Haarhoff-Van Der Linde Function Research Articles

Influence of theoretical and semi-empirical peak models on the efficiency calculation in chiral chromatography

Height equivalent to theoretical plate (H) equations, such as the van Deemter or Knox-Saleem equations, and other efficiency vs. linear velocity equations (u), provide kinetic insights into chromatographic separations phenomena and column performance. In enantioselective separations, the peak shape of the two enantiomers can differ significantly and are often asymmetric. The peak efficiency calculations heavily impact these efficiency-flow profiles, leading to erroneous estimations of eddy diffusion, longitudinal diffusion, and mass transfer terms. In this work, new asymmetric peak functions are employed for modeling enantiomer peaks based on the Haarhoff-Van der Linde function, its generalized variant (GHVL), once Generalized Asymmetric Gaussian (AGN), and Twice Generalized Gaussian (TGN). The new models (AGN, TGN, and GHVL) incorporate higher statistical moments besides the zeroth, first, and second moments to account for two-sided asymmetry (fronting or tailing). The fit results are compared with the traditional efficiency calculation methods endorsed by official pharmacopeia and numerical estimation of moments from the raw data. Enantiomeric separations of ibuprofen and dl-homophenylalanine were chosen as probe molecules. The results demonstrate that non-linear least squares fitted functions provide better estimations of peak efficiency data even in the presence of high noise. In particular, the generalized models consistently offered the best quality fits for various peak shapes in chiral separations. Conversely, the half-height Gaussian method greatly overpredicted skewed peak efficiencies. This investigation reveals that the commonly held assumptions of peak shape and numerical integration of raw data are highly insufficient for chiral chromatography. The impact of asymmetry on plate height should not be overlooked when accurate data from efficiency-flow rate curves is derived. We advocate for the broader adoption of these new generalized peak (AGN, TGN, GHVL) models because they provide robustness at various SNRs that account for right or left asymmetry while accurately representing peak geometry.

Boundary values of binding constants determined by affinity capillary electrophoresis.

This study shows that the upper limit of binding (stability) constants determined by mobility shift affinity capillary electrophoresis can be increased from 104 to 106 -109 L/mol if the lowest possible analyte concentration in samples is used (for example, the concentration that gives electrophoretic peaks with a signal-to-noise ratio of 10) and the effective electrophoretic mobility of the analyte is calculated via the parameter a1 of the Haarhoff-Van der Linde function. The equation to calculate the boundary values of binding constants for 1:1 complexes was derived for the case when the constants cannot be calculated in the usual way. These values are obtained from the inequality: the difference between the ionic mobility of the analyte-ligand complex and the effective electrophoretic mobility of the analyte determined at the lowest ligand concentration in the background electrolyte at which the analyte appears as an undistorted peak in electropherograms is less than or equal to the absolute error in mobility measurements. The application of the equation was illustrated by the example of electrophoretic data for a complex between betulin 3,28-diphthalate and (2-hydroxypropyl)-γ-cyclodextrin. An algorithm to determine the binding constants for strong complexation by mobility shift affinity capillary electrophoresis was suggested.

Determining binding constants for 1:1 and 1:2 inclusion complexes of ester betulin derivatives with (2-hydroxypropyl)-β-cyclodextrin by affinity capillary electrophoresis.

