Numerical algorithms for capillary electrophoresis

High-resolution numerical simulations of electrophoresis using the Fourier pseudo-spectral method

We present the formulation, implementation, and performance evaluation of the Fourier pseudo-spectral method for performing fast and accurate simulations of electrophoresis. We demonstrate the applicability of this method for simulating a wide variety of electrophoretic processes such as capillary zone electrophoresis, transient-isotachophoresis, field amplified sample stacking, and oscillating electrolytes. Through these simulations, we show that the Fourier pseudo-spectral method yields accurate and stable solutions on coarser computational grids compared with other nondissipative spatial discretization schemes. Moreover, due to the use of coarser grids, the Fourier pseudo-spectral method requires lower computational time to achieve the same degree of accuracy. We have demonstrated the application of the Fourier pseudo-spectral method for simulating realistic electrophoresis problems with current densities as high as 5000 A/m² with over tenfold speed-up compared to the commonly used second-order central difference scheme, to achieve a given degree of accuracy. The Fourier pseudo-spectral method is also suitable for simulating electrophoretic processes involving a large number of concentration gradients, which render the adaptive grid-refinement techniques ineffective. We have integrated the numerical scheme in a new electrophoresis simulator named SPYCE, which we offer to the community as open-source code.

High-resolution dynamic computer simulation of electrophoresis using a multiphysics software platform

The modeling and simulation software COMSOL Multiphysics^® was recently extended with an electrophoretic transport interface. Its performance was investigated by comparison to results obtained using the 1D dynamic electrophoresis simulators GENTRANS and SIMUL5. Simulations of zone electrophoresis, isotachophoresis, isoelectric focusing and of an oscillating electrolyte system were performed. Smooth profiles were essentially identical indicating that the COMSOL electrophoretic transport interface is able to reproduce results of the 1D simulators. Differences in the way the respective numerical schemes handle steep concentration gradients and associated instabilities were observed. The COMSOL electrophoretic transport interface is expected to be useful as a general model for simulations in 1D, 2D or 3D geometries, as well as for simulations combining electrophoresis with other physical phenomena.

A Semiempirical Approach for a Rapid Comprehensive Evaluation of the Electrophoretic Behaviors of Small Molecules in Free Zone Electrophoresis

A phenomenological model is proposed for the evaluation of relative electrophoretic migration of charged substances present in mixtures and for the rapid pH optimization prior CZE method development. The simple and robust model is based on the Offord model that takes account of the chemical structure. The effective charge and the molecular mass of the molecule are needed; the charge can easily be calculated from pK a obtained from known sources or simulated with existing pK-calculation programs. A first example was chosen with the separation of hydroxy-s-triazines to illustrate the applicability of this simple approach for determination of the first buffer-pH conditions prior experimental method optimization when separation of different ions is needed. In a second example, the confirmation of aminoalcohols in the CZE method development of unsaturated hexahydro-triazines and oxasolidines.

A semi-empirical approach for a rapid comprehensive evaluation of the electrophoretic behaviors of small molecules in free-zone electrophoresis

A phenomenological model is proposed for the evaluation of relative electrophoretic migration of charged substances present in mixtures and for the rapid pH optimization prior to capillary zone electrophoresis method development. The simple and robust model is based on the Offord model, which takes account of the chemical structure. The effective charge and the molecular mass of the molecule are needed; the charge can easily be calculated from pKa obtained from known sources or simulated with existing pK-calculation programs. A first example was chosen with the separation of hydroxy-s-triazines to illustrate the applicability of this simple approach for determination of the first buffer-pH conditions prior experimental method optimization when separation of different ions is needed. In a second example, the confirmation of aminialcohols in the CZE method development of unsaturated hexahydro-triazines and oxasolidines.

