Steady-state kinetics of one-substrate enzymic mechanisms involving two enzyme conformations

1. Theoret. Biol. (1969) 23, 53-71

Steady-state Kinetics of One-substrate Enzymic Mechanisms involving Two Enzyme Conformations II. Kinetic Treatment of a Reversible Mechanism: Modifiers and Product Inhibition

Effects of

C. CENNAMO Institute of Biological Chemistry and Institute of Human Physiolog!. University of Modena, Via Campi, Modena, Ital) (Received 2 August 1968) Steady-state kinetics have been worked out for an one-substrateoneproduct reversiblemechanism,involving two interconvertible conformations of the enzyme: one able to bind the substrateand the other able to bind the product. Modifiers, assumedto bind to both enzyme conformations with exclusion of the substrate(and the product) may give different inhibition and activation patternson the forward and the reversereactions. Product inhibition by the reverse reaction may be partly uncompetitive and can be antagonized by a suitable modifier. The model is discussed with referenceto someexperimental results, which are known for some enzymic reactions. 1. Introduction In the previous paper (Cennamo, 1968), the steady-state kinetic treatment of an enzymic mechanism (mechanism I) involving two interconvertible enzyme conformations, one able to bind the substrate and the other regenerated by the reaction, was reported; an irreversible one-substrate reaction S -+ P was considered, kfi E’ E d;> ES k3-+ E’+p. (mechanism I) 1 k-1 It was shown that modifiers, assumedto bind to both enzyme conformations in the place of the substrate or at least with exclusion of the substrate. may give uncompetitive or non-competitive inhibition patterns or may even behave as activators; such effects, according to the Michaelis-Menten mechanism must be explained by the existence of separate sites for the

simultaneous

binding of the substrate and the modifier.

The present paper deals with the steady-state kinetic treatment of the 53

54 mechanism

C.

II,

concerning

CENNAMO

a reversible one-substrate E’P +-

-

1 k-s(P) /k3

?

reaction S i

P:

ES

k-zIkz(S)

(mechanism II)

i -+- EI E’ I

As already pointed out (Cennamo, 1968), according to such model conformations E and E’ can be considered as the fitted forms of the enzyme for the binding of S and P, respectively; this interpretation bears some relation with the induced-fit hypothesis of Koshland (1964). The essential feature of both mechanisms I and II is the conformation change which takes place when the reaction occurs at the active site. As in the previous paper (Cennamo, 1968), the effects of two types of modifiers, A and B, will be considered. These modifiers are supposed to be able to bind to both enzyme conformations with exclusion of the substrate. However, it is assumed that the enzymic species EA and E’A cannot transform into one another, while the interconversion is considered possible for the species EB and E’B: the main purpose of the assumption about the species EA and E’A is to simplify the kinetic treatment: it is not a necessary requirement. The following reaction mechanisms derive from mechanism II, in the presence of modifier A (mechanism IIa) or of modifier B (mechanism IIb): E’P m--ES

1 I E’A -

(A) K’A

1

1 E’ -~

-E

and : E’P -

-p-

I

E’ -

1

-g-

\ I

1 k-sMB) ks k-5

EA

ES

E

1 h(B)k-4 E’B -

i

(mechanism IIa)

1

EB

(mechanism IIb)

MECHANISMS

INVOLVING

TWO

KA and Ki are the dissociation respectively.

ENZYME

CONFORMATIONS

55

constants of the complexes EA and E’A,

2. Kinetic Treatment

Steady-state rate equations have been derived according to the method of linear simultaneous equations and the schematic method of King & Altman (1956). The enzyme concentration is supposed to be very low compared to the concentrations of the substrate and the modifiers. In the numerical examples the values of rate constants and concentrations are expressed in arbitrary units. In the double reciprocal plot of rate equations the different inhibition types are formally defined as: competitive, when the modifier influences only the slope; uncompetitive, when the modifier influences only the intercept (this definition is also applied to activation); mixed, when there is an effect both on the intercept and the slope; and, particularly, non-competitive, when both the intercept and the slope are affected by a same factor.7 According to mechanism II, the steady-state rate equation for the reaction in the direction S -+ P is I k,(E’P)- k-,(E’)(P) W-7 -=1 (i), = dt (E)e (E)+(E’)+(ES)+(E’P) k,k,k3k,(S)-k-,k-,k-3k-7(P)

(E)+(E’)+(ES)+(E’P) and the rate equation for the reaction in the direction P -+ S is: V” (E),

d(S)

1

k-

dt (E)O

- k,(E)

6)

(E)+(E’)+(ES)+(E’P) =

The denominator

z(W

(1)

k-,k-Zk-3k-7(P)-klk2k3k7(S)

(E)+(E’)+(ES)+(E’P) of equations (1) and (2) is

-’

(2)

k-,k-,k-,(P)+klk3k7+klk-2k3+klk-2k-,+ +k,k,k,(S)+k-,k,k,+k-,k-,k,+k-,k-,k-,+ +k,kzk,(S)+k,k,k-,(S)+k-Ik-3k-7(P)+kZk-,k-,(S)(P)+ +k,k,k,(S)+k_,k_,k-3(P)+k-Ik-3k,(P)+kzk_3k,(S)(P).

