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On the application of Job's method of continuous variation to the stoichiometry of protein-ligand complexes
ANALYTICAL
BIOCHEMISTRY
68, 660-663
(
1975)
On the Application
of Job’s
Continuous Variation of Protein-Ligand
Method
of
to the Stoichiometry Complexes
Job’s method of continuous variation is a technique for determining the stoichiometry of molecular complexes (l-3). Originally developed for metal-ligand complexes, the method has more recently received scattered application in the biochemical literature for the determination of protein-ligand complexes (4-9). It offers the potential advantage of being able to determine the stoichiometry of complexation in a single experiment without actually determining the absolute concentration of bound ligand. However, certain conditions must be satisfied in order to avoid erroneous results, a fact that is not uniformly appreciated. The purpose of this communication is to point out the limitations of the method and the conditions under which it might be successfully applied. Consider a metal ion (M) in equilibrium with a ligand (L) according to the following equation: mM + nL ti M,L,.
Job’s method involves the preparation of a series of solutions in which the sum of the total molar concentration of M and L is constant while the ligand mole fraction X is varied from O-l. The concentration of the complex, or a quantity proportional to it, is then measured and plotted against X. Job (1) showed that, if certain assumptions are valid, such a plot will exhibit a maximum at a value of X = n/( m + n) . Some of the more important assumptions of the method are: 1) Neither M nor L selfassociate, 2) the law of mass action is obeyed, and 3) only one complex is formed, i.e., the concentration of intermediate species is negligible. Of these assumptions, the latter is most restrictive, especially when the method is applied to protein-ligand complexes, where it is equivalent to assuming that the ligands are bound with infinite cooperativity. Since many proteins, especially enzymes, have been shown to possess multiple independent and equivalent sites, the latter assumption is often unjustified. However, the method can still be useful when certain conditions are met. Consider the case of a protein, P, in equilibrium with n ligands which bind independently with dissociation constants Kd which are identical except for statistical factors. The equilibrium expression for such a case is given by 660 copyright AU rights
@ 1975 by Academic Press, Inc. of reproduction in any form reserved.
SHORT
661
COMMUNICATIONS
[II where P, and L, are total protein and ligand concentrations, and Lb is the concentration of bound ligand. An experiment is performed in which P, and L, are varied while their sum, A0 = PO + L,, is kept constant. The quantities P,, and L, can then be expressed in terms of A, and the ligand mole fraction X, such that L, = XAO and P,, = (1 - X)Ao. If one further defines the ratio R = A,,/Kd, Eq. [ 11 can then be solved for Lb in terms of n, R, and X. Taking the first derivative and equating to zero yields (I + n)‘RX’
- 3(n”R
+ nR + n - 1)X + n”R + tz - 1 = 0,
[I]
a quadratic in X, one of whose roots corresponds to the Job-plot maximum. Note that the position of the maximum depends not only on II but on the concentration through the quantity R. This is illustrated in Table 1 where the R dependence of the Job-plot maximum is tabulated for various values of II. For the special case where n = 1, the maximum occurs at X = 0.5 regardless of the value of R. However, at low values of R, it is difficult to distinguish between II = 1 and higher values of n. As R increases, this discrimination improves, and, at infinite concentration, the maxima approach the positions expected from the classical application of Job’s method, i.e., X,,, = l/z, %, 3/4 etc. The increased ability to distinguish between various values of II at higher values of R is illustrated in Fig. 1. It can be seen that increasing the value of R not only causes a greater separation between the theoretical maxima but also increases the accuracy with which the position of the maximum can be determined. In Fig. 1, the curves have been arbitrarily normalized to the same height since many measurements give not the absolute concentration of bound ligand but a quantity proportional to L,. The shaded areas reflect the uncertainty produced by a t 10% error in PO. TABLE DEPENDENCE
OF THE
R
POSITION
1
OF JOB’S-PLOT
MAXIMA
UPON
CONCENTRATION”
1
2
3
4
0.500 0.500
0.511 0.555
10 100 5
0.500 0.500 0.500
0.616 0.649 0.666
0.521 0.595 0.682 0.726 0.750
0.530 0.626 0.726 0.774 0.800
a The numbers give the A,IKd = (L, e P,)IK,.
value
0.1 1.0
of X,,,
for
different
values
of n as a function
of R =
662
SHORT
z” 9 i-
COMMUNICATIONS
R=IO N=1.2.4
tt
R=IO N=l.2,4
I
1
I
1
FIG. I. Theoretical Job plots illustrating the effect of concentration on the sharpness as well as the position (arrows) of the maxima. The quantity R is the ratio of the total concentration (ligand + protein) to the dissociation constant, assuming n equivalent and independent sites. The shaded area illustrates the effect of 2 10% error in protein concentration.
