Eur. J . Biochem. 102, 449-465 (1979)

Ligand Competition Curves as a Diagnostic Tool for Delineating the Nature of Site-Site Interactions : Theory Yoav I. HENIS and Alexander LEVITZKI Department of Biological Chemistry, Institute of Life Sciences, The Hebrew University of Jerusalem (Received December 18, 1978/July 10, 1979)

A few molecular models have been developed in recent years to explain the mechanism of cooperative ligand binding. The concerted model of Monod, Wyman and Changeux and the sequential model of Koshland, Nemethy and Filmer were formulated to account for positively cooperative binding. The pre-existent asymmetry model and the sequential model can account for negatively cooperative ligand binding. In most cases, however, it is virtually impossible to deduce the molecular mechanism of ligand binding solely from the shape of the binding isotherm. In the present study we suggest a new strategy for delineating the molecular mechanism responsible for cooperative ligand binding from binding isotherms. In this approach one examines the effect of one ligand on the cooperativity observed in the binding of another ligand, where the two ligands compete for the same set of binding sites. It is demonstrated that the cooperativity of ligand binding can be modulated when a competitive ligdnd is present in the protein-ligand binding mixture. A general mathematical formulation of this modulation is presented in thermodynamic terms, using modelindependent parameters. The relation between the Hill coefficient at 50 % ligand saturation with respect to ligand X in the absence, h(x), and in the presence of a competing ligand Z, h(x,z), is expressed in terms of the thermodynamic parameters characterizing the binding of the two ligands. Then the relationship between h(x) and h(x,z), in terms of the molecular parameters of the different allosteric models, is explored. This analysis reveals that the different allosteric models predict different relationships between h(x,z) and h(x). These differences are especially focused when Z binds non-cooperatively. Thus, it becomes possible, on the basis of ligand binding experiments alone, to decide which of the allosteric models best fits a set of experimental data. The binding of a ligand to a set of binding sites can either be non-cooperative or cooperative in nature. Two main types of cooperativity have been found : positive and negative. Mixed cooperativity (positive at a certain part of the binding curve and negative at another) has also been observed. Three molecular models are the most widely used by experimentalists to analyze cooperative binding: the concerted model of Monod, Wyman and Changeux (MWC) [l], the sequential model of Koshland, Nemethy and Filmer (KNF) [2], and the pre-existent asymmetry model [3]. Of these models the MWC model and the K N F model can account for positive cooperativity, while the K N F model and the pre-existent asymmetry model can account for negative cooperativity. Mixed cooperativity can be accounted for only by the K N F model. A This study is part of a Ph. D. Thesis of one of us (Y.I.H.) submitted to the Senate of The Hebrew University of Jerusalem, Jerusalem, Israel (October 1978). This study was supported by a grant from the Israel Academy of Sciences, Jerusalem, Israel. Ahhreviu/iorz.r. MWC model, the Monod-Wyman-Changeux model; K N F model, the Koshland-Nemethy-Filmer model. Enzyme. Glyceraldehyde-3-phosphate dehydrogcnase (EC 1.2.1.12).

parent model that unifies all the models in current use has been formulated by Wyman [4]. On the basis of ligand binding curves alone, it is usually impossible to deduce which of the molecular models for cooperativity applies for a particular situation. In this communication we present an analytical approach allowing one to determine which of the allosteric models best fits a regulatory system, on the basis of binding experiments only. The approach proposed is based on the observation that the same set of ligand binding sites on a protein can bind different ligands with different extents of cooperativity. For example, the enzyme glyceraldehyde-3-phosphate dehydrogenase binds N AD+ in a negatively cooperative fashion [5] but binds ADP-ribose in a non-cooperative manner [6, 71. In this study we demonstrate that the binding pattern of a ligand which, by itself, binds in a cooperative fashion, can be modulated in the presence of a second competing ligand which binds to the same set of binding sites. The competing ligand itself can bind either cooperatively or non-cooperatively. The different molecular models for cooperativity make specific predictions on the nature and extent of this modulation.

450

Analysis of Cooperativity

Therefore, this approach can be used to establish the molecular model that should apply for a particular situation. One of the convenient methods to plot binding data is the Hill plot [8, 91. We have found that the most convenient means to follow the changes in the binding pattern of one ligand by a second competing ligand is by examination of the Hill coefficient at 50 % ligand occupancy, h, which is a function of all the ligand binding constants. The analysis presented, therefore, will be in terms of the changes in h for the binding of one ligand upon the introduction of a second competing ligand. As we shall demonstrate, the best way to establish clearly which of the molecular models mentioned above best describes a certain experimental system, is by the use of a noncooperative competing ligand. If such a ligand cannot be found, another cooperative ligand may be used as a competitor. In the latter case, however, a clear distinction between the models becomes more cumbersome. THEORY GENERAL CONSIDERATIONS

Throughout the analysis we explore ligand binding phenomena to a homogeneous protein that does not undergo ligand-dependent aggregation or dissociation. The saturation of such a protein by a ligand is described by E q n (1) [lo]: n

c

1

-

Y(X, =

i$'(X)[X]i

i = l

; 1

+

(1)

n

$i(X)[XIi i = l

where y(,, is the fraction of sites saturated by X, [XI is the free ligand concentration, n is the number of sites per protein molecule, and t,bi(x) represents the Adair constants :

K,, is the thermodynamic association constant for the binding of the ith X molecule to the oligomer. Eqn (1) is applicable to any binding process for a Iigand X to a macromolecule possessing n sites. A useful expression for the description of the nature of X binding is the Hill coefficient at 50% ligand saturation, h(x), which is given by Eqn (2) [ l l , 121: n

,X i2$i

h(x)

=4

n

[ "

(XI

[XO.,]'

' = I

-?] 4

(2)

1 + C $i(x>[XO.,Ii i = l

where [X,,,] is the free concentration of X yielding Y(x, = 1/2. The value of [X,,,] can be solved analytically only in certain cases by inserting y(x,= 'iZ into Eqn (1).

