Quantitative Analysis of Systematic Errors Originated from Wall

Two-dimensional (2D) simulation of capillary electrophoresis is developed to model affinity interaction and wall adsorption simultaneously. Finite dif...
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Anal. Chem. 2007, 79, 5343-5350

Quantitative Analysis of Systematic Errors Originated from Wall Adsorption and Sample Plug Lengths in Affinity Capillary Electrophoresis Using Two-Dimensional Simulation Ning Fang, Jiangwei Li, and Edward S. Yeung*

Ames LaboratorysUSDOE and Department of Chemistry, Iowa State University, Ames, Iowa 50011

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. High separation efficiency in capillary electrophoresis (CE) may be reduced if peak broadening and distortion caused by operating conditions are significant.1,2 Some of these deleterious effects, including electromigration dispersion,3-8 inhomogeneous electroosmotic flow,9,10 extracolumn factors,11,12 and Joule heating,13 are caused by inconsistent field, high electrical current, buffer compositions, or other unfavorable experimental conditions. These factors can often be eliminated or significantly reduced with careful experimental design. Other factors are inherent to the analytes. Longitudinal diffusion14,15 is always a contributor to symmetric * To whom correspondence should be addressed. E-mail: [email protected]. (1) Gasˇ, B.; Kenndler, E. Electrophoresis 2000, 21, 3888-3897. (2) Gasˇ, B.; Stedry, M.; Kenndler, E. Electrophoresis 1997, 18, 2123-2133. (3) Gebauer, P.; Bocek, P. Anal. Chem. 1997, 69, 1557-1563. (4) Gebauer, P.; Borecka, P.; Bocek, P. Anal. Chem. 1998, 70, 3397-3406. (5) Beckers, J. L. J. Chromatogr., A 1995, 696, 285-294. (6) Beckers, J. L. J. Chromatogr., A 1995, 693, 347-357. (7) Beckers, J. L. J. Chromatogr., A 1996, 741, 265-277. (8) Beckers, J. L. J. Chromatogr., A 1997, 764, 111-126. (9) Keely, C. A.; Vandegoor, T.; McManigill, D. Anal. Chem. 1994, 66, 42364242. (10) Towns, J. K.; Regnier, F. E. Anal. Chem. 1992, 64, 2473-2478. (11) Delinger, S. L.; Davis, J. M. Anal. Chem. 1992, 64, 1947-1959. (12) Peng, X. J.; Chen, D. D. Y. J. Chromatogr., A 1997, 767, 205-216. (13) Knox, J. H.; Grant, I. H. Chromatographia 1987, 24, 135-143. (14) Jorgenson, J. W.; Lukacs, K. D. Anal. Chem. 1981, 53, 1298-1302. (15) Kenndler, E.; Schwer, C. Anal. Chem. 1991, 63, 2499-2502. 10.1021/ac070412r CCC: $37.00 Published on Web 06/12/2007

© 2007 American Chemical Society

peak broadening. When two or more species interact during the CE process, the peak shape is also determined by the relative mobilities of the free and complexed forms, the magnitudes of equilibrium or kinetic constants, and the concentrations of interacting species.16 Adsorption of analytes to the capillary wall is another important contributing factor.17-20 Severe peak tailing is often observed when proteins are analyzed using an uncoated fused-silica capillary.10 The elution of analytes may also be delayed. A variety of affinity CE (ACE) methods have been developed to study equilibrium or kinetic properties of binding interactions.21-26 In normal ACE, one of the two binding species (analyte) is injected to form a narrow plug into the capillary filled with a buffer containing the other binding species (additive) at varying concentrations. Linear and nonlinear regression methods have been developed for determining binding constants.27-29 The losses of analytes and changes in migration time due to wall adsorption could mar the accuracy of the estimated binding constants. It is therefore of great importance to examine affinity interactions in free solution together with wall adsorption/desorption kinetics and to estimate the associated systematic errors in order to interpret the experimental results more reliably. The often-overlooked source of systematic errors in ACE is the injected sample concentration and plug length. The regression methods are based on the assumption of instantaneous establishment of the steady state, i.e., the equivalence of the additive concentrations in the injected sample plug and in the background (16) Fang, N.; Chen, D. D. Y. Anal. Chem. 2006, 78, 1832-1840. (17) Ermakov, S. V.; Zhukov, M. Y.; Capelli, L.; Righetti, P. G. J. Chromatogr., A 1995, 699, 297-313. (18) Schure, M. R.; Lenhoff, A. M. Anal. Chem. 1993, 65, 3024-3037. (19) Ghosal, S. J. Fluid Mech. 2003, 491, 285-300. (20) Gasˇ, B.; Stedry, M.; Rizzi, A.; Kenndler, E. Electrophoresis 1995, 16, 958967. (21) Rundlett, K. L.; Armstrong, D. W. Electrophoresis 1997, 18, 2194-2202. (22) Rundlett, K. L.; Armstrong, D. W. Electrophoresis 2001, 22, 1419-1427. (23) Petrov, A.; Okhonin, V.; Berezovski, M.; Krylov, S. N. J. Am. Chem. Soc. 2005, 127, 17104-17110. (24) Galbusera, C.; Chen, D. D. Y. Curr. Opin. Biotechnol. 2003, 14, 126-130. (25) Heegaard, N. H. H.; Nissen, M. H.; Chen, D. D. Y. Electrophoresis 2002, 23, 815-822. (26) Busch, M. H. A.; Kraak, J. C.; Poppe, H. J. Chromatogr., A 1997, 777, 329353. (27) Rundlett, K. L.; Armstrong, D. W. J. Chromatogr., A 1996, 721, 173-186. (28) Bowser, M. T.; Chen, D. D. Y. J. Phys. Chem. A 1998, 102, 8063-8071. (29) Bowser, M. T.; Chen, D. D. Y. J. Phys. Chem. A 1999, 103, 197-202.

