Untangling the Diverse Interior and Multiple Exterior Guest Interactions

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Untangling the Diverse Interior and Multiple Exterior Guest Interactions of a Supramolecular Host by the Simultaneous Analysis of Complementary Observables Carmelo Sgarlata and Kenneth N. Raymond* Chemical Sciences Division, Lawrence Berkeley National Laboratory, and Department of Chemistry, University of California, Berkeley, California 94720, United States S Supporting Information *

ABSTRACT: The entropic and enthalpic driving forces for encapsulation versus sequential exterior guest binding to the [Ga4L6]12− supramolecular host in solution are very different, which significantly complicates the determination of these thermodynamic parameters. The simultaneous use of complementary techniques, such as NMR, UV−vis, and isothermal titration calorimetry, enables the disentanglement of such multiple host−guest interactions. Indeed, data collected by each technique measure different components of the host− guest equilibria and together provide a complete picture of the solution thermodynamics. Unfortunately, commercially available programs do not allow for global analysis of different physical observables. We thus resorted to a novel procedure for the simultaneous refinement of multiple parameters (ΔG°, ΔH°, and ΔS°) by treating different observables through a weighted nonlinear least-squares analysis of a constrained model. The refinement procedure is discussed for the multiple binding of the Et4N+ guest, but it is broadly applicable to the deconvolution of other intricate host−guest equilibria.

T

he design of artificial receptors containing well-defined binding pockets that may effectively mimic the properties of enzymes is a current intriguing challenge in supramolecular chemistry.1−5 Enzymes bind their substrates through hydrophobic and often sterically constrictive cavities and activate them through the concerted action of multiple cumulative noncovalent interactions. These interactions, such as hydrogen bonding, CH−π, π−π, and cation−π, are responsible for selfassembly and host−guest recognition processes,6−8 and their accurate evaluation is a key issue for the successful design and synthesis of supramolecular catalysts.9 However, the weak, reversible, and dynamic nature of such noncovalent forces makes their relative contribution to the overall free energy of binding difficult to dissect, which complicates the disentanglement of enthalpic and entropic contributions to binding.10 These thermodynamic parameters as well as the overall binding affinities are usually measured by solution NMR, UV−vis spectroscopy, or isothermal titration calorimetry (ITC), but each of these techniques intrinsically has unique issues related to their different time scales and physical observables. Multiple consecutive associations between the supramolecular host and guest molecules further complicate the determination of the species distribution and the analysis of thermodynamic parameters. The self-assembled supramolecular cluster [Ga4L6]12− [1, Figure 1; L = 1,5-bis(2,3-dihydroxybenzamido)naphthalene] can act as a host for suitable cationic and neutral guest © XXXX American Chemical Society

Figure 1. Left: schematic view of the supramolecular host 1; the bisbidentate ligands are represented by blue lines and the gallium atoms by red circles. Right: space-filling model of 1.

molecules.11−13 The metal−ligand framework of the host generates a large and hydrophobic interior cavity capable of encapsulating guest molecules of appropriate size14 and a highly anionic exterior surface which promotes external association of cationic molecules, and renders the host water-soluble.15−17 The interior microenvironment of host 1, which differs Received: April 28, 2016 Accepted: May 31, 2016

