J. theor. Biol. (1975) 49, 201-211

One-Substrate-One-Product Enzymic Reactions: The Relationship between Isotope-exchange Kinetic, Steady-state Kinetic and Equilibrium Parameters I. G. DARWY Department o f Biochemistry, University o f Sydney, Sydney, N.S.W'., 2006, Australia (Received 19 December 1973, and in revised f o r m 3 June 1974)

Expressions are derived for the parameters that can be obtained from (1) steady-state kinetics, (2) isotope-exchange kinetics at equilibrium, and (3) equilibrium binding experiments for the following two one-substrateone-product enzymic mechanisms: E + S .~-- X1 ~ X2 ~- • • • X , . ~ E + P (mechanism I) Em ~-~.Em_ l .~-~-Em_ 2 "~

. E2 .~-..E 1

J

(mechanism II)

It is shown that the exchange constants for both mechanisms are identical with the reciprocals of the equilibrium parameters (Darvey, 1973) that can be obtained from binding experiments. Therefore, expressions relating isotope exchange constants and steady-state kinetic parameters are analogous to those relating equilibrium parameters and steady-state kinetic parameters. The paper also presents, for mechanisms I and II, relationships between the maximum rate of isotope-exchange at equilibrium and the steady-state kinetic parameters. These relationships provide a new criterion for distinguishing experimentally between these two mechanisms. 1. Introduction Boyer (1959) has discussed steady-state kinetics of isotope-exchange at equilibrium. He analysed the enzyme-catalysed reaction S ~ P (for notation, see below) in terms of mechanism I: k+l

k÷2

k+3

E + S . ~ - X I ~ - X2 ~ k-I

k-2

k+(l+l)

X~

~-~ X~+I k-(l+ l)

k+t

• • • -,~ X t

k-a

k+(f+2)

~

...

k-(i+2)

k-~ k+n

~X, k-n

k+(n+l)

~

E4-P

(mechanismI)

k-(n+l)

~nd obtained the following expressions for R (i.e., the rate of exchange of 2Ol

202

I. G. D A R V E Y

labelled atoms between S and P at equilibrium):

R =

V*[~

(1)

R =

V*[P]

(2)

The constants K* and K~ are "exchange constants" for S and P respectively; V* is the maximum value of R when S or P is present at an infinitely large concentration. Apart from stating that:

k_~

(3)

K~ ~< k+l

K* ~< k+(,,+t) k-(.+l)

(4)

he did not explore the physical meaning of the parameters K~ and K*, nor did he investigate whether these parameters could be related to the steady-state kinetic parameters for mechanism I. It is shown that K* and K~ for mechanism I (see above) and also for mechanism II (see below) are identical with the reciprocals of the equilibrium parameters Ks and K'j, (Darvey, 1973) for these two mechanisms; i.e., K ] and K* can be expressed in terms of the equilibrium constants for the various steps in the mechanism describing the reaction S ~ P. Expressions relating K ] and K~ to the steady-state kinetic parameters of mechanisms I and II can therefore readily be obtained from analogous expressions relating Ks and Kp to the steady-state kinetic parameters for these two mechanisms. Boyer stated "that for any enzymic conversion of S ~ - P , the exchange constant for a substance will be equal to or less than its dissociation constant". The analysis presented here shows that, while inequalities (3) and (4) hold for mechanism I, they do not hold for mechanism II: k+t

k+2

k+3

k+i

EI + S . ~ X t ~-~ X 2 ~---" . . . ~-'-"X t k-t k+(t+l)

Xe

E.

~

k-2 k-3 k+U+2 )

X~+I

~--(|+1) k+(n+2)

...

k--(l+2) k+(n+3)

E. -1 ",-- ' ' -

~ k--(n+m--J+2)

k+(n+l)

~-X~

.~

Em+P

k--n ~--(n+l) k+(n+m-J+l)

- "

k-(n+2) k-(n+m-J+ 2)

Ej

~

k-~ k+n

~-

--"

k-(n+3) k-(n+m-d+l) ~--(n+m--J+3) k+(n+m--1)

Ej-I

~

"~

k-(n+m-J+?)

