Anal. Chem. 1997, 69, 4790-4798
Effects of Ligand Heterogeneity in the Characterization of Affinity Columns by Frontal Analysis Stacey A. Tweed, Bounthon Loun, and David S. Hage*
Department of Chemistry, University of Nebraska, Lincoln, Nebraska 68588-0304
This study examined how the binding capacities and equilibrium constants measured by frontal analysis are affected by ligand heterogeneity in affinity columns. Equations derived for n- and two-site systems gave good agreement with results obtained for the binding of Lthyroxine to a column containing human serum albumin (HSA) and for the binding of (R)-warfarin to coupled columns containing HSA or pigeon serum albumin. The same equations were used to examine how different degrees of ligand heterogeneity affected the apparent binding capacities or equilibrium constants measured using the linear range of double-reciprocal frontal analysis plots. A large proportion of two-site systems gave good estimates (i.e., less than 10-20% error) for the true total column capacity and for the association constant of the highest affinity ligand in the column. A smaller, but still appreciable, fraction of all three- and four-site cases also produced good estimates of these values. The results of this work are not limited to protein-based affinity columns but should be applicable to any type of stationary phase that has well-defined binding regions and relatively fast, reversible interactions with solutes. Frontal analysis is a technique that is commonly used to estimate the capacities and binding strengths of chromatographic columns.1-6 In this method, a solution of analyte is continuously applied to a column while the amount eluting from the end of the column is monitored. As binding sites in the column become saturated, the concentration of analyte eluting from the column gradually increases until it reaches a plateau, forming a characteristic breakthrough curve. By examining the position of this curve as a function of analyte concentration, it is often possible to gain information on both the amount of binding sites that are located in the column and the strength of analyte binding to these sites.1-6 Affinity chromatography is one method in which frontal analysis is often employed for column evaluation. Examples include the use of frontal analysis for binding capacity measure(1) Sebille, B.; Zin, R.; Madjar, C. V.; Thuaud, N.; Tillement, J. P. J. Chromatogr. 1990, 521, 51. (2) Dunn, B. M.; Chaiken, I. M. Proc. Natl. Acad. Sci. U.S.A. 1974, 71, 2382. (3) Loun, B.; Hage, D. S. J. Chromatogr. 1992, 579, 225. (4) Yang, J.; Hage, D. S. J. Chromatogr. 1993, 645, 241. (5) Chaiken, I. M., Ed. Analytical Affinity Chromatography; CRC Press: Boca Raton, FL, 1987. (6) Turkova, J. Affinity Chromatography; Elsevier: Amsterdam, the Netherlands, 1978.
4790 Analytical Chemistry, Vol. 69, No. 23, December 1, 1997
ments or for the study of biological interactions.2-7 The evaluation of frontal analysis results for columns that contain a single type of ligand is straight forward,3,4 but relatively few studies have considered how to process such data in the case where heterogeneous binding sites are present in the column.8-11 This issue is of particular concern when working with affinity columns, where binding site heterogeneity can be the result of several factors. These factors include the natural heterogeneity of the starting ligand (e.g., the use of polyclonal antibodies in an immunoaffinity column) or the creation of ligand heterogeneity due to the immobilization process (e.g., random ligand orientation or multisite attachment).6,11 In some cases, column heterogeneity can also be produced by the presence of both specific binding regions (due to the immobilized ligand) and nonspecific regions (due to solute interactions with the support material).9 This study will specifically examine the effects of ligand heterogeneity on the response expected for frontal analysis when using double-reciprocal plots for data treatment (i.e., graphs that plot the inverse number of moles of solute bound to the column at equilibrium versus the inverse concentration of applied solute). Such plots are commonly used in determining the total binding capacity and/or effective binding strength for affinity columns and have the advantage of yielding a linear relationship for systems with 1:1 solute-ligand interactions.3-7 One goal of this work will be to examine the changes in response for these plots when multiple types of ligands are present in the column. Another goal will be to determine the effect that ligand heterogeneity has on the apparent column capacities or binding affinities that are estimated by this approach. These goals will be met by using a combination of work with chromatographic theory and model solute-ligand systems (e.g., the binding of small drugs and hormones to serum albumins). From the data generated in this study, general guidelines will be developed regarding the role played by ligand heterogeneity in frontal analysis studies of other affinity columns or additional types of liquid chromatographic supports. THEORY During frontal analysis, the fast reversible binding of an analyte (A) to a column that contains a single type of immobilized ligand (L) can be described by the following relationships:3-7 (7) Kasai, K.-I.; Ishii, S.-I. J. Biochem. 1975, 78, 653. (8) Anderson, D. J.; Walters, R. R. J. Chromatogr. 1986, 376, 69. (9) Muller, A. J.; Carr, P. W. J. Chromatogr. 1984, 284, 33. (10) Johnson, R. D.; Arnold, F. H. Biotechnol. Bioeng. 1995, 48, 437. (11) Walters, R. R. Anal. Chem. 1985, 57, 1099A. S0003-2700(97)00565-9 CCC: $14.00
© 1997 American Chemical Society
mLapp ) mLtot(Ka[A])/(1 + Ka[A])
(1)
lim
[A]f0
1/mLapp ) 1/(mLtotKa[A]) + 1/mLtot
(2)
1 1 ) + mLapp mLtot (R1 + β2 - R1β2)Ka1[A] (R1 + β22 - R1β22) mLtot(R1 + β2 - R1β2)2
In eqs 1 and 2, [A] is the molar concentration of analyte applied to the column, Ka is the association equilibrium constant for the binding of A to L, mLtot is the total number of moles of active L present in the column, and mLapp represents the amount of L that is bound to A at the mean position of the resulting breakthrough curve.3-7 In the case of single-site binding, it is relatively easy to use the above equations to determine the equilibrium constant and binding capacity for A on an affinity column. For instance, eq 2 predicts that a plot of 1/mLapp versus 1/[A] should give a linear response with an intercept equal to 1/mLtot and a slope equal to 1/(KamLtot); in other words, the inverse of the intercept provides the total number of moles of active binding sites in the column, and the ratio of the intercept and slope provides the value of Ka for the analyte and ligand.3,4 Expressions analogous to eqs 1 and 2 can also be derived for columns with multiple types of ligands. For example, an expression similar to eq 2 but for a system with two classes of ligands, L1 and L2, is shown in eq 3 (see Appendix for derivation). In eq
1 mLapp
)
Although eq 5 indicates that a plot of 1/mLapp versus 1/[A] at low [A] values should give a linear relationship, there are now more factors that affect the slope and intercept than there are for the one-site case. Like the one-site expression given in eq 2, the intercept of eq 5 depends on mLtot; however, the amount of each ligand and their relative affinities (i.e., R1 and β2) are now important as well. Likewise, the ratio of the intercept to the slope in eq 5 can still provide an apparent association constant for the analyte and ligand, but the value of this constant will also depend on both R1 and β2. Expressions similar to those in eqs 3-5 can be derived for columns with more than two types of binding regions. The results predicted for a column with n types of sites are shown below, where all of the summations and products are over the interval i or j ) 1 to n.
