Evolution of the theoretical description of the isoelectric focusing experiment: I. The path from Svensson's steady-state model to the current two-stage model of isoelectric focusing.

In 1961, Svensson described isoelectric focusing (IEF), the separation of ampholytic compounds in a stationary, natural pH gradient that was formed by passing current through a sucrose density gradient-stabilized ampholyte mixture in a constant cross-section apparatus, free of mixing. Stable pH gradients were formed as the electrophoretic transport built up a series of isoelectric ampholyte zones-the concentration of which decreased with their distance from the electrodes-and a diffusive flux which balanced the generating electrophoretic flux. When polyacrylamide gel replaced the sucrose density gradient as the stabilizing medium, the spatial and temporal stability of Svensson's pH gradient became lost, igniting a search for the explanation and mitigation of the loss. Over time, through a series of insightful suggestions, the currently held notion emerged that in the modern IEF experiment-where the carrier ampholyte (CA) mixture is placed between the anolyte- and catholyte-containing large-volume electrode vessels (open-system IEF)-a two-stage process operates that comprises a rapid first phase during which a linear pH gradient develops, and a subsequent slow, second stage, during which the pH gradient decays as isotachophoretic processes move the extreme pI CAs into the electrode vessels. Here we trace the development of the two-stage IEF model using quotes from the original publications and point out critical results that the IEF community should have embraced but missed. This manuscript sets the foundation for the companion papers, Parts 2 and 3, in which an alternative model, transient bidirectional isotachophoresis is presented to describe the open-system IEF experiment.

Evolution of the theoretical description of the isoelectric focusing experiment: III. Carrier ampholyte behavior in transient, bidirectional isotachophoresis.

In modern isoelectric focusing (IEF) systems, where (i) convective mixing is prevented by gels or small cross-sectional area separation channels, (ii) current densities vary spatially due to the presence of electrode vessels with much larger cross-sectional areas than those of the gels or separation channels, and (iii) electrophoretic and diffusive fluxes do not balance each other, stationary, steady-state pH gradients cannot form (open-system IEF). Open-system IEF is currently described as a two-stage process: A rapid IEF process forms the pH gradient from the carrier ampholytes (CAs) in the first stage, then isotachophoretic processes degrade the pH gradient in the second stage as the extreme pI CAs are moved into the electrode vessels where they become diluted. Based on the ratios of the local effective mobilities and the local conductivities ( µ L eff ( x ) $\mu _{\rm{L}}^{{\rm{eff}}}( x )$ / κ ( x ) $\kappa ( x )$ values) of the anolyte, catholyte, and the CAs, we pointed out in the preceding paper (Vigh G, Gas B, Electrophoresis 2023, 44, 675-88) that in open-system IEF, a single process, transient, bidirectional isotachophoresis (tbdITP) operates from the moment current is turned on. In this paper, we demonstrate some of the operational features of the tbdITP model using the new ITP/IEF version of Simul 6.

Evolution of the theoretical description of the isoelectric focusing experiment: II. An open system isoelectric focusing experiment is a transient, bidirectional isotachophoretic experiment.

The carrier ampholytes-based (CA-based) isoelectric focusing (IEF) experiment evolved from Svensson's closed system IEF (constant spatial current density, absence of convective mixing, counter-balancing electrophoretic and diffusive fluxes yielding a steady state pH gradient) to the contemporary open system IEF (absence of convective mixing, large cross-sectional area electrode vessels, lack of counter-balancing electrophoretic- and diffusive fluxes leading to transient pH gradients). Open system IEF currently is described by a two-stage model: In the first stage, a rapid IEF process forms the pH gradient which, in the second stage, is slowly degraded by isotachophoretic processes that move the most acidic and most basic CAs into the electrode vessels. An analysis of the effective mobilities and the effective mobility to conductivity ratios of the anolyte, catholyte, and the CAs indicates that in open system IEF experiments a single process, transient bidirectional isotachophoresis (tbdITP) operates from the moment current is turned on until it is turned off. In tbdITP, the anolyte and catholyte provide the leading ions and the pI 7 CA or the reactive boundary of the counter-migrating H3 O+ and OH- ions serves as the shared terminator. The outcome of the tbdITP process is determined by the ionic mobilities, pKa values, and loaded amounts of all ionic and ionizable components: It is constrained by both the transmitted amount of charge and the migration space available for the leading ions. tbdITP and the resulting pH gradient can never reach steady state with respect to the spatial coordinate of the separation channel.

