- Email: [email protected]
Relationship between ligand binding and conformational change of macromolecules in the two-state allosteric model
J. theor. Biol. (1973) 38, 579-585
Relationship Between Ligand Binding and Conformational Change of Macromolecules in the Two-state Allosteric Model H. WILL AND G. DAMASCHUN Central Institute of Heart and Circulatory Regulation Research and Central Institute of Molecular Biology, German Academy of Sciences, 1115 Berlin-Buch, German Democratic Republic (Received 28 March 1972, and in revisedform
10 July 1972)
The relationship between the binding function p and the state function W of an oligomeric protein has been analysed for the general two-state allosteric model. It is shown that this relation is determined by the numerical values of the inherent parameters of the model. The shape of the function P = f(W) can therefore be strictly concave, strictly convex or inverse sigrnoidal according to the conditions. In the two-state allosteric model only a dimeric protein can display a linear relationship between ?‘and 8. In the paper general criteria for the estimation of the state function i? from experimentally obtained conformational parameters are discussed.
1. Introduction
For the description of cooperative ligand binding by macromolecules a number of models has been proposed, allowing a mathematical adaption to experimentally investigated saturation curves (Atkinson, Hathaway & Smith, 1965; Koshland, Nemethy & Filmer, 1966; Monod, Wyman & Changeux, 1965; Changeux & Rubin, 1968). It is generally accepted that saturation curves alone are not sufficient for defining the fundamental mechanism of ligand binding. Models of quite different assumptions may be valid for the same binding curve, if the introduced parameters are chosen correspondingly. Additional information is obtained when the ligand binding is related to the accompanying conformational change of the macromolecule. This is emphasized by the fact that conformational changes are responsible for most519
580
H.
WILL
AND
G.
DAMASCHUN
if not all-allosteric interactions. Considering binding change we have to oppose two fundamental models: (i) the sequential model, formation occur only et al., 1966); (ii) the two-state model operativity is due to more) conformations (Monod et al., 1965;
and conformational
based on induced fit, where changes in conas a consequence of ligand binding (Koshland (in general the shift of which also Changeux et
multi-state model) where coan equilibrium between two (or exist in the absence of ligands al., 1968).
A fundamental property of the latter model is that, in contrast to the sequential model, the conformation shift as a function of free ligand concentration is not necessarily proportional to the quantity of bound ligand (Changeux et al., 1968). Since the simple sequential model predicts linearity in the response of some structural parameters to the addition of ligand, deviations from linearity were interpreted to imply the two-state mechanism (Durchschlag, Puchwein, Kratky, Schuster & Kirschner, 1971). The following investigation shows to which extent the relationship between ligand binding and conformational shift depends on the inherent parameters of the latter model. This completes the earlier studies by McClintock & Markus (1969) which compared only special numerical assumptions. 2. Relationship Between the State Function R and the Saturation Function B in the Two-state Allosteric Model In the two-state model homotropic interactions are described by two functions (Monod et al., 1965; Changeux et al., 1968). If NR is the number of molecules in the R state, NT the number of molecules in the T state and NY the number of binding sites occupied with ligands of an oligomer consisting of n protomers, then i? = iVR/(NR+NT) is the fraction of molecules in the conformation state R, which is characterized by its higher affinity for the allosteric ligand S, and H = Nr/n(N, +iVr) is the fraction of binding sites occupied with ligand. It follows: (1-l-a) (1) R = 1-T = (l+~~+L(l+clx)“’ y=
cc(1+,>,-I
+Lca(l+ca)“-’ (l+a)“+L(l+ca)”
*
(2)
Here L = T/R represents the allosteric equilibrium constant, a = S/K, the normalized free ligand concentration and c = KJK,. KR and KT are the
LIGAND
BINDING
AND
CONFORMATIONAL
CHANGE
581
dissociation constants of the protein-ligand complexes for the conformations R and T, respectively. By elimination of a from equations (1) and (2), Y = (6
[l-
(gy”]
[l+Lc(go’-“‘“],
(3)
where 1 -
OlYIl.
Obviously the relation between P and K is determined by the values of the constants L, c and n. For n > 2, P as a function of i? in the considered region may have a strictly concave shape, a strictly convex shape or an inflexion point. If the first derivative ri is continuous, the behaviour of the second derivative 7;; discriminates between these possibilities : n-l ‘i7h = (1 -+qql
-jQ
(q-l’” ,g
[(gy-&
-;)I.