These cases include a dimer or a tetramer that obeys the condition K,, K,, = K,, K x 3 ,namely the Hill plot and the plot of Y(,, vs log [XI are symmetric about the midpoint. In the general case, [X,.,] depends on the complex polynomial expression of the t,bi(x)[X,,,]' values, and even for a tetramer, no analytical expression can be derived for the positive root of the quartic involved (unless the condition K,, K,, = K x 2K,, is met). In the presence of a competing ligand Z, the terms $i(x) and [X,,,] for the ligand X are changed to t,bi(x,z) and [X:bs], which represent the apparent Adair constants and the half-saturating X concentration for the binding of X in the presence of a constant concentration of Z, respectively. Thus, in the presence of Z, h for X may change too. We shall deal only with cases in which Z is added in a large excess over the concentration of binding sites, such that the free Z concentration remains constant during the titration with X. Otherwise the t+hi values, treated as constants in the derivation of Eqns (2) and (4), will no longer be constants but become dependent on [XI. As [X,,,] cannot be found analytically for the general case in a tetrameric (or a higher assembly) system, one has to define another mathematical condition which will allow one to elucidate whether h(x) does or does not change upon the addition of Z, even when the expression of [X,,,] cannot be solved analytically. Examination of Eqn (2), written with t,bi(x,z), [X;b;], and h(x, z) instead of $i(x), [X,,,], and h(x) respectively, reveals that h(x,z) = h(x) only if $i(x,z)[XOgbS]i = $i(x) [XO,,li. An essential condition required for this relationship to hold is: $i(x*Z)

(3)

= a'$i(x)

where a is a constant which equals the ratio $l(x, Z)/$~(X). However, Eqn (3) is not only an essential but also a sufficient condition for h(x) constancy. This can be seen by inserting Eqn (3) and Y(x,= into Eqn (l), which then yields: --

2

____ n

I1

1

+

.

(4)

, I $i(X) ia[X:bSlY i = 1

Comparing Eqn (4) with Eqn (1) in which Y(x,= and [XI = [X,,,] have been substituted, it is evident that whenever Eqn ( 3 ) holds, a[XOgb;] = [X,,,]. However, insertion of a[X:b;] = [X,,,] in Eqn (3) yields $i(x, z) [X:bS]' = $'(x) [XO.,li,which is the condition for h(x) = h(x,z). Thus, when Eqn (3) holds, it immediately follows that a[X$b;] = [X,,,] and that h(x,z) = h(x). The condition expressed in Eqn (3) also means that: tji(x,z) = KZp . KgY . K::' ... Kobs = (aKx,)(aKx2)( ~ K x x.).. (aKx,) (5)

Y. I. Henis and A. Levitzki

45 1

namely, that

molecule to EZ,- 1, the ith X molecule to EX,- ,Zj, and t h e j t h Z molecule to EXiZj- 1, are given by: [EX,]

Kxi=

[EX, - 11 [XI These considerations lead to three conclusions. (a) h can be used as a measure for the cooperativity of the system, as it remains constant only if the relations between all the binding constants for X remain unchanged. (b) Addition of a competing ligand Z into the system will always change $,(x) values to $,(x,z) and [X,.,] to [X;?]; h for X, therefore, may change too, unless Eqn (3) is obeyed. (c) The condition defined in Eqn (3) can be used to explore whether or not h for X changes upon addition of Z without the need for an analytical expression of [X,,,].

THE DIMER AND THE FUNCTIONAL DIMER

Let us consider a protein dimer where the two subunits are either identical or different one from the other. The analysis presented for the dimer case is also valid for the case of a multimer of dimers [ l l , 121. Introducing n = 2 and Y(x,= into Eqn (1) and solving for [XI, one obtains that [X,.,] = $;‘I.. Inserting this quantity into Eqn (2) yields:

&. ’=

[EZjl [Ezj- 1I [ZI [EX,Zjl

Kzjxi =

[EX,- 1Zjl [XI

Inserting these definitions in Eqn (7) and rearranging, one obtains:

Comparing Eqn (8) with Eqn (I), where n obtains :

=

2, one

’/’

and

In the presence of a competing ligand Z, the saturation function for X, y(x,,obtains the form:

[EX]

+ [EXZ] + 2 [EX,]

where [EX] refers to the concentration of all enzyme species possessing one X molecule bound and no Z bound, [EXZ] refers to the concentration of all enzyme species possessing one X molecule and one Z molecule bound, etc. The thermodynamic association constants for the binding of the ith X molecule to EX, - thejth Z

Inserting t,hl(x,z) and $’(x, z) into Eqn (6) instead of $I~(X and ) IC/’(x), yields the observed Hill coefficient for X in the presence of Z, h(x, z), at 50 % saturation with X: h(x,z) = 4

Analysis of Cooperativity

452 ZEX

The higher [Z] is, the larger the effect it will have on h for X. In the case where [Z] % l / K x , z , , I / K z , , 1/Kz2, Eqn. ( 1 1 ) simplifies to: 4

Without defining any particular model, one can define conditions under which h(x,z) = h(x). Examination of Eqn (12) reveals that h(x,z) = h(x) when :

1

+ K x l Z[Z] l = ( 1 + Kzl [Z] + K z , Kz2[Z]2)'12. (13)

At high concentrations of Z ([Z] % I / K z , , I/Kz,, l / K x l z , ) ,Eqn (13) simplifies to K,,,, = (KzlKZ2)"2. The allosteric models which we are about to consider differ in their predictions as to whether h(x, z) differs from h(x), and to what direction and extent h for X will change upon the addition of Z.

Dimer with Pre-Existent Asymmetry Possessing Two non-Interacting Sites

XEZ

Fig. 1. The binding of X lo an asymmetric dimer in the presence of a competing ligund Z . EX and XE are two different monobound species, both containing one bound X molecule. So are EZ and ZE. K ; and K;' are the intrinsic association constants of X to the two different sites, whereas K : and K;' are the intrinsic association constants of Z to these two sites. K ; = [EX]/[E][X]; Kg' = [XE]/ [E][X]; K ; = [EZ]/[E][Z]; K:' = [ZE]/[E][Z]. As there are no interactions between the sites, these constants are sufficient to describe all the binding steps. Thus, for example, [ZEX]/[ZE][X] is also given by K ;

At high Z concentrations, Eqn (16) simplifies to: 4

A dimer possessing two non-interacting sites can exhibit negative cooperativity but cannot exhibit positive cooperativity. In this case, $l(x) = K,, = K ; + K:' and $z(x) = K,, K,, = K ; K;, where K: and K:' are the intrinsic association constants for the binding of X to the two different sites on the dimer, respectively. Inserting these equalities into Eqn (6), yields: A

From Eqns (16) or (17) it is clear that if Z binds in a non-cooperative fashion (KL = K r ) , h(x,z) is identical to h(x) [given by Eqn (14)].Namely, according to the pre-existent asymmetry model, addition of a noncooperative Z cannot change h(x). Since the expression + 2 2 also applies for Eqn (17 ) ,which describes the maximal effect of Z on the cooperativity of X binding, it is clear that even in the presence of Z, the binding of X cannot become positively cooperative under any circumstances, i. e. h(x,z) I 1.0. A situation where h(x,z) = 1.0 (noncooperative binding of X in the presence of Z) will be obtained when KLIK; = K:/K:', namely, when the cooperativity in the binding of Z and X is identical. When K:/K; Ki/K:', i.e. X and Z exhibit different extents of negative cooperativity, the negative cooperativity in X binding can be either weakened or strengthened. For convenience, let us define K ; as representing the binding of X to the tighter site ( K ; > KF). Inspection of Eqns (14)and (17)reveals that the effect of Z depends on the relation between K;K;/K:'KL and KiIK:'. Two cases can be distinguished. a) KL/K:' < K: K:'. When K:' L K;, h(x, z) < h ( x ) , and when KL > KF, h(x,z) > h(x). b) K:/K; > K:/K:'. In this case one must compare K:'K:IK:K; with K!JK:', as the addition of Z

fi

fi

As + 2 2, it is clear that h(x) I 1.0. Upon the addition of a competing ligand Z to the system, the binding of X in the presence of Z occurs according to the scheme shown in Fig. 1 . K x I z ,is given by :

Inserting Eqn (15) and the t,bi(x) and $i(z) expressions into Eqn (1 I), one obtains: h(x,z) = 4 K: I - L

+ K:' + ( K ; K ; + K ; K;)

[Z]

-.

