Anal. Chem. 1998, 70, 966-970
Understanding Molecular Association and Isomers Recognition in Isomer-Cyclodextrin Multiple Complex Formation by Improved Liquid Chromatographic Studies Yannis L. Loukas*
Riga Ferreou 21, Ano Ilioupolis, 163 43 Athens, Greece
Equations are derived that allow the determination of guest-cyclodextrin primary (K11), secondary (K12), and higher order (K1n) binding constants as well as the degree of complexation n j (the percent of complexed guest), by monitoring the observed capacity factor k′obs for guests that can be structural, geometric, or optical isomers. The retention behavior (elution order) in a mixture of isomers is dependent on the cyclodextrin concentration in the mobile phase as well as on the stoichiometry and the binding constant of the guest-cyclodextrin (G-CD) complex. The simplification of higher order complexes (GCDn) to that with 1:1 stoichiometry can lead to erroneous results; therefore, it is important for the stoichiometry to be determined accurately, prior to any binding constant calculations, following the continuous variation method. The proposed models are solved iteratively using nonlinear least-squares analysis and following specific algorithms.
Cyclodextrins (CDs) and their derivatives are known to form inclusion complexes with a variety of molecules. CDs have been used widely as stabilizing and solubilizing systems, enzyme models, and catalysts. In the field of chromatography, CDs play an important role, especially for chiral and isomeric separations, either as a stationary phase or as additives in the mobile phase. The chirality of CDs is a known phenomenon and has been used successfully in many cases of isomer separations. Calculated stoichiometries and binding constants are used to make general conclusions and to predict or interpret the retention behavior and the elution order of the isomers.1,2 For instance, the elution order with a cyclodextrin stationary phase is expected to be indirectly proportional to the binding constant value; the more stable the complex (higher binding constant), the longer the elution time. Conversely, when CDs are present in the mobile phase, the elution order is expected to be proportional to the binding constant value (the complex with the highest binding constant will be eluted first). * Corresponding author: (e-mail)
[email protected]. (1) Saenger, W. Angew. Chem., Int. Ed. Engl. 1980, 19, 344-362. (2) Dalgliesh, C. E. J. Chem. Soc. 1952, 3940-3946.
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It is usually assumed in guest-cyclodextrin (G-CD) complexation studies that the stoichiometry of the complex is 1:1 and a series of linear procedures are solved graphically (like the Benesi-Hildebrand model) for the calculation of the binding constant. Most of these linear models suffer from theoretical and practical drawbacks,3 including assumed concentrations of the interacting moieties and products, poor solubility of certain compounds, a boundary condition (saturation binding) with respect to the ratio of the concentrations of the two binding partners, and the occasional formation of dimmers. Diederich4 suggested that nonlinear procedures are free of the above assumptions and have much broader applicability, potentially displacing the evaluations done according to Benesi-Hildebrand and to Scott or Scatchard linear models. Several inconsistencies in the assignment of binding stoichiometries appear in the literature. For example, p-nitroaniline is reported to form either a 1:25 or a 1:16 inclusion complex with R-CD. It is reported that o-xylene forms a 1:2 inclusion complex7 with β-CD while m-xylene and p-xylene form complexes with a 1:1 stoichiometry. In another study,8 it is assumed that a series of disubstituted benzenes form 1:1 complexes with R- and β-CD. Finally, prostaglandin B1 is reported to form either a 1:19 or 1:25 complex with R-CD. Such differences in assignments induce significant deviation in the results. In the present study, another way for assigning the complex stoichiometry is proposed. Specifically, the retention behavior of a guest in the presence of a cyclodextrin can be used to determine all the relevant binding constants for higher order complexes (G-CDn), as well as the percent of the guest that contributes to the formation of the intermediate complexes (degree of complexation). (3) Djedaini, F.; Perly, B. New Trends in Cyclodextrins and Derivatives, 1st ed.; Edition de Sante: Paris, 1993 (Chapter on Nuclear Magnetic Resonance of Cyclodextrins, Derivatives and Inclusion Compounds). (4) Diederich, F. Angew. Chem., Int. Ed. Engl. 1988, 27, 362-386. (5) Armstrong, D.; Nome, F.; Spino, L.; Golden T. J. Am. Chem. Soc. 1986, 108, 1418-1422. (6) Wong, A. B.; Lin S. F.; Connors, K. A. J. Pharm. Sci. 1983, 72, 388-390. (7) Debowski, J.; Sybilska, D. J. Chromatogr. 