(4)
For 7;; = 0 (and Y’RKK ” # 0) there is an inflexion point. Its position is given by 1 - (cL)w -n) 1 R, = --~ pw= ___ and 1 +(c~p)1/(2-“) (1 -c)[l +(c”Ly2-“)]’ In the region l/(1 + L) < R -c R,, YiI; < 0, therefore P as function of R is concave. In the region R, < R < l/(1 +Lf) is Yii > 0 and P is a convex function of R. For Lc = 1 the abscissae of the inflexion point and the initial value of R = l/(1 +L) are identical, so the curve is convex in the whole region. Convexity holds also for Lc < 1. In analogy Y vs. R is strictly concave for all Lc”- ’ 2 1. A linear dependence of the saturation function P on the state function R is impossible for n > 2. For n = 2 the function (3) has no inflexion point. In this case the following conditions hold: for Lc = 1, P is a linear function of i? with the slope (1 +LJ/(L- 1) = (1 + c)/(l -c), Lc < 1 characterizes a strictly convex, Lc > 1 a strictly concave course of the curve. The analysed functional dependence P = f(R) is demonstrated by simulated curves in Fig. I. Such theoretical curves illustrate to what extent a change of L or c (as it may be produced by changes of temperature, pH value, ionic strength or additional ligands) may lead to qualitatively different relations between Y and i? for the two state allosteric model. Of particular importance is the effect of heterotropic ligands whose influence on the homotropic interactions is mediated exclusively by the shift of the conformation
582
H.
WILL
AND
G.
DAMASCHUN
08
0.6
02
0.4
06
O-8
0.8
0.6
0
0.2
0 4
0.6
0.8
FIG. 1. Theoreticalcurvesfor the functionaldependence of the saturationfunction Ton the statefunction2 in thetwo-stateallostericmodel.(a) Dimericprotein,n = 2, c = 0.01. (b) Tetramericprotein,n = 4, c = 0.01.(c) Tetramericprotein,n = 4, L = 1000.
equilibrium
(Monod et al., 1965; Changeux et al., 1968). L is then replaced by (5)
where are the normalized
y = A/K; B = m, inhibitor and activator concentrations, d = K;/K; and e = K$K&
respectively,
3. Relationship Between Experimental Parameters and the State Function R In general, the experimentally determined “change of conformation” of macromolecules as related to ligand binding is not identical with the fraction
LIGAND
BINDING
AND
CONFORMATIONAL
583
CHANGE
w but is rather reflected by the relative function RF (Rubin & Changeux, 1966) : R.-R.=0 RF = (6) -IT& L, This normalized function reflects the variation of W within its respective limits as measured in the absence of a specified ligand and in the presence of saturating concentrations of this ligand. Since Rr; and R are related by equation (6) the dependence 5 = f(%r) can likewise be demonstrated (Fig. 2). a and K? are not directly observable quantities. Furthermore for a polydisperse system with elements of different conformation states R and T, as postulated here, in principle only average conformation parameters P can be obtained by physical experiments. If PR is the corresponding parameter of I o0 R ‘-
0.2
I I
02
I
I
0.4
O-6
04
06
00
I.0
(4
0,0
FIG. 2. Theoretical curves for the functional dependence of the saturation function Ton the relative state function E7 in the two-state allosteric model. (a) Dimeric protein, n = 2, c = 0.01. (b) Tetrameric protein, n = 4, c =i @Ol. (c) Tetrameric protein, n = 4, L = 1000. T.B. 39
584
H.
WILL
AND
G.
DAMASCHUN
the molecules in the R state and PT the parameter of the molecules in the T state, then P = P(P,,P,,R) (7) holds for such an average conformation parameter. Analogously to equation (6) a relative conformation measure is defined by Pa-I$,, Pr; = p (8) a+m-~.=O’ R and P (or R? and iV) are not necessarily proportional to each other. In order to define R (or RF) by experiment the functions (7) and (8) must be calculated for the special physical method. A possible example might be the adequate relation for the correlation volume V, (Durchschlag et al., 1971; Damaschun & Ptirschel, 1971) as measured by X-ray small-angle scattering according to Porod. For a monodisperse system of the conformation R one has (jf) = ~$2: v (91 c R o12> RHere V, stands for the volume of the molecules in the R state and r](x) for the difference between the electronic density of the molecule and the average electronic density of the solvent. For a bidisperse system of R and T molecules with the corresponding volumes V, and V, one would obtain
.
