© Copyright 2004 American Chemical Society
AUGUST 3, 2004 VOLUME 20, NUMBER 16
Letters Experimental Validation of an Affinity Energy Distribution Calculated with the Expectation Maximization Method Gustaf Go¨tmar and Georges Guiochon* Department of Chemistry, The University of Tennessee, Knoxville, Tennessee 37996-1600, and Division of Chemical Sciences, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831 Received February 10, 2004. In Final Form: May 12, 2004 The difference between the average energies of the high-energy modes of the adsorption energy distributions of (S)-alprenolol and (R)-alprenolol on a chiral stationary phase calculated by the expectation maximization method agree well with the difference between the adsorption energies of these two compounds measured by isothermal titration calorimetry.
For more than 20 years, considerable attention and many publications have been devoted to the study of the heterogeneity of adsorbent surfaces.1-4 An important characteristic of this heterogeneity is the adsorption energy distribution (AED) of selected probe compounds. Many methods of derivation of the AED from adsorption equilibrium data have been derived, discussed, and used. Unfortunately, there are no truly homogeneous surfaces. There are no surfaces with an independently known AED either. So, the performance of these different methods of derivation of the AED of a probe on a surface cannot be assessed nor compared, and it is difficult to decide on a best approach. There are yet no validation procedures of AED determinations available. This is a serious hindrance in the investigation of many problems of adsorption on actual, that is, heterogeneous, surfaces. We attempted to * To whom correspondence should be addressed. Fax: 1-865974-2667. E-mail:
[email protected]. (1) Rudzinski, W.; Everett, D. H. Adsorption of Gases on Heterogeneous Surfaces; Academic Press: New York, 1992. (2) Jaroniec, M.; Madey, R. Physical Adsorption on Heterogeneous Solids; Elsevier: Amsterdam, The Netherlands, 1988. (3) Umpleby, R. J., II; Baxter, S. C.; Chen, Y.; Shah, R. N.; Shimizu, K. D. Anal. Chem. 2001, 73, 4584. (4) Stanley, B. J.; Krance, J.; Roy, A. J. Chromatogr., A 1999, 865, 97.
prepare and use a surface for which at least part of the AED would be known. Chiral chromatography also has attracted much attention in recent times and made considerable progress.5 A popular method to prepare chiral stationary phases (CSPs) consists of chemically bonding a pure chiral ligand to the surface of a porous silica. Two enantiomers are separated if a strong three-point interaction forms between the chiral center of one of the enantiomers and that of the ligand and if the similar interaction of the other enantiomer is weaker or absent.6 The CSP has sufficient chiral selectivity only if the formation of the complex involves strong molecular interactions such as hydrogen bonding and strongly polar interactions. This has two important consequences. First, the adsorption energies are much higher on the enantioselective sites than on the nonselective sites. So, the difference between the AEDs of two enantiomers involves only the high-energy end of these AEDs because all low-energy interactions are nonselective. Second, the enantioselective interactions being strong and highly selective, the complexation constants of the two enantiomers with the chiral selector of the CSP will be close when measured with the selector in solution or (5) Haginaka, J. J. Pharm. Biomed. Anal. 2002, 27, 357. (6) Pirkle, W. H.; Pochapsky, T. C. Chem. Rev. 1989, 89, 347.
10.1021/la040023h CCC: $27.50 © 2004 American Chemical Society Published on Web 07/01/2004
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Letters
Table 1. Values of the Equilibrium Constants (at 25 °C and I of 0.1 M) for Alprenolol at the Enantioselective Site, Obtained by Fitting the Experimental Isotherm to the Bi-Langmuir Modela, by Applying the EM Calculation,band by ITCc pH
method
5.0 bi-Langmuir EM
bS (mM-1) RSDd (%) bR (mM-1) RSDd (%) 8.6 8.8
2.7
8.2 8.8
13
1.9
6.1 6.2
13
5.5 bi-Langmuir EM ITC
21 19 17
7.8
5.8 ITC
26
1.3
6.0 bi-Langmuir EM
42 38
1.9
6.8 ITC
52
24
6.5 6.2
9.9
5.4
7.8
a-c
Values taken from refs (a) 12, (b) 7, and (c) 11. The equation of the bi-Langmuir isotherm is qI,sbIC qII,sbIIC q) + 1 + bIC 1 + bIIC where q and C are the concentrations in the stationary and mobile phases at equilibrium, respectively, qI,s and qII,s are the saturation capacities of the nonselective and the enantioselective types of interaction sites, respectively, and the equilibrium constants for the two types of sites are denoted bI and bII, respectively. d RSD: Relative standard deviation of the mean.
