Transient-phase kinetics of enzyme inactivation induced by suicide substrates: enzymes involving two substrates

Transient-phase kinetics of enzyme inactivation induced by suicide substrates: enzymes involving two substrates

Joumul of Molecular Catalysis, 59 (1990) 97-118 97 TRANSIENT-PHASE KINETICS OF ENZYME INACTIVATION INDUCED BY SUICIDE SUBSTRATES: ENZYMES INVOLVING...

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Joumul of Molecular Catalysis,

59 (1990) 97-118

97

TRANSIENT-PHASE KINETICS OF ENZYME INACTIVATION INDUCED BY SUICIDE SUBSTRATES: ENZYMES INVOLVING Two SUBSTRATES R. VARON, M. GARCIA Departumento o!e Quimica, Escwla Universitaria Politt?cnica & Albacete, Universidad de Caatilla, La Man&a (Spain) FRANCISCO GARCIA-CANOVAS* and JOSE TUDELA Lkpartamento de Bioquimica y Bidogia Molecular, Universidad de Murcia, 30001 Murk (Spain ) (Received April 5,1989; accepted July 7,1989)

This paper reports the kinetic study of reaction mechanisms with enzyme inactivation induced by a suicide substrate for enzymes involving two substrates. A transient-phase approach has been developed that enables explicit equations with one or more significant exponentials to be obtained, thereby showing the dependence of product concentration on time. Methods for discrimination between different mechanisms for substrate binding sequences have been proposed, as well as an experimental design to determine the corresponding parameters and kinetic constants.

Introduction The inactivation of enzymes induced by a substrate takes place in enzymes which act on a substrate following a branched mechanism consisting of a catalytic route and an enzyme-inactivation route. These substrates are called suicide inhibitors, mechanism-based inhibitors, inactivating substrates and suicide substrates ( 1). The importance of enzyme suicide inactivation is gaining increased recognition, for both naturally occurring and totally synthetic suicide substrates 111. There is a wide range of enzymes of physiological interest with suicide substrates. Examples of enzymes with natural suicide substrates are tyrosinase [2-61, ascorbic acid oxidase [71 and ATPase [81; examples of enzyme with synthetic suicide substrates are /3-lactamases [g-12] and general acyl-CoA dehydrogenase [131; for a review see Walsh 111. The reaction scheme proposed by Walsh et al. 1141 in order to explain the suicide substrate action refers to a monosubstrate reaction. It has been studied by the steady state approach 115-171, obtaining implicit solutions for *Author to whom correspondence should be addressed. 0304-5102/90/$3.50

@) Elaevier Sequoia/Printed

in The Netherlands

98

time us. product concentration. More recently, the solution at final time of the reaction (i.e. t+ 00) has been described [l&31.We have studied this same reaction mechanism from the viewpoint of the transient-phase, obtaining explicit equations showing the dependence of product concentration on time 14, 51. Since most enzymes involve two substrates and a great number of them show suicide inactivation 11I, it seems to be convenient to develop a detailed kinetic study of this type of system. It is the purpose of the present paper to examine the kinetics of suicide inactivation for enzymes involving two substrates. The mechanisms under study are: WBI

hIA

EeEAT--,EAB k-1 k-2

k’ --s

Yk*-E+Q

(1)

P 1k, Ei

Ordered Bi Bi (B is the suicide substrate) ktlA1

EeEAe

kzIJf’l

EAB --% Ykl‘E+P+Q

(11)

Ordered Bi Bi (A is the suicide substrate)

(III)

Random Bi Bi (A or B is the suicide substrate)

Ping Pang Bi Bi (A is the suicide substrate)

EeEA k-1

Bi PingPang Bi Bi (B is the suicide substrate)

Notation In this paper we use the following notation: Species and concentrations initial initial initial eo: any of x:

2;

Kinetic parameters n: &(h=1,2,...,n):

concentration concentration concentration products P or

of substrate A, of substrate B; of free enzyme E; Q.

in mechanisms (I) and (II), n = 4; in mechanisms (III)-(V), n = 5. arguments of the respective exponential terms corresponding to appearance equation of products. They are the roots of equation:

2 py-i = 0

(1)

i=O

F. is equal to unity and 4 (i = 1,2, . . . , n) are functions of the rate constants and of a,, and bo. Their expressions for each mechanism are summarized in Appendix A. yxs (h = 1,2,. . . , n): amplitude of the exponential terms corresponding to accumulation equation of product X (X = P or Q). [Xl value for t ---* CQ.Concentration of products obtained [Xl,: at the final moment of the reaction.

c(i=O,l,...,n):

Kinetic constants T:

partition ratio; number of catalytic events (turnovers) per inactivation event. k4/k5 in mechanisms (I)-(III) r = KS/k4 in mechanism (IV) ks/k6 in mechanism (V>

r>>l: Ki and Ki:

(2) 1

r much higher than unity. dissociation constants: Ki = k-i/ki

(i = 1,2,3,4)

Ki = kI_Jk: (i = 1,2)

(3) (4)

Half-saturation constant of E by A for cases in which F >>

K&Z K&Z K$ k cat:

1.

Half-saturation constant of E by B for cases in which r >> 1. Michaelis constant for substrate A. Michaelis constant for substrate B. for mechanisms (I)-(III) K$ = K1 and for mechanisms (IV) and (V), K$ = 0. catalytic constant.

