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.

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).

Online preconcentration of weak electrolytes at the pH boundary induced by a system zone in capillary zone electrophoresis.

Within the frame of the dynamic pH junction preconcentration technique in capillary electrophoresis, we introduce a novel approach based on the use of the pH boundary of a system zone for the preconcentration of general, multivalent, weak analytes in a system of binary, uni-univalent, background electrolytes (BGE). For such purpose, in addition to presenting a comprehensive flowchart for the development of a method for BGE preconcentration, we showed several model cases using acidic, basic and ampholytic analytes. Furthermore, we combined the flowchart with calculations in electrophoretic software PeakMaster to determine all necessary information such as analyte mobility, system zones and the amplitude of the pH boundary of a system zone as a function of the sample matrix. For an even more detailed understanding of the process, we also investigated changes in the pH boundary through computer simulations with Simul 5, providing an in-depth characterization of all model analytes according to the steps of the flowchart and to PeakMaster calculations for experimental verification of the final BGE preconcentration.

Generalized model of the linear theory of electromigration and its application to electrokinetic chromatography: Theory and software PeakMaster 6-Next Generation.

The linear theory of electromigration, including the first-order nonlinear approximation, is generalized to systems with any equilibria fast enough to be considered instantaneous in comparison with the timescale of peak movement. For example, this theory is practically applied in the electrokinetic chromatography (EKC) mode of the CZE. The model enables the calculation of positions and shapes of analyte and system peaks without restricting the number of selectors, the complexation stoichiometry, or simultaneous acid-base equilibria. The latest version of our PeakMaster software, PeakMaster 6-Next Generation, implements the theory in a user-friendly way. It is a free and open-source software that performs all calculations and shows the properties of the background electrolyte and the expected electropherogram within a few seconds. In this paper, we mathematically derive the model, discuss its applicability to EKC systems, and introduce the PeakMaster 6 software.

Computer simulation and enantioselective capillary electrophoresis to characterize isomer mixtures of sulfated ß-cyclodextrins.

The enantiomeric separation of methadone in the presence of multiple isomer mixtures of sulfated ß-cyclodextrin (S-ß-CD) was studied experimentally with CZE and theoretically using computer simulation. Experiments were performed over many years with several lots of S-ß-CD from the same manufacturer with a specified degree of substitution of 7-11. Large differences in the migration patterns were observed between certain lots and it was concluded that the extent of labelling in lots released after a transition time was higher than originally specified. The migration pattern was observed to be associated with (i) the ionic strength increase resulting from using S-ß-CDs with a higher charge state and (ii) differences in buffer composition. Apparent binding constants between methadone and the S-ß-CD and complex mobilities were determined for different lots of S-ß-CD at varying ionic strength using phosphate and 3-morpholino-2-hydroxypropanesulfonic acid buffers. The obtained values were used as input for simulations. For a given ionic strength, agreement between predicted and experimentally observed behavior was obtained for different buffers. R-methadone has a stronger interaction with S-ß-CD than S-methadone. For any given configuration there is a distinct S-ß-CD concentration range which results in the cationic migration of S-methadone while the migration direction of R-methadone is reversed. This configuration was demonstrated to be applicable for micropreparative CZE separations.

Isolation and purification of glycoconjugates from complex biological sources by recycling high-performance liquid chromatography.

Among of the most urgent needs of the glycobiology community is to generate libraries of pure carbohydrate standards. While many oligosaccharides have recently been synthesized, some glycans of biomedical importance are still missing in existing collections or are available in only limited amounts. To address this need, we demonstrate the use of the relatively unexplored technique of recycling high-performance liquid chromatography (R-HPLC) to isolate and purify glycoconjugates from several natural sources. We were able to routinely achieve purities greater than 98%. In several cases, we were able to obtain isomerically pure substances, particularly for glycans with different positional isomerism. These purified substances can then be used in different analytical applications, for example, as standards for mass spectrometry (MS) and capillary-based separations. Moreover, using a bifunctional aromatic amine, the same derivatization agent can be used to enable UV detection of oligosaccharides during their purification and link the isolated molecules to functionalized surfaces and potentially create glycan arrays.

A nonlinear electrophoretic model for PeakMaster: part III. Electromigration dispersion in systems that contain a neutral complex-forming agent and a fully charged analyte. Theory.