1986, 353, 409-416. (8) Zukowski, J.; Sybilska D.; Jurczak, J. Anal. Chem. 1985, 57, 2215-2219. (9) Kawaguchi, Y.; Tanaka, M.; Nakae, M.; Funazo, K.; Shono, T. Anal. Chem. 1983, 55, 1852-1857. S0003-2700(97)00807-X CCC: $15.00
© 1998 American Chemical Society Published on Web 02/04/1998
THEORETICAL SECTION The method of isomer separation with a CD in the mobile phase is based on the competitive interaction of the solute with the hydrophobic cavity of the CD and the stationary phase. It is assumed that the binding is more rapid than the chromatographic exchange so that the system is always in equilibrium with respect to the complex formation. Consider a reversed-phase liquid chromatographic system (Scheme 1) in which the solute G (any guest isomer) at a total concentration [Gt] and the CD at a total concentration of [CDt] form a multiple a-b complex (Ga-CDb). This complex could be further defined as a G-CDn complex, where n ) b/a. The apparent distribution coefficient Dobs becomes
Dobs )
[G](s) + [G-CD](s) + ... + [G-CDn](s) [G](m) + [G-CD](m) + ... + [G-CDn](m)
(1)
where the suffixes s and m denote the stationary and the mobile phases, respectively. The apparent capacity factor k′obs is related to Dobs as k′obs ) Dobs/a, where a ) Vst/V0 is the phase volume ratio of the column, V0 being the dead volume (column mobilephase volume). Replacing Dobs with its equal from eq 1, k′obs becomes
Scheme 1
intermediate complexes G-CD, ..., G-CDn are substituted with their equals from eqs 3-5. After these transformations, k′obs becomes
k′obs ) (k′G[G] + k′G-CDK11[G][CD] + k′G-CD2K11K12[G][CD]2 + ... + k′G-CDnK11K12 ... K1n × [G][CD]n)/([G] + K11[G][CD] + K11K12[G][CD]2 + ... + K11K12 ... K1n[G][CD]n) (6)
k′obs ) [G](s) + [G-CD](s) + ... + [G-CDn](s) [G](m) + [G-CD](m) + ... + [G-CDn](m)
a-1 )
DG[G](m) + DG-CD[G-CD](m) + ... + DG-CDn[G-CDn](m) [G](m) + [G-CD](m) + ... + [G-CDn](m) )
ak′G[G](m) + ak′G-CD[G-CD](m) + ... + ak′G-CDn[G-CDn](m) [G](m) + [G-CD](m) + ... + [G-CDn](m)
a-1
KG-CD2 )
[G-CD2]
)
[G-CD2] KG-CD[G][CD]2
a
(2)
w [G-CD2] )
KG-CDKG-CD2[G][CD]2 (4)
[G-CDn] [G-CDn-1][CD]
)
[G-CDn] KG-CD ... KG-CDn-1[G][CD]n
where k′G, k′G-CD, and k′G-CDn are the capacity factors for the free G, the 1:1 complex (G-CD), and the 1:n complex (G-CDn). Equation 7 is the general polynomial model of the binding isotherm in chromatographic studies, which denotes the nonlinear dependence of the free (unbound) cyclodextrin concentration [CD] on the apparent capacity factor k′obs. At this point, the known linear models adopt a simple transformation: the total cyclodextrin concentration [CDt] is equal to the concentration of free cyclodextrin [CD]. Avoiding this simplification, which has been criticized,10 the system can proceed as follows: The average number of CD molecules bound per G molecule (nj ) is
n j)
l KG-CDn )
k′G-CDnK11K12 ... K1n[CD]n)/(1 + K11[CD] + K11K12[CD]2 + ... + K11K12 ... K1n[CD]n) (7)
[G-CD] w [G-CD] ) KG-CD[G][CD] (3) [G][CD]
[G-CD][CD]
k′obs ) (k′G + k′G-CDK11[CD] + k′G-CD2K11K12[CD]2 + ... +
-1
the concentrations of the intermediate complexes G-CD, ..., G-CDn in eq 2 are related to the corresponding binding constants (KG-CD, ..., KG-CDn) as follows
KG-CD )
elimination of [G] in eq 6 gives
w [G-CDn] )
KG-CD ... KG-CDn-1KG-CDn[G][CD]n (5) the term a in eq 2 is eliminated and the concentrations of the
∑(CD bound to G)/∑(all G)
(8)
Further, by defining the quantity ∑(CD bound to G) as CDb, the “total bound” cyclodextrin concentration [CDb], can be defined as
[CDb] ) [G-CD] + 2[G-CD2] + ... + n[G-CDn]
(9)
(10) Connors, K. A. Binding Constants. The measurement of molecular complex stability, 1st ed.; John Willey and Sons: New York, 1987.
