually slow reactions compared to the set up of the steady state of the enzymatic reaction, applying the steady-state assumption to E and ES, we have the Michaelis relations;
[E] [ES] =
Km[ ET]
(1)
Km+[S]' [S][ET] Km+[S]
(2) '
where Km is the Michaelis constant. As is usually the case, both IS] and [Y] are >>[E]o, the rate of inhibition of the enzyme can be written as; d[Er] dt
k+o[ Y][E] + k'o[ Y][ES] =
k+oKm+ k'+o[S] [ Y][ET]. Km+[S]
(3)
If the change of substrate concentration cannot be neglected, (k+oKm+k'o[S])/ (KIn+IS]) is no longer a constant and eqn (3) cannot therefore be treated as before (Tsou, 1965a, b; 1988). The rate of disappearance of substrate is;
d[S] d[P] . . . . dt
dt
k2[ ES]
k2[S][ET] Km+[S] "
(4)
Combining eqns (3) and (4), d[Er] -
k+oK~ + k'o[S] [ Y] d[S], k2[S]
(5)
with the boundary condition [S] = [S]o and [ET] = [E]o, this integrates to;
[E]o_[ET]_k+oKm[Y] k2
[S]o+k'o[Y]
In [S]
k2
([S]o- [S]).
(6)
METHOD FOR DETERMINING INHIBITION RATE CONSTANTS
533
When t approaches infinity [ E r ] = 0 , if the substrate has not been completely consumed and its remaining concentration is I S l e , eqn (6) can be written as: [E]o
k÷oKm[ Y] In [S]o + k~-o[ Y] k2 [S]~ k_, ([S]o - [ S]~.) k+oKm[ Y] In [S]o ~-k'~o[ Y] [p]~. k2 [ S ] o - [P]~ k2
(7)
For noncompetitive, k+o= k'o, competitive, k +' o-- 0, and uncompetitive k+o = 0, inhibitions, eqn (7) becomes respectively; Noncompetitive
[E]o
Competitive
[E]o
Uncompetitive
[E]o-
k+oKm[ Y] k2
In
[S]o k+o[ Y] + [P]~, [S]o-[P]~ k2
k+oKm[ Y] In [S]o k2 [S]o-[P]~' k~-o[ r][P]oo k2
(8) (9) (10)
Laidler & Bunting (1973) also discussed the substrate reaction during enzyme inactivation where substrate provides complete protection (competitive) or has no effect (noncompetitive) on the inhibition of enzyme activity. The case of uncompetitive irreversible inhibition (sensitization by substrate was the terminology) i.e. only ES complex was susceptible for inhibition was not considered kinetically. However, it should be pointed out in this connection that although [ES] does decrease during the course this does not prevent a similar kinetic treatment as in the cases for competitive and noncompetitive inhibition in which as [E-c] decreases, [El and [ES] will likewise decrease. Substitution of eqn (6) into eqn (4) yields; -
d[ d ~S -] - Kin+ k2[S][S] ( [E]o k+oKm[ k 2 Y] In [[S]o S~ k----7([S]o-[S]) k'o[ Y] ].
(11)
From eqn (11) it can be easily shown that when t ~ 0, [S] ~ [S]o. The initial rate of enzyme catalyzed reaction is the same as the uninhibited rate. Equation (7) may be rearranged to give: [P]~ [E]o
k+oKm - - I n k~-o[E]o
[S]o k2 +- I S ] o - [P]~ k~-o[Y ]
(12)
By keeping [S]o constant and changing [E]o, a series of [P]~ will be obtained and from eqn (12), it can be seen that a plot of [P]oo/[E]o against (1/[E]o) ln{[S]o/([S]o-[P]~)} should give a straight line with a slope of (-k+oKm/k'o[E]o) and intercepts on the vertical and horizontal axes of k2/k'o[ Y] and k2/k+o[ Y]Km respectively, as shown in Fig. 1. As Kin, k2 can be obtained from experiment without the inhibitor, the microscopic rate constants, k÷o and k'0 can then be determined. For competitive inhibition, k + o - 0, the plot becomes a vertical straight line, and for uncompetitive inhibition, k+o=O, a horizontal line will be obtained. t
--
534
Z.-X. W A N G
AND
C.-L. T S O U
kz
k.~o [Y]
This point corresponds [El 0 = O, [P]oo=O
/ /
/
/
/
/
,\
I
k+oKm
k~o
/ / /
///''"~Slope = [5"1o / /
/
kz k+oK,.[Y]
I
--
I
[£]o FIG. 1. Schematic plot of the Scheme 1.