We introduce a new nonlinear electrophoretic model for complex-forming systems with a fully charged analyte and a neutral ligand. The background electrolyte is supposed to be composed of two constituents, which do not interact with the ligand. In order to characterize the electromigration dispersion (EMD) of the analyte zone we define a new parameter, the nonlinear electromigration mobility slope, S(EMD,A). The parameter can be easily utilized for quantitative prediction of the EMD and for comparisons of the model with the simulated and experimental profiles. We implemented the model to the new version of PeakMaster 5.3 Complex that can calculate some characteristic parameters of the electrophoretic system and can plot the dependence of S(EMD,A) on the concentration of the ligand. Besides S(EMD,A), also the relative velocity slope, S(X), can be calculated. It is commonly used as a measure of EMD in electrophoretic systems. PeakMaster 5.3 Complex software can be advantageously used for optimization of the separation conditions to avoid high EMD in complexing systems. Based on the theoretical model we analyze the S(EMD,A) and reveal that this parameter is composed of six terms. We show that the major factor responsible for the electromigration dispersion in complex-forming electrophoretic systems is the complexation equilibrium and particularly its impact on the effective mobility of the analyte. To prove the appropriateness of the model we showed that there is a very good agreement between peak shapes calculated by PeakMaster 5.3 Complex (plotted using the HVLR function) and the profiles simulated by means of Simul 5 Complex. The detailed experimental verification of the new mode of PeakMaster 5.3 Complex is in the next part IV of the series.

A nonlinear electrophoretic model for PeakMaster: part IV. Electromigration dispersion in systems that contain a neutral complex-forming agent and a fully charged analyte. Experimental verification.

The complete mathematical model of electromigration dispersion in systems that contain a neutral complex forming agent and a fully charged analyte was introduced in the previous part of this series of papers (Part III - Theory). The model was implemented in the newest version of our simulation program PeakMaster 5.3 that calculates the effective mobility of the analyte and its nonlinear electromigration mobility slope, S(EMD), in the presence of a complex forming agent in the background electrolyte. The mathematical model was verified by both experiments and simulations, which were performed by our dynamic simulator Simul 5 Complex. Three separation systems differing in the chiral selector used (having different values for the complexation constant and the mobility of the complex) were chosen for the verification. The nonlinear electromigration mobility slope values were calculated from the simulations and the experiments that were performed at different complex forming agent concentrations. These data agree very well with those predicted by the mathematical model and provided the foundation for the discussion and explanation of the electromigration dispersion process that occurs in systems which contain a complex forming agent. The new version of PeakMaster 5.3 was shown to be a powerful tool for optimization of the separation conditions by minimizing electromigration dispersion which improves the symmetry of the analyte peaks and their resolution.

A nonlinear electrophoretic model for PeakMaster: II. experimental verification.

We introduce a computer implementation of the mathematical model of capillary zone electrophoresis described in the previous paper in this issue (Hruska et al., Electrophoresis 2012, 33), the program PeakMaster 5.3. The computer model calculates eigenmobilities, which are the eigenvalues of the Jacobian matrix of the electromigration system, and which are responsible for the presence of system eigenzones (system zones, system peaks). The model also calculates parameters of the background electrolyte: pH, conductivity, buffer capacity, ionic strength, etc., and parameters of the separated analytes: effective mobility, transfer ratio, molar conductivity detection response, and relative velocity slope. In addition to what was possible in the previous versions of PeakMaster, Version 5.3 can predict the shapes of the system peaks even for a complex injected sample profile, such as a rectangular plug. PeakMaster 5.3 can replace numerical simulation in many practically important configurations and the results are obtained in a very short time (within seconds). We demonstrate that the results obtained in real experiments agree well with those calculated by PeakMaster 5.3.

A nonlinear electrophoretic model for PeakMaster: I. mathematical model.