(3) The relative concentrations of E, E’, ES and E’P, respectively, are proportional to the summations of the terms in each separate line of denominator (3). t Symbols used in this paper are consistent with those used in the previous one (Cennamo, 1968). v’ and v” represent the velocities for the reactions S -+ P and P + S, respectively. v1 and v2 are the corresponding initial velocities.

56

C.

The Haldane relationship

CENNAMO

for mechanism II is

k,k,k,k,

K,K,, ~~~ K,K,

= K = (!b’, (4) o c%, where KE = k- ,lk, ; KEs = k,/k- 7; KSand K,, are the dissociation constants of the complexes ES and E’P, respectively; K, is the equilibrium constant for the reaction S+ P and (P), and (S), represent the concentrations at the equilibrium. When initial velocities are considered [that is, when (P) is zero in equation (1) and (S) is zero in equation (2)], equation (1) becomes k-,k-,k-,k-,

k,k2k,k,(S) 1’1 k,k,[k,+k,+k-,](S)+k2k3k7(S)+k,Q[1+KF]’ (El, = where Q = k,k, + k-,[k,+ k-:]: equation (2) becomes c k-,k-,k_,k-,(P) k-,k-,[k-,+k,+k-,](P)+k-,k-,k-,(P)+k,Q[l+K,j’ (E;, =

(3

(6)

Equation (5) has the sameform as the rate equation for mechanism I, which is concerned with an irreversible reaction S -+ P. The reciprocal forms of equation (5) and (6) are. respectively,

(E),,

1’1

k,[kj+k7+k-,]+k3k7 ~~ -~~~-k,k,k,

_ + ~~

(71

and

(El, -= km,[kL,+k,+k-,]+k-,k-, ~~~~~L’?

k_,k-&,

3. Effects of Modifier A Modifier A is an inhibitor of the reaction. According to mechanism Ila. the initial steady-state rate equation for the reaction in the direction S --+ P is 1:1

(E),,

= k,k&,k,(S)

k,k,[k3+k,+k-,](S)+k2k3k7(S)+ +k,Q[l+K,]+k,Q

(Al (A) r + K, K, A

A

1

(A)

+ k2k,k7(S)K,

, (9)

A

and the initial rate equation for the reaction in the direction P + S is .“z

= k-,k-,k-,k-,(P)

(E>o

k-Ik-3[k-2+k7+k-7](P)+k_zk-jk-.,(P)+ + k-,kw,kw,(P)~;

. (10)

MECHANISMS

INVOLVING

TWO

ENZYME

57

CONFORMATIONS

The reciprocal forms of equation (9) and (10) are, respectively,

t’1

(E),

k [k +k +k-

1 3

’

’

]+k

k

3 ’

k,k,k,

[ 1 l+@j

K;,

+

-L 1 (S)

(l"

and (E), -=-I‘ 2

-~~

k-,ke2k-,

16).

(12)

In Table 1, the various conditions are reported under which the inhibition is essentially competitive, non-competitive or uncompetitive for the reaction TABLE Inhibition

1 Conditions

type

for the reaction P + S

for the reaction S + P

Competitive

(A) in < I (hence,

(“? < 1 (hence 3.4K’ __ K A ) K; _

9

or

or

k-zk-7

Non-competitive

KA .& KA)

k,k, > kl[k, KE > 1

i

[I + g]

k, + k -7]

Uncompetitive (hence, KE 4: 1)

(hence, KE 2, 1)

58

C.

