ANALYTICAL
BIOCHEMISTRY
Boundary
90,
844-848 (1978)
Conditions
for the Hildebrand-Benesi
Equation
In a recent series of investigations focused on the thermodynamics of cycloamylose-substrate binding (l-4), we had the occasion to generate binding constants using two separate methods of data analysis, the Hildebrand-Benesi (5) treatment and a computer-facilitated curve fit. The resulting binding constants were found to differ significantly in a few instances. These apparent anomalies led us to reconsider the boundary conditions required for the Hildebrand-Benesi analysis of binding constant data. There are three forms of an equation originally used by Hildebrand and Benesi (5) to simultaneously determine the binding constant and extinction coefficient of a 1: 1 complex by spectrophotometric methods. In the derivation of these equations the linearization of a quadratic expression imposes certain restrictions on the experimental conditions. If these boundary conditions are not properly applied, large errors can result in the determination of binding constants. This note focuses on such systematic errors as applies to one of these equations. Assume a solution containing two species, A and B, in equilibrium with their 1: 1 complex, AB. If at a given wavelength the extinction coefficient ofAB is significantly different from the sum of the extinction coefficients of A and B, then the following three equations can be employed:’
[Blo _ AAbs
I
[A],,*AE
[AldBlo _ AAbs
Kdiss
Kdiss
I
1
([Al, + [Blo)
AC
[Alo
AE
([Al, + [Blo)
A.E
IA ldB10 =-+Kdiss [Al, + [Blo AAbs
AE
AE
The first two of these equations
Ktiiss=
[Ref. WI.
[II
[Ref. (6,711. PI [Ref. (lo)].
[3]
assume that
[Ald[Blo - [ABI) LABI ’
which is satisfied when [A] Z+[AB] or [A], b [B],. The last of these ’ (AAbs = Abs bound) concentrations
ca[A],, - E,,[L&,). (AE = 0.999999 for all simulated plots contributing to curve F) as well as adherence to the condition that [A],[B], % JABI (satisfied whenever [A], 9 [B], or [B], S [A],). This goes
SHORT
TABLE MOLAR
RATIOS
OF [A],,I[B],,
RESPECTIVE
CURVES ERROR
A
[Al,,M-%
0.050
mean
K,,s,m, = 3 K,,i,,@% = 10 ” See Fig.
B
AB EQUILIBRIUM
SYSTEM
FOR THY
SELECTED VALUES OF PERCENTAGE FOR EAT H CURVE.”
C
D
E
F
0.250 0.400
0.500
1.oo
5.00
0.080
0.800
1.60
8.00
50 80
0.100 0.125
0.500 0.625
1.00 1.25
2.00 2.50
10.00 12.5
100 125
0.200
1.00
2.00
4.00
20.0
200
0.100
0.500
1.00
2.00
10.0
100
A
B
C
D
E
F
Percentage error in calculated Kctisq values
= 30
F AND
of
[Al,M4,,
Kdisci[Bla
I
A + B :
IN JY,,~,, (CAL 10, which is quite different from the conventional condition that [A]o[B]o S [Al?]‘. Therefore, the number of instances in which the condition [A],[B], % [AB]’ alone is mentioned (8,9) should be cause for concern. Of greater concern, however, is the misuse of Eq. [3] either by failure to adhere to the condition K,iss/[B], > 10 (12,13) or by failure to report all concentrations used in cases where Eq. [3] is applied ( 14). In view of the great deal of attention that has been paid to the results of random errors and their effects on binding constants determined from
848
SHORT COMMUNICATIONS
linear plots (14- 16), it would be a shame for a source of serious systematic errors to go unnoticed. ACKNOWLEDGMENTS Acknowledgment is made to the Alfred P. Sloan Foundation for support of this research. We also wish to acknowledge Dr. Peter McPhie of the National Institutes of Health for his helpful suggestions.
