Kinetics of the two-sited enzyme

J. theor. Biol. (1971) 32, 405-414

Kinetics of the Two-sited Enzyme I. Activation

and Inhibition

by Substrate

EDWIN T. HARPER+ Department

qf Chemistry, Clarkson College qf Technology. Potsdam, N. Y. 13676, U.S.A.

(Received 29 July 1969. and in revised,form

18 January 1971)

Equations have been derived and plotted to describe apparent modifiel effects of a single substrate which is randomly bound, in rapid binding equilibria. at two sites of an enzyme. Three special cases have been considered: independent, non-equivalent catalytic sites; equivalent, interacting catalytic sites; one catalytic site and one modifier site. In each case, the curvature of Lineweaver-Burk plots has been determined by evaluating the limits of the derivatives, d( l/o,)/d( l/S ) and d(S/co)jdS. The direction of curvature has been correlated with modifier effects by distinguishing between activating and inhibiting effects on maximal velocities (V), or on dissociation constants of enzyme-substrate complexes (K). Upward curvature, with a minimum in the plot, corresponds to V-inhibition. Upward curvature without a minimum corresponds to various combinations of activating effects. Downward curvature represents either K-inhibition, with or without simultaneous l’-activation, or no interaction

at all.

1. Introduction The fundamental equation of enzyme kinetics (cf. Segal. 1959) expressesthe initial reaction rate, at constant enzyme concentration. by L’()= as/( 1+ bS),

(1)

where Q, is the rate in the absenceof product, S is the (invariant) concentration of substrate, and a and b are combinations of constants, a including the enzyme concentration. However, the equation is not applicable to enzymes with more than one active site, unless the sites are both equivalent and independent (Botts & Morales, 1953). The equations for initial reaction rates of multi-sited enzymes are ratios of t Present address, Department of Biochemistry, lndiana University School of Medicine. 1100 West Michigan Street. Indianapolis, Indiana 46202. L!.S.A. 405

406

E. T. HARPER

polynomials in S (Botts & Morales, 1962), of the form o,=-&S’

1953; Botts, 1958; Wong & Hanes. (2)

l+p+s’. i

ii

1

1

In the absence of other modifiers, the degree of the polynomial in the denominator is equal to the number of kinds of enzyme-substrate complexes. The degree of the polynomial in the numerator is equal to the number of kinds of complexes which lead to products. Thus, enzymes with two active sites may have initial reaction rates that are quadratic or cubic functions of S. The dependence of rate on higher powers of substrate concentration appears in experimental data as curvature in the normally linear LineweaverBurk (Lineweaver & Burk, 1934), or equivalent (Haldane & Stern, 1932; Haldane, 1957), plots. The direction of curvature has sometimes suggested the following interpretation: downward curvature (Fig. 1) indicates a rcaction going too fast with increasing substrate concentration, and thus represents activation of the enzyme by the substrate: upward curvature (Fig. 2, curve a) similarly represents substrate inhibition. -

_----

I

(A,)'

~

(6) /

!

IO ‘/S FIG. 1. Downwardly curving Lineweaver-Burk plots for various cases (4 = 10 -3, Ka = lo-‘). (A,) Enzyme A, VI = 1, Vz = 2; (A,) enzyme A, VI = 2, V, = I; (B) enzyme B. VI = 1, Va = 2; (C) enzyme C, K’, = 10m2, VI = 1, VZ = 2.

This interpretation cannot be completely correct. One mechanism of substrate activation could involve co-operative binding effects, similar to those postulated for hemoglobin and some other proteins (Wyman, 1964; Stadtman, 1966). However, Lineweaver-Burk plots showing co-operative binding effects curve upward (Fig. 2, curve b), not downward (Frieden, 1964; Dixon & Webb, 1964). Furthermore, downwardly curving Lineweaver-

KINETICS

OF SUBSTRATE

ACTIVATION

407 / /

I

FIG. 2. Lineweaver-Burk plots for enzyme B. Curve a: KI =T 2 :< 10e3, Kz = 2 ..: 10-4, Vl=lO, V2-I. Curves b to d: &=4~10-~, VI I/, == 5, Kl ~-. 8, 4, and 2
Burk plots can be obtained in the absence of any interactions between sites (Kistiakowsky & Rosenberg, 1952). The aim of this paper is to clarify the mathematical and graphical properties of substrate activation and inhibition for two-sited enzymes which bind a single substrate in a series of rapid binding equilibria. 2. Model and Rate Equations

I would like to choose a model which demonstrates the substrate modifier effects that are made possible merely by the presence of more than one substrate-binding site per enzyme molecule. The simplest suitable model is shown in equation (3). The model represents an enzyme with two substratebinding sites which can form two isomeric ES complexes and a ternary ES, complex. The model is general, in that all of the complexes may yield product, and limitations on the reactivities of the complexes, or on the degree of interaction between sites, can be introduced by merely specifying the values of appropriate rate constants. It is limited, however, to enzymes with rapid binding equilibria. This may not severely limit its generality, since substrate binding equilibria do appear to be rapid in most cases.

