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Population Characteristics of Cyclodextrin Complex Stabilities in Aqueous Solution
Population Characteristics of Cyclodextrin Complex Stabilities in Aqueous Solution KENNETH A. CONNORS Received December 30, 1994,from the School of Pharmacy, University of Wisconsin, Madison, WI 53706. publication February 24, 1995@. Abstract 0 Binding constants (41)of 1:l complexes of a-cyclodextrin, /3-cyclodextrin, and y-cyclodextrin with many substrates (guests) were collected from published sources and subjected to statistical analysis. All systems refer to 25 k 5 "C and aqueous solution. The frequency distributions of log 61are satisfactorily described by normal distributions with the following parameters (n = number of complexes, p = population mean, u = population standard deviation): a-cyclodextrin, n = 663, p = 2.11,u = 0.90;/3-cyclodextrin, n = 721,p = 2.69,u = 0.89; y-cyclodextrin, n = 166,p = 2.55,CJ = 0.93. Stabilities of pairs of cyclodextrin complexes with a common substrate are not precisely correlated, but they do not appear to be wholly independent quantities. The stabilities of a-cyclodextrin complexes are consistent with a recent interpretation of solvent effects on a-cyclodextrin complex stabilities.
The cyclodextrins (cycloamyloses) are cyclic oligomers of a-D-glucose produced by enzymatic cyclization of starch; the so-called "native)) cyclodextrins (subsequently abbreviated CyD) are a-CyD (cyclohexaamylose),which has six glucose units; B-CyD (cycloheptaamylose), which has seven glucose units; and y-CyD (cyclooctaamylose),which has eight glucose units. As a consequence of their ability to serve as "hosts" in host-guest inclusion complex formation, the CyDs have attracted much experimental and theoretical attention, and studies have been directed both to fundamental issues and to practical applications. Book-length reviews are Among the important issues are the stabilities in solution of CyD complexes. Complex stabilities have been measured for hundreds of these complexes, and much has been learned; yet, at the present time, we are unable to answer in a general way the following two important questions: ( I ) Given the identities of a CyD host, a guest, and a solvent, what will be the stability of the complex(es) formed in this system? and (2) What are the energetic sources of the complex stability? It is true that many correlations of limited scope have been described between complex stabilities and physical-chemical "descriptors", and that much discussion has been devoted to the forces involved in CyD complexation; nevertheless, the two questions still stand as general propositions. The following experimental and theoretical approaches have been applied to this problem: ( a ) molecular structure-complex stability relationships; ( b ) temperature effects on complex stability, leading to enthalpies and entropies of complexation; (c) pressure effects on complexation, giving volume changes; ( d ) medium effects; ( e ) kinetics of associatioddissociation; cf, complex structure determination; (g) theoretical calculations, by both quantum mechanics and molecular mechanics; and ( h )empirical correlations of the QSAR type. In the present paper, we adopt a very different point of view by asking how CyD complex stabilities behave as a class or population. This approach cannot yield specific answers to questions about individual complexes, but it can provide a context for further analysis. This viewpoint rephrases the aforementioned questions in the following forms: ( I ) What @
Abstract published in Advance ACS Abstracts, May 1, 1995.
0 1995, American Chemical Society and American Pharmaceutical Association
Accepted for
are the typical stabilities of CyD complexes? and ( 2 ) Why are these stabilities seen, and not others? The concept of a population of stabilities raises the following further questions that are explored in this paper: (a) Does complexation by a CyD in fact create a population in the statistical sense?; ( b )If a population exists, what is the nature of its frequency distribution?; (c) Can statistical parameters, the mean and standard deviation in particular, be defined?; (d)Can criteria be formulated for questioning or rejecting experimental findings; that is, for identifylng systematic errors?; and (e) Are the complex-forming abilities of the three CyDs essentially independent or are they substantially related?
Theoretical Section Data Collection and Interpretation-Definitions and PrincipZes-The CyD is called the host or Zigand, L, and the second interactant is the guest or substrate, S. The substrate and ligand interact to form a complex SL with a 1 : l stoichiometry: XI1
S+L===SL The equilibrium constant K11 (often called the binding constant or stability constant) is defined by eq 1, where brackets signify molar concentrations:
[SLI K,, = [S"I The units of K11 are M-l. We recognize that a t the molecular level most substrate molecules are too large to be completely included within a CyD cavity, so we consider that the substrate possesses one or more binding sites. For example, the interaction of one a-CyD with 4-nitrobenzoic acid might take place a t either the nitro-substituted site or the carboxysubstituted site on the substrate. For such a 2-site substrate, Scheme 1can be drawn, where X Y represents the 2-site substrate and a prime indicates CyD complexation at that site. The four equilibrium constants in Scheme 1are microscopic binding constants. The species X'Y' is the 1:2 complex SL2, whose stoichiometric binding constant is The stoichiometric binding constants defined as K12 = [SL&'[SL][L]. K11 and KIZare related to the microscopic binding constants by eqs 2 and 34:
K,, = K,
+K ,
(2)
where axy, called the interaction parameter, is given by axy = Pry/ K m = KuxrfK.yr, a r y measures the extent of interaction between sitesX and Y in 1:2 complex formation. If the sites are independent, then am = 1. If the two binding sites are identical, as in a sym-1,4disubstituted benzene, eqs 2 and 3 become?