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electrolyte (BGE).16,30 This assumption implies that the sample plug length should be much smaller than the capillary length to the detector. This is usually not a problem for regular CE experiments in which 0.1-0.3-cm-long sample plugs are introduced into 40-100-cm-long capillary columns. However, when ultrashort capillaries or microchips are employed for rapid CE analysis, the injected sample can fill a non-negligible portion of the capillary and produces systematic errors in the estimated binding constant. Here, it will be shown with computer simulation and actual ACE experiments that the magnitudes of these errors can be fairly large if the regression methods are used directly without further corrections. A number of computer simulation models of CE without the consideration of wall adsorption have been developed.31-40 All of these models are one-dimensional (1D); i.e., the concentration distribution along the longitudinal (axial) direction is calculated while the radial distributions is assumed to be constant. However, when the analytes are adsorbed on the capillary wall, the radial distribution and the concentrations at the wall have to be included explicitly in the calculation. Several two-dimensional (2D) or pseudo-2D models were developed to account for both wall adsorption and axial electrophoretic migration. Ermakov et al. used a pseudo-2D approach: wall adsorption and radial diffusion were considered only for a very thin adsorbing layer, and the concentration distribution in the rest of the radial dimension was considered constant.17 With the assumption of slow axial variation, Ghosal’s model used a similar pseudo-2D approach, but emphasized on the nonuniform ζ-potential as a consequence of wall adsorption.19 Gasˇ et al. developed a true 2D model and demonstrated simulated peak shapes and peak variances under different conditions for capillary zone electrophoresis (CZE).20 Computer simulation is probably the most convenient way of studying the impacts of single (or a group of) condition(s) because all other conditions can be readily kept constant. SimDCCE (SIMulation of Dynamic Complexation Capillary Electrophoresis) is a recently developed, efficient simulation program of affinity interactions in CE and has the unique ability of animating the migrational and interacting behaviors of ACE experiments in real time or faster to give a comprehensive visual presentation.40 A SimDCCE-based 2D simulation model of ACE with consideration of wall adsorption is introduced in this paper. While the environmental factors are assumed to be wellcontrolled and consistent throughout the entire capillary column and the separation process, the inherent properties of the analytes, namely, longitudinal and radial diffusion, affinity interaction, and (30) Galbusera, C.; Thachuk, M.; De Lorenzi, E.; Chen, D. D. Y. Anal. Chem. 2002, 74, 1903-1914. (31) Saville, D. A.; Palusinski, O. A. AIChE J. 1986, 32, 207-214. (32) Palusinski, O. A.; Graham, A.; Mosher, R. A.; Bier, M.; Saville, D. A. AIChE J. 1986, 32, 215-223. (33) Dose, E. V.; Guiochon, G. A. Anal. Chem. 1991, 63, 1063-1072. (34) Ikuta, N.; Hirokawa, T. J. Chromatogr., A 1998, 802, 49-57. (35) Ikuta, N.; Sakamoto, H.; Yamada, Y.; Hirokawa, T. J. Chromatogr., A 1999, 838, 19-29. (36) Ermakov, S. V.; Bello, M. S.; Righetti, P. G. J. Chromatogr., A 1994, 661, 265-278. (37) Ermakov, S.; Mazhorova, O.; Popov, Y. Informatica 1992, 3, 173-197. (38) Gas, B.; Vacı´k, J.; Zelensky´, I. J. Chromatogr., A 1991, 545, 225-237. (39) Dubroa´kova´, E.; Gasˇ, B.; J. V.; Smolkova´-Keulemansova´, E. J. Chromatogr., A 1992, 623, 337-344. (40) Fang, N.; Chen, D. D. Y. Anal. Chem. 2005, 77, 840-847.