Computer simulation of different modes of ACE based on the dynamic complexation model

Several modes of the often used ACE processes are simulated based on the principle of dynamic complexation of interacting species in a capillary column. The model is built on the mass transfer equation, to provide insight into the detailed analyte migration and interaction processes in CE. Normal ACE, Hummel-Dreyer method, vacancy affinity CE, vacancy peak method, and CE frontal analysis are simulated based on typical ACE conditions, and the results are compared with the detector responses of real CE processes using BSA and warfarin as a model system. Remarkable resemblance between the simulated results and the experimental observations was demonstrated for well-buffered ACE systems.

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.

Quantitative analysis of systematic errors originated from wall adsorption and sample plug lengths in affinity capillary electrophoresis using two-dimensional simulation

Two-dimensional (2D) simulation of capillary electrophoresis is developed to model affinity interaction and wall adsorption simultaneously. Finite difference schemes are used to evaluate the mass-transfer equation in cylindrical coordinates. A Langmuir second-order kinetic law is applied to regulate the wall adsorption and desorption processes. Contributions from the simulation parameters are investigated extensively, and parameters for accurate and efficient simulation are identified. With the 2D model, capillary zone electrophoresis and affinity capillary electrophoresis (ACE) in the presence of strong or weak wall adsorption are simulated to elucidate peak distortions. Finite sample injection length/amount and wall adsorption that lead to systematic errors in the estimated binding constants in ACE are quantified for the first time with both actual experiments and computer simulation. Methods for correcting the estimated binding constants are proposed to extend the usefulness of ACE.

Simul 5 – Free dynamic simulator of electrophoresis

We introduce the mathematical model of electromigration of electrolytes in free solution together with free software Simul, version 5, designed for simulation of electrophoresis. The mathematical model is based on principles of mass conservation, acid-base equilibria, and electroneutrality. It accounts for any number of multivalent electrolytes or ampholytes and yields a complete picture about dynamics of electromigration and diffusion in the separation channel. Additionally, the model accounts for the influence of ionic strength on ionic mobilities and electrolyte activities. The typical use of Simul is: inspection of system peaks (zones), stacking and preconcentrating analytes, resonance phenomena, and optimization of separation conditions, in either CZE, ITP, or IEF.

Numerical simulation of DNA sample preconcentration in microdevice electrophoresis

A numerical model is presented for the accurate and efficient prediction of preconcentration and transport of DNA during sample introduction and injection in microcapillary electrophoresis. The model incorporates conservation laws for the different buffer ions, salt ions, and DNA sample, coupled through a Gaussian electric field to account for the field modifications that cause electromigration. The accuracy and efficiency required to capture the physics associated with such a complex transient problem are realized by the use of the finite element-flux corrected transport (FE-FCT) algorithm in two dimensions. The model has been employed for the prediction of DNA sample preconcentration and transport during electrophoresis in a double-T injector microdevice. To test its validity, the numerical results have been compared with the corresponding experimental data under similar conditions, and excellent agreement has been found. Finally, detailed results from a simulation of DNA sample preconcentration in electrophoretic microdevices are presented using as parameters the electric field strength and the other species concentrations. The effect of the Tris concentration on sample stacking is also investigated. These results demonstrate the great potential offered by the model for future optimization of such microchip devices with respect to significantly enhanced speed and resolution of sample separation.

General Approach to High-Efficiency Simulation of Affinity Capillary Electrophoresis

The differential equation describing electrophoretic migration can be evaluated with various finite difference schemes (FDSs). However, the accuracy and efficiency can be dramatically different depending on the FDS chosen and the way the algorithm is implemented in a computer simulation program. The monotonic transport scheme is used as the algorithm for the hyperbolic part of the differential equation, and the first-order fully explicit scheme is used for the parabolic part of the equation. The combination of these algorithms minimizes the errors and maintains high efficiency. A circular arrangement of the cells in the computer's memory is used in the implementation of the algorithms, and the use of concentration thresholds to enable and disable cells along the capillary makes the new algorithm highly efficient. Either thermodynamic or kinetic constants can be used in this program to simulate binding interactions between two species for equilibrium and nonequilibrium affinity CE. Simulation results with various parameters are presented. The simulated peak with proper parameters for an equilibrium affinity CE experiment has shape and position similar to that of the experimental peak. The simulated electropherograms for a nonequilibrium affinity CE experiment also show characteristics of the experimental electropherograms.

