ANALYTICAL
BIOCHEMISTRY
1%&203-206
(1991)
Calorimetric Determination of Cooperative in High Affinity Binding Processes’ Gabrielle
Bains
Department
of Biology and Biocalorimetry
Received
August
and Ernest0
Freire Center, The Johns Hopkins
The association of ligands to macromolecules possessing multiple binding sites is often accompanied by cooperative effects. These effects are either positive or negative depending on whether the binding of the first ligand molecule increases or decreases the binding affinity of subsequent ones. In fact, a large number of fundamental biological processes are regulated by cooperative interactions, and as such they have been the focus of numerous studies throughout the years (see Ref. ( 1) for a recent review).
0003sx97/91 Copyright All rights
University,
Baltimore,
Maryland
21218
14,199O
It is demonstrated that isothermal titration calorimetry can be used to determine cooperative interaction energetics even for extremely tight binding processes in which the binding affinity constants are beyond the limits of experimental determination. The approach is based on the capability of calorimetry to measure the apparent binding enthalpy at any degree of ligand saturation. When calorimetric measurements are performed under conditions of total association at partial saturation, the dependence of the apparent binding enthalpy on the degree of saturation is a function only of the cooperative binding interactions. The method developed in this paper allows an independent estimation of cooperative energetic parameters without the need to simultaneously estimate or precisely know the value of the association constants. Since total ligand association at partial saturation is achieved only at macromolecular concentrations much larger than the dissociation constants, the method is especially suited for high and very high affinity processes. Biological associations in this category include fundamental cellular processes like cell surface receptor binding or protein-DNA interactions. 0 1991 Academic Press, Inc.
’ Supported
Interactions
by NIH
Grants
RR-04328
$3.00 0 1991 by Academic Press, of reproduction in any form
and NS-24520.
In the past, the study of cooperative interactions has primarily relied upon the analysis of ligand binding isotherms. Usually, variations in the shape of these isotherms, from what is expected for independent binding sites, have been used to draw conclusions regarding the sign and magnitude of the cooperative interactions. Early methods depended primarily on different linearized representations of binding isotherms, e.g., Hill plot, Scatchard plot (2,3), whereas modern methods are dependent primarily on nonlinear least-squares minimization schemes (4). In both cases, the objective has been to separate the binding energetics from the cooperative energetics. In many cases, this has proven to be a difficult task due to the high degree of statistical correlation existing between the various fitting parameters ( 5 ) . The situation is even more complex if the enthalpic and entropic contributions to the binding and cooperative parameters need to be determined. The purpose of this paper is to present a protocol aimed at experimentally determining cooperative interaction parameters independently of binding parameters. Many specific cellular interactions like the binding of hormones to cell surface receptors, e.g., insulin-insulin receptor (6)) or protein-DNA interactions, e.g., the right operator Oa of the bacteriophage X ( 7)) are characterized by extremely tight association constants (K > lOlo M-’ ) . These processes are often highly cooperative (7) ; however, at the extremely low concentrations required to perform binding experiments under those conditions, it is not always possible to perform accurate dissections of their binding and cooperative energetics. The experimental determination of cooperative interactions is of fundamental importance because these interactions are usually linked to conformational changes central to the molecular mechanisms of signal transduction in complex macromolecular assemblies. Because isothermal titration calorimetry allows a direct measurement of the binding enthalpy at any degree of saturation, it is possible to develop analytical meth203
Inc. reserved.