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wall adsorption, are simultaneously accounted for by this simulation model. Using the simulation model, the systematic errors originating from finite sample plug length and wall adsorption are quantified for the first time. Correction methods are proposed for more reliable determination of binding constants in short capillaries or in microchip channels. MATHEMATICAL MODEL AND IMPLEMENTATION Description of Simulated ACE Systems. The discussions in this paper are based on a 1:1 interaction between two species. A and P are used throughout to denote the two interacting species, and AP is their complex. A is often referred to the analyte present in the injected sample plug, and P is used for the additive present in the BGE. However, in some modes of ACE, such as the vacancy CE method and the nonequilibrium CE method, the injected sample solution or the BGE can contain both A and P.26,41 An uncoated fused-silica capillary (total length Lt, length to detector Ld, inner radius rc) is first conditioned and filled with the BGE. Then a sample plug of length ls is introduced into the capillary by low-pressure hydrodynamic injection. Rectangular injection plugs are used in all simulation runs presented in the current work. Other shapes of the initial sample profiles can also be implemented. Following sample injection, a voltage (V) is applied across the capillary. Each species migrates at a different velocity (v), which is determined by the electric field strength (E), the electroosmotic mobility (EOF, µeo), and its own electrophoretic mobility (µep): v ) (µep + µeo)E. In the simulated ACE runs, the analytes are always prepared in the same buffer as the BGE but without the additive, and the analyte and additive concentrations are always much smaller than the BGE concentration. Therefore, an evenly distributed electric field along the entire capillary can be assumed, and electromigration dispersion is minimized.3 Moreover, the ζ-potential is not considered in the present model because the peak distortion caused by nonuniform ζ-potential is usually much less significant than that caused by affinity interaction and wall adsorption, especially when sample concentrations and injection plug lengths are small. Two types of interactions are considered in the current model: a 1:1 interaction between two species in solution and wall adsorption/desorption. The interacting species and their complex are denoted As, Ps, and APs if they are in solution, and Aw, Pw, and APw if adsorbed to the capillary wall. The binding sites on the capillary wall are denoted as W. The concentrations in solution ([A]s, [P]s, and [AP]s) are in unit of mol m-3, and the concentrations on the wall ([A]w, [P]w, [AP]w, and [W]) are in the unit of mol m-2. The two interactions can be expressed as follows: (1) the 1:1 interaction in the solution, As + Ps ) APs. (2)the wall interactions, As + W ) Aw; Ps + W ) Pw; APs + W ) APw. Each interacting species can exist in multiple forms: free, complexed, and wall-adsorbed. The average mobility is calculated as the sum of the fractions of different forms multiplied by their individual mobilities: (41) Busch, M. H. A.; Carels, L. B.; Boelens, H. F. M.; Kraak, J. C.; Poppe, H. J. Chromatogr., A 1997, 777, 311-328.

µA ) f AsµAs + f AwµAw + f APsµAPs + f APwµAPw

(1)

where µA is the average apparent mobility of A, f As, f Aw, f APs, and f APw are the fractions of the different forms (f As + f Aw + f APs + f APw ) 1), µAsand µAPs are the apparent mobilities of A and AP in solution (µAs ) µep,As + µeo, µAPs ) µep,APs + µeo), and µAw and µAPw are the mobilities on the capillary wall. An adsorbed molecule does not move on the wall and can only migrate in the solution after being released from the wall; therefore, µAw ) µAPw ) 0, and µep,Aw ) µep,APw ) -µeo. An equation similar to eq 1 can be written for the other interacting species P. Description of Simulated CZE Systems. When only one species (A) is injected to form a narrow sample plug in a capillary filled with the BGE without additive, the system is reduced to capillary zone electrophoresis. Although the present simulation model is designed for more complicated ACE experiments, CZE simulations can be readily performed by setting the concentrations of the other species (P) in the sample and the BGE to zero. CZE simulations are used to demonstrate and verify the effects of wall adsorption of a single analyte. These results can be found in the Supporting Information. Mass Transport Equation. The electrophoretic migration processes can be described by the mass transport equation,42 which is expressed as eq 2 in cylindrical coordinates.

