I
V. L. Hess and Attila Szabol Indiana University Bloomington 47401
I
Ligand Binding to Macromolecules Allosteric and sequential models of cooperativity
Many biochemical processes involve the binding of ligands to macromolecules; for example, substrates or inhibitors to enzymes, and oxygen to myoglobin or hemoglobin. The simplest cases of ligaud binding can be handled easily by application of elementary thermodynamics, but more complicated problems, such as those involving cooperativity, can become extremely tedious. Our purpose is to describe a simpler way of attacking such problems and to demonstrate its utility by applying it to several situations in order of increasing complexity. Application will be made to cooperative hinding by hemoglobin and asoartate transcarbamvlase. and the sequential and allosteric models of cooperativebinding will be onsidered. The method is based on the construction of a function, actually the macroscopic analogue of the grand canonical partition function (I), from which the hinding properties of the molecule may be obtained. This function was introduced in the biochemical literature by Wyman (2) as the "binding polynomial" and used extensively by one of us (3-5). under the name "generating fuuction," to treat various aspects of coooerative lieand bindine to hemoelobiu. Our desire is to make this approach avoilahle to those who are interested in treatine similar urohlerns but who are without knowledge of statistGal mechanics.
.~ ~
~
General Formulation Consider a macromolecule MN, with N binding sites for a ligand, X. This system could, for example, be substrate binding to a multisubunit enzyme or protons binding to amino acid residues in a protein. The property of interest is y, the fractional saturation of the molecule: that is, the ratio of the number of occupied binding sites to the total number of binding sites, expressed in terms of X, the equilibrium concentration of free ligand. If the total concentration of the ligand is assumed to be much larger than the number of available binding sites, then Xis approximately equal to the total concentration of ligand, and the calculations are greatly simplified. This restriction can be removed if necessary. The system is completely described by N equilibrium constants Ai,which determine the equilibrium concentrations of species in the reactions
-
.,
(3)
where [ M ~ x o ] [MN].From eqn. (2) note that A. = 1. Now construct the function
1 Alfred
P. Sloan Fellow.
The fractional saturation is this average divided by the total number of binding sites
I t is more convenient to write this equation in the equivalent form A a =--1 JlnZ(A) y=--ln&(A) N JlnA NJA Thus, the fractional saturation can be calculated easily once the generating function is known. The key to a problem is the generating fuuction; however, i t is fairly simple to translate a thermodynamic scheme or a well-formulated physical model into a generating function, as will be demonstrated in a series of examnles. Independent Binding The simplest case is clearly the binding of one type of ligand t o a molecule with only one binding site; for example, the binding of oxygen to myoglobin. T o demonstrate the equivalence of generating functions to thermodynamic development, the problem will be approached both ways. The reaction is described by
-
First, by definition,
and
..,
where X = [XI. Since A,X' is proportional to [MNX,], i t is proportional to probability of binding precisely i ligands, which is just [MNXI p, =
AjA' p. = (5) ' Z(A) The definition of probability requires that &Pi = 1;this is indeed the case. At a given ligand concentration, the average number of ligands hound is
y _ = [MXI (14) [MI + [MXI Solving eqn. (13) for [MX] and substituting into eqn. (14) gives
By definition of the equilibrium constant
.
