Global analysis for pD-variation NMR titration of organic electrolyte complexes

ABSTRACT This is an addendum to our preceding article, ‘NMR pD-variation method to determine the proper stabilities of organic electrolyte complexes: case of histidine complexes with a cyclophane acid’ Supramolecular Chemistry https://doi.org/10.1080/10610278.2022.2134017, which proposes that the species distribution of complexes between organic electrolytes is determined by observing the pD dependence of NMR signals for a mixture of the reactants; the species distribution can be replotted in a pH scale. This addendum demonstrates that the process of data analysis is simplified even more reliably by employing a global fitting method in place of a local, successive method used previously; comparative evaluation was made using Excel® spreadsheets appended. The straightforward global analysis will facilitate the practice of pD-variation NMR titration. The application range is potentially extendable by the use of 13C NMR, which overcomes some weak points of the 1H NMR method. Grapical abstract


Introduction
Our preceding paper has shown that pD-variation NMR (nuclear magnetic resonance) titration makes it possible to determine the species distribution of organic electrolyte complexes as a function of pD and then replot the distribution in the pH scale [1]. When two species are formed in different pD ranges, the true species are identified by examining the δ versus pD plots of selected NMR signals, and the proper stability constants are determined by fitting successively the relevant titration curves. This rather complicated process of data analysis is expected to be simplified, even in a more reliable manner, by the so-called global analysis in which titration curves selected in a pair are fitted simultaneously [2]. This addendum explains the use of the global analysis in pD-variation NMR titration using appended Excel® spreadsheets, as compared with the local, successive fitting method employed in the preceding paper. The two methods have been examined in a comparative manner for the typical 1:1-complexes of monobasic acid AH and monoacidic base B; the tested titration curves were generated by simulation. Then, the global method has been extendedly applied to experimental data reported for histidine complexes with a cyclophane acid [1]. These examples have demonstrated that the global analysis is much more convenient for the practice of pD-variation NMR titration.