(

)

∂Cz,r,t,i ∂Cz,r,t,i ∂2 ∂2 1 ∂ ) -µiEz + Di 2 + + 2 Cz,r,t,i ∂t ∂z r ∂r ∂r ∂z

(2)

where Cz,r,t,i is the concentration of species i at axial position z, radial position r, and time t, Ez is the local electric field strength, µi is the apparent mobility of i, and Di is the diffusion coefficient of i. The partial differential terms on the axial space (z) and time (t) are evaluated by using the same finite difference schemes discussed previously.40 The general idea is to divide the entire axial space, radial space, and time domains into small steps (also known as increments or grids) and then to carry out calculations on these discrete points in space and time. The finite steps in axial space, radial space, and time are denoted ∆z, ∆r, and ∆t, respectively. Each divided piece of space will be referred to as a cell in this paper. Figure 1 depicts a capillary divided into cells. The axial space is divided evenly in terms of both length and volume. The cross section of the capillary is divided by a set of evenly spaced concentric circles; however, the radial space is not divided evenly in volume due to the cylindrical shape of the capillary. The radial position (r) has to be considered in order to preserve mass balance in radial diffusion and wall adsorption/desorption and to calculate the detected signals. The cells from the wall to the center of the capillary can be identified by cell’s central position:

rc -

1 ∆r, 2

rc -

3 ∆r, ... , 2

3 ∆r, 2

1 ∆r 2

(42) Giddings, J. C. Unified Separation Science; Wiley-Interscience: New York, 1991.

Figure 1. Capillary column that is divided in radial (∆r) and longitudinal (∆z) spaces. The central position of each cell is indicated by a black dot. The innermost and outermost radial coordinates are displayed in the figure.

The radial diffusion term Di ((1/r)(∂/∂r) + ∂2/∂r2)Cz,r,t,i can be evaluated in a way similar to the other differential terms. It can be written as the following finite difference term:

Di

1 Cz,r +∆r,t,i - Cz,r -∆r,t,i r 2∆r + Di

Cz,r+∆r,t,i - 2Cz,r,t,i + Cz,r-∆r,t,i (∆r)2

(3)

Higher-order forms can also be used; however, because the number of radial steps is usually small (1-25), as demonstrated in the Supporting Information, there is no need to use higherorder finite difference equations. Wall Adsorption. A Langmuir second-order kinetic law is applied in the present model to regulate the adsorption and desorption processes. The wall interactions are considered between the monolayer of available binding sites on the wall and the layer of solution that is closest to the wall. The change in concentration of adsorbed species can be calculated using eq 4,

∂Cz,w,t,i ∂t

or

Cz,w,t+∆t,i - Cz,w,t,i ) ∆t kaCz,rc-(1/2)∆r,t,i [W] - kdCz,w,t,i (4)

where Cz,w,t,i is the concentration of species i on the wall, at axial position z, and time t, Cz,rc-(1/2)∆r,t,i is the concentration of species i in the cell closest to the wall, ka and kd are the association and dissociation rate constants, respectively, and [W] is the concentration of free binding sites on the wall. The concentration change in the cell rc - 1/2∆r is calculated using eq 5, in which the result is modified according to the radius and the defined radial step.

Cz,rc-(1/2)∆r,t+∆t,i - Cz,rc-(1/2)∆r,t,i ∆t

)

- (kaCz,rc-(1/2)∆r,t,i [W] - kdCz,w,t,i)

2rc ∆r (2rc - ∆r)

(5)

If all three species can be adsorbed to the capillary wall, competition takes place. Moreover, these species may have different sizes, charge distributions, and numbers of available Analytical Chemistry, Vol. 79, No. 14, July 15, 2007

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binding sites on the wall, which makes it difficult for accurate simulation. However, in many ACE systems, only the protein (receptor) and the complex can be adsorbed to the capillary wall, while the drug (ligand, usually smaller) does not show significant adsorption. For simplicity, all simulation runs demonstrated in the current work assume the latter case. Flow of Simulation. Four processes are involved in an ACE experiment: longitudinal (axial) electromigration and diffusion, binding interaction in the solution, radial diffusion, and wall adsorption/desorption. It is possible to implement a one-step calculation module for all four processes; however, this approach is a little complicated due to the involvement of many variables. The present model implements a three-step approach. For any given time step, the following three computations are executed: (1) Calculate the concentration changes for all species in all cells due to electrophoretic migration and longitudinal diffusion. (2) The interaction takes place in solution, and the new concentrations of interacting species are calculated based on equilibrium or kinetic constants. (3) Wall interaction and radial diffusion are considered simultaneously to obtain the final concentrations for the current time step. No matter which approach (one-step or three-step) is used, the key to preserve accuracy is the proper selection of space and time steps. A small time step is usually required. If the time step is too large, the concentration changes due to electrophoretic migration and diffusion would be too large; this, in turn, would lead to significant overestimation or underestimation of the concentration changes due to interactions. Circular arrangement of cells and user-defined concentration thresholds for the active cells are once again implemented to accelerate the simulation.40 The calculations are carried out only for the cells containing enough amounts (more than the userdefined concentration thresholds) of monitored species. In the present model, the concentrations of adsorbed species become the key concentration thresholds for defining active cells. Simulation Parameters. Space (∆z, ∆r) and time (∆t) steps are the most important parameters. The requirements for ∆z and ∆t have been discussed previously.40 Briefly, ∆z and ∆t must satisfy the Courant condition (∆z g |µiEz| ∆t) to eliminate or minimize numerical instability, dispersion, and diffusion. In the current work, ∆r and its relations with ∆z, ∆t, and diffusion coefficients are discussed in the Supporting Information. Online Snapshot and Concentration Detection. Online snapshots of the capillary at any given moment can be taken easily as the concentrations of all species in the divided cells are stored in the computer’s memory. This simulation program can display multiple concentration profiles (in the solution or on the wall) as movies. Simulated electropherograms can be generated by calculating the simulated signal from the concentrations of all species at the position of the detector at constant time intervals. Two detection modes are implemented in the simulation program: over the entire cross section or at the center spot of the detector window. Simulated signals are calculated as the sum of the product of each species’ normalized concentration in the light path and the corresponding signal multiplier that is assigned to each species to compensate for the differences in molar absorptivity. 5346