(10)
This is a non-linear relation between (V,),. r and I?. 4. Discussion A non-linear relation between conformation shift (W) and saturation function (Y) excludes a mechanism which consists only of elementary second-order steps like the simple sequential mechanism (Durchschlag et al., 1971). For such mechanisms conformation change and binding must be proportional to each other. By way of contrast the Monod-model does generally not allow for a linear relationship between B and R (exception: n = 2 and Lc = 1). The functional dependence Y = f(R) may be concave, convex, or inversely sigmoid according to the values of the parameters n, c and L. There were reports in the literature on deviations from linearity between 7 and 8. For instance, the conformation change accompanying the binding of NAD to glyceraldehyde-3-phosphate dehydrogenase as demonstrated by Ord, sedimentation measurements and X-my small-angle scattering was shown to be a concave function of P (Jaenicke & Gratzer, 1969; Durch-
LIGAND
BINDING
AND
CONFORMATIONAL
CHANGE
585
schlag et al., 1971). The non-linearity of this relationship was interpreted as evidence for the two-state mechanism. It should be pointed out that the difference between a linear and a nonlinear relationship in some cases becomes vague when experimental data are used to distinguish between them. A strict linearity between Y and a in the two-state model is only possible for the exceptional case II = 2 and Lc = I, but the shape of the curve, as shown in Figs 1 and 2 may sometimes be so smooth that by experiment the curves are hardly distinguished from a straight line. For example, the linear relationship between the fractional change of the spinmarker spectrum and the fractional change of the optical spectrum (the latter is identical with the fraction of the oxygenated heme groups, P), which was demonstrated by Ogawa & McConnell (1967), on haemoglobin is not only consistent with the sequential model but may also be predicted from the two-state model and the numerical values of L and c given by Edelstein (1971). Haemoglobin, however, is a rather special case because of its non-equivalent subunits. According to recent studies (Ogata & McConnell, 1971, 1972; Ogata, McConnell & Jones, 1972) the R P T transition is much more affected by binding of ligands to the fi-heme than to the c+heme. The experimental evidence of a non-linear relationship between 7 and R represents not a sufficient condition for an allosteric two-state mechanism. In addition to the simple sequential model further mechanisms cannot be excluded. In order to find sufficient conditions for the allosteric two-state model principally new criteria must be worked out. We are greatly indebted to Mrs M. Falck from the Division of Methodics and Theory Berlin-Buch for the calculations of the theoretical curves and to Dr J. G. Reich and H.-V. Ptirschel for helpful discussions. REFERENCES ATKINSON, D. E., HATHAWAY, J. A. & SMITH, D. C. (1965). J. biol. Chem. 240, 2682. CHANGEUX, J.-P. & RLJBIN, M. M. (1968). Biochemistry, N. Y. 7, 553. DAMASXUN, G. & PDRSCHEL, H.-V. (1971). Mh. Chemie 102, 1146. DURCHSCHLAG, H., PIJCHWEIN, G., KRATKY, O., SCHUSTER, J. & KIRSCHNER, K. (1971). Eur. J. Biochem. 19, 9. EDELSTEIN, S. J. (1971). Nature, Lond. 230, 224. JAENICKE, R. & GRATZER, W. B. (1969). Eur. J. Biochem. 10, 158. KOSHLAND, D. E., NEMETHY, G. & FILMER, D. (1966). Biochemistry, N. Y. 5, 365. MCCLINTOCK, D. K. & MARKUS, G. (1969). J. biol. Chem. 244, 36. MONOD, J., WYMAN, J. & CHANGEVX, J.-P. (1965). J. molec. Biol. 12, 88. OGATA, R. T. & MCCONNELL, H. M. (1971). Proc. Cold Spring Harb. Symp. quant. Biol. 36, 325. OGATA, R. T. & MCCONNELL, H. M. (1972). Proc. natn. Acad. Sci. U.S.A. 69, 335. OGATA, R. T., MCCONNELL, H. M. & JONES, R. T. (1972). Biochem. biophys. Rex. Commun. 47, 273. OGAWA, S. & MCCONNELL, H. M. (1967). Proc. natn. Acad. Sci. U.S.A. 58, 19. RUBIN, M. M. & CHANGEUX, J.-P. (1966). J. molec. Biof. 21, 265.