100

maximum value of the steady state rate when the system operates in a restricted steady state, for saturating of E for A and B. Maximum value of the apparent inactivation constant for systems with only one significant exponential term, equivalent to the number of inactivation events per time unit. Combination of constants equivalent to the number of catalytic events per time unit, equivalent to &at when the system operates in a restricted steady state.

v_: LX:

;E,r:

Other symbols Ti,,z = l/{WA,

- 4,))

T,,,=l/{~~~~~(~~-~~)}

(h,p =1, 2;p Zh)

(5)

Cm>21

(6)

p+h

In this section we derive the kinetic equations corresponding to mechanisms (I)-(V) under the initial conditions ao and bO>>e,, E being the only enxyme species present at the start of the reaction. We will distinguish between three situations: (A) the general case, without assuming any simplifying relation between the rate constants; (B) cases in which the partition ratio, r, is not much higher than unity and rapid equilibrium conditions prevail; (Cl cases in which the partition ratio, r, is much higher than unity. Cases (Bl and (C) are, obviously, particular cases of (A), and will be treated as such. (A) General case Applying the method developed for the transient phase of enzyme systems [19-211 we obtain for P, Q:

[Xl = [Xl, + i: yxjr exp(Ad)

(X=P

or Q)

(7)

h=l

In Table 1 are summarized the expressions of [Xl, and y- (h = 1,2, . . . , n) for each mechanism under study. The )Chare the roots of eqn (1) and they are real and negative or complex with the real part negative 122, 231: In these cases the transient phases of both the catalytic and the inactivation routes of the enzyme occur on the same time-scale (cases with two or more significant exponential terms in eqn. (7)). Nevertheless, in some cases the transient phase of the catalytic route is much slower than that corresponding to the inactivation route, that phase determining the kinetic behaviour of the system (cases with one signticant exponential term in eqn. (7)). In any case, the catalytic route never reaches the steady state.

t111 [III1 IIVI WI

[II

. _

Mechanism

_

r r r

r r

._ _

r r

1+r r

l+r

tQL.h

[PI.&

- -

h,+k4)(k5+kdP,

kJeo

appear in eqn. (7) corresponding to the relation between the

-

_ -

r{F~+k,k,(k;k6+k,k~)~b~,}T*,~ rklk&t~{ksk~bo+ (k,bo+k,+k-&i +MT,.s k,kzaobo[kgk,(k,+k,)b,+ {k,(k,+k,+k,)b,,+ +(k_,+k,+k,+k,+k,b,)A~+A~lT,,~ k~k6hoboWG.~ rKJ’t.,e rF&s

-r{Z$+k,k,(kjk,+

- rFdTh.4

. . , n ) which

- rF1T’.4

I,%.

- rF,T,.,,

=

-F~(k,+k,+hJG,,Jk,

ypi/eo

Expressions of [PI,, [QJ,, yph (h = 1,2,. . . , n) and yQ& (h concentration of producta and time for mehnisms W-W

TABLE 1

102

For simplicity we will assume that the binding of the substrates occurs in rapid equilibrium, which is, in fact, quite probable for the suicide substrate, because these are poor substrates. If we apply the procedure described in the literature [24, 261 in order to obtain, from the transient phase kinetic equations of a mechanism, the corresponding ones if rapid equilibrium conditions prevail, we obtain:

[Xl = [Xl, +

2 yKh exp(Aht)

(X= P or Q; u = 2 for mechanisms WXIII); A,

(8)

h=l

and & for mechanisms (IHIII)

u = 3 for mechanisms (IV) and (V))

being the roots of eqn. (9):

122+MiE+N=0

(9)

and kl, AZand lLafor mechanisms (IV), (V) being the roots of eqn. (10): A3+JA2+Gh+H=0

(10)

The expressions for M, N, J, G and H in eqns. (9) and (10) and for yxh in

eqn. (8) are summarized in Appendix B. [Xl, for all mechanisms remains as indicated in Table 1. According to the polynomial theory, it is verified for mechanisms (I&(111) that: rl1+ A2= -M

(11)

iz13c2 =N

(12)

and for mechanisms (IV) and (V) that: A,+A,+3c,=

-J

(13)

12& + A1A3+ &A3 = G

(14)

311AZj13 = -H

(15)

Reduction of the number of significant exponential terms. From an experimental point of view it is interesting to consider the possibility of reducing the number of significant exponential terms in eqn. (8). If between the roots of eqn. (9) yields the relation IAll<
(16) The expressions for )Lmax, K?, Kt and K$, corresponding to these mechanisms are summarized in Table 2.

103 TABLE 2 Expressions for parameters A,,, Kf, KF and K$ of eqn. ( 16) Mechanism

A_

KAI

KBA

(I) and (II)

k3k5 k,+k,+k,

0

K&+k,) --

k,+k,+k,

K;(k,+k,) k,+k,+k,

(k,+k,)(k,+k,)+k,k,

k&,(k, +k,) (k,+k,)(k,+k,)+k,k,

(k,+k,)(k,+k,)+k,k,

k,k,ks (k,+k,)(k,+k,)+k,k,

k,K,(k,+k,) (k,+k,)&+k,)+k,k,

(k,+k,)(k,+k,)+k,k,

k,+k,+k,

W,k,

(Iv) (VI

Kl

k,+k,+k,

K&+&J

ksk,

(III)

KAs

Kl

k,K,(k,+k,) ’

k,K,(k,+k,) ’

On the other hand, in the two following cases of relations between the roots of eqn. (101, denoted as cases 1 and 2, the number of exponential terms of eqn. (8), corresponding to mechanisms (IV) and (V), can be reduced. Case 1:

IhI, I&l <
(17)