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EXPERIMENTAL SECTION Apparatus and Solutions. The 1:1 interaction between p-nitrophenol (analyte, A) and β-cyclodextrin (additive, P) to form a complex (AP) is used as a model system to study the systematic errors caused by different injection lengths. The experiments were carried out on a MDQ system (Beckman Coulter Inc., Fullerton, CA) with a photodiode array detector. A 48.5 cm long (38.2 cm to detector) × 50 µm i.d., fused-silica capillary (Polymicro Technologies, Phoenix, AZ) was used. β-Cyclodextrin (Sigma, St. Louis, MO) was dissolved in 160 mM borate buffer (pH 9.1) at various concentrations ranging from 1.0 to 15.0 mM. p-Nitrophenol (Fisher, Fair Lawn, NJ) was dissolved in the same borate buffer at a concentration of 1.0 mM, and acetone (1% v/v) was added as the EOF marker. All solutions were filtered through 0.22-µm filters. Capillary Electrophoresis Procedure. Prior to use, the capillary column was treated with 1 M NaOH, methanol, and purified water for 15 min each. Then, the capillary was flushed with the borate buffer for 30 min and was conditioned overnight. Between CE runs, the capillary column was rinsed with 0.1 M NaOH for 5 min, water for 2 min, and the appropriate buffer for 12 min. In each ACE run, a p-nitrophenol solution was injected at 0.5 psi for 3, 5, 10, or 20 s to form an analyte plug in the inlet of the capillary that was filled with the β-cyclodextrin solutions at various concentrations. A 20-kV potential of normal polarity was applied on the capillary, and a temperature of 20.0 °C was maintained throughout the experiments. Each ACE run under the same conditions was repeated three times. Stable current and good reproducibility were observed. The electrophoretic mobility of p-nitrophenol in the BGE without additive was measured by separate CE runs: the capillary column was flushed with the borate buffer for 12 min, followed by a pressure injection of p-nitrophenol for 3 s at 0.5 psi, and finally applying a voltage of 20 kV at the normal polarity. This procedure was repeated ∼20 times during the course of the ACE experiments. The time for pushing a short plug of p-nitrophenol solution from the inlet to the detector by a constant low pressure (0.5 psi) was also measured, and the sample plug lengths were estimated to be 0.15, 0.24, 0.48, and 0.95 cm for 3, 5, 10, and 20 s injections, respectively. With the presence of β-cyclodextrin in the BGE, the buffer viscosity increased slightly: ∼6% for the 15 mM β-cyclodextrin buffer over the BGE without additive, which could slightly change the injection plug length when the capillary was filled with different β-cyclodextrin buffers. This slight difference is ignored in all discussions. Simulated CE Runs. In addition to the actual CE experiments, several hundred ACE and CZE runs were performed by the simulation program. The most common conditions are as follows: capillary length, 5 cm; length to detector, 4 cm; capillary inner radius, 25 µm; applied voltage, 1000 V; and the diffusion coefficients of all species, 10-10 m2 s-1. A short capillary was used to greatly reduce the computing time. Other (or altered) conditions will be mentioned when the simulation results are discussed.

