Spectroscopic Determination of Protein-Ligand Binding Constants Aydin Orstan Department of Biochemistry, Mount Sinai School of Medicine, New York, NY 10029 John F. Wojcik Department of Chemistry, Villanova University, Villanova, PA 19085
The study of protein-ligand interactions is an important research area aimed a t understanding protein structure and function. Spectroscopy is commonly used to study proteinligand interactions and one application of it is in the determination of hinding constants of ligands to proteins. The theory and practice of the determination of hinding constants do not, however, receive enough attention in most nhvsical chemistrv and hiochemistrv courses. This paper willprovide a brief th&etical background for the s~ectrosconictreatment of wrotein-liaand interactions and procedure to determine binding constants using ahsorption or fluorescence measurements.
a
Proleln-Ligand Multiple Equilibria
briefly discuss two formulations that are used to treat multiple protein-ligand equilibria (I). The first formulation defines a microscopic hinding constant for each hinding site on the protein. In the general case the hinding sites are interdependent, that is, the affinity of a hinding site for a specific ligand, represented by a microscopic binding constant, depends on whether or not the other sites are occuwied. As an examwle. consider a nrotein with two hinding sites, A and B, as shb& in Figure i.If site B is empty the microscopic hinding constant of site A is kl;if B is occupied the binding constant becomes k4. Similarly, the microscopic binding constants of site B are kz and ks when site A is empty and occupied, respectively. The equilibria and the microscopic hinding constants are
In general, a protein molecule will have more than one hinding site for a specific ligand. Since the ligands will bind to the protein in a stepwise manner, there will he multiple equilibria between the protein and ligand molecules. We will
where P and L stand for free protein and ligand, respectively. The subscript of L refers to the site of hinding. Note that if the hinding sites are independent of each other, then k , = kp and k2 = kg. The second formulation uses the total concentrations of protein molecules with 1, 2,. . . , n ligands hound to them and defines stoichiometric binding constants. For example, the concentration of protein molecules with one ligand hound, (PL), is equal to (PL. + PLb). The example in Figure 2 is represented as
where K1 and K2 are stoichiometric binding constants. By comhining eqs 1-6 we get K,
Figure 1. Schematic representation of a protein molecule wim two imerdependent binding sites. A and 0. The binding of ligand molecules (omitted from the fioure for claritvl to the oratein is exoressed wino microscooic bindino canslanls (ha) Snaaea squares mdicale the accup ed o namg sates The wide oi the hlnd ng conslanl oelreen a band ng sae and a spec f c llgand depends on whether or not the other site is occupied.
814
Journal of Chemical Education
= k,
+ k,
(7)
As discussed in detail by Klotz (21, analysis of ligand binding data is usually easier using stoichiometric constants than with microscopic constants. One reason is that the number of stoichiometric constants needed to define a protein-ligand system is smaller thah the number of microscop-
ic constants. For example, for a protein with n interdependent bindine sites there are (2"-'n) different microsco~ic constants b i t only n stoichioketric'constants ( I ) . We will therefore develon method to determine stoi. a s~ectrosco~ic . chiometric binding constants. Spectroscoplc Treatment ot Blndlng Equlllbrla According to Beer's law, total light ahsorbed, At, by a solution containing more than one absorbing species is
where ti and C; stand for the extinction coefficient and concentration of the ith species, respectively, and 1 is the path length. Likewise, the total fluorescence intensity, Ft, of a solution containing more than one fluorescent species is