General Expression for All Analyte Concentrations: 1
1 + (β1 + β2)Ka1[A] + β1β2Ka12[A]2 mLtot{(R1β1 + R2β2)Ka1[A] + β1β2Ka12[A]2}
(5)
mLapp
(3)
) mLtot
∏(1 + β K [A]) ∏(1 + β K [A]) K Rβ
[
∑
i
a1 j
i i
a1
(1 + βiKa1[A])
]
(6)
a1[A]
Linear Approximation at Low Analyte Concentrations: 3, Ka1 is the association equilibrium constant for the class of highest affinity sites in the column, and R1 and R2 are the fractions of mLtot due to the high or low affinity sites, respectively (e.g., R1 ) mL1,tot/mLtot, where mL1,tot is the total number of moles of L1 in the column). The terms β1 and β2 represent the ratios for the association constants of each class of sites versus the association constant for the highest affinity region (e.g., β2 ) Ka2/Ka1, where 0 < Ka2 < Ka1). All other terms are the same as in eqs 1 and 2. Note that eq 3 can be simplified by using the fact that β1 must be equal to 1, since β1 ) Ka1/Ka1. Also, since there are only two types of sites in this system, the term R2 can be replaced by (1 R1). Using these facts gives rise to the following alternative form of eq 3:
1 + Ka1[A] + β2Ka1[A] + β2Ka12[A]2
1 mLapp
)
mLtot{(R1 + β2 - R1β2)Ka1[A] + β2Ka1 [A] } 2
2
(4)
It is this form that will be used throughout the rest of this study to examine the effects of binding heterogeneity in a two-site system. Unlike the one-site case represented by eq 2, the two-site case in eqs 3 and 4 would not be expected to produce a linear response over all concentrations of analyte in plots made of 1/mLapp versus 1/[A]. However, it is possible to approach a linear relationship for a multisite case when working at low analyte concentrations (see Results and Discussion for examples). Under such conditions, eq 4 approaches the following form, as derived in the Appendix:
lim
[A]f0
1 + Riβi)Ka1[A]
1 ) mLapp m
∑ (∑β )(∑R β ) - [∑R {β (∑β ) - β m (∑R β ) Ltot(
i
i i
i
i
j
2
Ltot
}]
2
i
(7)
i i
In the two-site case (n ) 2), eqs 6 and 7 reduce to the same expressions as shown in eqs 3-5. For convenience, eqs 6 and 7 have been written to include all β and R terms, but the number of variables in these expressions could be reduced in the same fashion as employed in eq 4 by using the fact that β1 is always equal to 1 and by replacing Rn with the equivalent expression 1 - (R1 + R2 + ... + Rn-1). EXPERIMENTAL SECTION Reagents. The HSA (Cohn fraction V, 96-99%), PSA (Cohn fraction V), and L-thyroxine were from Sigma (St. Louis, MO). The (R)-(+)-warfarin was generously provided by DuPont Pharmaceuticals (Wilmington, DE). Nucleosil Si-1000 (7 µm particle diameter, 1000 Å pore size) was obtained from Alltech (Deerfield, IL). Reagents for the bicinchoninic acid (BCA) protein assay were from Pierce (Rockford, IL). Other chemicals and biochemicals were of the purest grades available. All solutions were prepared using water from a Nanopure water system (Barnstead, Dubuque, IA). Apparatus. The chromatographic system consisted of two CM3000 isocratic pumps and one SM3100 UV/visible variablewavelength absorbance detector from LDC/Milton Roy (Riviera Beach, FL). Solvents from the two pumps were alternatively passed through the test columns by using a Rheodyne 7010 Analytical Chemistry, Vol. 69, No. 23, December 1, 1997
4791
switching valve (Cotati, CA). Data were collected by Winner-onWindows software from Thermoseparations (Fremont, CA). The data processing and numerical calculations were conducted by in-house programs written in Microsoft QuickBASIC (Redmond, WA) using double-precision logic. Columns and mobile phases were temperature-controlled by using an Isotemp 9100 circulating water bath (Fisher Scientific, Pittsburgh, PA). All columns were downward slurry-packed at 4000 psi (28 MPa) by an HPLC column-packer from Alltech. Columns. Diol-bonded silica was prepared from Nucleosil Si1000 according to a previously published procedure.12 The diol coverage of the Nucleosil prior to the attachment of HSA or PSA was 62 ( 2 ((1 SD) µmol/g of silica, as determined in duplicate by a capillary electrophoresis-based periodate oxidation method.13 The HSA and PSA were immobilized onto the diol-bonded silica by using the Schiff base method, as described earlier.3,4 In each case, the final support was centrifuged, washed with 2 N sodium chloride and pH 7.4, 0.067 M phosphate buffer, and stored in the pH 7.4 phosphate buffer at 4 °C until use. A portion of each support was further washed with deionized water, vacuum-dried at room temperature, and assayed in duplicate for protein content by a BCA assay,14 using HSA or PSA as the standard and diolbonded silica as the blank. The protein content measured for the HSA support used in the warfarin studies was 250 ( 20 ((1 SD) nmol/g of silica, the protein content of the PSA support was 200 ( 10 nmol/g of silica, and the content of the HSA support used in the thyroxine work was 196 ( 1 nmol/g of silica. Chromatography. The HSA, PSA, and diol-bonded silica matrices were placed into separate 3.5 cm × 4.2 mm i.d. columns of a previously published design.15 The columns and solvents were kept in a water jacket or water bath for temperature control. Each study was performed at 37 ( 0.1 °C. All mobile phases, analyte solutions, and packing solvents used in this work were prepared using pH 7.4, 0.067 M potassium phosphate buffer, according to previously reported procedures.3,16-18 Prior to use, the mobile phases were filtered through 0.45 µm cellulose acetate filters (Gelman Sciences, Ann Arbor, MI) and degassed under vacuum for 10 min. Elution of (R)-warfarin was detected by monitoring the absorbance of the column eluent at 310 nm; the elution of L-thyroxine was monitored at 300 nm. The concentrations of applied (R)-warfarin ranged from 2 × 10-7 to 4 × 10-5 M, and the concentrations of L-thyroxine varied from 7 × 10-7 to 5 × 10-5 M. The frontal analysis studies that took place when using single HSA or PSA columns were performed in the same format as described previously.3,4,16 A similar scheme was used for the experiments in which the HSA and PSA columns were connected in series. The void volume for each column (or set of columns) was determined by performing frontal analysis experiments with (R)-warfarin and L-thyroxine on 3.5 cm × 4.2 mm i.d. columns that contained diol-bonded Nucleosil Si-1000 with no HSA or PSA present. The breakthrough times obtained in these latter experiments were within (2-5% of the calculated void time of the (12) Ruhn, P. F.; Garver, S.; Hage, D. S. J. Chromatogr. A 1994, 669, 9. (13) Chattopadhyay, A.; Hage, D. S. J. Chromatogr. A 1997, 758, 255. (14) Smith, P. K.; Krohn, R. I.; Hermanson, G. T.; Mallia, A. K.; Gartner, F. H.; Provenzano, M. D.; Fujimoto, E. K.; Goeke, N. M.; Olson, B. J.; Klenk, D. C. Anal. Biochem. 1985, 150, 76. (15) Walters, R. R. Anal. Chem. 1983, 55, 591. (16) Loun, B.; Hage, D. S. Anal. Chem. 1994, 66, 3814. (17) Loun, B.; Hage, D. S. Anal. Chem. 1996, 68, 1218. (18) Loun, B.; Hage, D. S. J. Chromatogr. B 1995, 665, 303.
4792 Analytical Chemistry, Vol. 69, No. 23, December 1, 1997
Figure 1. Double-reciprocal frontal analysis plots for the application of (R)-warfarin to an immobilized HSA column (]), an immobilized PSA column (b), or a series of HSA and PSA columns (9) at pH 7.4 and 37 °C. Other experimental conditions are given in the text.
systems, indicating that little or no detectable nonspecific binding of (R)-warfarin or L-thyroxine was occurring to the supports used in the HSA and PSA columns. The frontal analysis experiments involving (R)-warfarin were performed at calibrated flow rates ranging from 0.15 to 0.25 mL/ min, and the L-thyroxine studies were done over flow rates of 0.10.5 mL/min. It is known from previous rate constant and frontal analysis measurements that the association and dissociation of (R)-warfarin and other agents (e.g., D/L-tryptophan) with immobilized HSA is sufficiently rapid to allow the establishment of a local equilibrium in the column under these flow rate conditions.16,17,19,20 The same situation was also found to be true for the L-thyroxine/HSA and (R)-warfarin/PSA systems, as indicated by the fact that no detectable changes in the number of moles of bound analyte or the shape of the frontal analysis curves were noted under the various flow rate conditions that were tested in this work. RESULTS AND DISCUSSION General Effects of Binding Site Heterogeneity. One of the first items considered in this work was the overall change in frontal analysis results that would be expected to occur in the presence of columns with single versus multiple types of binding sites. Figure 1 shows typical results obtained in double-reciprocal plots for single-site systems, such as the binding of (R)-warfarin to HSA or PSA, where both ligands are known or believed to have 1:1 interactions with this particular solute.21-23 According to eq 2, these systems should show a linear relationship between 1/mLapp and 1/[(R)-warfarin], as demonstrated in Figure 1. However, in heterogeneous systems, very different behavior can be obtained. In some cases, such as the binding of (R)-warfarin to a mixture of HSA and PSA (see Figure 1), linear behavior can still be observed, giving a situation that mimics that of a true one-site case. However, a closer examination of similar curves generated for other multisite systems (e.g., the binding of L-thyroxine to (19) Yang, J.; Hage, D. S. J. Chromatogr. A 1996, 725, 273. (20) Yang, J.; Hage, D. S. J. Chromatogr. A 1997, 755, 15. (21) Sudlow, G.; Birkett, D. J.; Wade, D. N. Mol. Pharmacol. 1976, 12, 1052. (22) Tillement, J. P.; Zini, R.; D’Athis, P.; Vassent, G. Eur. J. Clin. Pharmacol. 1974, 7, 307. (23) McMenamy, R. H.; Seder, R. H. J. Biol. Chem. 1963, 238, 3241.