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The free (unbound) cyclodextrin concentration [CD] then becomes
[CD] ) [CDt] - [CDb] ) [CDt] - n j [Gt]
(10)
Finally, substitution of [CD] in eq 7 with its equal from eq 10 provides a polynomial equation, which describes the behavior of guest complexing n cyclodextrin molecules. If K12, ..., K1n ) 0, the final formula describes the formation of 1:1 complexes and is reduced to the following equation:
k′obs )
k′G + k′G-CDK11([CDt] - n j [Gt]) 1 + K11([CDt] - n j [Gt])
RESULTS AND DISCUSSION Binding Constant Calculations. To evaluate the use of eq 11, the binding constant was calculated using experimental data from the literature.7 These data provide the capacity factors k′ of different groups, such as xylenes, ethyltoluenes, trimethylbenzenes, and propylbenzenes, as a function of increasing concentrations of β-CD (Tables 1 and 2) and R-CD (Table 3). The published values for the binding constants calculated from a linear model as well as the calculated ones from eq 11 appear in Tables 1-3. It is becoming evident from the values in Tables 2 and 3 that although the linear model failed to calculate the binding constant values for all the compounds with R-CD and for the compound 1,3,5-trimethylbenzene with β-CD, the proposed nonlinear model calculated the binding constant values for all the compounds with
968 Analytical Chemistry, Vol. 70, No. 5, March 1, 1998
compounds
KG-CDa,b (M-1)
n j
R2 b
o-xylene m-xylene p-xylene
84 (107) 66 (100) 122 (140)
0.83 0.74 0.89
0.999 (0.971) 0.997 (0.952) 0.999 (0.985)
a The model described in the Appendix gives K ) 92 M-1, K ) o m 77 M-1, and Kp ) 144 M-1. b Numbers in parentheses denote the calculated values from ref 7.
(11)
Similarly, from the general eq 7, descriptive models can be derived for higher order complexes such as 1:2. The proposed nonlinear model involves no approximation of the concentrations of the two species (G and CD) and correlates their initial total concentrations [Gt] and [CDt] with the capacity factors of the free (k′G) and complexed (k′G-CD) isomer. The unknown binding constant K11 in the simple case of a 1:1 complex or K11 and K12 in 1:2 complexes can be calculated according to this model by using the nonlinear least-squares regression analysis. To use eq 11, the term nj must be defined. For a system forming only a 1:1 (G-CD) complex, the quantity nj ) CDb/Gt can range from 0 to 1 (0 < nj 0.99). All the other cases in Tables 2 and 3 presented the same curve-fitting behavior (not shown). Comparing the published values with those obtained in this study, it is becoming evident that the order of the binding constant values is the same while the absolute values are different to some extent. The correlations obtained using the present model are statistically improved versus the ones calculated from the linear model. Furthermore, by examining the behavior
Figure 1. Graphical representation of eq 11 for the interaction of β-CD with (a) o-xylene, (b) m-xylene, and (c) p-xylene and showing the nonlinear (hyperbolic) dependence of β-CD concentration to the apparent capacity factor k′obs.
of eq 11, it becomes clear that changing the [Gt] value and assuming that all the guest is complexed (avoiding the term nj ), the K11 value changes significantly. On the contrary, when the term nj was used, a balance was kept between the [Gt] value and the calculated K11 value; for instance, when [Gt] was doubled, the value of nj was halved with the K11 value remaining almost unchanged. It is also observed from Table 1 that the nj values exhibit the same order as the binding constant values; similar behavior was noticed for all the examined cases (not shown). This is consistent with the observation that the higher the binding constant, the more favorable the complexation is and more G is involved in
the complexation procedure. One also notes that the complex of β-CD with p-xylene is 2 times more stable than that with m-xylene (Table 1) and consistent with the prediction of p < o < m for the retention behavior. In eq 11, the total concentrations of G and CD ([Gt] and [CDt]) appear, without any approximation, as well as the factor nj (degree of complexation) which describes the percent of G molecules involved in the complexation (Scheme 1). G as a member of the equilibrium in Scheme 1 it does not affect the binding constant value but it affects the time needed for the equilibrium to be reached; the more G that is present the faster the equilibrium is shifted toward G-CD and the faster the complex G-CD is formed. Consequently, the time for reaching the equilibrium is altered significantly, thus altering the retention behavior of the formed complex. Furthermore, the more G is added, the more CD is complexed, and hence the [CD] value does not remain always the same, as it is usually assumed in the simplified linear model. In experiments with isomer separation, methanol is the most commonly used mobile-phase component; however, due to its polar character a possible interaction or complexation with CD is considered negligible (for β-CD-MeOH, K11 ) 0.37 M-1). In cases where a more hydrophobic modifier is added in the mobile phase, a complexation or a possible ternary complex formation with CD could be