comparison with eqn. (3) gives
T o solve the problem by using a generating function, first obtain the generating function from eqn. (4) &(A)
= 1 +AX
(16)
Then, from eqn. (a), This example is too trivial for the second method to offer any real advantage. When more complicated problems are considered, and direct application of thermodynamics gets out of hand, the obtainment and use of generating functions remains almost as simple as it is here. Volume 56, Number 5, May 1979 / 289
PA'
422
ea22
4 a ~
Figure 1. Diagrammatic representation at independent binding to a macromalecule win? four binding sites. Unliganded sites are squares and iigsnded sites are circles The carmibution of each diagram to Uw generating functionis shown below the diagrams. The Corresponding generating function is :(A) = (1 + Ah)'. 1
2
0 0 q 3
4
1
2
1
0 0 q 3
2
1
0 0 0 3
4
2
0 0 q 0 3
4
0- bound rite -unbound rite
4
Figure 2. Combinationsgiving rise to a configuration with one bound site. The model can he generalized to account for the hinding of one type of ligand to a molecule with several identical, independent binding sites, such as the binding of reactants to catalyzing resins. If A is the binding constant of a lieand to a single site, we must find Aj, the equilibrium conitant for binding i ligands to the macromolecule. A series of diagrams like those in Figure 1is useful for this. T o determine the expressions for A; it is necessary to remember that AiA' is proportional to the probability of forming a species with precisely i bound ligands. This probability is related to three senarate ~hvsicalfactors: the amount of lieand "~ available, the eq&ibriuk constant for binding a ligand to a single site. and the number of comhinations of bound and unhaund sices in which a total of i sites is hound (that is the numhrr of ways of choming i out of N sites.) The first of these is accounted for by A'; theother two must he included in the expression for A;. When there is only one binding site there are only two configurations, bound and unbound, and only one way of formine each. (We use the word confirmration to refer to a distingkhahle ipecies which is specifLd completely by the number of bound sites of a given twe). When there are several hinding sites, as in Figure 1, there is only one way each of having all sites hound or all free, but there are several combinations which yield each of the intermediate configurations. For instance, by numbering the sites one can see that there are four ways of obtaining a configuration with precisely one bound and three unbound sites (Fig. 2). In general, the number of combinations giving i hound sites out of a total of N sites is given by ~~
(9
N!
(18) which is just the ithcoefficient in the binomial expansion of (1 + x ) ~ . Thus, = 1-
5 . r
The exponent arises from the rule that the probability of several independent events occurring simultaneously is given by the product of the individual probabilities. From the definition of the generating function, &(A)
=
,
1-0
[(q
+
A'] hi = (1 Ah)"
(20)
The fractional saturation is ..
...
..
which is identical to the result for one site. This is not surprising, since the molecule with N equivalent sites is not conce~tuallvdifferent from N molecules, each containing one site. since tile numher of macromolecules does not ente; into the calculations, thr fractimnl saturation is indel~endentof the number of sites available, as long as these sites do not interact. 290 / Journal of Chemical Edocation
Figure 3. Diagrams corresponding to a molecule with two differenttypes of sites for the same ligand. The contribution made to the generating f ~ n ~ l i oisnshown below each diagram. The corresponding generating function is :(A) = (1 + A,h)9t + binding
Comparing eqn. (20) with eqn. (16) gives (22) EN(M = [&(A)]" From eqn. (51, setting i = 0,we see that the generating function is the reciprocal of the probability that &bing is bound. Equation (22), then, states that the probability of having N unbound sites simultaneouslv is iust the &duct of t h e probabilities that the individui sit& are unbound. In general, the generatine function describing a number of indenendent events is just the pruduct of the individual generating functions. The ~ d m b i l i t v - l i k ebehavior of these functions tremendously~implifiescalculations involving large numbers of hinding sites. This can be generalized to the case of independent hut nonequivalent binding sites. Suppose the molecule has N1 sites whose binding constant for X is A1 and Nz sites whose hinding constant for X is Ap. The generating function is
-
ZN,.NJN = (1+ A,h)"l(I + A2A)"2 When N1 = Np = 2, the generating function, which is
(23)
is derived diagramnti(ally in Figure 3. A further peneralization of the rcnerntinc! function is necessary to account for macromolecuies with two different types of binding sites for two different types of ligands. Consider a system containing a macromolecule with N1 sites binding ligand 1with binding constant A1 and Nz sitcs binding ligand 2 with binding constant Az, and free ligand concentrations X1 and Ap. If the ligands bind independently, the two sites act like two separate macromolecules, and EN,,N~(A~,AJ = (1 + AlhdN'(l + A z X ~ ) " ~ (25) To obtain the fractional saturation of the two types of sites, eqn. (8) is generalized to Y, = A, a I n S ( A A ) =- 1 + A,A, ' , _ I , z (26) Again, the two types of sites behave as though they were on two separate molecules. Related to this is the hinding of twodifferent types of ligand to the same hinding site; for example, reversible inhibition of an enzyme. For a single hinding site, there are three possible reactions M + XI?%MX~
xx
A2Az
Figure 4. Binding ta a macromoleculewhich can bind up to two molecules of ligand 1 a one molecule of ligand 2, but not both. The conbibution made to the generating tunction is shown below each dlagam. carespwding genRating function is Z(h) = (I Alhd2 A2h2.