(16)

+

Y. I . Henis and A. Levitzki

453

makes K:' KLIK: K:' larger than K: KYIK; KL (the higher-affinity site for X becomes the lower-affinity site in the presence of Z ) . Under these conditions, KLiK; > (K:/K:')2 yields h(x,z) < h(x), and KL/Kg < (KiiK;)' yields h(x,z) > h(x). KLIK:' = (K:/K;)2 is a special situation where h(x,z) = h(x). The Symmetric Dimer Possessing Two Interacting Sites

In the case of a symmetric oligomer, where the binding sites exhibit identical affinity for the ligand, it is useful to express the binding in terms of the statistically corrected (intrinsic) association constants which are defined by [13 - 151: Kx, =

n-i+l .

K:,

1

Kqz, =

Kz, =

n-i-j+l

KkJ,

i n-j+l

i

K:,

where K,,, K,,, and KxLZ,are the thermodynamic association constants defined previously for the binding of the ith X molecule or thejth Z molecule, and KC,, K;,, and K:,,, are the corresponding statistically corrected association constants. n is the number of sites per protein molecule. Inserting these definitions in Eqns (6), (ll), and (12), one obtains h(x) and h(x,z) in terms of the statistically corrected association constants. Unlike the asymmetric dimer with non-interacting sites, h(x,z) can be different from h(x), even when Z binds in a non-cooperative fashion; namely, when K:, = K:, = KL. Whether h(x,z) differs from h(x) for a non-cooperative Z , depends on whether the ratios KL,zl/K; or (1 + K L l z l [ Z ] ) / ( l+ KL[Z]) differ from unity. This can be readily seen by comparing Eqns (12) or (1 l ) , in which Kh, = K;, = KL has been introduced, with Eqn ( 6 ) , all in terms of the statistically corrected association constants. If the affinity of the noncooperative Z is affected by the binding of X to the neighboring subunit, h(x, z) # h(x). If it is not affected, however, h(x,z) = h(x). Thus, the presence of a noncooperative Z can induce a change in the cooperativity of X binding in a symmetric dimer with interacting sites. Both the MWC and the K N F models assume a symmetric dimer with two identical sites. We shall now examine the relationship between h(x) and h(x,z), as predicted by these two models. The Monodl WymanlChangeux Dimer In the non-exclusive case of this model it is assumed [l] that the protein exists in two conformations, i.e. the

Fig.2. Non-exclusive binding of two ligands to a dimer accorciinx to the concerted Monodl Wymarz/Chaizgeux model. L is the allosteric constant rcpresenting the equilibrium between the T and the R states in the absence of ligands ( L = [T]/[R]) Ki, and Ki, are the intrinsic association constants for the binding of X and 2 to the T state, while Kk. and KR,are the intrinsic association constants for the binding of X and 2 to the R state

conformation exhibiting high affinity towards X binding, R, and the low-affinity conformation, T. The two conformations are in equilibrium. The binding pattern of two competing ligands to a MWC dimer is depicted in Fig. 2, along with the parameters of the model. For a protein possessing n subunits, one can write the relationships [I]:

and

where KR,, K+x,and L are defined in Fig.2. From the definition of *i(x), it is clear that:

454

Analysis of Cooperativity

Inserting [TI = L[R] and Eqns (18) and (19) into Eqn (20), one obtains:

Kk,‘

n! $i(x) =

(n - i) ! i !

X

+ LKi-,‘

I+L

(21)

.

Inserting $,(x) and $2(x) in a dimer, obtained from Eqn (21), into Eqn (6), one obtains [15]:

2 h(x)

=

/

+

(Kk, + LKi,)’

‘

(22)

I!(1 + L) (Ki(,2 + LKi.,2)

An inspection of Eqn (22) reveals that h(x) can vary only between 1.0 and 2.0 [12, 13, 151, namely, only positive cooperativity is possible. In the presence of a competing ligand Z, K,,,, in a MWC dimer is given by:

Inserting Eqn (23) and the expressions for Kxl, Kxz, K,,, and K,, into Eqn (ll), yields:

+

+

> Land the ratio Ki.,/(l Ki, [Z]):Kk,/(l Kk, [Z]) is smaller than KiJKk,. Thus, h(x,z) > h(x). It should be noted that in this case Z might bind to the dimer noncooperatively. The cooperative binding of X, which binds preferentially to the R conformation, indicates that a significant fraction of the dimer is found in the T conformation. When L % 1, most of the protein is found in the T conformation in the absence of ligands. Therefore, when Ki,> Kk,, the binding of Z to the dimer will be non-cooperative. Unlike the noncooperative Z discussed in case (a), a non-cooperative Z of this type (Ki, > Kk,, where L % 1) enhances the cooperativity of X binding. (c) K+,< Kk,. Z binding must be cooperative, since it binds with a higher affinity to the conformation that binds X with a higher affinity (R). In this case L‘< L, and the ratio Ki,/(l + Ki, [Z]):Kk,/(l Kk, [Z]) is higher than Ki,/Kk,. Thus, h(x, z) < h(x). A special case of the MWC model is the exclusive binding case in which X binds only to the R state. In this case KiJKk, = 0. Introducing this equality in Eqns (22) and (24), yields the expressions for h(x) and h(x,z) that are given in Table 1. In the exclusive binding case too,

+

KR,

1 where [11: L‘ = L( 1 + Ki, [Z])/(I + Kk, [Z]). L’ describes the equilibrium between the conformations R and T in the presence of Z but in the absence of X. Eqn (24) has the same analytical form as Eqn (22), where L, Kk, and Ki, in Eqn (22) are replaced by L’, Kk,/(l + Kk, [Z]) and Ki.,/(l + Ki, [Z]), respectively. Therefore, h(x,z) can vary only between 1 and 2, as in the case of h(x). At high Z concentrations, Eqn (24) is transformed into : h(x,z) = 9

__

L

I+

where L = L(Ki,/Kk,)’. Three cases can be distinguished. (a) KR, = KT,. In this case, Z binds non-cooperatively, and L’ = L. Therefore, Eqns (24) and (25) reduce to Eqn (22), namely, h(x,z) = h(x). (b) Ki,> KR,. Z binds with higher affinity to the T conformation. In this case, L’

+ Ki,[Z]

h (x) and h (x, z) can assume only values between 1.O and 2.0. As in the general case discussed above, h(x,z) > h(x) when Ki, > KR,, and h(x, z) < h(x) when Ki, < Kk,. When Ki, = KR,, h(x,z) = h(x).

