Anal. Chem. 2000, 72, 3272-3279
A Cluster Expansion Method for the Complete Resolution of Microscopic Ionization Equilibria from NMR Titrations Michal Borkovec*
Department of Chemistry, Clarkson University, Box 5814, Potsdam, New York 13699 Ger J. M. Koper
Laboratory of Physical Chemistry, Delft University of Technology, Julianalaan 136, 2628 BL Delft, The Netherlands
The NMR chemical shift of spin 1/2 nuclei in a polyprotic molecule represents a sensitive probe of microscopic protonation equilibria. However, these equilibria are commonly parametrized in terms of microscopic equilibrium constants, whose number increases very rapidly with the number of ionizable groups. For that reason their determination was considered to be basically impossible except for the cases of the simplest molecules. On the basis of a cluster expansion of the free energy of a microstate, we propose a novel parametrization of this problem that drastically reduces the number of necessary parameters needed to specify the microscopic equilibria. Such cluster parameters can be extracted from NMR titration data in a straightforward way. Once these parameters are known, all microscopic equilibrium constants can be obtained. Traditionally, protonation of a polyprotic molecule is approached within the macroscopic picture. Each protonation step is characterized by a reaction equilibrium, and the corresponding equilibrium constants are commonly reported as the (macroscopic) pK values. However, this description contains little information about the actual protonation states of the individual ionizable groups. In many modern analytical applications, this information is essential, and therefore the more complex microscopic picture must be considered. Thereby, protonation of each group is treated individually. The microscopic equilibria are quantified in terms of microscopic pK values, which are being assigned to each individual site. These microscopic equilibrium constants, or briefly microconstants, depend on the protonation state of the other ionizable groups within the molecule. While the macroscopic pK values of a polyprotic molecule can be obtained from a classical potentiometric titration in a straightforward fashion, the evaluation of microscopic pK values represents a lasting problem in analytical chemistry.1-4 In spite all the effort devoted to this problem in the past, a practicable procedure * Corresponding author. Phone: (315) 268-6621. Fax: (315) 268-6567. E-mail:
[email protected]. Web: http://www.clarkson.edu/∼borkovec. (1) King, E. J. Acid-Base Equilibria; Pergamon Press: Oxford, U.K., 1965. (2) Martell, A. E.; Motekaitis, R. J. The Determination and Use of Stability Constants; VCH Publishers: New York, 1988. (3) Nosza´l, B. In Biocoordination Chemistry; Burger, K., Ed.; Ellis Horwood: New York, 1990.
3272 Analytical Chemistry, Vol. 72, No. 14, July 15, 2000
is lacking. Previously, the problem was approached with optical spectroscopy and with comparative studies of similar compounds.5,6 However, these techniques were barely satisfactory, as optical spectra of polyprotic molecules are difficult to analyze and ad hoc assumptions about similarities to other molecules may prove erroneous. The situation has substantially changed with the availability of high-resolution multidimensional nuclear magnetic resonance (NMR) techniques.7 The chemical shift of a nucleus with spin 1/2 represents a sensitive probe of the protonation state of any ionizable group in its ultimate proximity. These techniques were extensively used to probe the protonation behavior in proteins,8-11 as well as that in other molecules.12-17 Since proton exchange is sufficiently fast, the chemical shift is a linear function of the degree of protonation of the neighboring ionizable sites. With the assumption that a particular nucleus responds merely to a single site, recording NMR spectra as a function of pH opens the unique possibility to measure the titration curves of individual ionizable sites in a polyprotic molecule.7 In contrast to a macroscopic titration curve (as measured by potentiometry), the site-specific titration curves contain much information about microscopic equilibria, and one would expect that all microconstants could be simply extracted from such data. (4) Borkovec, M.; Jo¨nsson, B.; Koper, G. J. M. In Surface and Colloid Science; Matijevic, E., Ed.; Plenum Press: New York, 2000; Vol. 16, in press. (5) Edsall, J. T.; Martin, R. B.; Hollingworth, B. R. Proc. Natl. Acad. Sci. U.S.A. 1958, 44, 505. (6) Ishimitsu, T.; Hirose, S.; Sakurai, H. Talanta 1977, 24, 555. (7) Slichter, C. P. Principles of Magnetic Resonance; Springer: Berlin, 1980. (8) Oda, Y.; Yamazaki, T.; Nagayama, K.; Kanaya, S.; Kuroda, Y.; Nakamura, H. Biochemistry 1994, 33, 5275. (9) Zhang, O.; Lewis, E. K.; Olivier, J. P.; Forman-Kay, J. D. J. Biomol. NMR 1994, 4, 845. (10) Kesvatera, T.; Jo¨nsson, B.; Thulin, E.; Linse, S. J. Mol. Biol. 1996, 259, 828. (11) Baker, W. R.; Kintanar, A. Arch. Biochem. Biophys. 