CENNAMO

in each direction. In the most general case, the inhibition will be mixed in each direction. In most cases, the type of inhibition depends on the value of (A) and will change if (A) is varied; however, non-competitive inhibition type may be independent of (A). According to some of the conditions in (b)

32

‘p 0

2-

iii1 : 0 $2

& I-

L

,n4 IV

i/(S)

!P

!,(i)

FIG. 1. Effects of modifier A: competitive inhibition of both reactions, (a) S + P and (b) P --f S. Double reciprocal plots in the absence (curves 1 and 3) and in the presence of A (curves 2 and 4, (A) = 10e3). Numerical values of the constants: kl = k-, = 10-l; k, = 1; ka = 10e4; k, 1: 10-5: k-3 = 1O-3; k7 = k-, = 102; KA = K; = lo-*. (01

(b) 4

2

3

I

I IO

IO 3 l/W

IlIP)

2. Effects of modifier A: non-competitive inhibition of both reactions, (a) S + P and (b) P + S. Double reciprocal plots in the absence (curves 1 and 3) and in the presence of A (curves 2 and 4, (A) = 10-a). Numerical values of the constants: k, == k-l = 10w2; k, =m 105; k-2 ~7 10’; k, -= 10: km, = k, = k-, = 1; K., = K; = 10-4. FIG.

MECHANISMS

INVOLVING

TWO

(01

ENZYME

CONFORMATIONS

59

tb)

3-

3-

I/W

IO’ I/(P)

FIG. 3. Effects of modifier A: (a) competitive inhibition of reaction S + P and (b) noncompetitive inhibition of reaction P + S. Double reciprocal plots in the absence (curves 1 and 3) and in the presence of A (curves 2 and 4, (A) = lOMa). Numericalvaluesoftheconstants:k, =.Ll = IO-2:k2 = 105;ka = 102;ka = lo-“: k..3 = k, = k-, = 1; Ka = Q = 10-4.

Table I, the inhibition may be essentially competitive or non-competitive in both directions of the reaction. The inhibition cannot be essentially uncompetitive in both directions. On the other hand, the inhibition may be competitive in one direction and non-competitive in the other one; or, competitive or non-competitive in one direction and uncompetitive in the other one. The numerical examples of Figs 1 to 5 concern these different cases. (0)

I

(bi

FIG. 4. Effects of modifier A: (a) uncompetitive inhibition of reaction S -+ P and (b) competitive inhibition of reaction P + S. Double reciprocal plots in the absence (curves 1 and 3) and in the presence of A (curve 2, (A) = 10-4; curve 4, (A) = IO- 2). Numerical values of the constants: kl = 10ea; k-1 = lo-*; k, = 105; k-, = 102; ka = 1O-5; km3 = k, = k-, = 1; K,, = 1; K; = 10-O.

60

C‘.

(‘IiNNAMO

FIG. 5. Effects of modifier A: (a) uncompetitive inhibition of reaction S --f P and (b) non-competitive inhibition of reaction P + S. Double reciprocal plots in the absence (curves 1 and 3) and in the presence of A (curve 2, (A) = 10m4: curve 4, (A) = 10-2). Numerical values of the constants: k, = 1O-2; kel = 10-l”; k, mm= lo”: k-? -- lo’: k, = 10-5; k-R = k, = k-, = 1; KA = 1O-2; K; = 1O-8.

4. Effects of Modifier B According to mechanism lib, the steady-state rate equation for the reaction in the direction S * P is V’

US, where

k,k,k,k,N(S)+k2k3k4k5k-6k,(S)(B)-k,,k-,k-,k-,N(P)-k-,k-3k-4k-5k6k-,(P)(B) (E)+(E’)+(ES)+(E’P)+(EB)+(E’B)

(13j

N = k-,k-,+k,k-,+k-,k-,, and the rate equation for the reaction in the direction P + S is ,? cK,,=

k-,k-2k-3k-7N(P)+k-2k-3k-4k-sksk-,(P)(B)-kLk2k3k7N(S)-k2k3k4k5k-bk7(S)(B) (E)+(E’)+(ES)+(E’P)+(EB)+(E’B)

’

(14)

The denominator of equations (13) and (14) is {k,k,[k,+k_,]N(S)+k,k,k,k_,[k,+k-,](S)(B)+k,k_,k_,N(S)(P)+ +k-,k~,k_,N(P)+k_,k_,k~,k,k_,(P)(B))+{k,k,k,N(S)+ +k,k,k,k~,k,(S)(B)+k,k-,k,N(S)(P)+k_,k_,[k_,+~,]N(P)+ +k-,k-,k-,k,[kz+k,](P)(B)j+{k,QN+k,k,k-,Q(B)+ +k-,k-3k-,N(P)}+(k-lQN+k-ok-Sk,Q(B)+k,k,k,N(S))+ +{k~k,[k-,+k,]Q(B)+k-,k,k,Q(B)+k,k,k,Q(B)Z +k~,li~,k,k~,[k~,+k,](P)(B)}+{k,k-,k,Q(B)+k_,k,[k-,+ +k-,]Q(B)+k,k-,k,Q(B)2+k,k,k,k,[k-,+k-,](S)(B)+ +k-,k-,k-,k,k-,(P)(B);.