as
K11= =xx
(4)
K12= a,K,,f4
(5)
These considerations show that measured binding constants such K11 and K12 are in general composite quantities of microscopic
0022-3549/953184-0843$09.00/0
Journal of Pharmaceutical Sciences / 843 Vol. 84, No, 7, July 1995
i
X'Y
8
1
4-Nitrophenol
4-Nitrophenolate
I
X'Y'
XY
log K,,
XY' Scheme 1-Scheme for a 2-site substrate. binding constants. It is really the microscopic constants that we wish to study, yet, except in special cases (such as those described by eqs 4 and 5), we do not routinely have access to these. Nevertheless, we can proceed as follows. For a symmetrical 2-site substrate AX,we have, from eq 4, that the microscopic constant is K d 2 . For a general 2-site substrate X Y , then, the microscopic constant must lie between K11 and K11/2. Thus, the maximum difference between the stoichiometric constant and the microscopic constant is 0.3 log unit, and we will use values directly in t h e following (though when it h a s been possible, microscopic constants have been evaluated for comparison). Moreover, because of the likelihood of non-independence of binding sites (the am parameter in eq 3), only 1:l binding constants have been considered in the following treatment. Criteria for Inclusion-CyD binding constants were collected subject to the following criteria: (a) All binding constants were drawn from the primary literature. ( b ) Only K11 values are included, for the reason just described. (c) The study temperature is 25 & 5 "C. Actually, most of the constants included were measured at 25 "C,and none lie outside the 20-30 "C range. Unreported temperatures could usually be inferred to fall in this range, as could the description 'room temperature'. In some instances, constants were calculated to 25 "C from thermodynamic data or by graphical interpolation. ( d )Only the native CyDs are included. (e) The substrate ionic state is known and dominant. Thus, if the solution pH was close to the substrate pK,, a mixture of species was present, and the system was excluded. Different ionic states constitute different substrates; for example, benzoic acid is a substrate and benzoate ion is another substrate. (f, The solvent is water, or a n aqueous solution containing no more than 1%of organic cosolvent. This collection of data from the literature is certainly not complete, but it probably contains most of the reported results that satisfy these criteria. Selection of Best Value-When only a single K11 value is reported in the literature, it was usually accepted as the best estimate, except for a very few instances of highly suspect values. When multiplicate values were available, usually the K11 values were averaged. Occasionally, however, duplicate values (from different laboratories) differed by one or more orders of magnitude. In this case, one of them was selected and the other was rejected on the basis of internal evidence (appropriateness of the experimental method, control of conditions, choice of stoichiometric model) or consistency with similar systems. All values, including rejected ones, are listed in the tables available as the Supplementary Material to this article.
Results and Discussion Binding Constants-A few examples drawn from the listings in the Supplementary Material will illustrate the agreement seen in literature reports of CyD binding constants. All of the following numbers are K11IM-l values, except as indicated. For the mephobarbital (acid form): P-CyD system, K11 is reported as 1500, 1630, 1460, 1800, and 1660. These data are excellent agreement. Another very satisfactory system is the flufenamic acid anion: b-CyD system that has K11 values of 1340, 1270, 1484, and 1330. The methyl orange 844 /Journal of Pharmaceutical Sciences Vol. 84, No. 7, July 1995
Figure 1-Frequency distributions (points) and normal distributions (curves) for reported binding constant values for 4-nitrophenol (open circles) and 4-nitrophenolate (closed circles) with a-CyD. Statistical parameters are given in the text.
zwitterion: a-CyD system also has yielded data in close agreement: 673; 664; 682; 893; 670; and 800. Not all results are so satisfactory. Thus, the two reported results for 2-nitrophenol (acid form): a-CyD are 8 and 5000. Obviously a t least one of these must be seriously wrong as a consequence of a systematic error. Uncertainties reported by experimentalists usually measure only random error, so such uncertainties are of no use in distinguishing between these numbers. The disagreement in this particular case is extreme and unusual. More commonly we can consider as acceptable agreement, a t the present time, a factor of -2 in K11 values. Clearly there is scope for much more experimental work in this field. (A very few laboratories have paid great attention to the detection of systematic errors, so some data of high reliability are available.) A further example of poor agreement is provided by the pyrene: p-CyD system that has Kl1 values of 128, 160, 910, 493, 44, 494, 190, 120, and 7.6. This has proved to be a difficult system to study, being complicated by low substrate solubility, the possibility of substrate selfassociation, and uncertainty in the choice of stoichiometric model. The well-studied systems 4-nitropheno1:a-CyDand 4-nitropheno1ate:a-CyD provide excellent opportunities for the evaluation of precision. Sixteen values for 4-nitrophenol and 17 for 4-nitrophenolate are listed in Supplementary Material. The highest value in each set was rejected, and the following statistical parameters (in units of log Kl1) were evaluated: for 4-nitrophenol, mean @) = 2.32, standard deviation (a)= 0.10; for 4-nitrophenolate, p = 3.36, u = 0.11. Frequency distributions of the reported values with superimposed normal distribution curves calculated with these statistical parameters are shown in Figure 1. The precision of these results appears to be reasonably described by the normal distribution applied to log K11. The Supplementary Material consists of three tables, one each for complexes of a-CyD, P-CyD, and y-CyD. The tables contain the following numbers of entries (binding constant estimates) and systems (complexes): for a-CyD, 960 entries for 667 systems; for P-CyD, 1142 entries and 722 systems; for y-CyD, 188 entries and 166 systems. References to the original sources are given. Frequency Distributions-The best estimates of the complex binding constants were converted t o log K11 values, and, within each presumed population (that is, separately for a-CyD, P-CyD, and y-CyD), these values were grouped into "cells" using a cell interval of 0.30 log K l l unit. This interval size is small enough to generate an appropriate number of cells," and large enough to provide some smoothing effect. Frequency distributions were plotted and are shown in Figures 2,3, and 4 for a-CyD, P-CyD, and y-CyD, respectively. The statistical parameters p (mean) and u (standard deviation) for these populations, calculated from the log Kll values in the usual way, are listed in Table 1. With these parameter
ICOt
I
d -cyclodextrin 0
t
40 t - c y c l o d e x t r i n
80-
60 -
E a t 3 w
I=
-
40-
log K,, Figure 4-Frequency distribution (points) and normal distribution (curve) calculated with the parameters n = 166, p = 2.55, and u = 0.93 for y-CyD complex stabilities with both uncharged and ionic substrates.