1996, 327, 189. (12) Takeda, Y.; Samejima, K.; Nagano, K.; Watanabe, M.; Sugeta, H.; Kyogoku, Y. Eur. J. Biochem. 1983, 130, 383. (13) Delfini, M.; Segre, A. L.; Conti, F.; Barbucci, R.; Barone, V.; Ferruti, P. J. Chem Soc., Perkin Trans. 2 1980, 900. (14) Hague, D. N.; Moreton, A. D. J. Chem. Soc., Perkin. Trans. 2 1994, 265. (15) Mernissi-Arifi, K.; Schmitt, L.; Schlewer, G.; Spiess, B. Anal. Chem. 1995, 67, 2567. (16) Felemez, M.; Bernard, P.; Schlewer, G.; Spiess, B. J. Am. Chem. Soc., in press. (17) Koper, G. J. M.; van Genderen, M. H. P.; Elissen-Roman, C.; Baars, M. W. P. L.; Meier, E. J.; Borkovec, M. J. Am. Chem. Soc. 1997, 119, 6512. 10.1021/ac991494p CCC: $19.00
© 2000 American Chemical Society Published on Web 06/01/2000
However, this conclusion is not quite appropriate. Microscopic ionization equilibria of polyprotic molecules are usually parametrized in terms of all possible microconstants, but as the number of ionizable sites increases, their number becomes quickly overwhelming. Already for a simple triprotic molecule, 12 microconstants have to be determinedsthis problem was recently tackled for a triphosphate by using 31P NMR data.15 In the general case of N ionizable sites, there are 2N-1N unknown microconstants.18 In spite the availability of individual site-titration curves from NMR, one may question whether all constants can ever be reliably determined for larger molecules. In this article, we demonstrate the potential of cluster expansions18 for the determination of all microconstants from NMR titration data. Due to a completely different parametrization of the problem, the number of unknowns is reduced drastically. The approach is analogous to cluster expansions commonly used in statistical mechanics.19 Not only can the description be systematically improved by including higher order contributions, but it also incorporates inherent symmetries, which are usually not taken into account in the classical description of microscopic equilibria. The feasibility of the method is demonstrated by presenting the complete resolution of microscopic equilibria for various polyprotic molecules based on previously published NMR titration data. While this task is accomplished in a straightforward fashion, the common assumption that only the nearest nucleus responds to a single ionizable site must be revised in the sense that usually more distant sites also contribute to the NMR line shift. CLUSTER EXPANSION TECHNIQUE The protonation state of a polyprotic molecule is often characterized by the total number of bound protons n (n ) 0, ..., N, where N is the total number of ionizable sites). There are N + 1 macrostates. This macroscopic description does not discriminate between the protonation states of the individual ionizable sites. If this information is of interest, one must invoke a microscopic description, which distinguishes between the protonation states of all individual groups. These different states are best specified by introducing a two-valued state variable si for each individual site i (i ) 1, 2, ..., N) such that si ) 1 if the site is protonated and si ) 0 if the site is deprotonated. The protonation states of all the groups within the molecule, which is referred to as one particular microstate, are then specified by the set of state variables {s1, s2, ..., sN}, abbreviated as {si}. There are 2N different microstates. To each microstate {si} we can assign a standard free energy of formation F({si}), which is defined with respect to the fully deprotonated state and for unit activity of protons. Once this free energy is known, we can evaluate the probability of a microstate, as it is proportional to the Boltzmann factor. Since the chemical potential of the protons is fixed, the actual expression reads4,18
p({si}) ) Ξ-1aHn e-βF({si})
(1)
where we have introduced the activity of protons aH (where pH ) -log aH), the inverse thermal energy β, the total number of bound protons (18) Borkovec, M.; Koper, G. J. M. J. Phys. Chem. 1994, 98, 6038. (19) Robertson, H. S. Statistical Thermophysics; Prentice Hall: New Jersey, 1993.