[n
[P]~o/[E]o against
IE]o
[S]o -[P]®
( l / [ E ] o ) In
{[S]o/([$]o-[P]~o)} for
a reaction obeying
As in the case of the Foster plot (Foster & Niemann, 1953) a straight line from the origin with a slope o f [S]o will strike the line at a point that corresponds to the limit condition of [E]o = 0 and [P]oo = 0. This may be seen from the fact that the slope of the line connecting the origin to the point in which [E]o = 0, [P]oo = 0 is equal to
[P]~/[E]o
lim - [S]o. [p]=-o (1/[E]o) In {[S]o/([S]o-[P]=)} Of course, this point is only a limit point approached with the decrease of [E]o and is unobtainable from experiments directly. However, it is important for choosing suitable values of [S]o. If the value of [S]o is too small, however change the value of [E]o, all of experimental points will be centered under the limit point. Hence, in order to obtain a more proper distribution for experiment points, the value of [S]o should be suitable increased. (b) Complexing inhibition For inhibitors complexing reversibly with the enzyme to form E Y before the irreversible modification step leading to EY' as shown in Scheme 2. We have now [ET] = [ E ] + [ES] + [EY] and [E]o = [ET] + [EY']. The complexing inhibitors are usually sufficiently similar to the substrate in structure, they most
METHOD
FOR
DETERMINING
INHIBITION
RATE
CONSTANTS
535
Y +
E+S.
kl
k2
"ES
k_l
=E+P
EY EY' SCHEME 2
probably occupy the substrate binding site so that ternary complex EYS does not form and for the same reason, it is also assumed in the following discussion that the complexing forming step is fast relatively to the subsequent modification step. Applying steady-state assumption to the reaction enclosed by dashed lines in Scheme 2, from the [ ES] and [EY] derived, the rate of inactivation and of product formation can be written as; d[Er]
dt
,
(13)
Km ( I + [ Y]~ + [S] Kt /
d[S]
- --
k,[ Y]Km[ET]/K,
k,[EY] -
kE[S][ ET]
= k2[ES] =
(14)
Km( I + [ Y]'~ +[ S] K~ ]
dt
From the above, we have
k,Km[ Y] diS]. KIk2[ S]
d i E t ] - -
(15)
When t = 0, IS] = [S]o and [ET] = [E]o, the above equation can be integrated to give; [ E ] o - [ET] =
k,Km[ Y] [S]o In K,k2 [ S ] o - [P]"
(16)
When t ~ co, [ E T ] = 0 and [ P ] = [ P ] ~ , eqn (16) becomes k,Km[ Y] [E]o
[S]o
K,k--------~In [S]o-[P]oo"
(17)
By plotting [E]o against In {[S]o/([S]o-[P]~)}, at constant [S]o, a straight line can be obtained with a slope of k;[ Y]Km/Ktk: from which the ratio of kffK~ can be calculated. Substitution of eqn (16) into eqn (14) gives;
-
d[S] d--~=v=
k2[S] {[E]o4 k,Km[ Y] [S] ( +[Y]~+[S] K , k - - - ~ l n [ - ~ o J.
Kml-kS,/
(18)
536
Z.-X. WANG
AND
C.-L. TSOU
When t -~ O, [S] -~ [S]o, eqn (18) becomes, after rearrangement, 1 Km+[S]o Km[Y] ÷ Vo k2[S]o[E]o k2[S]o[E]oK,"
(19)
As [S]o, Km and k 2 a r e known, Kz can be obtained from the plot of 1/Vo against [ Y]. The expressions for the initial rates of enzyme reaction, Vo, are different for complexing and noncomplexing inhibitors in that eqn (19) contain the term [ Y], whereas the expression for the noncomplexing inhibition does not. Hence, not only can K~ be obtained from the plot of 1/Vo against [ Y], but also noncomplexing and complexing types of irreversible inhibition can be differentiated.
2. I N H I B I T I O N
BY P R O D U C T
(a) Noncomplexing inhibition First, let us consider the case in which the product of the reaction is a competitive inhibitor. The modification of enzymes in the presence of the irreversible inhibitor can be represented as in Scheme 3. Y +
E + S.
kl '
Y
Y
Y
+
+
+
ES
k, "
~
Kp E+P
E P .
"
r: EY
+ S .