is observed for the entire additive concentration range even when a large analyte plug is introduced. When the K value increases to 500 M-1, peak fronting is obvious for low additive concentrations. With a high K (5000 M-1), the fronting is significant in the low end of the additive concentration range even with a small (0.01 cm) analyte plug. However, no matter which binding constant is used or how long the analyte plug is, at the high end of the additive concentration range, the analyte peaks are sharp and symmetric. These observations have been explained in detail in a previous paper.16 The migration times (tm) of the simulated analyte profiles are measured at the peak position or center of the plateau. The average analyte mobility under the influence of complexation with a certain amount of additive is calculated using eq 6 to account

A

µ )

µAep

+ µeo )

(

Lt Ld -

1 l 2 s

)

Vtm

(6)

for the different injection lengths. Due to different behaviors of the binding species and peak shapes caused by the mobility differences and other experimental conditions,16 eq 6 may not give the most accurate result. In the present work, the errors associated with eq 6 are treated as part of the systematic errors that will be quantified as a whole. Simulated Binding Isotherms. For ACE without wall adsorption, the average analyte electrophoretic mobility is calculated by the following equation:

µAep )

Figure 2. Simulated sample peak profiles. The binding constant and the injected sample plug length are displayed on each panel. The injected analyte concentration is 1 mM for all simulation runs. Each panel group (A, B, or C) has its own set of additive concentrations in the BGE ([P]), which are displayed on top of each panel group. These additive concentrations are in the same order as the analyte peaks, from left to right.

RESULTS AND DISCUSSION Simulated ACE Experiments with Different Injection Lengths. Three groups of ACE runs were simulated while wall adsorption was neglected. Each group used one of the following three binding constants (K): 0.05, 0.5, and 5 m3 mol-1 (or 50, 500, and 5000 M-1). Within each group, there were five to seven sets of runs using different sample injection lengths (ls), ranging from 0.001 to 0.18 cm or 0.025 to 4.5% of the capillary length to detector (4 cm in all simulation runs). For each set, a series of additive concentrations (0-100 mM) were used to produce a complete binding isotherm. In all three groups, the injected analyte concentration was kept constant at 1 mM and the apparent mobilities of free analyte, free additive, and complex are 2 × 10-8, 3.5 × 10-8, and 3 × 10-8 m2 V-1 s-1, respectively. The three sets of simulated analyte peak profiles in each group are shown in Figure 2. With a low K (50 M-1), little peak distortion

1 k′ + ) µ µ 1 + k′ ep,As 1 + k′ ep,APs K[P] 1 µep,As + µ (7) 1 + K[P] 1 + K[P] ep,APs

in which k′ t nAPs/nAs ) K[P] (nAPs and nAs are the amounts of complexed and free analyte, respectively) is the capacity factor.43 Equation 7 can be rearranged to give a nonlinear regression equation:

(µAep - µep,As) )

k′ (µ - µep,As) ) 1 + k′ ep,APs K[P] - µep,As) (8) (µ 1 + K[P] ep,APs

According to eq 8, the simulated binding isotherms are plotted as [P] versus µAep - µep,As for each set of experiments, and the binding constants are estimated using nonlinear regression. Systematic Errors in the Estimated Binding Constants. These estimated binding constants are all smaller than the K values input into the simulation program. The ratio of estimated versus true binding constants are plotted against the fraction of sample plug length to the capillary length (to detector) (ls/Ld) in Figure 3. At low K (50 M-1), all estimated values are within 90% of the true value. However, at high K (5000 M-1), the accuracy of the estimated K drops very quickly as the sample plug length (43) Bowser, M. T.; Bebault, G. M.; Peng, X. J.; Chen, D. D. Y. Electrophoresis 1997, 18, 2928-2934.

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Figure 3. Ratio of estimated to true binding constants plotted against the fraction of the sample plug length to the capillary length (to detector) (ls/Ld). Each data set was fitted with a three-parameter (y0, a, b) nonlinear regression equation: y ) y0 + ab/(b + x). All curves start from the point (0, 1).

