Behavior of Interacting Species in Capillary Electrophoresis Described

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Anal. Chem. 2006, 78, 1832-1840

Behavior of Interacting Species in Capillary Electrophoresis Described by Mass Transfer Equation Ning Fang and David D. Y. Chen*

Department of Chemistry, University of British Columbia, Vancouver, BC, Canada V6T 1Z1

Affinity capillary electrophoresis (ACE) has been used to estimate thermodynamic constants of binding interactions with linear or nonlinear regression methods. The accuracy of this approach relies heavily on the binding interaction mechanism, which is controlled by both the nature of the interaction and the experimental conditions. The development of a highly efficient computer-simulated ACE system makes it possible to demonstrate the detailed behavior of any interacting species of a given interaction under any conditions. The order of the mobilities of the complex and the two binding species in their free forms is a key factor to determine what molecules in what locations of the column are involved in the interaction, and the peak shape resulting from such interactions, of a given ACE experiment. In this paper and the supporting materials, 18 scenarios in 6 different combinations of migration orders of the free analyte, free additive, and complex formed are studied by a computer simulation program based on the mass transfer equation. From the study of these situations, we conclude high additive concentration (ensuring high capacity factor) and low analyte concentration (ensuring fast fill-in of the free additive in the analyte plug) are crucial for obtaining accurate results when using the regression methods. On the other hand, the approach to estimate binding constants with computer simulation can be much more accurate as long as accurate and efficient simulation models can be developed, especially when the ratio of the additive and analyte concentrations is not large enough. The mass transfer equation is the governing principle of analyte migration in all separation techniques.1 As for capillary electrophoresis (CE), this equation accounts for molecular transport driven by electric field and analyte diffusion. In our earlier work, we demonstrated efficient ways of interpreting and implementing the mass transfer equation into efficient algorithms so that modern computers can simulate the migration behavior of analytes in real time or faster.2 The dynamic complexation of solutes, while migrating at different velocities in a single phase, is a fundamental and crucial * To whom correspondence should be addressed; Tel. +01 (604) 822-0878. Fax +01 (604) 822-2847. E-mail: [email protected]. (1) Giddings, J. C. Unified Separation Science; Wiley-Interscience Publication: New York, 1991.

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assumption for developing practical theories for CE. The elution peaks in electropherograms do not always appear in perfect Gaussian shapes; instead, peak broadening, fronting, tailing, splitting, or other shapes are often observed. There are many sources that can contribute to peak distortion, including extracolumn factors,3,4 longitudinal diffusion,5,6 different path (eddy diffusion),7 wall adsorption,8 nonhomogeneous electroosmotic flow,9,10 nonhomogeneous field,11 Joule heating,12 and electromigration dispersion and anomalous electromigration dispersion.13 Gasˇ and his colleagues have given comprehensive reviews on the effect of these factors.14,15 In practice, all of the possible deteriorating phenomena except molecular diffusion can be either eliminated or significantly reduced with carefully designed experiments. However, some fundamental and unavoidable factors can also determine the peak shape and peak width even though most of the aforementioned factors are eliminated. In our previous paper, we proposed that the migration behavior of individual solutes can produce peak shapes that are characteristic of the relative mobilities of the analyte (µA), additive (µP), and complex (µC) formed in the separation process, and the peak maximums can be determined in each situation.16 There are six possible orders of the three mobilities: µP > µC > µA, µP > µA > µC, µC > µA > µP, µA > µC > µP, µC > µP > µA, and µA > µP > µC. The initial stages of some processes have been demonstrated by Busch et al.,17 but the recent progress in algorithm implementation and the optimization of computer simulation made it possible to study the whole process in real time or faster. In this paper, we will discuss the mechanisms of all six cases in more detail and demonstrate that not only the orders of the three mobilities but (2) Fang, N.; Chen, D. D. Y. Anal. Chem. 2005, 77, 840-847. (3) Delinger, S. L.; Davis, J. M. Anal. Chem. 1992, 64, 1947-1959. (4) Peng, X.; Chen, D. D. Y. J. Chromatogr., A 1997, 767, 205-216. (5) Jorgenson, J. W.; Lukacs, K. D. Anal. Chem. 1981, 53, 1298-1302. (6) Kenndler, E.; Schwer, C. Anal. Chem. 1991, 63, 2499-2502. (7) Kasicka, V.; Prusik, Z.; Gas, B.; Stedry, M. Electrophoresis 1995, 16, 20342038. (8) Schure, M. R.; Lenhoff, A. M. Anal. Chem. 1993, 65, 3204-3237. (9) Towns, J. K.; Regnier, F. E. Anal. Chem. 1992, 64, 2473-2478. (10) Kok, W. T. Anal. Chem. 1993, 65, 1853-1860. (11) Keely, C. A.; Vandegoor, T.; McManigill, D. Anal. Chem. 1994, 66, 42364242. (12) Knox, J. H.; Grant, I. H. Chromatographia 1987, 24, 135-143. (13) Gebauer, P.; Bocek, P. Anal. Chem. 1997, 69, 1557-1563. (14) Gasˇ, B.; Stedry, M.; Kenndler, E. Electrophoresis 1997, 18, 2123-2133. (15) Gasˇ, B.; Kenndler, E. Electrophoresis 2000, 21, 3888-3897. (16) Fang, N.; Ting, E.; Chen, D. D. Y. Anal. Chem. 2004, 76, 1708-1714. (17) Busch, M. H. A.; Kraak, J. C.; Poppe, H. J. Chromatogr., A 1997, 777, 329353. 10.1021/ac051822n CCC: $33.50