N
n)
∑s
(2)
i
i)1
and the normalization constant
Ξ)
∑a
n H
e-βF({si})
(3)
{si}
This normalization constant can be interpreted as a partition function. So far everything is rigorous, and the only assumption is that one ionizable site can bind a single proton. However, we can simplify the situation substantially by expanding the free energy F({si}), which is a function of the discrete state variables si, into a power series. (This expansion is similar to a Taylor expansion of a function of continuous variables.) The coefficients of this expansion can be used to parametrize the free energy as
βF({si}) ln 10
1
1
∑pKˆ s + 2! ∑ s s + 3! ∑λ
)-
i i
i
ij i j
i,j
ijksisjsk
+ ...
(4)
i,j,k
where the sums run over all the sites, pK ˆ i is the negative logarithm of the microscopic dissociation constant of site i, given all other sites are deprotonated, and ij and λijk are pair and triplet interaction parameters, respectively. The pair interactions obey the symmetry relation ij ) ji, and without loss of generality we set ii ) 0. For the triple interactions, one similarly has λijk ) λjik ) λikj and λiij ) λiji ) λijj ) 0. The parameter ij was also referred to as the interactivity parameter, but the symmetry relation was not considered.3,15 In eq 4 the individual terms correspond to sums over different groups (or clusters) of sites; the first runs over all individual sites, the second over all pairs, and the third over all triplets. Such a cluster expansion can be systematically improved by considering higher order terms. For molecules with a finite number of sites, the cluster expansion terminates and is exact. As we shall see, however, the cluster expansion converges rapidly, and in many situations, consideration of pair interactions is sufficient. The pair interactions decrease quickly with increasing distance between the ionizable groups, and interactions beyond the nearest neighbors can be frequently neglected. Molecular symmetry can be used to reduce the number of independent parameters further. The microscopic equilibria can thus be fully parametrized by specifying the microconstants pK ˆ i and the interaction parameters ij (and eventually λijk). These cluster parameters are most convenient to describe the microscopic equilibria. Even for a complex molecule, their number can be moderate. The NMR chemical shift of a nucleus originates from the screening of the magnetic field by its electronic environment. Protonation of a nearby group changes this environment and leads to a change in the chemical shift. For example, protonation of a phosphate group causes the chemical shift of the associated 31P nucleus to decrease from about 5 to 1 ppm.15 For a primary amine, the chemical shift of the 15N nucleus changes from about 23 to 33 ppm upon protonation.17 Analytical Chemistry, Vol. 72, No. 14, July 15, 2000
3273
The average degree of protonation of site m can be evaluated by averaging the state variable sm over all microstates, namely
θm )
1
∑s
N
mp({si})
(6)
{si}
Using eqs 1 and 4, the above expression is readily evaluated. For the numerical implementation, it is helpful to write the microstate probabilities as a product:
p({si}) ) πn({si}) Pn(aH)
(7)
where Pn(aH) is the (pH dependent) macrostate probability and πn({si}) is the (constant) conditional probability of finding a particular microstate within its macrostate n (cf. eq 2). The macrostate probability turns out to be
Pn(aH) ) Ξ-1K h naHn
(8)
where we have introduced the cumulative association constant Figure 1. Chemical shifts of 31P NMR resonances as a function of pH for D-myo-inositol 1,2,6-triphosphate in 0.1 M (C2H5)4NBr at 25 °C.15 The labels of the P atoms are shown in Figure 2a. Solid lines are best fit results including pair interactions only. Conditions: (a) the 31P nucleus of the phosphate group responds only to its own protonation; (b) the response of the 31P nucleus includes effects of cross-terms.