• ESY
EPY
_
• EY
+ P
SCHEME 3
Assuming that the irreversible modification reactions are relatively slow reactions compared to the rapid establishment of the steady state of enzymatic reaction and the binding of product to the enzyme. Let [ E r ] = [ E ] + [ E S ] + [ E P ] , [E*]= [ E Y ] + [ E S Y ] + [ E P Y ] and [E]o=[ET]+[E*r], it can be shown that the rates of modification of the enzyme and the appearance of product are, respectively; d[Er] dt
(k+o[E] + k'o[ ES] + k'~o[ EP ])[ Y] k+oKm + k~-o[S] + k'~o[ P]Kml Kp [ Y][ Er],
Km(I+[P]~÷[S] Kp] d[S] - --= dt
k2[ES] =
k2[ S][ Er] K m ( l + ~ p ]) + I S ]
,
(20)
(21)
M E T H O D FOR D E T E R M I N I N G
INHIBITION
537
RATE C O N S T A N T S
and, d[ET]-
k+oKm+ k~o[S] + k~oKm([S]o- [S])/Kp [ y ] d[S]. k2[S] (22)
With the boundary conditions [S] = [S]o, [Er] = [E]o, this integrates to; [ E ] o - [Er] - k+o[ Y]Km In IS]o+ k~-o[ Y] ( I S ] o - IS]) k2 [S] k2
+k~°[Y]Km{[S]oln[S]°-([S]o-[S])}. Kpk2
(23)
--~
Substitution of eqn (23) into eqn (21) gives; I)=--
k2[S] { k+oKp+ k~o[S]o [ Y]Km In [S]o dt =Km{I+([P]/Kp)I+[S] [E]o Kpk2 [S---]
d[S]
-
) k '+oKp k~oKm [ Y]([S]o- [S])[. ) Kpk2
(24)
When t ~ 0, [S] ~ [S]o, eqn (24) becomes; Vo= lim
d[S]
k2[S]o[E]o
dt
Km+ [S]o"
,~o
(25)
The enzyme is completely inactive when t ~ co, we therefore have [ E r ] = 0, [S] = [S]~o, and eqn (23) can be written as; [E]o = k+oKp+ k~o[S]o [ Y]Km In
Kpk2
[S]o + k'+oKp- kgoKm [ y ] [ p ] ~ . [S]o - [P]~ Kpkz
(26)
Equation (26) can be rearranged to give; r ku [E]o_k+oKp+k'o[S]o[ n y]Kml [S]o ~_k.oKp+oKra[y], [P]~ Kpk2[P]oo [ S ] o - [P]~ Kpk2
(27)
or
[P]oo (k+oKp+k~o[S]o)Km [S]° 1 -I Kpk2 [E]o = - (k'+oKp- k'~oKm)-----~[E]---o In [ S ] o - [ P / ~ (k'+oKp- k~-ogm)[ Y]"
(28)
It can be seen from eqn (27) that [P]~ changes with the values of [E]o employed, and plots of [E]o/[P]oo vs. (1/[P]oo)In {[S]o/([S]o-[P]~)} at a series fixed [S]o values give a group of straight lines (Fig. 2) with slopes of; s-
k÷o[ Y]Km
k,
÷
k~-o[Y]Km
k2Kp
IS]o,
(29)
538
Z.-X. W A N G
AND
C.-L. T S O U
\
_ _ k + o I5'Io"~ KIn[Y] (A.I.O+ "~-~p J
k~.o[rl
k.~'OKm[ Y]
k2
k2%
I --In IP}=
FIG. 2. Plots of [E]o/[P]~ against (1/[P]~o) In tively by the product.
Increasing [S] 0
IS] o - [S]o - [ P ] =
{[S]o/([S]o-[P]~)} for
a reaction inhibited competi-
and ordinate intercepts of, k~-o[Y] i- - -
k2
k~o[ Y]Km
k2Kp
(30)
The slope, as can be seen from eqn (29) is linear function of [S]o. Secondary plot of s vs. [S]o gives the ordinate intercept and the slope as (Fig. 3): intercept - k~o[ Y] Km,
k2Kp
(31)
k+o[ Y]Km (32) k2 As Kin, Kp and k 2 c a n be obtained in experiments without the inhibitor, the microscopic rate constants, k+a, k+o and k+o can then be calculated from eqns (30-32). Alternative method of plotting, which is similar to that proposed by Foster et ai. (1953) may also be used to get k+o, k'o and k-~o. According to eqn (28), a plot of [P]~/[E]o against (1/[E]o)ln{[S]o/([S]o-[P]~)} at constant [S]o will give a straight line with the slope and intercept indicated in Fig. 4. As in the case without slope =
M E T H O D FOR D E T E R M I N I N G
INHIBITION
RATE C O N S T A N T S
539
k+'oK,.tYI J o
k*oKm[ kz Yl [S]o FIG. 3. Replot of the slopes against [S]o.