Under conditions ( 17 ), the exponential term in eqn. (8) in the exponent of which 3c3is involved can be neglected, resulting in a release of products through a bi-exponential equation. From eqns. (13)-(X), (17) and from Appendix B, the two following equations can be easily obtained for mechanism (IV); Al + AZ= _ {(k, +kd(k,

+ kd + kzkshbo +&(k,

(12s+ k3 + k, + kthbo

+ kz(ks + kX+o + kdks + kXlbo + k3 + k,h, + K,(k, + k4 + kdbo + K&(ks + k4) (18)

AlA2 =

WJwobo

(122+ k3 + k4 + k,hob,

+K,(k,

+ K,(k,

(19)

+ ks + k&o

+ k4 + kdbo + K&(ks

+ k,)

and for mechanism (V); II + Lz = _ {(k, + k,)(k,

+ k,) + kA}a,bo

(kz + k4 + ks + k&oh,

+&(k,

+ k&(kxs + K,(k,

M&saobo

‘lA2 = (k, + k4 + k5 + k&,bo + &(kz +

+ ks + k&o

Kl(ka + k, + Who + K&(kE

+ k&o

+ k&l(ks

+ kdbo

+ kr, + k&o

+ kc4

+ k5 + kc.)bo + KJW,

+ k,) (20) (21)

Case 2: IhI << IU lhil I&l not very different from I&l

(22)

Under conditions (221, the exponential terms in eqn. (8) corresponding to & and & can be neglected, resulting in a release of products through a uni-exponential equation. From eqns. (13)-(X), (22) and from Appendix B, an expression for 3L1can be easily obtained for mechanisms [IV] and [VI which coincides with eqn. (16); the corresponding expressions for A,,,,, Kf, Kf and K$ are given in Table 2. (C) Cases for which r >> 1

In these cases the catalytic route attains the steady state, the transient phase being controlled by the suicide inactivation. Particular cases of this situation have been previously described [271. On the other hand, the evolution of products corresponds to a uniexponential equation such as: [PI = [&I = reO{l - exp(l2t))

(23)

Effectively, let us consider, as an example, mechanism (I). Since r >> 1 then 1 + r = r, and therefore [PL = [&I_ = rco. In turn, because k,>>k,, eqns. (Al )-(A31 become: F,=k,a,+k_,+k,bo+k_z+kS+kl

(24)

F2=klkza,,bo + k1(k_2 + KS + k,)a, + kz(k3 + k,)b, + (k_, + k,)(k_, F’=klkz(ks

+ k&b,,

+ k,(k_,

+ k,)k,a,,

+ k2k9k4b0 + k_,k,(k_,

+ k,) + k,)

(25) (26)

Z$ remains as given by eqn. (A4). After an examination of eqns. (24)-(26), (A4), (1) and from the polynomial theory, we deduce that, in the considered example, three roots, Ax, 3Lgand 3c*, from among the four corresponding to eqn. (11, are ksindependent, and much higher in absolute value than the absolute value of the fourth root rZ1,that is: ?bi- A-12( Ai

(i = 2,3,4)

(27)

Moreover, &, As and & coincide approximately with the roots of equation: A~+FJ2+F2~+FFs=o

(281

whereas Ai is: kM&saobo

Al= klkdks

+ k,hbo

+ kMk-2

+ k&o

+ kzk3k4bo + k_-lk4(k_2 + k,)

(29) From eqn. (29) and taking into account that k, >>kg, it is easily shown that: k4+ll-k4 which will be useful below.

(30)

105

Result (27) allows us to neglect in eqn. (7) the terms in the exponent in which )Lz, & and A4 are involved, and we have, taking into account that 1+r-r, [PI = reo -

Fdh + k5 + ~l)eoTl,4 ks

(31)

exp&t)

(32)

[&I = reo{l -F,Tl,4exp(~l~)) Since in this case, due to result (27) it is verified that: G,4=

(33)

1 LW&

and because, from polynomial theory k1&&3L4= F4, we have: 7’1.4= I/F,

(34)

Finally, if we insert eqn. (33) in eqns. (31) and (32) and take into account that k, >>k5 and ill we obtain, assuming 3L1 = A, eqn. (23). Equation (23) is suitable for all the mechanisms under study, as it can be shown that through similar arguments that Iz for mechanisms (I), (II), (IV) and (V) is given by: A=

-

Lasobo

(35)

a,,bo + K&a, + Kib, + K*&:

The expressions for A,,, KE, K& and K$ for each mechanism are indicated in Table 3. Note that A,, = k,,Jr and since V,, = k,,e,, we obtain: V E

LX

= re, = [PI= = [&I_

(36)

In turn, the A.expression for mechanism (1II)is: A= -

Clhbo + C2agbo+ Cs%bg C&b0 + C,a,-,b: + C,d + C,b: + Csaob,, + C,a,, + Cl,,bo + Cl1

(37)

The coefficients Ci (i = 1,2, . . . , 11) depend on the rate constants, and are summarized in Appendix C. TABLE 3 Expressions for parameters A,, Mechanism

LlX

K&, K& and Ki of eqn. (36) K*M

KB M

K:: Kl

k,+k,

k,(k,+kJ

k,(k_,+k,) k,(ka + k,)

w,

k,kdk, kJk,+k,)+k,k,

k,k,(k_,+kz) kl{k,(k,+k,)+kzks)

k,k,(k-,+k,) k,{ks(k,+k,)+kzk,l



W)

k,k,ks ks(k,+kJ+k2k4

k,k,(k_, +k,) k~{k,(k,+k,)+k,k,)

k,k,(k_,+k,) k.q{k,(k, + k,) +k,k,l



(I) and (II)

ksk,

ksk,

106 TABLE 4 Expressions of the parameters A,, KG, K& and Kg of eqn. (35) if in mechanisms (I)-(V) rapid equilibrium conditions prevail Mechanism (I) and