+
+
ZIAAI~ -
s4 F i g m 6. Seqwntial birding. T k conhibution made to the generating function is shown below each diagram. me corresponding generating function is
See text far discussion
of oxygen to hemoglobin. Figure 5 shows fractional saturation of hemoglohin and myoglohiu, plotted as a function of A, the partial pressure of oxygen. The a and @ chains of hemoglobin differ, hut the general characteristics of the hinding of oxygen by myoglohin are similar to those of free a and @ chains. The binding by single chains is described hy the equation
y=- A X
1+AX
Figure 5. Schematic plots of ?versus p02. Me partial pressure of axygen. (a) Hemoglobin (b) Myoglabin (analogous to free a and chains).
0
The generating function for this scheme is S ( h ~ , h z l= 1 + AIAI
+ A2hs + A1ZhlA2
(28)
If A12 = 0, hinding is mutually exclusive. A trivial extension of this accounts for sites which can hind two molecules of ligand 1 or one molecule of ligand 2, hut cannot hind the two at once. (See Fig. 4.) The generating function is Z(AI,X~) = 1
+ 2A1Alf
(A1X1)2
+ AZXZ = (1
+ A1A1)2 + A d 2
(29)
This model can describe the competitive binding of COz and organic phosphates to hemoglohin (6). Hemoglobin is a tetramer of approximate molecular weight 64000. Its subunits, two a-type and two @type, are so arranged that there is a crevice between the @ chains. The N-terminal residues of the @chainslie in this crevice, and each can hind one Con molecule. Alternatively, the crevice can hind one organic phosphate molecule by means of electrostatic interactions. Because of their size, the phosphates, once in place, appear to prevent the hinding of COz. lnteractlng Sites Generating functions show their advantages even in dealing with independent hinding sites, when the problem involves several possible configurations, hut they really prove their worth when cooperativity is added to the picture. When the sites are independent, fractional saturation is independent of the number of hinding sites; when they interact, this is no longer true. The most familiar example of cooperative hinding is that
(30)
which would also describe the hemoglohin molecule if binding were independent. Cooperativity is represented by a sigmoidal hinding curve, as shown in Figure 5. At low partial pressure, oxygen is bound more weakly to hemoglohin than to the free chains. This part of the curve represents the hinding of the first of the four possihle oxygens. Once the first oxygen is hound i t becomes easier to hind the second, and easier still to hind the third and fourth. We consider two different models for cooperativity: the sequential model of Koshland, Nemethy, and Filmer (7) (KNF), and the allosteric model of Monod, Wyman, and Changeux (8)(MWC). The stereochemical mechanism of Perutz (9)for cooperativity in hemoglohin belongs to the class of MWC-type models. T o illustrate clearly the physical difference between the MWC and KNF models, we describe both in a language based on the Perutz mechanism. Paullng or KNF Models The oldest model of cooperative binding of oxygen to hemoglobin was formulated by Pauling in 1935 ( l o ) ,and subsequently developed by Koshland, Nemethy, and Filmer. Actually, there are several KNF models, geometric variations of a mechanism in which oxygen hinding to a given site directly influences the affinity of the remaining sites. In particular, consider the "square model," so called because the four subunits are assumed to lie a t the corners of asauare. The unliganded sites are connected by four stabilizing Erosslinks' as shown in Fieure 6. When a site hinds a linand. .. . the existing crosslinks brtwcen it and the 1,ther suhunits must he brokm. Thus. tu bind the t h s t limnd, two cnaslinks must hr broken. The second ligand can bind &a site which already lost one of its crosslinks. and the last linand can hind without having t o break any c.rr,s*links at all. Sinre it i:oits r n c r n to hreak a crosslink. thr affinity of the. tmolccule inrrt,nsri in the course of oxygenation. T o calculate the generating function for the square model,
2Phy~ically these correspond to salt-bridges,which are electrostatic bonds between protonated nitrogens and ionized carboxyl grwps of differentamino acids (9). Volume 56, Number 5, May 1979 / 291