bonded to the surface of porous silica particles. The chiral selector is bulky and is bonded to the surface with a leash. So it is little affected by the electric field around the surface, a field that decreases most rapidly with increasing distance from this surface. We have little information on the heterogeneity of the surface of a CSP or on what should be the AEDs of different compounds on this surface. However, we know (1) that the AEDs of two enantiomers on any CSP should be identical as far as the nonselective sites are concerned and that they should differ only in the part of that distribution that accounts for the enantioselective interactions of the CSP with the two enantiomers (Pasteur principle), the high-energy end of the distribution and (2) that the interaction energy corresponding to this highenergy end should be closely correlated to the interaction energy measured in solution with the free ligand. Therefore, we can prepare a heterogeneous surface for which we will have an independent estimate of the difference between the average energies of the high-energy modes of the AED of two enantiomers. This approach can provide a check of the validity of the results provided by the method of determination of the AED used. In a recent study, we used the expectation maximization (EM) method to calculate the AEDs of the alprenolol and propranolol enantiomers on a cellulase CSP (Cel7A immobilized on silica) at different eluent pHs.7 The EM method8-10 is based on a robust algorithm that uses directly the raw experimental isotherm data measured, in this case by frontal analysis, and inverts this set of data into a distribution of adsorption constants without introducing any other arbitrary information than the assumption that the local adsorption isotherm is Langmuir. In this previous study,7 and for all the pHs considered, the EM method gave bimodal AEDs, a result consistent with these compounds exhibiting bi-Langmuir isotherm model behavior (see isotherm equation in Table (7) Go¨tmar, G.; Stanley, B. J.; Fornstedt, T.; Guiochon, G. Langmuir 2003, 19, 6950. (8) Stanley, B. J.; Guiochon, G. J. Phys. Chem. 1993, 97, 8098. (9) Stanley, B. J.; Guiochon, G. Langmuir 1994, 10, 4278. (10) Stanley, B. J.; Bialkowski, S. E.; Marshal, D. B. Anal. Chem. 1993, 65, 5, 259.
Figure 1. ITC titration data describing the complex formation of (S)-alprenolol and Cel7A at 25 °C in a sodium acetate buffer at pH 5.8 and ionic strength (I) 0.1 M. Part a shows the differential power signal recorded in the experiment. After integration with respect to time and normalization per mole of added ligand, the titration curve (part b) was obtained and the enthalpy and the equilibrium constant could be calculated by nonlinear regression analysis.(a) Record from a calorimetric titration experiment where 3 mM (S)-alprenolol solution is added to 32 consecutive 9 µL injections to the cell (1.4 mL) containing Cel7A, initially 0.2 mM. (b) Corresponding titration curve. Comparison of measured reaction heats (O) to the calculated 1:1 binding isotherm (solid line). The calculated association constant in this particular experiment was 26 mM-1, and the complexation enthalpy was 4.9 kCal mol-1. The total heat content (Q) of the solution after the ith injection is Q(i) )
[
CM,i∆HV0 CL,i 1 1+ + 2 CM,i bCM,i
x(
1+
CL,i 1 + CM,i bCM,i
)
2
-
]
4CL,i CM,i
where CM,i and CL,i are the protein and the ligand concentrations after the ith injection, respectively, H is the enthalpy change, V0 is the working volume, and b is the equilibrium constant. The change in heat content from the completion of the i - 1 injection to completion of the i injection is dVi Q(i) + Q(i - 1) ∆Q(i) ) Q(i) - Q(i - 1) + V0 2
(
)
where dVi is the displaced volume from the working volume upon injection (i.e., the volume injected).