408

E.

T.

HARPER

The equation for the initial rate of reaction in this system is

(Vz+li;)+(~~K*+li;K;)/S Do= -1(K,+K;)/S+K,XZ/S2

’

where z’. is the rate in the absence of products; VI = k, Eo, V; = I\;E:,. V2 = k,E,,V; = kkE,; E, is the total enzyme concentration; and the constants K are dissociation constants corresponding to reaction steps with matching notations. There are three special cases of model (3) in which only one parameter is independently variable : Enzyme A involves active sites which are independent, but not equivalent. Rate constants for reaction at each site are thus not influenced by the degree of saturation of the enzyme. Since the same site is filled when either E is converted to C,, or C; is converted to C,, K, = K;, K, = K;, I;, = k;, etc. Making this group of substitutions in equation (4) yields the initial rate equation for independent, non-equivalent sites, [‘O = 1 +(K,+K;)/S+-K, which may also be obtained of different enzymes acting presented by Kistiakowsky (1959), and Dixon & Webb Enzyme B involves active interacting. The complexes model (3) can be simplified

K2/S2 ’

(5)

by considering enzyme A equivalent to a mixture on one substrate. Similar equations have been & Rosenberg (1952), Alberty ( 1956), Reiner (1964). sites which are initially equivalent, but mutually C, and C; are thus indistinguishable, and to a “linear” scheme: kJ 2k4 ii t. ~ 0, 2k,S

kzS

E * c, e c,. (6) Zk-2 k-1 Statistical factors of two are used to retain the definitions of rate constants in terms of velocity per active site. The corresponding rate equation (7) is obtained by deleting the primes from equation (4). (7) Equivalent equations have also been derived by Kistiakowsky & Rosenberg (1952), Botts (1958) and Reiner (1959). Enzyme C is a form of the well-known “single substrate-single modifier” case (Segal, Kachmar & Boyer, 1952; Botts & Morales, 1953; King, 1956: Laidler, 1956; Dixon & Webb, 1964; Frieden, 1964) with the substrate as

KINETICS

OF

SUBSTRATE

409

ACTIVATION

modifier, in which one of the sites has no catalytic capacity but does influence the reactivity of the other site. Some hydrolytic enzymes may be of this type (Trowbridge, Krehbiel & Laskowski, 1963). The rate equation (8) is obtained from equation (4) by setting k; = k, = 0. V; + I”, K,/S ” = l+(K,+K;)/S+K,

(8)

K,/S”

3. Shapes of the Lineweaver-Burk

Plots

Enzymes A, B and C can all yield non-linear Lineweaver-Burk plots (Botts, 1958). The curves are predicted by partial derivatives of the reciprocals of equations (5), (7) and (8) with respect to I/S, and are shown in Figs 1 and 2. The curves are bounded by an intersecting pair of asymptotes (Fig. I), which are described by the limiting parameters of the Lineweaver-Burk plots (Table 1). Slopes of the asymptotes were obtained by evaluating the limits, at zero and infinite substrate concentration, of the partial derivatives, d(l/L>,)/d(l/S) of equations (5), (7) and (8). Limiting slopes have been similarly evaluated by Segal et al. (1952), Reiner (1959). and Trowbridge et uf. (1963). The .v-intercepts were obtained from the limits of the partial derivatives, d(S/a,)/dS, since the two sets of partial derivatives are based on linear transformations of equation (1) in which the slopes and intercepts are complementary to each other. The J.-intercepts of the asymptotes at high substrate concentration could also be obtained directly from equations (5). (7) and (8). It is important to note, in Table 1, that lillziting parameters of the Lineweaver-Burk plots do not necessarily provide the constants of’ individual sites in any of the enzymes. In particular, the x-intercepts often used to determine apparent dissociation (or Michaelis) constants are complex combinations ot all four constants characterizing the two sites. The equations are simpler if V, = VJ, but, in general, apparent Michaelis constants of multi-sited enzymes are not subject to straightforward interpretation. Another point to be noted is that very large ranges of substrate concentration must be covered to reach the limiting segments of the curves. If there is detectable curvature, each limb of the curve must represent at least a 30-fold extension of substrate concentration from the apparent point ot intersection in order to accurately measure the limiting parameters. The minimal overall range of substrate concentration required is thus of the order of lOOO-fold. Semilogarithmic plots have advantages in representing such wide concentration ranges, and permit direct estimation of both constants if K2 % K, (Trowbridge et al., 1963).