The complexation of ester betulin derivatives with (2-hydroxypropyl)-β-cyclodextrin (HP-β-CD) was studied by mobility shift affinity CE. Electrophoretic mobility for triangular peaks was calculated using the parameter a1 of the Haarhoff-Van der Linde function instead of the peak top time. Dependences of the viscosity corrected electrophoretic mobility on HP-β-CD concentration were not described on the basis of only complexes with 1:1 stoichiometry due to the fact that these binding curves did not reach a plateau. However, the dependences were well described taking into account both 1:1 and 1:2 complexes. The presence of higher order equilibria was also revealed by x-reciprocal plots. The values of apparent binding constant logarithm, obtained for the first time, for 1:1 and 1:2 HP-β-CD complexes of betulin 3,28-diphthalate and betulin 3,28-disuccinate with 95% confidence interval limits in brackets are the same within error and are equal to 4.85 (4.73-4.95), 8.56 (7.75-8.82), 4.92 (4.86-4.97), and 8.54 (8.23-8.72) at 25°C, respectively. These values for 1:1 and 1:2 HP-β-CD complexes of betulin 3,28-disulfate at 25°C are 4.61 (4.57-4.64) and 7.11 (6.57-7.34), respectively. The binding constants for betulin 3,28-disulfate agree with the previously obtained results from the separation in the thermostatted capillary segment.

Investigation of the enantioselective interaction between selected drug enantiomers and human serum albumin by mobility shift-affinity capillary electrophoresis.

Mobility shift-affinity capillary electrophoresis was employed for enantioseparation and simultaneous binding constant determination. Human serum albumin was used as a chiral selector in the background electrolyte composed of 20mM phosphate buffer, pH 7.4. The applied setup supports a high mobility shift since albumin and the drug-albumin complex hold negative net charges, while model compounds of amlodipine and verapamil are positively charged. In order to have an accurate effective mobility determination, the Haarhoff-van der Linde function was utilized. Subsequently, the association constant was determined by nonlinear regression analysis of the dependence of effective mobilities on the total protein concentration. Differences in the apparent binding status between the enantiomers lead to mobility shifts of different extends (α). This resulted in enantioresolutions of Rs=1.05-3.63 for both drug models. R-(+)-Verapamil (KA 1844M-1 ) proved to bind stronger to human serum albumin compared to S-(-)-verapamil (KA 6.6M-1 ). The association constant of S-(-)-amlodipine (KA 25073M-1 ) was found to be slightly higher compared to its antipode (KA 22620M-1 ) when applying the racemic mixture. The low measurement uncertainty of this approach was demonstrated by the close agreement of the association constant of the enantiopure S-(-)-form (KA 25101M-1 ).

Determination of binding constants for strong complexation by affinity capillary electrophoresis: the example of complexes of ester betulin derivatives with (2-hydroxypropyl)-γ-cyclodextrin.

Complexation plays an important role in many biological phenomena, the analysis of different samples, optimization of separation processes, and increasing the pharmacological activity of drugs. This paper discusses the features of using mobility shift affinity capillary electrophoresis for studying strong complexation. Electrophoretic peaks for this case are often triangular. It was shown that the use of electrophoretic mobility obtained from the peak apex time to calculate binding constants leads to significant systematic and random errors, and the parameter a1 of the Haarhoff-Van der Linde function should be used instead of the apex time. Distorted triangular peaks with dips were shown to be observed at too high a ratio of analyte concentration in the sample to ligand concentration in the background electrolyte, and the peaks and parameter a1 significantly shifted. It was found that the permissible excess of analyte concentration over ligand concentration was approximately 10-35, provided that the parameter a1 was used, but the peak shape should be used as a landmark, and only triangular peaks without dips should be fitted with the function. The lowest possible analyte concentration should be utilized, which allows the use of a wider range of ligand concentration leading to higher precision of determining the binding constants values. Kinetically labile 1:1 complexes between (2-hydroxypropyl)-γ-cyclodextrin (HP-γ-CD) and betulin 3,28-diphthalate (DPhB) and betulin 3,28-disuccinate (DScB) were studied as an example. The binding constant logarithms at 25°C are 7.23 ± 0.03 and 7.13 ± 0.10 for the HP-γ-CD complexes of DPhB and DScB, respectively. Graphical Abstract.

Determination of the correct migration time and other parameters of the Haarhoff-van der Linde function from the peak geometry characteristics.