Here IQ denotes the intensity of the incident light and q5; is the auantum vield of the ith snecies. Eauation 10 is an approximatioivalid if the absorption of t6e solution at the excitation waveleneth is less than about 0.05 (3). Flnorescence is emitted o v k a range of wavelengths in all directions. During an experiment, however, only a fraction, represented by f in eq 10, of the total light emitted is measured. To keep f constant, the fluorescence of each sample must be measured using the same instrumental geometj. Equations 9 and 10 can be combined into a general equation,
where Vt represents either the total absorbance or the total fluorescence. Since the intensity of absorption or fluorescence denends on the value of a. we will call a the intensity eoeffikent. For absorbance a = cjl with the units 1iter.mole-1. For fluorescence a = fI&rjl, where @i is unitless
and IQ is usually expressed as quanta per square centimeter per second. Therefore, a has the units of quanta-liter. second-kentimeter-2mole-'. If the absorption or fluorescence of a protein-ligand solution is measured at a wavelength where only the free and bound ligand molecules absorb or fluoresce, Vt will be where the subscripts f and p refer to free and bound ligands, respectively. As a model we will start with the system given in Figure 1but our final equations will be applicable to the stoichiometric system in Figure 2. Since the microscopic binding constant of aparticular binding site and the intensity coefficient of a ligand in that site both depend on the structure of the site, we will assign a microscopic intensity coefficient, a', to each microscopic binding constant, k. The total ligand concentration on protein molecules with both sites occupied is Z(PL.Lb), and the ligand concentration in each site is 2(PL&b)/2 = (PL,Lb). Expansion of eq 12 gives where ar is the intensity coefficient of free ligand, L. We will rearrange eq 13 so that it can be used to calculate the stoichiometric binding constants. Using the relationship, (PL,Lb) = PL2 and eqs 1,2,5, and 7, eq 13 is rearranged to
The constant terms in eq 14 have the same units as the intensity coefficients in eq 11; therefore, they can be replaced by a new set of intensity coefficients to get
V, = afL + a,(PL) + az(PL,)
+
(15)
+
+
where al = (or;kl a;kz)l(kl k2) and a 2 = (a; a;). For a protein with n mterdependent binding sites eq 15 becomes
In the special case when becomes
d, = a,
=
&, =
a&= a,, eq 14
The term in brackets gives the total bound ligand concentration. In the literature absorption or fluorescence of proteinligand solutions are usually expressed in the form of eq 17, without any explanation or justification. (For a recent example, see the paper by Marty e t al. in this Journal (4)J In the absence of any evidence indicating that all the a's are the same, this practice should he avoided, however, and eq 16 should instead be used. A Spectroscoplc Method To Determine K,
For a complete characterization of a protein-linand system it is necessary todetermine the totalnurnber~fbinding sites on the protein, n , and the values of all or most of the binding constants. For most educational purposes, however, it is enough to determine only the first binding constant, K,. We will now discuss a simple npectn~scopicmethod to determine K1. If the total nrotein concentration. P,.is lareer than the total ligand concentration, Lt, then one can asszme that the concentrations of nrotein molecules with more than one ligand hound will b; negligible. Under these conditions total concentrations can be expressed as Figure 2. Treatment of the protein-ligand system given in Figure 1 using stoichiometric binding constants (K& The individual binding sites are ignored and the first stoichiometric binding constant. K,, is expressed using the total Concentration of protein molecules d m one llgand bound.