© 2006 American Chemical Society Published on Web 02/16/2006

also the magnitude of the difference between the three mobilities, the concentrations of the analyte and the additive, the value of binding constants, and other factors can also determine peak shapes. The peak shapes determined by these factors are the result of fundamental limitations of electrophoretic migration and cannot be eliminated unless some experimental conditions are changed. If the studies of peak shapes are conducted based on real experiments, as we did in a previous paper,16 it is impracticable to cover all possible cases because there are too many of them and it is sometimes difficult to find real systems for certain cases. A correct theory should not only explain the phenomena already observed but also provide guidance to what could be observed in the future. Therefore, in this paper, a computer simulation program based on the fundamental mass transfer equation is used to simulate CE experiments with various experimental conditions. The foremost requirement to enable this study is that the simulation program must be accurate and efficient. Using the simulation program of affinity CE (ACE) we developed earlier,2 we are able to control most of the experimental conditions, as well as having simulation outputs in terms of plotting any and all components at any given simulated time. Not only can the final peak shapes be obtained but at any given moment the snapshots of the solute migration (the concentration profiles of the analyte and the additive) can be stored and retrieved. The mechanism can be illustrated directly by a series of snapshots. We will demonstrate that the analyte plug undergoes interesting processes of expanding or shrinking in each case, which has not been shown by others. The well-established linear and nonlinear regression methods can be used to estimate binding constants from ACE experiments.18-20 Bowser and Chen pointed out from a mathematical standpoint that when the concentration of the analyte is much smaller than the concentration of the additive, an appropriate range of additive concentrations should be chosen to calculate binding or dissociation constants.20 An enumeration method using the appropriate simulation model, however, can be used in nonideal situations.21 In this paper, by revealing the mechanisms under various experimental conditions, the deviation of the regression equations in some cases will be demonstrated. MECHANISMS AND DISCUSSION In a typical ACE experiment, a background electrolyte (BGE) with an additive (P) at various concentrations fills the capillary, and the analyte (A) is injected into the capillary to form a narrow plug. Once an electric field is applied, the additive enters the analyte plug, forming a complex (C). The migration rate of an analyte at any given instant can be described by the sum of the fractions of different analyte species multiplied by their individual velocities migrating in a separation system, and the capacity factor of an additive (k′ )[C]/[A]), is the product of the binding constant (K) and the free additive concentration ([P]).22 The regression methods for calculating binding constants are based on the assumption of the instant establishment of steady(18) Rundlett, K. L.; Armstrong, D. W. J. Chromatogr., A 1996, 721, 173-186. (19) Peng, X. J.; Bowser, M. T.; Britz-McKibbin, P.; Bebault, G. M.; Morris, J. R.; Chen, D. D. Y. Electrophoresis 1997, 18, 706-716. (20) Bowser, M. T.; Chen, D. D. Y. J. Phys. Chem. A 1998, 102, 8063-8071. (21) Fang, N.; Chen, D. D. Y. Anal. Chem. 2005, 77, 2415-2420. (22) Bowser, M. T.; Chen, D. D. Y. Electrophoresis 1997, 18, 2928-2934.

