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Principles and applications of affinity capillary electrophoresis based on mass transfer equation Fang, Ning 2006

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PRINCIPLES  AND APPLICATIONS  ELECTROPHORESIS  BASED  OF A F F I N I T Y  ON MASS  TRANSFER  CAPILLARY EQUATION  by  NING FANG  M.Sc,  B . S c , Xiamen U n i v e r s i t y , 1998 The U n i v e r s i t y of B r i t i s h Columbia, 2002  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF  PHILOSOPHY  in THE FACULTY OF GRADUATE STUDIES (CHEMISTRY)  THE UNIVERSITY OF BRITISH COLUMBIA J u l y 2006 ® Ning Fang, 2006  Abstract Unified separation science not only describes the separation process o f each technique, but is also instrumental to the further development o f separation science. Previous developments o f the theory have focused on macroscopic (often average) properties of separation systems: the average analyte migration rate, the steady state, resolution, sensitivity, precision, etc. O n the other hand, microscopic/instantaneous behaviors are essential to understand complex phenomena, such as dynamic complexation, sweeping/stacking o f sample analytes, and buffer depletion. Computer simulation is one o f the best ways to visualize the instantaneous behaviors of chemical/physical systems. The simulation model o f dynamic complexation capillary electrophoresis ( S i m D C C E ) is based on the differential mass transfer equation, the governing principle o f analyte migration in all separation techniques. S i m D C C E is highly efficient, and is the first to demonstrate the affinity interactions in capillary electrophoresis ( C E ) in real time or faster. S i m D C C E is one big step towards the ultimate goal: the unified computer simulation o f all separation techniques. Using S i m D C C E , a thorough study o f affinity capillary electrophoresis ( A C E ) mechanisms was carried out. The regression methods for determining binding constants from A C E experiments require the assumption o f instant establishment o f the steady state condition which was examined in a variety of scenarios in the six cases defined by the order o f the mobilities o f the analyte, additive, and complex. The enumeration algorithm built upon computer simulation was developed to provide a fast and accurate alternative for determining binding constants when the assumption is invalid. The enumeration approach is equally  applicable to binding studies using techniques such as NMR, chromatography, and optical methods. The second part of my research is the application of C E in other research fields, including biochemistry, anesthesia, and forensic chemistry. A systematic optimization of exhaustive electrokinetic injection and sweeping processes was carried out to improve the reproducibility and sensitivity for the detection of amphetamine and its derivatives.  Table of C o n t e n t s Abstract Table of Contents List of Tables List of Figures Abbreviations Publications Acknowledgments  ii iv vii viii x xi xii  Chapter 1 Introduction to Dynamic Complexation Capillary Electrophoresis and Unified Separation Science 1 §1.1 Capillary Electrophoresis 2 §1.2 Fundamental Theory of Capillary Electrophoresis 4 §1.2.1 Electrophoretic Mobility, ju 4 §1.2.2 Electroosmotic Mobility, ju 5 §1.2.3 Joule Heating 10 § 1.3 Dynamic Complexation in Capillary Electrophoresis 11 §1.3.1 Affinity Capillary Electrophoresis (ACE) 11 §1.3.2 Micellar Electro-Kinetic Capillary Chromatography (MEKCC) 14 §1.3.3 Dynamic Complexation Capillary Electrophoresis (DCCE) .15 §1.4 Limit of Detection 17 §1.5 Unified Separation Science 18 §1.5.1 Di splacement and Transport 18 §1.5.2 Redefined Separation Factors 19 §1.6 Research Objectives 22 § 1.6.1 Simulation of Instantaneous Properties 22 §1.6.2 Equilibrium and Nonequilibrium 23 §1.6.3 Enumeration Method to Overcome the Breakdown of Steady State Conditions ..24 §1.6.4 Achieving Better Sensitivity and Reproducibility 24 §1.7 References 26 ep  eo  Chapter 2 Determination of Shapes and Maximums of Analyte Peaks Based on Solute Mobilities in Capillary Electrophoresis 27 §2.1 Introduction 28 §2.2 Experimental Section 30 §2.3 Results and Discussion 32 §2.3.1 Peak Shapes and Relative Mobilities 32 §2.3.2 Migration of the Peak Maximums 40 §2.4 Conclusions 49 §2.5 References 50 Chapter 3 High-Efficiency Simulation of Dynamic Complexation Capillary Electrophoresis Based on Mass Transfer Equation §3.1 Introduction  51 52  iv  §3.2 Experimental Section §3.3 Algorithm and Implementation §3.3.1 Electrophoretic Migration Process §3.3.2 Association and Dissociation Processes §3.3.3 Implementation of Circular Cell Arrangement §3.4 Simulation of Equilibrium C E §3.5 Simulation of Nonequilibrium C E §3.6 Simulation of D C C E §3.6.1 Initial Experimental Conditions §3.6.2 User-Defined Concentration Thresholds §3.6.3 Outputs of SimDCCE §3.7 Conclusions §3.8 References  57 59 59 65 66 70 76 79 80 80 81 83 84  Chapter 4 Behavior of Interacting Species in Capillary Electrophoresis Described by Mass Transfer Equation 85 §4.1 Introduction 86 §4.2 Mechanisms and Discussion 89 §4.2.1 Simulation Conditions 89 §4.2.2 C A S E A: ju >/u > // ..92 p  c  A  §4.2.3 C A S E B: M?> M > M A  9  §4.2.4 C A S E C: ji > / / > ju c  A  §4.2.5 C A S E D:  JU >JU >  §4.2.6 C A S E E :  M A > M  X  r  101  ju  104  p  c  P  8  C  >  §4.2.7 C A S E F: M >Mv>  MC  •  1  0  6  1  1  1  c  §4.2.8 Regression Methods §4.3 Conclusions §4.4 References  114 115 116  Chapter 5 Enumeration Algorithm for Determination of Binding Constants in Capillary Electrophoresis 117 §5.1 Introduction 118 §5.2 Experimental Section 120 §5.2.1 A C E Experiments 120 §5.2.2 Mobility of the Analyte 120 §5.3 Enumeration Method 122 §5.3.1 Overview 122 §5.3.2 Step 1: 3-D Surfaces 124 §5.3.3 Step 2: 2-D Curves 127 §5.3.4 Step 3: Intersections of the 2-D Curves 127 §5.3.5 Complete Procedure Using SimDCCE 133 §5.3.6 Enumeration Method vs Regression Methods 135 §5.4 Conclusions 137 §5.5 References 138  v  Chapter 6 Systematic Optimization of Exhaustive Electrokinetic Injection Combined with Micellar Sweeping in Capillary Electrophoresis §6.1 Introduction §6.2 Experimental Section §6.3 Results and Discussion §6.4 Conclusions §6.5 References  139 140 144 147 161 162  Concluding Remarks  163  Appendix A SimDCCE Guide §A.l Introduction to User Interface §A.2 Tutorial for Simulation with the Multi-Cell Model §A.2.1 Step 1: Set Experimental Conditions §A.2.2 Step 2: Set Simulation Parameters for the Calculation Module §A.2.3 Step 3: Set Simulation Parameters for Outputs §A.2.4 Step 4: Start the Simulation Run §A.3 Load and Plot Data Files in SimDCCE §A.3.1 Browse Data Files §A.3.2 Curve Settings §A.3.3 Plot Curves §A.3.4 Zoom In and Out  166 167 168 168 170 172 176 180 180 181 183 185  Appendix B Business Plan  186  vi  List of Tables Table 2.1 Comparison of experimental and simulated migration times for case A using the interaction between /?-nitrophenol and B - C D Table 2.2 Comparison of experimental and simulated migration times for case B using the interaction between flurbiprofen and T T R Table 4.1 Experimental conditions for 18 scenarios Table 5.1 Solutions, ^(M" ) and 1  J U ^ A / M ^ C  >  shown in Figure 5.3 Table 6.1 Composition of the buffers Table A. 1 List of the experimental conditions  o  f e  v  e  r  v  t  w  0  1:1 46 2:1 48 91  equations representing the curves 131 145 168  vii  List of Figures Figure 1.1 Schematic representation of a C E setup 3 Figure 1.2 Forces acting on charged particles in a solution 4 Figure 1.3 Representation of the silanol groups inside a capillary 6 Figure 1.4 Electrical double layer formed at the inner capillary wall in contact with an electrolyte solution 7 Figure 1.5 Flow profiles of an electrically-driven system and a pressure-driven system 9 Figure 1.6 Routes taken to develop the simulation models 23 Figure 2.1 Migration process of case A with the 1:1 interaction between the analyte (pnitrophenol) and the additive (/3-CD) 33 Figure 2.2 /?-Nitrophenol peaks in the presence of /3-CD in the B G E 35 Figure 2.3 The migration process of case B with a 2:1 interaction between the analyte (flurbiprofen) and the additive (TTR) 37 Figure 2.4 Flurbiprofen peak in the presence of TTR in the B G E 39 Figure 3.1 Circular arrangement of cells in the memory of the PC 68 Figure 3.2 Simulated peaks with the implementation of eq 3.19 71 Figure 3.3 Simulated peaks with the implementation of eq 3.20 72 Figure 3.4 Experimental and simulated peaks 73 Figure 3.5 Simulated peaks for nonequilibrium C E experiments 77 Figure 3.6 User interface of SimDCCE 79 Figure 4.1 Three-dimensional line plot demonstrates the change of peak shape (total concentration of analyte) over the first 14 seconds of the electrophoretic migration process of Scenario A - l 93 Figure 4.2 Simulated concentration profiles for Scenario A - l . (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 95 Figure 4.3 Simulated concentration profiles for (A) Scenario A-2, (B) Scenario A-3, (C) Scenario A-4, and (D) Scenario A-5 97 Figure 4.4 Simulated concentration profiles for Scenario B - l 99 Figure 4.5 Simulated concentration profiles for Scenario B-2 101 Figure 4.6 Simulated concentration profiles for Scenario C - l 102 Figure 4.7 Simulated concentration profiles for Scenario C-2 103 Figure 4.8 Simulated concentration profiles for Scenario D - l 105 Figure 4.9 Simulated concentration profiles for (A) Scenario D-2 and (B) Scenario D-3 106 Figure 4.10 Simulated concentration profiles for Scenario E - l 107 Figure 4.11 Simulated concentration profiles for (A) Scenario E-2, (B) Scenario E-3, (C) Scenario E-4, and (D) Scenario E-5 110 Figure 4.12 Simulated concentration profiles for Scenario F - l 112 Figure 5.1 Flowchart of the procedure for determining binding constants 124 Figure 5.2 Three-dimensional surface which is cut through by a plane to produce a 2-D curve which is then projected onto the bottom plane 126 Figure 5.3 Two-dimensional graph generated from SimDCCE  with ju //u epA  epC  as the x-axis 128  viii  Figure 5.4 Intersections of two sets of curves. (A) 1 and 15 mM, (B) 2 and 12.5 mM, (C) 3 and 10 mM, and (D) 4 and 5 mM 130 Figure 5.5 Binding isotherm 131 Figure 5.6 Intersection of three sets of curves: 1, 4, and 15 m M 132 Figure 5.7 Screenshot of SimDCCE for Steps 1 and 2 133 Figure 5.8 Screenshot of SimDCCE for Steps 3 134 Figure 6.1 Molecular structures of 3 amine drugs studied 144 Figure 6.2 Current (proportional to sample conductivity) vs. electrokinetic injection efficiency 150 Figure 6.3 Current curves are used as signals for reproducible runs 152 Figure 6.4 Effect of HCB fraction 155 Figure 6.5 Electropherograms of the mixture of the three amine drugs 157 Figure 6.6 Calibration curves for the three amine drugs 160 Figure A . l User interface of SimDCCE is divided into three regions: 1. the settings panel defines the experimental conditions and simulation parameters; 2. the simulation control panel; 3. the display panel shows the animation of the simulated concentration profiles 167 Figure A.2 Screenshot of the settings panel after all the experimental conditions were set in 169 Figure A.3 Screenshot of the settings panel after the simulation parameters used in the calculation module were set 172 Figure A.4 Three-dimensional mesh representation of the peak shape changes during the first 14 minutes of a simulation 173 Figure A.5 Contour plot representation of the peak shape changes during the first 14 minutes of a simulation 174 Figure A.6 Parameters for generating electropherograms and automatic snapshots 175 Figure A.7 Screenshot of the control panel and the display panel 176 Figure A.8 Screenshot after the simulation is started 178 Figure A.9 Screenshot for the window asking for 3-D mesh settings 179 Figure A. 10 Screenshot showing the "Load and Plot Data File" workplace 180 Figure A. 11 Screenshot showing the curve settings 183 Figure A. 12 Two curves are plotted at the same time using different colors. The origin of the scale and range is determined by the drop box above the curves 184 Figure A. 13 Snapshot of the Zoom function 185  ix  Abbreviations A C E : Affinity Capillary Electrophoresis ASEI: Anion-Selective Exhaustive Injection BGE: Background Electrolyte CD: Cyclodextrin CE: Capillary Electrophoresis CEC: Capillary Electrophoresis Chromatography CE-FA: Capillary Electrophoresis - Frontal Analysis C M C : Critical Micelle Concentration CSEI: Cation-Selective Exhaustive Injection DCCE: Dynamic Complexation Capillary Electrophoresis EOF: ElectroOsmotic Flow FASS: Field-Amplified Sample Stacking GC: Gas Chromatography HCB: High-Conductivity Buffer HD: Hummel-Dreyer HPLC: High Performance Liquid Chromatography LCB: Low-Conductivity Buffer LIF: Laser Induced Fluorescence LOD: Limit Of Detection LOQ: Limit Of Quantitation MCB: Medium-Conductivity Buffer M E K C : Micellar ElectroKinetic Chromatography M E K C C : Micellar Electro-Kinetic Capillary Chromatography N E C E E M : Non-Equilibrium Capillary Electrophoresis of Equilibrium Mixtures RSD: Relative Standard Deviation SDS: Sodium Dodecyl Sulfate SimDCCE: Simulation model of Dynamic Complexation Capillary Electrophoresis V A C E : Vacancy Affinity Capillary Electrophoresis VP: Vacancy Peak  Publications Portions of the following chapters have been previously published, accepted for publication, or submitted for publication elsewhere. Chapter 2: Fang, N . ; Ting, E.; Chen, D.D.Y. Determination of shapes and maximums of analyte peaks based on solute mobilities in capillary electrophoresis. Analytical Chemistry, 2004, 76(6), 1708-1714. Chapter 3: Fang, N.; Chen, D.D.Y. General approach to high-efficiency simulation of affinity capillary electrophoresis. Analytical Chemistry, 2005, 77(3), 840-847. Chapter 4: Fang, N.; Chen, D.D.Y. Behavior of interacting species in capillary electrophoresis described by mass transfer equation. Analytical Chemistry, 2006, 78(6), 1832-1840. Chapter 5: Fang, N.; Chen, D.D.Y. Enumeration algorithm for determination of binding constants in capillary electrophoresis. Analytical Chemistry, 2005, 77(8), 2415-2420. Chapter 6: Fang, N.; Meng, P.J.; Hong, Z.; Chen, D.D.Y. Systematic optimization of exhaustive electrokinetic injection combined with micellar sweeping in capillary electrophoresis, The Analyst, 2006, submitted. Meng, P.J.; Fang, N.; Wang, M . ; Liu, H.; Chen, D.D.Y. Analysis of amphetamine, methamphetamine and methylenedioxymethamphetamine by micellar electrokinetic chromatography using cationselective exhaustive injection. Electrophoresis, 2006, in press. Other publications: Cressman S.; Fang, N . ; Chen, D.D.Y.; Cullis P. R. A whole-cell binding assay for testing the targeting potential of cyclic peptide ligands, Proceedings of the American Peptide Society, 2005.  xi  Acknowledgments I would like to thank my supervisor, Dr. David Chen, for sponsoring this work and for his advice and encouragement throughout my research. I am grateful to Dr. Lionel G. Harrison, Dr. Mark Thachuk, and Dr. Y . Alexander Wang for helpful discussions. I want to thank my former and current lab mates and coworkers, Wuyi, Rong, Irina, Brad, Cindy, Dave, Chiara, Gwen, Junji, Eon, Kevin, Pingjia, Alison, Hong, Kingsley, Sonya, Sharon, Koen, Jane, and Mark for sharing their knowledge and experience about C E . I also appreciate the generosity of Beckman Coulter Inc., for supplying a second C E instrument from 2000 to 2005. Finally, I would like to thank my wife for her invaluable support and tremendous belief in me, and to thank my parents for all of their encouragement throughout the years, for teaching me the meaning of hard work and endurance and for showing me how to appreciate and enjoy life through their example!  