204
BAINS
AND
ods and to design experiments under conditions in which only cooperative interactions are reflected in the measured data. This paper develops a general protocol for the calorimetric determination of cooperative parameters (cooperative free energy, cooperative enthalpy, and cooperative entropy) independently of binding constants. This is achieved by considering the saturation dependence of the apparent binding enthalpy measured under conditions in which all the ligand added binds to the macromolecule. This situation is experimentally satisfied by performing the titration experiments at concentrations much larger than the dissociation constant (usually two orders of magnitude or more). For the same reason this method is valid even for extremely tight associations in which the binding constants are beyond the limits of experimental resolution. THEORY
Isothermal titration calorimetry measures the amount of heat absorbed or released by a chemical or biochemical reaction initiated by the mixing of two or more reactants. In the case of a ligand binding reaction to a macromolecule, the amount of heat, 4, absorbed or released after addition of a ligand, X, is proportional to the amount of ligand bound, X,, and the mean binding enthalpy, AH; i.e., 9 = Af%,
PI
where X,, is the molar amount of ligand bound. If the experiment is performed at macromolecular concentrations much higher than the dissociation constant, essentially all of the ligand added, Xi, to the calorimeter cell will bind the macromolecule (X, = Xi). Under those conditions, it is possible to define an apparent binding enthalpy, AHaP,, as *Ha,, = q/Xi,
[23
where Xi is the molar amount of ligand injected into the calorimeter cell. AHaP, can be measured experimentally and represents the quantity of interest in this paper. For a macromolecular system having a single binding site the apparent binding enthalpy will be a constant independent of the degree of saturation. The same is true for a system composed of multiple, identical, and independent binding sites. For a system characterized by multiple, interacting binding sites the situation is different will vary with the degree of saturation in a and A&, manner dependent on the magnitude of the cooperative enthalpy and cooperative free energy interaction parameters. The above ideas can be illustrated using the case of a macromolecule possessing two binding sites. Figure 1 schematically depicts the various ligancy states, the free
FREIRE STATE
LIGANCY
FREEENERGY
2
ZAG+
Ag
FIG. 1. Schematic representation of the three states and corresponding free energies, degeneracies tical weights for a macromolecular system exhibiting binding sites.
0
1
S.W.
k K’IXI
’
different ligancy (w ) , and statistwo interacting
energies, degeneracies, and statistical weights associated with this binding model. According to the figure, the binding of the first ligand is characterized by a free energy change, AG, whereas the binding of the second ligand contains the additional cooperative free energy term Ag. A positive Ag will lower the affinity toward the second ligand resulting in negative cooperativity, whereas a negative Ag has the opposite effect and results in positive cooperativity. The case in which Ag = 0 results in two independent binding sites. The partition function, Z, for this binding model is given by z
= 1+
2K[X]
+ kKZ[X]2,
[31
where K = ePACIRTis the intrinsic association constant and k = e-rSg’RTis the cooperative interaction parameter. The fractional populations of the three ligancy states are given by F, = l/Z
[4al
F, = 2K[X]/Z
[4bl
F, = kK2[ X12/Z.
[4cl
The ligand saturation function, Y, defined as the ratio of the bound ligand concentration to the macromolecule concentration is given by Y = Fl + 2F2.
151
Figure 2 presents typical ligand saturation curves obtained for different values of the cooperative interaction parameter, k = e -&IRT . For illustration purposes, standard saturation plots as well as their Scatchard representations are shown in the figure. For k = 1 the plots correspond to the situation of two independent binding sites, whereas for k < 1 or k > 1 the binding curves are those characteristic of negative or positive cooperativity respectively.
COOPERATIVE
BINDING
ENERGETICS
Bound
KKI
FIG. 2. Ligand binding isotherms (left) plotted in terms of the dimensionless quantity K[ Xl and their Scatchard representations (right) to k values of a = 10, b = 2, c = 1, for different values of the cooperative interaction parameter k = e -ag’RT . The curves in this figure correspond d = 0.5, e = 0.1, and f = 0.01. For better illustration of the low saturation region, the binding curves on the left have been truncated before complete saturation.
It must be noted that for a macromolecule with two binding sites for a ligand, the case in which the two sites are different is mathematically equivalent to the situation existing for k < 1 (negative cooperativity) for the model in Fig. 1. In this case, the cooperative interaction parameter k is related to the association constants Kl and K2 for the two binding sites by the equation k = 4K,K,/(K, + K2)‘. Calorimetric Titrations In calorimetric experiments the heat absorbed or released which is directly proportional At any degree of saturation thalpy, (AH), is given by the (AH)
the quantity of interest is by the binding process, to the binding enthalpy. the av,erage binding engeneral expression
= c FiAHi,
[61
where Fi is the macromolecular population in the ith state and AHi the enthalpy change associated with that state. For the two-site system in Fig. 1, Eq. [ 61 reduces to (AH)
= F,AH
+ F,(2AH
+ Ah),
[71
where AH is the intrinsic binding enthalpy and Ah the additional enthalpy arising from the cooperative interactions. The amount of heat measured after the addition of ligand into the calorimeter cell can be written as
4 = Wfl(AH),
[al
where 2) is the reactive volume of the calorimeter cell and [M ] is the macromolecule concentration in the calorimeter cell. The dependence of 4 on the total ligand concentration defines the traditional binding isotherm. This function, however, does not allow an independent determination of the cooperative energetics. The situa-
tion is different if the quantity considered is the apparent binding enthalpy. By combining Eqs. [ 21 and [ 81, the expression obtained for the apparent binding enthalpy is
AHwP= (AH)/Y.