/g--(N+m--I)

k+ (n4-n't)

E2

~

El.

k-(n÷m)

(mechanism II)

ISOTOPE EXCHANGE KINETIC PARAMETERS

203

Mechanism II is a general mechanism describing the enzyme catalysed conversion of S to P by an enzyme which exists in m distinct conformational forms (E x, E2, E3, . • . , Era); the conformational form that reacts with S(E1) is different from that which reacts with P(Em). Special cases of mechanism II with m = 2 have been discussed previously (Darvey, 1972, 1973). It is also shown that the relationship between the isotope-exchange parameter V* and the steady-state kinetic parameters for the reaction S ~ P is different for mechanisms I and II. This offers a new criterion for distinguishing these two mechanisms. (Other criteria have been presented by Darvey, I972, 1973.) The paper begins with an outline of the basic notation used. It then presents expressions (in terms of the velocity constants for mechanisms I and I1) for steady-state kinetic, equilibrium and isotope-exchange parameters. Finally, the relationships between these various parameters are explored. 2. N o t a t i o n

A combination of the notation of Alberty and co-workers (Peller & Alberty, 1959; Alberty, Bloomfield, Peller & King, 1962) and of Darvey (1972, 1973) is used in this text. It is summarized as follows:

S , E , X , P = substrate, enzyme, enzyme-intermediate and product; when more than one enzyme form, or more than one enzyme intermediate, are involved in a mechanism, these are denoted by El, E2, E3 . . . . or X1, X2, X3, • • • , respectively. S*, P* = radioactively-labelled S and P. =

steady-state concentration of S and P at time t.

[~, [P] ---- equilibrium concentration of S and P. [gr] = V

total concentration of all species containing enzyme. steady-state velocity the ratio of the total concentration of aU the various enzyme complexes containing bound substrate or product at equihbrium to [E-r].

R =

rate of exchange of labelled atoms between S and P at equilibrium.

Ksv

Kv

Vp

Ks

Vs

Parameter

i=1

l=l

s

~ Ks s=O k+(s+t)

i-I

s=O k+ (s+l)

k+ (s+l)

not appropriate

K~+ 1

k+0+l)

EET]

s=e

i: s=O z

1=1

i=1

I=1

s=o

K~

j=n+2

EEr]

.+m +E

s=n+l

k+ 0+1)

Kj

($+1)

j=n+2

~-,

afn+l

n+mj--1

(s+l)

K~

.+,,, K ~ +m

k.

k+

k+(s+l)

j=n~+l

sffin+l

j-1 ~'

sfj

X

--J

+ E K ".+m --J a

K"+I.

j=n+t

(s+l)

K~

k+ "+=

n+m--1

K~

$=0

~"

i- i

k+ 0+1)

t

"

X

j=n+l

jfn+l

E sfj

"+~ K".+m

s=O k+0+l)

~

k+ (s+l)

• K,

K~ + k+ (s+l)

. [~=1

s=O

i-1 ~.,

all

K~+1 ~

K~+"

i=1

i=1

K~

s=o k+ (s+ 1)

~.

__K~ + E s=l

k+(s+l) jfn+l

i=1

K~

k+o+l)

s=o k+ 0+1)

s=i

lET]

[Er]

i=1

Mechanism II

Mechanism I

k+ O+1)J

Steady-state kinetic, equilibrium and isotope-exchange parameters for mechanisms I and H

TABLE 1

V*

1 K~

----

K~

=

Ks

K*

1

n

1=1

i=1

I

s=o . k+O+l)

EE~]

K~+1

1

i=1

K7 +1

s=O

KO ~

1=1

~=n+l

2 K~+'

n+m

j=n+l

Z K; *~

n+m

K~ k+ (s+l)

206

I.G.

k+l, k - i , k+2, k-2 . . . .

DARVEY

velocity constants for the various steps in an enzyme mechanism. 'k-(i+l)k-(t+2)

• • • k-s,

k+o+l)k+(l+2 ) 1, i = s

K~=

i < s

k+s

qK',, i > s

3. Steady-state Kinetic Parameters These are derived by the procedure of PeUer & Alberty (1959). For mechanism I:

Vs

Vp

Kss [S] - ~ [P]

IS]

o=

[P]"

l+'~s +

(5)

Kp

For mechanism II:

~[s]v:

v, p k-;[]

[s] 1+ ~

[s][e___]]"

+

(6)