We extended the linearized model of electromigration, which is used by PeakMaster, by calculation of nonlinear dispersion and diffusion of zones. The model results in the continuity equation for the shape function ϕ(x,t) of the zone: ϕ(t) = -(v(0) + v(EMD) ϕ)ϕ(x) + δϕ(xx) that contains linear (v(0)) and nonlinear migration (v(EMD)), diffusion (δ), and subscripts x and t stand for partial derivatives. It is valid for both analyte and system zones, and we present equations how to calculate characteristic zone parameters. We solved the continuity equation by Hopf-Cole transformation and applied it for two different initial conditions-the Dirac function resulting in the Haarhoff-van der Linde (HVL) function and the rectangular pulse function, which resulted in a function that we denote as the HVLR function. The nonlinear model was implemented in PeakMaster 5.3, which uses the HVLR function to predict the electropherogram for a given background electrolyte and a composition of the sample. HVLR function also enables to draw electropherograms with significantly wide injection zones, which was not possible before. The nonlinear model was tested by a comparison with a simulation by Simul 5, which solves the complete nonlinear model of electromigration numerically.

Simulation of the effects of complex- formation equilibria in electrophoresis: I. mathematical model.

Simul 5 Complex is a one-dimensional dynamic simulation software designed for electrophoresis, and it is based on a numerical solution of the governing equations, which include electromigration, diffusion and acid-base equilibria. A new mathematical model has been derived and implemented that extends the simulation capabilities of the program by complexation equilibria. The simulation can be set up with any number of constituents (analytes), which are complexed by one complex-forming agent (ligand). The complexation stoichiometry is 1:1, which is typical for systems containing cyclodextrins as the ligand. Both the analytes and the ligand can have multiple dissociation states. Simul 5 Complex with the complexation mode runs under Windows and can be freely downloaded from our web page http://natur.cuni.cz/gas. The article has two separate parts. Here, the mathematical model is derived and tested by simulating the published results obtained by several methods used for the determination of complexation equilibrium constants: affinity capillary electrophoresis, vacancy affinity capillary electrophoresis, Hummel-Dreyer method, vacancy peak method, frontal analysis, and frontal analysis continuous capillary electrophoresis. In the second part of the paper, the agreement of the simulated and the experimental data is shown and discussed.

Simulation of the effects of complex- formation equilibria in electrophoresis: II. experimental verification.

The complete mathematical model of electromigration in systems with complexation agents introduced in the Part I of this article (V. Hruska et al., Eletrophoresis, 2012, 33, this issue), which was implemented into our simulation program Simul 5, was verified experimentally. Three different chiral selector (CS) systems differing in the type of the CS, the magnitude of the complexation constants as well as in the experimental conditions were selected for verification. The experiments and simulations were performed at various concentrations of the CSs in order to discuss the influence of the concentration of the CS on the separation. The simulated and experimental electropherograms show very good agreement in the position, shape and amplitude of the analyte peaks. The new Simul 5 Complex offers a deep insight into electrophoretical separations that take place in systems containing complexing agents, for example into enantiomer separations. Using Simul 5 Complex we were able to predict and explain the significant electromigration dispersion of analyte peaks. It was clarified that the electromigration dispersion in these systems results directly from complexation. The new Simul 5 Complex was also shown to be a useful and powerful tool for the prediction of the results of enantioseparations.

Electromigration oscillations occurring in ternary electrolyte systems with complex eigenmobilities, as predicted by theory and ascertained by capillary electrophoresis.

Chemical oscillations are driven by a gradient of chemical potential and can only develop in systems where the substances are far from chemical equilibrium. We have discovered a new analogous type of oscillations in ternary electrolyte mixtures, which we call electromigration oscillations. They appear in liquid solutions of electrolytes and are associated with the electromigration movement of ions when conducting an electric current. These electromigration oscillations are driven by the electric potential gradient, while the system can be close to chemical equilibrium. The unequivocal criterion for the instability of the electrolyte solution and its ability to oscillate is the existence of complex system eigenmobilities. We show how to calculate the system eigenmobilities by utilizing the linear theory of electromigration and how to identify the complex system eigenmobilities to predict electromigration oscillations. To experimentally prove these electromigration oscillations, we employ a commercially available instrument for capillary electrophoresis. The oscillations start a certain period of time after switching on the driving electric current. The axial concentration profiles of the electrolytes in the capillary attain a nearly periodic pattern with a spatial period in the range of 1-4 mm, with almost constant amplitude. This periodic pattern moves in the electric field with mobility that is equal to the real part of the complex eigenmobility pair. We have found several ternary oscillating electrolytes composed of a base and two acids, of which at least one has higher valence than one in absolute value. All the systems have three system eigenmobilities: one is real and close to zero, and the two others form the complex conjugate pair, the real part of which is far from zero.