20 -
Table 1-Population Statistical Parameters for CyD Complex Stabilitiesa CYD
Substrate Class
n
P
U
a a a
Combined Uncharged Ionic Combined Uncharged Ionic Combined
663 392 271 721 379 342 166
2.1 1 1.99 2.28 2.69 2.53 2.87 2.55
0.90 0.83 0.98 0.89 0.90 0.85 0.93
P P P
i
Y
- 0 -cyclodextrln OOnO
log K,,
Figure 3-Frequency distribution (points) and normal distribution (curve) calculated with the parameters n = 721, p = 2.69, and u = 0.89 for PCyD complex stabilities with both uncharged and ionic substrates.
values, normal distribution curves were calculated7 and plotted on the empirical frequency distributions; these are the smooth curves in Figures 2-4. These figures include data for all substrates, irrespective of ionic state. The data collections for a-CyD and p-CyD were further divided into subsets of uncharged substrates and ionic substrates (the latter including anions, cations, and zwitterions). Statistical parameters for these populations are given in Table 1. Their frequency distributions are not shown because they are similar in appearance to Figures 2 and 3. The ionic populations are dominated by anions; too few cations have been studied to justify further subdivision. Considering first the parameters gathered in Table 1, we learn that the standard deviation of all populations is -0.9 log Kll unit. This is sufficiently large that the population
a
Mean (LL) and standard deviation (a)in units of log K , .
means are not statistically different. This conclusion is certainly correct if it is interpreted to mean that, given an unidentified binding constant, one cannot state from which population the constant was drawn. On the other hand, there are indications of chemical significance, such as the mean values of a-CyD (KII= 129 M-l) and p-CyD (Kll = 490 M-I) complexes and the twofold greater stability of ionic (mainly anionic) complexes compared with uncharged complexes. Turning to Figures 2-4 and the calculated normal distributions, it appears that the experimental results are consistent with a normal distribution of log K l l , or equivalently, a log normal distribution of K11. We can conclude that complexation with a CyD does create a population that can be subjected to statistical analysis. However, as revealed in Figures 2-4, the considerable scatter in the experimental frequency distributions indicates that more rigorous tests of adherence to the normal distribution are not justified. (Use of microscopic binding constants, where they could be calculated from eq 4, did not significantly alter the frequency distributions.) Should we be surprised that, with such large sample numbers, the frequency distributions show so much scatter? The reason appears to be chemical, not statistical. Cyclodextrin experimentalists do not select their substrates randomly; they choose them systematically to test hypotheses, to search for structure-stability relationships, or to solve practical problems. The published data, accordingly, include many studies on substrates closely related by chemical structure, whose CyD complex stabilities are likewise often similar, leading to non-random occupancy of a few cells in the frequency distribution. For example, Uekama et a1.* studied 18 steroids with a-CyD, P-CyD, and y-CyD. In the a-CyD frequency distribution, all 18 points fall in contiguous cells, with nine of them in a single cell. Other examples of closely related substrates studied in large numbers are barbiturates (21 compounds studied in uncharged and ionic forms with a-CyD and P-CYD)~and cinnamic acids (20 compounds, uncharged and ionic, with a-CyD and P-CyD).I0 Journal of Pharmaceutical Sciences / 845 Vol. 84, No. 7, July 1995
We have found that log K11 is normally distributed, so the best estimate of a CyD binding constant is obtained by averaging log K11 rather than by averaging K11. This is a useful result. (In the present work K11 values were averaged, but this procedural decision did not significantly bias the populations because of the large sample numbers.) When precision is good, the two calculational routes yield similar results. With the information contained in Table 1 and Figures 2-4, we have criteria for judging "outliers" in experimental log Kll values, but these criteria apply only to one tail, the high stability end, of the distributions. Experimental values lying more than about three standard deviations to the low stability side of the mean will commonly be interpreted as negligible binding. Values greater than the mean by about three standard deviations call for careful examination as to possible systematic error before they are accepted as members of the population. Comparison of Cyclodextrins-Our data collection provides an opportunity to compare the complexing tendencies of a pair of CyDs with a common substrate. Here is a helpful way to interpret the comparison. Consider this hypothetical exchange process, where S is the substrate and CyD/S represents the 1:l complex: a-Cyd/S B-CyD =t p-CyDIS a-CyD. The exchange constant for this reaction is defined by eq 6:
+
"I 4-
-
3-
Y m
-
2-
I-
+
'09
[P-CyD/Sl[a-CyDl Kpla= [a-CyD/SI[P-CyDl
Figure 5-Plot of log 41 for P-CyD against log Kl1 for a-CyD with a common substrate; 409 substrates are plotted, and the line has unit slope.
It is easily seen that Kpla = Kll,j/Klla, where Kllp and Kl1a are the 1:l binding constants for the substrate binding to B-CyD and to a-CyD, respectively. By analogy with eq 6, the constants KyIp and Kylacan also be defined. Although K1la, Kllg, and Klly are, a priori, all independent quantities, the three constants KpIa,K,,lp, and Kylaare not independent, and can be related by eq 7:
This treatment suggests two ways to compare CyD complexing abilities. One of these is to examine the statistical behavior of log Kpla, log KY/b,and log Kyla. (The identity of eq 7 is exact only for a single substrate; it is unlikely to be satisfied for experimental samples, because the three populations will usually have been sampled differently.) The following statistical parameters were found: for log Kpla, 6 n - = - 409, ,u = 0.50, u = 0.93; for log Ky/p,n = 140, ,u = -0.45, u = 0.73; for log Kyla,n = 117, ,u = 0.57, u = 1.07. The more intuitive way to view the results is to plot log Kllp against log Klla, according to eq. (8). A priori, we could anticipate two extreme forms of behavior in such a plot. If KbIa = 1for all substrates, the points will fall on a line of unit slope. If, on the other hand, the two CyDs are completely independent in their behavior, a random distribution of points in the log Klla, log Kl1p plane will be seen. These plots for the pla, y/B, and y l a comparisons are shown in Figures 5-7. As the statistical parameters will have suggested, the distributions are not random. The correlations are not sufficiently precise to have quantitative predictive value, but they suggest that, averaged across a large number of substrates, the three CyDs are more alike than they are different in their complexing abilities. Some of the most interesting points in Figures 5-7 are the extreme outliers, which invite further experimental work to determine if they 846 /Journal of Pharmaceutical Sciences Vol. 84, No. 7, Juk 1995
I
-
i -1 l
I
/
Kt00
Figure 6-log K, for y-CyD against log K,I for P-CyD for 140 substrates. The line has unit slope.