Relationship Between Ligand Binding and Conformational Change of Macromolecules in the Two-state Allosteric Model H. WILL AND G. DAMASCHUN Central Institute of Heart and Circulatory Regulation Research and Central Institute of Molecular Biology, German Academy of Sciences, 1115 Berlin-Buch, German Democratic Republic (Received 28 March 1972, and in revisedform
10 July 1972)
The relationship between the binding function p and the state function W of an oligomeric protein has been analysed for the general two-state allosteric model. It is shown that this relation is determined by the numerical values of the inherent parameters of the model. The shape of the function P = f(W) can therefore be strictly concave, strictly convex or inverse sigrnoidal according to the conditions. In the two-state allosteric model only a dimeric protein can display a linear relationship between ?‘and 8. In the paper general criteria for the estimation of the state function i? from experimentally obtained conformational parameters are discussed.
1. Introduction
For the description of cooperative ligand binding by macromolecules a number of models has been proposed, allowing a mathematical adaption to experimentally investigated saturation curves (Atkinson, Hathaway & Smith, 1965; Koshland, Nemethy & Filmer, 1966; Monod, Wyman & Changeux, 1965; Changeux & Rubin, 1968). It is generally accepted that saturation curves alone are not sufficient for defining the fundamental mechanism of ligand binding. Models of quite different assumptions may be valid for the same binding curve, if the introduced parameters are chosen correspondingly. Additional information is obtained when the ligand binding is related to the accompanying conformational change of the macromolecule. This is emphasized by the fact that conformational changes are responsible for most519
580
H.
WILL
AND
G.
DAMASCHUN
if not all-allosteric interactions. Considering binding change we have to oppose two fundamental models: (i) the sequential model, formation occur only et al., 1966); (ii) the two-state model operativity is due to more) conformations (Monod et al., 1965;
and conformational
based on induced fit, where changes in conas a consequence of ligand binding (Koshland (in general the shift of which also Changeux et
multi-state model) where coan equilibrium between two (or exist in the absence of ligands al., 1968).
A fundamental property of the latter model is that, in contrast to the sequential model, the conformation shift as a function of free ligand concentration is not necessarily proportional to the quantity of bound ligand (Changeux et al., 1968). Since the simple sequential model predicts linearity in the response of some structural parameters to the addition of ligand, deviations from linearity were interpreted to imply the two-state mechanism (Durchschlag, Puchwein, Kratky, Schuster & Kirschner, 1971). The following investigation shows to which extent the relationship between ligand binding and conformational shift depends on the inherent parameters of the latter model. This completes the earlier studies by McClintock & Markus (1969) which compared only special numerical assumptions. 2. Relationship Between the State Function R and the Saturation Function B in the Two-state Allosteric Model In the two-state model homotropic interactions are described by two functions (Monod et al., 1965; Changeux et al., 1968). If NR is the number of molecules in the R state, NT the number of molecules in the T state and NY the number of binding sites occupied with ligands of an oligomer consisting of n protomers, then i? = iVR/(NR+NT) is the fraction of molecules in the conformation state R, which is characterized by its higher affinity for the allosteric ligand S, and H = Nr/n(N, +iVr) is the fraction of binding sites occupied with ligand. It follows: (1-l-a) (1) R = 1-T = (l+~~+L(l+clx)“’ y=
cc(1+,>,-I
+Lca(l+ca)“-’ (l+a)“+L(l+ca)”
*
(2)
Here L = T/R represents the allosteric equilibrium constant, a = S/K, the normalized free ligand concentration and c = KJK,. KR and KT are the
LIGAND
BINDING
AND
CONFORMATIONAL
CHANGE
581
dissociation constants of the protein-ligand complexes for the conformations R and T, respectively. By elimination of a from equations (1) and (2), Y = (6
[l-
(gy”]
[l+Lc(go’-“‘“],
(3)
where 1 -
OlYIl.
Obviously the relation between P and K is determined by the values of the constants L, c and n. For n > 2, P as a function of i? in the considered region may have a strictly concave shape, a strictly convex shape or an inflexion point. If the first derivative ri is continuous, the behaviour of the second derivative 7;; discriminates between these possibilities : n-l ‘i7h = (1 -+qql
-jQ
(q-l’” ,g
[(gy-&
-;)I.