Occurrence and behavior of system peaks in RP HPLC with solely aqueous mobile phases.

System peaks are important but often also disturbing phenomena occurring in separation systems. Behavior of system peaks was studied in reversed phase high performance liquid chromatography (RP HPLC) systems consisting of an RP Amide C16 column and aqueous solutions of organic acids with alkaline metal hydroxides as mobile phases. Binary mobile phases, composed of benzoic acid and lithium hydroxide (LiOH) or cesium hydroxide (CsOH), yielded two system peaks. The first peak was stationary and the second one moved with dilution of the mobile phase or with changes of the alkaline metal hydroxide concentration. The latter changes affected dissociation of the benzoic acid present in the mobile phase and thereby its retention. The presumption that the first system peak is not influenced by the type of alkaline metal cation and that it is related to the non-adsorbed component of the mobile phase was confirmed by a cyclic procedure. Three-component mobile phases composed of benzoic acid, tropic acid, and a hydroxide gave rise to three system peaks as expected. The first peak was again stationary and the two others shifted depending on the concentration variation of both acids. Resonance causing a zigzag peak, well described in capillary zone electrophoresis (CZE), was observed if 1-pentanol was injected into a chromatographic system with one-component mobile phase.

Determination of the surface heat-transfer coefficient in CE.

A knowledge of the heat-transfer coefficient, h(s), for the external surface of the capillary or the overall heat coefficient, h(OA), is of great value in predicting the mean increase in temperature of the electrolyte, DeltaT(Mean), during electrokinetic separations. For CE, traditional indirect methods of determining h(s) were time-consuming and tended to overestimate cooling efficiency; a novel method is introduced, which is based on curve-fitting of plots of conductance versus voltage to calculate several important parameters including DeltaT(Mean), h(s), the conductance free of Joule heating effects (G(0)) and the voltage that causes autothermal runaway, V(lim). The new method is superior to previously published methods in that it can be performed more quickly and that it corrects for systematic errors in the measurement of electric current for voltages <5 kV. These errors tended to exaggerate the cooling efficiency of commercial instruments so that the calculated increases in electrolyte temperature were smaller than their actual values. Axially averaged values for h(s) were determined for three different commercial CE instruments ranging from 164 W m(-2) K(-1) for a passively cooled instrument in a drafty environment to 460 W m(-2) K(-1) for a liquid-cooled instrument.

Simulation of desalting that occurs during isoelectric trapping separations.

Simul 5, the simulation program based on the 1-D model of electrophoresis has been extended to simulate isoelectric trapping (IET) separations that take place in recirculating multicompartmental electrolyzers (MCEs). In the extended Simul 5, the simulated separation space between the anode and cathode can be divided into multiple segments to represent the anode compartment, separation compartment(s) and the cathode compartment. The compartments may have identical or different cross sections. A new algorithm simulates convective mixing that occurs in the recirculating MCEs where the distances between the buffering membranes are short and the velocities of tangential flows through the compartments, orthogonal to the electric field, are high. The intensity of simulated mixing can be independently controlled in each compartment. pH transients that were reported to occur during the desalting step in IET separations were simulated with the extended Simul 5 program: the main features of the experimental results were reproduced by the simulations. The simulations shed light on the possible causes of uneven anion and cation removal rates, pH transients and the transient invasion of the separation compartment by components of the electrode solutions that might occur during the desalting step.

System peaks in micellar electrophoresis: I. Utilization of system peaks for determination of critical micelle concentration.

A new way to determine the critical micelle concentration (CMC) based on the mobilities of system peaks is presented. A general approach for the CMC determination is based on the change of the slope or on finding the inflection point in the plot of a physical property of solution as a function of surfactant concentration. The determination of CMC by system peaks in CE utilizes a "jump" instead of a continuous change in the measured quantity. This phenomenon was predicted by the program PeakMaster, which was modified for simulation of micellar systems. The simulation of the steep change in mobilities of the anionic system peaks showing the CMC value was verified experimentally in a set of measurements, where the concentration of the surfactant was varied while the ionic strength was kept constant. The experimental work fully proved our model. A comparative electric current measurement was carried out. The proposed method seems to offer easier CMC determination as compared to the standard methods.

Reliable electrophoretic mobilities free from Joule heating effects using CE.