are in error. If the outliers are experimentally valid, they constitute systems in which great selectivity is shown by a substrate for one CyD relative to another. Energetics of Cyclodextrin Complex Stability-The preceding sections have dealt with the first question posed in the introduction; namely, What are the typical stabilities of CyD complexes? We now turn attention to the second question: Why are these stabilities seen, and not others? Our laboratory has developed a phenomenological model of solvent effects that bears on this question. We consider the free energy change of a process to receive contributions from solvent-solvent interactions (giving rise t o the general medium or solvophobic effect), solvent-solute interactions (the
Now we call on specific experimental results. Our studied4 of solvent effects on the stabilities of a-CyD complexes of 4-nitroaniline and of methyl orange led to the conclusion that these complexes in water essentially owe their entire stability to the general medium effect; that is, the 2 term in eq 12 is negligible. Moreover, the averaged value of AgA was -58 A2 molecule-l, with a standard deviation of 10 A2 molecule-l ( n = 5). Using 2 = 0 and AgA = -58 f 10 A2 molecule-l in eq 12 yields log Kll = 2.66 f 0.75. Reference to Figure 2, the frequency distribution for a-CyD complex stabilities, shows that this simple calculation gives a result that coincides quite well with observed stabilities; in fact, the range of 2.66 f 0.75 includes 52%of the area under the normal curve. The implication of the calculation is that experimentally observed a-CyD complex stabilities in the approximate range of 1.9 to 3.4 (in log Kl1 units) can be fully accounted for by invoking the general medium effect alone. This conclusion must, of course, be an oversimplification. Let us examine the condition 2 = 0 that was used in applying eq 12. For the 4-nitroaniline and methyl orange systems, this condition was justified on the basis that the observed complex stability in water could be essentially completely accounted for by the AgA values obtained from solvent effect studies. That 2 will oRen be of negligible magnitude seems reasonable because it is composed of four intermolecular energy terms, two preceded by positive and two by negative signs, so substantial compensation can be expected. Yet, there are certainly many a-CyD complex stabilities that lie outside the range just calculated. Such experimental results mean, on the one hand, that stabilities greater than those calculated implicate a complex-strengtheningcontribution from 2,almost certainly from a very favorable AG:trasol term, or, on the other hand, that weaker complexes receive a less than maximal contribution from the AgAyl term, or a destabilizing effect from the 2 term (through the solvation energies), or both. The reasonableness of the prediction of log K11 just made can be assessed by an independent calculation. The quantity AgA can be estimated as the product of the internal area of ~ ) ~an ~ indepenthe a-CyD cavity (125 f 25 A2m ~ l e c u l e - and dent estimate of g (0.41),12giving AgA = -51 f 10 A2 molecule-I, a result that is in very reasonable agreement with the value obtained from the solvent effect studies. To go further than this will require additional experimental work to refine the estimate of the maximum AgA value, to extend the observations to substrates of many structural types, and to examine P-CyD and y-CyD systems. Theoretical study may be helpful by defining the maximum possible contributions that the intrasolute and solvation components of 2 can contribute.
1
- 1 -I
109 K I M
Figure 7-log Kll for y-CyD against log K1for a-CyD for 117 substrates. The line has unit slope.
solvation effect), and solute-solute interactions (the intersolute or intrasolute effect).11J2 The theory has been applied to complex formation,13including CyD ~omp1exation.l~ In the fully aqueous solution, the condition of present concern, the theory expresses the free energy change of complex formation by eq 914:
+
AG,*,,, = AGEtrasol (AG: - AG; - AG;)
+ MAYl
(9)
In this equation, AG,*,,, is given by the following:
AG~o,, = -kT In K,,
(10)
where Kmfis the binding constant on the mole fraction scale, and the free energy change is on a per molecule basis; k and T have their usual meanings. AG&:ntrasol describes substrateligand interaction within the complex. The AG:, AGE, AGL quantities are solvation energies for the complex C, the substrate S, and the ligand L, respectively. The AgAyl term describes the general medium effect, y1 being the surface tension of water and AgA being given by AgA = gAL - gAs gAL. Each quantity A is a molecular surface area (actually the nonpolar molecular surface area),15and g is an empirical factor that corrects for the effect of curvature on the surface tension. In complex formation, AgA is negative, and this term constitutes a driving force for complex formation; in fact, it is one way to describe the hydrophobic effect. The binding constant K,f on the mole fraction scale is related to Kl1 on the molar scale by eq 11, where e is the solvent density and M* is the number of moles of solvent per kg of solvent:
+
For convenience, we write 2 = AGitrasol (AG; - AGE AG;). Then, combination of eqs 9-11 gives eq 12. log Kll = -log eM* - -- -
2.3 kT
2.3 kT
References and Notes 1. Bender, M. L.; Komiyama, M. Cyclodextrin Chemistry; Springer-Verlag: Berlin, Germany, 1978. 2. Szejtli, J . Cyclodextrins and their Inclusion Complexes; Akademiai Kiado: Budapest, Hungary, 1982. 3. Cyclodextrins and their Industrial Uses; Duchhe, D., Ed.; Editions de Sant6: Paris, France, 1987. 4. Connors, K.A.; Lin, S.-F., Wong, A. B. J. Pharm. Sci. 1982, 71, 217 -222. 5. Connors, K. A.; Pendergast, D. D. J. A m . Chem. SOC.1984,106, 7607-7614. 6. Bennett. C. A,: Franklin. N. L. Statistical Analvsis in Chemi.strv and the 'Chemical Industry; Wiley: New Yo&, 1954;-p 11. 7. Hoel, P. G. Introduction to Mathematrcal Statistics, 2nd ed.; Wilev: New York. 1954: D 81. 8. Uekama, K.; Fujinaga, T.IHirayama,F.; Otagiri, M.; Yamasaki, M. Int. J. Pharm. 1982,10, 1-15. 9. Uekama, K.; Hirayama, F.; Nasu, S.; Matsuo, N.;Irie, T. Chem. Pharm. Bull. 1978,26,3477-3484. . I
Journal of Pharmaceutical Sciences / 847 Vol. 84, No. 7, July 1995
10. Uekama, K.; Otagiri, M.; Kanie, Y.; Tanaka, S.; Ikeda, K. Chem. Pharm. Bull. 1975,23, 1421-1430. 11. Khossravi, D.; Connors, K. A. J . Pharm. Sci. 1992, 81, 371319. 12. LePree, J. M.; Mulski, M. J.; Connors, K. A. J . Chem. Soc., Perkin Trans. 2 1994, 1491-1497. 13. Connors, K,A.; Khossravi, D. J . sol, Chem. 199322, 677-694, 14. Mulski, M. J.; Connors, K. A. Supramol. Chem., in press. 15. Khossravi, D.; Connors, K. A. J . Pharm. Sci. 1993, 82, 817820.