(II)

(III) uv) (V)

KBM

A,

KAM

k,k, k,+k,

0

W, k,+k, k&&s

Wz k,+k, M&

k& k,+k, * k,+k, W@&

ks(k,+k,)+k,k,

k,(k,+k,)+k,k,

k,(k2+k,)+k,k,

kzk,k, k,(k,+k,)+k,k,

k,(k,

k,kdG +k,) +k,k,

k,k,K, k,(k,+k,)+k,k,

the

K2

K, KX 0 0

If, in addition to r >> 1, rapid equilibrium conditions prevail, the equation for accumulation of the products remains (23) in all cases, the expression of A for all mechanisms, (III) included, having the same form as indicated by eqn. 1351, but il,, Ki, Kg and K$ being now those indicated in Table 4. Continuous and discontinuous methods dealing with Ei

In all cases (A), (B) and (C) studied in this section, it is obviously verified that:

=[xl

[&I

r

In cases (A) and (B), X = P in mechanism (IV) and X = Q in the other ones; in case (C) X = P = Q. Equation (38) is useful for following the kinetic behaviour of the system by measuring [Ei] in a continuous way. The data analysis would be carried out in a manner similar to that followed for P or Q. From eqn. (38) may be designed a discontinuous method to follow the suicide inactivation kinetics, regrouping the equation so that:

rxi

eo-[$I

e.

=

“‘-7 e.

=-

A,

A0

(39)

where A, is the residual activity at time t and & is the initial activity. The information obtained by this method is equivalent to the measurement of [Eil in the continuous procedure described earlier. It is easy to show, substituting in eqns. (38) and (39) [Xl by its corresponding expression, that [Ei] is r-independent; therefore, the determination of parameter r when [Ei] is measured must be made using a discontinuous method described by Knight and Waley [121 and Tudela et al.

[51.

107

Results and discussion In this section we propose an experimental design procedure for discriminating between the cases with r >>1 and r > 1, as well as the type of mechanism. Moreover, it allows the determination of the corresponding kinetic parameters and constants which characterize the action of enzyme on the suicide substrate. Our experimental design consists of the following three steps. First step: inactivation assays This step consists of assaying the inactivation of the target enzyme. The experimental recording of these assays can be fitted to a multiexponential equation of the type [Xl = [Xl, + CiCl yxJleAht(s = 1, 2 or 3), yielding the number of significant exponential terms (one, two or three) and the corresponding values of their kinetic parameters ([Xl,, 3L1,A,, A,). If the assay is discontinuous, that is, on measuring the residual activity, the fit would be to eqn. (39). In all cases, data fitting to both explicit equations by non-linear regression 128, 291 gives the corresponding values of the x2 parameters which can be compared with the F-test [30]; Thus, fitting to urn-exponential behaviour can correspond to the cases with r > 1 or r >>1, described by the kinetic parameters P,, Qm and A. However, the fitting to more than one exponential term indicates that the system is included in cases with r S 1, and gives [PI_ or [&I_ and Ah (h = 1,2 or h = 1,2,3). Once the most appropriate equation has been identified, it is used for the fitting of all inactivation curves obtained below. Second step: effect of e, This step consists of the choice of an e, value that provides [PI, and [&I_ values, such that the condition [PI, <>l. In the cases in which the kinetics are followed by continuous recording of [Ei] (eqn. (38)) or by the discontinuous method of measuring the residual activity with time (eqn. (39)), the r value must be determined by the method described by Knight and Waley [121 and Tudela et al. 151. Third step: effect of ao and b. Application of the methods described in the former step to assays carried out with different b. values at a fixed concentration of h, and repeating the former approach to different h values, provides discrimination between the different mechanisms and determination of the corresponding kinetic constants, in the same way as that described for the suicide

108 TABLE 5 Information available when step 2 of the experimental design is completed. Cases (a)-(d) coincide with those studied in the third in the text. Boldface type indicates the mechanisms where rapid equilibrium conditions prevail Possible mechanisms

Information obtained number of signilicant exponential terms

r value

3 2 1 1

I?51” I+ la $1 >>1

WI,

(a)

WI

(b)

[II-WI [II-WI [II-WI,

CaSe

[II-WI

(c) (d)

*If the number of significant exponential terms is 3 or 2, it is necessarily verified that r r> 1, according to the analysis developed.

inactivation of monosubstrate reactions [51, as well as in a method parallel to that applied for the irreversible inhibition of two-substrate reactions [311. The first and second steps of the experimental design reveal the number of significant exponential terms and if r >>1 or ry 1. Table 5 (cases (a), (b), (c) and (d)) indicates the possible mechanisms for each result. Case (a): three significant exponential terms This result is compatible with mechanisms (IV) and (V). Discrimination between these mechanisms can be carried out using either of the following two experimental criteria: (1) According to schemes (IV) and (VI, an incubation of the enzyme with the substrate A over a long time, resulting in a decrease in the residual activity at the end of this time, is compatible with mechanism (IV), whereas a constancy of this activity is compatible with mechanism (V). (2) According to Table 1, a ratio [Pl,/[[Ql, equal to or greater than unity is compatible with mechanism (IV) or (VI, respectively. On the other hand, from eqns. (131, (15) and Appendix B we have for mechanism (IV): (40) 1