+k,k,k,k,k,(S)(B)+

(15)

MECHANISMS

INVOLVING

TWO

ENZYME

CONFORMATIONS

61

The relative concentrations of ES, E’P, E, E’, EB and E’B, respectively, are proportional to the successive summations of the bracketed groups of terms of denominator (15). Since mechanism IIb involves the cycle of reversible reactions: E -+ EB + E’B + E’-+ E, the corresponding rate constants must satisfy the equality: k-,k,k,k-, = k,kw4k-,k,, which can be written as k-,k-,k,

= k,k,k-,K,.

(16)

When the equality (16) is used, the same Haldane relationship (4) is obtained for mechanism IIb as for mechanism II. When initial velocities are considered, equation (13) becomes

"1

--{ iE>o

+ kzk3k4~k-6k7(S)(B)}/jk,k,[k3+k,+k-7](S)+

= k,k,k,k,(S) + k2WAV

+

k2k4k,k-,[k,

+ k,k3k4k7[k5+k-,+k-,] ~__----N

+ +“[L+K,]Q(B)

+ k, + km71

~~__--N

(S)(B) + +KE] t

(S)(B)+k,Q[l + k$k6[k-,+ks+k-,]+

+k4[k5+k-s+k-JKE}(B)

(17)

+ -

and equation (14) becomes %= 0%

k-lk-Zk-3k-7(P)

+

k-2k-3k-4k-5k6k-, -~~N

(P)(B)

+ k_,k-4k_,k,[k_,+k,+k_,] N

O’)(B) t

+ k-Sk-3ksk-7[k-4+k5+k-5] N

(P)(B) -t

+ k,Q[l+K,]

+ 5!+ +k4[k,+k-s+k-,]K,}(B)

[l+K,]Q(B)

+ $f(k,[k~,+C,+k~,]

b’d3s + k- ,lQ (B)2}. ____~----N

+ ---

+

(18)

62

C.

The reciprocal

forms of equations (17) and (18) are, respectively

k,[k,+k,+k-,]+k3k,

Q[l+&]

CENNAMO

+

k,k5k-,[k,

+ k, + --.k- .,]...(B) + N

[k, + s+(B)]

+ !${k6[k-,fki+k-s]+

b’dk,+k-,lQ~B~,

+k,jk,+k-,+k-,-)-C,)(B) + ~~--_-~__.___~. k2k,k,

+ ---N----

1 w

k, + t+L;G(B)]

(19) and I;-,[k-,+k,+k-,]+k-,k-,+

k-,k-, Q[l+&]

[k, + SF

k-4k-,k6[k-2+k7+k-,] -__ -m-~---(B)+ N + I<-zk6k-,[k-4+k,+k-5] -~ ~-09 N ~-. - + km, + k-+$~(B) I (B) ] + !${k6[k-4+k,+k-5]+

+k4[k5+k-,+k-,-j&}(B)

+

+ ~_...____~..~ k-,k-&,

k-,

+ k2$&

‘%@s +k- JQ N

(B)]

WZ

1

m

(20) Modifier B may be an inhibitor of the reaction as modifier A, but it may also act as an uncompetitive activator. It can be shown that intercept of equation (19) is lower than intercept of equation (7) when k,[k5+k-5+k-6] < k,k-,. (21) This is the same condition deduced previously (Cennamo, 1968) in the case of the irreversible reaction S + P. Similarly, intercept of equation (20) is lower than intercept of equation (8) when k-,[ke4+k5+k-J < k-,k-,. (2.3 If the slopes of equations (19) and (20) are compared with the slopes of equations (7) and (8) respectively, it can be seen that B in any case has the

MECHANISMS

INVOLVING

TWO

ENZYME

63

CONFORMATIONS

effect of increasing the slope. This effect may be almost negligible and thus B can act essentially as an uncompetitive activator. However, in the most general case, B is an activator at high substrate concentrations and an inhibitor at low substrate concentrations. If the reactions in the two directions are considered separately, it can be shown that B may be an activator in both cases (Fig. 6) or an activator for one reaction and an inhibitor for the reverse one (Fig. 7).

FIG. 6. Effects of modifier B: uncompetitive activation of both reactions, (a) S --f P and (b) P + S. Double reciprocal plots in the absence (curves 1 and 3) and in the presence of B (curves 2 and 4, (B) = 10-l). Numerical values of the constants: k, = 1; k. 1 = 10e2; kz = 106; kz = 10; k, -: 102; k., = log; k, = 104; k-? = 102; k, = k-4 = k, = k-, = ke6 = 105; ks = 10”.