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The simultaneous presence of multiple species in solution, the high correlation among the many parameters to be refined, and the clear benefit to combining data from different complementary techniques make commercially available programs inadequate for the accurate quantitative analysis of intricate host−guest binding equilibria. Indeed, any attempt to refine multiple stepwise binding constants and thermodynamic parameters by independent analysis of different sets of observables through commonplace software26−32 always failed or provided unreliable results. Since no available program fulfills the above requirements, we developed a novel procedure for the simultaneous refinement of multiple complementary observables through a weighted nonlinear least-squares analysis, which makes use of constraints.33 This optimization process is used to find the regression coefficients (or thermodynamic parameters) that give the best fit of a function to a set of data points. The best fit of the curve is found when the sum of the squares of the weighted deviations (or residuals) of the data points from the calculated curve is minimized. For most chemical systems, the equations relating the dependent (the observable) and independent (the concentration) variables are known, and therefore, the final goal of the analysis is to obtain the values of the parameters of that equation. When this procedure is appropriately applied to different sets of observables which look at different components of multiple and often competing host−guest equilibria, the determination of the binding constants and the enthalpy and entropy change for the interior and multiple exterior guest binding to the host can be achieved. Estimation of the errors and correlation coefficients in this simultaneous minimization technique is a key part of the calculation.34 The simultaneous curve-fitting analysis described in detail for the Et4N+−1 complexes is of general applicability and can be readily applied to other complicated multiple equilibria that can be deconvoluted only by collecting and simultaneously analyzing complementary observables.

dramatically from the bulk solvent environment, has been successfully used to modify the physical properties and reactivity of encapsulated guest molecules18,19 and to catalyze chemical transformations20−22 with rate accelerations of up 2 × 107.23 This catalytic activity is achieved by transition state stabilization in a manner similar to the commonly accepted mechanism of action for enzymatic catalysis. However, the specific noncovalent interactions responsible for this host−guest chemistry are difficult to study directly due to the different enthalpic and entropic contributions to the overall free energy of binding. We have previously reported on the deconvolution of the very different driving forces for internal and external binding of Et4N+ by host 1 in water.24 This was achieved analyzing data obtained from three complementary techniques (NMR, ITC, and UV−vis) using suitable commercially available software packages. Each NMR, ITC, and UV−vis experiment yielded a set of observables, which is sensitive to a different aspect of the host− guest binding equilibria.25 However, the analysis of the data is complicated by the variety of responses provided by each technique. In NMR experiments, the observable for a guest bound to the interior of the cluster (in the slow exchange regime) is a set of resolved resonances, with the peak corresponding to internal association shifted significantly upfield by close contact with the ring current of the naphthalene walls. External ion-associated and nonassociated (free) guests are seen as one average species because of fast exchange on the time scale of the experiment. The calorimetric observable is the total heat change for any (exterior and interior) reaction occurring, and the UV−vis observable is the change in the host absorbance due to external guest association. The treatment of different observables implies that many different parameters (integral and chemical shift for NMR, molar absorption for UV−vis, enthalpy change for ITC) have to be refined to obtain both the binding constants and the other relevant thermodynamic information. This analysis is further complicated by the fact that a guest (G), such as Et4N+, may interact with both the internal and external space of host 1 (H) by a stepwise series of equilibria, which are mutually linked (Table 1).

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RESULTS AND DISCUSSION Simultaneous Analysis of Different Observables. The simultaneous treatment of titration data having different physical meaning and units (often ranging over several orders of magnitude) for the determination of stability constants or other thermodynamic parameters is achieved in the present study by using a weighted nonlinear least-squares analysis. The function that is minimized is the sum of the squared residuals, R2, defined as

Table 1. Multiple Internal and External Guest Equilibria Occurring in Solution with the Supramolecular Host 1 equilibria G G G G G

+ + + + +

H ⇆ G⊂H G⊂H ⇆ (G⊂H)G (G⊂H)G ⇆ (G⊂H)G2 (G⊂H)G2 ⇆ (G⊂H)G3 (G⊂H)G3 ⇆ (G⊂H)G4

event encapsulation 1st exterior association 2nd exterior association 3rd exterior association 4th exterior association

n

R2 =

∑ wi(yiobs − yicalc )2 i=1

yobs i

(1)

ycalc i

where and are the observed and calculated value for each ith data point, wi = 1/σi2 is the weighting factor, and σi is the uncertainty assigned to the observation yobs i . The residuals corresponding to different classes of observables are weighted by their experimental errors. The weighting procedure causes the residuals to be unitless numbers which can be appropriately refined together within the same analysis in spite of their originally diverse units and meaning. This is a key point of our fitting procedure as it allows for the global refinement of data probing different aspects of the host−guest equilibria. Since some observations are more accurate than others, the use of the weighting factor emphasizes the contribution to the R2 sum of