A,+122+IZQ=kz+kr(l+r)+ka+K,{k,(l+r)+k,) k&&6

Wz&

KS + k&s (

k&&e

k4

a,

1

k,+kd(l+r)+KI(l+r) a,

> bo

(41)

us. l/b, at constant a, will be linear, with a Therefore, a plot of - l/&I& slope of K,{l + (KI/a,J}/k2k4k6 and intercept on the -1/3c11& axis of K,{l + (l/q)}/k2k4k6. From the plots of the slope and the intercept us. l/& it is possible to determine K,, K5 and kzkak6. Proceeding analogously, from the linear plot of (A, + &. + A3)/A1&A3 us. l/b, at constant a, and from the

109

corresponding replot of the slope us. l/a,,, KS{& +k,(l+ r)}/K2k4K6 and K&(1 + r&k, can be determined. From all the previous experimental data and from the r value, the constants k2, kS, k4, k6, K1 and KS can be calculated. It is interesting to note that, generally, the plots which are proposed in this paper in order to obtain the kinetic parameters have in particular a diagnostic value, because they have a distinctive appearance which depends on the particular mechanism. However, the rate constants which are estimated graphically are used as initial estimates of the values, which are then refined by non-linear regression fitting 128, 291 in which the appropriate theoretical function is fitted, without prior transformation, to the experimental data. From eqns. (131, (15) and Appendix B, the expressions for -l/A1il&, and (A, + &. + &)/3L11Lzj13 corresponding to mechanism (V) can be derived, and from them and the experimental data the corresponding kinetic parameters can be calculated, similarly to those made for mechanism (IV). Case 65): two significant exponential terms In this case the mechanism can be any of the mechanisms under study (see Table 5). From eqns. (121, (B21, (B7), (19) and (21) we deduce that for mechanisms (I)-(V) a plot of l/&A, us. l/b0 at constant a~, is linear with intercept of fil + (g,,/~) and slope of fi2 + (g,,/%,). On the other hand, from eqns. (ll), (12), (18)-(21), (Bl) and (B2), a plot of -(A, + 3L2)/A13c2 us. l/b,, at constant a, will be linear with intercept of fzl + (gJa,J and slope of fz2 + (gz2/~). The coefficients fti and g, (i,j = 1,2) in the previous intercepts and slopes depend only on the rate constants, and their expressions for the different mechanisms are summarized in Table 6. Observe that g,, = 0 for mechanisms (I) and (II) and g,, = 0 for mechanisms (IV) and (V). Therefore, an intercept of the plot -(A, + A2)/&3L2us. l/b, being a,,-independent, together with a slope decreasing with increasing a~, is compatible with both mechanisms (I) and (II) (Fig. lA), whereas an intercept and slope decreasing with increasing a, are compatible with mechanism (III) (Fig. 1B). In turn, a slope of the same plot q-independent, together with an intercept decreasing with increasing a~, is compatible with both mechanisms (IV) and (V) (Fig. 1C). In turn, the criteria (1) of the previous case (a) can be used here to discriminate between mechanisms (IV) and (V). On the other hand, the ratio U’MQlm al so P ermits discrimination between mechanisms (I) and (II) in this case: a [P],/[Q], value higher than unity is compatible with mechanism (I), whereas a [Pl,/[QL equal to unity is compatible with mechanism (II) (see Table 1). By secondary plots of the intercepts and the slopes corresponding to plots of l/&L2 and -(A, + ;12)/L,n2 us. l/ho, the coefficients fti (i,j = 1,2) and g, (i, j = 1,2) can be determined. From the values of fti and gti and their corresponding expressions in Table 6, all the rate constants can be calculated.

(VI

0-V)

kAk,

k,+k,+k,

k,kcke

k,+k,+k,+k,

k,k,k,

k,k,ka

K,(k,-tk,i-k,) kzkrks

K,(k,+k,+k,)

k,k,kci

K,(k,+k,+k,)

k3ks

k,k,k,

K,K,O,+k,)

kzk, +k,

K~K,Os+k,I

ksks

KA

hk,

jG_

K&

4&s

g12

aK

f12

j = 1,s1

K,tk,+k,+k,)

2

&

(III)

35

0

1 ksks

(I) and (II)

K

811

&

Masse

Expreesione for coefficienta & and gti f i,

TABUE 6

k&k,

(kz+kd(k,+k,)+k,k,

k,k,ks

tkz+k,)lk,+k,)+k,k,

k3ki

k,+tl+r)k,

k&s

R,+(l+r)k,

f2%

kzk,

K,(ks+kd

kzkd

K,(k,+U

k3

&tl+r) ~

0

g21

kake

Kdk,+kd

kdk,

Kdks+kJ

k3

K!Jl +r) ~

k,

k&(1 +r)

fn

o

o

h

KiKh(l+r)

ks

K*K& +r1

822

111

,S’

14

l/b,

Fig. 1. The -(A, + I.JA,I, vs. l/b, plota for the discrimination between mechanisms (I)-(V) when the progress curves for the accumulation of products can be fitted to a two-exponential equation (case b) of third step in Results and discussion section. (A) mechanism (I) or (II), (B) mechanism (III), (C) mechanism (IV) or (V).

Case cc): one signifiant exponential term and

ry/l

According to eqn. (16), a plot of - l/& us. l/b, at constant a, will be linear with a slope of K?{l + (Kg/a,,,)}/&,,, and an intercept on the -l/A1 It can be seen from the previous expressions for axis of (1 + (K3a&llz,,. the slope and the intercept that the constancy of the intercept (but not of the slope) with varying a, is compatible with both mechanisms (I) and (II)

-

VA,

A

- VA,

‘\I a0

lncrerrlng

\ b a,

IncrcarlnfJ

>I’

A

I’ ,

-

I_

l/b,

B

x:x:.

