Multiparametric curve fitting-IX Simultaneous regression estimation of stoichiometry and stability constants of complexes

A chemical model (i.e., the number of complexes, their stoichiometry and stability constants with molar absorptivities) in solution equilibria may be established by (i) the trial-and-error method in which stability constants are estimated for an assumed set of complexes in the mixture and a fitness test is used to resolve a choice of plausible models to find the true one; (ii) the simultaneous estimation of the stoichiometry and stability constants for species divided into "certain" species for which the parameters beta(pqr), (p, q, r) are known and held constant, and "uncertain" species with unknown parameters which are determined by regression analysis. The interdependence of stability constants and particular sets of stoichiometric indices requires that the computational strategy should be chosen carefully for each particular case. The benefits and limitations of both approaches are compared by means of examples of potentiometric titration data analysis by the POLET(84) program and of spectrophotometric data analysis by the SQUAD(84) program. A strategy for efficient computation is suggested.     

Tutorial on a chemical model building by least-squares non-linear regression of multiwavelength spectrophotometric pH-titration data

Although the modern instrumentation enables for the increased amount of data to be delivered in shorter time, computer-assisted spectra analysis is limited by the intelligence and by the programmed logic tool applications. Proposed tutorial covers all the main steps of the data processing which involve the chemical model building, from calculating the concentration profiles and, using spectra regression, fitting the protonation constants of the chemical model to multiwavelength and multivariate data measured. Suggested diagnostics are examined to see whether the chemical model hypothesis can be accepted, as an incorrect model with false stoichiometric indices may lead to slow convergence, cyclization or divergence of the regression process minimization. Diagnostics concern the physical meaning of unknown parameters beta(qr) and epsilon(qr), physical sense of associated species concentrations, parametric correlation coefficients, goodness-of-fit tests, error analyses and spectra deconvolution, and the correct number of light-absorbing species determination. All of the benefits of spectrophotometric data analysis are demonstrated on the protonation constants of the ionizable anticancer drug 7-ethyl-10-hydroxycamptothecine, using data double checked with the SQUAD(84) and SPECFIT/32 regression programs and with factor analysis of the INDICES program. The experimental determination of protonation constants with their computational prediction based on a knowledge of chemical structures of the drug was through the combined MARVIN and PALLAS programs. If the proposed model adequately represents the data, the residuals should form a random pattern with a normal distribution N(0, s2), with the residual mean equal to zero, and the standard deviation of residuals being near to experimental noise. Examination of residual plots may be assisted by a graphical analysis of residuals, and systematic departures from randomness indicate that the model and parameter estimates are not satisfactory.     

Multiparametric curve fitting-XI: POLET computer program for estimation of formation constants and stoichiometric indices from normalized potentiometric data

The FORTRAN computer program POLET(84) analyses a set of normalized potentiometric titration curves to find a chemical model, i.e., the number of species present and their stoichiometry, and to determine the corresponding stability constants log beta(pqrs) and unknown stoichiometric indices p, q, r, and s of up to quaternary M(p)L(q)Y(r)H(s) complexes. The program belongs to the ABLET family, based on the LETAG subroutine, and can use an algorithmic and/or heuristic minimization strategy, or a beneficial combination of both. The data, a set of potentiometric titration curves plotted as volume and potential, are converted into normalized variables (formation function, pH) and then a computer-assisted search for a chemical model by POLET(84) is applied. The procedure for efficient application of POLET(84) in an equilibrium analysis is described and the program is validated by use of literature and simulated data. The reliability of the chemical model and its parameters is established by the degree-of-fit achieved, and the closeness of the stoichiometric indices to integral values.     

Spectrophotometric determination of acidity constants by two-rank annihilation factor analysis

Annihilation of the contribution of one chemical component from the original data matrix is a general method in rank annihilation factor analysis (RAFA). However, RAFA is not applicable for studying the protonation equilibria of multiprotic acids. In this work, a two-rank annihilation factor analysis (TRAFA) method was proposed for determination of the acidity constants of diprotic acids. After recording the electronic absorbance spectra of the acids at different pH, the contributions of both H₂A and A²⁻ were annihilated from the absorbance data, which made feasible the determination of two successive acidity constants. The method was validated by analysis of simulated data and its application to the determination of the acidity constants of calmagite, as a reference compound. A close agreement was obtained between the resulted values by TRAFA and the declared values. Indeed, the method was used for determination of the acidity constants of two new chromenone derivatives in binary solvents mixtures of methanol and water. The effects of changing solvent composition on acidity constant data were explained by linear solvation free energy relationships (LSFER) utilizing solvatochromic parameters.     