P, = P
+ (PL) + (PL,) + ... + (PL,,)
and L, = L
-
+ (PL) + 2(PL,) + ... + tt(PL,)
Volume 64
Number 9
P + (PL) L + (PL)
September 1987
(18)
(19) 815
Using eqs 18 and 19 Kl and Vl become
If the absorbance or fluorescence of a series of solutions with the same concentration of lieand and increasine concentrations of protein are measure;, then the two unkgowns in eas 20 and 21. K, and a,. can be calculated usine a curvesolving fitting procedure. ~ o t that e (PL) can be ohtained iy eq 20 for a given value of K1 and ar can be calculated by dividing the Vl value of a ligand solution with no protein by the total ligand concentration. If fluorescence is used, the emission intensities of all the solutions must be measured relative to each other. T o calculate these unknowns we bave used a so-called "brute-force" least-squares curve-fitting procedure (5). Our computer program calculates the values of (PL) and Vt a t each protein concentration using eqs 20 and 21, respectively, for a large number of combinations of K1 and a1 in a given range of K1 and ul values. (The ranges of K1 and ul can be set by trial and error.) Agreement between the calculated and measured Vl values are judged (5) by calculating x2, x2 = Z: [V,(ealc) - V,(meas)12
(22)
The values of K1 and al that give the smallest x2 are chosen as the experimental results. We have used this procedure with absorption measurements to determine K1 for the hinding of methyl orange (MO) to bovine serum albumin (BSA). Experimental Experiments were done in phosphate buffer, pH 7.2,1 = 0.15, at 22 + 0.5 'C. At this pH, MO is practically 100% ionized. (If the ligand exists in more than one ionic form the hinding equations will be more complicated since each ionic form is likely to have a different affinity for the protein.) Stock solutions of BSA and the sodium salt of MO were prepared in water. BSA stock was filtered through a fine sintered-glass filter. Since the BSA powder contained about 8%water, concentration of the stock was determined from the ahsorhances of dilutions at 279 nm. An extinction coefficient of 44,162 Lmol-'ax', calculated from an A& of 6.67 and amolecular weight of 66,210 (6),was used. For absorption measurements six solutions containing 5.7 X 10-6 M MO were prepared. Five of the solutions also contained BSA in a range from 8.0 X 10W to 4.0 X M. The lowest BSA concentration was higher than the MO concentration so that eqs 18 and 19 would be valid. Absorption spectrum of each solution was recorded on a Cary 219 speetrophatometerusing 2-cm cuvettes. To correct for the turhidity of the solutions containing protein, blank solutions containing only BSA were used. The largest change in the absorbance of MO as a result of hinding to BSA was at about 490 nm. K1 was calculated at 472 and 490 nm, using the curve-fittingprocedure explained above and an average value of 5.25 X l(r M-' was ahtained. In comparison,using an equilibrium dialysis method, Klotz etal. obtained avalue of4.9 X 10%-' for K1at 5 "C,pH 5.7 (7).Atitration curve, drawn using eq 21 with the values of K1 and a, obtained at 490 nm, is shown in Figure 3. In certain cases, when nz >> a,,eq 21 may not be valid even if the concentration of (PL2)is much smaller than that of (PL). In the case of BSA and MO, however, the
816
Journal of Chemical Educatlon
Figure 3. A titration curve far the binding of MO to BSA. Absorbance differences (AA per cm at 490 nm) induced by the binding of MO to BSA are plotted against the logarithm of the total BSA concentration. Equation 21 was rearranged to express the absorbance differences, mat is. A V = (4 a,&)= (PLKa, - a,). The curve was constructed using mis equation wim K, = 5.38 X 10' M-' and (a, - ad = -7775 L.moi-' obtained from the data at 490 nm. Black dots are the experimental points. The dashed line indicates A A that would be observed if all of MO were bound to BSA.
-
agreement between the values of K1 ohtained by us and that reported by Klotz et al. (7) indicates that eq 21 is valid.
Conclusion An undereraduate laboraton, exneriment based on the topics discussed in this paper ;ill i&oduce students t o the theorv and nractice of nrotein-lieand interactions and binding constant rneasurentencs. T~ BSA-MO system, which has been studied in drtail bv Klotz and co-workers 17-9). . .. is especially suitable for such a n experiment. BSA, easily available in pure form, is highly water-soluble and stable against denaturation. Its structure and properties bave been reviewed by Peters (6).In addition, MO, also water-soluble in the form of its sodium salt, apparently is not carcinogenic (10). It should, however, be handled with care. Literature Clted 1. KloU.1. M.Acr. Chem. Re*. 1914.7.162 2. KloU, I. M.Schnee 1982,217, 12k7:
3. Parker, C. A. Photolumin~sconmofSolulions: Elsevier: New York. 1968:p. 20. 1. Marty,A.: Boiref, M.; Deumi6, M . J. Chem. Edue. 1986.63, 365. 5. Etevington, P. R. Dnto Rsduclion ond E ~ m rAnalysis for the Physical Sciancea: McCraw-Hill: New York, 1969:p 206. 6. Pebrs, T . In The Plasma Proteins, 2nded.; Putnsm, F. W., Ed.;Academic: New York.