xii  Chapter 1  Introduction to Dynamic Complexation Capillary Electrophoresis and Unified Separation Science  Chapter 1 Introduction to Dynamic Complexation Capillary Electrophoresis and Unified Separation Science §1.1 Capillary Electrophoresis Capillary electrophoresis (CE), a separation technique based on differential migration of charged substances in a solution under the influence of an electric field in a narrow bore silica capillary, has become one of the most useful separation techniques since 1981, when Jorgenson and Lukacs first performed electrophoresis in glass capillaries [1]. High speed, high mass sensitivity, high resolution, precision, and the ease of automation are among the advantages usually observed for C E over the conventional formats of gel and paper electrophoresis. Separations are generally considered a slow step in any analytical methodology. High performance liquid chromatography (HPLC) and electrophoresis, the mainstays for separation of nonvolatile compounds in modern laboratories, usually take anywhere from tens of minutes to hours to complete depending upon the complexity of the sample. This time requirement has limited the types of problems that can be solved by these separation techniques. Performing electrophoresis separations in capillaries allows much higher electric fields to be applied than in conventional gel electrophoresis; this, in turn, allows separations to be performed much more rapidly. With typical capillary dimensions ranging from 20 to 100 urn in inner diameter and from 20 to 100 cm in length, the volumes required to fill the capillary are in the microliter (ul) range. C E is ideal for studying small samples, such as D N A / R N A and proteins, when only extremely small amounts of analyte are available.  2  Commercial C E instruments are highly automated. When carrying out a regular experiment, one usually spends minutes to a couple of hours setting up the machine and program, and then the instrument could run automatically for hours or days. Whether C E is carried out using a commercial or a "homemade" instrument, the basic instrumental design is the same. Figure 1.1 shows a schematic representation of a CE instrument. The basic components include a high voltage power supply, a fused silica capillary, buffer reservoirs and a detector. The electric field applied across the capillary is generated by a high voltage power supply that usually provides voltages up to 30 kV. Detection of the separated analytes is often achieved on-column using various types of detectors, such as ultraviolet absorbance (UV), photo diode array (PDA), laser induced florescence (LIF) and conductivity.  Computer Data Collector  Capillary  -O-C  Detector  * 0 »  High Voltage Power (5-30kV)  Sample Vial  Inlet Buffer Vial  Lamp  v j Outlet Buffer Vial  Figure 1.1. Schematic representation of a C E setup.  3  §1.2 Fundamental Theory of Capillary Electrophoresis §1.2.1 Electrophoretic Mobility, /4  P  There are two forces acting on a charged particle in a homogenous electric field. The electrostatic force (F ) accelerates the charged particle, and the frictional force (Fy) counteracts e  the electrostatic force and decelerates the charged particle (Figure 1.2). F = Q-E = Z-e -E  (1.1)  F =-f-v  (1.2)  e  0  f  in which Q is the net effective charge of the ion (Coulombs), E is the electric field strength (V • cm"'), Z is the number of charges on the particle, eo is the elementary charge (1.602xlCT Coulombs),/is the frictional coefficient (g-s"') and vis the ion velocity 19  (cm-s~'). For a hard sphere falling through a liquid in a velocity regime where viscosity is the dominant effect and turbulence can be neglected, the frictional force is given by: / -6-n-rj-r  (1.3)  in which r is the radius of the particle (cm) and t] is the viscosity of the medium (Pa • s). During electrophoresis, a balance between these two counteractive forces is obtained, at which point the forces are equal. Therefore,  + + + + +  ©  F/  Fe  Figure 1.2 Forces acting on charged particles in a solution.  4  F =-F e  f  (1.4)  Q-E - 6-7r-r]-r-v  The electrophoretic mobility is a characteristic of each individual ion and is determined by the velocity per unit electric field, that is,  M =j  (1.5)  ep  where / 4 is the electrophoretic mobility (cm •s 2  P  -1  -V"').  Substituting eq 1.4 into eq 1.5 gives Q  /4p=7-^  (1-6)  This equation implies that the electrophoretic mobility of each species in a solution depends only on its charge and size.  §1.2.2 Electroosmotic Mobility, /4  0  In electrophoresis, electroosmosis is a phenomenon in which the bulk liquid migrates relative to the stationary solid phase because of the imposed electric field. Similarly, the electroosmotic flow (EOF) in C E , the bulk flow of separation buffer through the capillary, is generated due to the nature of the capillary wall [2]. The fused silica capillary used in C E is made from bridged silicon dioxide compounds (Si-O-Si). At the surface of a fused silica capillary, the silanol groups start to deprotonate when the pH of a solution in contact with the wall is higher than 2. More silanol groups deprotonate as the pH increases up to pH 9. Therefore, negatively charged Si-O" groups are formed on the wall of the capillary (Figure 1.3).  5  OH  OH  O"  O " inner surface of a capillary  Si  o I o o oI o oI o o o o  o  o  o  capillary wall  o outer surface of a capillary  Figure 1.3  Representation of the silanol groups inside a capillary.  Positively charged particles in the solution are attracted to this layer of negative charges on the capillary wall, thus forming an electrical double layer. A portion o f the cationic counterions w i l l adsorb to the wall, giving rise to an immobilized compact layer called the Stern layer. The charge and potential distribution in the Stern layer are mainly determined by the ionic strength o f the solution and the geometric restrictions o f ions. The Stern layer only neutralizes part of the excessive charge at the capillary wall. Due to the residual negative charges, there are additional cations present in the adjacent area, forming a diffuse layer. The electrical double layer at the capillary wall is illustrated i n Figure 1.4. The zeta potential is the potential drop across the diffuse layer. When an electric field is applied across the capillary, the cations in the diffuse layer begin to migrate towards the cathode. These cations drag the solvent molecules with them. This movement of solution is called electroosmotic flow (EOF).  6  Excess Negative Charge  Excess Positive Charge  Even Charge Distribution  Distance from the wall F i g u r e 1.4. Electrical double layer formed at the inner capillary wall in contact with electrolyte solution.  The relationship between the velocity of the EOF (y ) and the zeta potential (Q at the eo  capillary/solution interface is given by [2]  v =- -E eo  Mto  =- ^ - E =- ^ ^ - E 1 V  (1.7)  in which eis the static permittivity of the solution, e is the relative static permittivity, and eo x  is the permittivity of vacuum. The negative sign means that when Q is negative, the electroosmotic flow is towards the cathode; this is defined as a positive flow. There are several factors that can affect the EOF: 1. At higher pH, a greater number of negative charges are present at the capillary wall, resulting in a higher C, potential, and thus higher velocity of the EOF. 2. At higher ionic strength, more cations are present in the Stern layer, resulting in a smaller t, potential, and thus lower velocity of EOF. 3. At higher temperature, the Stern layer becomes less stable, and the viscosity is reduced. This results in a higher C, potential and less resistance, and thus higher velocity of EOF. Therefore, by various chemical or physical methods, one can control the EOF in order to achieve better separation. Since the electroosmotic mobility is often larger than the electrophoretic mobility of most analytes in an aqueous system, nearly all species, regardless of charge, move in the same direction. Smaller and more positively charged ions migrate faster, larger and more negatively charged ions migrate more slowly, and small and negative ions migrate the slowest. As the EOF originates from the wall, the driving force of the EOF is uniformly distributed along the entire length of the capillary. As a result, there is no pressure drop, and  8  the flow velocity is uniform across the entire tubing diameter except very close to the wall where the velocity approaches zero (Figure 1.5A). In contrast, when employing a pressuredriven system such as a liquid chromatograph, the frictional forces at the liquid-solid interfaces, such as the packing and the walls of the tubing, result in substantial pressure drops. In an open tube, the frictional forces are severe enough even at low flow rates to result in laminar or parabolic flow profiles (Figure 1.5B). This velocity gradient results in substantial band-broadening.  B  Figure 1.5. Flow profiles of an electrically-driven system (A) and a pressure-driven system (B).  9  §1.2.3 Joule Heating The application of high field strength leads to the production of heat in CE. This heat then causes two major problems: temperature gradients across the capillary and temperature changes with time due to ineffective heat dissipation. The rate of heat generation can be approximated as dH  kV  dt  L  1  1  = tcE  2  (1.8)  in which K is the conductivity, V is the voltage applied on the capillary, and L is the capillary length. The amount of heat generated is proportional to the square of the field strength. To reduce the heat generated, one can increase the capillary length, decrease the voltage, or use low conductivity buffers. Temperature gradients across the capillary are a consequence of heat dissipation. Since heat is dissipated by diffusion, it follows that the temperature at the center of the capillary should be greater than at the capillary walls, which in turn leads to lower viscosity and higher electrophoretic mobilities at the center of the capillary. Therefore, a flow profile that resembles hydrodynamic flow (Figure 1.5B) is obtained, and band-broadening occurs. Narrow-diameter capillaries help heat dissipation, but effective cooling systems are required to ensure heat removal. Liquid cooling is the most effective means of removing heat and controlling capillary temperature.  10  §1.3 Dynamic Complexation in Capillary Electrophoresis §1.3.1 Affinity Capillary Electrophoresis (ACE) Affinity capillary electrophoresis (ACE) is a powerful tool for studying protein-ligand interactions. A fundamental and crucial concept for A C E is that the additive is considered a solute, not a separate phase, in the same solution as the analyte. With this concept in mind, we can derive the equations for A C E . When 1:1 stoichiometry is present, the equilibrium is represented by A + P ===== C  (1.9)  in which A is the analyte, P is the additive (such as proteins, surfactants, etc.), and C is the analyte-additive complex. The equilibrium constant (K) for eq 1.9 can be written as K =- ^ [A][P]  (1.10)  in which [A], [P], and [C] are the concentrations of free analyte, free additive and complex, respectively. The capacity factor (k') can be described as the ratio of the amount of complex to the amount of free analyte, which is similar to the definition of the capacity factor in chromatography (defined as the ratio of the amount of analyte in the stationary phase to the amount of analyte in the mobile phase), namely k' = ^ = «A  [  £ \ = K[V]  (1.11)  [ ] A  in which n is the amount of complex and n is the amount of free analyte. c  A  If the fraction of free analyte (/) is defined as [A]/[A] ([A] is the total analyte t  t  concentration, [A] = [A] + [C]), this gives t  11  (  The electrophoretic mobility of the free analyte is represented by  //  e  p  ,A-  L  1  2  )  The  electrophoretic mobility of complex is represented by /%,,<:• The net electrophoretic mobility of the analyte, represented by ju* , can be written as p  K  =  Mp A+( -/K ,c 1  >  P  1 i  k'  ^  +  A  +  ! 7 I ^  ,  c  (  L  1  3  )  ,  1  \ + K[?]  MepA  \+  +  K[P]  MepC  Equation 1.13 actually represents the case of an ideal state, in which additives do not affect the physicochemical properties (such as viscosity) of the buffer. However, in reality, as shown in eq 1.6, the electrophoretic mobility is inversely proportional to the viscosity of the solution, and the viscosity will change when the concentration of the additive is changed to give different k' values (eq 1.11). So if the changes in viscosity cannot be ignored, a correction factor (v) has to be introduced to eq 1.13  V  ^  =  IT^^  A  +  I ^ P 1 ^  C  (  }  The correction factor (v) is defined as rjf rj , where rj is the viscosity of the buffer at the 0  concentration of additive in question and rf is the viscosity of the buffer when [P] is equal to or approaching zero. The use of the correction factor converts the apparent electrophoretic mobility of the analyte to an ideal state. From the migration time of the analyte and the EOF marker, one can obtain the effective electrophoretic mobility (ju* ) of the analyte by using the equations  12  K=V -M  (1-15)  A  eo  A _ Aietector ' Aotal  Q | g\  .. _ Aietector ' Aotal  / I 1 -7\  in which  is the apparent mobility, detector *  end to the detector (cm), Z of the analyte (s), t  eo  tota]  sm  e  length of the capillary from the injection  is the total length of the capillary (cm), t  A  is the migration time  is the migration time of the EOF marker (s) and V is the voltage applied  through the whole capillary (volt). Equation 1.14 can be rearranged to give  «-^)= ^L<^-,,.)  0..8)  T  Equation 1.18 can then be linearized using one of the following three forms [3, 4]:  1  1  ™  =  Mc - M ,A P  ep  .A M ~M  V  ep  epA  t i  +  V - ^ep,A  p  A  (1.20)  (/"ep.C ~ ^ep.A ) '  _  = -K-(v^ -^ )  (1.19)  1  !  P  /  V  +  1  +  .(^ -  K  c  K  )  MtpA  (1.21)  [P] These three equations can be referred to as double-reciprocal (eq 1.19), ^-reciprocal (eq 1.20), and x-reciprocal (eq 1.21) regression equations, respectively. Since all three equations are derived from eq 1.18, one would expect them to generate identical results. However, this is rarely the case due to different error propagation associated  13  with the variables during the three different transformations. A thorough study of this has been done previously in this lab [3-5].  §1.3.2 Micellar Electro-Kinetic Capillary Chromatography (MEKCC) Micellar electrokinetic capillary chromatography (MEKCC) was first introduced by Terabe et al. in the mid-1980s [6, 7]. In M E K C C , surfactants are added to the buffer solution. Surfactants are molecules that exhibit both hydrophobic and hydrophilic characteristics. They have polar "head" groups that can be cationic, anionic, neutral, or zwitterionic and they have nonpolar, hydrocarbon tails. The surfactant molecules can self-aggregate to form micelles if the surfactant concentration exceeds a certain critical micelle concentration (CMC). The hydrocarbon tails will then be oriented toward the center of the aggregated molecules, whereas the polar head groups will point outward. Micellar solutions may solubilize hydrophobic compounds which otherwise would be insoluble in water. The separation principle of M E K C C is based on the differential partitioning of the analytes between the micelle and water. The micelle acts as a pseudo-stationary phase. The interaction between the micelle and the analytes can be treated in a way similar to A C E . The capacity factor (k') is now defined as  k' = ^ -  in which n  mc  and «  (1.22)  a q  are the amount of the analyte incorporated into the micelle and that in the  aqueous phase, respectively. The effective mobility of an analyte (//*)  c  a  n  be calculated from:  14  .  A  where ju  = =  m c  1  T7TT ^ P , A  +  k' 77TT / W  (1.23)  is the electrophoretic mobility of the micelle.  §1.3.3 Dynamic Complexation Capillary Electrophoresis (DCCE) The dynamic complexation of solutes while migrating at different velocities in one single phase is a fundamental and crucial assumption for C E separations. The theory of dynamic complexation capillary electrophoresis (DCCE) [8] can be applied to all open tubular C E including A C E , M E K C C , chiral CE, and other types of complexation. The migration behavior of all types of analytes and additives, whether charged or neutral, is fully encompassed by D C C E theory. The type of complexation is not the only substantial difference among D C C E systems. Many different modes or formats of D C C E have been developed; these are mainly identified by the length of the injected sample plug and various combinations of binding species present in the injection plug and the background electrolyte (BGE). In A C E and the Hummel-Dreyer (HD) method, one of the two binding species (analyte) is injected to form a narrow plug and the capillary is filled with the other binding species (additive) at varying concentrations. In HD, the running buffer can also be a mixture of the analyte and the additive at varying concentrations when the internal calibration method is used. In the vacancy affinity C E (VACE) method and the vacancy peak (VP) method, the capillary is filled with buffer containing both binding species; then, a small amount of plain buffer is injected. The determination of binding constants in A C E and V A C E is based on the changes in electrophoretic mobility of the analyte due to complexation, whereas in HD and  15  VP, the amount (peak area) of bound (HD) or free (VP) additive is measured to determine binding constants. The frontal analysis (CE-FA) method uses a much different scheme. In CE-FA, a large amount of pre-equilibrated additive and analyte mixture is injected into the capillary, which is filled with plain buffer. The binding constant can be determined with C E FA by measuring the height of plateaus. All methods mentioned so far are equilibrium C E methods that can be applied to systems with fast association and dissociation rates. If the dissociation of the complex is relatively slow compared to the time required to separate the free analyte from the complex, nonequilibrium C E methods can be used. In nonequilibrium C E , a small amount of pre-equilibrated additive and analyte mixture is injected into the capillary, which is filled with plain buffer. The binding constant can be determined by measuring the areas of the free analyte peak and the complex peak. Berezovski et al. introduced a method to determine not only binding constants but also rate constants using nonequilibrium C E of equilibrium mixtures (NECEEM) [9, 10].  16  §1.4 Limit of Detection Capillary electrophoresis methods can be used to complement gas chromatography (GC) and high-performance liquid chromatography (HPLC) methods because of its high efficiency, accuracy, very high resolution, and tolerance to biological matrices. However, poor concentration sensitivity of C E with on-column U V detection due to low sample injection volume and short optical path-length limits the use of C E as an effective method to analyze trace amounts of compounds in biological samples. Off-line preconcentration procedures have been used to obtain low detection limits but usually involve long analysis times and laborious sample handling. Efforts to improve online concentration sensitivity usually fall into three categories: (1) Increase the detection path-length by modifying the capillary shape [11-13]; (2) Use more sensitive detection methods, such as laser induced fluorescence (LIF) and linear photodiode array detector [14]; (3) Introduce a large volume of sample into the capillary, but still obtain a narrow sample zone at the detector by stacking or sweeping based on the differences in the velocity of analyte molecules at different locations of the capillary. Online concentration techniques have been increasingly employed, thanks to its simple experimental setup, two to three orders of magnitude improvement in concentration sensitivity, and the ability to concentrate cationic, anionic and neutral molecules. Several on-line concentration methods developed by Quirino and Terabe et al. [15-21] provide promising options for improving the sensitivity of CE. More detailed introduction to online concentration techniques is provided in Chapter 6.  17  §1.5 Unified Separation Science §1.5.1 Displacement and Transport Prof. John Calvin Giddings was among the first scientists to put great efforts into unifying the numerous separation techniques under one set of theories. In his 1991 book titled "Unified Separation Science" [22], Prof. Giddings proposed "a theoretical consideration of displacement and transport exerts a unifying influence on separation science, bringing diverse methods under a common descriptive umbrella." In general, there are two forms of separative transport: flow transport and relative displacement. Flow transport is a form of bulk displacement in which every component in a flowing medium is carried along nonselectively with the medium. Flow is an integral part of many separation techniques, including chromatography, ultrafiltration, and elutriation. The flow process itself is not selective, but the existence, orientation, and nature of flow often determine the quality of separation. In liquid chromatography, the flow is created by applying a high pressure; in C E , the bulk flow is the result of electroosmosis originating from the capillary wall under the application of a strong electric field. Relative displacement and the state of equilibrium are controlled by the physicochemical potential, which can be varied in an enormous number of ways along the coordinate(s) of a separation system by using different phases and fields. In a common chromatographic system, the analyte is distributed between two immiscible phases (the stationary phase and the mobile phase), and its transport is determined by the fractions of analyte present in the two phases. In carrier-free C E , where only free analytes, but no stationary phases or free additives, are present, the transport of each species (velocity) is determined by its intrinsic migration rate in the electric field. In D C C E , the transport of an  18  analyte is governed by both the intrinsic migration rates for each species and the fraction of analyte present as each species, as described by eq 1.13.  §1.5.2 Redefined Separation Factors In the late 1990s, Bowser et al. went one step further towards a unified separation science by comparing fundamental equations in the major areas of separation science, clarifying the meaning of the capacity factor, and redefining the separation factors [23]. Their unified separation science encompasses the following three equations. First, the average migration rate of an analyte ( U ) is determined by A  m  u =x/;u,. A  (i.24)  /=i  in which m is the number of analyte species, f  t  is the fraction of analyte present as species i,  and U is the migration rate of species i. Equation 1.24 is generally true for all types of ;  equilibria in all areas of separation science, with a key assumption that the equilibria between species are much faster than the overall separation process. Second, the definition of capacity factor (k') can be generalized to include any equilibrium, that is, ^, _ the amount of analyte species i  ^  ^  the amount of free analyte Using this definition, there is a capacity factor for each analyte species. Equations 1.11 and 1.22 are examples of eq 1.25 for two different systems. According to this definition, the fraction of analyte present as each species can be calculated with  19  f.—U  (1-26)  1=1  Substituting eq 1.26 into eq 1.24 yields m U  A  = ^  (1.27) 1=1  Equation 1.27 is a general equation that can be used to describe the migration behavior of any analyte in any separation technique without limitation as to the number or type of equilibria, as long as the equilibria can be established much faster than the separation process. Third, the generalized resolution equation is  R  — ! R ± R | ( W  A  + W  =  B  )  2^L _i) ( 7  ( L 2  8)  4  in which t^ , t , and W A , W B are the migration times and peak widths of the analytes A and A  B, respectively, N is the number of theoretical plates, and 7, the separation factor, is equal to the ratio of the fractions of the analytes in the mobile phase: 2^ k of species in the mobile phasae A  ,_7A  ^ k of all species A  _ V  7 = ^f = ; _ , ~ k ^ k of species in the mobile phasae JB ^ k of all species  (1-29)  s  B  The resolution equation (eq 1.28) is identical for all separation techniques with the same definition (eq 1.29), but different forms, of 7. The separation factor (7) accounts for all possible analyte species with different mobilities in a physicochemical potential field (e.g., an electric or centrifugal field). The resolution of a separation system can be predicted regardless of the number or the type of the equilibria involved.  20  Recently, Newman and McGuffin published a theoretical development of plate height for C E separations in which solute molecules undergo reactive separations [24]. The development is consistent with chromatographic theory, and a generalized equation for calculating plate height (H) in chromatographic and electrophoretic separations was derived. The generalized form of plate height from diffusion and mass transfer for a solute present as two species, A and B, is defined by  in which D is the weighted average of the mobile phase diffusion coefficients of species A m  and B (D  m  = YjfP i  mi  =f D A  mA  + f D ), B  mB  v is the weighted average of the individual  velocities of species A and B (v = ^ / v , . =f v A  A  + f v ), B  B  Av is the velocity difference  between species A and B (Av = v - v ), and &BA is the rate constant for transfer of solute A  B  from species B to A. The first term on the right-hand side of eq 1.30 accounts for diffusion, and the second term shows the dependence of plate height on equilibrium constant, kinetic rate constant, and mobility difference. When v = 0, the second term describes the B  contribution in liquid chromatography from solute partitioning in the mobile and stationary phases and the contribution in capillary electrophoresis chromatography (CEC) from interactions at the wall or at a particle surface. Additional terms that account for eddy diffusion (multiple paths) and Joule heating may be added where appropriate.  21  §1.6 Research Objectives §1.6.1 Simulation of Instantaneous Properties The previous developments of unified separation science have focused on macroscopic (often average) properties of separation systems: the average analyte migration rate (eqs 1.24 and 1.27), the steady state, resolution, sensitivity, precision, etc. On the other hand, microscopic/instantaneous behaviors are essential in understanding complex phenomena, such as dynamic complexation, sweeping/stacking of sample analytes, and buffer depletion. Computer simulation is one of the best ways to visualize instantaneous behaviors of chemical/physical systems. The driving force for chemical separation in chromatography is chemical equilibrium, and that in electrophoresis and centrifugation is physical field. C E uses the driving forces of both chemical equilibrium and physical field in its separation process, and it therefore has the advantage of being a starting point for the development of a unified theory for separation science. A computer program capable of simulating dynamic complexation in C E would be very helpful in understanding the underlying principles and mechanisms of the unified separation theory. Two simulation models of D C C E , the One-Cell model and the Multi-Cell model, have been developed using different approaches. The One-Cell model was built following the conventional approach: the real A C E experiments were studied to discover the mechanisms, and then the simulation model was created based on the mechanisms. The development of the Multi-Cell model, on the other hand, was carried out based on the unified separation theory, and then was used to study the mechanisms of dynamic complexation in C E under a variety of experimental conditions.  22  Predict  Real Experiments  Theory for THE Experiments  Unified Separation Theory  One-Cell Model  Simulated Experiments  Simulated Experiments Predict Create Theory  More Real Experiments  Figure 1.6. Routes taken to develop the simulation models.  Since the One-Cell model was built upon the understanding of one kind of experiments (ACE), its application is limited to A C E . In contrast, the Multi-Cell model was built upon the unified separation theory, and, therefore, has the potential to simulate all separation techniques. It can be used to predict experimental results, reveal the mechanisms, or even create new theory for real experiments as long as the unified separation theory stands true. Currently, the Multi-Cell model is capable of simulating all D C C E modes. The One-Cell model is discussed in Chapter 2, and the Multi-Cell model is demonstrated in Chapter 3. In Chapter 4, the mechanisms of A C E in six cases were studied using the Multi-Cell model.  §1.6.2 Equilibrium and Nonequilibrium The equations for the average mobility of an analyte (eqs 1.24 and 1.27) stand true only for systems featuring fast interactions between two or more species, that is, the equilibrium is reached so fast that the system can be assumed equilibrated at any given position and moment In A C E , M E K C C , and most D C C E systems, the assumption of fast interaction and much  23  slower separation is valid. However, for slower interactions, such as the dissociation of drugcarrier complexes, the system is not equilibrated and therefore has to be treated with chemical kinetics. The Multi-Cell model is capable of using either equilibrium or kinetic parameters.  §1.6.3 Enumeration Method to Overcome the Breakdown of Steady State Conditions Regression methods for calculating binding constants, as described in Section 1.3.1, are based on the assumption of instant establishment of steady 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. However, this assumption is valid only if the free additive present in the B G E is much higher than the concentration of the free additive in the analyte plug. Detailed discussion can be found in Chapter 4. The enumeration algorithm, borrowed from computer science and built upon the simulation models, is proposed in Chapter 5 to handle the cases when steady state conditions cannot be reached fast enough. The other advantage of the enumeration algorithm over the regression methods is that fewer experiments are required to get one binding constant.  §1.6.4 Achieving Better Sensitivity and Reproducibility Chapter 6 demonstrates a technique that combines exhaustive electrokinetic injection and sweeping micellar electrokinetic chromatography (sweeping-MEKC) to provide a several thousand-fold improvement in concentration detection limit. This technique can be systematically optimized with five key factors: the conductivity of the sample solution, the  24  conductivities of the separation buffers, the fraction of the capillary that is filled with the high conductivity buffer, the electrokinetic injection time, and the micelle concentration. At the optimal conditions, the concentration detection limits for amphetamines are lowered to ppt (parts per trillion) level, and reproducibility is greatly improved.  25  §1.7 R e f e r e n c e s [I] Jorgenson, J. W., Lukacs, K. D., Anal. Chem. 1981, 53, 1298-1302. [2] Rice, C. L., Whitehea.R, J. Phys. Chem. 1965, 69, 4017-&. [3] Bowser, M . T., Chen, D. D. Y . , J. Phys. Chem. A 1998,102, 8063-8071. [4] Bowser, M . T., Chen, D. D. Y., J. Phys. Chem. A 1999, 103, 197-202. [5] Bowser, M . T., Sternberg, E. D., Chen, D. D. Y . , Electrophoresis 1997, 18, 82-91. [6] Terabe, S., Otsuka, K., Ichikawa, K., Tsuchiya, A., Ando, T., Anal. Chem. 1984, 56, 111-113. [7] Terabe, S., Otsuka, K., Ando, T., Anal. Chem. 1985, 57, 834-841. [8] Peng, X. J., Bowser, M . T., Britz-McKibbin, P., Bebault, G. M . , Morris, J. R., Chen, D. D. Y'., Electrophoresis 1997, 18, 706-716. [9] Berezovski, M . , Krylov, S. N., J. Am. Chem. Soc. 2002, 124, 13674-13675. [10] Berezovski, M . , Nutiu, R., Li, Y. F., Krylov, S. N., Anal. Chem. 2003, 75, 1382-1386. [II] Moring, S. E., Reel, R. T., Vansoest, R. E. J., Anal. Chem. 1993, 65, 3454-3459. [12] Xue, Y. J., Yeung, E. S., Anal. Chem. 1994, 66, 3575-3580. [13] Wang, T. S., Aiken, J. PL, Huie, C. W., Hartwick, R. A., Anal. Chem. 1991, 63, 13721376. [14] Culbertson, C. T., Jorgenson, J. W., Anal. Chem. 1998, 70, 2629-2638. [15] Quirino, J. P., Terabe, S., J. Chromatogr. A 1997, 781, 119-128. [16] Quirino, J. P., Terabe, S., J. Chromatogr. A 1997, 791, 255-267. [17] Quirino, J. P., Terabe, S.,Anal. Chem. 1998, 70, 149-157. [18] Quirino, J. P., Terabe, S., J. Chromatogr. A 1998, 798, 251-257. [19] Quirino, J. P., Terabe, S.,Anal. Chem. 1998, 70, 1893-1901. [20] Quirino, J. P., Otsuka, K., Terabe, S., J. Chromatogr. B 1998, 714, 29-38. [21] Quirino, J. P., Terabe, S., Science 1998, 282, 465-468. [22] Giddings, J. C , Unified Separation Science, Wiley-Interscience Publication: New York 1991. [23] Bowser, M . T., Bebault, G. M . , Peng, X. J., Chen, D. D. Y . , Electrophoresis 1997, 18, 2928-2934. [24] Newman, C. I. D., McGuffm, V. L., Electrophoresis 2005, 26, 4016-4025.  