Properly handling negative values in the calculation of binding constants by physicochemical modeling of spectroscopic titration data

Abstract

To implement equilibrium hard‐modeling of spectroscopic titration data, the analyst must make a variety of crucial data processing choices that address negative absorbance and molar absorptivity values. The efficacy of three such methodological options is evaluated via high‐throughput Monte Carlo simulations, root‐mean‐square error surface mapping, and two mathematical theorems. Accuracy of the calculated binding constant values constitutes the key figure of merit used to compare different data analysis approaches. First, using singular value decomposition to filter the raw absorbance data prior to modeling often reduces the number of negative values involved but has little effect on the calculated binding constant despite its ability to address spectrometer noise. Second, both truncation of negative molar absorptivity values and the fast nonnegative least squares algorithms are superior to unconstrained regression because they avoid local minima; however, they introduce bias into the calculated binding constants in the presence of negative baseline offsets. Finally, we establish two theorems showing that negative values are best addressed when all the chemical solutions leading to the raw absorbance data are the result of mixing exactly two distinct stock solutions. This allows the raw absorbance data to be shifted up, eliminating negative baseline offsets, without affecting the concentration matrix, residual matrix, or calculated binding constants. Otherwise, the data cannot be safely upshifted. A comprehensive protocol for analyzing experimental absorbance datasets with is included.

Sensitivity limits for determining 1:1 binding constants from spectrophotometric titrations via global analysis

“…In these simulations, spiked profiles are run to 1.5 equivalents to ensure that the 1:1 equivalence is reached near the same point during the titration. Spiked profiles should be employed when the guest species does not absorb so that the dataset may be safely shifted upwards, eliminating negative values and any associated bias without introducing new errors . These profiles are additionally employed in the existing experimental literature to reduce the possibility of aggregation side reactions, to avoid unnecessary dilution factors in the data analysis, and to permit easier visual identification of the titration endpoint.…”

“…This technique achieves superior resolution over averaging results from single‐wavelength vectors by making the global minimum more sharply defined on the error surface . The basic implementation of global analysis in physicochemical modeling (also known as hard modeling) has been explained in detail elsewhere . Briefly, the idea is to decompose the absorbance matrix, D , into the matrix product of smaller matrices of molar absorptivities, R , and equilibrium concentrations, C , with an additive residual error matrix: D = RC + E .…”

A reliable algorithm for calculating stoichiometry parameters in the hard modeling of spectrophotometric titration data

“…While statistical methods such as the Hamilton R-factor, Pearson R-factor, mean error of prediction, and Akaike information criteria can be employed, 20,21 previous work has shown these methods frequently fail to predict the correct model owing to experimental error in the initial concentration (predictor variables) and the many degrees of freedom in the fitted absorptivity curves. 16 Furthermore, the number of reactions considered does not vary in a stoichiometry optimization calculation, so the use of information criteria metrics does not alter the model rankings. 43,44 We therefore take the view that the RMSE metric is sufficient, and we employ chemical intuition based on inspection of the stoichiometry ratios and absorptivity curves whenever it is necessary to distinguish between models with different degrees of freedom.…”

“…The mathematical procedure of hard modeling has been described in great detail in previous literature 6 . When the exact equilibrium binding model is known, the hard‐modeling process is in general very time efficient, requiring at most a couple minutes to obtain results even for complex equilibrium models 16 …”

“…6 When the exact equilibrium binding model is known, the hard-modeling process is in general very time efficient, requiring at most a couple minutes to obtain results even for complex equilibrium models. 16 However, in many chemical systems of interest, the chemical reactions comprising the binding model are unknown because the stoichiometry of the chemical species involved is unknown 17,18 ; in some cases, it may be challenging to even narrow down the possibilities to a range of reasonable guesses. 19 In such cases, the analyst using traditional hardmodeling techniques must resort to trial and error in optimizing multiple different models, judging their chemical and spectrochemical feasibility, and choosing between them on the basis of the root-mean-square error (RMSE).…”