5. Product Inhibition by the Reverse Reaction

The reaction in the direction S + P will be considered, but the conclusions can, of course, be applied to the reverse reaction also. It is admitted that a constant concentration of P is present, and initial rates are considered. Equation (1) can be written --Vt = k,k,k,k,(S)-k-lk-zk-,k-,(P) k,k,[k,+k,+k-,-j(S)+

(J%

+k,k,k,(S)+kZk+[k,+kq](S)(P)+klQ[l

+K,]+

+k-Ik-,[k-2+k7+k-7](P)+k-2k-3k-7(P)}.

-7 (23)

64

C.

CENNAMO

FIG. 7. Effects of modifier B: (a) activation of reaction S -- P and (b) inhibition of reaction P + S. Double reciprocal plots in the absence (curves 1 and 3) and in the presence of B (curves 2 and 4, (B) = 10-l). Numerical values of the constants: /il I ;k 1 -:= 10-z; k, 2 IO”. k -., IO;/\., 102; ke3 = 108; k7 =Y 104; ,I~, z 10”; k, 7 k5 =z k, z 105; h- 4 :- 108: km, :- IO ‘; ke, := 103.

The reciprocal form of equation (23) is

03, klk2[k~+k7+k-i]+kZk3k,+kZk-3[k7+k-7](P)+ ~-- = - --t’

I

k k k k -k1

2

3

7

I-- km k- ‘p’ 1’2

3

7tS)

__.._ + klQ[1+KE]+k-Ik-3[k-2+k7+k-7](P)+k-2k-3k-7(P) _-k,k,k,k7-k-,k~,k-3k-7~;;

I 6)

(24)

Equation (24) gives a curve concave upward. a behaviour similar to that obtained in the case of the Michaelis-Menten mechanism. However, the intercept (for (S) -+ co) of equation (24). k,[k,+k,+k-,]+k,k,+k-,[k,+k&J(P) k,k,k,

(25)

is higher than the intercept of equation (7) and increasesby enhancing (P). This effect is due to the presencein the denominator of equation (23) of a term containing (S)(P), and is not possible with the Michaelis-Menten

MECHANISMS

INVOLVING

TWO

ENZYME

65

CONFORMATIONS

mechanism. Therefore, even in the case of product inhibition, the mechanism admitting the existence of two interconvertible enzyme conformations allows the prediction of an effect of uncompetitive type. The effect of a modifier of type B on the product inhibition will be now considered. The reciprocal form of equation (13) is

(Eh ---- =

k2k4k,k-,[k3 + ___-

((k,kJk,+k,+k-,]+k,k,

1’1

+ k,MJ4k,+k-,+k-61

+ k, + k-71 N

09 +

(B)+k&-Jk,+k-,I@‘)

N

k4k$+B)]-k-,k~,k-,[k-,+k-4;5k6(B)]~)}+

k-,k-4k-5k6[k-2+k,+k-,] + ----Iv-

__~

+ Cc_zk-,k6k-,[k-4+k5+k-5] N + Q[l +K,]

(P)(B) + --m(P)(B)+

kl + k4L;L5

(B)

1

+

+ ii~~{k6[k-4+k5+k-5]+k4[k5+k-5+k-6]~E}(~)+

+ @d35 +k- JQ ~ (B)‘)/(k,k,k7 N

[kl + %+(B)]

-

k-, +!.-Z+&(B)

(P> L. (26) 1 (9 )> (8 The intercept of equation (26) is smaller than the intercept of equation (24) when k,[k,+k-,+k-,]

< k,k-,

CP)].

(27)

On the other hand, if it is also k,[k,+k-,+k-,] z k5k-6, B wih not be an activator in the absence of P according to condition (21). A numerical example is reported in Fig. 8: the effect of B is essentially that of removing the uncompetitive part of the product inhibition, although B, when present alone, is a competitive inhibitor of the reaction. If the relative concentrations of the different enzyme species (for (S) + co) are calculated, it appears that the presence of B results in a decrease of the relative concentration of E’ and consequently of the rate of the reverse reaction P + S. In fact, in the presence of B the enzyme is predominantly in the form EB, from which the 7.8. 5

66

C.