the appropriate hinding constants must he ohtained. If they were independent, each site would have binding constant A. Here. however, the hindinz constant is reduced by a factor of S foreach cnaslink that must be hroken when anoxygen is hound. As before, statistical factors must he included. Thus, the bindinr" constant tor the first lirand is 4A/S2. Tltecenerating function is ohtained by summing the contributions of the diagrams in Figure 6, where each diagram contributes a term of the fnrm f , ( A h ) ~ s - where ~, f, is the numher of combinations renresented bv the diamam, i the numher of ligands bound, and k the number of cro&nks hroken. (Note teat, in obtainine the contribution of a confirmration to the generating function, the path to that configuration is not c&sideredL Thus Z&)=1+-+-
4AA SZ
4(AAI2 S"
2(AM2 +-+-+S4
4(AAP S4
( A W (31) S4
One simple variation locates the sites on the corneraof a tetrahedron. so that each site interacts with all the others. The reader should convince himself that
Another variation is the rectangular model, where two "long" and two "short" bonds decrease A by factors of S, and Sz, respectively. By drawing a series of diagrams similar to those of Figure 6, one can show that
-
4AA A,.,(A)=I+-+-(-+-+-) SISZ
2(AW 1 SISZ ~1
1 sz
1
SLSP
4(AAP + ( A N 4 (33) + ---S,2S22 S12S22
In a similar way, the model can be generalized to any combination of nonequivalent interacting sites. As one would expect, if all the inceractim parameters are net. equal to one, the independent generating function is recovered Z(A) = ( I +AX)'
(34)
MWC Models
to the binding of oxwen A somewhat more subtle anoroach .. ... to hemoglobin and related pn)l,lems was present~dhyMonod, Wvman. and Chnneeux cd,. 'l'hpir allosteric model is based on-the observation t h a t two different quaternary structures exist for hemoglobin, one for the oxygenated form, the other for the deoxygenated form. In fact, crystals of deoxygenated hemoglobin shatter when exposed to oxygen. In the allosteric model, hinding within each quaternary structure is independent; there are no direct interactions between sites. In the deoxy form (the tense, or T form), there are crosslinks between the sites which have two effects-they stabilize the T structure, and they reduce the affinity of this structure for oxygen. Because of the absence of these crosslinks, the oxygenated (or relaxed, or R) structure has the higher affinity. The T structure can exist in the absence of crosslinks, hut the R structure is much more stable. When an oxygen is hound in the T state, one of the two crosslinks associated with its binding site is broken. Unlike the breaking of crosslinks in the KNF models, this does not directly affect the affinity of the other binding sites, but it does shift the equilihrium toward the R state, and thus, indirectly, make it easier to hind the next oxygen. Thus, cooperativity is built into this model by linking two reaction schemes within each of which oxygen hinds independently, in such a way that transition from one scheme to the other depends on the numher of oxygens already bound. This is shown diagrammatically in Figure 7. The generating function, as before, is the sum of the contributions from each diagram. We assume that the equilihrium constant between unliganded tetramers in the R and T states in the absence of crosslinks is Q ~
..
Adding the crosslinks stabilizes the T structure, increasing the equilibrium constant for eqn. (35) by a factor of S for each crosslink, to give QS4.Thus, each R diagram contributes a term of the fnrm
(q)
(AX)'
(37)
where i is the numher of circles, and each T diagram contributes a term of the form
(4)
QS4-'(A A)'
(38)
with a factor of S for each remaining crosslink. When these terms are added, the generating function ohtained is
+ + ( 7) The probability-like behavior of the generating functions E(A) = QS4 1
(1
AX)'
(39)
has already been mentioned, and eqn. (39) is another example of this. If transitions between R and T were impossible, each would have its own generating function for the independent binding of hemoblogin
where the S in &(A) accounts for the effect of crosslinks on oxygen affinity, as explained above. The probability of the occurrence of one of several mutually exclusive events is given hy the sum of the individual probabilities; generating functions also obey this rule. Thus, except for the factor of QS4, which accounts for the transition between R and T , eqn. (39) is the sum of the independent generating functions given in eqn. (40). Monod, Wyman, and Changeux (8)use a different notation, which is related to ours as follows
~
The R structure is intrinsically more stable; hence 292 1 Journal of Chemical Education
By analogy with the independent case, the generating function for N subunits is
+
Z ~ w d a=) L ( l + c a I N (1 + a ) N (42) T h e nonequivalenceiof a and 0 chains can be handled in the same way as were different types of independent hinding sites, since within each quaternary structure the sites are assumed to be independent.