Mathematical model of electromigration allowing the deviation from electroneutrality.

The structure of the double layer on the boundary between solid and liquid phases is described by various models, of which the Stern-Gouy-Chapman model is still commonly accepted. Generally, the solid phase is charged, which also causes the distribution of the electric charge in the adjacent diffuse layer in the liquid phase. We propose a new mathematical model of electromigration considering the high deviation from electroneutrality in the diffuse layer of the double layer when the liquid phase is composed of solution of weak multivalent electrolytes of any valence and of any complexity. The mathematical model joins together the Poisson equation, the continuity equation for electric charge, the mass continuity equations, and the modified G-function. The model is able to calculate the volume charge density, electric potential, and concentration profiles of all ionic forms of all electrolytes in the diffuse part of the double layer, which consequently enables to calculate conductivity, pH, and deviation from electroneutrality. The model can easily be implemented into the numerical simulation software such as Comsol. Its outcome is demonstrated by the numerical simulation of the double layer composed of a charged silica surface and an adjacent liquid solution composed of weak multivalent electrolytes. The validity of the model is not limited only to the diffuse part of the double layer but is valid for electromigration of electrolytes in general.

Simul 6: A fast dynamic simulator of electromigration.

Simul 6 is a 1D dynamic simulator of electromigration based on the mathematical model of electromigration in free solutions. The model consists of continuity equations for the movement of electrolytes in a separation channel, acid-base equilibria of weak electrolytes, and the electroneutrality condition. It accounts for any number of multivalent electrolytes or ampholytes and provides a complete picture about dynamics of electromigration and diffusion in the separation channel. The equations are solved numerically using software means which allow for parallelization and multithreaded computation. Simul 6 has a user-friendly graphical interface. It is typically used for inspection of system peaks (zones) in electrophoresis, stacking and preconcentrating analytes, optimization of separation conditions, method development in either capillary zone electrophoresis, isotachophoresis, and isoelectric focusing. Simul 6 is the successor of Simul 5, and has been launched as a free software available for download at https://simul6.app/.

Electrolysis phenomena in electrophoresis.

We present a new theoretical approach for calculating changes in the physico-chemical properties of BGEs for measurements by CZE due to the electrolysis in electrode vials (vessels). Electrolysis is an inevitable phenomenon in any measurement in CZE. Water electrolysis, which occurs in most measurements, can significantly alter the composition of the BGE in electrode vials and in the separation capillary and has a negative influence on the robustness and quality of separations. The ability to predict changes in the composition of the BGE is important for evaluation of the suitability of the BGEs for repeating electrophoretic runs. We compared theoretically calculated changes in the physico-chemical properties (pH, conductivity) with those measured using pH-microelectrode and contactless conductivity detection of the BGE after the electrophoretic run. We confirmed the validity of our theoretical approach with a common BGE composed of acid-base pair, where one constituent is fully dissociated while the second constituent is dissociated by only half, and with Good's buffer. As predicted by theoretical approach, the changes in the physico-chemical properties of the Good's buffer after the electrophoretic run were several times lower than in the case of a common BGE composed of a weak acid - strong base pair.

The dynamics of band (peak) shape development in capillary zone electrophoresis in the case of two co-migrating analytes: The displacement and the tag-along effects.