Determination of the number of light-absorbing species in the protonation equilibra of selected drugs

The determination of the number of components in a mixture is an important tool for qualitative and quantitative analysis in spectroscopy. The accuracy of nine selected indices for an estimation of the number of components that contribute to a set of spectra was critically tested on experimental data sets of protonation equilibria of four drugs using the INDICES algorithm in S-Plus. Methods are classified into two categories: precise methods based on a knowledge of the instrumental error of the sabsorbance data, s_inst(A), and approximate methods requiring no such knowledge. Indices of precise methods predict the correct number of components, even the presence of a minor one, when the quality of data is high and instrumental error is known. Improved identification of the number of species uses the second or third derivative function for some indices, namely when the number of species in the mixture is higher than four and when, due to large variations in the indicator values even at logarithmic scale, the indicator curve does not reach an obvious point where the slope changes. The number of variously protonated components and their dissociation constants for four drugs—mycophenolate, ambroxol, silybin and silydianin—at 25 °C were determined using SQUAD(84) regression and INDICES principal component analysis of the pH-spectrophotometric data. A proposed strategy of efficient experimentation in protonation constants determination, followed by a computational strategy, is presented with the goodness-of-fit tests for various regression diagnostics enabling the reliability of parameter estimates to be accessed.     

Application of experimental design for investigating the differential spectrophotometric determination of large amounts of niobium with 4-(2-pyridylazo)resorcinol

Differential spectrophotometry is used to determine large amounts of niobium in the alkali metal/niobium/fluoride mixes involved in the electrolytic separation of pure niobium. In tartrate medium, niobium forms a purple mixed-ligand complex with 4- (2-pyridylazo) resorcinol (PAR). The pH range for constant absorbance found, for two tartrate concentrations, is 6.1 ± 0.05; outside this pH range, the absorbance is strongly affected by small changes of pH. The optimum concentrations of tartrate and PAR for the differential method were found by means of experimental design. A quadratic polynomial model was applied; the results obtained were also used to calculate the stability constants of the coloured complexes. Comparison of the variation of these constants with changes in the absorption spectra of the solutions and with changes in the empirical response surfaces provided an explanation of the inadequacy of the calculated polynomial; several tartrate complexes and mixed ligand complexes of niobium appeared in some parts of the investigated factor space. The differential spectrophotometric procedure developed allows determination of 10–30% niobium with a relative standard deviation of 0.0025; its accuracy is proved by the determination of niobium in the stoichiometric compound K₂NbF₇.     

Investigation of equilibria in solution. Determination of equilibrium constants with the HYPERQUAD suite of programs

A new suite of 10 programs concerned with equilibrium constants and solution equilibria is described. The suite includes data preparation programs, pretreatment programs, equilibrium constant refinement and post-run analysis. Data preparation is facilitated by a customized data editor. The pretreatment programs include manual trial and error data fitting, speciation diagrams, end-point determination, absorbance error determination, spectral baseline corrections, factor analysis and determination of molar absorbance spectra. Equilibrium constants can be determined from potentiometric data and/or spectrophotometric data. A new data structure is also described in which information on the model and on experimental measurements are kept in separate files.     

Competitive ion complexation to polyelectrolytes: determination of the stepwise stability constants. The Ca2+/H+/polyacrylate system

This work presents a new methodology aimed at obtaining the stepwise stability constants corresponding to the binding of ions (or other small molecules) to macromolecular ligands having a large number of sites. For complexing agents with a large number of sites, very simple expressions for the stepwise stability constants arise. Such expressions are model-independent; that is, they allow the determination of the stepwise stability constants without making any previous assumption of the detailed complexation mechanism. The formalism is first presented for a single complexing ion and further extended to competitive systems where the competing ions can display, in general, different stoichiometric relationships. These ideas are applied to the analysis of experimental titrations corresponding to competitive binding of calcium ions to poly(acrylic acid) for different pH values and ionic strengths. Intrinsic stability constants were estimated from the stepwise stability constants (by removing the corresponding statistical factor), and split into specific and electrostatic contributions (by means of the Poisson-Boltzmann equation). After this treatment, the specific proton binding energies showed almost no dependence on the coverage and ionic strength. Likewise, for the range of concentrations studied, the specific component of the intrinsic stability constants of the calcium ions, calculated assuming bidentate binding of Ca to neighboring groups of a linear chain, is almost independent of the calcium and proton coverage and ionic strength.     