(4)
For 7;; = 0 (and Y’RKK ” # 0) there is an inflexion point. Its position is given by 1 - (cL)w -n) 1 R, = --~ pw= ___ and 1 +(c~p)1/(2-“) (1 -c)[l +(c”Ly2-“)]’ In the region l/(1 + L) < R -c R,, YiI; < 0, therefore P as function of R is concave. In the region R, < R < l/(1 +Lf) is Yii > 0 and P is a convex function of R. For Lc = 1 the abscissae of the inflexion point and the initial value of R = l/(1 +L) are identical, so the curve is convex in the whole region. Convexity holds also for Lc < 1. In analogy Y vs. R is strictly concave for all Lc”- ’ 2 1. A linear dependence of the saturation function P on the state function R is impossible for n > 2. For n = 2 the function (3) has no inflexion point. In this case the following conditions hold: for Lc = 1, P is a linear function of i? with the slope (1 +LJ/(L- 1) = (1 + c)/(l -c), Lc < 1 characterizes a strictly convex, Lc > 1 a strictly concave course of the curve. The analysed functional dependence P = f(R) is demonstrated by simulated curves in Fig. I. Such theoretical curves illustrate to what extent a change of L or c (as it may be produced by changes of temperature, pH value, ionic strength or additional ligands) may lead to qualitatively different relations between Y and i? for the two state allosteric model. Of particular importance is the effect of heterotropic ligands whose influence on the homotropic interactions is mediated exclusively by the shift of the conformation
582
H.
WILL
AND
G.
DAMASCHUN
08
0.6
02
0.4
06
O-8
0.8
0.6
0
0.2
0 4
0.6
0.8
FIG. 1. Theoreticalcurvesfor the functionaldependence of the saturationfunction Ton the statefunction2 in thetwo-stateallostericmodel.(a) Dimericprotein,n = 2, c = 0.01. (b) Tetramericprotein,n = 4, c = 0.01.(c) Tetramericprotein,n = 4, L = 1000.
equilibrium
(Monod et al., 1965; Changeux et al., 1968). L is then replaced by (5)
where are the normalized
y = A/K; B = m, inhibitor and activator concentrations, d = K;/K; and e = K$K&
respectively,
3. Relationship Between Experimental Parameters and the State Function R In general, the experimentally determined “change of conformation” of macromolecules as related to ligand binding is not identical with the fraction
LIGAND
BINDING
AND
CONFORMATIONAL
583
CHANGE
w but is rather reflected by the relative function RF (Rubin & Changeux, 1966) : R.-R.=0 RF = (6) -IT& L, This normalized function reflects the variation of W within its respective limits as measured in the absence of a specified ligand and in the presence of saturating concentrations of this ligand. Since Rr; and R are related by equation (6) the dependence 5 = f(%r) can likewise be demonstrated (Fig. 2). a and K? are not directly observable quantities. Furthermore for a polydisperse system with elements of different conformation states R and T, as postulated here, in principle only average conformation parameters P can be obtained by physical experiments. If PR is the corresponding parameter of I o0 R ‘-
0.2
I I
02
I
I
0.4
O-6
04
06
00
I.0
(4
0,0
FIG. 2. Theoretical curves for the functional dependence of the saturation function Ton the relative state function E7 in the two-state allosteric model. (a) Dimeric protein, n = 2, c = 0.01. (b) Tetrameric protein, n = 4, c =i @Ol. (c) Tetrameric protein, n = 4, L = 1000. T.B. 39
584
H.
WILL
AND
G.
DAMASCHUN
the molecules in the R state and PT the parameter of the molecules in the T state, then P = P(P,,P,,R) (7) holds for such an average conformation parameter. Analogously to equation (6) a relative conformation measure is defined by Pa-I$,, Pr; = p (8) a+m-~.=O’ R and P (or R? and iV) are not necessarily proportional to each other. In order to define R (or RF) by experiment the functions (7) and (8) must be calculated for the special physical method. A possible example might be the adequate relation for the correlation volume V, (Durchschlag et al., 1971; Damaschun & Ptirschel, 1971) as measured by X-ray small-angle scattering according to Porod. For a monodisperse system of the conformation R one has (jf) = ~$2: v (91 c R o12> RHere V, stands for the volume of the molecules in the R state and r](x) for the difference between the electronic density of the molecule and the average electronic density of the solvent. For a bidisperse system of R and T molecules with the corresponding volumes V, and V, one would obtain
.