Within this framework, the treatment of different observables for the refinement of multiple parameters, each looking at different components of the reactions shown in Table 1, is challenging, since these parameters are strongly correlated with one another and are often indistinguishable within their experimental error [for example, the entropic and enthalpic contributions to the ith and (i + 1)th external association are strongly correlated and therefore difficult to reliably disentangle]. Furthermore, each unique observable often requires the use of a different software package capable of processing the different sets of raw data. B

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those experimental data that have a smaller uncertainty. If the standard deviations are correctly assigned for different types of data, then the result is an unbiased estimate of the model parameters. The overall assessment of the correctness of the standard deviations can be evaluated by the “goodness of the fit” (GOF) within each class of data

GOF =

1 n−p

⎛ y obs − y calc ⎞2 i ⎟ ∑ ⎜⎜ i ⎟ σ i ⎠ i=1 ⎝

Δ = Kext(i)/Kext(i − 1)

ratio of the binding constants of two sequential exterior binding equilibria (i > 1)

The imposition of the constraint implies that the value of the binding constant for each successive exterior ion-association step is forced to decrease by the same factor Δ (with Δ < 1). This assumption makes the least-squares refinement better conditioned while providing the association constant for any sequential exterior binding reaction simply combining the refined parameters Kenc, Kext(1), and Δ (see Supporting Information). Since the successive exterior binding steps are physically similar to one another and the affinities associated with each of them decrease by the same constant amount Δ, we made the further assumption that the corresponding ΔH° and ΔS° values have to decrease accordingly41 in such a way that the ratio ΔH°/TΔS° remains constant as well. In other words, the consistent linear decrease of the ΔG° value for subsequent exterior ion-association equilibria accounts for the application of a further constraint by which the ΔH°/TΔS° ratio is required to have a constant value (α). Since

n

(2)

where n is the number of points and p is the number of parameters to be refined. GOF should be close to 1 for all the various classes of data if all the weights are properly assigned. If this is not the case, the errors of each class of observables (which might differ by orders of magnitude) can be multiplied by its GOF in order to obtain a more uniform set of weighed residuals to be summed in the final R2 function. The uncertainties associated with each different class of observables measured by using different techniques (NMR, UV−vis, ITC) were calculated and assigned based on various secondary literature sources35,36 and relevant papers.37,38 The equations employed for the calculation of ycalc for each different i technique are based on the familiar relationships between the observables (NMR integral, NMR chemical shift, absorbance, and heat values) and the concentration of the species in solution. In general, such observables are linearly related to the concentration (molar fraction, molarity, changes in the number of moles, etc.) with the coefficient usually being a parameter to be refined (δ, ε, ΔH°, etc.). Each observable represents the combined contribution of consecutive association stoichiometries, and thus the determination of the species distribution is a crucial point of the calculation. Species are deconvoluted by solving the massbalance equations and arranging the expressions for the calculation of ycalc for each different technique as a function i of the (unknown) concentration of the free guest, the (known) total concentrations of host and guest, and eventually the stability constant value (to be refined). Relevant equations are illustrated in the Supporting Information. The application of some constraints to the parameters of the least-squares refinement is required to account for the interdependence of correlated consecutive equilibria.39 The constraints usually derive from the imposition of known conditions or suitable relationships involving the parameters that are free to vary. In the present host−guest system, our model assumes that the binding affinities for sequential exterior associations would decrease by a given constant factor, given that, with each consecutive association, the host decreases in overall negative charge and unoccupied external binding sites. Similar behavior has been reported for series of complexes resulting from a certain number of identical ligands and a given metal ion for which their successive stability constants are found to decrease consistently in magnitude40,41 absent confounding issues of cooperativity.42 Equations proposed by different authors43,44 have proved to be quite satisfactory to predict the values of steadily decreasing stepwise stability constants. Our assumption dramatically reduced the number of regression coefficients to be refined for the internal and multiple external equilibria to the following: Kenc

binding constant for the first exterior binding equilibrium (i = 1)