,:‘,’

/”

lh

- VA, ,

Fig. 2. The -l/A, vs. l/b, plots for the dkrimination between mechanisms (I)-(V) when r 3 1 and the progress curves for the accumulation of products can be fitted to a uni-exponential equation (case c) of the third step in Results and discussion section. (A) mechanism (I) or (II), (B) mechanism (III), (C) Mechanism (IV) or (V).

112

(Fig. 2A); a decrease of both the slope and the intercept with increasing a, is compatible with mechanism (III) (Fig. 2B); the constancy of the slope (but not of the intercept) with varying a,, is compatible with both mechanisms (IV) and (V) (Fig. 2C). Moreover, from secondary plots of both the slope and the intercept us. l/a,,, the kinetic parameters II,.,,,, Kt, KF and Kg can be easily obtained. In turn, one can discriminate between mechanisms (I), (II), (IV) and (V) proceeding as in the previous case (b). Case (d): one signifiant exponential term and r >> 1

In these conditions it is necessary to discriminate not only between the mechanisms, but also if in them rapid equilibrium conditions prevail. From eqn. (37) it results that a plot of -l/J. us. l/ho, being non linear (Fig. 3A), is compatible with mechanism (III), provided that there are no rapid equilibrium conditions. In turn, from eqn. (35) it is seen that the same previous plot will be linear for mechanisms (I), (II), (IV) and (V), with or without rapid equilibrium conditions, and for mechanism (III) in rapid equilibrium conditions, with a slope of Kz{l + (K,$/ao)}/&,, and an intercept on the From the plot of -l/k us. l/b0 and Tables 3 -1/n axis of {1+ (K$&ilJ}ln,,. and 4, the following procedure for discrimination is available. If both the slope and the intercept decrease with increasing a,,, then mechanism (I) or (II) is involved in steady state conditions and with mechanism (III) in rapid

(

A l/b,

l/b,

;;1 ,G’ I‘

l/b,

l/b,

Fig. 3. The -l/n vs. l/b,, plots for the discrimin ation between mechanisms (I)-(V) when r >>1 (case d) of the third step in Results and discussion section. (A) mechanism (III) is not in rapid equilibrium conditions (non-linear plot), (B) mechanism (I) or (II) is not in rapid equilibrium conditions or mechanism (III) is in rapid equilibrium conditions, (C) Mechanism (IV) or (V) with or without rapid equilibrium conditions, (D) mechanism (I) or (II) in rapid equilibrium conditions.

113

equilibrium conditions (Fig. 3B). Note that it is not possible to discriminate between the three previous mechanisms, but is necessary to resort, as done in the steady state approach, to the effect of substrate analogs, product analogs, etc. [321. The constancy of the slope (but not of the intercept) with varying a,, is compatible with mechanism (IV) or (V), with or without rapid equilibrium conditions (Fig. 3C). The constancy of the intercept (but not of the slope) with varying a, is compatible with mechanisms (I) and (II) in rapid equilibrium conditions (Fig. 3D). Between mechanisms (IV) and (V) one can discriminate as in criteria ( 1) of case (a). Once the possible mechanism is identified, replots of both the slope and the intercept us. l/~ allow the kinetic parameters A,,.,,, Kg, K& and K& to be obtained easily. At the end of this section we want to emphasize how the experimental design used here from the point of view of transient phase kinetics of enzymes involving two substrates (one being suicide) serves in most cases two purposes: discrimination between the different mechanisms and determination of the kinetic parameters. Moreover, the effects of the product inhibition may be negligible, since its concentration, in spite of the progress curve followed, is negligible and experimentally controllable by means of the appropriate selection of enzyme concentration. Acknowledgement This work has been partially supported by grant 287/85 from CAICXT, Spain. References 1 C. Walsh, Annu. Rev. Biachem., 53 (1984) 493. 2 L. L. Ingraham, J. Come and J. B. Makower, J. Am. Chem. Sot., 74 (1952) 2623. 3 L. L. lngmham, J. Am. Chem. Sot., 77 (1955) 2875. 4 F. Garcia-Cbovas, J. Tudela, C. Martinez Madrid, R. Var6n, F. Garcia-Carmona Lozano, Biochim. Biophys. Acta, 912 (1987) 417. 5 J. Tudela,

F. Garcta-Cbovas,

R. Var-611, F. Garcia-Carmona,

J. Gglvez

and J. A.

and J. A. Lezano,

Biochim. Bbphys. Acta, 912 (1987) 408. 6 J. Tudela, F. Garcia-Canovas, R. Var6n, M. Jimenez, F. Garcfa-Carmona and J. A. Lezano, J. Enzyme Znhib., 2 (1987) 47. 7 K Tokuyama and C. R. Dawson, B&him. Biophys. Acta, 56 (1962) 427. 8 F. J. Femandez-Belda, F. G&a-Cannona, F. Garcfa-Cbnovas, J. A. Lezano and J. C. Gomez-Femandez, Arch. Biochem. Biophys., 215 (1982) 40. 9 J. Fisher, R. L. Chamas and J. R. Knowles, Biochemistry, 17 (1978) 2180. 10 J.-M. Frere, C. Dormans, V. M. Lenzini and C. Duyckaerts, Biochem. J., 207 (1982) 429. 11 W. S. Faraci and R. F. Pratt, Biochemistry, 24 (1985) 903. 12 G. C. Knight and S. G. Waley, Biochem. J.. 225 (1985) 435. 13 J. W. Baldwin and D. W. Parker, Biochem. Biophys. Res. Commun., 146 (1987) 1277. 14 C. Walsh, T. Cromartie, P. Marcott and R. Spencer, Methods Enzymol., 53 (1978) 437. 15 S. G. Waley, Biochem. J., 185 (1980) 771. 16 S. Tatsumani, N. Yago and M. Hosoe, Biachim. Biophys. Acta, 662 (1981) 226.