/i/ ~
k;oK~k+oKm
/ Line of slope [S]o ~
I (5"1o t n - IE] o IS]o - [P]=
FIG. 4. Schematic plot o f
[P]=/[E]o
against ( 1 / [ E ] o ) In
2Kp
{[S]e/([S]o-[P]~o)}
by the product. The experimental points shown (---0---) are hypothetical.
for a reaction inhibited
540
Z.-X. W A N G A N D C.-L. T S O U
product inhibition, a line drawn through the origin with a slope equal to the initial substrate concentration, IS]o, will again strike the line at a point corresponding to the condition of [E]o = 0, [ P ] ~ = 0. At various initial substrate concentrations the results will therefore be as represented schematically in Fig. 5. The points of intersection will fall on a straight line of slope -k+oKm/k'+o and of intercept k2/k'o[ Y] and k2/k+o[Y]Km, as in the uninhibited case, since the initial points are uninfluenced by the products.
k2 / Ko[×l
/E% I t, ]o3
!/[s]~ /
1//
/
,
IS] o
~-'oo~n IS]o- [p]= FIG. 5. The determination of the microscopic inhibition rate constants during enzyme modification reaction subject to competitive product inhibition. [S]o ,-[S]o4 represent different initial substrate concentrations. The experimental points shown (----43--) are hypothetical. For each value of [S]o, the full circle is obtained by extrapolating the line through the experimental points back to the limit point of [ P ] ~ = 0, [ E ] o = 0 . These extrapolated points lie on a straight line with slope of -k+oKm/k'+ o and intercepts of k2/k+o[ Y]Km and k2/k'+o[Y] on the abscissa and ordinate, respectively.
From the line through the limit points k+o and k~-o can be obtained when the values of k2, Km and Kp are known, and from the intercepts through the line for the individual runs (see Fig. 4) k2Kp/(k'oKp-k~oKm)[Y] can be found, it is therefore possible to separate k+o, k'o and k~-o. If the inhibition by a product is of the mixed type the reaction scheme is shown in Scheme 4.
METHOD
FOR
DETERMINING
INHIBITION K,
E . k*O[KY]
Ke EP ~
"
RATE
CONSTANTS
541
k2
ES
, E + P
,.
ESP
k.-,,[ r}
~:~{ r
EPY
ESPY SCHEME 4
The corresponding equation is;
k'o
[E]o =
k'oKs] [ Y] rp~ + k~'o[ Y] rp.2 K v I k 2 [ Joo 2k2K'p I Jo~ •
.
+Ks[Y]k2 k+o+
k" rs1JO/\
[S]o
+OL
Kp ]ln[s]o_[p]~.
(33)
In this case the plot of[P]oo/[E]o against (1/[E]o) In {[S]o/([S]o- [P]~)} at constant [S]o does not give a straight line. However, as in the case of the competitive product inhibition, the values of k÷o and k'o can still be obtained by the extrapolating method given above (see also the discussion of Comish-Bowden on Foster plot, Cornish-Bowden, 1976). Because the slope of the graph is no longer a constant but changes with [P]~, it is difficult to get the values of k~-o and k~o from the plot of [P]oo/[E]o against (1/[E]o){ln [S]o/([S]o-[P]oo)}. If the inhibitor cannot add on to ESP to form ESPY during the reaction, that is k~o = 0, eqn (33) reduces to; [E]o =
k'+o
k : o K ~ [ Y ] r p ~ + K , [ Y I [ k +k~-o[S]o'~l [S]o Kp / k : " jo~ k2 \ +o Kp ] n [ s ] o - [ P ] ~ '
(34)
which is identical with eqn (26). Hence, the same method can be used to obtain the values of k+o, k~-o and k~o. (b) Complexing inhibition Consider the following scheme in which the product is a competitive inhibitor with respect to substrate. Y + E+S.