d

1)

Downward (r>l)

(r=

sets of conditions

2b

? Alternative

Impossible

2a

Minimum (O>r) Monotonic upward (1 >r>O)

LiIlear

Impossible

Figure

Shape

(a) VI/Vi (b) VI/V;

sufficient.

c 2, and

= 2

are independently

(a) VI/V, (b) V,/Vz

Enzyme type B

Suficient conditionsfor shapesof Lineweaver-Burk plots?

TABLE 2

1 +K,‘/K, i 1 +K;[Kz, and =

412

E.

T.

HARPER

The shapes of Lineweaver-Burk plots which result from various combinations of catalytic and dissociation constants may be seen from the values of r, the ratios of the limiting slopes at high and low substrate concentrations. Ratios greater than one signify downward curvature; ratios less than one represent upward curvature. The relationships are summarized in Table 2. Enzyme A is characterized by linear or downwardly curving LineweaverBurk plots. Upward curvature is not possible because r > 1 for all values of K, except K, = I. It should be noted that, by definition, K2 cannot be less than K,. since the first site filled is the one with the smaller dissociation constant. Downward curvature occurs if Kz > K, [Fig. l(A,)]. The curvature is exaggerated as K,/K, increases, and is diminished as V, and VZ diverge. It can be shown, by equating to zero the partial derivative of r with respect to V,/V,, that the curvature for any value of K,/K, is greatest when I’, = VZ. Thus, if identifiable curvature requires a 5072 change in slope, K2 must be at least a factor of five larger than K,. It is interesting that, in contrast to enzymes B and C, downwardly curving plots may be obtained even when V, > V,, although higher substrate concentrations are required to observe it [Fig. l(A,)]. With enzyme B, downwardly curving Lineweaver-Burk plots also require K, > K, [Fig. l(B): Fig. 2, curve d]; the curvature is greatest when Vi = 1;. and diminishes as V, becomes larger than V,. Linear plots result if (a) K, = K, and I’, = I’, (Fig. 2, curve c; the trivial casein which the sites are both independent and equivalent), or (b) t;2 > K,, and (I - Vi/V,)’ = (I -K,/K& which means, in effect, that downwardly curving plots become linear when V,/V, r 2KJK, (unless both ratios are near unity). With further increases in the value of V,/Vi, the plots turn upward. Upwardly curving plots are also obtained if either V,/V, < 0.5. or KJK, < 0*5VJV,. or K2 < K, (Fig. 2, curve b), and contain a minimum (corresponding to a maximum in the saturation curve) if 2Vz < V, (Fig. 2, curve a) (cf. Reiner, 1959). With enzyme C, downward curvature results if either, or both, K, and K; are greater than K, [Fig. l(C)]. Maximum downward curvature occurs when V, = V; only if K, = K;; otherwise the condition for maximum curvature is V,/V; = l/2(1 +K;/K,). As with enzyme B, the plots may be coincidentally linear, or upwardly curving, and may have slopes near the ordinate that are negative (creating a minimum in the curve), zero, or positive. Botts (1958) has also correlated shapes of Lineweaver-Burk plots with relative magnitudes of coefficients in a generalized equation for enzymes A, B and C. My conclusions are similar, but hers were not directed toward the mechanistic distinctions discussedhere.