CENNAMO

FIG. 8. Product inhibition and effect of modifier B, for the reaction S ~~_P. Double reciprocal plots in the absence of B and P (curve I), in the presence of B (curve 2, (B) = 10e3), in the presence of P (curve 3, (P) :~m3 x 10h4; curve 4, (P) == 10e3), and in the presence of B and P (curve 5, (B) IO-“, (P) m-~3 10m4;curve 6, (B) -~ IO-“, (P) = 10-Z). Numerical values of the constants: k, : 1; k-, y-7lo-“; k, z 106; k-, ..- 102; k3 = 102; k-, = 106; k7 = 1O4; k-, = 102: k, = k-, -- k5 = 105; k-s = k-, = 1; k, = 103.

species E is regenerated at a rate equal to that of the direct conversion E’ -+ E (since k- 6 = k,); this step is the rate determining one for the forward reaction S + P, in the absence of B. This effect of B is similar to that shown previously (Cennamo, 1968) of removing the uncompetitive inhibition of modifier A. 6. Discussion Mechanism I has been considered in detail in the previous paper (Cennamo, 1968); the interpretation given can also be applied to mechanism II which represents a more general formulation. Mechanism I, considering a reaction S + P, corresponds formally to that proposed by Medwedew (1937). As already pointed out by King (1956), the difference with the Michaelis-Menten mechanism is that. at high substrate concentrations, the rate determining step is represented by the con-

MECHANISMS

INVOLVING

TWO

ENZYME

67

CONFORMATIONS

version E’ --f E. Therefore, the actual maximum velocity does not correspond to the complete saturation of the enzyme by the substrate; when this velocity is attained the enzyme can be predominantly present as E’. The MichaelisMenten dissimilarity of the mechanism can be exaggerated by a modifier of type A because of the formation of the complex E’A; on the other hand. a modifier of type B can reduce this dissimilarity, and therefore increase the maximum velocity, by accelerating the conversion of E’ to E through the side pathway E’ + E’B + EB + E. These effects concern the kinetic behaviour at high substrate concentrations and are therefore essentially uncompetitive. If the reversible mechanism II is now considered, the same interpretation can be applied to the case of the reverse reaction P -t S. In this case, however, it is the step E + E’ which becomes the rate determining one at high concentrations of P, because E’ is the enzymic form which gives rise to the active complex for the reverse reaction. The present paper shows that, according to mechanism II, modifiers which bind to both enzyme conformations at the active site in the place of the substrate, or at least with exclusion of the substrate, may exhibit different effects on the reactions in the two directions. Inhibitory effects of different type on the two reactions are possible, as well as activation for the reaction in one direction and inhibition for the reverse reaction. Further, the mechanism allows the prediction of a product inhibition by the reverse reaction which is partly of uncompetitive type. There are many examples of such effects, which have been observed experimentally with some enzymic reactions, and which could be explained in a simple way by the present model. Otherwise, these effects must be interpreted by assuming the existence of separate binding sites for the substrate and the modifiers (cf. Frieden, 19641, or for the substrate and the product, and this hypothesis is not always plausible; other explanations are possible in some cases only. According to the present model, assuming the existence of two interconvertible enzyme conformations which are considered to be different in kinetic properties and the catalytic ability of the active site. it is possible to give a general explanation admitting proper values for some kinetic constants. Some examples of such effects will be reported now, under separate heads : they are not intended to represent a complete list. (A)

STRUCTURALLY

RELATED PATTERNS

INHIBITORS WITH

THE

GIVING SAME

DIFFERENT

INHIBITION

ENZYME

The inhibition of xanthine oxidase by purine compounds structurally similar to the substrate is not always competitive; the milk enzyme is competitively inhibited by adenine, 6-chloropurine, 6-mercaptopurine, 2-aza-

68

c.

CENNAMO

adenine and other compounds, and non-competitively by 8-azaguanine; the enzyme from Clostridium cylindrosporum is competitively inhibited by 2-hydroxy-6-aminopurine, but mixed inhibition is observed with adenine and 8-azaxanthine (references in Webb, 1966, p. 282). oc-Amilase of Candida tropicalis is inhibited competitively by maltose and cr-phenylglucoside, and non-competitively by sucrose and P-phenylglucoside (reference in Webb, 1966, p. 420). L-Lactate dehydrogenase of beef heart is inhibited competitively with respect to NAD by thionicotinamide-NAD, nicotinylhydroxamate-NAD and nicotinylhydrazide-NAD, and uncompetitively by 3-benzoylpiridineNAD (Anderson & Kaplan, 1959). Nicotinamide deamidase of Torula cremoris is inhibited competitively by 3-acetylpyridine, and non-competitively by NAD, NADP and the 3-acetylpyridine analogue of NAD (Joshi & Handler, 1962). Kidney D-amino acid oxidase is inhibited competitively, with respect to FAD, by FMN and riboflavin-5’-sulphate, while the inhibition by riboflavin is not reduced by increasing the concentration of FAD (references in Webb. 1966, pp. 540-541). (B)