114 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

S. G. Waley, Biochem. J., 227(1985) 843. R. G. Duggleby, J. Theor. Bid., 123 (1986) 67. J. Gdlvez and R. Varon, J. Theor. Biol., 89 (1981) 1. J. GBlvez, R. Var6n and F. Garcia-Canovas, J. Theor. Biol., 89 (1981) 19. J. Gblvez, R. Var6n and F. Garcia-Carmona, J. Theor. Bid., 89 (1981) 37. J. Z. Hearon, Ann. N.Y. Acud. Sci., 108 (1963) 36. I. G. Darvey, J. Theor. BtiZ., 65 (1977) 465. R. Var6n, F. Garcia-CBnovas, F. Garcia-Carmona, J. Tudela, M. Garcia, A. Vdzquez and E. Valero, Math. Biosci., 87 (1987) 31. R. Var6n, F. Garcia-C&novas, F. Garcia-Carmona, J. Tudela, A. Roman and A. Vazquez, J. Theor. BioZ. 132 (1988) 51. F. Garcia-Cbnovas, R. Var6n, J. Galvez, F. Garcia-Carmona, J. Tudela and M. Garcfa, An. Quim., 83 CC), (1987) 219. P. V. Brzheshch and S. D. Varfolomeev, Biochemistry USSR. (Engl. transl.), 50 (1985) 125. G. N. Wilkinson, Biochem. J., 80 (1961) 324. D. W. Marquardt, J. Sot. Id AppZ. Math., 11 (1963) 431. W. G. Bardsley, P. B. McGinlay and A. J. Wright, Biometrika, 73 (1986) 501. Z. 2. Wang and C.-L. Tsou, J. Theor. Biol., 127(1987) 253. I. H. Segel, Enzyme Kinetics, Wiley, New York, 1975, p. 793.

Appendix

A

Expressions for coefficients J$ (i = 1,2, . . . , n ) of eqn. (1) for each of mechanisms (I)-(V): Mechanisms (I) and (II):

Fl = k,a, + k_, + kzbo + k-, + k3 + k, + k5

(Al)

F2 = k,k,a,,b, + k,(k_, + k3 + k, + k,)h + kz(k3 + k4 + kdb,, + (k_, + k,)(k_, + k, + k5) F’ = klkAk3 + k, + k&b,

W)

+ k,(k_, + k&k, + k,)a,

+ k,k,(k, + kdb, + k_,(k_,

+ k&k4 + k5)

(A3) (A4)

K = k,k,k,k,%b, Mechanism (III): Fl = (k, + k,)a, + (k; + k;>b, + k_, + kI_, + k+ + kI_, + k, + k, + It5 F2 = klk&

+ k;k;b;

+ (k,k;

(A5)

+ k;kn + k,k;)a,,b,,

+ {k,(kI_,

+ ki_, + k-2 + k, + k4 + kh) + kz(k_,

+ kLz + k3 -t k4 + k,)}a,

+ {k;(k_,

+ k-p, + k’,

+ k_2 + k3 + kg + k,)}b,

+ k3 + k4 + k5) + k;(kL,

+ (kI_, + k_2 + k3 + k, + k,)(k_,

+ kl_,) + k_,kL, + k,Jk+ + kLz + k3) W)

115

F3= k&&a%c, + k;k,k;at,b; + k;k;(k_,

+ klkz(kl_2 + k3 + k, + k,)&

+ k3 + k4 + ks>b”,

+ {klk;(k1_1+ k-z + ks + k4 + k,) + k;k2(k_1 + k,k&

+ k, + k,)}aob,

+ k,(k,

+ k&k_,

+ k&I_,

+ {kIkl_l(kl_z + k+ + KS + k4 + k,)

+ kl_z + k3) + k_,kz(kl-2

+ k,)(k,

+ kick-2

+ k,)}b,

+ k_lkLI(k1-2

+ ks + k4 + ks) + kJk_lkl_z

F4 = k&k3

+ kLl{(kI_s

+ k&k,

+ kLlk-d

+ k&k,

+ k4 + ks)(klao

+ klkdkl-2

+ k_z

+ k_I{(k_z

+ KS)

+ k5) + k_,k,}

(A7)

+ It&

+ k;k;(k_,

+ k,)(k,

+ k5)b;

+ kS) + k,(kl_, + k5)}

+ k;k,{(k-,

+ kl_,)(k,

+ k&k&4

+ k,)lq,b,

+ k-lkz(kI-2

+ k,)}a,

+ (k, + k,){k;k_,(k_,

+ kllk;(k_a

+ k,)}b,

+ k_IkI_l(kI_, + k--2 + k,)(k,

Fs = Wdk~kikkl-~

+ k&k,

+ k;b,h,b,

+ [klkl?{(kl_l + k-&k,

Mechanism

+ k--2 + k,)

+ k3 + k, + k6)

+ k,)(k,

+ k&i}

+ k, + k4 + k,)

+ ks)}a,, + {k;k_,(k_,

+ kL + k3 + k, + k,) + kick, + k&k:, + kI_,k;(k_,

+ k& + k3 + k, + k,)