state conditions. The steady-state condition in this case is that the concentration of the free additive in the analyte plug is equal to the concentration of the additive present in the BGE ([P]0), that is, [P] ) [P]0.23 However, in many CE processes, [P] is not constant throughout the CE process. In fact, it is often significantly higher or lower than [P]0. The average mobility of the analyte, rather than the mobility at any given moment, has to be used in the regression equations. Computer simulation based on the mass transfer equation can illustrate the mechanisms without performing real experiments. It can be used to evaluate experimental designs in order to obtain more accurate results. The key to a successful computer-simulated CE model is to implement a stable and accurate finite difference scheme to solve the following mass transfer equation efficiently.2,24-26

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

(1)

where Cz,t,i is the concentration of species i at position z and time t, Ez is the total local electric field at position z, µi is the apparent mobility of the ion i, and Di is the diffusion coefficient of ion i. The simulation program used in this paper implemented the firstorder forward-space scheme, the second-order monotonic transport scheme, and the first-order fully explicit scheme to evaluate the three partial differential terms, ∂Cz,t,i/∂t, ∂Cz,t,i/∂z, and ∂2Cz,t,i/ ∂z2, respectively.27,28 The mechanism of ACE is studied under the following experimental conditions. The local electric field is maintained constant throughout the capillary. With this condition, the simulation program does not need to keep track of concentrations of other ions in the background electrolyte, and only the ions of analytes and additives, as well as complexes, are monitored. In practice, H+, OH-, and other ion concentrations are not the point of interest in a well-buffered system where the analyte and additive concentrations are much less than the BGE concentration. Other factors that can distort the ideal peak shapes, such as Joule heating and wall adsorption, will not be considered. Only diffusion is considered because its effect is always present during the analyte migration, and its contribution to mass transfer is evaluated by eq 1. ACE experiments can be categorized into six cases according to the combinations of the three mobilities, and in each case, the analyte plug goes through a unique sequence of stages and results in a different peak shape. Furthermore, as the experimental conditions vary within each case, the time for the analyte peak to stay in a certain stage changes and the characteristics of the analyte profile on the column may also change, which leads to different peak shapes. The number of possible peak shapes is large, and we can only discuss the most typical examples in each of the six cases in this paper. The parameters used in all simulation runs are listed in Table 1. (23) Galbusera, C.; Thachuk, M.; De Lorenzi, E.; Chen, D. D. Y. Anal. Chem. 2002, 74, 1903-1914. (24) Bier, M.; Palusinski, O. A.; Mosher, R. A.; Saville, D. A. Science 1983, 219, 1281-1287. (25) Saville, D. A.; Palusinski, O. A. AIChE J. 1986, 32, 207-214. (26) Palusinski, O. A.; Graham, A.; Mosher, R. A.; Bier, M.; Saville, D. A. AIChE J. 1986, 32, 215-223. (27) Hawley, J. F.; Smarr, L. L. Astrophys. J. Suppl. 1984, 55, 221-246. (28) Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical Recipes in C++; Cambridge University Press: New York, 2002.

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Table 1. Experimental Conditions for 18 Scenariosa mobilities (×10-4cm2/V‚s)

length (cm) scenario

[A]0 (mM)

[P]0 (mM)

of capillary

to detector

of injection plug

µA

µP

µC

voltage (kV)

K (M-1)

A-1 A-2 A-3 A-4 A-5 B-1 B-2 C-1 C-2 D-1 D-2 D-3 E-1 E-2 E-3 E-4 E-5 F-1

2.0 2.0 0.2 2.0 2.0 0.1 0.1 5.0 5.0 2.0 2.0 20 2.0 2.0 0.2 2.0 2.0 4.0

5.0 50 5.0 5.0 5.0 0.036 0.36 1.5 15 5.0 50 5.0 5.0 50 5.0 5.0 50 5.0

64.5 64.5 64.5 64.5 64.5 47 47 47 47 47 47 47 47 47 47 47 47 47

54.3 54.3 54.3 54.3 54.3 40 40 40 40 40 40 40 40 40 40 40 40 40

0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18

1.364 1.364 1.364 1.364 1.364 2.10 2.10 2.10 2.10 2.80 2.80 2.80 2.80 2.80 2.80 2.80 2.80 1.80