Diprotonation process of meso-tetraphenylporphyrin derivatives designed for photodynamic therapy of cancers: from multivariate curve resolution to predictive QSPR modeling

Tetrapyrrole rings possess four nitrogen atoms, two of which act as Br?ndsted bases in acidic media. The two protonation steps occur on a close pH range, particularly in the case of meso-tetraphenylporphyrin (TPP) derivatives. If the cause of this phenomenon is well known--a protonation-induced distortion of the porphyrin ring--data on stepwise protonation constants and on electronic absorption spectra of monoprotonated TPPs are sparse. A multivariate approach has been systematically applied to a series of glycoconjugated and hydroxylated TPPs, potential anticancer drugs usable in Photodynamic Therapy. The dual purpose was determination of protonation constants and linking substitution with basicity. Hard-modeling version of MCR-ALS (Multivariate Curve Resolution Alternating Least Squares) has given access to spectra and distribution profile of pure components. Spectra of monoprotonated species (H(3)TPP(+)) in solution resemble those of diprotonated species (H(4)TPP(2+)), mainly differing by a slight blue-shift of bands. Overlap of H(3)TPP(+) and H(4)TPP(2+) spectra reinforces the difficulty to evidence an intermediate form only present in low relative abundance. Depending on macrocycle substitution, pK values ranged from 3.5±0.1 to 5.1±0.1 for the first protonation and from 3.2±0.2 to 4.9±0.1 for the second one. Inner nitrogens' basicity is affected by position, number and nature of peripheral substituents depending on their electrodonating character. pK values have been used to establish a predictive Multiple Linear Regression (MLR) model, relying on atom-type electrotopological indices. This model accurately describes our results and should be applied to new TPP derivatives in a drug-design perspective.     

Potentiometric and conductometric studies of malonyl bis(salicyloylhydrazone) and divalent metal complexes

A series of complexes of divalent transition metal ions with malonyl bis(salicyloylhydrazone) (H₄MSH) have been prepared and characterized with the help of conductometric, potentiometric methods. The proton–ligand and metal–ligand stability constants were obtained pH-metrically. The electrical conductivity of solid complexes was measured at 289 K. The low molar conductance values observed for these complexes indicate that, they are non-electrolytes. They are soluble to a limited extent in DMF and DMSO. The elemental analyses of the complexes indicate that the complexes have 1:1 and 2:1 (M:L) stoichiometry with the existence of water, chloride, acetone molecules inside the coordination sphere as evidence from the IR spectral studies. Further, the complexes have been formulated by comparing C, H, N & metal analysis data, and UV–visible spectra of the complexes have been discussed. The protonation constants of the ligand and the stability constants of their metal complexes will be evaluated potentiometrically. The stoichiometric ratios of the complexes formed in solution will be evaluated applying the molar ratio (spectrophotometric) method and confirmed conductometrically.     

Optical constants and integrated intensities of liquid hexafluorobenzene between 4000 and 250 cm at 25 °C

The optical constants (real and imaginary refractive indices) of hexafluorobenzene were determined at 25 °C via transmission measurements. Experimental absorbance spectra measured on a Nicolet Impact 410 FTIR were converted to imaginary refractive indices using methods described in the literature. The real refractive indices were obtained by Kramers–Kronig transformation of the imaginary refractive indices. From the complex refractive indices, the molar absorption coefficient (E_m) and complex molar polarizability spectra were calculated. The integrated intensities for the E_1u fundamentals were obtained from the areas under the bands in the spectrum. These integrated intensities are compared to those for benzene and benzene-d₆ in the literature.     

Multiparametric curve fitting--VIII: The reliability of dissociation constants estimated by analysis of absorbance-pH curves

The program SPOPT estimates stability constants beta(pqr) and molar absorptivities epsilon(pqr) of all light-absorbing species M(p)L(q)H(r) by analysis of the absorbance-concentration (or absorbance-pH) curve. The program DCMINUIT estimates dissociation constants and molar absorptivities of protonated species. Both programs have been tested and compared with DCLET and LETAGROP-SPEFO for analysis of the overlapping equilibria of a triprotic acid. Computer plots of the residual-square-sum function are used to test the conditioning of parameters. Two approaches are made to formulation of the mathematical model, and several optimization algorithms are tested to find a reliable minimization procedure. The accuracy of ill-conditioned parameters is shown to be dependent on the precision of the absorbance measurements. General rules for investigation of A -pH curves are recommended.     