k~
k_j
k, • ES
Kp ~ EP~
EY
EY' SCHEME 5
'
E+P
542
Z.-X. W A N G
AND
C.-L. T S O U
As complexing inhibitors are usually similar in structure to substrate, they very probably occupy the substrate binding site so that the ternary complexes E S Y and E P Y do not form. Let lET] = [ E ] + [ E S ] + [ E P ] + [ E Y ] and [E]o=[ET]+[EY']. Again applying steady-state assumption it can be shown that when t -~ oo, [Er] --- 0 and [S] = [S]~,
k,[Y]Km [S]o In Ktk2 [S]o-[P]~"
[E]o
(35)
This equation is identical with eqn (17) hence, the same method can be used for determination of kJ K~. Similarly, when t --> 0, IS] ~ [S]0, [P] --> 0, we have; d[S] -
v°=! i m - dt
k2[S]o[E]o Km(I+[Y]~+[S]o
(36)
K, / It can be seen that Kt can again be obtained by the plot of 1/Vo against [ Y].
3. R E V E R S I B L E
ONE-SUBSTRATE
REACTION
For a simple single-substrate, single-product reversible reaction, modification of the enzyme can be represented by Scheme 6. Y +
k~ E + S ~ k ,
Y + E
S
k, ~ k-z
Y + E
k? EY + S .
" ESY
EPY
k*l
P
k~ ~ k3
Y + E+P
" k* . ' EY + P k*~
SCHEME 6
Let [ E T ] = [ E ] + [ E S ] + [ E P ] , [ E * ] = [ E Y ] + [ E Y S ] + [ E Y P ] [E*]. According to steady-state assumption; [E]
-
[ES] [EP] -
(k-~k-2+ k-lk3+ k2k3)[ET]
E {kin(k_2 + k 3 ) [ S ] + E
and [ E ] o = [ E T ] +
,
(37)
k_2k-3[P]}[Er]
{k,k2[S] + k_3( k_, + k2)[ P ]}[ ET ]
,
,
where ~ = k_~ k_2 + k_l k3 + k2k3 + k, (k2 + k-2 + k3)[ S] + k-3(k-~ + k2 + k-2)[P].
(38) (39)
METHOD
FOR
DETERMINING
INHIBITION
RATE
CONSTANTS
543
The rates of enzyme modification and product appearance are respectively; d[Er] dt
(k+o[E] + k'o[ ES] + k'~o[EP])[ Y] =
{k+o(k_,k_2+ k-lk3+ k2k3)-~ k'o{k,(k_2+ k3)[S] + k_:k_3[P]} + k+o{k,k2[S] + k_3(k_,~ + k2)[P]}} [Y][ET]
diS] dt
diP] --= dt
k2EES]
-
(40)
k_2[EP]
( k,k2k3[ S] - k_,k_2k_3[ P ])[ Er]
E
V,Kp[ S] - VpK,[ P ] [G], K~Kv + Kp[ S] + K~[ P ]
(41)
where;
KKs--
k2 k3 k2 + k-2 + k3'
k-lk-2 + k-lk3 + k2k3 kl( k2 + k-2+ k3)
k_lk_ 2
Vo - k_l + k2 + k_2' k_lk-2 + k-lk3 + k~k3 Kp= k-3(k-t+k2+k-2)
From eqns (40) and (41), we have, d[Er] ..[k+o(k_,k_2+k_,k3+k2k3) k'o{k,(k_2+k3)[S]+k_2k_3[P]} d[S] - [ r j / ~ [ - ~ - )c--~k--2;_3-~ ÷ ktk2k3tS] - k-,k-2k-3[P] 4
k+o{k,k2[ S] + k-3( k-i + k2)[ P ]} ~ ~ - - ~ k-3--~ J'
(42)
If initially [ P ] = 0 , we have [ P ] = [ S ] o - [ S ] at any time. Equation (42) can be integrated with [ET] ----[ E ] o , [ S ] = [ S ] o to give: [E]o-[ET] -
k~-o(kl k_2 + kl k3 + k_2k_3) + k~-o(kl k 2 - k_ i k-3 - k2k_3) [ Y][P] klk2k3 + k-lk-2k-3
,, + k_,k3+k2k3)[S]o[Y] { (k'+ok_2+k+ok2)k,k_a(k_,k_,_ [P] ] (klk2k3+ k_ik_2k-3) 2 In 1 - [P]eqJ~ k+°(k-'k-2+k-'k3+k2k3)[Y] { ~ } klk2k3+k_lk_2k_3 In 1 -
,
(43)
where [Pica is the concentration of product present at equilibrium as defined by the Haldane relationship
[ P]eq kz k2k3 V,Ke _ [ S ] o - [P],q - k-~-~-2k-3- VpK, - K,q.