Supplementary Material Available-The supplementary material described in this article is available from an
848 /Journal of Pharmaceutical Sciences Vol. 84, No. 7, July 1995
anonymous FTP site a t the University of Wisconsin. This material contains the following information for a number of different substrates which bind to cyclodextrins: (1)stability constant (Kl1)values, (2) log K11 values, (3) charge, and (4) references* Readers wishing to Obtain the supplementa~ material should access the following FTP site: site, phml.phmacy.wisc.=du; login, anonymous; dopment name, connorssup JS940702K
The cyclodextrins (cycloamyloses) are cyclic oligomers of a-D-glucose produced by enzymatic cyclization of starch; the so-called "native)) cyclodextrins (subsequently abbreviated CyD) are a-CyD (cyclohexaamylose),which has six glucose units; B-CyD (cycloheptaamylose), which has seven glucose units; and y-CyD (cyclooctaamylose),which has eight glucose units. As a consequence of their ability to serve as "hosts" in host-guest inclusion complex formation, the CyDs have attracted much experimental and theoretical attention, and studies have been directed both to fundamental issues and to practical applications. Book-length reviews are Among the important issues are the stabilities in solution of CyD complexes. Complex stabilities have been measured for hundreds of these complexes, and much has been learned; yet, at the present time, we are unable to answer in a general way the following two important questions: ( I ) Given the identities of a CyD host, a guest, and a solvent, what will be the stability of the complex(es) formed in this system? and (2) What are the energetic sources of the complex stability? It is true that many correlations of limited scope have been described between complex stabilities and physical-chemical "descriptors", and that much discussion has been devoted to the forces involved in CyD complexation; nevertheless, the two questions still stand as general propositions. The following experimental and theoretical approaches have been applied to this problem: ( a ) molecular structure-complex stability relationships; ( b ) temperature effects on complex stability, leading to enthalpies and entropies of complexation; (c) pressure effects on complexation, giving volume changes; ( d ) medium effects; ( e ) kinetics of associatioddissociation; cf, complex structure determination; (g) theoretical calculations, by both quantum mechanics and molecular mechanics; and ( h )empirical correlations of the QSAR type. In the present paper, we adopt a very different point of view by asking how CyD complex stabilities behave as a class or population. This approach cannot yield specific answers to questions about individual complexes, but it can provide a context for further analysis. This viewpoint rephrases the aforementioned questions in the following forms: ( I ) What @
Abstract published in Advance ACS Abstracts, May 1, 1995.
0 1995, American Chemical Society and American Pharmaceutical Association
Accepted for
are the typical stabilities of CyD complexes? and ( 2 ) Why are these stabilities seen, and not others? The concept of a population of stabilities raises the following further questions that are explored in this paper: (a) Does complexation by a CyD in fact create a population in the statistical sense?; ( b )If a population exists, what is the nature of its frequency distribution?; (c) Can statistical parameters, the mean and standard deviation in particular, be defined?; (d)Can criteria be formulated for questioning or rejecting experimental findings; that is, for identifylng systematic errors?; and (e) Are the complex-forming abilities of the three CyDs essentially independent or are they substantially related?
Theoretical Section Data Collection and Interpretation-Definitions and PrincipZes-The CyD is called the host or Zigand, L, and the second interactant is the guest or substrate, S. The substrate and ligand interact to form a complex SL with a 1 : l stoichiometry: XI1
S+L===SL The equilibrium constant K11 (often called the binding constant or stability constant) is defined by eq 1, where brackets signify molar concentrations:
[SLI K,, = [S"I The units of K11 are M-l. We recognize that a t the molecular level most substrate molecules are too large to be completely included within a CyD cavity, so we consider that the substrate possesses one or more binding sites. For example, the interaction of one a-CyD with 4-nitrobenzoic acid might take place a t either the nitro-substituted site or the carboxysubstituted site on the substrate. For such a 2-site substrate, Scheme 1can be drawn, where X Y represents the 2-site substrate and a prime indicates CyD complexation at that site. The four equilibrium constants in Scheme 1are microscopic binding constants. The species X'Y' is the 1:2 complex SL2, whose stoichiometric binding constant is The stoichiometric binding constants defined as K12 = [SL&'[SL][L]. K11 and KIZare related to the microscopic binding constants by eqs 2 and 34:
K,, = K,
+K ,
(2)
where axy, called the interaction parameter, is given by axy = Pry/ K m = KuxrfK.yr, a r y measures the extent of interaction between sitesX and Y in 1:2 complex formation. If the sites are independent, then am = 1. If the two binding sites are identical, as in a sym-1,4disubstituted benzene, eqs 2 and 3 become?
as
K11= =xx
(4)
K12= a,K,,f4
(5)
These considerations show that measured binding constants such K11 and K12 are in general composite quantities of microscopic
0022-3549/953184-0843$09.00/0
Journal of Pharmaceutical Sciences / 843 Vol. 84, No, 7, July 1995
i
X'Y
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1
4-Nitrophenol
4-Nitrophenolate
I
X'Y'
XY
log K,,
XY' Scheme 1-Scheme for a 2-site substrate. binding constants. It is really the microscopic constants that we wish to study, yet, except in special cases (such as those described by eqs 4 and 5), we do not routinely have access to these. Nevertheless, we can proceed as follows. For a symmetrical 2-site substrate AX,we have, from eq 4, that the microscopic constant is K d 2 . For a general 2-site substrate X Y , then, the microscopic constant must lie between K11 and K11/2. Thus, the maximum difference between the stoichiometric constant and the microscopic constant is 0.3 log unit, and we will use values directly in t h e following (though when it h a s been possible, microscopic constants have been evaluated for comparison). Moreover, because of the likelihood of non-independence of binding sites (the am parameter in eq 3), only 1:l binding constants have been considered in the following treatment. Criteria for Inclusion-CyD binding constants were collected subject to the following criteria: (a) All binding constants were drawn from the primary literature. ( b ) Only K11 values are included, for the reason just described. (c) The study temperature is 25 & 5 "C. Actually, most of the constants included were measured at 25 "C,and none lie outside the 20-30 "C range. Unreported temperatures could usually be inferred to fall in this range, as could the description 'room temperature'. In some instances, constants were calculated to 25 "C from thermodynamic data or by graphical interpolation. ( d )Only the native CyDs are included. (e) The substrate ionic state is known and dominant. Thus, if the solution pH was close to the substrate pK,, a mixture of species was present, and the system was excluded. Different ionic states constitute different substrates; for example, benzoic acid is a substrate and benzoate ion is another substrate. (f, The solvent is water, or a n aqueous solution containing no more than 1%of organic cosolvent. This collection of data from the literature is certainly not complete, but it probably contains most of the reported results that satisfy these criteria. Selection of Best Value-When only a single K11 value is reported in the literature, it was usually accepted as the best estimate, except for a very few instances of highly suspect values. When multiplicate values were available, usually the K11 values were averaged. Occasionally, however, duplicate values (from different laboratories) differed by one or more orders of magnitude. In this case, one of them was selected and the other was rejected on the basis of internal evidence (appropriateness of the experimental method, control of conditions, choice of stoichiometric model) or consistency with similar systems. All values, including rejected ones, are listed in the tables available as the Supplementary Material to this article.