(3)

ΔG° = ΔH ° − T ΔS°

and we assume that the ratio ΔH ° = α is constant for all the T ΔS ° exterior ion-association equilibria, then eq 3 becomes ΔH ° = ΔG° +

ΔH ° α

(4)

from which

ΔH ° =

α ΔG° α−1

(5)

and

ΔS° =

ΔH ° αT

(6)

The equation for the calculation of the enthalpy change of a reaction obtainable by calorimetric measurements (shown in the Supporting Information) can be modified as in eq 7 n

−Q = ΔH °enc δnenc +

∑ ΔH °ext(i) δnext(i) i=1

(7)

where ΔH°enc and ΔH°ext(i) are the enthalpy changes for the encapsulation and any ith exterior binding equilibria, respectively, and δnenc and δnext(i) are the change of number of moles of the encapsulated and any ith exteriorly bound species, respectively. ΔH°enc is a parameter to be refined, whereas ΔH°ext(i) can be calculated combining eq 5 with eq 8 (8)

ΔG = −RT ln K

to obtain ΔH °ext(i) = −

α 2.303RT log Kext(i) α−1

(9)

Similarly, the entropy change for each exterior binding equilibria, ΔS°ext(i), can be obtained by eqs 6 and 9 ΔS°ext(i) = −

binding constant for the encapsulation equilibrium C

2.303R log Kext(i) α−1

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Table 2. Thermodynamic Parameters for the Interior and Exterior Binding of Et4N+ to Host 1 at 25 °C in Water (0.1 M KCl)a

a Parameters for the exterior binding equilibria are obtained as a combination of the refined parameters Kext(1), Δ, and α. The refined values for the latter two parameters are Δ = 0.29(2) and α = 1.89(1).

Figure 2. Curve-fitting (observed/calculated) obtained from the simultaneous refinement of NMR, UV−vis, and ITC data for the Et4N+−1 system in water. Although fits to data are shown in different plots, they have been processed simultaneously within the same weighed least-squares analysis.

The use of the constraints Δ and α effectively simplified the least-squares analysis of the many different observables as it severely reduced the number of parameters to be refined. All the thermodynamic parameters dealing with the interior and multiple exterior ion-association equilibria of host 1 can be easily calculated by combining the refined parameters Kenc and ΔH°enc (for the encapsulation reaction) as well as Kext(1), Δ, and α (for the external binding events).

Encapsulation and Multiple External Ion-Association of Et4N+. The simultaneous refinement of the Et4N+ titration data24 with this procedure yielded the binding constants and the thermodynamic parameters shown in Table 2. Figure 2 shows the fit of the observed and calculated values for observables collected by three complementary techniques (NMR, UV−vis, and ITC). Although these data are displayed in different plots, it is worth noting that they have been processed simultaneously within the same weighed leastD

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Table 3. Calculated Thermodynamic Parameters for the External Ion-Association of Et4N+ to the Empty Host and the Successive Encapsulation of This Ion-Associated Guest into the Host Cavity