3.699 3.699 3.699 3.699 3.699 2.50 2.50 1.80 1.80 1.60 1.60 1.60 2.10 2.10 2.10 2.10 2.10 2.10

2.994 2.994 2.994 1.600 3.500 1.80 1.80 2.50 2.50 2.10 2.10 2.10 1.60 1.60 1.60 1.60 1.60 2.40

10 10 10 10 10 20 20 20 20 20 20 20 20 20 20 20 20 20

533 533 533 533 533 20000 20000 500 500 100 100 100 500 500 500 100 100 500

a [A] and [P] are the initial analyte and additive concentrations, respectively, µ , µ , and µ are the mobilities of free analyte, free additive, and 0 0 A P C the complex formed, respectively, and K is the binding constant. Note: The diffusion coefficients of all species are set to 1.0 × 10-6 cm2/s.

Simulation Program. A newly developed computer simulation system of CE is used in this study of ACE mechanisms.2 This program is highly efficient, due to the circular arrangement of cells representing a capillary column and the use of concentration thresholds to deactivate the majority of cells during the calculation. As a result, the electrophoretic migration process in a full-length (47 or 64.5 cm) capillary column can be simulated with this program in a few minutes with a laptop PC powered by an Intel Centrino 1.8-GHz CPU. The program is also accurate enough for our purpose because of the stability and convergence of the implemented finite difference schemes. It is important to set proper simulation parameters to obtain the best simulation results. For all 18 scenarios, the time increment (∆t) is set to 0.001 s, and the space increment (∆z) is set to 0.001 cm. More details of the simulation program can be found in a paper published earlier.2 The simulation program can display the concentration profiles at a user-defined interval, and the electrophoretic migration process can be observed in real time or faster. The snapshots at selected moments are exported, and the concentration profiles of the total analyte ([At] ) [A] + [C], solid line), the complex (dash line), and the free additive (dash-dot line) are shown in the figures. CASE A, µP > µC > µA. The analyte and additive concentration profiles generated by the simulation program for five scenarios of case A are presented to illustrate the mechanism of the electrophoretic migration processes. The conditions in scenario A-1 are obtained from ACE experiments using p-nitrophenol as the analyte and β-cyclodextrin as the additive, as listed in Table 1. The mechanism of this interaction in ACE has been discussed based on the experimental peak shapes.16 The change of the analyte concentration profile, which includes both free and complexed forms, is illustrated in Figure 1. During the first 14 s of analyte migration, a high-concentration region is first formed at the back of the analyte plug, because the complex 1834 Analytical Chemistry, Vol. 78, No. 6, March 15, 2006

migrates faster than the free analyte. After a certain concentration is reached, the higher concentration region expands along the analyte plug and results in a new profile with higher concentration and shorter plug length. Different stages of this process are illustrated in Figure 2. Stage 1: The injected analyte initially forms a rectangular plug on the inlet of the capillary. Because the free additive migrates fastest in this case, it enters the analyte band from behind. The analyte at the rear (left) edge interacts with the additive and forms the complex (Figure 2A). The length of the analyte plug was estimated to be 0.18 cm according to Beckman Coulter P/ACE MDQ injection parameters provided by the manufacturer. This length is used as the length of the injection plugs for all scenarios presented in this paper.

Figure 1. 3-D line plot demonstrating the change of peak shape (total concentration of analyte) over the first 14 s of the electrophoretic migration process of scenario A-1. Because of the idle time and the ramp time, only the profiles of the analyte plug after the first 2 s are plotted.

Figure 2. Simulated concentration profiles for scenario A-1. (A-E) Three concentration profiles at the location of the analyte plug are shown. (F) The additive trough is displayed together with the analyte plug.

Stage 2: The free additive keeps moving into the analyte plug, and a sweeping effect29 takes place (Figure 2B and C). The additive picks up the free analyte to form a faster-migrating complex. The portion of the analyte plug that has been swept by the additive has a higher total concentration of analyte. The length of the analyte plug is reduced from the initial length of 0.18 cm to 0.10 cm at the end of the sweeping process (Figure 2C), and the entire analyte plug except the edges now has the same [At] (4.30 mM or 215% of [A]0) and [P] (3.75 mM). Stage 3: The complex, which migrates faster than the free analyte, continues to move forward, resulting in an extension of the analyte plug at the front (Figure 2D). The front (right-hand side) of the analyte plug morphs into a slope, and the rear (left) edge remains as a steep cliff. After the analyte plug travels more than 50 cm in the capillary, the front slope extends to ∼0.85 cm in length, and the rear cliff is just 0.05 cm (mainly due to diffusion) (Figure 2E). At the same time, the peak concentration is reduced from 4.30 to 0.89 mM (46% of [A]0). The free additive concentration ([P]) in the front of the analyte plug continues to increase and becomes closer to the initial additive concentration in the BGE (29) Quirino, J. P.; Terabe, S. Anal. Chem. 1999, 71, 1638-1644.