Figure 2. Double-reciprocal frontal analysis plot for the application of L-thyroxine to an immobilized HSA column at pH 7.4 and 37 °C. Other conditions are given in the Experimental Section.
multiple sites on HSA, as shown in Figure 2),3,24-28 indicates that the response for heterogeneous systems in double-reciprocal plots will give linear regions only at low analyte concentrations, followed by curvature in the results at higher solute levels. This behavior can be explained by the fact that, at low solute concentrations, only the highest affinity sites on the column will tend to bind to solute; but as higher solute concentrations are used, a significant amount of the lower affinity sites will also begin to take part in solute retention. The frontal analysis behavior for heterogeneous systems was examined further by using eq 4 to determine what changes in double-reciprocal plots would be expected for a two-site system when changing either the distribution of binding sites (R1) or the relative size of one association constant versus the other (β2). Examples of the results that were obtained are shown in Figure 3. In each case, normalized, dimensionless plots were used to study the general effects of column heterogeneity by plotting mLtot/ mLapp (instead of 1/mLapp) as a function of 1/Ka1[A] (instead of 1/[A]). When using a value of R1 ) 1.0 or β2 ) 1.0, a homogeneous system was obtained, and the linear behavior predicted by eq 2 was seen. But curvature began to appear at high solute concentrations, or low 1/Ka1[A] values, as either R1 or β2 was decreased (i.e., as a greater degree of heterogeneity was introduced into the column). The data in Figure 3 indicate that every multisite case eventually showed linear behavior once sufficiently low solute concentrations were used (i.e., as 1/ Ka1[A] was increased past a certain minimum value). These trends were the same as those seen for the L-thyroxine/HSA and (R)warfarin/HSA+PSA data in Figures 1 and 2, indicating that there was good qualitative agreement between eq 4 and the results generated for these actual heterogeneous systems. The theoretical data in Figure 3 indicate that column heterogeneity will always lower the apparent number of moles of binding sites that are measured by frontal analysis when working in the linear region of double-reciprocal plots. For example, when the linear range for these plots is used to estimate mLtot, the result will be an estimated intercept for graphs of 1/mLapp versus 1/[A] (24) Tabachnick, M. J. Biol. Chem. 1964, 239, 1242. (25) Steiner, R. F.; Roth, J.; Robbins, J. J. Biol. Chem. 1966, 241, 560. (26) Trisch, G. L.; Rathke, C. E.; Tritsch, N. E.; Weiss, C. M. J. Biol. Chem. 1961, 236, 3163. (27) Sterling, K. J. Clin. Invest. 1964, 43, 1721. (28) Tabachnick, M. J. Biol. Chem. 1967, 242, 1646.
Figure 3. Predicted effects in double-reciprocal frontal analysis plots for a two-site system (a) when changing the value of β2, or the relative size of the association constant for the second ligand versus the highest affinity binding region, and (b) when varying R1, or the relative amount of the highest affinity ligand in the column. The results shown by the solid lines were generated using eq 4; the dashed lines were generated using eq 5.
that is higher than the true intercept value (e.g., compare dotted and solid lines in Figure 3). This, in turn, produces a low apparent value for mLtot. The extent of these deviations will be examined in more detail later over a wide range of heterogeneity indexes. The effect of column heterogeneity on the apparent values measured for Ka from the linear region of double-reciprocal plots will also be examined in a later section. Validation of Equations Used To Describe Multisite Binding. The next item considered was the quantitative fit of experimental data to the expressions developed in this work to describe the linear region of frontal analysis double-reciprocal plots for heterogeneous ligands (see eqs 5 and 7 in the Theory). This was examined by preparing a set of affinity columns that contained two different ligands (HSA and PSA), each of which had a single binding site for a common test analyte ((R)-warfarin) but different association constants for these interactions. As already mentioned, both HSA and PSA have been reported to have only one binding site for (R)-warfarin21-23 but differ by about 5-fold in their affinity for this agent.29 Such properties made these ligands useful as models for studying the effects of column heterogeneity. Two separate columns were initially prepared in this work, one containing only immobilized HSA and the other containing only immobilized PSA. A series of frontal analysis experiments were then performed on each column, and double-reciprocal plots of 1/mLapp versus 1/[warfarin] were made, as shown in Figure 1. Equation 2 was then applied to these plots to evaluate the total number of moles of binding sites and the association equilibrium constant (Ka) for (R)-warfarin on each column. The plots generated for both the PSA and HSA columns were linear over the entire (29) Lagercrantz, C.; Larsson, T.; Denfors, I. Comp. Biochem. Physiol. 1981, 690, 375.