endoentropic) due to strong attractive interactions, and the values compare well with those previously reported for the exterior binding.24 On the other hand, the encapsulation of the ion-associated Et4N+ into the cavity is endothermic and highly exoentropic due to desolvation. Extrapolation of parameters earlier determined11 through the temperature dependence of the binding constant for the encapsulation of cationic guests, such as Pr4N+ and Me2Pr2N+, seems to be consistent with the entropic and enthalpic contributions to encapsulation of the smaller Et4N+ into host 1. On the basis of the smaller size and length of the alkyl chains of the guest, ΔH° and ΔS° of ∼30 kJ mol−1 and ∼140 J deg−1 mol−1 may be reasonably estimated for the inclusion of Et4N+ into 1. Remarkably, these values nicely match with those shown in Table 3 for the encapsulation of the ion-associated guest (i.e., the second event) and further prove the key role played by the entropic drive to the effective guest encapsulation into the host interior space. Conversely, the multiple exterior association of Et4N+ is confirmed to be an enthalpy-driven process ascribable to enthalpically favorable attractive forces (such as electrostatic, cation−π, and CH−π interactions)6−8 between the guest and the aromatic exterior walls of the host. Such exothermic external association events are also entropically disfavored since only partial desolvation of the guests, which progressively bind an increasingly crowded host exterior, is required.50,51 Results in Tables 2 and 3 unambiguously show that the solution thermodynamics of multiple and competing host− guest events can be successfully deconvoluted through the multiobservables curve-fitting procedure described.

squares analysis. The excellent agreement between measured and calculated values verifies the assumptions of the model and the correct assignment of the relative errors (weights) to the observables. Table 2 shows that one encapsulated and up to four exterior bound guest species for the Et4N+−1 system were refined by imposing the constraints Δ and α which dramatically reduced the correlation among the regression coefficients (see Table S1) and improved the reliability of the final results. Data in Table 2 are in excellent accord with those previously reported24 for the encapsulation and the first exterior binding event by independently analyzing NMR, ITC, and UV−vis data using commercially available software packages. Data in Table 2 show that the encapsulation of Et4N+ by host 1 is a predominantly entropy-driven process. The release of solvent molecules from the interior of the solvent-filled cavity of 1 and the desolvation of the cationic guest account for the large entropic gain.45−49 The encapsulation is an enthalpyfavored process despite the cost that has to be paid for the desolvation of both the of host and guest. We suggest that the encapsulation process is the combination of two reactions:11,12,24 (1) the association of the positively charged guest to the exterior of the empty 12− host (similarly to the external ion-association steps illustrated in Table 2) and (2) the encapsulation of this ion-associated Et4N+ into the host cavity. The thermodynamic parameters of these two reactions cannot be determined by direct measurement but can be easily calculated through the refined parameters Kext(1), Δ, and α. Empty host indicates that the cavity of 1 has no encapsulated guest but is expectedly occupied by solvent molecules. Cavity volumes previously measured (∼250 Å3) indicated that about 8−10 water molecules can occupy the host interior,17 and this estimate was found consistent with the weight loss observed in the thermogravimetric analysis.24 Indeed, assuming that the guest binding to the exterior of the empty host is a process basically similar to the linear stepwise ion-association equilibria shown in Tables 1 and 2, the affinity constant for this first event, Kext(empty), can be determined by the Kext(1)/Δ ratio. Accordingly, the corresponding ΔH°ext(empty) and ΔS°ext(empty) values can be calculated from Kext(empty) and α using eqs 9 and 10. With these values in hand, the affinity as well as the enthalpic and entropic terms for the next step (the encapsulation of the ion-associated guest into the host cavity) can be evaluated. Values calculated for the two equilibria contributing to the overall encapsulation process are shown in Table 3. The association of the positively charged guest to the exterior walls of the empty anionic host is highly exothermic (and

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CONCLUSION The combination of NMR, UV−vis, and ITC data within the same weighted nonlinear least-squares analysis enabled the accurate determination of the ΔG°, ΔH°, and ΔS° parameters for the encapsulation and multiple exterior ion-association of Et4N+ to host 1 in water. The assignment of the appropriate uncertainty to each observable made possible the simultaneous refinement of a model using data from different types of observation. The use of suitable constraints, which impose successive stability constants to decrease consistently in magnitude, dramatically reduced the number of parameters to be refined and provided a well-conditioned refinement. Data obtained by different techniques are sensitive to different aspects of multiple host−guest equilibria, and only when globally refined can they provide an accurate overall picture of the solution thermodynamics. The separate analysis of each individual observable is like the famous story of the E

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Mugridge, David Kaphan, and Professor Giuseppe Arena for helpful discussions.

blind men and an elephant, where men examining the elephant concluded that the elephant is like very different things, depending upon what part they touch.52 We conclude that our multiobservable analysis is able to yield reliable thermodynamic parameters and can be of use to a broad audience dealing with host−guest or metal−ligand solution equilibria. We encourage the development of a userfriendly version of this spreadsheet to suitably expand its functionalities and area of application.