The KoshlandlN~methylFi~me~ Dimer In this model it is assumed [2] that the unliganded protein exists in one conformation, A, and all the binding sites are identical. In the simple sequential model, the conformation of the subunit which binds the ligand X is converted from A to B, whereas the unbound subunits remain in the original conformation A [2]. The interactions between the subunits at the interfaces are considered to determine the relative stabilities of the various protein-ligand species and, therefore, determine the type and extent of the cooperativity exhibited by the system. However, it has become apparent that the simple K N F model is insufficient to account for many allosteric phenomena. Therefore, a modified model that includes the situation in which a vacant subunit changes its conformation (from A to D) in response to the binding of X to a neighboring subunit, has been formulated [14].

v,

In

h (XI

-

_

(1

4

_

~

+ L)(Kkf + LKi-3

(KR. + L a , ) *

l + L

2 _

2

2

h(x,z)

+ 2

1

2

+ L‘ + KR, [ZI

+ KR,[Z] 1

2

1

1

+ Ki,[Z]

Ki.

+ Ki,[ZI

K ; + K;‘ + (KiK:’ + K:K;)[Z] I/K,K:I {I + (K: + KZ) [ Z ]+ K;K; [Z]’)’ *

cooperative Z

2

4

1+L

2 _____ 1

+ LKi3

Type 1: h(x) can change in any direction and in any amount

Trpe 2: 2

T y ~ Ie:

(1 + L ) (Kkf

( K R ,+ L K ~ , ) ~

2

+F

4

non-cooperative Z

/z(x,z)

=

/1(x)

h(x.2) = h(x) or h(x,z) >h(x)

h(X,Z) = h(x) or h(x,z) >h(x)

Type 2 : h(x,z) = h(x)

=

Kz,,

Relation between h(x) and h(x,z) when Z binding is non-cooperative

Table 1. Effect o j ’ a compering IiLqandZ on h j b r X in a dinzrr The parameters ofthe different models are defined in the text as well as in Fig. 1 (the pre-existent asymmetry modcl), Fig.2 (the MWC modcl). and Fig. 3 (the K N F model). Note that a multi-dimer, where the subunit interactions occur only within the dimers, behaves as adimer and thus obeys the equations developed for the dimer case. L’ = L( 1+ K i , [Z])’!(l+ KR, [Z])’. In the K N F model two distinct types of non-cooperative Z are treated: type 1, which induces a conformational change from A to C upon binding (with K:,/K,-- = l ) , and type 2, which does not induce any conformational change. A non-cooperative Z of type 1 can behave in a similar manner to a non-cooperative Z of type 2, providing that the conformational change it induces does not affect (and is not affected by) the and KAA are identical, and so are K,, and KAc.According to the MWC model, Z binds non-cooperatively when KR,=K+,, or when Ki,>KR, bindingofx. In this case, the interactionconstants KAc, GC, and L % 1. The case where K i , = K i , is shown in the column of ‘non-cooperative Z’, whereas the effect of a non-cooperative Z of the second type is described by the expression shown in the column of ‘cooperative Z’, after introducing K i , > Kb,. In the latter case h(x, z) > h(x), as is indicated in this table Model

Pre-existent asymmetry

MWC, exclusive binding

MWC, nonexclusive binding

Simplest K N F

h(x) can change in any direction and in any amount. h(x,z) > 1.0 can be obtained even when h(x) < 1.0 and vice versa

Typo 2 : h(x,z) # h(x)

Special case: when K,,, then: h(x,z) = h ( x )

Analysis of Cooperativity

456

DI

K'As=m

(a ) Simple sequential model Binding o f X alone;

Binding of X and 2:

( b) The more general sequential mdel Binding of X alone:

~

K ' ,X

K

~

~

Binding of X and Z ( Z non-cooperative)

Fig. 3. The binding o j t w o competirig liganrls to a dimer accordinK to the sequential KoshlarldlNk.methylFilmermodel. The definitions of the various constants follow the original definitions of Koshland et al. [2] and are shown in the figure. A circle designates a subunit in conformation A ; a square, conformation B; a triangle, conformation C; and a hexagon, conformation D. (a) The simple sequential model (the conformational change is limited to the ligand binding subunit). Both ligands induce a conformational change upon binding: X fromconformation A to B, and Z from conformation A to C. (b) The general K N F model (the induced conformational change propagates to vacant sites). The ligand X induccs a conformational change (from A to B) in the subunit to which it binds, and a change from conformation A to D in the neighbouring unoccupied subunit. For simplicity (sce text), a case where Z binds non-cooperatively and does not induce any conformational change upon binding is treated

In the simplest K N F model (Fig. 3a) one obtains for the binding of X to a dimer [12, 13, 151:

2 h(x) =

-

(26)

KBB where the various parameters of the K N F model are defined in Fig. 3. Since Ki,/KBB can acquire any value, h(x) can assume values between 0 and 2.0, and either positive or negative cooperativity can be observed. According to the simple K N F model, the binding of a

competitive ligand Z (which induces a conformational change upon binding) changes the conformation of the bound subunit from A to C (Fig.3a), whereas the neighboring subunit remains in its previous conformation A. Such a Z can be either a cooperative ligand or a non-cooperative ligand inducing a conformation change but not being affected by this change ( K i C / K C = 1). The binding of Z to a dimer thus yields [14]: $ 1 ( ~ )= K z , = 2 K A c K ~ ~ $K2 ~( ~~) = ~ ; Kz,Kz, = K C C ( K ~ ~ K ~K,,,, , , ~ ) ~is. given by K B C K ~ ~ K ~ , , ~ / K A B . Inserting these expressions, along with $l(x) and $2(x), [which are similar to ~ / i ~ ( zand ) ] ~)~(z),with X and B instead of Z and C, respectively, into Eqn (1 l), yields: 2

Y. I. Henis and A. Levitzki

Comparing Eqns (27) and (26), it is evident that the addition of a competing ligand Z which induces a conformational change upon binding, can change h for X in any direction and amount (between the limits of 0 and 2.0), depending on the relations between all the interaction constants KAB, KBB, KBc, KAC, and GC. Only if the interactions of the conformation C are identical to those of A with respect to the binding of X, will h(x,z) equal h(x). When the free Z concentration is high, Eqn (27) reduces to : 2 h(x,z) = (28)