[91
This equation relates the experimentally accessible AH,,, to the macromolecular population of states for any degree of saturation. In general, for a given degree of saturation, the distribution of ligand molecules among the binding sites is determined solely by the magnitude of the cooperative interaction parameters. The dependence of AH,,, on the degree of saturation is independent of the actual value of the association constant and depends only on the cooperative interaction parameters. Thus, these parameters can be estimated from the dependence of the apparent binding enthalpy on the degree of saturation. This is illustrated in Fig. 3 in which AHaP, has been plotted as a function of Y for different values of the cooperative interaction parameter k. For illustration purposes, the intrinsic binding enthalpy has been set equal to -30 kcal/mol and the interaction enthalpy equal to -10 kcal/mol. As shown in the figure, the intercept of AHapP at zero saturation is equal to the intrinsic binding enthalpy, AH, whereas the intercept at full saturation is equal to (2AH + Ah) /2. The actual shape of the curve is completely determined by the value of k. For negative cooperativity ( k < 1) , AH,,, remains close to AH up to relatively high degrees of saturation, reflecting the fact that under those conditions the probability of occupying the second binding site is diminished. On the contrary, for positive cooperativity (12 > 1) , AH,,, changes rapidly at low degrees of saturation reflecting the increased probability of the ligand to occupy the second binding site once the first one is occupied.
206
BAINS
0
0.4
0.8
1.2
1.6
AND
2
Y FIG. 3.
Dependence of the the ligand saturation function, tive interaction parameter k = spond to k values of a = 1000, = 0.5, h = 0.1, i = 0.5, j = 0.01,
apparent binding enthalpy, AHam, on Y, for different values of the cooperae -u’RT. The curves in the figure correb = 100, c = 50, d = 10, e = 5, f = 1, g and k = 0.001.
Figure 3 clearly indicates that the AH,,, dependence on the saturation function is extremely sensitive to k, and that this plot can be used to determine the entire cooperativity energetics. Total Association at Partial Saturation The best experimental condition to measure AHfipr, is when the degree of saturation is accurately known. This can be easily achieved by performing the calorimetric measurements at concentrations much higher than the dissociation constant. Under these circumstances, all of the ligand injected into the calorimeter cell binds to the macromolecule, effectively satisfying the condition of total association at partial saturation (TAPS) .2 One of the main advantages of working under those conditions is that the cooperative parameters become statistically uncorrelated from the binding parameters and can be accurately determined in an independent fashion. This should be contrasted with the situation existing in standard calorimetric titrations where the binding constant, cooperative free energy, binding enthalpy, and cooperative enthalpy need to be estimated simultaneously by nonlinear least-squares fitting of the titration data.
the dependence of AHa,,p on the degree of saturation becomes an experimental function of the cooperative energetics alone, and as such it can be used to evaluate cooperative parameters independently of binding constants. Even though the approach has been illustrated for the case of two binding sites, the generalization to other situations is straightforward. As is the case with calorimetry in general, the experimental observable is the amount of heat absorbed or released with a reaction, and as such only reactions exhibiting an enthalpy change can be measured. Fortunately, cooperative interactions usually involve macromolecular changes in conformation and therefore they are accompanied by enthalpy changes (8). In other cases, cooperative effects are linked to protonation reactions (1) and their energetics can be easily determined by performing calorimetric experiments in buffer systems with high heats of ionization. Another major advantage of the method described in this paper is that it can be performed at high macromolecular concentrations thus facilitating the analysis of reactions characterized by relatively small enthalpy changes. Modern isothermal titration calorimeters (8-11) have sensitivities that permit measurement of total heat effects on the order of 1 peal. With this sensitivity range, it is in principle possible to measure the cooperative energetics of a binding reaction characterized by an interaction enthalpy of 10 kcal/mol with as little as O.l0.5 nmol of material per data point. For a protein of M, 100,000 this figure translates into 10-50 pg of protein per data point. These modern experimental and analytical capabilities open the door to a direct thermodynamic analysis of systems like cell surface receptors or DNAbinding proteins, that until now have eluded a calorimetric characterization. REFERENCES 1. Wyman, tional Science 2. Hill,
In this paper we have shown that isothermal titration calorimetry permits a determination of the energetics of the cooperative interactions associated with ligand binding in an independent fashion. In the past, cooperative interaction parameters and binding constants have been estimated simultaneously from the analysis of ligand binding isotherms (8). Under conditions of TAPS, used:
TAPS,
total
association
at partial
saturation.