More distant groups may cause a change opposite to that of the closest group.16 The rate of exchange of protons with the molecule will determine the resulting shape of the NMR spectra. Here we shall consider spectra that are taken after delays much larger than the T1 relaxation times of the nuclei, namely, under fully relaxed conditions.7 Then, the observed chemical shift of a nucleus l within a polyprotic molecule responds linearly to the protonation of neighboring sites and can be represented as20
K hn )
∑ e-βF({s })δn,∑ s i
j j
{si}
(9)
The macrostate probability is the mole fraction of all species with n bound protons. The commonly used macroscopic stepwise dissociation constants can be expressed in terms of the cumulative constants as pKn ) log K h n/K h n-1. The partition function Ξ given in eq 3 can be simplified to N
Ξ)
∑Kh a
n
(10)
n H
n)0
The conditional probability is given by
πn({si}) ) K h n-1 e-βF({si})
(11)
N
δl ) δl(0) +
∑∆
lmθm
(5)
m)1
Here δl(0) is the chemical shift of nucleus l when all groups are deprotonated (baseline), ∆lm is the chemical shift increment representing the change in the chemical shift of nucleus l, given site m changes its state from deprotonated to protonated, and θm is the average degree of protonation for this site (site-specific titration curve). We shall refer to δl(0) and ∆lm as the chemical shift parameters. Since the interaction between ionizable sites and the probed nuclear spin is rather short ranged, for a nucleus l in the ultimate proximity of ionizable site m one usually sets ∆lm ) ∆mmδlm, where δlm is the Kronecker δ. While this approximation has frequently been used in the past,8-17 it neglects all cross-terms, and as we shall see later, it is not warranted. (20) Witanowski, M.; Stefaniak, L.; Webb, G. A. Annu. Rep. NMR Spectrosc. 1986, 18, 1.
3274
Analytical Chemistry, Vol. 72, No. 14, July 15, 2000
and represents the mole fraction of a particular microstate within all microstates with n bound protons. Using eq 7, we can write eq 6 as N
θm )
∑A
mnPn(aH)
(12)
n)0
where the coefficients are given by
Amn )
∑smπn({si}) δn,∑ s
j j
{si}
(13)
Inserting eq 12 into eq 5, we recover the expression for the NMR chemical shifts as proposed earlier12 N
δl ) δl(0) +
∑B
n)1
lnPn(aH)
(14)
Figure 2. Cluster and chemical shift parameters for D-myo-inositol 1,2,6-triphosphate corresponding to fits shown in Figure 1b: (a) chemical structure; (b) cluster; (c) chemical shift parameters. Note that the phosphate group attached to the C6 carbon is labeled as 3.
Figure 3. All microconstants of D-myo-inositol 1,2,6-triphosphate calculated from the cluster parameters given in Figure 2b. The conditional probabilities of each microstate are indicated as well.
where the coefficients can be expressed as N
Bln )
∑∆
lmAmn
(15)
m)1
While Takeda et al.12 have successfully applied eq 14 to determine macroscopic pK values from NMR data, no expressions for the coefficients Bln were available to them. The present approach not only eliminates this ambiguity but also significantly reduces the number of unknown parameters. NMR titration curves can now be evaluated by direct summation of the sums over the state variables. The problem is fully parametrized by specifying the chemical shift parameters specifying the NMR response of the molecule (i.e., baselines δl(0) and chemical shift increments ∆lm) and the cluster parameters specifying the microscopic equilibria (i.e., microconstants pK ˆ i and interaction parameters ij and λijk). The major advantage of this approach is that on these parameters one can easily impose any molecular symmetries or use various simplifications based on structural considerations. We have implemented this prescription in a nonlinear least-squares regression procedure,22 and as we shall
Figure 4. Chemical shifts of 31C NMR resonances as a function of pH for 1,4,7,10,13-pentaazatridecane in D2O at total concentration of 0.2 M.14 The labels of the C atoms are defined in Figure 5a. Solid lines are best fit results including nearest neighbor pair and triplet interactions only. Conditions: (a) the amine groups respond only to the 13C nucleus in the β-position; (b) the responses of the amine groups include effects of cross-terms.