REFERENCES I. Bergeron. R. J.. Channing, M. A., Gibeily. G. J., and Pillor, D. M. (1977)5. Amer. Chcm. SW. 99, 5146. 2. Bergeron, R. J.. and Channing, M. (1976) Bioorg. Chum. 5, 289. 3. Bergeron. R. J.. and McPhie. P. (1977) Bioorg. Chem. 6, 465-471. 4. Bergeron. R. J., and Rowan. R. (1976) Bioorg. Chem. 5, 290. 5. Hildebrand, J. A.. and Benesi. H. A. (1949) J. Amer. Chem. SW. 71, 2703. 6. McConnel, H.. and Davidson, N. (1950) J. Amer. Chem. Sot. 72, 3164. 7. Scott. R. L. (1956) Rec. Truv. Chim. Pays-Bus 75, 787. 8. Biggs, A. I., Patton, H. N.. and Robinson. R. A. ( 1955) .I. Amer. Chem. Sot. 77, 5844. 9. Kustin, K. (1970) Inor,q. Chem. 9, 1536. IO. Davis, W. H.. and Pryor, W. A. (1976) J. Chem. Ed. 53, 285. I I. Cohen, S. R., and Plane, R. A. (1957) J. Phys. Chem. 61, 1096. 12. Kloosterboer, J. G. (1975) Inorg. Chem. 14. 536. 13. Maggio, F., Romano, V., and Cefalu, R. (1966) J. Inorg. Nucl. Chrm. 28, 1979. 14. Person, W. B. (1965) J. Amer. Ckem. Sot. 82, 167. 15. Deranleau. D. A. (1969) J. Amer. Chem. SW. 91, 4044. 16. Cole, S. J.. Curthoys. G. C.. and Magnusson, E. A. (1970)5. Amer. Chem. Sot. 92,299l.
RAYMOND J. BERGERON WILLIAM P. ROBERTS Department of Chemistry University of Marylund College Park. Maryland 20742 Received May 15. 1978
BIOCHEMISTRY
Boundary
90,
844-848 (1978)
Conditions
for the Hildebrand-Benesi
Equation
In a recent series of investigations focused on the thermodynamics of cycloamylose-substrate binding (l-4), we had the occasion to generate binding constants using two separate methods of data analysis, the Hildebrand-Benesi (5) treatment and a computer-facilitated curve fit. The resulting binding constants were found to differ significantly in a few instances. These apparent anomalies led us to reconsider the boundary conditions required for the Hildebrand-Benesi analysis of binding constant data. There are three forms of an equation originally used by Hildebrand and Benesi (5) to simultaneously determine the binding constant and extinction coefficient of a 1: 1 complex by spectrophotometric methods. In the derivation of these equations the linearization of a quadratic expression imposes certain restrictions on the experimental conditions. If these boundary conditions are not properly applied, large errors can result in the determination of binding constants. This note focuses on such systematic errors as applies to one of these equations. Assume a solution containing two species, A and B, in equilibrium with their 1: 1 complex, AB. If at a given wavelength the extinction coefficient ofAB is significantly different from the sum of the extinction coefficients of A and B, then the following three equations can be employed:’
[Blo _ AAbs
I
[A],,*AE
[AldBlo _ AAbs
Kdiss
Kdiss
I
1
([Al, + [Blo)
AC
[Alo
AE
([Al, + [Blo)
A.E
IA ldB10 =-+Kdiss [Al, + [Blo AAbs
AE
AE
The first two of these equations
Ktiiss=
[Ref. WI.
[II
[Ref. (6,711. PI [Ref. (lo)].