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(1) Yoshizawa, M.; Klosterman, J. K.; Fujita, M. Angew. Chem., Int. Ed. 2009, 48, 3418−3438. (2) Pluth, M. D.; Bergman, R. G.; Raymond, K. N. Acc. Chem. Res. 2009, 42, 1650−1659. (3) Avram, L.; Cohen, Y.; Rebek, J., Jr. Chem. Commun. 2011, 47, 5368−5375. (4) Wiester, M. J.; Ulmann, P. A.; Mirkin, C. A. Angew. Chem., Int. Ed. 2011, 50, 114−137. (5) Castilla, A. M.; Ramsay, W. J.; Nitschke, J. R. Acc. Chem. Res. 2014, 47, 2063−2073. (6) Meyer, E. A.; Castellano, R. K.; Diederich, F. Angew. Chem., Int. Ed. 2003, 42, 1210−1250. (7) Ma, J. C.; Dougherty, D. A. Chem. Rev. 1997, 97, 1303−1324. (8) Nishio, M. Tetrahedron 2005, 61, 6923−6950. (9) Brown, C. J.; Toste, F. D.; Bergman, R. G.; Raymond, K. N. Chem. Rev. 2015, 115, 3012−3035. (10) Schmidtchen, F. P. Chem. Soc. Rev. 2010, 39, 3916−3935. (11) Parac, T. N.; Caulder, D. L.; Raymond, K. N. J. Am. Chem. Soc. 1998, 120, 8003−8004. (12) Caulder, D. L.; Powers, R. E.; Parac, T. N.; Raymond, K. N. Angew. Chem., Int. Ed. 1998, 37, 1840−1843. (13) Biros, S. M.; Bergman, R. G.; Raymond, K. N. J. Am. Chem. Soc. 2007, 129, 12094−12095. (14) Mugridge, J. S.; Zahl, A.; van Eldik, R.; Bergman, R. G.; Raymond, K. N. J. Am. Chem. Soc. 2013, 135, 4299−4306. (15) Fiedler, D.; van Halbeek, H.; Bergman, R. G.; Raymond, K. N. J. Am. Chem. Soc. 2006, 128, 10240−10252. (16) Pluth, M. D.; Tiedemann, B. E. F.; van Halbeek, H.; Nunlist, R.; Raymond, K. N. Inorg. Chem. 2008, 47, 1411−1413. (17) Pluth, M. D.; Johnson, D. W.; Szigethy, G. S.; Davis, A. V.; Teat, S. J.; Oliver, A. G.; Bergman, R. G.; Raymond, K. N. Inorg. Chem. 2009, 48, 111−120. (18) Pluth, M. D.; Bergman, R. G.; Raymond, K. N. Science 2007, 316, 85−88. (19) Hart-Cooper, W. M.; Sgarlata, C.; Perrin, C. L.; Toste, F. D.; Bergman, R. G.; Raymond, K. N. Proc. Natl. Acad. Sci. U. S. A. 2015, 112, 15303−15307. (20) Wang, Z. J.; Brown, C. J.; Bergman, R. G.; Raymond, K. N.; Toste, F. D. J. Am. Chem. Soc. 2011, 133, 7358−7360. (21) Wang, Z. J.; Clary, K. N.; Bergman, R. G.; Raymond, K. N.; Toste, F. D. Nat. Chem. 2013, 5, 100−103. (22) Hastings, C. J.; Pluth, M. D.; Bergman, R. G.; Raymond, K. N. J. Am. Chem. Soc. 2010, 132, 6938−6940. (23) Kaphan, D. M.; Levin, M. D.; Bergman, R. G.; Raymond, K. N.; Toste, F. D. Science 2015, 350, 1235−1238. (24) Sgarlata, C.; Mugridge, J. S.; Pluth, M. D.; Tiedemann, B. E. F.; Zito, V.; Arena, G.; Raymond, K. N. J. Am. Chem. Soc. 2010, 132, 1005−1009. (25) Thordarson, P. Chem. Soc. Rev. 2011, 40, 1305−1323. (26) Frassineti, C.; Alderighi, L.; Gans, P.; Sabatini, A.; Vacca, A.; Ghelli, S. Anal. Bioanal. Chem. 2003, 376, 1041−1052. (27) Fielding, L. Tetrahedron 2000, 56, 6151−6170. (28) Gampp, H.; Maeder, M.; Meyer, C. J.; Zuberbühler, A. D. Talanta 1985, 32, 95−101. (29) Gans, P.; Sabatini, A.; Vacca, A. Talanta 1996, 43, 1739−1753. (30) Gans, P.; Sabatini, A.; Vacca, A. J. Solution Chem. 2008, 37, 467− 476. (31) ITC Data Analysis in Origin, Tutorial Guide; MicroCal: Northampton, MA, 2004. (32) NanoAnalyze Software, Getting Started Guide; TA Instruments: New Castle, DE, 2013. (33) Raymond, K. N. Acta Crystallogr., Sect. A: Cryst. Phys., Diffr., Theor. Gen. Crystallogr. 1972, 28, 163−166. (34) Hamilton, W. C. Statistics in Physical Science; The Ronald Press Company: New York, 1964.