26  Chapter 2  Determination of Shapes and Maximums of Analyte Peaks Based on Solute Mobilities in Capillary Electrophoresis  Chapter 2 Determination of Shapes and Maximums of Analyte Peaks Based on Solute Mobilities in Capillary Electrophoresis*  1  §2.1 Introduction The dynamic complexation of solutes during the analyte migration results in analyte species' migrating at different velocities [1], but the analyte could still migrate as one peak if the reaction is much faster than the rate of migration. However, the effect of the migration rates, or electrophoretic mobilities, of each individual species on the shapes of the analyte peak has not been systematically studied. In chromatography, peak shapes are often determined by the shapes of binding isotherms. A linear binding isotherm, the relationship between the concentrations of the analyte in the stationary phase and the mobile phase, produces Gaussian elution peaks; a convex isotherm results in peak tailing; and a concave isotherm results in peak fronting [2, 3]. The situation in chromatography is simpler because there are only two possible velocities for any analyte species to have. When the analyte is in the mobile phase, it migrates with the mobiles phase; when it is in the stationary phase, its velocity is zero. In C E , the equilibrium is established between the analyte and the migrating additives that have their own velocities, and the analyte species contribute to the peak shape in a much more complex manner. Previous studies on peak broadening in C E have been extensively reviewed by Gas et al. [4, 5]. The effects of extracolumn factors [6, 7], longitudinal diffusion [8, 9], different path  A version of this chapter has been published. Fang, N.; Ting, E.; Chen, D.D.Y. Determination of shapes and maximums of analyte peaks based on solute mobilities in capillary electrophoresis. Analytical Chemistry, 2004, 76(6), 1708-1714. Part of the material presented in this chapter, including the experimental section and the calculations, has been published in my master thesis. However, the focus of this study has shifted from the one-cell model (master thesis) to the study of peak shapes and maximums of different ACE experiments. t  28  (eddy diffusion) [10], wall adsorption [11], nonhomogeneous electroosmotic flow [12, 13], nonhomogeneous field [14], Joule heating [15], and electromigration dispersion and anomalous electromigration dispersion [16] have been studied. At optimum conditions, the contribution from most of these factors can be either eliminated or reduced; however, there are also fundamental and unavoidable factors that determine the peak shape and peak width even though most of the aforementioned band broadening factors are eliminated. This work shows a detailed description of how the migration behavior of individual solutes can produce peak shapes that are characteristic of the relative mobilities of the analyte, additive, and the complex formed in the separation process and how the peak maximums can be determined in each situation. It should be noted that without the presence of the additive, the analyte peaks obtained from our experiments were not distorted; therefore, the factors discussed in previous studies are not predominant in this case. With the more comprehensive understanding of the mechanism, the equations to describe the migration behavior of the peak maximums can be derived. A computer program is then written to predict the migration rate of the peak maximums throughout the separation column.  29  §2.2 E x p e r i m e n t a l S e c t i o n The first system is a 1:1 interaction between p-nitrophenol and /3-cyclodextrin (/3-CD). The experiments were carried out on a Beckman Coulter P/ACE System M D Q (Beckman Coulter Inc., Fullerton, CA) with a built-in photodiode array detector (PDA). A 30.3 cm long (20.2 cm to detector) x 50 um I.D., fused-silica capillary (Polymicro Technologies, Phoenix, AZ) was used. /3-CD (Sigma, St. Louis, MO) was dissolved in 160 mM borate buffer (pH 9.1) at various concentrations ranging from 1 to 15 mM.^-Nitrophenol (Fisher, Fair Lawn, NJ) was dissolved in the same borate buffer at 0.2 mM. In the experiments, the capillary was filled with (3-CD solution (additive, P) at various concentrations. /?-Nitrophenol (analyte, A) was then injected into the capillary to form a very narrow analyte plug. The analyte volume introduced into the capillary by applying a pressure of 0.5 psi (3447 Pa) for 3 s was calculated to be 3.6 nL, and the plug length was 0.18 cm. This calculation was based on Beckman Coulter P/ACE injection parameters provided by the manufacturer. Both inlet and outlet vials contained /3-CD solution. A voltage of 5 kV was applied on the capillary, and a temperature of 20 °C was maintained throughout the experiments. (3-CD is a neutral molecule and migrates with the same speed as the electroosmotic flow (EOF), which can be calculated directly from the time of the EOF peak. To determine the speed of EOF, methanol was used as an EOF marker. /7-Nitrophenol is negatively charged and migrates slower than EOF. The mobility of p-nitrophenol in neat buffer (/u ) has to be measured A  separately. The mobility of the analyte (ju ) is the sum of its electrophoretic mobility (// A  epA  )  30  and the electroosmotic mobility (jU  eo  In this measurement, />-nitrophenol was injected into  ).  the capillary filled with neat borate buffer under a pressure of 0.5 psi for 3 sec, and then a voltage of 5 kV was applied. The measurements were repeated multiple times before carrying out any single C E experiment. Under the experimental conditions, the free additive migrates the fastest (/u ), and the p  free analyte has a smaller mobility (ju ) than the complex (/u ), that is, / / > ju > jU . When A  c  p  c  A  the electric field is applied, the analyte plug migrates toward the detector. Free additive enters the analyte plug from behind, forming a complex with analyte. The procedure was repeated on a new capillary under the same conditions except that the concentration of />-nitrophenol was 2 mM (10 times higher). This second set of experiments was used for the comparison with the first set. The second system was carried out to study a 2:1 (analyte/additive) interaction between flurbiprofen (analyte, A) and transthyretin (TTR in short, additive, P). The experimental conditions have been published elsewhere [17]. The relative mobilities of each species is jUp > u > /Uq , in contrast to the first system, and the complex migrates slower than free P and r  A  free A.  31  §2.3 Results and Discussion §2.3.1 Peak Shapes and Relative Mobilities Steady state plays an important role in describing analyte migration behavior. The electrophoretic mobility of the analyte in the presence of the additive  ) depends on the  fractions of the species in which the analyte is present, that is K  =fM ,  V  in which f  A  A  + fcM ,  ep A  and f  ep c  c  (-) 2  1  are the fractions of free analyte and the complex, respectively, and v is a  correction factor that converts the observed mobility to the ideal state mobility where the additive concentration approaches 0 [1, 18]. When the concentration of the additive in the analyte plug becomes constant, the steady state is reached, and ju* becomes a constant as well [17]. For a very narrow plug, the steady state can be reached instantly. On the other hand, for the entire analyte plug to reach the steady state, it requires a certain amount of time, depending on the length of the analyte plug and the ratio of the concentration of the analyte to the concentration of the additive. As the analyte plug becomes longer and the ratio of the initial concentration of the analyte ([A]o) to the concentration of the additive ([P]o) in the background electrolyte (BGE) becomes larger, a longer time is required for the system to reach steady state. The shapes of the analyte peaks for the 1:1 interaction and the 2:1 interaction are different: the analyte peaks for the 2:1 interaction have similar front and rear regions, whereas the peaks for the 1:1 interaction have a long and extended front region and a very steep rear region. This difference results mostly from the relative mobilities of the free analyte, the free  32  additive, and the complex in the individual cases. There are six different cases to consider because the number o f possible combinations of these three elements (/J , /u , and ju ) is six. p  Case A , p  v  A  c  > / / > / / . This is the case in the affinity C E of p-nitrophenol and /3-CD c  A  [18]. The migration process is illustrated in Figure 2.1.  Figure 2.1, The migration process o f case A with the 1:1 interaction between the analyte (pnitrophenol) and the additive (/3-CD). The black boxes indicate the position o f the injection plug (0.18 cm in length). ( A ) The analyte plug is injected into the capillary. (B) The additive, which migrates faster than the free analyte, enters the analyte plug from the rear edge. (C) The analyte at the rear edge interacts with the passing additive and migrates, on average, faster than that at the front edge of the plug, reducing the plug length from behind. The steady state region migrates at a constant velocity that is faster than the free analyte. (D) The complex migrates faster than the free analyte, bringing the analyte out o f the front edge o f the original analyte plug. (E) The resulting analyte zone at the steady state, where the equilibrium is reached for the entire analyte plug. The amounts o f additive entering and leaving the analyte zone become equal.  33  The following discussion assumes a rectangular injection (Figure 2.1 A). The free additive (/3-CD) migrates the fastest in this case, and it enters the analyte (/?-nitrophenol) plug from behind. The analyte at the rear edge of the plug interacts with the additive and forms the complex (Figure 2. IB). Because the mobility of the complex is larger than the mobility of free analyte, the complex at the rear edge migrates faster than the free analyte to produce a most concentrated region just ahead of the rear edge. The most concentrated region then shifts from the rear edge to the front edge of the analyte plug (Figure 2.1C and D), as free analyte molecules are picked up by the faster moving additive and stacked to the front of the analyte zone. The final on-column peak (Figure 2. IE) has a large front but no significant tail. Figure 2.2A and B shows two real peaks from electropherograms for this 1:1 interaction. The only difference is that [A]o in panel A is 0.2 mM, and [A] in panel B is 2 mM. It should be noted 0  that the front of the analyte band appears on the left side of the peak in an electropherogram because it reaches the detector first. The peaks in Figure 2.2A and B have fronting characteristics, and the extent of fronting is determined by the difference in solute mobilities, the ratio of the initial analyte concentration and the additive concentration, and the time required for the analyte plug to reach the steady state where equilibrium is established in the entire plug.  34  Figure 2.2./?-Nitrophenol peaks in the presence of /3-CD in the BGE. (A) [ A ] = 0.2 mM, [P] = 5 mM in this 1:1 interaction system. (B) [ A ] = 2 mM, [P] = 5 mM. Other conditions are the same as in A. Because they are elution peaks, the direction of the peaks is opposite to the one shown in Figure 2. IE. The position of the initial injection plug is shown by a black box (0.18 cm). The location of the cell is indicated by the narrow gray box at the left edge of the injection plug. 0  0  0  0  35  The difference between the complex mobility and the mobility of free analyte is the driving force to produce the concentration gradient at the front of the peak (fronting). The width of the front is determined by the ratio of [A]o to [P]o and the difference between ju  A  and ju . A higher analyte-to-additive concentration ratio or a larger difference between /u c  A  and /u results in a larger front. If the ratio [A]o/[P]o is small, the time for the entire plug to c  reach the steady state is very short, so there is not much time for the front to develop. The length of the front in Figure 2.2A (0.38 cm) is less than half of the length in Figure 2.2B (0.86 cm), because the ratio for Figure 2.2A (0.2/5) is 10 times smaller than the ratio for Figure 2.2B (2/5). The concentration gradients keep getting longer until the entire plug reaches the steady state equilibrium. Because at the steady state, the entire analyte plug migrates at a constant speed, the difference between mobilities can no longer contribute to band broadening, and only the diffusion of analyte and complex continues to cause the band broadening.  Case B, fi > fi > /j . This is the case in the affinity C E of flurbiprofen and TTR [17]. p  A  c  The migration process is illustrated in Figure 2.3. In this case, the additive (TTR) also enters the analyte (flurbiprofen) plug from behind, forming complex at the rear edge of the analyte plug right away (Figure 2.3B), as in case A. Considering the rear edge as an infinitely narrow plug, this edge reaches the steady state equilibrium right away because a relatively large amount of additive comes into the rear edge of the analyte plug at the very first moment after the C E voltage is turned on, despite the fact that the rest of the analyte plug is still far from reaching equilibrium.  36  A  Figure 2.3. The migration process o f case B with a 2:1 interaction between the analyte (flurbiprofen) and the additive (TTR). (A) A rectangular analyte plug is injected into the capillary. (B) The additive migrates into the plug from the rear edge. (C) The complex falls out o f the plug because it has the smallest mobility, then dissociates. The free analyte then interacts with the passing additive until a new equilibrium is reached. The free additive migrates the fastest, interacting with analyte in the rest o f the plug, replenishing the rear edge with the more slowly moving complex molecules. The front edge free analyte molecules migrate faster than the rear edge complex, extending the analyte zone. (D) The equilibrium is reached throughout the analyte plug, and a near symmetrical peak is produced.  