increases: dropping to ∼50% of the true value when the injected sample plug occupies 1% of the capillary from inlet to detector. In reality, the sample plug length in a short channel (e.g., microfluidic device) could be well over 1% of the distance from inlet to detector. It is important to note that it is the systematic errors that are quantified in the current work, not the random errors analyzed by Bowser and Chen using the Monte Carlo simulation.28 The deviations from the true binding constant are also related to the analyte concentration and the relative mobilities of the analyte, additive, and complex. More ACE experiments were simulated with different concentrations and mobilities. The results showed that the accuracy of the regression methods improves with lower analyte concentrations or larger mobility differences between free analyte and additive. This is because these conditions favor the rapid establishment of the steady-state conditions. The best strategies for getting accurate results with the ACE regression method are as follows: (1) The analyte concentration is as low as the detection limit allows; (2) the additive concentrations cover the full range of a nonlinear regression curve; (3) the sample injection length is as small as possible provided that the injection method is reproducible. However, it is not always easy to meet these conditions. In the case that a large binding constant is to be determined, or a relatively narrow analyte plug is not achievable, the estimated binding constant may not be accurate enough without further corrections. Actual ACE Experiments with Varying Injection Lengths and “True” Binding Constant. The well-characterized 1:1 binding interaction between p-nitrophenol and β-cyclodextrin44,45 was chosen as the model system to verify the simulation results. Four sets of ACE experiments were carried out with different sample injection lengths, and four estimated binding constants were calculated according to eq 8 and plotted in Figure 4, which shows a trend similar to those simulated data sets in Figure 3. Nonlinear curve fitting was performed on the data points and error bars, and the intercepts at the y-axis (infinitely narrow sample (44) Fang, N.; Chen, D. D. Y. Anal. Chem. 2005, 77, 2415-2420. (45) Fang, N.; Ting, E.; Chen, D. D. Y. Anal. Chem. 2004, 76, 1708-1714.

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Figure 4. Experimentally estimated binding constant vs the fraction of the sample plug length in the capillary length to detector. Each data point was calculated from one set of ACE experiments with the corresponding sample plug length. The error bars were generated from the nonlinear regression of the binding isotherms. The solid curve gives a good fit of the data points, and the dotted curves give the upper and lower limits. The intercepts of the curves at the y-axis give the true binding constant at 630 ( 30 M-1.

plug) give the true binding constant and its associated errors: 630 ( 30 M-1. This value is certainly more accurate than any of the four values calculated directly with the regression methods. This true binding constant describes this interaction under the selected solvent and capillary conditions, devoid of the influences of injection parameters and sample concentrations. Simulation of ACE with Wall Adsorption. The other type of systematic error is wall adsorption. The impacts of weak and strong wall adsorption on the analyte peak shape will be demonstrated by simulation. In addition to the common conditions listed in the Experimental Section, these simulated ACE runs have the following conditions: the initial concentration of the sample A (1 mol m-3), the initial concentration of P in the BGE (5 mol m-3), the binding constant for the A + P interaction (500 M-1), the injection plug length (0.18 cm), the electrophoretic mobilities of free A, free P, and complex AP (2, 3.5, 3 × 10-8 m2 V-1 s-1, respectively), no EOF (µeo ) 0), and the maximum concentration of available binding sites on the wall for A (10-5 mol m-2). The three key simulation parameters are ∆t ) 0.001 s, ∆z ) 1.0 × 10-5 m, and ∆r ) 1.25 × 10-6m. The additive P has no wall interaction, and the free analyte A and complex AP have identical adsorption/desorption rates. The electropherograms generated with four sets of adsorption/ desorption rates are compared to the one without wall adsorption in Figure 5. The concentration profiles of the total analyte concentration (the sum of free A and complex AP concentrations in the solution and on the wall for the entire cross section) are displayed in Figure 5. When there is no wall interaction, the resulted analyte peak has a front slope and a larger peak height (2.0 mol m-3, twice as large as the initial concentration). This analyte stacking is caused mainly by the mobility differences.16 When there is wall adsorption, the front slopes completely disappear, and the peaks become much broader. Peak distortion is not significant with faster rates; however, when the desorption rate gets smaller, the expected long tails are evident.

ACE Binding Isotherms in the Presence of Wall Adsorption. Figure 6 shows three sets of simulated ACE runs with and without wall adsorption. Most conditions are similar to the simulation runs shown in Figure 5. The analyte peaks are delayed longer as wall adsorption becomes stronger. It is also evident that wall adsorption and affinity interaction compete in molding the final peak shapes. In the presence of wall adsorption (assuming identical adsorption/desorption abilities for the free and the complexed analyte), the capacity factor for wall adsorption (k′′ t (nAw + nAPw)/nAs, the ratio of the total amount of analyte adsorbed to the wall to the amount of free analyte in the solution) should be added into eq 7:

µAep ) Figure 5. Simulation of ACE runs with different values of ka and kd. The two peaks on the left-hand side are also shown in the blownup picture. Other conditions are listed in the text.

The relative mobilities of free and complexed species, the binding constant, the concentrations, and other conditions can be varied easily in the simulation program, and different peak profiles can be generated to reflect the impact of the changed conditions. Other CE modes involving dynamic complexation of two species, such as vacancy methods and wall adsorption, can also been simulated with the present simulation model by changing the initial conditions.