In conclusion, the usual application of Job’s method to simple protein-ligand equilibria gives an unambiguous estimate of the stoichiometry only when the sum of protein plus ligand concentrations is kept large relative to the dissociation constant. It should be emphasized that the above treatment applies only to protein-ligand systems in which all the sites are equivalent and independent. If heterogeneity or protein selfassociation occurs, Job’s method ceases to be appropriate.
SHORT
663
COMMUNICATIONS
Summary. The application of the method of continuous variation to protein-ligand complexes is critically examined for the case of a nonassociating protein which has n equivalent and independent ligand-binding sites. It is shown that in order to obtain a reliable estimate of the stoichiometry, the sum of protein and ligand concentrations must be kept large relative to the dissociation constant. REFERENCES I, Job, P. (1928) Ann. Chem. (Paris) 9, 1 13. 2. Rossotti, F. J. C., and Rossotti, H. (1961) in The Determination of Stability Constants and Other Equilibrium Constants in Solutions, McGraw-Hill, New York. 3. Meites, L., and Thomas, H. C. (1958) Advanced Analytical Chemistry, McGraw-Hill, New York. 4. Hammes, G. G., Porter, R. W., and Wu, Cheng-Wen (1970). Biochemistry 9, 2992. 5. Mooser, G., Schulman, H., and Sigman, D. S. (I 972) Biochemistry 11, 1595. 6. D-Anna, J. A., and Isenberg, 1. i 1973) Biochemistry 12, 1035. 7. Kosakowski, H. M., and Holler. E. (I 973) Eur. J. Biochem. 38, 274. 8. Dunn, M. F., and Hutchison, J. S., ( 1973) Biochemistry 12, 4882. 9. Guttenplan, J. B., and Calvin, M. (1973) Biochim. Biophys. Acta 322, 301.
K. C. Clinical Endocrinology Branch National Institute of Arthritis, Metabolism. National Institutes qf Health Bethesda, Maryland 20014 Received March 12, 1975; accepted May
and Digestive
2. I975
Diseases
INGHAM
BIOCHEMISTRY
68, 660-663
(
1975)
On the Application
of Job’s
Continuous Variation of Protein-Ligand
Method
of
to the Stoichiometry Complexes
Job’s method of continuous variation is a technique for determining the stoichiometry of molecular complexes (l-3). Originally developed for metal-ligand complexes, the method has more recently received scattered application in the biochemical literature for the determination of protein-ligand complexes (4-9). It offers the potential advantage of being able to determine the stoichiometry of complexation in a single experiment without actually determining the absolute concentration of bound ligand. However, certain conditions must be satisfied in order to avoid erroneous results, a fact that is not uniformly appreciated. The purpose of this communication is to point out the limitations of the method and the conditions under which it might be successfully applied. Consider a metal ion (M) in equilibrium with a ligand (L) according to the following equation: mM + nL ti M,L,.
Job’s method involves the preparation of a series of solutions in which the sum of the total molar concentration of M and L is constant while the ligand mole fraction X is varied from O-l. The concentration of the complex, or a quantity proportional to it, is then measured and plotted against X. Job (1) showed that, if certain assumptions are valid, such a plot will exhibit a maximum at a value of X = n/( m + n) . Some of the more important assumptions of the method are: 1) Neither M nor L selfassociate, 2) the law of mass action is obeyed, and 3) only one complex is formed, i.e., the concentration of intermediate species is negligible. Of these assumptions, the latter is most restrictive, especially when the method is applied to protein-ligand complexes, where it is equivalent to assuming that the ligands are bound with infinite cooperativity. Since many proteins, especially enzymes, have been shown to possess multiple independent and equivalent sites, the latter assumption is often unjustified. However, the method can still be useful when certain conditions are met. Consider the case of a protein, P, in equilibrium with n ligands which bind independently with dissociation constants Kd which are identical except for statistical factors. The equilibrium expression for such a case is given by 660 copyright AU rights
@ 1975 by Academic Press, Inc. of reproduction in any form reserved.