The free additive continues to move forward and forms the complex with the analyte in the plug. Because the complex migrates slower than the free analyte and the free additive, the relative movement o f the complex inside the analyte plug is toward the rear edge. Thus, there is an accumulation o f analyte at the rear edge o f the analyte plug. The complex at the rear edge o f the analyte plug w i l l lag behind. Because the concentration o f analyte outside of the plug is much lower, most o f the complex falling out o f the plug dissociates immediately and establishes a new equilibrium. The disassociation products (free analyte and free additive)  37  continue to migrate: the free additive migrates faster and reenters the analyte plug, and the free analyte migrates behind the edge. The rear edge of the plug is constantly replenished with the additive from the B G E ; therefore, it remains as the most concentrated region for the analyte throughout the entire process (Figure 2.3C). Two significant concentration gradients are produced on both sides of the peak (Figure 2.3D). Because the front edge (more free analyte) migrates faster than the rear edge (more complex), the distance between the front edge and the rear edge increases, which leads to the front concentration gradient; and the rear concentration gradient results mainly from the fact that the complex at the rear edge moves out of the analyte plug. The concentration gradients keep getting longer until the entire plug reaches the steady state equilibrium. Figure 2.4 shows a real peak for this 2:1 interaction, which agrees well with this proposed mechanism. Because the rear edge of the initial analyte plug remains as the most concentrated region for the analyte and has reached the steady state from the very beginning, no matter how long the analyte plug is, theoretically the migration behavior of this edge is always the same, and its migration time is the experimental migration time of the entire analyte peak. The lengths of the front in Figure 2.2A and B are significantly larger than both the front and the tail in Figure 2.4 (<0.23 cm) partly due to much larger \ju - ju \ for the 1:1 A  c  interaction (~1.6xlO" cm V"'s" ) than the 2:1 interaction (~0.2xlO" cm V s"'). 4  2  1  4  2  -l  38  Figure 2.4. Aflurbiprofenpeak in the presence of TTR in the B G E ([A] = 3.24//M, [P] = 0  0  100/M).  Other Four Cases. In the case of ju > // c  A  > ju , the analyte plug catches up to the p  additive in front, forming the complex at the front edge. The front edge remains as the most concentrated section. As the complex, having a larger mobility than the free analyte, can migrate out of the initial analyte plug, a front concentration gradient can be expected. The velocity difference between the front edge and the rear edge of the analyte plug can produce a rear concentration gradient. The resulting peaks in electropherograms should show both fronting and tailing characteristics. In the case of /u > /u > /u , the analyte plug catches up to the additives in the front, and A  c  p  the complex is first formed at the front edge. As the run progresses, it shifts to the rear edge of  39  the analyte plug. Therefore, both front and rear concentration gradients will be formed. More symmetrical peaks should be expected. The last two cases (ju > /u > ju and / / > ju > /u ) are very different from the four c  p  A  A  p  c  cases already discussed. Because the mobility of free additive is between the complex mobility and the mobility of free analyte, it may lead to two (instead of one) highly concentrated regions during the analyte migration, resulting in peak splitting. This prediction will be confirmed in Chapter 4 with computer simulation based on the mass transfer equation.  §2.3.2 Migration of the Peak Maximums The migration time of the analyte plug is recorded by the detector according to the most concentrated section of the analyte plug (peak maximum), which also gives information about the peak height. Therefore, this most concentrated section of the analyte zone is the most important part of the entire analyte plug. If the mechanism of the C E experiment is understood, the behavior of the peak maximums can be mathematically described and predicted. From the discussion on the peak shape, we realize that either the rear or front edge of the initial analyte plug holds the most information about C E experiments. The rear or front edge, which will be called the "cell" in the following discussion, is the most important region of the entire analyte plug. It is important to describe the behavior of the cell mathematically. Furthermore, the derived equations can be used in computer simulation programs to simulate the migration process of the cell. In the derivation of the equations for this cell, the electrophoretic migration and the equilibrium are considered separately. In other words, the changes in concentration due to the  40  electrophoretic migration are first calculated; then the equilibrium is considered in order to calculate the resulting concentrations o f all species in the cell at the end o f the current step. Case A . The system used to demonstrate the peak shape when /u > ju > /u is the 1:1 ?  c  A  interaction between p-nitrophenol and /3-CD. The equilibrium is presented as follows:  A  +  -^-^  P  AP  Start  [Ah  [P]i+A[P]  [Cji  End:  [A]  [P]  [C]  2  2  2  The subscript 1 means the beginning o f the current step, and the subscript 2 indicates the end o f the current step. A[P] is the change in the concentration o f free additive due to the electrophoretic migration. The change in the concentration o f additive in the cell, in an infinitesimal time span dt, can be expressed as [17]:  4P] = —  a  ([P] - [P],)  (2.2)  0  in which a is the length o f the cell, E is the electric field strength, and  , is the  electrophoretic mobility o f the cell at the beginning o f the current step. The absolute value of (n  p  -  |) is used, because the additive can either enter the cell from behind or be caught  up to by the analyte with the front edge o f the analyte plug. The cell can be infinitely narrow, but infinitely small numbers cannot be used in simulation programs. Therefore, the length o f the cell (a) has to be treated as a finite small value, and eq 2.2 has to be transformed into a finite-difference equation. The infinitely small time span dt is replaced by a time increment (A/), which essentially defines a step in the  41  calculation cycles. Thus, the change in the concentration of additive in the cell in one step can be calculated as  A[P] =  ^ a  ([P] - [ P I )  (2.3)  0  After the electrophoretic migration and before the equilibrium is reached, the total concentrations of the analyte ([A] ) and the additive ([P] ) in the cell are t  t  [ A l ^ A ^ C l ^ A ] ^ ^  (2.4)  [Pl^Ctn+AIPD + Cq^tP^+tC],  (2.5)  After one time interval (At), the concentrations of all species in the cell are updated, and the distance that the analyte plug migrates during this time interval is calculated. Following this, the equilibrium is taken into account. The binding constant K is given by K-  ™> = [A],-[P]  -  (2.6)  [A] x([P],-([A],-[A] ))  2  2  2  Rearranging this equation, we have K[A]\+(K[?] -K[A] l)[A] -[A] =0 t  t+  2  (2.7)  t  The value of [A]2 can be obtained by solving this quadratic equation. The electrophoretic mobility of the cell at the end of the current time interval (//  A 2  ) is  given by v \ M M e p  2  = V± [A], ^ ' M  e p  [  A  A+ A  ]  ' ~ ^ [A] ^ ' [  A  ]  c e p  C  (2.8)  t  The distance (/) the cell migrates during the current time interval is A l  =  (  ^  +  A  / ^  +  M  j .  E  .  A  t  ( 2  .9)  42  The average electrophoretic mobility o f the cell at the beginning o f the current step ( / / * , ) and the end o f the current step (ju^ ) is used in this calculation. The current position p2  of the analyte plug is the sum o f / for every step (time interval). So far, the calculations for one step are finished, and all end-step variables with a subscript 2 w i l l become the starting conditions for the next step. The calculation cycles continue until the cell reaches the detector. In case A , the peak maximum shifts from the rear edge to the front edge. To describe the shifting of the peak maximum mathematically, an assumption is made so that before the front edge becomes the most concentrated region, only pure free analyte exists at this region. With this assumption, the shifting process can be divided into two separate steps: initially, the front edge migrates at a constant speed which is the speed o f the free analyte until the additive reaches the front edge; following this, an equilibrium takes place, the steady state is reached, and the front edge migrates at a new speed determined by eq 2.1. This assumption is valid when the concentration o f additive is high enough compared to the concentration o f analyte so that it is not too hard for the additive to reach the analyte at the front edge. The time spent in the first step is  f, = .  in which Z  '  Z p  .  u s  (2.10)  is the length o f injection plug.  p l u g  The distance that the cell migrates during the first step can also be calculated as: l =M -E-t +I i  A  l  Vag  (2.11)  Several C E simulation models have been published [19-23]. A similar scheme was used by all these models. The capillary was divided into thousands o f very narrow cells, and the  43  concentrations of all species in.each cell were calculated step by step. This scheme required extremely long computational time for a single simulation run. Either a powerful mainframe computer was used [19] or a very long time (tens to 100+ hours) was taken for a single run with a personal computer [20]. In the simulation program based on the equations we have just derived, instead of monitoring the process on the entire capillary, only the peak maximums are monitored, so the simulation speed is extremely fast. The inputs of the simulation program are experimental conditions and simulation parameters. The output is the simulated migration time as well as other detailed information about C E experiments. The required experimental conditions include the initial concentrations of analyte and additive, the mobilities of free analyte and free additive, the viscosity correction factor (v), the binding constant, the complex mobility, and so forth. The electrophoretic mobility of the free analyte (ju  A  ) has to be measured by separate  C E runs in which the analyte is injected into the capillary filled with neat buffer ([P]o = 0). The conditions of the capillary can change from one run to another, so the separately measured p.  A  may not be exactly the same as the real ones in affinity C E runs. One way to  minimize this effect is by measuring // using the average //  A  A  many times prior to each affinity C E run, and then  is the simulation of the following affinity C E run. The accuracy of the  simulation depends on the size of the difference is between the average /j. individual /u  epA  A  and the  . Thirty-two consecutive measurements of the mobility of />nitrophenol in  44  .  J  neat borate buffer were carried out, and the average ju  was - 2.508 x 10~ cm V~'s~' with a 4  A  2  standard deviation o f 0.008 x l O " c m V " ' s " ' . 4  2  The correction factors (v) are also needed as indicated by eq 2.1. Because the change in viscosity is the major concern in this case, the relative viscosity (rj/r/ , 0  where TJ is the  viscosity o f the running buffer at each [P] , rf is the viscosity o f the solution when [P] is 0  0  equal to 0) is used as the correction factor [18]. The binding constant and the complex mobility can be obtained from either regression methods [18, 24, 25] or the analytical solution o f the differential equation that describes the analyte migration behavior [17]. The binding constant for the 1:1 interaction between pnitrophenol and /3-CD in the 160 m M borate buffer at 20 °C was determined using the nonlinear regression method at 5 9 6 ± 1 0 M " , and (ju 1  A  -//  e p C  ) was ( - 1 . 6 2 ± 0 . 0 5 ) x l 0 "  4  cm V~'sec~'. 2  The simulation parameters include the time increment and the length o f the cell. It is found that after decreasing the cell length beyond a certain point, the simulation results stopped changing. Thus, 0.01 cm was chosen as the length o f the cell in all simulation runs. The time increment was set to be 0.05 s for the same reason. When all the experimental conditions and simulation parameters are in place, the simulation begins and the simulated results are output into data files. Table 2.1 shows the simulation results for the first series of experiments, the 1:1 interaction between p-nitrophenol and fi-CD. The errors shown in the table are the percentage difference between each experiment and its corresponding simulated result. The percentage errors are < 1%, mainly due to the uncertainties in // _ . ep  A  45  Table 2.1. Comparison of experimental and simulated migration times for case A using the 1:1 interaction between />nitrophenol and /3-CD.  Run 1  Run 2  [P]o  Texp  Tsim  mM  min  min  Error %  1.0  10.16  10.10  2.0  8.65  3.0  Run 3  Texp  Tsim  Tsim  min  Error %  Texp  min  min  min  Error %  -0.63  10.01  9.90  -1.10  9.99  9.82  -1.69  8.59  -0.70  8.65  8.66  0.08  8.70  8.69  -0.05  8.29  8.22  -0.83  8.23  8.18  -0.67  8.88  8.83  -0.57  4.0  8.00  8.01  0.09  7.95  7.99  0.51  8.00  8.04  0.52  5.0  7.85  7.88  0.37  7.83  7.87  0.44  7.86  7.90  0.57  10.0  7.45  7.43  -0.28  7.46  7.43  -0.44  7.45  7.46  0.12  12.5  7.34  7.28  -0.73  7.40  7.35  -0.67  7.43  7.36  -0.83  15.0  7.36  7.34  -0.24  7.37  7.35  -0.22  7.36  7.33  -0.30  Case B. The system used to demonstrate the peak shape when  ju  p  > ju > A  /u  c  is the 2:1  interaction between flurbiprofen and TTR. Equations 2.2 and 2.3 are generally applicable to all cases, but the fractions of analyte species have to be determined with the consideration of stoichiometry. p  The equations for 2:1 interactions can be derived in a similar way. The equivalent of eq 2.6 in this case is K  [C]  =  2  ([E] -[E) )/2  =  t  2  [Agx([P] -([A] -[A] )/2)  [A]*-[P]  2  t  t  2  And the equivalent of eq 2.7 is  A1A] + K(2[?) -[A] )[A) 3  2  2  t  t  2  + [A] - [A], = 0 2  (2.13)  46  From eq 2.13, [A] can be solved using the Newton-Raphson algorithm [26], in which 2  [A]i is used as a good initial approximation. In case B, the rear edge remains as the peak maximum, so no special treatment (eqs 2.10 and 2.11) is required. For the 2:1 interaction between flurbiprofen and TTR, K and fi (2.00±0.08)xlO'°M"  2  and ( - 2 . 1 9 ± 0 . 0 3 ) x l 0 "  4  c  are  cr^V'sec" , respectively, using the 1  analytical solution of the differential equation [17]. The simulation results are shown in Table 2.2. The percentage errors of the simulation results are ~0 to 5%. The simulation results for the second (2:1) series have larger percentage errors. In the first (1:1) series of experiments, the measurements of ju  A  were carried out before each  experiment, but in the second series, the measurements of ju  epA  set of experiments. It is obvious that the measurements of //  A  were carried out before each in the first series are more  accurate, and can reflect the current conditions of the capillary. The other reason could be that the viscosity correction factor was ignored for the second series, because the concentrations of the additive were smaller (10~ M). 6  47  Table 2.2. Comparison o f experimental and simulated migration times for case B using the 2:1 interaction between flurbiprofen and T T R . SET  1  SET  Texp  Tsim  Error  juM  min  min  %  0  100  6.51  6.51  0.00  2  0.36  100  8.91  9.11  3  0.72  50  8.21  4  1.08  33.3  5  1.44  6  2  Texp  Tsim  Error  fiM  min  min  %  0  100  6.40  6.40  0.00  2.28  0.36  500  9.13  9.38  2.72  8.39  2.25  0.72  250  8.34  8.75  5.00  8.60  8.81  2.44  1.08  167  9.05  9.50  5.01  25  11.69  11.91  1.88  1.44  125  12.46  13.12  5.36  1.8  20  11.45  11.60  1.31  1.8  100  12.23  12.82  4.75  7  2.16  16.7  11.58  11.75  1.44  2.16  83.3  12.31  12.77  3.74  8  2.52  14.3  11.75  11.92  1.41  2.52  71.4  12.96  13.31  2.69  9  2.88  12.5  13.19  13.21  0.12  2.88  62.5  15.66  15.68  0.14  10  3.24  11.1  20.39  20.38  -0.08  3.24  55.6  22.19  22.12  -0.28  11  3.6  10  22.83  22.83  0.00  3.6  50  23.25  23.16  -0.38  [PJo  [A]  juM  1  No.  0  [P]o  [A]  0  48  §2.4 Conclusions The peak shapes in capillary electrophoresis can be solely determined by the relative mobilities of the solutes in the system when other factors affecting peak shapes are minimized and proper C E conditions are used. This phenomenon is similar to how analyte peaks are affected by the shapes of binding isotherms in chromatography. Because the situation is much more complicated in C E , different orders in the mobilities of analyte, additive, and their complex will result in different peak shapes for the observed analyte peak. When the theory for analyte migration in C E is presented correctly, it should be able to describe the migration behavior of analyte in other separation techniques, such as chromatography and other forms of electrophoresis. The driving force for chemical separation in chromatography is chemical equilibrium, and that in electrophoresis and centrifugation is physical field. C E uses the driving forces of both chemical equilibrium and physical field in its separation process; therefore, it has the advantage of being a starting point for the development of a unified theory for separation science. This work is another proof that separation science is a unified science rather than a far-flung collection of unrelated techniques [27]. The simulation of peak maximums is based on the calculations on a single cell; therefore, this model is named the One-Cell model.  