Results and Discussion Binding Constants-A few examples drawn from the listings in the Supplementary Material will illustrate the agreement seen in literature reports of CyD binding constants. All of the following numbers are K11IM-l values, except as indicated. For the mephobarbital (acid form): P-CyD system, K11 is reported as 1500, 1630, 1460, 1800, and 1660. These data are excellent agreement. Another very satisfactory system is the flufenamic acid anion: b-CyD system that has K11 values of 1340, 1270, 1484, and 1330. The methyl orange 844 /Journal of Pharmaceutical Sciences Vol. 84, No. 7, July 1995
Figure 1-Frequency distributions (points) and normal distributions (curves) for reported binding constant values for 4-nitrophenol (open circles) and 4-nitrophenolate (closed circles) with a-CyD. Statistical parameters are given in the text.
zwitterion: a-CyD system also has yielded data in close agreement: 673; 664; 682; 893; 670; and 800. Not all results are so satisfactory. Thus, the two reported results for 2-nitrophenol (acid form): a-CyD are 8 and 5000. Obviously a t least one of these must be seriously wrong as a consequence of a systematic error. Uncertainties reported by experimentalists usually measure only random error, so such uncertainties are of no use in distinguishing between these numbers. The disagreement in this particular case is extreme and unusual. More commonly we can consider as acceptable agreement, a t the present time, a factor of -2 in K11 values. Clearly there is scope for much more experimental work in this field. (A very few laboratories have paid great attention to the detection of systematic errors, so some data of high reliability are available.) A further example of poor agreement is provided by the pyrene: p-CyD system that has Kl1 values of 128, 160, 910, 493, 44, 494, 190, 120, and 7.6. This has proved to be a difficult system to study, being complicated by low substrate solubility, the possibility of substrate selfassociation, and uncertainty in the choice of stoichiometric model. The well-studied systems 4-nitropheno1:a-CyDand 4-nitropheno1ate:a-CyD provide excellent opportunities for the evaluation of precision. Sixteen values for 4-nitrophenol and 17 for 4-nitrophenolate are listed in Supplementary Material. The highest value in each set was rejected, and the following statistical parameters (in units of log Kl1) were evaluated: for 4-nitrophenol, mean @) = 2.32, standard deviation (a)= 0.10; for 4-nitrophenolate, p = 3.36, u = 0.11. Frequency distributions of the reported values with superimposed normal distribution curves calculated with these statistical parameters are shown in Figure 1. The precision of these results appears to be reasonably described by the normal distribution applied to log K11. The Supplementary Material consists of three tables, one each for complexes of a-CyD, P-CyD, and y-CyD. The tables contain the following numbers of entries (binding constant estimates) and systems (complexes): for a-CyD, 960 entries for 667 systems; for P-CyD, 1142 entries and 722 systems; for y-CyD, 188 entries and 166 systems. References to the original sources are given. Frequency Distributions-The best estimates of the complex binding constants were converted t o log K11 values, and, within each presumed population (that is, separately for a-CyD, P-CyD, and y-CyD), these values were grouped into "cells" using a cell interval of 0.30 log K l l unit. This interval size is small enough to generate an appropriate number of cells," and large enough to provide some smoothing effect. Frequency distributions were plotted and are shown in Figures 2,3, and 4 for a-CyD, P-CyD, and y-CyD, respectively. The statistical parameters p (mean) and u (standard deviation) for these populations, calculated from the log Kll values in the usual way, are listed in Table 1. With these parameter
ICOt
I
d -cyclodextrin 0
t
40 t - c y c l o d e x t r i n
80-
60 -
E a t 3 w
I=
-
40-
log K,, Figure 4-Frequency distribution (points) and normal distribution (curve) calculated with the parameters n = 166, p = 2.55, and u = 0.93 for y-CyD complex stabilities with both uncharged and ionic substrates.
20 -
Table 1-Population Statistical Parameters for CyD Complex Stabilitiesa CYD
Substrate Class
n
P
U
a a a
Combined Uncharged Ionic Combined Uncharged Ionic Combined
663 392 271 721 379 342 166
2.1 1 1.99 2.28 2.69 2.53 2.87 2.55
0.90 0.83 0.98 0.89 0.90 0.85 0.93
P P P
i
Y
- 0 -cyclodextrln OOnO
log K,,
Figure 3-Frequency distribution (points) and normal distribution (curve) calculated with the parameters n = 721, p = 2.69, and u = 0.89 for PCyD complex stabilities with both uncharged and ionic substrates.
values, normal distribution curves were calculated7 and plotted on the empirical frequency distributions; these are the smooth curves in Figures 2-4. These figures include data for all substrates, irrespective of ionic state. The data collections for a-CyD and p-CyD were further divided into subsets of uncharged substrates and ionic substrates (the latter including anions, cations, and zwitterions). Statistical parameters for these populations are given in Table 1. Their frequency distributions are not shown because they are similar in appearance to Figures 2 and 3. The ionic populations are dominated by anions; too few cations have been studied to justify further subdivision. Considering first the parameters gathered in Table 1, we learn that the standard deviation of all populations is -0.9 log Kll unit. This is sufficiently large that the population
a
Mean (LL) and standard deviation (a)in units of log K , .