see later, meaningful parameter sets can be extracted from NMR titration data. Once all cluster parameters specifying the microscopic equilibria are known, all microscopic pK values can be calculated in a straightforward fashion. This equilibrium constant refers to a hypothetical protonation of one particular site, keeping the protonation state of all other sites the same. If we label the site to Analytical Chemistry, Vol. 72, No. 14, July 15, 2000
3275
Figure 5. Cluster and chemical shift parameters for 1,4,7,10,13-pentaazatridecane (tetren) corresponding to fits shown in Figure 4b: (a) chemical structure; (b) cluster; (c) chemical shift parameters.
be protonated with j, the protonation equilibrium can be written as
A{si} + H h A{si′}
(16)
where si ) si′ for all i * j but sj ) 0 and sj′ ) 1. Using the free energy equation (4), the microscopic pK value for the reaction given by eq 16 reads18
pK ˆ A{si} ) pK ˆi -
1
λijksjsk - ... ∑j ijsj - 2 ∑ j,k
(17)
Neglecting triplet contributions, this relation reflects the additivity of group contributions as commonly used for the estimation of pK values by Perrin et al.21 This relation also provides an intuitive interpretation of the pair interaction parameter ij; it represents the change in the microscopic pK value of site i when site j is being protonated. RESOLVING MICROSCOPIC EQUILIBRIA FROM NMR TITRATIONS We shall consider three molecules with increasing numbers of sites. From a least-squares fit of the NMR titration data, we (21) Perrin, D. D.; Dempsey, B.; Serjeant, E. P. pKa Prediction for Organic Acids and Bases, Chapman & Hall: London, 1981.
3276
Analytical Chemistry, Vol. 72, No. 14, July 15, 2000
obtain the cluster and chemical shift parameters. The cluster parameters describe the microscopic equilibria, while the chemical shift parameters describe the NMR response of the molecule. From the cluster parameters, all microconstants can be calculated. Inositol Triphosphate. Mernissi-Arifi et al.15 have presented a complete 31P NMR titration study of D-myo-inositol 1,2,6triphosphate in 0.1 M (C2H5)4NBr at 25 °C. Their data points are shown in Figure 1, and the molecular structure is displayed in Figure 2a. The full cluster expansion introduces seven parameters, ˆ 2, pK ˆ 3, 12, 23, 13, and λ123. Note that this model namely, pK ˆ 1, pK is exact for a triprotic molecule, as the cluster expansion terminates after the triplet term. We first analyzed the data above pH 3.8 by assuming that only the 31P nucleus of the phosphate group responds to its own protonation. (The experimental data at lower pH indicate the second protonation step of the phosphate group; this effect will not be considered here.) The best fit including this model is shown in Figure 1a. While all general features of the data are reproduced, the best fit is not fully satisfactory. The description can be improved by assuming that the 31P nuclei respond to the protonation of the other phosphate groups. This effect should be strongest between the most closely spaced groups, namely P1 and P2. Including these cross-terms in the leastsquares fit, we find that the diagonal chemical shift increments ∆ii for each phosphate group can be taken to be the same, as one
Figure 6. Microconstants and conditional probabilities for the most important microstates of 1,4,7,10,13-pentaazatridecane calculated from the parameters given in Figure 5b.