+ Ksp

The steady-state kinetic parameters Vs, Ks, Vp, Ke and Ks1, for mechanisms I and II are given in Table 1. Two other steady-state kinetic parameters (Darvey, 1973) for mechanism II, U and W, are expressed by equations (7) and (8): u = i=1

n+m

w=

2 j=n+l

(7)

X

K,

s=o

k+(s+l)

n+m-1

E

r~

s=n+t

k+(s+l)"

(8)

U can be expressed in terms of the rate constants for those steps in which X~, X2, t'z . . . . undergo unimolecular reactions. W is expressed in terms of the rate constants for those steps in which El, E2, E3 . . . . undergo unimolecular reactions. The use of the parameters U and W in relating steady-state kinetic and equilibrium parameters has been discussed by Darvey (1973) for the case n=m=2. Note that U, W and Ksr do not occur in mechanism I.

ISOTOPE EXCHANGE KINETIC PARAMETERS

207

4. Equilibrium Parameters Equations (9) and (10), which relate ~, [~ and [P], can be obtained by extending the derivations given by Darvey (1973) to mechanisms I and II: 1 + Ks[$]

(9)

K,[~

=

1 + Kp[P]

(10)

Expressions for the equilibrium parameters Ks and K"v in terms of the velocity constants of mechanisms I and II are given in Table 1. 5. Isotope-exchange Parameters By the procedure of Alberty et al. (1962) one can show that, for both mechanisms I and II, R is expressed by equation (1) or (2). Comparison of the expressions for K~ and K~, with those for Ks and/~e shows that, for both mechanisms I and II, K* = 1~Ks and K~, = 1/~e. The expression for V* is the same for both mechanisms. The expressions for the isotopeexchange parameters in terms of the velocity constants for both mechanisms are given in Table 1. 6. Relationship between Exchange Constants and Steady-state Kinetic Parameters Since the exchange constants are reciprocals of equilibrium parameters, they too can be interpreted physically as equilibrium parameters. From the expressions in equation (7) and Table 1, it can be seen that the relationships between the equilibrium parameters and the steady-state kinetic parameters derived previously (Darvey, 1973) for special cases of mechanisms I (n = 2) and II (n = m = 2), also hold for the more general mechanisms discussed in this paper. Thus, for mechanismsI: 1 KsV P K~ = K~ = v s + vp 1

KpVs

(11) (12)

K * = K~ = V s + v,,

while for mechanism II:

1

K~[~-I

(13)

1

KvCEr]

(14)

K * = K-~ =

vsv

208

I . G . DARVEY

Equations (11)-(14) can be used as criteria for deciding whether data are consistent with mechanisms I or II. They should prove useful in systems where direct measurement of ~ (and thus of K s and Kv) is difficult or impossible. From equations,(11) and (12), it follows that, for mechanism I:

K~ < Ks K* < K v.

(15) (16) However, inequalities (15) and (16) do not hold for mechanism II. For mechanism II, K ] and K* may be greater than, less than or equal to Ks and Kp respectively. Therefore, if experimental data indicate that K* > Ks or K~ > Kv, then the data are not consistent with mechanism I; however if KS < Ks and K* < Kv, then the data may be consistent with either mechanism I or II. A comparison of the magnitudes of K ] and K* (or Ks and Kv) with Ks and Kv may be useful in distinguishing mechanisms I and II, even if product inhibition studies (Darvey, 1972) have not been performed. 7. Relationship between Exchange Constants and Dissociation Constants for Steps where Enzyme combines with S and P For mechanism I, K~' and K* may be expressed as: K~' -

(17) 1+ i=2

K* =

K~+ t II--1

(18)

1+ EK" t=l

Equations (3) and (4) follow from equations (17) and (18). Inequalities (3) and (4) do not hold for mechanism II, since K~ and K~ may be greater than, less than or equal to K~ ( = k_~/k+l) and Knn+l ( = k+o+l)/k_o+l) ), respectively. In fact, if n -- 1 and m = 2, then: K~ = K~(1 +Kz3) K ; = K~(I+K~) and K* > K~ and K* > K~. Thus, Boyer's general statement concerning the relationship between the exchange constants and the dissociation constants for the steps where enzyme combines with S and P does not hold for the reaction S ~--P if the process follows mechanism II.