([P]0 ) 5 mM). At the end, the concentration profile of the free additive results in a “V” shape (Figure 2E), in which the lowest value at the peak position is 4.75 mM. Because [P] is significantly different from [P]0 at the peak position, no true steady-state condition is reached at any point during the CE run. The negative additive peaks illustrated in Figure 2C-E should not be confused with the large additive trough shown on the right side of Figure 2F, which shows a greater region on the capillary column, at a time between Figure 2D and E. The additive trough is formed when the additive is used up by the analyte to form the complex in the process of sweeping (stage 2). On the other hand, the dip in additive concentration within the analyte plug is created due to the equilibrium between the analyte and the additive. The concentration profiles are not only determined by the order of the three mobilities but also determined by the concentrations of the analyte and the additive and by the differences between the three mobilities. The process and the trend of the profile change are similar for all rectangular-shaped injection plugs, even though the time required for the process to complete is different. Four more scenarios are simulated to demonstrate the effects of the latter two factors. Although the final peak shapes Analytical Chemistry, Vol. 78, No. 6, March 15, 2006

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Figure 3. Simulated concentration profiles for (A) scenario A-2, (B) scenario A-3, (C) scenario A-4, and (D) scenario A-5. All curves are shown with the same scale on the x-axis.

change, the three stages remain unchanged for the following scenarios. Scenarios A-2 and A-3 show the influence of different concentrations. The conditions for these two scenarios are also listed in Table 1. In scenario A-2, [P]0 is increased by 10 times from 5 to 50 mM, and all other conditions are identical to those in scenario A-1. The capacity factor also increases by ∼10 times. In other words, most of the analyte will be complexed with the additive not long after the CE process is started. After the sweeping effect (stage 2), there is not enough free analyte in the analyte plug to produce the front slope (stage 3). Therefore, the resulting peak (Figure 3A) is symmetrical. Because [P] is now nearly equal to [P]0, the steady-state condition is fulfilled. In scenario A-3, the only condition changed is [A]0, which is reduced from 2 to 0.2 mM. Because much less free analyte is present in the analyte band, [P] increases faster and eventually becomes closer to [P]0 than in scenario A-1: the lowest [P] in Figure 3B is 4.94 mM, compared with 4.75 mM in Figure 2E. Although the ratio of [P]0 to [A]0 is increased by 10 times in both scenarios A-2 and A-3, the resulting peak shapes are not the same. The [C]/[A] ratio in scenario A-3 (∼2.5) is only 1/10 of that in scenario A-2 (∼25), which allows the free analyte to produce a small front slope in scenario A-3. Scenarios A-4 and A-5 demonstrate the effects of the differences between the three mobilities. In scenario A-4, µC is closer to µA than µP. Because the difference between µC and µA is smaller, the front slope is not visible (Figure 3C). In scenario A-5, µC is closer to µP than µA. Because the difference between µC and µA is larger, a larger front slope shows up in Figure 3D. CASE B, µP > µA > µC. The mechanism of this case is different from case A. The conditions in scenario B-1, as listed in Table 1, were created to illustrate the characteristics of this case. Stage 1: After the analyte plug is injected, the additive migrates fastest and enters the analyte plug from behind to form the 1836 Analytical Chemistry, Vol. 78, No. 6, March 15, 2006