Evaluation of the acidity constants of the 4-hidroxy-5-[salicylideneamino]-2-7-naphthalenedisulfonic acid (Azomethine-H) using UV-vis spectrophotometry

The time stability of the azomethine-H species was determined not to be better than 10 min in the absence of oxygen and light, however under phosphate buffered conditions the azomethine-H species remained stable for longer periods, as indicated by the spectrophotometric behaviour. Nevertheless, the analysis time still exceeded the stability allowance. Therefore, the determination of the acidity constants of the Azomethine-H species was studied by means of UV-vis spectrophotometry in buffered media by means of the point-by-point analysis and data processing with SQUAD to refine the resulting constants, which were: pK(a1) = 3.39, pK(a2) 7.36 and pK(a3) 8.73. The latter were associated to the corresponding acid-base equilibria of the amine and hydroxy groups constituting the molecule.     

Mid-infrared extinction spectra and optical constants of supercooled water droplets

Complex refractive indices of supercooled liquid water have been retrieved at 269, 258, 252, and 238 K in the 4500-1100 cm(-1) wavenumber regime from series of infrared extinction spectra of micron-sized water droplets. The spectra collection was recorded during expansion experiments in the large coolable aerosol chamber AIDA of Forschungszentrum Karlsruhe. A Mie inversion technique was applied to derive the low-temperature refractive index data sets by iteratively adjusting the room-temperature optical constants of liquid water until obtaining the best agreement between measured and calculated infrared spectra of the supercooled water droplets. The new optical constants, revealing significant temperature-induced spectral variations in comparison with the room-temperature refractive indices, proved to be in good agreement with data sets obtained in a recent study. A detailed analysis was performed to elaborate potential inaccuracies in the retrieval results when deriving optical constants from particle extinction spectra using an iterative procedure.     

Characterization and determination of stability constants of copper(II)-l-histidine complexation system by using multivariate curve resolution method of visible spectra and two hard modeling methods in aqueous solutions

Multivariate curve resolution, alternating least-squares is applied to spectra data obtained in the study of Cu(II) complexation by l-histidine. The combination of several chemometric techniques based on factor analysis (FA), singular value decomposition (SVD), evolving factor analysis (EFA), and multivariate curve resolution with constrained alternating least-squares (ALS) is used to determine the number of species and their distribution diagram. This multivariate analysis data treatment simultaneously reveals the species Cu, CuL, CuLH, CuL₂, CuL₂H, and CuLOH, through the calculated concentration profiles and allows the assignment of numerically obtained pure individual spectra. Formation constants of these species were calculated by hard-modeling methods applied potentiometric and spectrophotometric measurements.     

Number of species in complexation equilibria of SNAZOXS or Naphtylazoxine 6S and Cd(2+), Co(2+), Cu(2+), Ni(2+), Pb(2+) and Zn(2+) ions by PCA of UV-vis spectra

A critical comparison of the various PCA methods on the absorbance matrix data concerning the complexation equilibria between SNAZOXS and Cd²⁺, Co²⁺, Cu²⁺, Ni²⁺, Pb²⁺ and Zn²⁺ or Naphtylazoxine 6S and Cd²⁺, Cu²⁺, Ni²⁺ and Zn²⁺ at 25 °C is performed. The number of complex species in a complex-forming equilibria mixture is the first important step for further qualitative and quantitative analysis in all forms of spectral data treatment. Therefore, the accuracy of the nine selected index functions for the prediction of the number of light-absorbing components that contribute to a set of spectra is critically tested using the principal component PCA algorithm INDICES in S-Plus software. Four precise methods based upon a knowledge of the experimental error of the absorbance data and five approximate methods requiring no such knowledge are discussed. Precise methods always predict the correct number of components even a presence of the minor species in mixture. Due to the large variations in the index values and even at logarithmic scale they do not reach an obvious point where the slope changes. An improved identification with the second or third derivative and derivative ratio function for some indices is preferred. Behind the number of various complexes formed the stability constants of species ML, ML₂, (and ML₃, respectively) type log β₁₁, log β₁₂, (and log β₁₃, respectively) for the system of SNAZOXS (ligand L) with six metals (the standard deviation s(log β_pq) of the last valid digits are in brackets) Cd²⁺ (4.50(3), 8.36(7)), Co²⁺ (5.75(6), 9.79(9), 13.05(2)), Cu²⁺ (6.69(6), 11.40(7)), Ni²⁺ (6.44(8), 10.91(11), 15.07(10)), Pb²⁺ (5.63(5), 9.97(9)) and Zn²⁺ (5.11(3), 8.84(5)) and for system of Naphtylazoxine 6S with Cd²⁺ (6.08(4), 11.44(7), 16.06(11)), Cu²⁺ (7.80(8), 13.41(14)), Ni²⁺ (6.35(12), 11.43(19), 16.68(24)) and Zn²⁺ (7.01(8), 12.65(15)) at 25 °C are estimated with SQUAD(84) nonlinear regression of the mole-ratio spectrophotometric data. The proposed strategy of an efficient experimentation in a stability constants determination, followed by a computational strategy, is presented with goodness-of-fit tests and various regression diagnostics able to prove the reliability of the chemical model proposed.     