In this case, the relation between h(x,z) and h(x) depends on the ratio K i C / K B E G c as , compared with K i B / K B B .Unlike the cases of the dimer with preexistent asymmetry and the MWC dimer, there is no way of predicting the relationship between h(x, z) and h(x), even when the binding patterns of X and Z separately are known. This is due to the fact that KBC or its relationship to the other interaction constants cannot be obtained from the binding of X and Z separately. The ligand Z binds non-cooperatively when K i c / G c = 1. Two cases of non-cooperative binding can be distinguished. a) Z binding induces a conformational change in the binding subunit (from A to C), but this change does not affect further binding of Z (as Kic/K,-- = 1.0). Introducing K i C / G c = 1.0 in Eqns (27) or (28) does not reduce them to Eqn (26), and a situation where KiC/KBB&C # KiB/KBB and, therefore, h(X, Z) # h(X), can occur (Tablel). Only if the binding of X is not affected by the binding of Z will h(x,z) = h(x). b) The binding of Z does not induce any conformational change ; namely, the conformation C is identical to A. Under these conditions, KBc = K A B ;KAc = GC= KAA = 1.0; and Kt,, = Kt,, = 1.0. In this case, Eqns (27) and (28) reduce to Eqn (26), and h(x,z) = h(x). Thus, according to the simple K N F model, a situation where h(x,z) # h(x), upon the addition of a non-cooperative competing ligand Z , is allowed. Only when the binding of Z does not induce any conformational change, is the equality h(x, z) = h(x) obligatory. In the more general case of the K N F model, it is recognized (Fig. 3 b) that the conformational change induced by the binding of X propagates to the neighboring subunit. Let us therefore assume that, while the conformation of the binding subunit is changed from A to B, the conformation of the neighboring, vacant, subunit is changed from A t o D. The expressions for the binding of X in the absence of Z are [14]: $l(x) = K x , = 2 KBDK~,K~,,K~A, and $z(X) = Kx,Kx2 = KBB(KxBK~AB)’.

457

Inserting these expressions into Eqn (6), yields: 3 h(x) = (29) L

h(x) can acquire any value between 0 and 2.0, depending on the quantity KiDK?,,/KBB. In the case of the simple K N F model, the addition of a competing ligand Z which induces a conformational change, can cause a change in h for X in either direction. This is obviously true for the general K N F model where more interactions are allowed. We shall therefore examine the relationship between h(x,z) and h(x) only for the simple case in which Z binds in a non-cooperative fashion and does not, by itself, induce any conformational change. Under such conditions, K,, = 2 KzA; K,,K,, = K & ; and K,,,, = KzD.K,, and K,, are defined in Fig.3. Inserting these equalities into Eqn ( Il) , one obtains:

A comparison of Eqn (30) with Eqn (29) reveals that h(x) = h(x,z) only when K,, = K,,. Thus, in the general K N F model, even a competing ligand Z which binds without inducing a conformational change, can still affect the cooperativity of X binding. This is due to the fact that even when Z does not induce any conformational change, its binding can still be affected by the conformational change brought about by the binding of X to the neighboring subunit. Namely, Z competes differently for the A and D conformations (when K,, # KzA) and, in this way, affects the cooperativity of X binding. The results obtained for h(x) and h(x,z) in a dimer, according to the different models, are summarized in Table 1. THE TETRAMER CASE

The discussion of dimeric systems has outlined the principles of ligand competition experiments. Since many cooperative oligomers are tetramers, it is worthwhile analyzing the behavior of tetramers, especially since the mathematical analysis of tetrameric systems is much more cumbersome than that of dimeric systems. The saturation function of a tetramer by a ligand X is given by Eqn (1) with n = 4. Introducing Y(x,= and solving the quartic obtained for [XI, one obtains [X,,,]. An analytical solution for this quartic can only be obtained when the Hill plot or the y(,,vs log [XI plot are symmetric about the midpoint. The symmetry condition is equivalent to the condition K,,K,, = Kx2Kx3, or $:(x)$~(x) = $$(x) [12]. When the symmetry condition is met, one obtains [12]: [X,.,] = $ql’+(x). The symmetry condition is fulfilled in the case of a

458

Analysis of Cooperativity

tetrahedral or a square tetramer according to the simple KN F model [2], but the general K N F model [14], the MWC model, and the pre-existent asymmetry model (except for the dimer-of-dimers case) do not fulfil this condition. Another limitation for the use of the equality [X,,,] = $q1’4(x) is that even in the simple K N F model, the symmetry condition breaks upon the addition of a cooperative competing ligand Z. In all these cases it is therefore impossible to obtain an analytical expression for h, but it is possible t o test whether or not h for X changes upon the addition of Z by employing Eqn (3). For both symmetric and asymmetric Hill plots, h(x) is given by Eqn (2) with n = 4. If the binding obeys the symmetry conditions, it follows that [X,,,] = $q1/4(x), and the expression for h(x) reduces to [ l l , 121: h(x) =

8 $24 + 2 $,(XI 2 $W + 2 $3(4

+ $,”-1 + y c L ( 1 + cyy-’ (39) Y(X, = (1 y>” L(1 cy>” n!

$i(X)

=

X

+ +

+

where y = Kk, [XI and c = K i X / K k xThe . cooperativity in the binding of a ligand X, which binds with a higher affinity to the R conformation, depends only on L and c. The positive cooperativity of X binding is stronger when L is high and cis small [l], and X binding becomes non-cooperative when L = 0 or when c = 1. The presence of a competing ligand Z, which binds to the R and T states with the intrinsic association constants Kk, and K T ~has , two effects on X binding to the oligomer. The first effect is a change in the allosteric constant from L to L‘ [ l ] ,where L’ = L(l + K i , [Z])n/ (1 + Kii,[ZJ)”.The second effect is a change in the apparent affinity of X to the R and T states due to the competition of Z for the sites which bind X : KR:bs = Kk,/(l Kk, [Z]); Kebs = K+J(I Ki-, [Z]). Therefore, the value of c changes to cobs, where cobs = c(1 + Kk, [Z])/(l K+z[Z]). Introducing L‘, K6:bs, and cobsin place of L, Kk, and c in Eqns (38) and (39) yields an expression for $;(x, z) and for the saturation function of X in the presence of a constant Z concentration :