complex (Figure 4A). At the same time, a small amount of the additive mixes with the analyte plug at the front (right-hand side) despite µP > µA. One reason for such interaction to occur is diffusion of the additive from high-concentration region (BGE) to low-concentration region (the analyte plug), and the other reason is the artificial mixing caused by the plug boundaries located in the middle of a simulation cell. In reality, the later is analogous to the mixing caused by the injection process, either the parabolic profile caused by pressure injections or the mixing of boundaries during electrokinetic injections. Therefore, the complex is formed on both the front edge (peak 1 in Figure 4A) and the rear edge. This happens in all other cases, too. However, it is more significant in this scenario, because the difference between µA and µP is smaller (only 3 × 10-5 cm2 V-1 s-1), compared with 2.3 × 10-4 cm2 V-1 s-1 in scenario A-1. Stage 2: The free analyte migrates faster than the complex; therefore, as soon as the complex is formed, it lags behind, leading to a broadened analyte plug (Figure 4B). There are two plateaus in the migrating plug: one is the free analyte plateau ([A]t ) [A]0 ) 0.1 mM), and the other is the plateau of the mixture of the free analyte and the complex ([A]t ) 0.052 mM). The small bump on the free analyte plateau is caused by the dissociation of the complex in peak 1 in Figure 4A, because of the lower additive concentration at the location in the analyte plug. After a certain amount of time, the additive travels past the analyte plug (Figure 4C). The free analyte plateau no longer exists, and the length of the analyte plug is extended to 0.41 cm. Meanwhile, the extra analyte resulting from peak 1 in Figure 4A travels from the right-hand side of the analyte plug to the lefthand side due to slightly higher complex concentration caused by peak 1 than the rest of the analyte plug, which in turn leads to slightly smaller mobility at that location in the analyte plug. Stage 3: The faster free analyte continues to move ahead of the complex to produce a front slope (0.76 cm in length) at the end of the capillary (Figure 4D).

Figure 4. Simulated concentration profiles for scenario B-1. The peak labeled “1” is the complex formed at the front edge of the analyte plug before the plug started moving because of the contact between the analyte and additive at the boundary of the plug, as discussed in the text.

Although the final [At] profiles shown in Figures 2E (scenario A-1) and 4D (scenario B-1) have similar front slopes, there are two significant differences. First, in case A, the sweeping effect reduces the length of the analyte plug at first, and then the mobility difference between the free analyte and the complex extends the plug length during the rest of the migration process, while in case B, there is no sweeping effect, and the broadening of the analyte plug occurs from start to finish. Second, in case A, [P] inside the analyte plug is smaller than [P]0, while in case B, [P] inside the analyte plug is greater than [P]0, as shown by the Λ-shaped traces of [P] in Figure 4D. The highest [P] is 4.3 × 10-5 M or 119% of [P]0. In scenario B-2, the only change from scenario B-1 is the increase of [P]0 by 10 times to 3.6 × 10-4 M, as listed in Table 1. The figure and discussion can be found in the Supporting Information. CASE C, µC > µA > µP. The conditions of scenario C-1, as listed in Table 1, were created to demonstrate the mechanism in this situation. Stage 1: The analyte plug catches up with the additive in front and forms the complex. Meanwhile, a small amount of the additive mixes with the analyte plug at the rear edge to form a small peak 1 (Figure 5A) for the same reasons as mentioned in case B. Stage 2: Unlike in case A, there is no sweeping effect in this case. Because µC > µA, the complex on both edges migrates ahead of the free analyte (Figure 5B). The length of the analyte plug is extended at a relatively fast rate determined by the difference between µC and µA. At the same time, the bump resulting from peak 1 moves toward the outlet within the analyte plug. Stage 3: As the run goes on, more additive comes into the analyte plug, forming more complex. Figure 5C shows the concentration profiles just before the entire analyte plug becomes saturated with the free additive. The concentration of P is ∼1.83 mM, or 122% of [P]0, in the analyte plug. The large additive trough