For Gaussian peaks, the migration time of the analyte results as the position of the top of the peak and the zone variance is proportional to the peak width. Similar relations have not yet been derived for the Haarhoff-van der Linde (HVL) function, which appears as a fundamental peak-shape function in electrophoresis. We derive the relations between the geometrical measures of the HVL-shaped peak, that is the position of its maximum, its width and a measure of its asymmetry, and the respective parameters a1, a2, and a3, of the corresponding HVL function. Under the condition of the HVL-shaped peak, the a1 parameter reflects the true migration time of the analyte, which may differ from the peak top position significantly. Our procedure allows us to express the parameters without the need of any external data processing (nonlinear regression). We demonstrate our approach on simulated peaks and on experimental data integrated by the ChemStation software (delivered with the CE instrumentation by Agilent Technologies). A significant improvement is achieved reading the migration time of the experimental and simulated peaks, which draws the error of the HVL-shaped peak migration time evaluation down to the resolution of the data sampling rate.

Valence-Shell Excitations of Nitrous Oxide Studied by Fast Electron Impact

The valence-shell excitations of nitrous oxide are studied by fast electron energy loss spectroscopy. From the spectra measured at 2.5 keV and scattering angles of 3.5°–8.5°, it is found that the asymmetric peak of the transition B1Δ can be well fitted by Haarhoff-Van der Linde function, while the symmetric peaks of the transitions of C1Π and D1Σ+ can be well fitted by the Voigt function. The parameters of the peak profiles of B1Δ, C1Π and D1Σ+, i.e., their energy level positions and linewidths, are determined. With the aid of these parameters, the overlapping spectra measured at the low-energy electron impact can be deconvolved, which provides the possibility to determine the quantitative differential cross sections. The present results also show that the peak profiles of the transitions of B1Δ, C1Π and D1Σ+, are independent of the momentum transfer.

Elemental Speciation by Parallel Elemental and Molecular Mass Spectrometry and Peak Profile Matching

Trace elemental speciation in complex, real-world matrixes is a daunting task because of the low concentration of metals/metalloids and the correspondingly high molecular chemical noise. We constructed a liquid chromatography parallel elemental and molecular mass spectrometry (PEMMS) system and evaluated the use of peak elution profiles to identify trace molecular species containing specific heteroatoms, using the case of Se in yeast. We demonstrate that it is possible to use the HPLC-inductively coupled plasma (ICP)MS peak profile (retention time, width) to identify candidate ions with matching peak profiles in the molecular MS data. Proof of principle was demonstrated by C18 separation of three Se-amino acid standards (0.005-15 ppm as Se). The molecular MS (atmospheric pressure chemical ionization time-of-flight, APCI-TOF-MS) data set was converted into selected ion chromatograms of 0.05 Th width. ICPMS and APCI-TOF-MS ion chromatograms were fit by the Haarhoff-VanderLinde function using the following parameters: area, retention time, width, and skew. The ICPMS fit parameters were more reproducible than the APCI-TOF-MS fit parameters from run to run, and the APCI-TOF-MS signal was expected to limit correlation in most circumstances. Retention time and width were found to correlate well between the two MS systems for APCI-TOF-MS peaks with signal-to-fit-error (S/FE) of >25. Correction factors for differences in flow path length and peak broadening were required. The normalized correction factors were species and concentration independent and were stable from run to run. The skew parameter was found to be highly susceptible to noise and was not generally useful in matching ICPMS and APCI-TOF-MS peaks. An artificially noisy sample was generated by spiking 30 ppb Se-methionine (SeMet) and 5 ppb Se-methylselenocysteine (SeMSC) with unselenized yeast extract and run by PEMMS. The PEMMS software was able to detect four molecular MS peaks associated with SeMet and two for SeMSC, while filtering out >40 coeluting spectral peaks associated with chemical noise in each sample. In summary, we have demonstrated that correlated information in peak shape between parallel detectors can facilitate detection of trace elemental species in complex matrixes.