J., and Gill, S. J. (1990) Chemistry of Biological Books, Mill Valley, CA.
A. V. (1910)
J. Physiol.
in Binding and Linkage. FuncMacromolecules, University
(London)
40,4-7.
3. Scatchard, G. (1949) Ann. N. Y. Acod. Sci. 61,660-672. 4. Johnson, M. L., and Frasier, S. G. (1985) in Methods in Enzymology (Hirs, C. H. W., and Timasheff, S. N., Eds.), Vol. 11'7, pp. 301-342, Academic Press, San Diego, CA. 5. Johnson,
DISCUSSION
2 Abbreviation
FREIRE
M. L. (1983)
6. Fujita-Yamaguchi,
Biophys.
Y., and
J. 44,101-106.
Harmon,
J. T. (1988)
Biochemistry
27,3252-3259. 7. Senear, 6577. 8. Schon,
D. F., and Ackers, A., and Freire,
G. K. (1990)
E. (1989)
Biochemistry
Biochemistry
29,
656%
28,5019-5024.
9. McKinnon, I. R., Fall, L., Parody, A., and Gill, D. J. (1984) Anal. Biochem. 139,134-139. 10. Wiseman, T., Williston, S., Brandts, J. F., and Lin, L.-N. ( 1989) Anal. Biochem. 179, 131-135. 11. Freire, E., Mayorga, 0. L., and Straume, M. (1990) Anal. Chem.,
62,95OA-959A.
BIOCHEMISTRY
1%&203-206
(1991)
Calorimetric Determination of Cooperative in High Affinity Binding Processes’ Gabrielle
Bains
Department
of Biology and Biocalorimetry
Received
August
and Ernest0
Freire Center, The Johns Hopkins
The association of ligands to macromolecules possessing multiple binding sites is often accompanied by cooperative effects. These effects are either positive or negative depending on whether the binding of the first ligand molecule increases or decreases the binding affinity of subsequent ones. In fact, a large number of fundamental biological processes are regulated by cooperative interactions, and as such they have been the focus of numerous studies throughout the years (see Ref. ( 1) for a recent review).
0003sx97/91 Copyright All rights
University,
Baltimore,
Maryland
21218
14,199O
It is demonstrated that isothermal titration calorimetry can be used to determine cooperative interaction energetics even for extremely tight binding processes in which the binding affinity constants are beyond the limits of experimental determination. The approach is based on the capability of calorimetry to measure the apparent binding enthalpy at any degree of ligand saturation. When calorimetric measurements are performed under conditions of total association at partial saturation, the dependence of the apparent binding enthalpy on the degree of saturation is a function only of the cooperative binding interactions. The method developed in this paper allows an independent estimation of cooperative energetic parameters without the need to simultaneously estimate or precisely know the value of the association constants. Since total ligand association at partial saturation is achieved only at macromolecular concentrations much larger than the dissociation constants, the method is especially suited for high and very high affinity processes. Biological associations in this category include fundamental cellular processes like cell surface receptor binding or protein-DNA interactions. 0 1991 Academic Press, Inc.
’ Supported
Interactions
by NIH
Grants
RR-04328
$3.00 0 1991 by Academic Press, of reproduction in any form
and NS-24520.