+

+

+

y(x,= yobs

(1 + y b s ) n - 1 (1

+

yobscobsLr) (1

+ yobs)n + L‘ (1 +

+ cobsyobs)n

c0bsyobs)n

L & l ) . Since K+,> Kk,, it follows that L > L and cobs L and cobs< c. c) K;, < Kk,. In this case, L < L and cob’> c; namely, the addition of Z weakens the positive cooperativity in the binding of X. Eqn (3) is not obeyed, and h(x, z) # h(x). It should be noted that when K+, < Kk,, the binding of Z to the oligomer must be positively cooperative, since a considerable fraction of the oligomer is found initially in the T conformation, as indicated by the cooperative binding of X. In the exclusive binding case, where X binds only to the R conformation, a similar behavior is encountered. The equations for this extreme case are obtained from those of the non-exclusive binding case by inserting c = 0. The Simple Koshland/Nhethy/Filmer Model In the simplest sequential model [2] it is assumed that a conformational change (from A to B) occurs only in the ligand-binding subunit. Since according to this model the cooperativity is determined by the relative interactions between the subunits, the cooperativity depends in this model on the geometric arrangement of the subunits within the oligomer [2]. In the simplest KNF model, it is assumed that the intersubunit interactions are identical in all directions of space [2]. For a tetrahedral geometry, where all the subunits are equivalent and interact in the same manner with each other, one obtains [2]:$l(x) = 4 K b R , Kt,,; tj2(x) = 6 KBBK~B(KX,K~~,)*; I)~(x) = 4 K ~ B K ~ B ( K & , , ) and ~ ; t,b4(x) = K $ B ( K ~ , K ~ , It , ) ~can . be seen that the relation $ ~ ( x ) $ ~ ( x=) $”,x) holds, and thus Eqn (31) can be applied. Inserting the t,hi(x) expressions into Eqn (31), yields [Ill:

-1

(41)

where yobs = [XI. Since Eqn (41) has the same analytical form as Eqn (39), it is clear that only positive cooperativity can be obtained according to the MWC model, even in the presence of a competing ligand Z. As in the case of the MWC dimer, three cases can be distinguished. a) Kk, = Ki,.In this case, Z binds to the oligomer non-cooperatively. Thus, L’ = L and cobs= c. Introducing these equalities, along with the expression for K e b s ,in Eqn (40), one finds that t,bi(x,z) = $i(x)/(l + Kk, [Z])’,where $i(x) is given by Eqn (38). Thus, Eqn (3) is obeyed, and h(x,z) = h(x). b) K’r,> Kk2. In this case, Z binding is positively cooperative (when L = l ) , or non-cooperative (when

When the competing ligand Z induces a conformational change, the liganded subunit is converted to the conformation C (Fig. 3). One can insert the appropriate KXgzj expressions into Eqns (33 - 37) and obtain the $i(x, z) expressions. (For example, the product KxtKx,z,Kx,z,Kx,z, describes the formation of the various EXlZ3 species from free E, X, and Z. Therefore, the interactions in these species are being considered. The desired Kx1z1Kx1zzKx,z3 expression can be obtained by dividing the expression for K x , K x , z t K x , z , K xby ,z, KX1.1

Y. I. Henis and A . Levitzki

In the tetrahedral case one obtains:

Eqns (43 - 46) do not fulfil the symmetry condition {$:(x, z)$,(x, z) # $",x, z)). Therefore, Eqn (31) does not hold and an analytical equation for h(x, z) cannot be obtained. Checking the relationships between the $'(x, z) expressions and the corresponding tji(x) terms reveals that $'(x,z) # a$bi(x); namely, in the presence of a ligand Z which induces a conformational change, h(x, z) # h(x). This is true not only for a cooperative Z, but also for a non-cooperative Z which induces a conformational change upon binding (such that KiC/&, = I), since introduction of KiC/K,-- = 1 into Eqns (43 - 46) is not sufficient for Eqn (3) to hold. Since no analytical expression for h(x, z) can be obtained under these conditions, the change in h for X cannot be formulated algebraically. However, examination of Eqn (l), after insertion of $i(x,z) expressions [Eqns (43-46)], indicates that the presence of a competing ligand which induces a conformational change, can change the cooperativity in X binding in any direction and amount, including a reversal from negative to positive cooperativity and vice versa. As KBC and its relationships with the other interaction constants can assume any value, it is apparent from Eqns (43-46) that the relations between the $i(x, z) expressions can be completely different from the relations between the $i(x) values. Therefore, the saturation functions for X in the presence and in the absence of Z do not have to resemble one another, and any change in the cooperativity of X binding may occur. In the special case where Z binds non-cooperatively and does not induce any conformational change, the conformations C and A are identical. Under these conditions KBc = KAB; KAc = = KAA= 1 ; and Kt,, = KtAA= 1.InsertingtheseequalitiesintoEqns(43 - 46), one obtains that for each of the $i(x, z) values, the relationship with $i(x) is given by: Gi(x,z) = $i(x)/ (1 + KZ,KtAc[Z])'. This relationship obeys Eqn (3) and, thus, h(x,z) = h(x).

Analogous treatment of the tetrameric 'square' case, where the subunits interact along two axes only 121, yields the same qualitative results. In complete analogy to the tetrahedral case, one finds that in the case of a square tetramer h(x,z) # h(x), both for a cooperative Z and a non-cooperative Z which induces a conformational change affecting (and affected by) the binding of X to neighboring subunits. h(x,z) = h(x) only when Z binds non-cooperatively without inducing a conformational change, or when it induces a conformational change that does not affect the interactions affecting the binding of X. An extension of the simple K N F model, which takes into consideration the fact that a tetramer with a 2 :2 :2 tetrahedral symmetry has three different intersubunit binding domains, has been considered by CornishBowden and Koshland [16]. The main feature of the simple K N F model, i.e. a conformational change induced by the ligand only in the ligand-binding subunit, has been conserved in this modified model. Deriving the t,bi(x,z) expressions according to this model and comparing them with the $i(x) expressions computed by Cornish-Bowden and Koshland [16], one finds (derivation not shown) the same behavior encountered in the simplest K N F model, namely, h (x, z) # h(x), unless Z binds non-cooperatively and, at the same time, does not induce any change in the subunit interactions upon binding. The General Koshland/N&nethy/Filmer Model In the simple K N F model it is assumed that the only subunit which undergoes a conformational change is the one that binds the ligand. In the more general case [14] the conformational change is transmitted to the neighbouring subunits. We have already seen in the simple K N F model that the cooperativity in the binding of X can be modulated in any direction and amount in