System zones in capillary zone electrophoresis

When working with capillary zone electrophoresis (CZE), the analyst has to be aware that the separation system is not homogeneous anymore as soon as a sample is brought into the background electrolyte (BGE). Upon injection, the analyte creates a disturbance in the concentration of the BGE, and the system retains a kind of memory for this inhomogeneity, which is propagated with time and leads to so-called system zones (or system eigenzones) migrating in an electric field with a certain eigenmobility. If recordable by the detector, they appear in the electropherogram as system peaks (or system eigenpeaks). However, although their appearance can not be forecasted and explained easily, they are inherent for the separation system. The progress in the theory of electromigration (accompanied by development of computer software) allows to treat the phenomenon of system zones and system peaks now also in very complex BGE systems, consisting of several multivalent weak electrolytes, and at all pH ranges. It also allows to predict the existence of BGEs having no stationary injection zone (or water zone, EO zone, gap, dip). Our paper reviews the theoretical background of the origin of the system zones (system peaks, system eigenpeaks), discusses the validity of the Kohlrausch regulating function, and gives practical hints for preparing BGEs with good separation ability not deteriorated by the occurrence of system peaks and by excessive peak-broadening.

Software and internet resources for capillary electrophoresis and micellar electrokinetic capillary chromatography

The purpose of the contribution is to show how software and the internet have changed the way in which we carry out research, and what additional possibilities these new resources and tools provide. The internet and e-mail broaden our horizon for cooperation. There are no borders for information exchange: our library is a virtual one, electronic databases are at our fingertips. E-mail discussion groups provide an electronic community of CE users. Such forums have provided a basis for worldwide scientific cooperation amongst scientists; the present contribution is only one of several examples. Estimation software provides us with estimates of component properties in those cases where this information is not available from literature or from experiments (an estimated value is better than no value). Several examples will illustrate the use of estimation software in capillary zone electrophoresis and micellar electrokinetic capillary chromatography. Simulation software presents visualization of experimental results to be expected, both as a training tool and as the first step in method development. Other simulations yield valuable insights into phenomena that are not readily accessible experimentally for reasons of size or time-scale.

Simulation of electrophoretic separations by the flux-corrected transport method

Electrophoretic separations at typical experimental electric field strengths have been simulated by applying the flux-corrected transport (FCT) finite difference method to the transient, one-dimensional electrophoresis model. The performance of FCT on simulations of zone electrophoresis (ZE), isotachophoresis (ITP), and isoelectric focusing (IEF) has been evaluated. An FCT algorithm, with a three-point, central spatial discretization, yields numerical solutions without numerical oscillations or spurious peaks, which have plagued previously-published second-order solutions to benchmark ZE and ITP problems. Moreover, the FCT technique captures sharp zone boundaries and IEF peaks more accurately than previously-published, first-order upwind schemes.

Training software for electrophoresis

Computer programs, simulating electrophoretic separations, were evaluated and discussed with respect to their suitability for training purposes. Quite a number of them, mainly those dealing with steady-state phenomena, are sufficiently fast and user-friendly for the purpose of visualization of electrophoretic principles. Transient-state or dynamic models, however, are more suitable for the advanced user, mainly because of their inherent complexity and long calculation times.

A computer model for time-based simulation of electrophoresis systems with freely defined initial and boundary conditions

A PC-based program has been developed that allows the user to perform one-dimensional computer simulations of electrophoresis systems with freely definable initial- and boundary conditions. The program can handle n constituents with n pK-values and calculates constituent concentrations and derived parameters as a function of time. Results are displayed graphically on screen and are stored as data files for graphical hard-copy processing.