Case of simple acid-base complexation
To help the comparative examination of the successive and global fitting methods, the titration method is summarised here along with the definitions of the variables in least-squares calculations [1]. The chemical shifts δ of organic electrolytes AH j and BH k are dependent on pD, respectively, as follows.
Here, β DAj and β DBk are the overall protonation constants of A and B in D 2 O, respectively; δ AHj and δ BHk are the intrinsic shifts of monitored proton signals in AH j and BH k . Plots of δ ðAÞ ; δ ðBÞ versus pD are shown in Figure 1 (dotted lines) for monobasic acid AH ðlog K D ¼ 6:0Þ and a monoacidic base B ðlog K D ¼ 9:0Þ; the species distributions are presented in the top panel of the figure. For a 1:1-complex formed between the acid and base, the stability constant is defined by the following equation.
The chemical shifts of AH j and BH k are derived in the rapid exchange case as follows.
Here, δ jk ðAÞ and δ jk ðBÞ are the intrinsic chemical shifts of the monitored signals of AH j and BH k in the complex; C A and C B are the total concentrations of the reactants. In each equation, the first term presents the δ versus pD curve of protonation on each electrolyte, and the second term makes the curve displaced from the corresponding curve in the non-complexation case, responding to the type of complex and the stability constant K jk . (2) the formation of AH·BH or A·B displaces one titration curve but does not the other; (3) when specie A·BH is accompanied by a by-product A·B, for example, the net displacement of B01 is reduced due to the secondary species, whereas A01 is not affected; (4) AH·B is not detectable because of the low NMR chemical shift δ versus pD plots simulated for components A and B in 1:1-complexes AH j �BH k (j, k = 0 or 1): the plots are labelled Ajk and Bjk, respectively; C A �K jk ¼ 10; concentrations of the components in the pD range of complexation. On the basis of these features of curve displacements, the formed species AH j �BH k are identified, and the proper stability constants are determined by least-squares fitting of suitably selected curves; the experimental requirement is that the titration curves of each reactant and a mixture should be obtained under the same conditions including pH-pD conversion.
In the present examination of the methods of data analysis, titration curves are simulated for the formation of A·BH and A·B with the K 01 :K 00 ratio of 10:1 and the variables of 'true values' in Table 1. Figure 2 shows tested titration plots of δ ðAÞ and δ ðBÞ , which were generated by adding random noises of ±0.005 to δ and ±0.04 to pD (these high noise levels were intentionally chosen for examination in the worst scenarios). The stability constants are calculated step-by-step by reference to the above-described features: (1) the opposite displacements of δ ðAÞ and δ ðBÞ identify A·BH as the major species; (2) the stability constant K 01 of the complex is equated to the value determined by the local curve fitting of δ ðAÞ rather than that of δ ðBÞ because the former value is larger than the latter, and the minor species is identified to be A·B for the same reasoning; (3) the stability constant K 00 of the minor species is determined by successive curve fitting of δ ðBÞ under the constraint of K 01 to the value determined from δ ðAÞ . The parameter values obtained in each step are shown in Table 1.
In the global analysis, the process of steps 2 and 3 is performed more definitely in a single step. The simultaneous curve fitting of δ ðAÞ and δ ðBÞ for the formation of A·BH and A·B gave the reasonable results compared with the true values (cf. global fit 1 in Table 1). By contrast, another curve fitting by assuming A·BH and AH·BH yielded the illogical results like a negative value of log K jk (global fit 2), ruling out the formation of AH·BH. Thus, the global method makes the data analysis straightforward. In addition, the fitness of curve δ ðAÞ is improved over the entire pD range including the high pD range in which the minor species is formed, resulting in a slightly smaller standard deviation for the log K 01 of the major species and a larger deviation for the log K 00 of the minor species; the reliabilities of the constants may be more fairly evaluated in the global analysis.
One of the disadvantages of the pD-variation 1 H NMR method is that the formation of A·B complex, for example, displaces only the titration curve of reactant B; if this reactant does not carry a proton 3.177 (2) 3.026 (7) 3.023 (5) Global fit 2 sensitive to protonation, the titration method is useless. Exceptionally, if the complexation causes a significant δ change for A, the stability constant K 00 is possible to determine: Figure 3 examples the titration curves of A in A·B-complexation accompanied by selected δ changes, Δ 00 ¼ δ 00 ðAÞ À δ A , at a noise level of ±0.002 for δ and ±0.04 for pD. The stability constant was correctly determined with acceptable standard deviations as shown in the inset of the figure; obviously, the larger δ change results in the higher reliability. In this practice, a low degree of data scatterings is required: in the case of the present simulation at a noise level of ±0.005, K 00 =M À 1 obtained for Δ 00 ¼ 0:02 was 1720, which largely differed from the true value 1000 (with an unacceptable standard deviation of 1820), in contrast to the value K 00 ¼ 1000 576 ð Þ at the ±0.002 noise level; for Δ 00 ¼ 0:06; K 00 ¼ 650 220 ð Þ at the ±0.005 noise level, while K 00 ¼ 1052 190 ð Þ at the ±0.002 level. These weak points of the 1 H NMR method may be overcome by 13 C NMR. The potential advantage of 13 C NMR is that it may show additional signals sensitive to protonation and complexation, e.g., the signals of carbon atoms in carboxylate group and substituted heterocycles; hence, 13 C NMR has a better chance to present multiple signals suitable for being monitored, even in the A·B-complexation case. Another possible advantage is that H-decoupled sharp signals are  Table 1 and by adding random noises of ±0.005 to δ and ±0.04 to pD: the solid lines present least-squares fits based on the simultaneous method (or global analysis); the local, successive method gave practically the same goodness of fit; the obtained parameter values are shown in Table 1; the dotted lines are δ versus pD plots in the non-complexation case. generally observed with large chemical shifts, and consequent small relative errors and low data-scattering degrees are expectable. Moreover, a larger number of signals may be useful for a global analysis. Potentially, therefore, 13 C NMR can extend the application ranges of the pD-variation method, with the ready use of global analysis, despite a longer time necessary for experiments.

Case of histidine complexes
The NMR signal of ring proton CH(2) in histidine responds mainly to protonation at the ring nitrogen, and that of aliphatic proton CH(α) does to protonation at the side-arm nitrogen; Figure 4 presents the pD dependence of the proton signals and the distribution of protonation species. The δ versus pD plots of the proton signals are displaced in the titration of a mixture with a cyclophane acid (cf. Scheme 1); the protonation status of the cyclophane, the experimental process (including pD determination), and the observed spectral changes were described in the preceding paper [1]. The formed complexes were identified as hdH·cyH and hd·cyH, and the stability constants were determined by local, successive curve fittings of the titration curves of CH (2) and CH(α) protons, as shown in Table 2 [1]. The simultaneous fitting of the two curves gives the consistent results in a simpler calculation process. The standard deviations suggest that the global analysis evaluates the reliabilities of the stability constants more reasonably.  Table 2; the dotted lines are δ versus pD plots in the noncomplexation case.

Scheme 1.
Structures of histidine and cyclophane at pH ≈ 7 and the abbreviations.

Conclusion
The pD-variation NMR titration is advantageous for studies of organic electrolyte complexes. When two species are formed with different protonation statuses, they are identified by comparing the titration curves of suitably selected proton signals, and then the proper stability constants are determined by fitting the relevant titration curves. These processes of data analysis are made straightforward and more reliable by employing the global analysis, which may help the practice of the titration method.

Disclosure statement
No potential conflict of interest was reported by the author(s).

Funding
This work was supported by División de Ingeniería de la Universidad de Sonora (Grant no. USO316007870).