Vlastimil Hruška

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. Hruška 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.

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: ϕt = −(v0 + vEMDϕ)ϕx + δϕxx that contains linear (v0) and nonlinear migration (vEMD), 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.

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 (Hruška 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.

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.

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

The enantiomeric separation of methadone in the presence of multiple isomer mixtures of sulfated β‐cyclodextrin (S‐β‐CD) was studied experimentally with CZE and theoretically using computer simulation. Experiments were performed over many years with several lots of S‐β‐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‐β‐CDs with a higher charge state and (ii) differences in buffer composition. Apparent binding constants between methadone and the S‐β‐CD and complex mobilities were determined for different lots of S‐β‐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‐β‐CD than S‐methadone. For any given configuration there is a distinct S‐β‐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.

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.