49  §2.5 References [I] Bowser, M . T., Chen, D. D. Y . , Electrophoresis 1997, 18, 2928-2934. [2] Cantwell, F. F., Principles of Analytical Gas and Liquid Chromatography, Class notes, available upon request. [3] Snyder, L. R., Kirkland, J. J., Introduction to Modern Liquid Chromatography, Wiley, N.Y. 1979. [4] Gas, B., Stedry, M . , Kenndler, E., Electrophoresis 1997, 18, 2123-2133. [5] Gas, B., Kenndler, E., Electrophoresis 2000, 21, 3888-3897. [6] Delinger, S. L., Davis, J. M . , Anal. Chem. 1992, 64, 1947-1959. [7] Peng, X. J., Chen, D. D. Y., J. Chromatogr. A 1997, 767, 205-216. [8] Jorgenson, J. W., Lukacs, K. D.,Anal. Chem. 1981, 53, 1298-1302. [9] Kenndler, E., Schwer, C , Anal. Chem. 1991, 63, 2499-2502. [10] Kasicka, V., Prusik, Z., Gas, B., Stedry, M . , Electrophoresis 1995, 16, 2034-2038. [II] Schure, M . R., Lenhoff, A. M . , Anal. Chem. 1993, 65, 3024-3037. [12] Towns, J. K., Regnier, F. E., Anal. Chem. 1992, 64, 2473-2478. [13] Kok, W. T., Anal. Chem. 1993, 65, 1853-1860. [14] Keely, C. A., Vandegoor, T., McManigill, D., Anal. Chem. 1994, 66, 4236-4242. [15] Knox, J. H . , Grant, I. PL, Chromatographia 1987, 24, 135-143. [16] Gebauer, P., Bocek, P., Anal. Chem. 1997, 69, 1557-1563. [17] Galbusera, C , Thachuk, M . , De Lorenzi, E., Chen, D. D. Y . , Anal. Chem. 2002, 74, 1903-1914. [18] Peng, X. J., Bowser, M . T., Britz-McKibbin, P., Bebault, G. M . , Morris, J. R., Chen, D. D. Y . , Electrophoresis 1997, 18, 706-716. [19] Dose, E . V., Guiochon, G. A., Anal. Chem. 1991, 63, 1063-1072. [20] Hopkins, D. L., McGuffin, V. L., Anal. Chem. 1998, 70, 1066-1075. [21] Ikuta, N., Hirokawa, T., J. Chromatogr. A 1998, 802, 49-57. [22] Ikuta, N., Sakamoto, H., Yamada, Y . , Hirokawa, T., J. Chromatogr. A 1999, 838, 1929. [23] Andreev, V. P., Pliss, N. S., Righetti, P. G., Electrophoresis 2002, 23, 889-895. [24] Bowser, M . T., Sternberg, E. D., Chen, D. D. Y . , Electrophoresis 1997,18, 82-91. [25] Britz-McKibbin, P., Chen, D. D. Y., J. Chromatogr. A 1997, 757, 23-34. [26] Burden, R. L., Faires, J. D., Numerical Analysis, 1993. [27] Giddings, J. C , Unified Separation Science, Wiley-Interscience Publication: New York 1991.  50  Chapter 3  High-Efficiency Simulation of Dynamic Complexation Capillary Electrophoresis Based on Mass Transfer Equation  Chapter 3 High-Efficiency Simulation of Dynamic Complexation Capillary Electrophoresis Based on Mass Transfer Equation* 1  §3.1 Introduction A s the k n o w l e d g e o f analyte b e h a v i o r i n c a p i l l a r y electrophoresis ( C E ) accumulates, it becomes o b v i o u s that the theory based o n the observation o f retention t i m e and peak shape at the detector is inadequate for d e s c r i b i n g the analyte m i g r a t i o n a n d b a n d b r o a d e n i n g processes. T h e b e h a v i o r o f analyte i n a separation c o l u m n can be d e s c r i b e d i n detail b y appropriate differential equations [1]. H o w e v e r , it can be extremely c h a l l e n g i n g to e x p a n d the equations into solvable forms, and d e s i g n an experiment to meet the requirement o f the s i m p l i f i e d equations i n order to solve f o r useful parameters such as the b i n d i n g constant and c o m p l e x m o b i l i t i e s . C o m p u t e r s i m u l a t i o n p r o v i d e s a u n i q u e o p p o r t u n i t y f o r accurately d e s c r i b i n g the b e h a v i o r o f analytes i n a c o l u m n w h e n the differential equations are translated into finite difference schemes. H o w e v e r , even w i t h the same a l g o r i t h m , different i m p l e m e n t a t i o n s c o u l d result i n s i g n i f i c a n t l y different c o m p u t i n g times. B e c a u s e it u s u a l l y requires hundreds to thousands o f s i m u l a t e d runs to solve for t h e r m o d y n a m i c o r k i n e t i c parameters for the interaction o f a single p a i r o f c o m p o u n d s , c o m p u t a t i o n t i m e b e c o m e s c r u c i a l for the successful a p p l i c a t i o n o f c o m p u t e r s i m u l a t i o n i n high-throughput a p p l i c a t i o n s where tens o f thousands o f c o m p o u n d s need to be characterized i n a short time. W i t h appropriate implementations o f the a l g o r i t h m used for d e s c r i b i n g the analyte m i g r a t i o n b e h a v i o r i n C E , the c o m p u t i n g t i m e c a n be s i g n i f i c a n t l y reduced.  * A version of this chapter has been published. Fang, N.; Chen, D.D.Y. General approach to high-efficiency simulation of affinity capillary electrophoresis. Analytical Chemistry, 2005, 77(3), 840-847.  * The ideas of active cells and circular arrangement of cells were first proposed in my master thesis. These ideas were implemented successfully during my Ph.D. years.  52  Because C E has a well-controlled physical and chemical environment, it can be used to determine binding or dissociation constants. When C E is used for these studies, it is often referred to as affinity C E or A C E [2]. There are two strategies used in A C E : the equilibrium method and the nonequilibrium method. In the equilibrium method, a capillary is filled with a background electrolyte (BGE) with an additive at various concentrations, and the analyte is injected into the capillary to form a narrow plug. When a high voltage is applied on the capillary, the analyte plug migrates toward the detector at a speed determined by the interaction between the analyte and the additive. This method is suitable for studying fast protein-ligand interactions. In the nonequilibrium method, the capillary is filled with a B G E that does not have mobility altering additives, and the sample injected is a mixture of analyte and additive. When a high voltage is applied, the complex is separated from the free analyte and additive as they migrate at different velocities. Eventually, several portions of the analyte band will pass through the detector window rather than only one peak as in equilibrium CE. For traditional nonequilibrium C E , the key to a successful binding study is that the dissociation rate constant (k.\) must be small enough for the complex to be held together before it is completely separated from the starting materials. Berezovski et al. introduced nonequilibrium capillary electrophoresis of equilibrium mixtures (NECEEM) as a method to determine kinetic and equilibrium parameters for protein-DNA interactions [3,4]. In N E C E E M , k.\ is determined from the exponential part of the electropherogram by fitting the experimental data with a single-exponential function. Since Bier et al. used partial differential equations, including the mass transfer equation, to describe the electrophoretic migration process [5], several computer simulation models of CE have been developed [6-14]. The finite difference techniques used in these models can be  53  divided into two major categories: the Euler/Runge-Kutta integration methods [6-9, 12-14], and the finite difference scheme (FDS) with artificial dispersion [10, 11]. The Euler/RungeKutta methods are unstable due to spurious oscillations and numerical instability as demonstrated by Ermakov et al. [11]. The FDS with artificial dispersion is stable, but could produce undesired cusps at sharp gradients. In this work, various FDS's are compared, and a more robust and accurate one is implemented. Most simulation models of C E carefully account for electroneutrality and mass balance. All the ions (H , OFF, conjugate acid and base forms of buffer ions, and the ions of analytes +  and additives) in the system are considered in the calculation, and a large number of differential equations had to be used. As a consequence, the computing time is long, and the memory required to store simulation data is large. In practice, H , OH", and buffer ion +  concentrations are often kept near constant, and are not the point of interest in a well-buffered system where the analyte and additive concentrations are much less than the BGE. If only the ions of analytes and additives are monitored, as in the model proposed in this chapter, the local electric field strength can be assumed constant throughout the capillary. The resulting simulation data agree well with the experimental data with the assumption of constant local electric field strength. The computing time and the memory requirement can be significantly reduced. Dose and Guoichon implemented a simulation model, in which the calculations were carried out only for the active cells that contained ion concentrations above a set threshold [8]. Theoretically, this method could reduce the computing time by up to 90% because only a small portion of the capillary was monitored. However, their implementation of activation/deactivation of cells relied on complicated column segmentation, which required  54  frequent dynamic operations on the data, such as relocating, creating, merging, deleting, etc. These operations increased the complexity of the simulation program, as well as the computing time. Thus, the performance boost was only 3-30% [8]. A computer simulation model of A C E was developed by Andreev et al. [15]. In their model, the rate constants, instead of binding constants, were included in the differential equations to describe the electrophoretic migration and the binding interaction at the same time. For many systems, the rate constants are often not available, but the binding constants are. In equilibrium C E , the binding constants are sufficient for the simulation of fast binding interactions. Okhonin et al. developed a mathematical model for N E C E E M [16]. Their key assumptions are that interacting species are separated immediately, and that there is no new complex formed during the C E process. These assumptions made it possible to find solutions for the differential equations describing the mass transfer process with a Green's function. However, the application of this model is limited to N E C E E M . On the other hand, FDS's are still needed to obtain the precise solutions of the integrals describing the migration of the dissociated reactants. In this chapter, a new algorithm is proposed to simulate both equilibrium and nonequilibrium CE. A number of finite difference schemes are considered and the one with minimal errors was implemented to evaluate the differential equations describing the analyte behavior in A C E . The electrophoretic migration and the binding interaction are considered in two separate steps. The binding constants and rate constants are used for the simulation of equilibrium C E and nonequilibrium C E , respectively. To eliminate the laborious dynamic  55  operations on the data, a circular arrangement of cells is used to implement the algorithm. As a result, more than 90% decrease in the computing time is achieved. Petrov et al. developed a multigrid algorithm for numerical modelling of N E C E E M and other modes of kinetic capillary electrophoresis (KCE) [17]. The multigrid algorithm can eliminate numerical diffusion which may cause severe error propagations at sharp edges of electrophoretic peaks. However, a new error may be introduced due to the mismatch of the concentrations of interacting species in the calculation of equilibrium or kinetic, which may eventually lead to the disturbance of the mass balance. On the other hand, the computing time is longer than using a uniform grid length. In the implementation described in this chapter, an improved second-order upwind scheme was used to reduce numerical diffusion while still maintaining high efficiency. A C E and N E C E E M are two modes of D C C E . As mentioned in Section 1.3.3, there are a few other modes: HD, V A C E , VP, and CE-FA. The computer program is capable of simulating all these D C C E modes, and therefore, is named SimDCCE. SimDCCE  The functions of  will be explained in the final section of this chapter.  56  §3.2 Experimental Section A well characterized 1:1 interaction between p-nitrophenol and /3-cyclodextrin (/3-CD) was chosen to compare the experimental and simulation results. The experiments were carried out on a Beckman Coulter P/ACE System M D Q (Beckman Coulter Inc., Fullerton, CA) with a built-in photodiode array detector (PDA). A 64.5 cm long (54.3 cm to detector), 50 um i.d. fused-silica capillary (Polymicro Technologies, Phoenix, AZ) was used. /3-CD (Sigma, St. Louis, MO) was dissolved in 160 mM borate buffer (pH 9.1) at 5 mM as the B G E , and />nitrophenol (Fisher, Fair Lawn, NJ) was dissolved in the same borate buffer at 2.0 mM. The capillary was filled with the B G E with /3-CD, and p-nitrophenol was injected into the capillary to form a very narrow analyte plug. The analyte was introduced into the capillary by applying a pressure of 0.5 psi (3447 Pa) for 3 s, and the volume was calculated to be 3.6 nL, with a plug length of 0.18. cm. This calculation was based on Beckman Coulter P/ACE injection parameters provided by the manufacturer. The C E process was operated at 10 kV, and a temperature of 20 °C was maintained throughout the experiments. Methanol was used as the electroosmotic flow (EOF) marker. /3-CD is a neutral molecule, and migrates at the same speed as the EOF. The mobility of /?-nitrophenol in neat buffer (ju ) has to be measured A  separately. In this measurement, the sample was injected into the capillary filled with neat borate buffer with no additive under a pressure of 0.5 psi for 3 s, and the migration was driven by a voltage of 15 kV. The measurements were repeated multiple times before carrying out the A C E experiment.  57  /3-CD has no U V absorbance in the range of 190-500 nm, and the complex is assumed to have a U V absorbance similar to the analyte. The binding constant of 533 M" for this 1:1 interaction is determined using nonlinear 1  regression or linear regression methods [18]. This number was used as one of the inputs of the simulation program. The results of the simulation, including peak shapes and migration times, were compared with the experimental data. Nonequilibrium C E experiments, similar to the N E C E E M experiments reported by Berezovski et al. [4], were simulated with the same program. No real nonequilibrium C E experiments were carried out. Instead, the conditions emulating N E C E E M experiments were input into the simulation programs to demonstrate the ability of this program to simulate all A C E processes.  58  §3.3 A l g o r i t h m a n d I m p l e m e n t a t i o n The capillary is divided into thousands of narrow cells. Between adjacent cells, there is migrational flux of all species. At the same time, there are interactions between the additive and the analyte. The electrophoretic migration and the association/dissociation processes are considered separately: the changes in concentrations due to the electrophoretic migration are first calculated; then the association/dissociation processes of the analyte (A), the additive (P), and the complex (AP) are considered in order to calculate the final concentrations of all species in each cell at this moment before the next step of electrophoretic migration. i,  §3.3.1 Electrophoretic Migration Process The electrophoretic migration process for both equilibrium and nonequilibrium CE is identical, and can be described by the mass transfer equation [6, 7] 3C.,,  8t  dC dC =E — ^ + D --f+' ' dz dz l  2li  M  i  z  2ti  1 l  z  (3.1)  2  in which C ,,. is the concentration of species i at position z and time t, E is the total local z  z  electric field strength, // . is the apparent mobility of ion i (ju = /u (  i  epi  + ju , where ju eo  is the  electrophoretic mobility of the ion i, /u is the electroosmotic mobility (EOF)), and A is the eo  diffusion coefficient of the ion. In C E , the diffusion coefficient could include other symmetrical dispersion caused by other factors. The local electric field strength is assumed constant throughout the capillary.  