is about to be separated from the analyte plug at this moment. The analyte plug continues to grow in length, and the final shape of the analyte plug (Figure 5D) consists of a large slope and a plateau. The plateau exists because the difference between µC and µA is not big enough to transform the entire plug into a slope. Scenario C-2, in which [P]0 is increased by 10 times, is discussed in the Supporting Information. CASE D, µA > µC > µP. This case is similar to case A, except that case A’s front slope becomes a rear slope. The conditions in scenario D-1, as listed in Table 1, were created to illustrate the characteristics of this case. Stage 1: After the analyte plug is injected, and the CE process is started, the plug catches up with the additive in front to form the complex (Figure 6A). Stage 2: The faster migrating analyte continues to overlap with more additives in front of the plug, and the plateau of the mixture of the complex and the free analyte keeps extending to the lefthand side of the analyte plug (Figure 6B). As the additive sweeps backward through the analyte plug, the length of the analyte plug decreases. Eventually, the entire analyte plug becomes the plateau of the mixture (Figure 6C). Stage 3: The free analyte keeps moving ahead, which leaves the complex at the left side of the analyte plug to form a rear slope (Figure 6D). The length of the analyte plug increases to 0.38 cm. The concentration of P in the analyte plug is always smaller than [P]0. The [At] profiles shown in Figures 5D (scenario C-1) and 6D (scenario D-1) have similar rear slopes. However, the profiles of [P] are different. In scenario D-1, [P] is lower than [P]0, having a V-shaped profile. In scenario C-1, [P] is higher than [P]0, having a Λ-shaped profile. Scenario D-2, in which [P]0 is increased by 10 times, is discussed in the Supporting Information. Analytical Chemistry, Vol. 78, No. 6, March 15, 2006

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Figure 5. Simulated concentration profiles for scenario C-1. The peak labeled “1” is the comlex formed at the rear edge of the analyte plug boundary, as discussed in the text.

Figure 6. Simulated concentration profiles for scenario D-1.

CASE E, µA > µP > µC. This is one of the two cases where the mobility of the free additive is between the mobilities of the free analyte and the complex. Relative to the movement of the free additive, the free analyte and the complex move to the opposite directions. Therefore, peak splitting is expected. The mechanism of this scenario is shown in Figure 7, and the conditions for scenario E-1 are listed in Table 1. Stage 1: The free analyte catches up with the additive in front to form the complex. At the same time, a small amount of the free additive enters from the rear end of the analyte plug to form 1838

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a small peak 1 (Figure 7A) for the same reasons as mentioned in case B. Stage 2: The free additive keeps moving into the right side of the analyte plug to form a plateau with a mixture of free and complexed analyte, and the free analyte on the left side keeps moving into the plateau of the mixture. As a result, the length of the analyte plug is reduced (Figure 7B). The analyte plug is further narrowed until the region containing only the free analyte morphs completely into the plateau of the mixture (Figure 7C). Peak 1 is left outside of the analyte plug and migrates at a constant speed

Figure 7. Simulated concentration profiles for scenario E-1. The peak labeled “1” is the complex formed at the rear edge of the analyte plug boundary, as discussed in the text.

determined by the capacity factor (k′ ≈ K[P]0). Due to diffusion, peak 1 gets smaller over time and is hardly visible in Figure 7D-F. Stage 3: The analyte plug starts to extend on both sides, and there is a large gap containing no additive to the left of the analyte plug. The slowest migrating complex falls out of the analyte plug into the gap. Because there is no additive in the gap, the complex dissociates quickly to free analyte and additive again. The newly dissociated free analyte migrates back into the analyte band, and the newly dissociated free additive is left behind. Therefore, in Figure 7C-E, it can be observed that the edge of the additive

zone associated with the analyte plug is slightly behind the left edge of the analyte plug. When more complex falls out of the analyte plug, it does not move into the gap directly, but stays in the additive zone to establish a new equilibrium. Therefore, the analyte plug is rapidly extended to the left (Figure 7D and E). At the right edge of the analyte plug, the free analyte is moving out. However, [P] to the right of the analyte plug is equal to [P]0, which is much higher than [P] at the right edge of the analyte plug. Thus, most of the free analyte molecules moving out of the plug will be complexed and fall back into the analyte plug. Analytical Chemistry, Vol. 78, No. 6, March 15, 2006