would expect intuitively. Similarly, both cross-terms turn out to be same. As evident from Figure 1b, the data can be described very well with this extended model, which actually contains a smaller number of adjustable parameters than the model presented previously. There is no need for any further model refinements. The resulting cluster parameters describing the microscopic equilibria are summarized in Figure 2b, while Figure 2c shows the corresponding chemical shift parameters describing the NMR response. While the 31P nucleus of the phosphate group is coupled to its own protonation most strongly, the cross-terms are by no means negligible. From the cluster parameters, all pK values can be obtained. The macroscopic pK values turn out to be 9.31, 7.20, and 5.72. The microscopic pK values are calculated using eq 17. The results are shown in Figure 3 together with the microstate probabilities (cf. eq 11). The analysis yields the following protonation pattern. The most likely event is the protonation of site 1 with the highest microscopic pK of 9.19, in the second step site 3 with a pK around 7.03 is most likely to protonate, and finally in the third step site 2 with a pK around 6.01 protonates. However, due to the small differences in the pK values, the other microstates are important as well; they are arranged in the order of decreasing probability in Figure 3. A similar pattern was suggested by Mernissi-Arifi et al.15 The present description of the data was achieved by assuming a vanishing triplet interaction parameter (λ123 ) 0). A small value for this parameter is expected due to the moderate values of the
Figure 7. Chemical shifts of 15N NMR resonances as a function of pH for a 50% solution of DAB-dendr-(NH2)8 (by weight) in water.17 The labels of the N atoms are explained in Figure 8a. Solid lines are best fit results including nearest neighbor pair and triplet interactions only. Conditions: (a) the amine groups respond only to their own protonations; (b) the responses of the 15N nuclei include effects of cross-terms.
pair interaction parameters and the substantial separation between the ionizable groups. A simultaneous fit of the data including pair and triplet interactions does not lead to any significant improvements, and the λ123 value remains indeterminate. In particular, the best fit shown in Figure 2a neglecting the NMR cross-terms terms cannot be improved by assuming a nonzero λ123. The parameter errors can be estimated from standard extensions of least-squares procedures.22 The estimated standard deviations of the cluster parameters are about 0.05, while the corresponding values for the chemical shift parameters are about twice as high. However, the actual error estimates are likely to be somewhat higher, since the cluster expansion is already truncated after the second term and no meaningful estimate of the triplet interaction term is possible. Pentaazatridecane. Detailed 13C NMR titration studies were carried out for 1,4,7,10,13-pentaazatridecane. Figure 4 shows the data acquired in D2O at a total concentration of 0.2 M by Hague et al.;14 the molecular structure is given in Figure 5a. A model based on the cluster expansion was suggested earlier by us18 for these types of molecules. The simplest but accurate model involves two microscopic equilibrium constants pK ˆ (I) and pK ˆ (II) for the primary and secondary amine groups and two nearest neighbor interaction parameters and λ for pairs and triplets, respectively. The power of the present approach now becomes fully apparent; as for the description of all microscopic equilibria, only four cluster parameters are needed. (22) Wolfberg, J. G. Prediction Analysis; D. van Nostrand Co. Inc.: New York, 1967.
Analytical Chemistry, Vol. 72, No. 14, July 15, 2000
3277
Figure 8. Cluster and chemical shift parameters for DAB-dendr-(NH2)8 corresponding to the fits shown in Figure 7b: (a) chemical structure; (b) cluster; (c) chemical shift parameters. The superscripts R, β, and γ refer to any of the nuclei in the respective shells.