means are not statistically different. This conclusion is certainly correct if it is interpreted to mean that, given an unidentified binding constant, one cannot state from which population the constant was drawn. On the other hand, there are indications of chemical significance, such as the mean values of a-CyD (KII= 129 M-l) and p-CyD (Kll = 490 M-I) complexes and the twofold greater stability of ionic (mainly anionic) complexes compared with uncharged complexes. Turning to Figures 2-4 and the calculated normal distributions, it appears that the experimental results are consistent with a normal distribution of log K l l , or equivalently, a log normal distribution of K11. We can conclude that complexation with a CyD does create a population that can be subjected to statistical analysis. However, as revealed in Figures 2-4, the considerable scatter in the experimental frequency distributions indicates that more rigorous tests of adherence to the normal distribution are not justified. (Use of microscopic binding constants, where they could be calculated from eq 4, did not significantly alter the frequency distributions.) Should we be surprised that, with such large sample numbers, the frequency distributions show so much scatter? The reason appears to be chemical, not statistical. Cyclodextrin experimentalists do not select their substrates randomly; they choose them systematically to test hypotheses, to search for structure-stability relationships, or to solve practical problems. The published data, accordingly, include many studies on substrates closely related by chemical structure, whose CyD complex stabilities are likewise often similar, leading to non-random occupancy of a few cells in the frequency distribution. For example, Uekama et a1.* studied 18 steroids with a-CyD, P-CyD, and y-CyD. In the a-CyD frequency distribution, all 18 points fall in contiguous cells, with nine of them in a single cell. Other examples of closely related substrates studied in large numbers are barbiturates (21 compounds studied in uncharged and ionic forms with a-CyD and P-CYD)~and cinnamic acids (20 compounds, uncharged and ionic, with a-CyD and P-CyD).I0 Journal of Pharmaceutical Sciences / 845 Vol. 84, No. 7, July 1995
We have found that log K11 is normally distributed, so the best estimate of a CyD binding constant is obtained by averaging log K11 rather than by averaging K11. This is a useful result. (In the present work K11 values were averaged, but this procedural decision did not significantly bias the populations because of the large sample numbers.) When precision is good, the two calculational routes yield similar results. With the information contained in Table 1 and Figures 2-4, we have criteria for judging "outliers" in experimental log Kll values, but these criteria apply only to one tail, the high stability end, of the distributions. Experimental values lying more than about three standard deviations to the low stability side of the mean will commonly be interpreted as negligible binding. Values greater than the mean by about three standard deviations call for careful examination as to possible systematic error before they are accepted as members of the population. Comparison of Cyclodextrins-Our data collection provides an opportunity to compare the complexing tendencies of a pair of CyDs with a common substrate. Here is a helpful way to interpret the comparison. Consider this hypothetical exchange process, where S is the substrate and CyD/S represents the 1:l complex: a-Cyd/S B-CyD =t p-CyDIS a-CyD. The exchange constant for this reaction is defined by eq 6:
+
"I 4-
-
3-
Y m
-
2-
I-
+
'09
[P-CyD/Sl[a-CyDl Kpla= [a-CyD/SI[P-CyDl
Figure 5-Plot of log 41 for P-CyD against log Kl1 for a-CyD with a common substrate; 409 substrates are plotted, and the line has unit slope.
It is easily seen that Kpla = Kll,j/Klla, where Kllp and Kl1a are the 1:l binding constants for the substrate binding to B-CyD and to a-CyD, respectively. By analogy with eq 6, the constants KyIp and Kylacan also be defined. Although K1la, Kllg, and Klly are, a priori, all independent quantities, the three constants KpIa,K,,lp, and Kylaare not independent, and can be related by eq 7:
This treatment suggests two ways to compare CyD complexing abilities. One of these is to examine the statistical behavior of log Kpla, log KY/b,and log Kyla. (The identity of eq 7 is exact only for a single substrate; it is unlikely to be satisfied for experimental samples, because the three populations will usually have been sampled differently.) The following statistical parameters were found: for log Kpla, 6 n - = - 409, ,u = 0.50, u = 0.93; for log Ky/p,n = 140, ,u = -0.45, u = 0.73; for log Kyla,n = 117, ,u = 0.57, u = 1.07. The more intuitive way to view the results is to plot log Kllp against log Klla, according to eq. (8). A priori, we could anticipate two extreme forms of behavior in such a plot. If KbIa = 1for all substrates, the points will fall on a line of unit slope. If, on the other hand, the two CyDs are completely independent in their behavior, a random distribution of points in the log Klla, log Kl1p plane will be seen. These plots for the pla, y/B, and y l a comparisons are shown in Figures 5-7. As the statistical parameters will have suggested, the distributions are not random. The correlations are not sufficiently precise to have quantitative predictive value, but they suggest that, averaged across a large number of substrates, the three CyDs are more alike than they are different in their complexing abilities. Some of the most interesting points in Figures 5-7 are the extreme outliers, which invite further experimental work to determine if they 846 /Journal of Pharmaceutical Sciences Vol. 84, No. 7, Juk 1995
I
-
i -1 l
I
/
Kt00
Figure 6-log K, for y-CyD against log K,I for P-CyD for 140 substrates. The line has unit slope.