Analysis of Cooperativity

462

Unliganded tetramer

Fig. 5. The species obtained in the titration ofa tetrahedral tetramer with identical intersubunit domains, according to a simple case of the general Koshland/NPmethy/Filinevmodel. The definitions of the conformations and constants are as in Fig. 3. Statistical corrections have been included in the overall association constants shown as there are, for example, 12 identical EX,Z, species possible. (a) The species obtained in the titration of the tetramer with a cooperative ligand X. (b) The various EX,Z,, EX,Z,, and EX,Z, species obtained when a competing ligand Z (that binds non-cooperatively and does not induce any conformational change) is present. The other EX,Zj species are not shown

the presence of a competing ligand Z, even when the latter binds in a non-cooperative fashion. The cooperativity of X binding is not affected by Z only when the latter binds non-cooperatively and, in addition, does not induce a change in the subunit interactions upon binding. In the general K N F model Z has a more pronounced effect on the subunit interactions as it can change the conformation of more than one subunit upon binding. Therefore, it is clear that in this case a competing ligand Z that induces a conformational change, can affect the cooperativity of X binding in any direction and amount. Only, the effect of a noncooperative Z that does not induce a conformational change on the cooperativity of X binding be explored. The effect of Z depends on the binding pattern of X, and a large variety of such patterns exist in this model. For example, the binding of X can induce a certain conformational change in one of the neighbouring subunits and another change in the others. We thus chose to explore this effect in the simplest case of the general K N F model. This is the case of a simple tetrahedron where all the intersubunit domains are identical. Upon X binding, the binding subunit changes its conformation from A to B, whereas its empty

neighbours change from A to D. Binding of additional X molecules changes the binding subunits’ conformation from D to B, whereas the empty subunits (already in the D conformation) do not undergo further change. The species obtained in the titration of a tetramer obeying this model with X are shown in Fig. 5, along with the $i(x) expressions. The species EX,Z,, EX,Z,, and EX,Z,, which demonstrate the various EX,Z, species obtained in the presence of the competitor Z, are also shown in Fig.5. As the t,hi(x) expressions do not fulfil the symmetry condition {$~ ( X)$~ ( X) = $:(x)}, one can only test whether h ( x , z ) = h(x) solely by the detailed examination of the relations between $i(x, z) and t+hi(x)[Eqn (3)]. For simplicity, we shall only examine the case where Z binds non-cooperatively without inducing a conformational change. Inserting the appropriate K,,,, expressions into Eqns (33-37), one finds that the $i(x,z) expressions are given by:

where the t,hi(x)expressions are given in Fig. 5. K,, and K,, are the intrinsic association constants of Z to the A and D conformations respectively, as defined in Fig. 3.

Y. I. Henis and A. Levitzki

Only when K,, = K,, (Z binding is unaffected by the binding of X to the neighbouring subunits), is the relation $i(x,z) = u'$~(x)[Eqn (3)] fulfiled and h(x,z) = h(x). In the general case, however, K,, # K,, (Z binding is affected by the binding of X to neighbouring subunits). In such a case, Eqn (3) does not hold and h(x,z) # W ) . In conclusion, in the general K N F model a change in h for X can occur in the presence of a competing noncooperative ligand Z, even when Z binding does not induce a conformational change. As KzDIKzA can acquire any value, Eqn (47) implies that the relations between the $i(x, z) values (or between the apparent association constants for the binding of X in the presence of Z) can be totally different from the relations between the t,bi(x) values. Thus, the presence of Z can affect the cooperativity for X binding in any direction and amount. DISCUSSION General Considerations It is virtually impossible to deduce the mechanism of cooperative ligand binding solely from the shape of the binding isotherm [ll, 131. In the present study we have established an approach to determine which of the most widely used cooperativity models best fits a particular experimental situation. The approach is based on the study of the effect of one ligand on the binding of another, where both compete for the same set of sites. The parameter we chose to study is the Hill coefficient at 50 % ligand saturation (h). This parameter is a function of all the ligand association constants (see first section of Theory) and can therefore serve as a useful index of cooperativity in ligand binding. h can be expressed in terms of the thermodynamic constants characterizing ligand binding, and can also be expressed in terms of the parameters of the different allosteric models [11 - 13, 151. Thus, the modulation of X binding by a competing ligand Z can be studied by comparing the Hill coefficient for the binding of X alone, h(x), to the Hill coefficient characterizing the binding of X in the presence of Z, h(x, z). The most instructive situation to study was found to be when the competing ligand Z binds non-cooperatively. As shown throughout the theoretical section for the dimer and tetramer cases and summarized in Tables 1 and 2, the pre-existent asymmetry model, the MWC model, and the K N F model predict different relationships between h(x) and h(x, z). The Pre-existent Asymmetry Model vs the KoshlandlNhnethylFilmer Model Negative cooperativity can be accounted for either by the pre-existent asymmetry model or by the K N F

463

model. The pre-existent asymmetry model can account only for negatively cooperative binding [3], and the addition of a competing ligand Z cannot transfer the binding of X to positive cooperativity (see relevent sections in Theory). It is clearly demonstrated (Tables 1 and 2) that the pre-existent asymmetry model does not allow a change in the cooperativity of X binding when a non-cooperative competing ligand Z is added, namely h(x,z) = h(x). The K N F model, however, allows for a change in any direction and amount in the cooperative pattern of X binding (including a reversal from negative to positive cooperativity and vice versa),even when Z is a non-cooperative competitor. The simple K N F model (where the conformational change is limited to the ligand-binding subunit) predicts the equality h(x, z) = h(x) to hold only when Z binds non-cooperatively without inducing any changes in the subunit interactions affecting X binding. It also predicts that h(x,z) # h (x) when the binding of the non-cooperative ligand Z induces a conformational change that affects (and is affected by) the binding of X (Tables 1 and 2). In the general K N F model (where the conformational change propagates to vacant subunits), a situation where h(x, z) # h(x) can be obtained even when the competing ligand Z binds non-cooperatively without inducing a conformational change. According to the general KNF model, the equality h(x,z) = h(x) holds only when Z binding is not affected by the binding of X to the other subunits (KzD= KzA)(Tables 1 and 2). These findings are in accord with a study by Nari et al. [17], where a kinetic approach was employed to study the effect of a competitive inhibitor on the cooperativity towards the substrate in a dimer obeying the general K N F model. In this study [I71 it was shown that when no hybrid complex is formed, a competitive inhibitor which induces a conformational change can affect the negative cooperativity exhibited towards the substrate, and can even reverse it into positive cooperativity. This prediction has been found to occur experimentally [17]. In conclusion, a change in the negative cooperativity of X binding in the presence of a noncooperative competing ligand, h(x, z) # h(x), excludes the pre-existent asymmetry model but fits the K N F model. A clear choice between the two models can also be made employing a cooperative competing ligand Z, if the addition of Z transforms X binding from negative to positive cooperativity. Such a transformation is allowed by the K N F model but not by the pre-existent asymmetry model, which allows only for negative cooperativity. The Monodl WymanlChungeux Model vs the Koshland/N6methy/Filmer Model Positively cooperative ligand binding can be fitted either by the MWC model [l]or by the K N F model [2]. According to the former, the cooperativity of X binding