59  With a selected time increment At, which defines the step, and a selected space increment Az, which is the length of a cell, eq 3.1 can be evaluated with finite difference schemes (FDS's).  Finite Difference Analysis of Hyperbolic Partial Differential Equations. In the absence of diffusion, eq 3.1 can be simplified into a hyperbolic partial differential equation dC.,,  = -p E ' i  dt  z z  dC. ^ 8z  (3.2)  Equation 3.2 can be evaluated with one of the following finite difference schemes: r  C  zU  C  *-z,t+At,i  At  77  -C Z.t.i  = -MiE  z  C = -MiE  C  z,U  r-i  z+Az A ,i  -C  +C  L  —  1 -  C  C  ^z,l-Al,i  2At  (3-6)  -C  j - . ^z+Az,(,(  '•—- =  :  —  2Az  -C  ^z.l + Al.i  -C  =" M :  A?  :  \  (3-5)  2AZ  - (C  — —  c  ^  At  C  (3-4)  z-Az / /  = -Vt z E  C  /o A \  —  A  C  ^z,l+Al,i  z.t.i  ~ Az  z  r  (3-3)  -C z+Az,/,/  r-i  At  —  Az  r  0 \  Z~Az,tJ  ^z-Az,t,i  '  —  /o -r\  (3.7)  2Az  Equations 3.3-3.7 are the forward-time backward-space scheme (FTBS), forward-time forward-space scheme (FTFS), forward-time centered-space scheme (FTCS), Lax-Friedrichs scheme (an enhanced scheme over eq 3.5), and staggered leapfrog (centered-space centeredtime) scheme, respectively [19, 20]. Stability and convergence are required of FDS's. The Courant condition, which is described by the following equation, must be satisfied in all five schemes to avoid amplitude errors [20].  60  M . l fAz -^l  (3-8)  The solution of a hyperbolic problem at a point depends on information within some domain of dependency to the past. Equation 3.8 states that the space increment (Az) must not be smaller than the distance that ion / travels during Ar. In other words, Az defines the boundary of the spatial region that is allowed to calculate C  2 ( + A  , b y the differencing schemes.  Mathematically, the Lax-Friedrichs scheme (eq 3.6) and the staggered leapfrog scheme (eq 3.7) are more accurate than eqs 3.3 and 3.4, because associated errors for eqs 3.6 and 3.7 (proportional to Az ) are smaller than those for eqs 3.3 and 3.4 (proportional to Az), as Az 2  approaches zero. Using higher orders of centered differencing on the right-hand side (RHS) of eq 3.6 or 3.7, greater accuracy can be achieved. However, if there are sharp gradients in the system, the use of eq 3.6 or 3.7 will lead to numerical dispersion, or phase errors. Sharp gradients cannot be avoided either in the operation or the simulation of CE. For a rectangular injection profile, there are sharp gradients on both edges of the injection plug; while for a parabolic profile, there is a sharp gradient at the inlet of the capillary [21]. In addition to the sharp gradients resulting from the injection, the difference in concentrations between two adjacent cells does not approach 0 if the length of cell does not approach 0; i.e., there is always a small gradient between two adjacent cells. Because numerical dispersion is a propagation error, it becomes more significant as the simulation proceeds. On the other hand, the smaller the length of the cells (Az), the slower numerical dispersion becomes significant is. Numerical dispersion in various FDS's has been demonstrated by Ermakov et al. [10, 11]. In addition, eqs 3.6 and 3.7 do not describe the physical properties of C E systems correctly, because the equations describing C E systems are advective equations, not wave  61  equations. Without diffusion, any species migrates in only one direction due to the applied electric field. The migrational flux (F), which is the amount of a species migrating out of a cell, is defined as = C ^E zli  (3.9)  z  If a species migrates toward the detector (ju E > 0), only the migrational fluxes of the i  z  previous cell and the current cell determine the new concentration in the current cell. Therefore, the forward-time backward-space scheme (eq 3.3) should be used in this case. Substituting eq 3.9 into eq 3.3 gives C  - C  ^z,l + At,i  F z,t,i _ _  At  - F z,l,i  ^ 1  z-Az,l,i  Az  If a species migrates away from the detector (/u E < 0), eq 3.4 should be used to give C -C F -F j  z,l + At,i  At  z,t,i  _ _  z + Az,/,/  z  ^  z,l,i  11^  Az  Equations 3.3, 3.4, 3.10, and 3.11 are also called the first-order upwind difference schemes because the spatial differences are skewed in the "upwind" direction of  /u E . j  z  In most C E systems, because of the existence of a strong electroosmotic flow, all species migrate toward the detector. Thus, in the following discussion, only the equations describing the movements toward the detector ( u E > 0), such as eq 3.10, are given. J  j  z  Thefirst-orderupwind difference schemes not only provide the "fidelity" to C E systems which is lacking in other schemes, but also eliminate numerical dispersion. But the cost for stability and simplicity is a new type of error - numerical diffusion. In eqs 3.10 and 3.11, the ions always arrive only from one cell (Az) away. But in reality, the ions from a distance ju E At i  z  away arrive after a time interval At. The Courant condition (eq 3.8) states the  62  requirement of Az and At. If [/UjE^At = Az, the numerical diffusion error is eliminated. However, if |//.is |Af «  Az , numerical diffusion can cause large errors. This type of error can  z  be reduced by using higher-order upwind difference schemes. The monotonic transport scheme is an efficient second-order upwind difference scheme, which is derived by considering the analytic transport equation [22]. The equation for the case of /J E > 0 is: 1  C ,t+&t,i ~ C z  r E i  At  C  z  Az  in which d (defined  -C  °z,/,i  ,,  f  U  z t i  +—  ^z-Az,<,/  ^  Z7  A * \  (3.12)  Az  2  Z  by eq 13) and d _^ ,,. are the concentration gradients at position z and  z t  z  z-Az, respectively.  2  (  ^ "  C  =  d  z4j  ^  C  )  (  ^ "  C  ^  )  ,  if ( C , , - C _ , , ) ( C z  (  /  z  A z  (  ;  z + A z  , „-C , „)>0 (  z  /  = 0, otherwise.  With the consideration of both accuracy and efficiency, eq 3.12 is chosen as the algorithm for the hyperbolic part of the differential equation.  Finite Difference Analysis of Parabolic Partial Differential Equations. The following diffusion equation is a parabolic partial differential equation, dC ,. d C ,. —iiL=D. *M2  dt  '  (3.14)  dz  2  Equation 3.14 can be evaluated with the following three schemes: the first-order fully explicit scheme (eq 3.15), the first-order fully implicit scheme (eq 3.16), and the second-order Crank-Nicholson scheme (eq 3.17). C  -C At  C ~  '  -2C  +C  (Az)  2  63  C  -C  ^z,l+At,i  At C  A  t  C  -2C  ^z,t,i _ pj ^ z+Az,t+At,i  ~  '  +C  ^^z,t+At,i ^ ^ z-Az,t+At,i  (Az)  ^ < ^\  2  '  {  -C  '  (3.17)  _p^ (^ +&z,t+At,i 2C z  j  + C_  zl+Al  z  )  + (C  Azl+&lj  2(Az)  2C  r z + A z l j  zti  +  C_ ) z  Azll  2  Equation 3.15 (used in the simulation model of Dose el al. [8]) is stable only for sufficiently small time increments (At), and the stability criterion is  (3.18) Both eqs 3.16 and 3.17 are unconditionally stable, that is, arbitrarily large time increments can be used. Equation 3.17 is more accurate than eq 3.16 because of its secondorder expression on RHS. Although eqs 3.16 and 3.17 are better choices, eq 3.15 is used in this model for the following reasons. First, for most large molecules such as proteins, the diffusion coefficients are usually quite small (< 10~ cm /s). The simulation results are already very good using eq 6  2  3.15 as shown later in this chapter. Second, eq 3.15 can be solved in a way similar to eq 3.12, so that they can be easily incorporated together in a program evaluating both the hyperbolic and parabolic parts of the function for analyte migration. However, to solve eqs 3.16 and 3.17, a set of simultaneous linear equations must be solved at each time step for C  z  When u E J  i  z  , ,,. +A  > 0, the overall finite difference scheme for the evaluation of the partial  differential equation for electrophoretic migration in C E is the combination of eqs 3.10 and 3.15 or the combination of eqs 3.12 and 3.15 as the following  At  ~  Az  '  (Az)  2  64  c  ,- c • At  Az  C  C  fi E At t  z  Az  J  (3.20)  J  C (Az)  2  To achieve sufficient accuracy with eq 3.19, very small Az and At have to be used, which requires a long computing time to finish a single simulation run. But with the more accurate eq 3.20, it is not necessary to use extremely small Az and At, so that the simulation becomes much faster. The simulation results with both equations will be compared in the results and discussion section.  §3.3.2 Association and Dissociation Processes At this point, the electrophoretic migration process in this step is finished, and the new concentrations of all species in each cell are obtained. In the next step, the association/dissociation processes are taken into account, and the concentrations at the newly established equilibrium are calculated. The concentrations of free A , free P, and AP after the electrophoretic migration process of the current step are denoted as [ A  rree  ] , [P y  free  ] and ;  [AP] . The concentrations after the association/dissociation process are denoted as [ A ^ J * , y  [P J* and [AP]*. fre  The equations for equilibrium C E and nonequilibrium C E are different. Equilibrium C E . In this case, the binding interaction is assumed fast, and the equilibrium can be reached within At. Thus, the binding constant (K) is used in the calculations  65  Rearrange eq 3.21 by replacing [AP]* and [P J* with the total concentrations of A and fre  P ( [ A ] and [P,] •) which are obtained from the calculations for the electrophoretic migration t  y  process and remain constant at the equilibrium stage. K[A J/ fr  +(/qP ], - * [ A ] , + l)[A ]* - [ A ] = 0 t  t  free  t  y  (3.22)  [A J* can be easily solved from eq 3.22. fre  Nonequilibrium C E . Because slower interactions are studied with this technique, rate constants, instead of binding constants, have to be used. Different binding interactions may have different mechanisms, and each mechanism needs its own set of equations. For a simple interaction, A+P  AP, the forward rate constant and reverse rate constant are k  +i  and  ,  respectively. The final concentrations of the additive and the analyte in cell j are: [Pfree Ij = [Pfree ]j " [A  free  ]* = [A  free  [A  ], -  free  [A  ]j[P  free  free  ].[P  }j ~ k_ [AP],) X At  (3.23)  {  free  I - k_ [AP],) x A; x  (3.24)  Once we have [A J* and [P J*, the calculation cycle is finished for the current step fre  fre  (time t). We then move on to the next step (time t + At) and repeat the calculations.  §3.3.3 Implementation of Circular Cell Arrangement An efficient implementation is as important as developing appropriate mathematical algorithms in computer simulation. Conventionally, the entire capillary is equally divided into  66  m very narrow cells. A linear array of m elements is allocated in the computer's memory, and each element in this array stores the data for one cell. The way that the data for the divided cells are stored and manipulated in the computer's memory is the key to the efficiency improvement of the simulation programs. In this work, a circular cell arrangement and data manipulation algorithm is used to significantly reduce the computing time and the data storage. In equilibrium C E , the concentration in most cells does not change because only the cells within or near the analyte plug have a noticeable amount of the analyte. Therefore, there is no need to track the changes in concentration for every cell during the electrophoretic migration of the analyte. In this implementation, the capillary is also divided into m cells. The cells within and near the analyte plug are labeled as active cells, in which there is a noticeable amount of analyte. All other cells are called inactive cells, in which the analyte concentration is less than a threshold value. Only the concentration changes in active cells are tracked and stored. At the beginning of an equilibrium C E experiments, the analyte exists only in the analyte plug. Due to the applied electric field, the analyte plug moves toward the detector; due to diffusion, the analyte spreads to the adjacent cells in both directions. To determine whether a cell is active, a user-defined threshold concentration of the analyte is used. When the concentration of the analyte in a cell is greater than the threshold, the cell is activated; when the concentration of the analyte in a cell is smaller than the threshold, the cell is deactivated. An array with n elements is allocated in the memory. Although n can be much smaller than m, it has to be large enough to store the data for all active cells. This array is arranged as a circle as shown in Figure 3.1, which makes it possible to reuse the cells. Two pointers (pi  67  and p2) are used with pi pointing at the starting point of the active segment, and p2 pointing at the end of the active segment. Only the elements within pi and p2 are updated in the current step. At the beginning of a simulation, pi points at element 0 , and p2 points at element j (j = L  p /  I Az), which is the last cell of the analyte plug. Any cell outside this range (from  7 + 1 to n) has the same default conditions: the concentration of the additive is [P]o and the concentration of the analyte is 0 . In the first step, the changes in the concentrations of all species for the cell from pi ( 0 ) top2 (J) are calculated using eqs 3.19-3.26. Some analyte moves into cell7+1, so the pointer p2 is moved forward  to7+1  as well. At the same time, the concentration of analyte in cell  0  decreases.  PI  Figure 3.1. Circular arrangement of cells in the memory of the PC. Cell 0 is the starting point. Cell j is the last cell that contains an appreciable amount of the analyte. Cell n is the last cell allocated in the memory, but when the analyte migrates pass this cell, it automatically enters and reuses cell 0 .  68  In every following step, the pointer p2 is moved one cell forward. But the pointer pi still points at the first cell until the concentration of the analyte in the first cell is lower than the predefined threshold value. As the pointers pi and p2 keep being moved forward, p2 may finally bypass the last physical element in the computer's memory, No. n of the array. However, because the elements are arranged as a circle, logically there is no "last" element. The pointer p2 is moved to No. 0 again after No. n. This operation is valid also because the elements are released on the side of pointer pi as the analyte moves forward. This new algorithm is efficient for the simulation of equilibrium C E , because there is always just one analyte plug needed to be monitored. In nonequilibrium C E , the analyte band often becomes quite broad. Therefore, two or more significant portions of the analyte band are monitored in the electropherogram, and more cells are active at any given time, which leads to a longer computing time for running a simulation. Similar implementation strategy as described for the simulation of equilibrium C E is used to simulate nonequilibrium C E , which is an advantage of this new simulation model because it can be used on nearly any kind of C E application with increased speed. However, the initial conditions for the simulation of nonequilibrium C E are different: the equilibrated mixture of the additive and the analyte is injected into a capillary filled with a plain buffer. In C E applications for separation purposes, there are gaps between two adjacent, but completely separated peaks. If the cells within these gaps, having no analyte inside, are treated as inactive cells, the speed of the simulation can also be increased. In that case, more pointers (p3, p4, ...) are needed to track the start and the end of each active segment.  