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EXPERIMENTAL SECTION General. The supramolecular cluster 1 was synthesized as previously reported.12 NMR, UV−vis, and ITC titrations were carried out as described in ref 24. Data Analysis. Titration data (NMR integrals, chemical shift, absorbance, and heat values along with the host and guest total concentrations) were simultaneously handled by a weighted nonlinear least-squares analysis through Excel 2007 software. The spreadsheet columns contain, for each titration and set of observables, the total concentrations of host and guest, the polynomial function of the free guest concentration which allows for the calculation of the species distribution (see Supporting Information), the observed and the calculated values, the uncertainties associated with each observable, and the residuals. The sum of the weighed squared residuals n R2 = ∑i = 1 wi(yiobs − yicalc )2 is then minimized using the Solver, an Excel add-in that can find a minimum by changing the values of the cells containing the parameters to be refined. The process is repeated iteratively until a global minimum and satisfactory parameters are obtained. Different starting guess values are used to rule out local minima. Standard deviations of the refined parameters and the correlation coefficients, which are not provided by the Solver, are computed using the macros reported in the literature.53,54 The uncertainties on the calculated parameters [Kext(i) (i > 1), ΔH°ext(i), and ΔS°ext(i) (i ≥ 1) as well as ΔS°enc] are obtained by propagating the errors of the refined parameters (Kenc, ΔH°enc, Kext(1), Δ, and α).

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.analchem.6b01684. Observed and calculated values for NMR, UV−vis, and calorimetry experiments, equations for multiple host− guest binding equilibria, and correlation coefficients matrix of the refined parameters (PDF) Et4N+−host sample spreadsheet (XLS)

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REFERENCES

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research was supported by the Director, Office of Science, Office of Basic Energy Sciences, and the Division of Chemical Sciences, Geosciences, and Biosciences of the U.S. Department of Energy at LBNL (DE-AC02-05CH11231). We thank Dr. Jeff F

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DOI: 10.1021/acs.analchem.6b01684 Anal. Chem. XXXX, XXX, XXX−XXX