In the past, the study of cooperative interactions has primarily relied upon the analysis of ligand binding isotherms. Usually, variations in the shape of these isotherms, from what is expected for independent binding sites, have been used to draw conclusions regarding the sign and magnitude of the cooperative interactions. Early methods depended primarily on different linearized representations of binding isotherms, e.g., Hill plot, Scatchard plot (2,3), whereas modern methods are dependent primarily on nonlinear least-squares minimization schemes (4). In both cases, the objective has been to separate the binding energetics from the cooperative energetics. In many cases, this has proven to be a difficult task due to the high degree of statistical correlation existing between the various fitting parameters ( 5 ) . The situation is even more complex if the enthalpic and entropic contributions to the binding and cooperative parameters need to be determined. The purpose of this paper is to present a protocol aimed at experimentally determining cooperative interaction parameters independently of binding parameters. Many specific cellular interactions like the binding of hormones to cell surface receptors, e.g., insulin-insulin receptor (6)) or protein-DNA interactions, e.g., the right operator Oa of the bacteriophage X ( 7)) are characterized by extremely tight association constants (K > lOlo M-’ ) . These processes are often highly cooperative (7) ; however, at the extremely low concentrations required to perform binding experiments under those conditions, it is not always possible to perform accurate dissections of their binding and cooperative energetics. The experimental determination of cooperative interactions is of fundamental importance because these interactions are usually linked to conformational changes central to the molecular mechanisms of signal transduction in complex macromolecular assemblies. Because isothermal titration calorimetry allows a direct measurement of the binding enthalpy at any degree of saturation, it is possible to develop analytical meth203
Inc. reserved.
204
BAINS
AND
ods and to design experiments under conditions in which only cooperative interactions are reflected in the measured data. This paper develops a general protocol for the calorimetric determination of cooperative parameters (cooperative free energy, cooperative enthalpy, and cooperative entropy) independently of binding constants. This is achieved by considering the saturation dependence of the apparent binding enthalpy measured under conditions in which all the ligand added binds to the macromolecule. This situation is experimentally satisfied by performing the titration experiments at concentrations much larger than the dissociation constant (usually two orders of magnitude or more). For the same reason this method is valid even for extremely tight associations in which the binding constants are beyond the limits of experimental resolution. THEORY
Isothermal titration calorimetry measures the amount of heat absorbed or released by a chemical or biochemical reaction initiated by the mixing of two or more reactants. In the case of a ligand binding reaction to a macromolecule, the amount of heat, 4, absorbed or released after addition of a ligand, X, is proportional to the amount of ligand bound, X,, and the mean binding enthalpy, AH; i.e., 9 = Af%,
PI
where X,, is the molar amount of ligand bound. If the experiment is performed at macromolecular concentrations much higher than the dissociation constant, essentially all of the ligand added, Xi, to the calorimeter cell will bind the macromolecule (X, = Xi). Under those conditions, it is possible to define an apparent binding enthalpy, AHaP,, as *Ha,, = q/Xi,
[23
where Xi is the molar amount of ligand injected into the calorimeter cell. AHaP, can be measured experimentally and represents the quantity of interest in this paper. For a macromolecular system having a single binding site the apparent binding enthalpy will be a constant independent of the degree of saturation. The same is true for a system composed of multiple, identical, and independent binding sites. For a system characterized by multiple, interacting binding sites the situation is different will vary with the degree of saturation in a and A&, manner dependent on the magnitude of the cooperative enthalpy and cooperative free energy interaction parameters. The above ideas can be illustrated using the case of a macromolecule possessing two binding sites. Figure 1 schematically depicts the various ligancy states, the free
FREIRE STATE
LIGANCY
FREEENERGY
2
ZAG+
Ag
FIG. 1. Schematic representation of the three states and corresponding free energies, degeneracies tical weights for a macromolecular system exhibiting binding sites.
0
1
S.W.
k K’IXI
’
different ligancy (w ) , and statistwo interacting
energies, degeneracies, and statistical weights associated with this binding model. According to the figure, the binding of the first ligand is characterized by a free energy change, AG, whereas the binding of the second ligand contains the additional cooperative free energy term Ag. A positive Ag will lower the affinity toward the second ligand resulting in negative cooperativity, whereas a negative Ag has the opposite effect and results in positive cooperativity. The case in which Ag = 0 results in two independent binding sites. The partition function, Z, for this binding model is given by z
= 1+
2K[X]
+ kKZ[X]2,
[31
where K = ePACIRTis the intrinsic association constant and k = e-rSg’RTis the cooperative interaction parameter. The fractional populations of the three ligancy states are given by F, = l/Z
[4al
F, = 2K[X]/Z
[4bl
F, = kK2[ X12/Z.