Letters
1). The numerical values of the coefficients derived from the EM method and the trends observed were very similar to those obtained by fitting the analytical bi-Langmuir model to the data. This quantitative agreement between results derived from two completely different methods suggested that the EM method is valid. However, the same raw data are used in the two methods. So, this result could not be construed as an independent confirmation of the validity of the EM method. We present such a validation here by comparing to these earlier results the values measured by isothermal titration calorimetry (ITC), for the binding constants between Cel7A and the alprenolol enantiomers in solution.11 As just stated, obtaining similar results for the estimates of the equilibrium constant of alprenolol on the high-energy sites of the CSP and on the ligand in solution would constitute an independent check and a confirmation of the EM method. Figure 1a shows the calorimetric signals obtained during a titration of Cel7A with (S)-alprenolol in a sodium acetate buffer at pH ) 5.8, I ) 0.1 M, and 25°C. Titrations were made at 3-min intervals, the limit for baseline separation. Figure 1b shows a plot of the integrated, normalized areas of the peaks in Figure 1a versus the molar ratio (symbols) and the best 1:1 binding isotherm (solid line). The values of the equilibrium constants between Cel7A and alprenolol obtained by fitting the experimental isotherm data to the bi-Langmuir equilibrium model,12 by applying the EM method to the same data,7 and those derived from a series of isothermal titration microcalorimetric (ITC) determinations11 are compared in Table 1. (R)-alprenolol gave a very poor signal in the ITC experiments and a scattered titration curve that deviated significantly from a sigmoid, which is why ITC data are lacking for this enantiomer. By contrast, for (S)-alprenolol, the agreement between the values obtained from chromatographic determinations, fitting the bi-Langmuir model or calculating the AED using EM, and the values obtained by ITC is very good (cf. Table 1). The equilibrium constants of the selective site (high energy) obtained by chromatography are 19-21 and 3842 mM-1 at pH 5.5 and 6.0, respectively.7,12 The value at pH 5.5 is, thus, in good agreement with the one obtained by ITC (17 mM-1). Interpolating the values at different pHs also gives a quite good agreement with the one obtained by ITC at pH 5.8. Because these values are in good agreement, it must be considered that the bimodal AED for this CSP is not an artifact and that the values of the equilibrium constant are correctly determined. The resolution between the low, nonselective mode and the high, enantioselective modes of the AED of (R)alprenolol is very poor in the pH range studied.7 Therefore, only rough estimates of the enantioselective equilibrium constant (bR,II) and of the corresponding saturation capacity (qR,II,s) could be derived for this enantiomer. These estimates suggest that the interaction energy does not (11) Go¨tmar, G.; Ozen, C.; Serpersu, E.; Guiochon, G. Manuscript in preparation. (12) Go¨tmar, G.; Fornstedt, T.; Guiochon, G. Anal. Chem. 2000, 72, 3908.
Langmuir, Vol. 20, No. 16, 2004 6523
Figure 2. Plot of the dependence of the ratio bS,II/bR,II on pH. Symbols: experimental data, obtained from ITC (∆), from ref 13 (×), and ratio obtained from AED (9). Line: best linear fit to the AED data.
vary significantly with the solution pH while the saturation capacity increases with increasing pH. The value bR obtained by ITC at pH 6.8 was 5.4 (Table 1). Hedeland et al. has reported a value of 6.9,13 in excellent agreement. It seems, thus, that this value (bII) remains constant between pH ) 6.0 and pH ) 6.8. The ratio of the average equilibrium constants of the two enantiomers (bS,II/bR,II) increases with increasing pH, being 1.0 at pH ) 5.0 and 6.1 at pH ) 6.0. Extrapolating this trend to pH ) 6.8 (the only pH for which ITC data exist for both enantiomers) gives a value of 10 for the ratio (cf. Figure 2), a value in close agreement with the value of 9.7 derived from ITC measurements.11 However, extrapolation should be considered with caution, even in such a limited pH range. It should be noted that, as a result of the limited capacity of the acetate buffer, a phosphate buffer had to be used for the ITC measurements at pH ) 6.8. Protein structures and chiral recognition mechanisms may be affected by this change in the counteranion of the buffer. Nevertheless, the agreement between the values of the high-energy equilibrium constants derived from the AEDs calculated from chromatographic data by the EM method and those given by ITC validates the results of the EM method. Acknowledgment. This work was supported in part by Grant DE-FG05-88-ER-13869 of the U.S. Department of Energy and by the cooperative agreement between the University of Tennessee and Oak Ridge National Laboratory. LA040023H (13) Hedeland, M.; Henriksson, H.; Ba¨ckman, P.; Isaksson, R.; Pettersson, G. Thermochim. Acta 2000, 356, 153.