expected. The later case means that the coexistence of an isomer and a modifier in the CD cavity leads to changes to the binding constant values, reflecting also changes to the elution order. In these cases, nj can be used as an indication of the extent to which the addition of various modifiers changes the degree of the complexed isomer. Nonlinear Parameter Estimation. The nonlinear estimation of the parameters (K11 and nj ) was based on an iteration procedure following specific algorithms. The proposed mathematical model could be examined with different algorithms in order to observe any possible deviation on the calculated values. For instance, the Marquardt iterative algorithm was employed at first, which at every iteration evaluates the estimates against a set of control criteria. If successive iterations fail to change the sum of squares of the convergence criterion, the procedure stops. In addition, the sequential quadratic programming was used, which is a doubly iterative algorithm. Briefly, each major iteration sets up a quadratic program to determine the direction of search and the loss function is evaluated at any iterative point, until the search converges. In standard multiple regression, the regression coefficients were estimated by “finding” those coefficients that minimize the residual variance (sum of squared residuals) around the regression line. Any deviation of an observed score from a predicted score signifies some loss in the accuracy of our prediction, for example, due to random noise (error). Therefore, the goal of the least-squares estimation is to minimize a loss function. In this study, the loss function is defined as the sum of the squared deviations about the predicted values. To minimize the loss function (and thus finding the best fitting set of parameters) and to estimate the standard errors of the parameter estimates, a very efficient algorithm was used (quasi-Newton) that approximates the Analytical Chemistry, Vol. 70, No. 5, March 1, 1998
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second-order derivatives of the loss function and guides the search for the minimum. The present study focused on the proposal of an improved nonlinear model for the calculation of the binding constant and the degree of complexation in isomer-cyclodextrin complexes based on HPLC retention data (capacity factor). As such it does not add to the separation optimization of isomers but it can be of potential interest for the examination of mechanistic (chromatographic) studies and complex equilibrium in solution phase. More specifically, it serves as an alternative way that uses the retention data of the HPLC run in order to derive useful information about the binding affinity of CDs with the examined isomers, to predict the elution order from the binding values, to calculate the binding constants for all the intermediate complexes and the degree of complexation (the percentage of the included isomer), and to examine the effect of different modifiers on the degree of complexation. The proposed model has been examined with various isomer data sets using an RP-HPLC system in which CDs (R or β) were mobile-phase additives. The same model can potentially be used in general host-guest interactions involving other complexing agents, such as crown ethers or in thin-layer chromatography studies monitoring the behavior of the retention factors Rf of isomers (instead of their capacity factors) in the presence of CDs (or other complexing agents) as mobile-phase additives. In conclusion, the nonlinear model described was used for the calculation of the binding constants in multiple complex formation of isomers with CDs. The calculation is based on the chromatographic capacity factor of a guest being affected by the presence of CDs. As this model requires only the initial concentrations of the two species (G and CD) without any limitations and approximations, experimental and theoretical shortcomings are avoided. APPENDIX Following the mass balance for the two reacting species, G and CD, the unknown parameters [G] and [CD] can be written
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Analytical Chemistry, Vol. 70, No. 5, March 1, 1998
[G] ) [Gt] - ([CDt] - [CD]) [CD] ) [CDt] - [CD-G] ) [CDt] - K11[CD] [G] w [CD] + K11[CD] [G] ) [CDt] w [CD](1 + K11[G]) ) [CDt] w [CD] )
[CDt] 1 + K11[G]
[CDt] 1 + K11([Gt] - [CD-G])
)
)
[CDt] 1 + K11([Gt] - [CDt] + [CD])
w
After consecutive transformation, the following quadratic equation is obtained:
K11[CD]2 + (K11[Gt] - K11[CDt] + 1)[CD] - [CDt] ) 0 The value of [CD] can be calculated by solving this quadratic equation and discarding the erroneous solution (negative or redundant):
[CD1,2] ) (-(K11[Gt] - K11[CDt] + 1) (
x(K11[Gt] - K11[CDt] + 1)2 + 4K1:1[CDt])/2K11
(A1)
Substitution of [CD] in eq 7 with its value from eq A1 gives a nonlinear model, which can be solved iteratively using leastsquares regression analysis as described previously. The difference between this model and eq 7 is that it does not employ any stoichiometry calculations (i.e., continuous variation method) and thus could be used in systems in which the 1:1 stoichiometry is already established. Received for review July 28, 1997. Accepted December 15, 1997. AC970807I