1 µ + 1 + k′ + k′′ ep,As k′′ k′ + (9) µ µ 1 + k′ + k′′ ep,APs 1 + k′ + k′′ ep,Aw

If µeo ) 0, then µep,Aw ) µep,APw ) -µeo ) 0. Equation 9 can be simplified and rearranged to give

µAep -

1 ) µ 1 + k′′ ep,As k′ 1 µ µ (10) 1 + k′ + k′′ ep,APs 1 + k′′ ep,As

(

)

Note that 1/(1 + k′′)µep,As is the analyte mobility in the presence of only wall adsorption and the absence of additive in the BGE.

Figure 6. Three sets of analyte peak profiles generated with (A) no wall interaction, (B) fast adsorption and desorption, and (C) slower adsorption and desorption. (A) and (B) have identical time scale (66-113 s); (C) shows a longer time span (90-210 s). The additive concentrations (in mol m-3) in the BGE are displayed on top of the peaks.

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The success of this method heavily relies on the simulation program. The model presented in this paper is accurate and efficient as long as the assumption of constant electric field strength throughout the capillary is valid. Even when the exact (true) binding constant cannot be found under more complicated conditions, this approach would still give a good estimate of the associated systematic errors.

Figure 7. ACE binding isotherms. The y-axis is the difference between the analyte mobility at a given additive concentration and the analyte mobility in the absence of the additive, that is, µAep µep,As without wall adsorption or µAep - 1/(1 + k′′) µep,As with wall adsorption. The regression equation used for all three data sets is y ) ax/(1 + bx), where a and b are regression parameters.

Binding isotherms for all three sets of ACE runs are plotted as µAep - µep,As (no wall adsorption) or µAep - 1/(1 + k′′) µep,As (with wall adsorption) versus [P] in Figure 7. The estimated binding constants from nonlinear regression are 334, 362, and 537 M-1 for no wall adsorption, weak adsorption, and strong adsorption, respectively. While the long sample plug (4.5% of the capillary length from inlet to detector) leads to an underestimate of the binding constant, wall adsorption causes an overestimate. Corrections for Systematic Errors with Computer Simulation. Figure 3 demonstrates a method to find the “true” binding constant in the absence of wall adsorption; however, it requires multiple sets of actual ACE experiments, and it is not suitable for estimating systematic errors caused by wall adsorption. Another approach, which requires more simulation but fewer experiments, may be used to compensate for both types of systematic errors. The proposed procedure is as follows: (1) Run a typical set of ACE experiments. (2) Estimate a binding constant using the regression method. (3) Simulate the same set of experiments with real injection parameters, the electrophoretic mobilities of all species, and the adsorption properties. The binding constant used in the simulation is chosen from the range suggested by the estimated value above. (4) The regression method is used on the simulated data to obtain another estimated binding constant. If this second value is close enough to the one obtained from actual experiments, the binding constant used as the input to the simulation is the true value with certain uncertainties; otherwise, go back to step 3 to try a different K. It is not difficult to write a program to let the computer repeat steps 3 and 4 until the true binding constant is found based on internal consistency.

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CONCLUSIONS This 2D CE simulation model with consideration of binding interactions in the solution and wall adsorption is an efficient way to study the behaviors of interacting species and to explain the resulting peak profiles in well-buffered systems. The selection of the key parameters for achieving fast and accurate simulation is investigated extensively based on simple CZE experiments. Fast and slow nonlinear adsorption/desorption kinetics are studied using this model. Other types of kinetics of affinity interaction and wall adsorption can also be implemented. Other deteriorating effects with parabolic profiles, e.g., Joule heating, pressure-driven flow, and uneven ζ-potential-induced laminar flow, may also be simulated using the concepts behind this 2D simulation model. The model can be used to explain experimental observations and predict peak profiles for ACE experiments when wall adsorption is a contributing factor. As a result, the usefulness of ACE may be expanded. Binding isotherms have been generated from complete sets of simulated ACE experiments in the presence or absence of wall adsorption. Using computer simulation, the influences of wall adsorption, sample plug length, and capillary radius on analyte migration can be better understood. The proposed simulation-intense approach for compensating for the systematic errors is admittedly in its infancy and requires further advances in computing power and simulation accuracy to realize its full potential. ACKNOWLEDGMENT The authors thank Dr. David Chen of University of British Columbia for sharing the simulation program SimDCCE, which is the starting point for the new model presented in this paper. E.S.Y. thanks the Robert Allen Wright Endowment for Excellence for support. The Ames Laboratory is operated for the U.S. Department of Energy by Iowa State University under Contract DE-AC02-07CH11358. This work was supported by the Director of Science, Office of Basic Energy Sciences, Division of Chemical Sciences. SUPPORTING INFORMATION AVAILABLE Discussions on the simulation parameters and the simulation results of CZE with wall adsorption. This material is available free of charge via the Internet at http://pubs.acs.org. Received for review February 28, 2007. Accepted May 1, 2007. AC070412R