+ k,) + k,(k_,

+ ks)}

+ (k, + ks){klk’_l(kl-2

+ k;kzk_l

+ klk&h

+ k+ + k3) + kL, + k,) + k5)

W)

+ k;k,k;b,)hb,

(A91

(IV):

F,=k,a,,+k_,+k,+k,+k,bo+k_,+k, Fz = k,ksaob,, + k,(k, + (k-1 + k&k,

(AlO)

+ ks + k, + k,_i+ k_,)a, + k4 + k6 + k-,)

Fs = k1krdk2 + ks + k, + k,hb,

+ k,(k_I

+ (k--S + k&k,

+ kl{k2(k3

+ k,)}a,,

+ ks{kdk-I+

+ (k-1 + k,)(k,

+ k,)}b,

+ (k-1 + k,)(k_,

+ k,k,(k-,

+ k&k, + k,)(k,

Fs = klkzk4kEksaobo

+ k,)

(All)

+ k4 + k6 + kJ

+ (k-5 + k&k3

4 = k&d&

+ kz + k3 + k, + k,)b,,

kz + k3 + k4) f k,)(k,

+ k,)

(A12)

+ k,) + k,k,}a,,b, + k&o

+ k,k,(k_,

+ k&k,

+ k,)b,

(A131 (A14)

116

Mechanism

N):

F,=k,a,+k_,+k,+k,+k4+k3b,,+k-3+k6

(A15)

F3 = klk3a,,b0 + k,(k_, + k3 + k, + k5 + k,)a,

+ (k-, + k,)(k_,

+ k,(k_,

+ k, + k5 + k,) + (k-3 + k,)(k,

+ k2 + k3 + k3 + kdb,,

(A161

+ k3)

F3 = klk3(k2 + k4 + k5 + k,)a,,b, + k1{kp(k_3 + k4 + k5 + k,) + (k-3 + k,)(k,

+ k,)}ao

+ (k-1 + k,)(k,

+ ks)}bo + (k-1 + k,)(k-3

F“ = klk3{&

+ k,Hk,

+ k,k,(k-,

+ k3{k&--1+

+ k,) + k&&,,bo

kz + k3 + k,) + k&k,

+ k,k,(k-3

+ k,)

+ k,)(k,

(A171 + ksbo

+ ks)bo

+ k&k,

(A181 (A191

Fs = klkzkakhksaobo Appendix B

Expressions for the coefficients M, ZV, J, G and H of eqns. (9) and (10) and for the amplitudes ydxb of eqn. (8) corresponding to mechanisms (I)-(V). Mechanism

(I):

M = (k3+k4+k&bbo+K2(k4 +k,)a,+K,K,(k,+k,) aobo +&a, + KIKz

(Bl)

(I321 YP,hh= -

N(k, + k5 + b, )eJi,,

ks

Mechanism

(II):

The expressions for M and N are given by eqns. (Bl) respectively. YP,~,=

‘yQ,h

033)

=

Mechanism

-rNeJi,z

and (B2),

0%)

(III):

M=(k3+k,+k,)rbb,+K~(k,+k,)a,+K,(k,+k,)b,+K;K,(k,+k3)

(B6)

a,,bo + K&x,, + Kzbo + K,K; N=

k&do %bO + K&h + Kzbo + K,K;

YPJI =

‘/a,h

=

-

rNeoTh.2

037)

u38)

117

Mechanism

(IV):

(k, + k3 + k4 + k,)a&, + K,(k,

J=

+ K&.

+ k4 + kdb,

aobo + &a,

+ k3 + k,&

+ K,K,(ka

+ k4)

+ Klbo + K&s

039)

G = {(kz+k,)(k,+k,)+kzk~}aobo+k2(k3+k4)K~a0+k~(k3+k~)K~b~ aobo + &a0 + K,bo + K&s (BlO) (Bll)

YP,h=

rH{k&

+ (KS+ bdbJeoG,3

(B12)

k&o

0313)

YQ,h= rH%Th,3 Mechanism

(V):

(k, + k4 + ks + k,h,b,

+ Kdk, + ksyi+ k&o + Kl(kst + ks + Mb, + K&(ks + k,)

J=

aobo + 00

+ Klbo + KI&

G = {(k,+k,)(k,+k,)+k,k,)aobo+k,K,(k,+kG)ao+k,K,(k,+k,)b, aobo +&a, H=

(B15j

+ Klbo + K&s

kzk,kGaobo aobo + &a,

(I3141

u316)

+ Klbo + KS,

H[k,(k,+k,)b,+{(k,+k,+k,)b,+K,(k,+k,)}A,+(K,+bo>~~le,Th,, YP,h

= W&o

(I3171 YQ,h

=

W8)

rfkJh,s

Appendix

C

Expressions for coefficients Ci (i = 1,2, . . . , 11)of eqn. (37). Cl = k,k,(k,k@Ll

+ k;k2k--lj

(Cl)

C2 = kBksklk&

(C2)

C3 = k3k5klk&

((23)

C4 = klk2k;(k3

+ k4j

Cg=k;k&(k3+k4j C6 = klk,k,(kL,

(C4) (C5)

+ k,)

(C6)

118

c, = k;ka*(k_,

+ k,)

+ K&l_,} + k;k,{k,(k_,

c, =kl{klkI_l(kLz

cc71

+ KL,) + kskq} + kp,k&k4

+ k-2 + k,) + k__lkJk1-2 + k,)}

Cm = k4{k;k_1(k_2

+ kLz + k3) + kI1k;(k_:!

Cl1= k_lkI-lkJkL2

+ k_2 + k3)

+ KS)}

((33) ((29) (ClO)

(Cll)