-
;(A)
= L(1 + c , A N (1
+ c,A,A)~ + (1 + A,A)2(1+
A,hj2
(43)
The MWC-type model can he generalized to handle mul-
Figure 7. Allosteric binding. The conhibution made to the generating function IS Shown below each diagram. The corresponding generating function is
See text tw discus&n
tisubunit enzymes catalyzing reactions involving two substrates. such as asDartate trauscarbamvlase. (ATCase). which r a t a l v k the reaction between mrl,amyl phosphate and aspartate in b:. coli pyrimidine biosynthesis ( 1 1 1 . A'l'Case has six catalytic subunits which hind the suhstratesand six regu l a t o r ~suhunits which hind such phos~hatesas rrstodine triphoiphate and adenosine triphosphate. At saturation concentrations of carbanyl phosphate, ATCase activity is sigmoidally dependent on aspartate concentration, indicating cooperative binding of this substrate. Using the formalism presented here, it is easy to generalize the MWC model to the binding of two ligands, so that it can be applied to this svstem. Assume that there are two possible quaternary structures, and to make the model as eeneral as nossible, N suhstratebinding subunits. Then, bianalogy &th eqn.(29) and eqn. (4%
..
fi(hl,hz) = L(1 + clAlhl + C Z A A+ C I ~ A ~ ~ A ~ X ~ ) ~ + (1+ A l h l t A& + A12XrXdN (44) where clAl and c2A2 are the binding constants of the two substrates in the T state:. A,. andA7. -. these bindine constants in the H stntc,; c1?:1 th1:ir sirnultanevus binding constant in the T state: and A,.,. their simultanwui binding runstant in the R state. By anzogy with the independent case, if A t %= 0, the ligand binding is mutually exclusive, if A2 = 0, the second ligand binds only after the first is hound; if c12 = clc2 and Al, = A1A2. binding of the two substrates is independent within each quaternary structure, giving
In summary, we have presented a powerful technique based on the erand canonical oartition function for describine lipand I~indinghy rnacn~m~,leculcs. L'siny this formalis~n,it is relatively pass t o calculate the frartionnl snturation r~redictedbv any conceivable model for cooperative ligand Linding by protein. Thus, the challenge is to give structural significance to the parameters. Examples of this kind of approach, as applied to hemoglobin, can be found in references (3-5).
-
..~~~-~
a
Acknowledgment We wish to thank Dr. G. I. H. Hanania for encouraging us to write this article. This work has been supported by Public Health Service Grant HZ-21483. Literature Cited 111 Hill. T. L., "Introduction to Statistical Thsrmodynamics,.' Addison-Wdey. Reading. 1960, p. 20-24. (Good explanation ofthe grand canonical partition function). (2) Wyman..l.,J. Mol. R i d , ll.631 11965). 13) S u h e , A . , a n d Karplus, M . . J. Mnl. R i d 12.168 (1972). 14 S~2aho.A.a.d Karplus, M . . B i o c h ~ m i a l r y .14.931 11976). 151 Slaho.A..and Karplur. M..Rhehernislry. 16,2869 11976). 161 Stlyer, L., "Riochemistgv," Fmeman, San Francirco. 191S.p. 46-92. Chapters 3 and I contain a good introduction to hemoglobin. 171 Koshland, 0. E.,Nemethy.G.,snd Filmor. D., Rmchsrnislry, 5,365 11966). 181 Mon0d.J..Wyman, J.,andChangeux, J. P., J. M o l . Riol.. 12,881 119651. 191 Perutr, M. F.,Notore ILondonl. 228,726 119701: f l r Med. R u l l , 32. 195 (19761. I101 Pauling, L.. P i o r Noll Arad. Sci. W o s h . 21,186 (1935). I l l 1 Lehninger, A. L.."Bioehemiriry" (2nd edition), Worth, New York. 197%.p. 287-340. Good introduction to ATCsse.
Volume 56, Number 5. May 1979 / 293