THE

SAME

INHIBITOR SAME

ENZYME

GIVES

DIFFERENT

OBTAINED

FROM

1NHIBITION DIFFERENT

PATTERNS

WITH

I‘HE

SOURCES

Adenine inhibits competitively xanthine oxidase of milk and non-competitively the enzyme of chicken kidney; mixed inhibition has been observed with the enzyme of Clostridium cylindrosporum. 6-Mercaptopurine inhibits competitively the enzyme of milk and non-competitively the chicken kidney enzyme (references in Webb, 1966, p. 282). D-Asparagine inhibits competitively asparaginase of Mycobacterium phlei (Grossowicz & Halpern, 1956), while the inhibition appears not to be competitive with the enzyme from Bacillus coagulans or Bacillus stearothermophilus (Manning & Campbell, 1957). Hexokinases of liver and yeast are inhibited non-competitively with respect to ATP by ADP, while the enzyme from Echinococcus is inhibited competitively (references in Webb, 1966, p. 383). NADases of rabbit erythrocytes (Alivisatos, Kashket & Denstedt, 1956) and lupin seedlings (Hasse & Schleyer, 1961) are inhibited competitively by nicotinamide. The inhibition is non-competitive in the case of NADases of brain and spleen (Zatman, Kaplan & Colowick, 1953); it has been proposed that in the case of the NADases of brain and spleen nicotinamide competes with water for the complex between the enzyme and the nicotinamide-missing moiety of NAD, while the NADases of rabbit erythrocytes and lupin seedlings would act by a different mechanism.

MECHANISMS (C)

THE

INVOLVING INHIBITOR

TWO

GIVES

FORWARD

ENZYME

DIFFERENT AND

THE

69

CONFORMATIONS

INHIBITION REVERSE

PATTERNS

ON

THE

REACTION

L-Lactate dehydrogenase from rabbit erythrocytes is inhibited by oxalate, tartronate and malonate, competitively with respect to lactate, but the inhibition is non-competitive with respect to pyruvate; the inhibition by phenoxyacetate was reported to be simultaneously competitive and noncompetitive towards both substrates (Ottolenghi & Denstedt, 1958). Ottolenghi & Denstedt (1958) proposed that pyruvate and lactate react with different sites on the enzyme; however, this hypothesis is not necessary according to Novoa, Winer, Glaid & Schwert (1959), since enzyme-NAD and enzyme-NADH complexes bear different conformations of the active site. Oxalate inhibition is competitive with respect to lactate with other dehydrogenases also, but it is uncompetitive in the case of the lactate dehydrogenase of Propionibacterium pentosaceum (references in Webb. 1966, p. 435). On the other hand, the inhibition by oxamate is almost cotnpetitive towards pyruvate and non-competitive towards lactate (Novoa et al., 1959). These effects on lactate dehydrogenases may also be placed under sections 6(A) and 6(B). Malic enzyme of pigeon liver is inhibited by tartronate; the inhibition is competitive with respect to malate and largely non-competitive towards pyruvate. Malonate, however, inhibits competitively in both cases (Stickland, 1959). L-Glutamate dehydrogenase of beef liver is inhibited by glutarate, isophtalate and 5-bromofuroate : competitively towards glutamate and non-competitively towards cr-ketoglutarate (Caughey, Smiley & Hellerman. 1957). (D)

THE

INHIBITION ENZYME

PATTERN IS ABLE

DEPENDS TO

ON

CATALYSE

THE

DIFFERENT

SUBSTRATE

WHEN

THh

REACTIONS

According to Bonsignore, Pontremoli, Grazi & Horecker (1960), transaldolase from yeast is inhibited by phosphate competitively with respect to fructose-6-P (when this compound reacts with glyceraldehyde), but the inhibition is non-competitive towards sedoheptulose-7-P (when the reaction of this compound with glyceraldehyde-3-P is considered). (E)

THE

INHIBITION EXPERIMENTAL

PATTERN

DEPENDS

ON

THE

CONDITIONS

L-Amino acid oxidase from the hepatopancreas of Cardium tuberculatum is inhibited by L-leucine competitively at pH 7-6, while the inhibition is not competitive at pH 9.2 (Roche, Glahn, Manchol & Thoai, 1959).

70

C.