Complex Formation Equilibria and Molecular Structure of Divalent Metal Ions–Vitamin B3–Glycine Oligopeptides Systems

Complex formation of divalent transition metal ions (copper(II), cobalt(II) and nickel(II)), vitamin B3 (nicotinic acid) and glycine oligopeptides (glycine, glycylglycine, glycyl-l-phenylalanine, and glycylglycylglycine) were studied at 298 K in aqueous solutions using the pH-potentiometric technique. The copper Cu(II), cobalt Co(II), and nickel Ni(II) complexing capacity of vitamin B3 in the absence and in the presence of glycine peptides and their overall stability constants in aqueous solutions were obtained and explained by the HYPERQUAD 2008 program using the potentiometric data. From the protonation and complex formation constants, representative complex species distribution diagrams were obtained using HYSS 2009 software. The UV–Vis spectroscopic, cyclic voltammeteric and conductometric titration measurements were carried out to give qualitative information about the conformation of the complexes formed in these solutions and their stoichiometric ratios. The Gibbs energies and the molecular structures of the complexes were evaluated and predicted using Gaussian 09 software molecular modeling and density functional theory calculations.     

Calculation of stability constants for the chiral selector-enantiomer interactions from electrophoretic mobilities

Interactions of chiral selectors with enantiomers or with other chiral analytes, underlying their electrophoretic separations, are particular cases of interactions of dissolved species. The interaction model that describes these reversible, fast interactions is simple. Equations describing the model are also simple and applying equations for the calculation of respective equilibrium constants to experimental data is easy using computers. Obtaining the experimental mobility data, necessary for the calculation, is the critical step on the way to stability constants quantitating the strengths of interactions of chiral selectors with enantiomers and other analytes. These data are decisive for both the accuracy and precision of the calculated constants. The meaning and applicability of the particular constant depend on the type of the constant. The common method for the determination of stability constants from electrophoretic migration data is reasonable for low and medium stability constants. For stronger complexing, characterised by stability constants of the order of 10(4) l/mol, typical of affinity chiral selectors, the method becomes unreliable. For strong complexing giving constants of the order of 10(5) or higher of or higher the method is not applicable in its commonly used form.     

Spectrophotometric data reduction by eigenvector analysis for equilibrium and kinetic studies and a new method of fitting exponentials

The investigation of multicomponent mixtures by spectrophotometry is described. A desk computer, APPLE II, is used to calculate the number of absorbing species in a series of measured spectra by matrix rank analysis. Representation of the observed spectra as linear combinations of eigenvectors leads to significant reduction of the data set, so that a nonlinear least-squares fit based on the Newton—Gauss—Marquardt algorithm is possible on a small computer. As an example, the complexation of copper(II) with 1,4,7-triazaheptane (dien) was studied by combined spectrophotometric and pH titration. The spectra of 62 mixtures at 26 wavelengths were analysed; the number of absorbing species, their spectra and the underlying equilibrium constants were determined. Representation of kinetic curves as linear combinations of eigenvectors is described. It is shown that instead of finding the minimum of the square sum in a multidimensional rate constant space, these minima can be found in a one-dimensional space. Two examples are given: the first is theoretical whereas the second is based on the kinetics of dissociation of the μ-peroxo complexes formed between cobalt(II), oxygen and 4,7,10-triazatridecanedioic acid in acidic solution.