Peak shapes in electrophoresis are often distorted from the ideal Gaussian shape due to disturbing phenomena, of which the most important is electromigration dispersion. For fully dissociated analytes, there is a tight analogy between nonlinear models describing a separation process in chromatography and electrophoresis. When the velocity of the separated analyte depends on the concentration of the co-analyte, the consequence is a mutual influence of the analytes couples, which distorts both analyte zones. In this paper, we introduce a nonlinear model of electromigration for the analysis of two co-migrating fully dissociated analytes. In the initial stages of separation, they influence each other, which causes much more complicated peak shapes. The analysis has revealed that the two most important phenomena-the displacement and the tag-along effects-are common both for nonlinear chromatography and electrophoresis, though their description is partly based on rather different phenomena. The comparison between the nonlinear model of electromigration we describe and the numerical computer solution of the original continuity equations has proven an almost perfect agreement. The predicted features in peak shapes in initial stages of separation have been fully confirmed by the experiments.

CE determination of the thermodynamic pKa values and limiting ionic mobilities of 14 low molecular mass UV absorbing ampholytes for accurate characterization of the pH gradient in carrier ampholytes-based IEF and its numeric simulation.

Fourteen low molecular mass UV absorbing ampholytes containing 1 or 2 weakly acidic and 1 or 2 weakly basic functional groups that best satisfy Rilbe's requirement for being good carrier ampholytes (ΔpKa = pKamonoanion - pKamonocation < 2) were selected from a large group of commercially readily available ampholytes in a computational study using two software packages (ChemSketch and SPARC). Their electrophoretic mobilities were measured in 10 mM ionic strength BGEs covering the 2 < pH < 12 range. Using our Debye-Hückel and Onsager-Fuoss laws-based new software, AnglerFish (freeware, https://echmet.natur.cuni.cz/software/download), the effective mobilities were recalculated to zero ionic strength from which the thermodynamic pKa values and limiting ionic mobilities of the ampholytes were directly calculated by Henderson-Hasselbalch equation-type nonlinear regression. The tabulated thermodynamic pKa values and limiting ionic mobilities of these ampholytes (pI markers) facilitate both the overall and the narrow-segment characterization of the pH gradients obtained in IEF in order to mitigate the errors of analyte ampholyte pI assignments caused by the usual (but rarely proven) assumption of pH gradient linearity. These thermodynamic pKa and limiting mobility values also enable the reality-based numeric simulation of the IEF process using, for example, Simul (freeware, https://echmet.natur.cuni.cz/software/download).

Determination of thermodynamic acidity constants and limiting ionic mobilities of weak electrolytes by capillary electrophoresis using a new free software AnglerFish.

Thermodynamic acidity constants (acid or acid-base dissociation constants, sometimes called also as ionization constants) and limiting ionic mobilities (both of them at defined temperature, usually 25°C) are the fundamental physicochemical characteristics of a weak electrolyte, that is, weak acid or weak base or ampholyte. We introduce a novel method for determining the data of a weak electrolyte by the nonlinear regression of effective electrophoretic mobility versus buffer composition dependence when measured in a set of BGEs with various pH. To correct the experimental data for zero ionic strength we use the extended Debye-Hückel model and Onsager-Fuoss law with no simplifications. Contrary to contemporary approaches, the nonlinear regression is performed on limiting mobility data calculated by PeakMaster's correction engine, not on the raw experimental mobility data. Therefore, there is no requirement to perform all measurements at a constant ionic strength of the set of BGEs. We devised the computer program AnglerFish that performs the necessary calculations in a user-friendly fashion. All thermodynamic pKa values and limiting electrophoretic mobilities for arbitrarily charged substances having any number of ionic forms are calculated by one fit. The user input consists of the buffer composition of the set of BGEs and experimentally measured effective mobilities of the inspected weak electrolyte.

The dynamics of band (peak) shape development in capillary zone electrophoresis in light of the linear theory of electromigration.