Ionic electrophoretic mobilities determined by means of CE experiments are sometimes different when compared to generally accepted values based on limiting ionic conductance measurements. While the effect of ionic strength on electrophoretic mobility has been long understood, the increase in the mobility that results from Joule heating (the resistive heating that occurs when a current passes through an electrolyte) has been largely overlooked. In this work, a simple method for obtaining reliable and reproducible values of electrophoretic mobility is described. The electrophoretic mobility is measured over a range of driving powers and the extrapolation to zero power dissipation is employed to eliminate the effect of Joule heating. These extrapolated values of electrophoretic mobility can then be used to calculate limiting ionic mobilities by making a correction for ionic strength; this somewhat complicated calculation is conveniently performed by using the freeware program PeakMaster 5. These straightforward procedures improve the agreement between experimentally determined and literature values of limiting ionic mobility by at least one order of magnitude. Using Tris-chromate BGE with a value of conductivity 0.34 S/m and ionic strength 59 mM at a modest dissipated power per unit length of 2.0 W/m, values of mobility for inorganic anions were increased by an average of 12.6% relative to their values free from the effects of Joule heating. These increases were accompanied by a reduction in mobilities due to the ionic strength effect, which was 11% for univalent and 28% for divalent inorganic ions compared to their limiting ionic mobilities. Additionally, it was possible to determine the limiting ionic mobility for a number of aromatic anions by using PeakMaster 5 to perform an ionic strength correction. A major significance of this work is in being able to use CE to obtain reliable and accurate values of electrophoretic mobilities with all its benefits, including understanding and interpretation of physicochemical phenomena and the ability to model and simulate such phenomena accurately.

Prediction and understanding system peaks in capillary zone electrophoresis.

Introduction of a sample into the separation column (microchip channel) in capillary zone electrophoresis (microchip electrophoresis) will cause a disturbance in the originally uniform composition of the background electrolyte. The disturbance, a system zone, can move in some electrolyte systems along the separation channel and, on reaching the position of the detector, cause a system peak. As shown by the linear theory of electromigration based on linearized continuity equations formulated in matrix form, the mobility of the system zone--the system eigenmobility--can be obtained as the eigenvalue of the matrix. Progress in the theory of electromigration allows us to predict the existence and mobilities of the system zones, even in very complex electrolyte systems consisting of several multivalent weak electrolytes, or in micellar systems (systems with SDS micelles) used for protein sizing in microchips. The theory is implemented in PeakMaster software, which is available as freeware (www.natur.cuni.cz/gas). The linearized theory also predicts background electrolytes having no stationary injection zone (water zone, water gap, water dip, EO zone) or unstable electrolyte systems exhibiting oscillations and creating periodic structures. The oscillating systems have complex system eigenmobilities (eigenvalues of the matrix are complex). This paper reviews the theoretical background of the system peaks (system eigenpeaks) and gives practical hints for their prediction and for preparing background electrolytes not perturbed by the occurrence of system peaks and by excessive peak broadening.

Kohlrausch regulating function and other conservation laws in electrophoresis.

The Kohlrausch regulating function (KRF) is a conservation law (conservation function), which is held in electrophoresis and which enables calculation of the so-called adjusted concentrations of constituents. The KRF is not the only conservation function and, depending on the complexity of the electrophoretic system, other conservation laws may be obeyed having a broader range of applicability. The conservation laws are tightly related to system eigenmobilities and system zones (system peaks). In principle, no system eigenmobility is exactly zero, but in most practical cases at least one system's eigenmobility is close to zero. The existence of the close-to-zero eigenmobility inherently points to the existence of a conservation function and a system zone which is stationary. The stationary system zone is called injection zone, stagnant zone, water peak, or solvent dip. Electrophoretic (electromigration) systems can be divided into two types: (i) conservation systems, in which the absolute value of at least one system eigenmobility is close to zero and where at least one conservation law is obeyed and (ii) nonconservation systems, where no system eigenmobility is close to zero and no conservation law is obeyed. The paper reviews work dealing with conservation functions in electromigration, derives some "historical" conservation functions in a new way, derives several conservation functions for systems of multivalent electrolytes, and discusses electrophoretic systems that have nonconservation behavior. In some typical instances, the conservation functions are simulated by means of a dynamic simulation tool and depicted graphically.