KINETICS

OF

SUBSTRATE

41.1

.ACTIVATIOI\;

4. Substrate Activation

and Inhibition

One would like to be able to interpret curved Lineweaver-Burk plots in terms of activation or inhibition by the substrate, assumingthat artifacts of the kinetic method have been excluded (Almond & Niemann, 1960). The difficulty is that substrate activation can be associated with either upwardly or downwardly curving plots. and upwardly curving plots can represent both activation and inhibition. The relationships become clearer with a finer classification of modifier effects of substrates. Modifiers may act by altering dissociation (Michaelis) constants or maximal velocities, or both. Modifier effects thus fall into three categories. as shown in Table 3: “partial” effects on K or P’ alone; -rABL.E

Classificatiotl -

3

qf’modljier

rjjkts

I Effect

on

K

:

None

Effect on I’ Increase (V, -. L’2)

Decrease (V, :> K2)

-~ None Decrease

(K,

C’-activation

C-inhibition

K-activation

Combined activation

K-activation,/ L’-inhibition

K-inhibition

C-activation! K-inhibition

Combined inhibition

>Kz)

Increase (k’, ’ Kz)

“combined” effects, with mutually reinforcing changes in K and I’; and “crossed” effects. with a stimulatory change opposed by an inhibitory change. There are eight possibilities, as shown in Table 3, which may be compared to the four types of observable Lineweaver-Burk plots of Table 2. V-inhibition, alone or in combination, is readily distinguished by upwardly curving Lineweaver-Burk plots with minima. and is the usually implied mechanism of substrate inhibition. It is also the dominant mechanism. K-inhibition tends to produce downward/~~curving plots. but its influence is apparent only if V-inhibition or strong V-activation are both absent. Note that this is the case usually referred to as “substrate activation”. However. real substrate activation (combined activation, K-activation, or even C’activation if K-inhibition is not too strong) produces upwardly curving Lineweaver-Burk plots (without minima). which correspond to sigmoid plots of I’() against S (Botts. 1958).

414

I:.

T.

UARPER

The downwardly curving Lineweaver-Burk plots of enzyme A may bc thought of as “apparent substrate activation”, since participation 01‘ 3 site which is normally significant only at high substrate concentration appears to be elicited by increasing concentrations of substrate. As indicated above. this apparent activation formally corresponds to K-inhibition, usually, but not necessarily, crossed with V-activation. Of course, since modifier effects are not possible between sites that are independent, no real activation is involved at all. Discrimination between the real and the apparent mechanisms of crossed V-activation/K-inhibition is a problem of long standing (Kistiakowsky & Rosenberg, 1952), and will be treated in a hubsequent paper. REFERENCES ALBEKTY.

ALMOND, BOTTS, BOTTS, DIXON, Press. FRIEDEN, HALDANE. HALDANE,

R. A. (1956).

Adv. E~?;ynlol. 17, 1. H. R. & NIEMANN, C. (1960). Biochin~. Dioplrys. .4c,ftr 44, 143. J. (1958). Trans. Faraday Sot. 54, 593. J. & MORALES, M. (1953). Trans. Forodoy Sot. 49, 696. M. & WEBB, E. C. (1964). O?i~~/~2es. 2nd ed.. pp. 81-98. Ne\v

York:

Academic

C. (1964). J. biol. Chew. J. B. S. (1957). Nutrrre,

239, 3522. Land. 179, 832. K. 11932). A//ge,~zeirfe

J. B. S. & STERN, C/rel>rir tier Ejli.p/r/r, pp. I IY-120. Leipzig and Berlin: Steinkopf. KING, E. L. (1956). J. phys. Chew?. 60, 1378. KISTIAKOWSKY, G. B. & ROSENBERG, A. J. (1952). ./. ,411~. &Z/U. Sot. 74, 5020. LAIDLER, K. J. (1956). Trans. Faraday Sot. 52, 1374. LINEWEAVER, H. & BUKK, D. (1934). J. Am. chum Sot. 56, 658. REINER, J. M. (1959). Behavior of Enzynze System.s, Chap. 4, pp. 98-107. Minneapolis: Burgess Publishing Company. &GAL, H. L. (1959). In The Enzymes, 2nd ed., Vol. I, Chap. I (P. D. Hoyer. H. Lardy 6i K. Myrbtick, eds.). New York: Academic Press. SEGAL, H. L., KACHMAR, J. F. & BOYER, P. D. (I 952). Euz.vnlo/ogia 15, 187. STADTMAN. E. R. (1966). A&. Enzpx~/. 28, 41. TROWBRIDG~, C. G., KREHRIEL, A. & LASKOWSKI, M., JR. (1963). WONG, J. T.-F. & HANES, C. S. (1962). Can. J. Biochenl. Ph.vsiol. WYMAN, J., JR. (1964). Adv. Protein Chem. 19, 223.

Bioc/remi.ctr~~ 40, 763.

2, 843.