(10)
This is a non-linear relation between (V,),. r and I?. 4. Discussion A non-linear relation between conformation shift (W) and saturation function (Y) excludes a mechanism which consists only of elementary second-order steps like the simple sequential mechanism (Durchschlag et al., 1971). For such mechanisms conformation change and binding must be proportional to each other. By way of contrast the Monod-model does generally not allow for a linear relationship between B and R (exception: n = 2 and Lc = 1). The functional dependence Y = f(R) may be concave, convex, or inversely sigmoid according to the values of the parameters n, c and L. There were reports in the literature on deviations from linearity between 7 and 8. For instance, the conformation change accompanying the binding of NAD to glyceraldehyde-3-phosphate dehydrogenase as demonstrated by Ord, sedimentation measurements and X-my small-angle scattering was shown to be a concave function of P (Jaenicke & Gratzer, 1969; Durch-
LIGAND
BINDING
AND
CONFORMATIONAL
CHANGE
585
schlag et al., 1971). The non-linearity of this relationship was interpreted as evidence for the two-state mechanism. It should be pointed out that the difference between a linear and a nonlinear relationship in some cases becomes vague when experimental data are used to distinguish between them. A strict linearity between Y and a in the two-state model is only possible for the exceptional case II = 2 and Lc = I, but the shape of the curve, as shown in Figs 1 and 2 may sometimes be so smooth that by experiment the curves are hardly distinguished from a straight line. For example, the linear relationship between the fractional change of the spinmarker spectrum and the fractional change of the optical spectrum (the latter is identical with the fraction of the oxygenated heme groups, P), which was demonstrated by Ogawa & McConnell (1967), on haemoglobin is not only consistent with the sequential model but may also be predicted from the two-state model and the numerical values of L and c given by Edelstein (1971). Haemoglobin, however, is a rather special case because of its non-equivalent subunits. According to recent studies (Ogata & McConnell, 1971, 1972; Ogata, McConnell & Jones, 1972) the R P T transition is much more affected by binding of ligands to the fi-heme than to the c+heme. The experimental evidence of a non-linear relationship between 7 and R represents not a sufficient condition for an allosteric two-state mechanism. In addition to the simple sequential model further mechanisms cannot be excluded. In order to find sufficient conditions for the allosteric two-state model principally new criteria must be worked out. We are greatly indebted to Mrs M. Falck from the Division of Methodics and Theory Berlin-Buch for the calculations of the theoretical curves and to Dr J. G. Reich and H.-V. Ptirschel for helpful discussions. REFERENCES ATKINSON, D. E., HATHAWAY, J. A. & SMITH, D. C. (1965). J. biol. Chem. 240, 2682. CHANGEUX, J.-P. & RLJBIN, M. M. (1968). Biochemistry, N. Y. 7, 553. DAMASXUN, G. & PDRSCHEL, H.-V. (1971). Mh. Chemie 102, 1146. DURCHSCHLAG, H., PIJCHWEIN, G., KRATKY, O., SCHUSTER, J. & KIRSCHNER, K. (1971). Eur. J. Biochem. 19, 9. EDELSTEIN, S. J. (1971). Nature, Lond. 230, 224. JAENICKE, R. & GRATZER, W. B. (1969). Eur. J. Biochem. 10, 158. KOSHLAND, D. E., NEMETHY, G. & FILMER, D. (1966). Biochemistry, N. Y. 5, 365. MCCLINTOCK, D. K. & MARKUS, G. (1969). J. biol. Chem. 244, 36. MONOD, J., WYMAN, J. & CHANGEVX, J.-P. (1965). J. molec. Biol. 12, 88. OGATA, R. T. & MCCONNELL, H. M. (1971). Proc. Cold Spring Harb. Symp. quant. Biol. 36, 325. OGATA, R. T. & MCCONNELL, H. M. (1972). Proc. natn. Acad. Sci. U.S.A. 69, 335. OGATA, R. T., MCCONNELL, H. M. & JONES, R. T. (1972). Biochem. biophys. Rex. Commun. 47, 273. OGAWA, S. & MCCONNELL, H. M. (1967). Proc. natn. Acad. Sci. U.S.A. 58, 19. RUBIN, M. M. & CHANGEUX, J.-P. (1966). J. molec. Biof. 21, 265.