ISOTOPE E X C H A N G E K I N E T I C PARAMETERS

209

8. Relationship between V* and Steady-state Kinetic Parameters Boyer's relationship V*=

1 +

(19)

can be shown to hold for.mechanism I, using the expressions for the various parameters given in Table I. For mechanism II, one obtains:

+

-

i=l

k+(s+l)

s=O

+

j=n+l

2 s=n+l

k + (s+l)

From the expression for V* in Table 1 and equation (20), it can be seen that, for mechanism II, equation (19) does not hold. For this mechanism:

V,>/

1

I \-I

The relationship between V* and the steady-state kinetic parameters-which follows from equation (7) and the expression for V* in Table 1--may be written as: v* = [eT] (21) U Equations (19) and (21) therefore provide an additional means of testing whether experimental data are consistent with mechanisms I or If. A further pair of relationships that hold for both mechanisms I and II is:

V__*= V__~s

K'~

(22)

Ks

V-- = Vp

g~

(23)

gp"

[Equations (22) and (23) follow from equations (1I), (12) and (19) for mechanism I and from equations (13), (14) and (21) for mechanism II.] The ratios given in equation (22) relate the initial steady-state velocity (studied in the absence of P), and R, when measurements are made at low substrate concentrations ([S],=o ,~ Ks and t-S] ,~ K~'). Similarly, equation (23) relates the initial steady-state velocity (studied in the absence of S) and R, when measurements are made at low product concentrations ([P]t=o "~ Kp and [P] ,~ K~,). Using data obtained only under these conditions, does not allow one to distinguish between mechanisms I and II. When the concentration of S and P is large ([$I=0 >> Ks and IS] >> K~'; [P]t=o >> Kr and [P] >> K~,), measurements of the initial steady-state T,n.

14

210

L G. DARVEY

velocity in the absence of P and in the absence of S yield estimates of Vs and Ve; a measurement of R under these conditions yields an estimate of V*. From equation (19) it follows that, for mechanism I: V* < Vs

(24)

V* < re.

(25)

Inequalities (24) and (25) do not hold for mechanism II. For mechanism II, V* may be greater than, less than or equal to Vs or Ve. Thus, if experimental data indicate that V* is greater than either Vs or Ve, then the data cannot be consistent with mechanism I; if V* is less than both Vs and Ve, then the data may be consistent with either mechanisms I or II. 9. Discussion

The relationships between the various parameters which are presented in this paper should hold provided all measurements are made under the same conditions of temperature, pH, ionic strength, etc. The measurements needed for obtaining estimates of the equilibrium parameters require conditions where the total concentration of S and P bound to the enzyme is not negligible in comparison with the concentrations of free S and P. Isotope-exchange and steady-state kinetic parameters are estimated from measurements made under conditions where the total concentration of all species containing substrate and product is much greater than [ET]. It may be difficult (or impossible) to perform equilibrium experiments under conditions identical to those used in steady-state kinetic experiments. Thus, the relations between the exchange constants and the steady-state kinetic parameters [equations (11)-(14)] may prove to be more useful experimentally than those between the equilibrium and steady-state kinetic parameters [equations (11)-(14)]. It is also an advantage to have the extra relationship [equations (19) and (21)] between V* and the steady-state kinetic parameters. In order to use equations (13), (14) and (21), it is necessary first to perform product inhibition studies (Darvey, 1972) on the reaction S~-~-P, since such studies are required for calculating the parameter U from the steadystate kinetic parameters Vs, Ks, Ve, Ke and Kse (the parameter Kse can only be obtained from product inhibition studies). If product inhibition studies have been performed, the relationships presented in this paper can be used as additional criteria for distinguishing between mechanisms I and II. On the other hand, equations (I1), (12) and (19) and inequalities (15), (16), (24) and (25) can be used without previous product inhibition experiments: i.e. it is possible to rule out mechanism I without product inhibition studies.

ISOTOPE E X C H A N G E K I N E T I C PARAMETERS

211

The author thanks Mr D. R. Woodward for his constructive criticisms of the manuscript. REFERENCES ALBERTY, R. A., BLOOMFIELD,V., PELLER,L. & K~O, E. L. (1962). J. Am. chem. $oc. 84, 4381. BoYER, P. D. (1959). Archs Biochem. Biophys. 82, 387. DARWY, I. G. (1972). Biochem. J. 128, 383. DARVEY,L G. (1973). J. theor. Biol. 41, 441. P~LLER,L. & ALeERTY,R. A. (1959). J. Am. chem. Soc. 81, 5907.