544
Z.-X. W A N G
AND
C.-L. T S O U
When the reaction time is sufficiently long, [ E r ] equal to zero. At this moment, if the substrate reaction does not reach equilibrium, we have [P] = [P]~ and eqn (43) can then be written as:
[E]o -
k~-o(k~k_ 2 + k~k 3 +
k_2k_3) +
k~o(k, k 2 - k_ ~k-3 - k2k-3)
k I k 2 k 3 + k_ 1k_2k_3
[ Y][e]o~
', (k'+ok_2+k+ok2)k,k_3(k ~k_2+ k_~k3+ k2k3)[S]o[Y] f _FPlo~I'_ - ~ (k~k2k3+k_,k_2k_3) 2 ln~.l [p]~qj
k+o(k-~k-2 + k-,k3 + k2k3)[ Y] In ~ 1 - [P]~'~ k l k 2 k 3 + k_l k-2k-3 [ [ P]eqJ"
(44)
Similarly, a plot of [E]o/[P]~ against (1/[P]~)ln{1-([P]oo/[P]eq)} will give a straight line. The slope of the straight line is slope = -
(k'+ok_2 + k+ok2)k, k_3(k_~ k_2 + k_l k3 + k2k3)[ S]o[ Y] ( k Ik2k 3+ k_ i k_2k_3)2 k+o(k_, k-a+ k - , k 3 + k2k3)[ Y] klk2k3+k ik_2k_3
(45)
As (k_,k_2+k_,k3+k2k3)/(k, k2k3+k_,k_2k_3)= V,Kd(VsG+ V~Ks) and the values of V,, Vp, Ks and Kp can be obtained from experiments of steady-state kinetics in the absence of the inhibitor, k+o can be determined from a plot of the slope against [S]o. In general, the individual constants k2, k3 etc can not be evaluated by steady-state investigations. Consequently, in order to obtain the values of k~-o and k"+o, it is necessary to use the special methods for determining k2, k3 etc. Equations for complexing irreversible inhibitors can be obtained by similar method. 4. REACTIONS WITH MORE T H A N O N E SUBSTRATE
Although similar equations can be derived for enzyme reactions involving two different substrates and two different products, the equations generally are too complicated for practical use, particularly where reversibility or product inhibition is taken into account. Hence, only the simplest case will be considered, in order to illustrate the general principles. If irreversibility and the absence of product inhibition is assumed, for a two substrate reaction following a rapid equilibrium random mechanism can be represented by Scheme 7. E R - K~ " E R S
'~
E ,' ESK, ,L k-.[ Y] I E R Y - I ~ . , E R S Y
l :lY
EY-
k3 , E + P
I~'°t~!/' I~:ot,'J
' ESY
SCHEME 7
METHOD
FOR
DETERMINING
INHIBITION
RATE
CONSTANTS
545
According to fast equilibrium assumption and [Er] = [E] + lEg] + [ ES] + lEgS], [E*] = [ EY] + l e g Y] + [ ESY] + [ ERSY], and [E]o = [ET] + [E~], we have, [El
-
[ER] -
K 'RKs[ Er ] K'R Ks + Ks[R] + KR IS] + [R][S]' Ks[ R ][ ET ]
(46) (47)
K'R Ks + Ks[R] + KR[S] + [R][S]'
[~s] -
KR[S][ET] K'RKs + Ks[R] + KR [S] + [R][S]'
(48)
[ERS] -
[S][R][ET] K'RKs + Ks[R] + KR[S] + [R][S]'
(49)
then, d[Er] - ~ = {k+o[E] + k'+o[ER ] + k+o[ES] + k~.o[ERS]}[ Y] tit {k+oK'RKs + k'oKs[ R ] + k%KR[ S] + k+o[R ][ S]}[ Y][ ET ] K ~Ks + Ks[R] + K . [ S ] + [R][S] diP]
dt
= k3[ERS] =
ka[R][S][Er] K'RKs + Ks[R] + KR[S] + JR]IS].