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Table 1. Linear Best-Fit Parameters for Frontal Analysis Studies Performed with (R)-Warfarin on Affinity Columns Containing Only HSA, PSA, or HSA plus PSAa ligand HSA PSA HSA + PSA measd pred
association constant, Ka (M-1)
binding capacity, mLtot (mol)
2.5 (( 0.2) × 105 5.1 (( 0.2) × 104
2.8 (( 0.2) × 10-9 1.6 (( 0.1) × 10-8
1.7 (( 0.2) × 105 2.0 (( 0.2) × 105
0.9 (( 0.2) × 10-8 1.3 (( 0.2) × 10-8
a The numbers in parentheses represent (2 SD. The values given for Ka represent the apparent association equilibrium constants determined from the slopes and intercepts of the plots in Figure 1. The values given for mLtot are the apparent binding capacities measured for the intercepts of the same plots. All of the above measurements were performed at pH 7.4 and 37 °C.
range of conditions studied, with correlation coefficients of 0.9885 and 0.9870 being obtained over six warfarin concentrations and 12-14 determinations. The values for Ka and mLtot determined from these plots are summarized in Table 1. The Ka value of 2.5 × 105 M-1 found for (R)-warfarin on the HSA column agreed well with the association constants of (2.1-3.3) × 105 M-1 that have been reported in previous studies with this system.3,29 The Ka measured on the PSA column was also similar to a value of 6.3 × 104 M-1 that was reported earlier for (R)-warfarin and PSA.29 After the HSA and PSA columns had been characterized separately, they were next placed in series and studied simultaneously by frontal analysis. Since the number of moles of applied analyte that is required in frontal analysis to reach the final breakthrough point is additive in the case of multiple sites (see eq A5), using these two columns in series was mathematically equivalent to preparing a single mixed-bed column that contained both types of ligands. The data obtained from the frontal analysis studies on both columns were then analyzed by making plots of 1/mLapp versus 1/[warfarin] (see Figure 1). Identical results were generated regardless of the order of the HSA and PSA columns. A linear relationship was again obtained over the entire range of concentrations tested, with a correlation coefficient of 0.9911 being found over six warfarin levels and 21 measurements. Such linear behavior was expected because of the dilute analyte concentrations that were being examined in this work; at slightly higher analyte levels (i.e., greater than 5 × 10-5 M warfarin) detectable deviations from a linear response would be expected to appear, as demonstrated by the L-thyroxine/HSA data in Figure 2. By treating the results for the coupled HSA and PSA columns as a pseudo-one-site case, eq 2 was applied to the plot for these coupled columns in Figure 1 in order to determine what apparent values for mLtot and Ka would be obtained for this heterogeneous system (see Table 1 for a summary of the results). Based on the binding capacities and association constants that were measured for the individual HSA and PSA columns, the combined HSA and PSA columns had known values for both R1 (i.e., mLtot,HSA/mLtot ) 0.23) and β2 (i.e., Ka,PSA/Ka,HSA ) 0.20), as well as known quantities for Ka1 and mLtot (i.e., Ka,HSA and mLtot,HSA + mLtot,PSA). With these data, eq 5 was used to predict what slope and intercept should have been obtained in the linear region of double-reciprocal plots for this test system; the same equation was used to predict the apparent values for mLtot and Ka that would be obtained if it was mistakenly assumed that the combined columns actually repre4794
Analytical Chemistry, Vol. 69, No. 23, December 1, 1997
sented a one-site case. It was found that the linear response predicted by eq 5 gave an excellent fit to actual HSA+PSA data (see best-fit line in Figure 1). There was also good agreement between the apparent Ka and mLtot values that were measured on the basis of eq 2 and those that were predicted by eq 5 (see Table 1). This indicated that the model used to derive eq 5 and related multisite expressions gave good quantitative, as well as qualitative, agreement with the frontal analysis behavior seen for heterogeneous ligands in affinity columns. Effects of Multisite Binding on Binding Capacity Measurements. After the validity of eq 5 had been verified, this relationship was used to determine how the experimental binding capacity (mLtot,exp) measured for an affinity column in the linear region of double-reciprocal plots will differ from the true binding capacity (mLtot) when multiple types of ligand sites are present in the column. For a two-site case examined at low analyte concentrations (i.e., conditions that produce a linear relationship between 1/mLapp and 1/[A]), the expected value for mLtot,exp can be obtained by taking the inverse of the intercept term in eq 5.
two-site case: mLtot,exp )
mLtot(R1 + β2 - R1β2)2 (R1 + β22 - R1β22)
(8)
or
mLtot,exp/mLtot )
(R1 + β2 - R1β2)2 (R1 + β22 - R1β22)
(9)
Note that, by rearranging eq 8 into the form shown in eq 9, it is possible to directly determine the accuracy of estimates made for mLtot,exp versus mLtot as a function of both the fractional distribution of the binding sites (R1) and the relative range of binding site affinities (β2). Figure 4 shows how the ratio of the experimental binding capacity and actual binding capacity (mLtot,exp/mLtot) would be expected to change with R1 and β2 for a two-site system, as calculated according to eq 9. The lower portion of this graph is a contour plot indicating which combinations of R1 and β2 fall within a given degree of accuracy for the apparent binding capacity measurements (e.g., a value of mLtot,exp/mLtot ) 0.95 represents a maximum error of 5% in mLtot,exp versus mLtot). These results show that about half (48.7%) of all possible combinations of R1 and β2 give less than a 5% error in binding capacities that are measured over the linear range of plots like those in Figures 1-3. Similarly, less than 61% of all two-site cases give a 10% error in the apparent binding capacity, and less than 76% give a 20% error under such conditions. The extent of this error is particularly small when the two binding sites differ by less than 2-fold in their binding affinities (β2 g 0.5). However, the error grows to 30% or greater when the majority of all binding regions in the column consist of the weaker site (R1 < 0.5) and the strong versus weak sites differ by more than 5-fold in their association constants (β2 < 0.2). The effects of ligand heterogeneity on the accuracy of the apparent binding capacity mLtot,exp was also examined for systems with more than two types of binding regions, as represented by Figure 5a. This graph shows the relative number of all R and β combinations for two, three, and four binding regions that provide
a value for mLtot,exp that lies within a given accuracy range versus the true column binding capacity mLtot. The results in Figure 5a were determined by using computer-based calculations and the following expression, derived from eq 7, to tabulate the number of cases occurring within each desired level of accuracy.