The continuity equations that describe the movement of ions in liquid solutions under the influence of an external stationary electric field, as it is utilized in electrophoresis, were introduced a long time ago starting with Kohlrausch in 1897. From that time on, there have been many attempts to solve the equations and to discuss the results. In electrophoresis, special attention has always been devoted to the peak shapes obtained by the detector since the shapes have a tight connection with the phenomena taking place during electromigration and influence the efficiency and selectivity of the separation. Among these phenomena, the most important is electromigration dispersion. In this commented review paper, we compare various models of electromigration, try to find points that connect them, and discuss the range of their validity in light of the linear and nonlinear theory of electromigration.

Enhancement of the conductivity detection signal in capillary electrophoresis systems using neutral cyclodextrins as sweeping agents.

Conductivity detection is a universal detection technique often encountered in electrophoretic separation systems, especially in modern chip-electrophoresis based devices. On the other hand, it is sparsely combined with another contemporary trend of enhancing limits of detection by means of various preconcentration strategies. This can be attributed to the fact that a preconcentration experimental setup usually brings about disturbances in a conductivity baseline. Sweeping with a neutral sweeping agent seems a good candidate for overcoming this problem. A neutral sweeping agent does not hinder the conductivity detection while a charged analyte may preconcentrate on its boundary due to a decrease in its effective mobility. This study investigates such sweeping systems theoretically, by means of computer simulations, and experimentally. A formula is provided for the reliable estimation of the preconcentration factor. Additionally, it is demonstrated that the conductivity signal can significantly benefit from slowing down the analyte and thus the overall signal enhancement can easily overweight amplification caused solely by the sweeping process. The overall enhancement factor can be deduced a priori from the linearized theory of electrophoresis implemented in the PeakMaster freeware. Sweeping by neutral cyclodextrin is demonstrated on an amplification of a conductivity signal of flurbiprofen in a real drug sample. Finally, a possible formation of unexpected system peaks in systems with a neutral sweeping agent is revealed by the computer simulation and confirmed experimentally.

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.

Equivalent peak resolution: characterization of the extent of separation for two components based on their relative peak overlap.

Although the classical formula of peak resolution was derived to characterize the extent of separation only for Gaussian peaks of equal areas, it is often used even when the peaks follow non-Gaussian distributions and/or have unequal areas. This practice can result in misleading information about the extent of separation in terms of the severity of peak overlap. We propose here the use of the equivalent peak resolution value, a term based on relative peak overlap, to characterize the extent of separation that had been achieved. The definition of equivalent peak resolution is not constrained either by the form(s) of the concentration distribution function(s) of the peaks (Gaussian or non-Gaussian) or the relative area of the peaks. The equivalent peak resolution value and the classically defined peak resolution value are numerically identical when the separated peaks are Gaussian and have identical areas and SDs. Using our new freeware program, Resolution Analyzer, one can calculate both the classically defined and the equivalent peak resolution values. With the help of this tool, we demonstrate here that the classical peak resolution values mischaracterize the extent of peak overlap even when the peaks are Gaussian but have different areas. We show that under ideal conditions of the separation process, the relative peak overlap value is easily accessible by fitting the overall peak profile as the sum of two Gaussian functions. The applicability of the new approach is demonstrated on real separations.

Twenty years of development of dual and multi-selector models in capillary electrophoresis: a review.

It has been 20 years since Lurie et al. first published their model of electromigration of an analyte under simultaneous interaction with two cyclodextrins as chiral selectors. Since then, the theory of (enantio)separation in dual and complex mixtures of (chiral) selectors is well understood. In spite of this, a trial-and-error approach still prevails in analytical practice. Such a situation is likely caused by the fact that the entire theory is spread over numerous papers and the relations between various models are not always clear. The present review condenses the theory for the first time. Available mathematical models and feasible practical approaches are summarized and their advantages and limitations discussed.