Peak shape modeling by Haarhoff‐Van der Linde function for the determination of correct migration times: A new insight into affinity capillary electrophoresis

Among the different experimental strategies available in capillary electrophoresis (CE) to determine binding parameters, affinity capillary electrophoresis (ACE) has been the most widely embraced due to its easiness of implementation and of data handling. Ligand-substrate binding constants are thus directly derived from the substrate migration time shifts resulting from the variation of ligand concentration introduced in a background electrolyte. Classically, the substrate migration time is measured on top of the electrophoretic peak, assuming symmetrical peak shape. Depending on both substrate and ligand concentrations that may be required to meet detection sensitivity or complexation conditions, zonal migrations in ACE may, however, produce triangular peak shape, most often due to pronounced electromigration dispersion (EMD), and this may result in positively or negatively erroneous migration time assessments. In this work, EMD distorted triangular peak shapes obtained in the course of host-guest complexation studies were fitted with the Haarhoff-Van der Linde function, allowing better estimation of migration time. The model systems studied were those of beta-cyclodextrin and naproxen, 2-naphthalenesulfonate, or 1-adamantanecarboxylate. The impact of this correction on binding isotherms and binding constant evaluation was exemplified. Furthermore, in situations where the substrate concentration injected by far overtakes that of the ligand in the electrolyte, the interest in this peak shape correction was discussed in connection with the question of whether the free ligand concentration can be still considered equal to the ligand concentration introduced, a question that still remains under debate nowadays.

A new mathematical function for describing electrophoretic peaks

A new model is proposed for characterizing skewed electrophoretic peaks, which is a combination of leading and trailing edge functions, empirically modified to get a rapid recovery of the baseline. The peak model is a sum of square roots and is called thereby "combined square roots (CSR) model". The flexibility of the model was checked on theoretical and experimental peaks with asymmetries in the range of 0-10 (expressed as the ratio of the distance between the center and the trailing edge, and the center and the leading edge of the chromatographic peak, measured at 10% of peak height). Excellent fits were found in all cases. The new model was compared with other three models that have shown good performance in modelling chromatographic peaks: the empirically transformed Gaussian, the parabolic Lorentzian-modified Gaussian, and the Haarhoff-van der Linde function. The latter model was proposed recently to describe electrophoretic peaks. The CSR model offered the highest flexibility to describe electrophoretic peak profiles, even those extremely asymmetrical with long tails. The new function has the advantage of using measurable parameters that allow the direct estimation of peak areas, which is useful for quantitative purposes.

Electromigration dispersion in capillary zone electrophoresis: Experimental validation of use of the Haarhoff–Van der Linde function

This paper provides experimental validation of the use of the Haarhoff–Van der Linde (HVL) peak fitting function to fit experimental capillary zone electrophoresis (CZE) electropherograms. The test mixtures were composed of paraquat over a five order of magnitude concentration range (1.2 μ M to 120 m M) and 4-aminopyridine at constant concentration (0.53 m M) as internal standard. Peak descriptors and electrophoresis parameters were extracted reliably by a Gaussian function from 4 to 40 μ M; by the HVL function from 120 μ M to 4 m M; and by a triangular function from 4 to 120 m M. The HVL function can be used where there is significant peak asymmetry due to electromigration distortion (EMD) and the Gaussian contribution toward the peak variance is greater than 25%. The peak centre ( a 1) and the Gaussian variance ( a 2) of the paraquat peak are shown to be independent of concentration. Diffusion coefficients obtained from a 2 for both analytes were found to be in good agreement with their theoretical values. For all peaks where the distortion coefficient ( a 3) can be extracted, this parameter is shown to be directly proportional to the sample loading, as predicted by EMD theory. For the 4-aminopyridinium ion, mobilities calculated from a 3 and measured independently are in excellent agreement. These results show that the HVL function accurately describes the two major processes, diffusion and EMD, contributing to the variance during a CZE separation.