n-site case: mLtot,exp/mLtot )
∑R β ) (∑β )(∑R β ) - [∑R {β (∑β ) - β (
i
Figure 4. Change in the ratio mLtot,exp/mLtot versus the mole fraction of the highest affinity site (R1) and the ratio of binding affinities (β2) for a system with two types of ligands in a column that is studied by frontal analysis at low analyte concentrations. These data were generated by using eq 9. The contours in the lower graph (in order of increasing value) represent mLtot,exp/mLtot ratios of 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 0.95.
i i
2
i i
i
i
j
}] (10) 2
i
For the two-site case, the results in Figure 5a simply represent the integrated areas for the various contour regions shown in Figure 4. As stated earlier, these data indicate that a large number of two-site cases produce only small differences between the actual and apparent column binding capacities when working in the linear range of double-reciprocal plots. For example, over 60% of all two-site cases produced less than a 10% error in mLtot,exp versus mLtot, as demonstrated earlier in Figure 4. When compared to the three- or four-site cases, the two-site data in Figure 5a always gave the closest agreement between mLtot,exp and mLtot, as indicated by its greater number of situations that fell with any given error range. This was not surprising, considering the larger number of R and β combinations that occurred in the three- or four-site systems, thus leading to the occurrence of more situations that produced significant deviations in the apparent binding capacity measurements. But even though more deviations were seen for the higher order systems, these systems still had a relatively large fraction of cases that resulted in only low to moderate errors in the apparent binding capacities. For instance, 45% and 36% of the three- and four-site systems, respectively, gave errors in mLtot,exp of 10% or less. Similarly, 67% and 60% of the three- and four-site cases had errors of less than 20%. All of the two-, three-, and four-site systems produced errors below 50% for more than 90% of all R and β combinations (i.e., nine-tenths of all tested cases had less than a 2-fold difference in mLtot,exp versus mLtot), and 97-99% of all combinations had an error smaller than 65-70% (i.e., had less than a 3-fold difference in the apparent and actual binding capacities). Effects of Multisite Binding on Association Constant Measurements. An approach similar to that used in generating Figure 4 was used to examine how the apparent association constant (Ka,exp) measured from the linear region of doublereciprocal plots for a two-site case would compare to the largest association constant for the test system (Ka1). As suggested earlier, the value of Ka,exp for a heterogeneous column studied by frontal analyis at low analyte concentrations would be obtained by taking the ratio of the intercept and the slope in eq 5, as shown in eqs 11 and 12:
two-site case: Ka,exp ) Figure 5. Fraction of all two-site (9), three-site (]) or four-site (4) cases that produce values within given error ranges for estimates of (a) mLtot and (b) Ka1 as determined by frontal analysis measurements of Ka,exp and mLtot,exp from the linear range of double-reciprocal plots. These results were calculated by using eq 10 or 13, respectively, as described in the text.
(R1 + β22 - R1β22)/{mLtot(R1 + β2 - R1β2)2} 1/{mLtotKa1(R1 + β2 - R1β2)}
(11)
or
Ka,exp/Ka1 )
(R1 + β22 - R1β22) (R1 + β2 - R1β2)
Analytical Chemistry, Vol. 69, No. 23, December 1, 1997
(12) 4795
Figure 6. Change in the ratio Ka,exp/Ka1 versus the mole fraction of the highest affinity site (R1) and the ratio of binding affinities (β2) for a system with two types of ligands in a column that is studied by frontal analysis at low analyte concentrations. These data were generated by using eq 11. The contours in the lower graph (in order of increasing value) represent Ka,exp/Ka1 ratios of 0.15, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 0.95.
Note that eq 12 allows the values of Ka,exp and Ka1 to be directly compared as a function of only the fractional distribution of the binding sites in the column (R1) and their relative affinities (β2). This makes it possible to predict the accuracy of Ka,exp for a variety of two-site systems, in a manner analogous to that used to study the accuracy of mLapp,exp in the previous section. Figure 6 shows how the ratio Ka,exp/Ka1 varies as a function of R1 and β2 for Ka,exp values obtained in the linear region of doublereciprocal plots for two-site systems, as calculated according to eq 12. As in Figure 4, the contour plots in the lower portion of Figure 6 show what degrees of accuracy are produced for Ka,exp versus Ka1. All tested conditions gave Ka,exp values that were less than or equal to Ka1. There was only a small difference (i.e., Ka,exp/ Ka1 g 0.90, or less than 10% variation) in these values when the highest affinity site made up at least half of the binding regions in the column (R1 g 0.5) and when the second site was close to the highest affinity site in its binding strength (β2 g 0.85) or had a much weaker association constant (β2 e 0.02). The greatest differences in Ka,exp versus Ka,1 occurred when these two types of sites varied by 3-10-fold in their affinities and the weaker site made up the majority of the binding regions in the column. To study the effect of higher order systems on the accuracy of Ka,exp versus Ka1, an expression similar to that in eq 12 was developed for an n-site case, but now based on the relationship given in eq 7. The resulting expression is shown in eq 13. Based
n-site case: Ka,exp/Ka1 )
∑R β ) (∑β )(∑R β ) - [∑R {β (∑β ) - β (
i
i i
i i
i
i
j
}]
2
i
(13)
on eq 13, computer calculations were used to determine the 4796