The present NMR data set can be used to test this model. The carbons in the β-position respond to the protonation of the nitrogen groups much more strongly than those in the R-position. As a first approximation, we have considered this leading contribution through the parameters ∆β and ∆β′. The best fit shown in Figure 4a is already quite acceptable, but some systematic deviations remain. These can be removed by introducing the effect of the R-carbons through the constants ∆R and ∆R′. The fit with this more involved model shown in Figure 4b is rather satisfactory, and the resulting parameters are summarized in Figure 5. Indeed, the four cluster parameters are sufficient to model the data reasonably well. The resulting parameters are quite comparable to a parameter set derived from an analysis of the macroscopic pK values in a homologous series, albeit in a different medium.18 On the basis of this data set, we have also tested more involved models, such as with next nearest neighbor pair interactions, different interactions along the chain, or a different pK ˆ value for the central amine group. Decreasing the triplet interaction parameter, which involves secondary amines only, leads to a slightly better fit, but the marginal improvement does not justify an additional adjustable parameter. From the cluster parameters all pK values can be calculated. The macroscopic pK values are 10.65, 10.11, 9.34, 5.00, and 3.13. All microscopic pK values can be obtained, and some of the results are summarized in Figure 6. The derived protonation pattern is rather interesting. In the first protonation step, the protonation of a primary amine is most likely; secondary groups are more acidic and thus protonate to a lesser extent. During the second step, the other primary group is expected to protonate, but protonantion of any other distant secondary group is almost equally probable. Only the neighboring group will hardly protonate, and the corresponding microstates have a very low probability (not shown in Figure 6). The third protonation step leads to the highly symmetric microstate, which is very prominent due to strong nearest neighbor pair interactions. For the fourth 3278 Analytical Chemistry, Vol. 72, No. 14, July 15, 2000
protonation step, the further protonation of the latter microstate is likely, but it is not the most probable event. The most relevant microstate is again a symmetric one. A Poly(propylene imine) Dendrimer. As a last example, consider the highly symmetric branched polyamine DAB-dendr(NH2)8 with 14 ionizable groups. The core is 1,4-diaminobutane (DAB) to which aminopropyl chains are grafted. The titration of this molecule was studied by Koper et al.17 using 15N NMR spectroscopy for a 50% (by weight) solution in water, and the data are shown in Figure 7. The molecular structure is given in Figure 8a. The cluster expansion involving nearest neighbor pair interactions was used to model the data. The authors assumed that each nitrogen responds only to its own protonation state, which led them to the conclusion that the microscopic constants for the tertiary amines should be different. Here we shall demonstrate that the data can be equally well described by a simpler model involving a common microconstant for the tertiary amines. Our four cluster parameters involve pK ˆ (I) and pK ˆ (III), which represent the microconstants of the primary and tertiary groups, and two nearest neighbor pair interaction parameters and ′. Taking only the most important chemical shift increments ∆(RR), ∆(ββ), and ∆(γγ) into account, the best fit of this data set is not satisfactory, as shown in Figure 7a. However, the data can be explained very well by including the leading cross-terms. These effects are parametrized by the chemical shift increments ∆(Rβ), ∆(βR), ∆(βγ), and ∆(γβ). The description of the data is now substantially better, as shown in Figure 7b. The resulting parameters are summarized in Figure 8. Once the four cluster parameters are known, the microconstants can be calculated, and the important ones are summarized together with the most probable microstates in Figure 9. The protonation follows a clear pattern. Initially, all primary amines protonate, as they are the most basic groups within the molecule. The most likely microstate with eight protons represents the only exception, whereby the innermost tertiary group is protonated
and both innermost tertiary groups. This pattern reflects the repulsive nature of the nearest neighbor interactions and leads to the odd-even shell protonation pattern of dendrimers.17 CONCLUSION The cluster expansion procedure proves to be an extremely effective means for parametrizing microscopic ionization equilibria in polyprotic molecules. Since these interactions are typically of short range, a limited set of parameters usually proves sufficient to parametrize the microscopic equilibria fully. This parametrization has substantial advantages over the commonly used microscopic equilibrium constants, whose number becomes overwhelming ever for moderately sized molecules. We have shown that such cluster parameters can be extracted very effectively from NMR titration data. However, this interpretation requires the consideration of the fact that the chemical shift of a given nucleus usually responds to a multiplicity of ionizable sites simultaneously. These cross-terms can be extracted from the NMR data as well. For larger molecules, the cluster parameters represent a more economic parametrization of the protonation equilibria than the classical macroscopic ionization constants. While the number of macroscopic ionization constants is the same as the number of ionizable sites, the number of cluster parameters may increase only marginally with increasing molecular size.
Figure 9. Microconstants and conditional probabilities for the most important microstates of DAB-dendr-(NH2)8 calculated from the parameters given in Figure 8b.
instead of the outermost primary group. The most probable microstate with 10 protons is very prominent, with its characteristic arrangement of fully protonated outer shell of primary amines
ACKNOWLEDGMENT We thank Danny Stam for pointing out the importance of NMR cross-terms and J. Brussee for illuminating discussions. Received for review December 30, 1999. Accepted April 20, 2000. AC991494P
Analytical Chemistry, Vol. 72, No. 14, July 15, 2000
3279