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Stage 4: The additive and the complex continue to travel backward and fill up the gap. While the analyte plug continues to extend to both sides, the height of the concentration profile drops as the plug migrate further (Figure 7E), and eventually, the gap completely disappears. The complex and the additive keep falling behind to create a region containing higher [P] than [P]0 on the left side of the analyte plug (Figure 7F). Although the gap with no additive has disappeared, there is still a big additive trough. After traveling 40 cm, the analyte plug is illustrated in Figure 7G, and significant peak splitting is observed. There are two peaks: one on the left with a steep left cliff and a long front slope, the other on the right with a steep right cliff and a smaller rear slope. The profile of [P] is shown in Figure 7H. Peak 1 migrates a little faster than the left edge of the analyte plug, because of the smaller concentration of the additive at the location of peak 1 than the left edge of the plug. Peak 1 can catch up and merge with the analyte plug. Therefore, peak 1 no longer exists in Figure 7G. By carefully examining Figure 7G and H, we find the similarities between this case and the others. The shape of the left peak is similar to that in case B: a front slope and a positive additive peak. The shape of the right peak is similar to that in case D: a rear slope and a negative additive trough. Therefore, case E can be considered as case B + case D. All three cases share one common property: µA > µC. This peak-splitting phenomenon can be further studied by changing the concentration of the analyte or the additive. Scenarios E-2, E-3, and E-4 are discussed in the Supporting Information. Peak splitting creates a difficult situation for the determination of migration time, which is the most important parameter used to determine binding constants in regression methods. Both peaks are in the constant expansion mode. In most cases, the right peak coexists with the additive trough, which makes the assumption of [P] ≈ [P]0 invalid. If a much higher additive concentration is used, as in scenario E-2, the right peak disappears completely. Therefore, the right peak can never be used for the regression methods. On the other hand, the left peak does not exist in the beginning of the CE process, which normally disqualifies it from representing the entire process. However, if the additive concentration is so high that it can fill the analyte plug in a short period of time (scenario E-2), the left peak, which is the only peak now, can give a relatively accurate data point for the regression methods. CASE F, µC > µP > µA. Case A, C, and F share one common property: µC > µA. Following our discussion of the similarities between the cases, we can expect case F to show the characteristics of case A + case C. Scenario F-1 is simulated with the conditions listed in Table 1, and the profiles agree with our prediction. More detailed discussion can be found in the Supporting Information. Regression Methods. Through the discussion of 6 cases and 18 scenarios, we can conclude that one relatively accurate data point for the regression methods can be generated when high [P]0 and low [A]0 are used for the first four cases. To obtain a binding constant with regression method using binding isotherms, one needs to have 5-10 data points depending on the range of

the isotherm covered by the experiment.20,30 High enough [P]0 ensures a high capacity factor, and low enough [A]0 leads to a short time required for the free additive to go through the analyte plug. For the last two cases, because of their complicated mechanisms, even though a high ratio of [P]0 to [A]0 is achieved, the error in the regression methods can still be fairly large if a wrong peak is chosen. As shown in Figure 7, the relative peak height resulting from the same analyte could change, and one of them could even disappear during the migration process. To know whether a particular set of conditions are valid, one can run the simulation to discover the analyte and additive concentration profiles. To use the regression methods, an appropriate range of additive concentrations has to be used. As we have proposed in another paper, a better way to determine binding constants is to use the enumeration algorithm with computer simulation of ACE, in which [P] in the analyte plug is not assumed constant.21 Therefore, it can generate accurate results under any given conditions as long as the simulation model is accurate.

(30) Bowser, M. T.; Chen, D. D. Y. J. Phys. Chem. A 1999, 103, 197-202.

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CONCLUSIONS The traditional way of studying the mechanism of a chemical/ physical process is to perform carefully controlled experiments and then to develop a theory to support as many experimental observations as possible. In this paper, we demonstrate ACE mechanisms based on the well-established mass transfer equation. This equation has not been extensively used by chemists because of the difficulties associated with finding the analytical solutions with given experimental conditions. Combining the power of modern computers and the proper implementation of an algorithm based on the finite difference schemes, we are able to simulate many practical scenarios on the CE to illustrate detailed analyte migration processes. The results of these simulations provide guidance for scientists who are interested in using the migration times obtained from CE experiments to extract physicochemical parameters, such as binding or dissociation constants. It is re-enforced with these results that, to obtain reliable constants based on the measured migration time, the additive concentration has to be much higher than the analyte concentration, if regression methods are to be used for data processing. If this condition cannot be satisfied for practical reasons, enumeration methods based on properly implemented simulation methods should be considered.21 The results shown in the paper should also provide insights into the phenomena observed in separation systems with much shorter columns, such as microfluidic devices. ACKNOWLEDGMENT This work is supported by the Natural Sciences and Engineering Research Council of Canada and the Department of Chemistry, University of British Columbia, Vancouver, BC, Canada. SUPPORTING INFORMATION AVAILABLE Additional discussions on scenarios B-2, C-2, D-2, D-3, E-2, E-3, E-4, and F-1, including five figures. This material is available free of charge via the Internet at http://pubs.acs.org. Received for review October 11, 2005. Accepted January 19, 2006.