[4cl
The ligand saturation function, Y, defined as the ratio of the bound ligand concentration to the macromolecule concentration is given by Y = Fl + 2F2.
151
Figure 2 presents typical ligand saturation curves obtained for different values of the cooperative interaction parameter, k = e -&IRT . For illustration purposes, standard saturation plots as well as their Scatchard representations are shown in the figure. For k = 1 the plots correspond to the situation of two independent binding sites, whereas for k < 1 or k > 1 the binding curves are those characteristic of negative or positive cooperativity respectively.
COOPERATIVE
BINDING
ENERGETICS
Bound
KKI
FIG. 2. Ligand binding isotherms (left) plotted in terms of the dimensionless quantity K[ Xl and their Scatchard representations (right) to k values of a = 10, b = 2, c = 1, for different values of the cooperative interaction parameter k = e -ag’RT . The curves in this figure correspond d = 0.5, e = 0.1, and f = 0.01. For better illustration of the low saturation region, the binding curves on the left have been truncated before complete saturation.
It must be noted that for a macromolecule with two binding sites for a ligand, the case in which the two sites are different is mathematically equivalent to the situation existing for k < 1 (negative cooperativity) for the model in Fig. 1. In this case, the cooperative interaction parameter k is related to the association constants Kl and K2 for the two binding sites by the equation k = 4K,K,/(K, + K2)‘. Calorimetric Titrations In calorimetric experiments the heat absorbed or released which is directly proportional At any degree of saturation thalpy, (AH), is given by the (AH)
the quantity of interest is by the binding process, to the binding enthalpy. the av,erage binding engeneral expression
= c FiAHi,
[61
where Fi is the macromolecular population in the ith state and AHi the enthalpy change associated with that state. For the two-site system in Fig. 1, Eq. [ 61 reduces to (AH)
= F,AH
+ F,(2AH
+ Ah),
[71
where AH is the intrinsic binding enthalpy and Ah the additional enthalpy arising from the cooperative interactions. The amount of heat measured after the addition of ligand into the calorimeter cell can be written as
4 = Wfl(AH),
[al
where 2) is the reactive volume of the calorimeter cell and [M ] is the macromolecule concentration in the calorimeter cell. The dependence of 4 on the total ligand concentration defines the traditional binding isotherm. This function, however, does not allow an independent determination of the cooperative energetics. The situa-
tion is different if the quantity considered is the apparent binding enthalpy. By combining Eqs. [ 21 and [ 81, the expression obtained for the apparent binding enthalpy is
AHwP= (AH)/Y.
[91
This equation relates the experimentally accessible AH,,, to the macromolecular population of states for any degree of saturation. In general, for a given degree of saturation, the distribution of ligand molecules among the binding sites is determined solely by the magnitude of the cooperative interaction parameters. The dependence of AH,,, on the degree of saturation is independent of the actual value of the association constant and depends only on the cooperative interaction parameters. Thus, these parameters can be estimated from the dependence of the apparent binding enthalpy on the degree of saturation. This is illustrated in Fig. 3 in which AHaP, has been plotted as a function of Y for different values of the cooperative interaction parameter k. For illustration purposes, the intrinsic binding enthalpy has been set equal to -30 kcal/mol and the interaction enthalpy equal to -10 kcal/mol. As shown in the figure, the intercept of AHapP at zero saturation is equal to the intrinsic binding enthalpy, AH, whereas the intercept at full saturation is equal to (2AH + Ah) /2. The actual shape of the curve is completely determined by the value of k. For negative cooperativity ( k < 1) , AH,,, remains close to AH up to relatively high degrees of saturation, reflecting the fact that under those conditions the probability of occupying the second binding site is diminished. On the contrary, for positive cooperativity (12 > 1) , AH,,, changes rapidly at low degrees of saturation reflecting the increased probability of the ligand to occupy the second binding site once the first one is occupied.
206
BAINS
0
0.4
0.8
1.2
1.6
AND
2
Y FIG. 3.
Dependence of the the ligand saturation function, tive interaction parameter k = spond to k values of a = 1000, = 0.5, h = 0.1, i = 0.5, j = 0.01,
apparent binding enthalpy, AHam, on Y, for different values of the cooperae -u’RT. The curves in the figure correb = 100, c = 50, d = 10, e = 5, f = 1, g and k = 0.001.