are in error. If the outliers are experimentally valid, they constitute systems in which great selectivity is shown by a substrate for one CyD relative to another. Energetics of Cyclodextrin Complex Stability-The preceding sections have dealt with the first question posed in the introduction; namely, What are the typical stabilities of CyD complexes? We now turn attention to the second question: Why are these stabilities seen, and not others? Our laboratory has developed a phenomenological model of solvent effects that bears on this question. We consider the free energy change of a process to receive contributions from solvent-solvent interactions (giving rise t o the general medium or solvophobic effect), solvent-solute interactions (the
Now we call on specific experimental results. Our studied4 of solvent effects on the stabilities of a-CyD complexes of 4-nitroaniline and of methyl orange led to the conclusion that these complexes in water essentially owe their entire stability to the general medium effect; that is, the 2 term in eq 12 is negligible. Moreover, the averaged value of AgA was -58 A2 molecule-l, with a standard deviation of 10 A2 molecule-l ( n = 5). Using 2 = 0 and AgA = -58 f 10 A2 molecule-l in eq 12 yields log Kll = 2.66 f 0.75. Reference to Figure 2, the frequency distribution for a-CyD complex stabilities, shows that this simple calculation gives a result that coincides quite well with observed stabilities; in fact, the range of 2.66 f 0.75 includes 52%of the area under the normal curve. The implication of the calculation is that experimentally observed a-CyD complex stabilities in the approximate range of 1.9 to 3.4 (in log Kl1 units) can be fully accounted for by invoking the general medium effect alone. This conclusion must, of course, be an oversimplification. Let us examine the condition 2 = 0 that was used in applying eq 12. For the 4-nitroaniline and methyl orange systems, this condition was justified on the basis that the observed complex stability in water could be essentially completely accounted for by the AgA values obtained from solvent effect studies. That 2 will oRen be of negligible magnitude seems reasonable because it is composed of four intermolecular energy terms, two preceded by positive and two by negative signs, so substantial compensation can be expected. Yet, there are certainly many a-CyD complex stabilities that lie outside the range just calculated. Such experimental results mean, on the one hand, that stabilities greater than those calculated implicate a complex-strengtheningcontribution from 2,almost certainly from a very favorable AG:trasol term, or, on the other hand, that weaker complexes receive a less than maximal contribution from the AgAyl term, or a destabilizing effect from the 2 term (through the solvation energies), or both. The reasonableness of the prediction of log K11 just made can be assessed by an independent calculation. The quantity AgA can be estimated as the product of the internal area of ~ ) ~an ~ indepenthe a-CyD cavity (125 f 25 A2m ~ l e c u l e - and dent estimate of g (0.41),12giving AgA = -51 f 10 A2 molecule-I, a result that is in very reasonable agreement with the value obtained from the solvent effect studies. To go further than this will require additional experimental work to refine the estimate of the maximum AgA value, to extend the observations to substrates of many structural types, and to examine P-CyD and y-CyD systems. Theoretical study may be helpful by defining the maximum possible contributions that the intrasolute and solvation components of 2 can contribute.
1
- 1 -I
109 K I M
Figure 7-log Kll for y-CyD against log K1for a-CyD for 117 substrates. The line has unit slope.
solvation effect), and solute-solute interactions (the intersolute or intrasolute effect).11J2 The theory has been applied to complex formation,13including CyD ~omp1exation.l~ In the fully aqueous solution, the condition of present concern, the theory expresses the free energy change of complex formation by eq 914:
+
AG,*,,, = AGEtrasol (AG: - AG; - AG;)
+ MAYl
(9)
In this equation, AG,*,,, is given by the following:
AG~o,, = -kT In K,,
(10)
where Kmfis the binding constant on the mole fraction scale, and the free energy change is on a per molecule basis; k and T have their usual meanings. AG&:ntrasol describes substrateligand interaction within the complex. The AG:, AGE, AGL quantities are solvation energies for the complex C, the substrate S, and the ligand L, respectively. The AgAyl term describes the general medium effect, y1 being the surface tension of water and AgA being given by AgA = gAL - gAs gAL. Each quantity A is a molecular surface area (actually the nonpolar molecular surface area),15and g is an empirical factor that corrects for the effect of curvature on the surface tension. In complex formation, AgA is negative, and this term constitutes a driving force for complex formation; in fact, it is one way to describe the hydrophobic effect. The binding constant K,f on the mole fraction scale is related to Kl1 on the molar scale by eq 11, where e is the solvent density and M* is the number of moles of solvent per kg of solvent:
+
For convenience, we write 2 = AGitrasol (AG; - AGE AG;). Then, combination of eqs 9-11 gives eq 12. log Kll = -log eM* - -- -
2.3 kT
2.3 kT
References and Notes 1. Bender, M. L.; Komiyama, M. Cyclodextrin Chemistry; Springer-Verlag: Berlin, Germany, 1978. 2. Szejtli, J . Cyclodextrins and their Inclusion Complexes; Akademiai Kiado: Budapest, Hungary, 1982. 3. Cyclodextrins and their Industrial Uses; Duchhe, D., Ed.; Editions de Sant6: Paris, France, 1987. 4. Connors, K.A.; Lin, S.-F., Wong, A. B. J. Pharm. Sci. 1982, 71, 217 -222. 5. Connors, K. A.; Pendergast, D. D. J. A m . Chem. SOC.1984,106, 7607-7614. 6. Bennett. C. A,: Franklin. N. L. Statistical Analvsis in Chemi.strv and the 'Chemical Industry; Wiley: New Yo&, 1954;-p 11. 7. Hoel, P. G. Introduction to Mathematrcal Statistics, 2nd ed.; Wilev: New York. 1954: D 81. 8. Uekama, K.; Fujinaga, T.IHirayama,F.; Otagiri, M.; Yamasaki, M. Int. J. Pharm. 1982,10, 1-15. 9. Uekama, K.; Hirayama, F.; Nasu, S.; Matsuo, N.;Irie, T. Chem. Pharm. Bull. 1978,26,3477-3484. . I
Journal of Pharmaceutical Sciences / 847 Vol. 84, No. 7, July 1995
10. Uekama, K.; Otagiri, M.; Kanie, Y.; Tanaka, S.; Ikeda, K. Chem. Pharm. Bull. 1975,23, 1421-1430. 11. Khossravi, D.; Connors, K. A. J . Pharm. Sci. 1992, 81, 371319. 12. LePree, J. M.; Mulski, M. J.; Connors, K. A. J . Chem. Soc., Perkin Trans. 2 1994, 1491-1497. 13. Connors, K,A.; Khossravi, D. J . sol, Chem. 199322, 677-694, 14. Mulski, M. J.; Connors, K. A. Supramol. Chem., in press. 15. Khossravi, D.; Connors, K. A. J . Pharm. Sci. 1993, 82, 817820.
Supplementary Material Available-The supplementary material described in this article is available from an
848 /Journal of Pharmaceutical Sciences Vol. 84, No. 7, July 1995
anonymous FTP site a t the University of Wisconsin. This material contains the following information for a number of different substrates which bind to cyclodextrins: (1)stability constant (Kl1)values, (2) log K11 values, (3) charge, and (4) references* Readers wishing to Obtain the supplementa~ material should access the following FTP site: site, phml.phmacy.wisc.=du; login, anonymous; dopment name, connorssup JS940702K