(50) (51)
From eqns (50) and (51), d[Er] k+oK'RKs[Y] k%oKs[ Y] d[P] - k3([R]o-[P])([S]o-[P]) ~ k3([S]o-[P]) -~ k+oge[ Y]
+ k+o[ - - Y]
k3([R]o-[P]) k3 With the boundary conditions [ET]----[E]o and [P] = 0, this integrates to, [E]o-[Er]
KR[Y]( k3
k+oK's +
\[S]o-[R]o
+ Ks[Y]( k3
(52)
) [R]o k+o l n [ R ] o _ [ p ]
k+oK'R +k'o)
\[R]o-[S]o
r q- r[S]° ~_k~o[Y][p]. In t S j o - t P J
k3
(53)
When t --> co, [ E T ] = 0 and [ P ] = [ P ] ~ , eq (53) can be written as, k ' [ E ] o - .Ku[r] [R]o . . ( . + o K s +k.+o"~In k3 \ [ S ] o - [ R ] o } [R]o-[P]~ +
+oK. [R]o-[S]o
, +k+o
)
In
[ S]o ~-k.o[ Y] [S]o-[P]~ k3 [P]oo.
(54)
546
z.-x.
WANG
AND
C.-L. T S O U
This is a nonlinear equation; however, it can be linearized if one o f the substrate is saturating at all stages; in this case, an equation identical to eqn (7) is obtained in the non-saturating substrate. This would enable k ' o , k+o and k~o to be obtained from experiments where each substrate in turn was saturating, but not k.o. When k~-'o= 0, eqn (54) simplifies to, [ E ] o - KR[Y] [ k+oK's k---'~ [,[ S]-~- [-R]o +
.,, \ [R]o I¢+o) In [R ] o - [ P]~
Ks[ Y] (. k+og'R ~-k+o~ In +
k3
\ [ g ] o - [S]o
/
[S]o [ S ] o - [P]oo"
(55)
Rearranging, we obtain, In
[S]o IS]o- [P]~
{k+oK~+ k~-o([R]o- [S]o)}KR In [R]o {k+oK'n+k'o([R]o-[S]o)}Ks [ R ] o - [P]oo +
A plot of In
[ E ]o([R ]o - [ S]o) k3 {k+oK ~ + k~-o([R ]o - [S]o)}Ks[ Y]"
{[S]o/([S]o-[P]o~)} against
In
{[R]o/([R]o-[P]~)}
/[P]'~
(56) will give a straight
I
t FIG. 6. Progress curve of enzyme-catalyzedreaction in the presence of an irreversibleinhibitor. Curve 1: actual progress curve. Curve 2: Theoretical progress curve described by equation of t)
[P] = .':7~... (1 - e-At~q'). At r j
where v is the initial velocity of the enzyme catalyzed reaction without the inhibitor and A the apparent rate constant of the irreversible inhibition, and it is assumed that the substrate concentration remains a constant during the reaction.
METHOD
FOR
DETERMINING
INHIBITION
RATE
CONSTANTS
547
line with a slope s, and an ordinate intercept, i, of {k+oK ~ + k'~o([ g ]o - [ S]o)} KR s - {k+oK ~ + k~-o([R]o- [S]o)}Ks' i-
(57)
[ E ] o ( [ g ] o - [S]o)k3 {k+oK ~ + k'o([R]o - [S]o)}Ks[ Y]"
(58)
It can be seen from eqn (58) that a replot of 1/i against 1/([R]o-[S]o) should give a straight line with a slope of k+oKsK'R[Y]/k3[E]o and an intercept of k'oKs[Y]/k3[E]o. As KR, K'R, Ks, k3 and [Y] are known, k+o and k'o can be calculated, and k+o can be obtained further from eqn (57). The microscopic inhibition rate constants for enzyme with other mechanisms can be obtained in a similar way. Discussion
It can be seen from the above that for the case where the effects of substrate consumption or product accumulation cannot be neglected, we can still determine the reaction rate constant of enzyme modification by following the substrate reaction. The present method not only keeps the advantages of Tsou's method, but also extends its range of application. However, although, in theory, the present method is more strict than that proposed by Tsou, in practice, it is less convenient, especially for enzyme catalyzed two-substrate reactions. If the consumption of substrate cannot be controlled within an appropriate range, we can use the initial part of the reaction curve and make the Guggenheim plot (Shoemaker & Carland, 1962) to determine the [ P ] * which is different from the actually measured value of [P]o~, and then obtain the inhibition kinetic parameters by the double reciprocal plot of 1 / [ P ] * against 1/[S]. However, as both the consumption of substrate and other complicated reaction mechanism can result in the curvature of Guggenheim plot (Zhou et al., 1988), in order to distinguish these two possibility, it is still necessary to plot [P]~/[E]o against (1/[E]o)In {[S]o/([S]o-[P]oo)} and see whether a straight line can be obtained. We will now discuss the conditions for which Tsou's method applies in the case of uncomplicated one-substrate reaction. Expanding the right hand side of eqn (11) to power series, we have,
[E]o_k~o[Y]
k+o[Y]Km ( -
k---~ [P]~
k2
In 1
_ 'k÷o[ Y] [ p ] ~ + k÷o[ V]Km ['[P]~
[P]o~'~ [S]o] 1 /[p]~\2
1 ([p]~s
+ When [P]oo/[S]o<<1, the term containing eqn (59) can then be written as,
1
(59)
[Sloj + .....