464

Analysis of Cooperativity

Tahle2. The qualitative eilect ? f a competing ligand Z on h for X in a tetramer The cooperativity models are defined in the text. In the pre-existent asymmetry model there is only one type of a non-cooperative ligand, a ligand that hinds with the same affinity to all sites. In the MWC and the KNF models there are two types of non-cooperative ligands. In the MWC model a non-cooperative ligand of type 1 is a ligand which hinds with a higher affinity to the T conformation in a system where L 9 1 (most of the protein is in the T conformation in the absence of ligands). A non-cooperative ligand of type 2 in the MWC model is a ligand which hinds with identical affinities to the conformations R and T. In the KNF model a non-cooperative Z of type 1 is a ligand whose binding induces a conformational change (from A to C) hut this change does not affect further 2 binding ( K i c / k C = 1). A non-cooperative 2 of type 2 is a ligand which does not induce a conformational change upon binding. As in the dime, case, a non-cooperative Z of type 1 can behave in the same manner as a noncooperative 2 of type 2, if the conformational change it induces does not affect X binding to the neighboring subunits Model

Effect of a cooperative 2 on h for X

Effect of a non-cooperative Z on h for X ~

~~

~-

type 1

~~

type 2

Preexistent asymmetry

h(x) changes hut always h(x,z) 1 1 . 0

h(x) does not change: h(x,z) = h(x)

MWC

h(x) changes hut always h(x,z) 2 1.0

h(x) changes hut only to higher values: h(x,z) >h(x)

h(x) does not change: h(x,z) = h(x)

Simplest KNF

Any change in h(x) is possible: h(x,z) > 1.0 can he obtained, even when h(x) < 2.0,and vice versa:

any change in h(x) is possible: m z ) # h(x)

h(x) does not change: h(x,z) = h(x)

h(x,z) #

w

Simple KNF with different intersubunit binding domains

same as in the simplest KNF model:

h(x,z) # h(x)

same as in the simplest KNF model : h(x,z) # 4 x 1

h(x) does not change: h ( x , z ) = h(x)

General KNF

similar to simple KNF model hut more variability is possible.

similar to simple KNF model hut more variability is possible:

h(x,z) # h(x)

h(x,z) # h(x)

any change in h for X is possible: h ( x , z ) # h(x); special case, when Z hinds with an identical affinity to all conformations, h ( x , z ) = h(x).

is either unaffected (when Kk, = KT,) or enhanced (if Ki,> K& and L $ 1) by the addition of a noncooperative competing ligand Z. Thus, a weakening in the positive cooperativity of X binding upon addition of a non-cooperative Z , h(x, z) < h(x), excludes the MWC model but can be accounted for by the KNF model (Tables 1 and 2). If curve-fitting of X binding yields an L value which is not much larger than 1, Z binding can be non-cooperative only if KR, = K+zand, in this case, any change in the cooperativity of X binding in the presence of a non-cooperative Z excludes the MWC model. A change from positive to negative cooperativity, i.e. h(x,z) < 1 whereas h(x) > 1 , upon the addition of a competing ligand Z (which in this case may also be a cooperative ligand) also excludes the MWC model, as this model allows only for positively cooperative binding (see relevant sections in Theory). Application to Real Situations

The approach outlined in the present study formulates a strategy for exploring which of the cooperativity models discussed above should be used for a set of binding data. For each case where cooperativity is observed, one should test the cooperativity in the binding of the primary ligand in the absence and in the

presence of competing ligands which bind to the same set of sites. The most efficient procedure for distinguishing between the cooperativity models is through the use of non-cooperative competing ligands. It is preferable to use a set of such ligands and not a single one, as ligand-induced conformational changes are ligand-specific [ 181. Therefore, some non-cooperative competing ligands may induce a change in the cooperativity of the binding of the primary ligand, while others will not. This approach was recently applied to establish the mechanism of the negative cooperativity in coenzyme binding to rabbit muscle glyceraldehyde-3-phosphate dehydrogenase [7]. In this study, the binding of NAD+ and its fluorescent analogue ENAD' (nicotinamide 1 :N6-ethenoadenine dinucleotide) which bind to this enzyme in a negatively cooperative manner [5, 191, was measured in the presence of the non-cooperative NAD ' analogues, ADP-ribose, AcPyAD+ (3acetylpyridine - adenine dinucleotide) and ATP. The negative cooperativity in coenzyme binding was found to be considerably weakened in the presence of saturating AcPyAD and ATP concentrations but remained unaffected by the presence of saturating ADP-ribose concentrations. These results exclude the pre-existent asymmetry model and provide direct proof that the +

Y . I. Henis and A. Levitzki

negatively cooperative mode of coenzyme binding to this enzyme is due to ligand-induced conformational transitions. This approach can also be used for the study of ligand binding to receptors. Thus, for example, it has been shown that the binding of insulin to its receptor is negatively cooperative [20]. It will be interesting to employ competition experiments in order to clarify the issue of whether this negative cooperativity is due to pre-existent asymmetry (heterogeneity) in the binding sites, or due to ligand-induced effects.

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465 6. Eby, D. & Kirtley, M. E. (1976) Biochemistry, 15, 2168-2171. 7. Henis, Y. I. & Levitzki, A. (1979) Eur. J . Bioc~hmi., in the press. 8. Barcroft, J. & Hill, A. V. (1910)J. Physiol. (Lond.) 39,411 -428. 9. Hill, A . V. (1913) Biochem. J . 7, 471-4480, 10. Adair, G. S. (1925) J . Biol. Chem. 63, 529-545. 11. Dahlquist, F. W. (1978) Methods Enzymol. 48, 270-299. 12. Levitzki, A. (1978) Quantitative Aspects of’ANostery in Molecular Biology, Biochemistry and Biophysics, vol. 28, SpringerVerlag, Heidelberg. 13. Levitzki, A. (1975) in Subunit Enzymes (Ebner, K. E., ed.) pp. 1 40, Marcel Dekker, New York. 14. Koshland, D. E., Jr (1970) in The Enzymes (Boyer, P. D., ed.) 3rd edn, vol. 1, pp. 342- 396, Academic Press, New York. 15. Levitzki, A. & Koshland, D. E., Jr (1976) Curr. TOP.Cell Regul. 10, 1-40. 16. Cornish-Bowden, A. J. & Koshland, D. E., Jr (1970) J . Bid. Chem. 245, 6241 - 6250. 17. Nari, J., Mouttet, C., Fouchier, F. & Ricard, J. (1974) Eur. J . Biochem. 41, 499-515. 18. Byers, L. D. & Koshland, D. E., Jr (1975) Biochemistry, 14, 3661 - 3669. 19. Schlessinger, J. & Levitzki, A. (1974) J . Mol. Biol. 82, 547- 561. 20. De-Meyts, P., Roth, J., Neville, D. M., Gavin, J. R., I11 & Lesniak, M. A. (1973) Biochem. Biophys. Res. Commun. 55, 154- 161.

Y. 1. Henis and A. Levitzki, Department of Biological Chemistry, Institute of Life Sciences, Hebrew University of Jerusalem, Givat Ram, Jerusalem, Israel