[3]
assume that
[Ald[Blo - [ABI) LABI ’
which is satisfied when [A] Z+[AB] or [A], b [B],. The last of these ’ (AAbs = Abs bound) concentrations
ca[A],, - E,,[L&,). (AE = 0.999999 for all simulated plots contributing to curve F) as well as adherence to the condition that [A],[B], % JABI (satisfied whenever [A], 9 [B], or [B], S [A],). This goes
SHORT
TABLE MOLAR
RATIOS
OF [A],,I[B],,
RESPECTIVE
CURVES ERROR
A
[Al,,M-%
0.050
mean
K,,s,m, = 3 K,,i,,@% = 10 ” See Fig.
B
AB EQUILIBRIUM
SYSTEM
FOR THY
SELECTED VALUES OF PERCENTAGE FOR EAT H CURVE.”
C
D
E
F
0.250 0.400
0.500
1.oo
5.00
0.080
0.800
1.60
8.00
50 80
0.100 0.125
0.500 0.625
1.00 1.25
2.00 2.50
10.00 12.5
100 125
0.200
1.00
2.00
4.00
20.0
200
0.100
0.500
1.00
2.00
10.0
100
A
B
C
D
E
F
Percentage error in calculated Kctisq values
= 30
F AND
of
[Al,M4,,
Kdisci[Bla
I
A + B :
IN JY,,~,, (CAL 10, which is quite different from the conventional condition that [A]o[B]o S [Al?]‘. Therefore, the number of instances in which the condition [A],[B], % [AB]’ alone is mentioned (8,9) should be cause for concern. Of greater concern, however, is the misuse of Eq. [3] either by failure to adhere to the condition K,iss/[B], > 10 (12,13) or by failure to report all concentrations used in cases where Eq. [3] is applied ( 14). In view of the great deal of attention that has been paid to the results of random errors and their effects on binding constants determined from
848
SHORT COMMUNICATIONS
linear plots (14- 16), it would be a shame for a source of serious systematic errors to go unnoticed. ACKNOWLEDGMENTS Acknowledgment is made to the Alfred P. Sloan Foundation for support of this research. We also wish to acknowledge Dr. Peter McPhie of the National Institutes of Health for his helpful suggestions.
REFERENCES I. Bergeron. R. J.. Channing, M. A., Gibeily. G. J., and Pillor, D. M. (1977)5. Amer. Chcm. SW. 99, 5146. 2. Bergeron, R. J.. and Channing, M. (1976) Bioorg. Chum. 5, 289. 3. Bergeron. R. J.. and McPhie. P. (1977) Bioorg. Chem. 6, 465-471. 4. Bergeron. R. J., and Rowan. R. (1976) Bioorg. Chem. 5, 290. 5. Hildebrand, J. A.. and Benesi. H. A. (1949) J. Amer. Chem. SW. 71, 2703. 6. McConnel, H.. and Davidson, N. (1950) J. Amer. Chem. Sot. 72, 3164. 7. Scott. R. L. (1956) Rec. Truv. Chim. Pays-Bus 75, 787. 8. Biggs, A. I., Patton, H. N.. and Robinson. R. A. ( 1955) .I. Amer. Chem. Sot. 77, 5844. 9. Kustin, K. (1970) Inor,q. Chem. 9, 1536. IO. Davis, W. H.. and Pryor, W. A. (1976) J. Chem. Ed. 53, 285. I I. Cohen, S. R., and Plane, R. A. (1957) J. Phys. Chem. 61, 1096. 12. Kloosterboer, J. G. (1975) Inorg. Chem. 14. 536. 13. Maggio, F., Romano, V., and Cefalu, R. (1966) J. Inorg. Nucl. Chrm. 28, 1979. 14. Person, W. B. (1965) J. Amer. Ckem. Sot. 82, 167. 15. Deranleau. D. A. (1969) J. Amer. Chem. SW. 91, 4044. 16. Cole, S. J.. Curthoys. G. C.. and Magnusson, E. A. (1970)5. Amer. Chem. Sot. 92,299l.
RAYMOND J. BERGERON WILLIAM P. ROBERTS Department of Chemistry University of Marylund College Park. Maryland 20742 Received May 15. 1978