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relative number of two-, three-, and four-site cases that produced Ka,exp values within given degrees accuracy versus Ka1. The results are plotted in Figure 5b. The overall trends in Figure 5b are similar to those seen for the apparent binding capacity data in Figure 5a, with the two-site systems having the greatest relative number of R and β combinations at any given degree of accuracy. As explained earlier in the binding capacity studies, this was expected due to the larger number of combinations present in the three- and four-site cases and the associated increase in cases that resulted in large deviations in Ka,exp versus Ka1. For small errors (i.e., less than a 20% difference in Ka,exp and Ka1), the accuracy of Ka,exp was affected more by ligand heterogeneity than were the apparent binding capacities, as can be seen by comparing Figure 5 parts a and b. However, a fairly large proportion of all heterogeneous cases still gave reasonably small differences between Ka,exp and Ka1. For example, less than a 10% error in Ka,exp was seen for 58%, 29%, and 17% of the two-, three-, and four-site cases, respectively, while less than a 20% error was noted for 74%, 58% and 49% of these systems. At larger error ranges (i.e., greater than 40% error in Ka,exp versus Ka1), the apparent association constants were affected less by heterogeneity than the apparent binding capacities, with 94-96% of all two- to four-site cases giving errors of less than 50% in Ka,exp and 98-99% producing errors of 60% or less. This indicates that relatively good estimates of Ka1 can be made from Ka,exp, even when a fairly large amount of ligand heterogeneity is present in a column. SUMMARY This study used chromatographic theory and work with model solute-ligand systems to examine the effects produced by ligand heterogeneity on frontal analysis measurements of affinity columns. It was found that equations developed for double-reciprocal plots in two-site systems gave good qualitative and quantitative agreement with results seen in the experimental models. These same equations were used to examine the accuracy of binding capacity or association constants measured from the linear region of double-reciprocal plots when multiple types of ligands were present in the column. A large proportion of two-site systems gave a good estimate (i.e., less than 10-20% error) of the true total column capacity and the association constant for the highest affinity ligand in the column. A smaller, but still appreciable, fraction of all three- and four-site cases also produced good estimates of these values. The results of this work have great potential practical value because of the widespread use of frontal analysis in characterizing affinity supports and the relatively common occurrence of ligand heterogeneity in affinity columns. For example, Figures 4-6 should be useful in determining the accuracy and robustness of frontal analysis when employed in the quality control of affinity support production. The same data should be of interest in more fundamental studies, such as those concerning the use of affinity columns as a means for examining biomolecular interactions. Regarding this last application, the results in Figure 6 indicate that frontal analysis can often be used to obtain reliable association constant estimates even if a fairly large proportion of the ligand is denatured or inactivated as a result of immobilization. This agrees with earlier observations made in the use of frontal analysis and HPLC to examine the binding of several chiral solutes to immobilized HSA, in which close agreement (i.e., typically less than a 2-16% difference) was seen between the HPLC-based and
solution-phase association constants, despite the fact that a significant fraction of the immobilized protein in the HPLC columns was known to be inactive or to have a greatly reduced binding activity.3,4,16 Although protein-based affinity columns were the specific experimental systems examined in this work, it should be noted that the results presented in this study apply to any chromatographic column that contains a well-defined ligand as the stationary phase and that has relatively fast, reversible interactions between the ligand and solutes. Some examples include ion-exchange resins, low molecular weight affinity ligands (e.g., triazine dyes or boronic acid), molecular imprints, and various non-proteinbased chiral stationary phases (e.g., cyclodextrins or Pirkle supports).6,10,30 As such, the guidelines and equations presented in this report are general ones that can be adapted for use in analyzing the effects of ligand heterogeneity in any of these other systems. This, in turn, should help in the development of improved optimization and characterization protocols for methods based on these various types of chromatographic materials. ACKNOWLEDGMENT This work was supported by the National Institutes of Health under grant No. GM44931. S.A.T. was supported in part through a 1992-1993 fellowship from the U.S. Department of Education under the Graduate Assistance in Areas of National Needs (GAANN) program. NOMENCLATURE Rn
fraction of total binding sites in the column represented by ligand n
βn
ratio of the association equilibrium constant for ligand site n versus the association equilibrium constant for the group of highest affinity sites in the column (Ka1), where βn ) Kan/Ka1
mLn
amount of ligand site n (in moles) that is not bound to analyte
APPENDIX Derivation of Equation 3. The reactions shown in eqs A1 and A2 can be used to describe the binding of a single analyte A to two immobilized ligand sites (L1 and L2), where A-L1 and A-L2 Ka1
A + L1 y\z A-L1 Ka2
A + L2 y\z A-L2
(A1) (A2)
represent the complexes formed between the analyte and each type of ligand. The terms Ka1 and Ka2 are the association equilibrium constants for A at these sites, as defined by the following expressions:
Ka1 ) {A-L1}/([A]{L1}) ) mA-L1/([A]mL1)
(A3)
Ka2 ) {A-L2}/([A]{L2}) ) mA-L2/([A]mL2)
(A4)
In eqs A3 and A4, { } represents a surface concentration of the given species, mA-L1 or mA-L2 is the number of moles of analyteligand complex in the column at equilibrium, mL1 or mL2 is the number of moles of nonbound ligand at equilibrium, and [A] is the corresponding concentration of analyte in solution. At the mean point of the breakthrough curve obtained during a frontal analysis experiment, the following mass balance equation relates the apparent amount of analyte that it takes to saturate the column (mLapp) to the amount of each ligand that is present:
[A]
molar concentration of applied analyte
{A-Ln}
surface concentration of ligand site n that is bound to analyte in the column
mLapp ) mA-L1 + mA-L2
Kan
association equilibrium constant for ligand site n, where n ) 1 for the group of sites with the largest equilibrium constant
In addition, the distribution of the binding sites at the breakthrough point can be described by the mass balance equations:
Ka,exp
apparent association equilibrium constant obtained from the linear region of a double-reciprocal frontal analysis plot
{Ln}
surface concentration of nonbound ligand site n in the column
mLapp
total amount of all ligand sites (in moles) that is bound to analyte at equilibrium under a given set of experimental conditions
mLtot
total amount (in moles) of all binding sites in the column
mLtot,exp
apparent total column binding capacity obtained from the linear region of a double-reciprocal frontal analysis plot
mLn,tot
total amount (in moles) of ligand site n in the column
mA-Ln
amount of ligand site n (in moles) that is bound to analyte
(30) Allenmark, S. Chromatographic Enantioseparation: Methods and Applications, 2nd ed.; Ellis Horwood: New York, NY, 1991.