CENNAMO

The inhibition of L-lactate dehydrogenase from beef heart by oxamate is almost competitive with respect to pyruvate, and the competitive character is more evident as pH is increased (Novoa et al., 1959). (F)

A

COMPOUND

CONCENTRATIONS

BEHAVES AND

AS

AN

AN

INHIBITOR

ACTIVATOR

AT

AT HIGH

LOW

SUBSTRATE

CONCENTRATIONS

This is the case when the double reciprocal plots intersect to the right of the ordinate; it is possible for a modifier of type B according to mechanisms I and 11. Such effect has been shown with D-asparagine on asparaginases of Bacillus coagulans and Bacillus stearothermophilus (Manning 8z Campbell, 1957). (G)

PRODUCT

INHIBITION

In many cases, product inhibition is not competitive. It is well known that hexokinases from various sources are inhibited non-competitively by glucose-6-P (Weil-Malherbe & Bone, 1951; Crane & Sols, 1953), while hexokinase from yeast is not inhibited (Colowick & Kalckar, 1943). Glutaminases from different sources are inhibited non-competitively by glutamate (references in Webb, 1966, p. 332). Glucose-6-phosphatase of rat liver is inhibited non-competitively by glucose (Segal, 1959); this inhibition has been explained by competition of glucose with water for its site. It could be noted that, if the concave shape of the curves obtained by the double reciprocal plots in the presence of the product, according to the present model (cf. Fig. S), is not very evident,? the inhibition could in practice appear to be of mixed or non-competitive type. The experimental demonstration of the occurrence of the model proposed here requires a specific approach in each definite case. King (1956) has suggested measurements at substrate concentrations comparable to that of the enzyme. Recently, Britton (1966) has proposed flux measurements as a tool for evidentiating the participation of more enzyme conformations to the reaction mechanism. Many thanks are due to Dr L. Razzoli for her help during the preparation of this paper. REFERENCES ALIVISATOS, S. G. A., KASHKET, S. & DENSTEDT, 0. F. (1956). Can. J. Biochem. Physiol. 34,46. ANDERSON, B. M. & KAPLAN, N. 0. (1959). J. biol. Chem. 234, 1226. BONSIGNORE, A., PONTREMOLI, S., GRAZI, E. & HORECKER, B. L. (1960). J. biol. Chem. 235, 1888. t For

values

of Cpj far from

those

corresponding

to the equilibrium.

MECHANISMS

INVOLVING

TWO

ENZYME

CONFORMATIONS

71

BRITTON, H. G. (1966). Archs Biochem. Biophys. 117, 167. CAUGHEY, W. S., SMILEY, J. D. & HELLERMAN, L. (1957). J. biol. Chem. 224, 591. CENNAMO, C. (1968). J. Theoref. Biol. 21, 260. COLOWICK, S. P. & KALCKAR, H. M. (1943). J. biol. Chem. 148, 117. CRANE, R. K. & Sons, A. (1953). J. biol. Chem. 203, 273. FRIEDEN, C. (1964). J. biol. Chem. 239, 3522. GROS~~WICZ, N. & HALPERN, Y. S. (1956). Nature, Land. 177, 623. HASSE, K. & SCHLEYER, H. (1961). Biochem. Z. 334,360. JOSHI, J. G. & HANDLER, P. (1962). J. biol. Chem. 237,929. KING, E. L. (1956). J. phys. Chem., Ithaca, 60, 1378. KING, E. L. & ALTMAN, C. (1956). J. phys. Chem., Ithaca, 60, 1375. KOSHLAND, JR., D. E. (1964). Fedn. Proc. Fedn. Am. Sots. exp. Biol. 23, 719. MANNING, G. B. & CAMPBELL, L. L. (1957). Can. J. Microbial. 3, 1001. MEDWEDEW, G. (1937). Enzymologia, 2, 53. NOVOA, W. B., WINER, A. D., GLAID, A, J. & SCHWERT, G. W. (1959). J. biol. Chem. 234, 1143. OTTOLENGHI, P. & DENSTEDT, 0. F. (1958). Can. J. Biochem. Physiol. 36, 1075. ROCHE, J., GLAHN, P. E., MANCHOL, P. &THOAI, N. (1959). Biochim. biophys. ACM, 35,111. SEGAL, H. L. (1959). J. am. them. Sot. 81,4047. STICKLAND, R. G. (1959). Biochem. J. 73, 646, 654. WEBB, J. L. (1966). “Enzyme and Metabolic Inhibitors”, Vol. II. New York: Academic Press. WEIL-MALHERBE, H. & BONE, A. D. (1951). Biochem. J. 49, 339. ZATMAN, L. J., KAPLAN, N. 0. & COL~WICK, S. P. (1953). J. biol. Chem. 200,197.