([P]~/[S]o)'- and higher can be neglected,
k'o[ Y] k+o[ Y]Km [E]o- - [P],+ [P]~, k2 k2[S]o
(60)
548
Z.-X. W A N G A N D C.-L. T S O U TABLE 1
Relative error caused by Tsou's method. For different values of [P]oo/[E]o [P]J[S]o
Relative error
0.01 0.03 0.05 0.10 0.15 0.20 0.25 0.30
<0.005 <0.015 <0.026 <0.056 <0-088 <0-125 <0.167 <0.214
[P]o~-
k2[ E]o[ S]o (k+oKm + k'o[S]o)[ Y]'
(61)
which is identical with eqn (12) in Tsou's paper. (Tsou, 1965b, 1988). By comparing eqns (59) and (60) it is obvious that the error caused by Tsou's method is,
oo 1 ([P]oo~" .~--2nt [S]oJ
I ( [ P ] = ~ 2 ~ ([P]oo~" 1 ([P]ool[S]o) 2 [P]~ 2 t [S]oJ .=o \ [-'~o/ =7 1-([P],~l[S]o) -2([S]o-[P]<~)[S]o"
When [P]= = 0.05[S]o,
[P]~/2([S]o- [ P]oo)[S]o
([P]J[S]o)
= 0.026.
That is to say, when the formation of product, [P]=, is 5% of [S]o, the relative error caused by Tsou's method is less than three percent, in general, it is negligible. The relative errors caused by Tsou's method for different values of [P]oo/[E]o is listed in Table 1. REFERENCES BIETH, J. G. (1984). Biochem. Med. 32, 387-397. CORNISH-BOWDEN, A. (1976). Principles o f Enzyme Kinetics. pp. 144-149. London: Butterworth. CRAWFORD, C., MASON, R. W., WIKSTROM, P. & SHAW, E. (1988). Biochem. J. 253, 751-758. FOSTER, R. J. & NIEMANN, C. (1953). Proc. hath. Acad. Sci. U.S.A. 39, 999-1003. HARPER, J. W. & POWERS, J. C. (1984). J. Am. Chem. Soc. 106, 7618-7619. HARPER, J. W., HEMMI, K. & POWERS, J. C. (1985). Biochemistry 24, 1831-1843. KAM, C. M., COPHER, J. C. & POWERS, J. C. (1987). J. Am. Chem. Soc. 109, 5044-5045. LAIDLER, K. J. & BUNTING, P. S. (1973). The Chemical Kinetics of Enzyme Action. pp. 163-180. London: Oxford University Press. LIu, W. & T s o u , C. L. (1986). Biochim. biophys. Acta 870, 185-190. MASON, R. W., GREEN, G. D. J. 8(. BARRETT, A. J. (1985). Biochem. J. 226, 233-241. SHOEMAKER,O. P. & CARLAND,C. W. (1962). Experiments in Physical Chemistry. p. 222. New York: McGraw-Hill. SOULIE, J. M., RIVIERE, M., BUC, J., GONTERO, B. & RICHARD, J. (1987). Eur. J. Biochem. 162, 271-284.
M E T H O D FOR D E T E R M I N I N G
INHIBITION
RATE C O N S T A N T S
TIAN, W. X. & TSOU, C. L. (1982). Biochemistry 21, 1028-1032. Tsou, C. L. (1965a). Acta. Biochim. biophys. Sin. 5, 398-408. TSOU, C. L. (1965b). Acta. Biochim. biophys. Sin. 5, 409-417. Tsou, C. L. (1988). Ado. Enzymol. Relat. Areas molec. Biol. 61, 381-436. WANG, Z. X. & TSOU, C. L. (1987). J. theor. Biol. 127, 253-270. WANG, Z. X., PREISS, B. & T s o u , C. L. (1988) Biochemistry. 27, 5095-5100. WIJNANDS, R. A., MULLER, F. & VISSER, A. J. W. G. (1987). Eur. J. Biochem. 163, 535-544. ZHOU, J. M., LIU, C. & T s o u , C. L. (1989). Biochemistry 28, 1070-1076.
549