SHORT
661
COMMUNICATIONS
[II where P, and L, are total protein and ligand concentrations, and Lb is the concentration of bound ligand. An experiment is performed in which P, and L, are varied while their sum, A0 = PO + L,, is kept constant. The quantities P,, and L, can then be expressed in terms of A, and the ligand mole fraction X, such that L, = XAO and P,, = (1 - X)Ao. If one further defines the ratio R = A,,/Kd, Eq. [ 11 can then be solved for Lb in terms of n, R, and X. Taking the first derivative and equating to zero yields (I + n)‘RX’
- 3(n”R
+ nR + n - 1)X + n”R + tz - 1 = 0,
[I]
a quadratic in X, one of whose roots corresponds to the Job-plot maximum. Note that the position of the maximum depends not only on II but on the concentration through the quantity R. This is illustrated in Table 1 where the R dependence of the Job-plot maximum is tabulated for various values of II. For the special case where n = 1, the maximum occurs at X = 0.5 regardless of the value of R. However, at low values of R, it is difficult to distinguish between II = 1 and higher values of n. As R increases, this discrimination improves, and, at infinite concentration, the maxima approach the positions expected from the classical application of Job’s method, i.e., X,,, = l/z, %, 3/4 etc. The increased ability to distinguish between various values of II at higher values of R is illustrated in Fig. 1. It can be seen that increasing the value of R not only causes a greater separation between the theoretical maxima but also increases the accuracy with which the position of the maximum can be determined. In Fig. 1, the curves have been arbitrarily normalized to the same height since many measurements give not the absolute concentration of bound ligand but a quantity proportional to L,. The shaded areas reflect the uncertainty produced by a t 10% error in PO. TABLE DEPENDENCE
OF THE
R
POSITION
1
OF JOB’S-PLOT
MAXIMA
UPON
CONCENTRATION”
1
2
3
4
0.500 0.500
0.511 0.555
10 100 5
0.500 0.500 0.500
0.616 0.649 0.666
0.521 0.595 0.682 0.726 0.750
0.530 0.626 0.726 0.774 0.800
a The numbers give the A,IKd = (L, e P,)IK,.
value
0.1 1.0
of X,,,
for
different
values
of n as a function
of R =
662
SHORT
z” 9 i-
COMMUNICATIONS
R=IO N=1.2.4
tt
R=IO N=l.2,4
I
1
I
1
FIG. I. Theoretical Job plots illustrating the effect of concentration on the sharpness as well as the position (arrows) of the maxima. The quantity R is the ratio of the total concentration (ligand + protein) to the dissociation constant, assuming n equivalent and independent sites. The shaded area illustrates the effect of 2 10% error in protein concentration.
In conclusion, the usual application of Job’s method to simple protein-ligand equilibria gives an unambiguous estimate of the stoichiometry only when the sum of protein plus ligand concentrations is kept large relative to the dissociation constant. It should be emphasized that the above treatment applies only to protein-ligand systems in which all the sites are equivalent and independent. If heterogeneity or protein selfassociation occurs, Job’s method ceases to be appropriate.
SHORT
663
COMMUNICATIONS
Summary. The application of the method of continuous variation to protein-ligand complexes is critically examined for the case of a nonassociating protein which has n equivalent and independent ligand-binding sites. It is shown that in order to obtain a reliable estimate of the stoichiometry, the sum of protein and ligand concentrations must be kept large relative to the dissociation constant. REFERENCES I, Job, P. (1928) Ann. Chem. (Paris) 9, 1 13. 2. Rossotti, F. J. C., and Rossotti, H. (1961) in The Determination of Stability Constants and Other Equilibrium Constants in Solutions, McGraw-Hill, New York. 3. Meites, L., and Thomas, H. C. (1958) Advanced Analytical Chemistry, McGraw-Hill, New York. 4. Hammes, G. G., Porter, R. W., and Wu, Cheng-Wen (1970). Biochemistry 9, 2992. 5. Mooser, G., Schulman, H., and Sigman, D. S. (I 972) Biochemistry 11, 1595. 6. D-Anna, J. A., and Isenberg, 1. i 1973) Biochemistry 12, 1035. 7. Kosakowski, H. M., and Holler. E. (I 973) Eur. J. Biochem. 38, 274. 8. Dunn, M. F., and Hutchison, J. S., ( 1973) Biochemistry 12, 4882. 9. Guttenplan, J. B., and Calvin, M. (1973) Biochim. Biophys. Acta 322, 301.
K. C. Clinical Endocrinology Branch National Institute of Arthritis, Metabolism. National Institutes qf Health Bethesda, Maryland 20014 Received March 12, 1975; accepted May
and Digestive
2. I975
Diseases
INGHAM