Figure 3 clearly indicates that the AH,,, dependence on the saturation function is extremely sensitive to k, and that this plot can be used to determine the entire cooperativity energetics. Total Association at Partial Saturation The best experimental condition to measure AHfipr, is when the degree of saturation is accurately known. This can be easily achieved by performing the calorimetric measurements at concentrations much higher than the dissociation constant. Under these circumstances, all of the ligand injected into the calorimeter cell binds to the macromolecule, effectively satisfying the condition of total association at partial saturation (TAPS) .2 One of the main advantages of working under those conditions is that the cooperative parameters become statistically uncorrelated from the binding parameters and can be accurately determined in an independent fashion. This should be contrasted with the situation existing in standard calorimetric titrations where the binding constant, cooperative free energy, binding enthalpy, and cooperative enthalpy need to be estimated simultaneously by nonlinear least-squares fitting of the titration data.
the dependence of AHa,,p on the degree of saturation becomes an experimental function of the cooperative energetics alone, and as such it can be used to evaluate cooperative parameters independently of binding constants. Even though the approach has been illustrated for the case of two binding sites, the generalization to other situations is straightforward. As is the case with calorimetry in general, the experimental observable is the amount of heat absorbed or released with a reaction, and as such only reactions exhibiting an enthalpy change can be measured. Fortunately, cooperative interactions usually involve macromolecular changes in conformation and therefore they are accompanied by enthalpy changes (8). In other cases, cooperative effects are linked to protonation reactions (1) and their energetics can be easily determined by performing calorimetric experiments in buffer systems with high heats of ionization. Another major advantage of the method described in this paper is that it can be performed at high macromolecular concentrations thus facilitating the analysis of reactions characterized by relatively small enthalpy changes. Modern isothermal titration calorimeters (8-11) have sensitivities that permit measurement of total heat effects on the order of 1 peal. With this sensitivity range, it is in principle possible to measure the cooperative energetics of a binding reaction characterized by an interaction enthalpy of 10 kcal/mol with as little as O.l0.5 nmol of material per data point. For a protein of M, 100,000 this figure translates into 10-50 pg of protein per data point. These modern experimental and analytical capabilities open the door to a direct thermodynamic analysis of systems like cell surface receptors or DNAbinding proteins, that until now have eluded a calorimetric characterization. REFERENCES 1. Wyman, tional Science 2. Hill,
In this paper we have shown that isothermal titration calorimetry permits a determination of the energetics of the cooperative interactions associated with ligand binding in an independent fashion. In the past, cooperative interaction parameters and binding constants have been estimated simultaneously from the analysis of ligand binding isotherms (8). Under conditions of TAPS, used:
TAPS,
total
association
at partial
saturation.
J., and Gill, S. J. (1990) Chemistry of Biological Books, Mill Valley, CA.
A. V. (1910)
J. Physiol.
in Binding and Linkage. FuncMacromolecules, University
(London)
40,4-7.
3. Scatchard, G. (1949) Ann. N. Y. Acod. Sci. 61,660-672. 4. Johnson, M. L., and Frasier, S. G. (1985) in Methods in Enzymology (Hirs, C. H. W., and Timasheff, S. N., Eds.), Vol. 11'7, pp. 301-342, Academic Press, San Diego, CA. 5. Johnson,
DISCUSSION
2 Abbreviation
FREIRE
M. L. (1983)
6. Fujita-Yamaguchi,
Biophys.
Y., and
J. 44,101-106.
Harmon,
J. T. (1988)
Biochemistry
27,3252-3259. 7. Senear, 6577. 8. Schon,
D. F., and Ackers, A., and Freire,
G. K. (1990)
E. (1989)
Biochemistry
Biochemistry
29,
656%
28,5019-5024.
9. McKinnon, I. R., Fall, L., Parody, A., and Gill, D. J. (1984) Anal. Biochem. 139,134-139. 10. Wiseman, T., Williston, S., Brandts, J. F., and Lin, L.-N. ( 1989) Anal. Biochem. 179, 131-135. 11. Freire, E., Mayorga, 0. L., and Straume, M. (1990) Anal. Chem.,
62,95OA-959A.