(A5)
mLtot ) mL1,tot + mL2,tot
(A6)
mL1,tot ) mL1 + mA-L1
(A7)
mL2,tot ) mL2 + mA-L2
(A8)
where mL1,tot and mL2,tot are the total number of moles of L1 and L2 within the column. By rearranging eqs A3 and A4 in terms of mA-L1 or mA-L2 and substituting the resulting expressions into eqs A5 and A6, the following equation can be obtained for the ratio mLtot/mLapp:
mL1,tot + mL2,tot mLtot ) mLapp mL1Ka1[A] + mL2Ka2[A]
(A9)
In a similar fashion, incorporating the expressions obtained for mA-L1 and mA-L2 from eqs A3 and A4 into eqs A7 and A8 and rearranging in terms of mL1 or mL2 gives the expressions shown Analytical Chemistry, Vol. 69, No. 23, December 1, 1997
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below:
mL1 ) mL1,tot/(1 + Ka1[A])
(A10)
mL2 ) mL2,tot/(1 + Ka2[A])
(A11)
or are independent of analyte concentration. To a first approximation, the limit of eq 3 at small values of [A] can be represented by eq A15.
lim
[A]f0
Substitution of eqs A10 and A11 into eq A9 then results in the following relationship:
mLtot ) (mL1,tot + mL2,tot)/{mL1,tot(Ka1[A])/(1 + Ka1[A]) + mLapp
1 mLapp
≈
1 + (β1 + β2)Ka1[A] mLtot(R1β1 + R2β2)Ka1[A]
Note that rearrangement of eq A15 leads to a linear relationship between 1/mLapp and 1/[A], as shown in eq A16.
lim
[A]f0
1 1 ≈ + mLapp mLtot(R1β1 + R2β2)Ka1[A]
mL2,tot(Ka2[A])/(1 + Ka2[A])} (A12)
(β1 + β2) mLtot(R1β1 + R2β2)
It is possible to rewrite eq A12 by using the term Rn to give the fraction of all binding sites in the column that are represented by a particular ligand Ln, where Rn ) mLn,tot/mLtot. Also, the association equilibrium constant for any given site (Kan) can be described in relation to the association constant for the highest affinity site on the column (Ka1) by using the ratio βn, where βn ) Kan/Ka1. Substituting these new terms into eq A12 produces eq A13.
mLtot ) (R1 + R2)mLtot/mLtot{R1(β1Ka1[A])/ mLapp
(1/mLapp)eq 3 - (1/mLapp)eq A15 ) β1β2{(R1β1 + R2β2 - β1β2)Ka1[A] - 1} mLtot(R1β1 + R2β2){(R1β1 + R2β2) + β1β2Ka1[A]}
1 ) (R1 + R2)(1 + β1Ka1[A])(1 + β2Ka1[A])/ mLapp mLtot{R1(β1Ka1[A])(1 + β2Ka1[A]) +
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(A17)
By taking the limit of this difference as [A] approaches zero, all terms on the right-hand side of eq A17 that contain the term Ka1[A] will be eliminated, leaving only the following constant:
lim (1/mLapp)eq 3 - (1/mLapp)eq A15 )
[A]f0
R2(β2Ka1[A])(1 + β1Ka1[A])} (A14) Finally, the relationship in eq 3 is obtained by multiplying through the right-hand side of eq A14, combining all common terms, and substituting in a value of 1 for (R1 + R2). The same approach can be used to generate a expression for a system with n types of ligands, as given in eq 6. Derivation of Equation 5. As the concentration of analyte approaches zero, the higher order terms in eq 3 will become much smaller than those that have only a first-order dependence on [A]
(A16)
A comparison of eqs 3 and A16 (or A15) as a function of [A] reveals that the exact and approximate expressions differ by a constant amount at small analyte concentrations, or large 1/[A] values. This is due to the net contribution of the higher order terms that were dropped during the derivation of eqs A15 and A16. The contribution of these other terms can be evaluated by examining the difference between eqs 3 and A15 as [A] approaches zero:
(1 + β1Ka1[A]) + R2(β2Ka1[A])/(1 + β2Ka1[A])} (A13) By canceling common terms and multiplying both the numerator and the denominator of the right-hand side of eq A13 by (1 + β1Ka1[A])(1 + β2Ka1[A]), the result is eq A14.
(A15)
-(β1 β2) mLtot(R1β1 + R2β2)2 (A18)
Combining eq A18 with the intercept of eq A16, plus substituting in β1 ) 1 and R2 ) (1 - R1), gives the final result shown in eq 5. The same method can be used to derive an expression for the linear approximation in an n-site case, as given in eq 7. Received for review June 2, 1997. Accepted September 18, 1997.X AC970565M X
Abstract published in Advance ACS Abstracts, November 1, 1997.