69  §3.4 S i m u l a t i o n o f E q u i l i b r i u m C E Equations 3.19 and 3.20 were both implemented in the simulation program SimDCCE*. Most of the discussion is based on the 1:1 equilibrium C E experiments. However, these properties are applicable to both equilibrium and nonequilibrium CE. Time and Space Increments. The time increment At and the space increment (the length of cell) Az are the most important simulation parameters. The implementation of eq 3.19 is first tested with different A; and Az. Figure 3.2A shows the simulated peaks for the equilibrium C E experiment described in the Experimental Section (initial conditions: [P]o = 5 mM, [A]o = 2 mM) with the same Az (0.0005 cm) but different At (0.0005 to 0.002 s). The three simulated curves only have small differences that can be ignored. Thus, there is no need to use extremely small values for At. The space increment Az plays a more important role than the time increment At in predicting peak shapes with eq 3.19. Figure 3.2B shows the simulated peaks with the same At (0.0005 s) but different Az (0.0001 to 0.003 cm). With a larger Az, the numerical diffusion is so significant that the simulated peaks are broad and very different from the real peak. As Az becomes smaller, the simulated peaks become narrower and closer to the real one. However, even if one can afford to spend longer computing time to integrate using extremely small At and Az, round-off errors, originated from machine errors, will accumulate and eventually become large enough to invalidate the solution.  The simulation program was initially called SimACE, and was later renamed to SimDCCE after the ability of simulating dynamic complexation capillary electrophoresis (DCCE) was successfully implemented. ACE is one of the many modes of DCCE.  70  Figure 3.2. Simulated peaks with the implementation of eq 3.19. (A) Same Az and different Ar. (B) Different Az and same At.  10 .  i  Az = 0.003 cm  i  —  - i  At = 0.003 s  1  1  1  1  1365 1370 1375 1380 1385 1390 1395 Time (second)  Figure 3.3. Simulated peaks with the implementation of eq 3.20. Even though different simulation parameters are used, the generated peaks have a very similar shape.  The implementation of eq 3.20 is then tested in a similar, way. Because the numerical diffusion is greatly reduced with eq 3.20, no extremely small At and Az are required. The simulated peaks shown in Figure 3.3 are all very similar even though very different values of At and Az are used. The simulated peaks generated using eq 3.19 and eq 3.20 are plotted together with the real peak in Figure 3.4. All three peaks have very similar shapes: a steep edge on the right and a gentle slope on the left. The cause of this kind of shape has been discussed in Chapter 2. However, the real peak is a little broader (having larger variance) than the simulated peaks. The total peak variance is the sum of the variances caused by injection, longitudinal diffusion, and other factors [21]. The variance caused by diffusion has been considered in SimDCCE.  72  Treating the parabolic profile caused by pressure injection as a rectangle and the inability to measure the length of the injection plug directly lead to an uncertainty in the variance caused by injection. In addition, other variances, such as band broadening caused by the length of the detector window and wall adsorption, are not considered. Therefore, the simulated peaks usually have smaller variances. The simulated peak positions (1389 s from eq 3.19 and 1390 s from eq 3.20) are slightly different from the real peak position (1386 s), -0.3%, mainly due to the errors associated with the measurements of the electrophoretic mobility of the free analyte (ju ) and the viscosity epA  correction factor (v).  Figure 3 . 4 . Experimental and simulated peaks. The y-axis for the simulated peaks is the concentration of the analyte, and the y-axis for the experimental peak is the absorption. In this figure, all peaks are normalized to the same scale. The experimental conditions for generating this experimental peak are described in Section 3.2.  73  The speed of the simulation is determined by Az and At. The computing time for generating the peak with At = 0.0001 s and Az = 0.0001 cm using eq 3.19 is more than 4 hours with a laptop PC powered by an Intel Centrino™ 1.8 GHz CPU. On the other hand, the computing time for generating the peak with At = 0.001 s and Az = 0.001 cm using eq 3.20 is only 4 minutes with the same PC. It is clear that eq 3.20 is superior to eq 3.19, because rather large increments can be used with eq 3.20 to achieve similar accuracy compared to much smaller increments used with eq 3.19. The speed of the simulation is improved close to 2 orders of magnitude. Active/Inactive Cells. Another important step of the simulation is to determine which cell should be activated or deactivated. This is achieved by using threshold values on both inlet and outlet sides of the capillary as described in the previous section. These threshold values play important roles in the simulation. If the threshold values are too large, a cell with a significant amount analyte could be deactivated if its total concentration of analyte is smaller than the threshold values, which leads to large errors in the simulation. If the threshold values are too small, a large number of cells remain active for a long time, which decreases the efficiency of the simulation program. The acceptable choices of threshold values are usually 1/10,000 to 1/1,000,000 of the concentration of injected analyte solution. Diffusion Coefficients. The longitudinal diffusion of all species in the capillary contributes to the peak shape in C E experiments. The more significant the diffusion is, the broader the final peak is, which leads to more active cells and longer computing time. According to the Stokes-Einstein equation for isolated hard spheres, the diffusion coefficient  74  depends on the size and the shape of the molecule, the interaction with the solvent, and the viscosity of the solvent. The diffusion coefficients for proteins in aqueous solution are usually on the order of 10~ cm V 6  or smaller. The influence of the longitudinal diffusion on the peak shape is not  significant for studying protein binding interactions. For all systems presented in this chapter, the diffusion coefficients are not measured experimentally, instead, estimated values are given to the simulation program.  75  §3.5 Simulation of Nonequilibrium CE The simulation model can also be used to simulate nonequilibrium CE. However, no real experimental data are compared to the simulation results in this section. Instead, the characteristics of Berezovski et al.'s N E C E E M experiments [4] will be demonstrated by simulation. The conditions given to the program are as follows: the binding constant (K  - 2.0xl0 M"'), 7  a  the dissociation rate constant (&_, =0.01s  -1  in Figure 3.5A;  k_ = 0.05 s" in Figure 3.5B; k_ = 0.001s' in Figure 3.5C), the initial concentration of the 1  1  {  x  additive (2 \M), the initial concentration of the analyte (61 nM), the length of the capillary (33 cm), the length to detector (33 cm), the length of the injected plug (0.036 cm), the applied voltage (13200 V), the diffusion coefficients (10" cm s 5  2  _1  for all species), the mobility of free  analyte (0.000357 c m V s " ) , and the complex mobility (0.000146cm V's" ). The 1  2  1  association rate is set to 0, because the re-formation of the complex is completely prevented as in Berezovski et al. 's N E C E E M experiments [4]. In addition to the experimental conditions, the simulation parameters are set as the following: the time increment (At = 0.003 s), the length of the cell (Az = 0.003 cm), the concentration threshold (10" M). 14  Figure 3.5 shows the simulated peaks under the given experimental conditions and simulation parameters. The peaks 1 and 3 are sharper than normal experimental peaks due to the incomplete consideration of various causes of peak variance.  76  5  3\ 2 1  3  0 10  B 2 6 '©  A 4 <  2  50 3l  40 30 20 10  200  300  400  500  Time (s)  600  Figure 3.5. Simulated peaks for nonequilibrium C E experiments. All three figures are generated with identical experimental conditions and simulation parameters except the dissociation constant (k.\). (A) k_ = 0.01s '. Peak 1 corresponds to the free analyte under the initial equilibrium conditions. The exponential part 2 corresponds to the dissociation of the complex, and the small peak 3 corresponds to the remaining complex. -  x  (B) k_i = 0.05s" . In this case, there is no complex peak because the complex has been 1  completely dissociated. (C)  = 0.001s" . In this case, the dissociation process is so slow that the exponential part 2 1  is not visible.  77  Because the entire peak region in nonequilibrium C E is much longer than that in equilibrium C E , the number of active cells in nonequilibrium C E is also much larger. The computing time for the simulation of nonequilibrium C E is usually longer than the simulation of equilibrium C E , i f the same Az and At values are used.  78  §3.6 S i m u l a t i o n o f D C C E The core of SimDCCE  is its calculation module which is used for the simulation of all  the D C C E modes. However, for different D C C E modes, the initial experimental conditions and some of the simulation parameters are different. Figure 3 . 6 shows the user interface of SimDCCE.  Some key parameters are explained in  this section, and a step-by-step instruction is included as Appendix A.  1(5:1X1  : A C E _ W 6 0 0 _ 3 . p a r - SimDCCE2 He  Wew  Li E ? I s«Sf«s  H  Help 3)  • Uae the Mitb -CeS model tc study the rr^echantsms cf arTinrty C E . *"  f Property  Value  •  '  Chemicals BSA  The Additive (P)  Warfaim  Reaction Type  AP.  [A] in  BGE M  S  Adjust Automatically l  SECONDS Specifies the MIGRATION T H E interval for refreshing the peaks  Y M a x (M) >  (1-800)  0 Refresh every  00006  (cm2^Ts)  jQ 1  Delay  0  Specifies the real-time delay after every shewing of the peaks  S h o w Curves  Free Analyte (A)  B  Zoom x  (concentration)  [ i/  Adjust Automatically  0  • enaoied the x and/or y axes are sutcmatcsily adjusted toftthe entire * »n - : y data ranges  •iiminiiiii  y-axis  1:1  0  [P] m B G E (M) MobiMies  limn  MIII.  4E-O05  n m Sample (M) IP] m Sample (M)  B  i  The Anatyte (A)  SIOP  £AUSE  A B  : XCli:  n e e Additive (P)  •0 00018  Comptei (C)  -0 000133  BectroosmotK: Row (E  0  Time of E O F Marker Cs)  270.87  Migration Time of A :s  0  Viscosity ComscDon A  1  Viscosity Correction P  1  Vrscojity Correction C  1  I JC IPfree)I a/Signal II  yij  [ * j(  [Complex] |  total! ]  t i< F ^ f . l  f| j I  Signal I  2 9 187  4 873e-005  [ «j  T i m e (s) = 4 4 3 4 4 0  36095  fdrvCapillary Position (cm)  3  flifft-tnon C o e f f i c i e n t s t*sc Length of Injection Piu  025  i  pq(M)  Binding Constant Association Rate C » / M )  0  Dissociation Rate (/%)  D  Diffusion Coefficients  v  (on"2/«) 00  3  Experimental Conditions  4  Smulauon Parameters  y  Figure 3.6. The user interface of  «  4  V  » w \ S i n g l e Run: Multi-Cell Model /  i  F-i^r"--,*t - i - A:;-it  ^  :-:»'^:;i:.:':  /  y  .  i- -  :  SimDCCE.  Region 1: the settings of the experimental conditions and simulation parameters; Region 2 : the settings for the animation of migrational processes; Region 3 : the panel on which the animation is displayed.  §3.6.1 Initial Experimental Conditions Two major differences amid various D C C E modes are ions present in the injected sample and the B G E , and the length of the injection plug. The injected sample can be plain buffer, buffer containing the analyte, or buffer containing the pre-equilibrated mixture of the additive and analyte. The B G E can be plain buffer, buffer containing the additive, or buffer containing the pre-equilibrated mixture of the additive and analyte. A large amount of sample is injected in C E - F A to maintain a sample plateau throughout the C E process; while in other modes, only a small amount of sample is injected to form a narrow injection plug. The simulation program must have the ability to handle all these initial conditions. This requirement can be fulfilled by letting users define all possible initial conditions on SimDCCE's  user interface.  In this section, [A]b and [P] denote the initial concentrations of the analyte and the D  additive in the B G E , respectively. [A] and [P]o denote the analyte and additive 0  concentrations in the injected sample, respectively.  §3.6.2 User-Defined Concentration Thresholds User-defined concentration thresholds are used to define active cells. Each D C C E mode has its own requirements for active cells. The requirements are often imposed on the total concentration of the analyte and/or the concentration of the free additive. In A C E and N E C E E M , the calculations are mostly based on the migration times (ACE) or the shapes (NECEEM) of the analyte peaks. A threshold of the total analyte concentration ([A ]) is required for determining active cells. A cell should be active when containing a t  noticeable amount of analyte. This threshold is usually set at 1/100,000 to 1/1,000,000 of [A]<  80  I Naturally, bigger thresholds lead to faster simulation, while smaller thresholds result in smoother, more accurate simulated peaks. The HD mode uses the identical experimental set-up as the A C E mode, except that the additive peak area is required in the calculations. Therefore, in addition to the requirement on the analyte concentration, an additive concentration threshold should be enforced on active cells. A cell will be activated if its additive concentration is changed noticeably from [P] , that b  is, the change in [P] is greater than 1/100,000 to 1/1,000,000 of [P] . b  Both V A C E and V P result in negative peaks. V A C E is similar to A C E , but the negative analyte peaks demand that a cell is active when [A] < [A]b. V P is analogous to HD, and its requirement for active cells are [A] < [A]b and [P] < [P]b. C E - F A is essentially an A C E / H D experimental setup with longer injection, and it shares the similar threshold settings with HD.  §3.6.3 Outputs of  SimDCCE  Simulated Signals. The simulated results are naturally the concentration profiles of the species of interest. However, experimental electropherograms display the sum of signals provided by all species under either U V or fluorescence detectors. For A C E and V A C E experiments, the additive does not have to provide significant signals to the detector because only the analyte peak is required for calculating binding constants. These methods are particularly useful for the additives with little or no U V absorption or fluorescence activity, such as cyclodextrins. On the other hand, for VP and HD experiments, the additive must give either U V or fluorescence signals to enable the quantification of bound additive (HD) or free additive (VP) for the determination of binding constants. Therefore, SimDCCE  must have the  81  flexibility to generate simulated peaks based on each species' ability to absorb U V light or emit laser induced fluorescence. This can be achieved by providing a multiplier to each species according to its U V or fluorescence ability. The simulated signal is the sum of product of each species' concentration and multiplier. If a species does not give any signal, or its concentration is uniform throughout the capillary, its multiplier can be set to zero. Online Snapshots of the capillary at any given moment can be taken easily as the current concentrations of all species in each cell are stored in the computer's memory. A series of snapshots taken at proper moments of the electrophoretic migration process can illustrate the mechanism of the C E process. Animation of C E Processes. SimDCCE  is able to display the snapshots of a simulation  run at a given time interval on screen to animate the migration process. Thanks to the highly efficient algorithm and implementation, it is the first time that C E processes can be seen in real-time or faster. Simulated Electropherograms can be acquired by taking the concentrations from the cell at the position of the detector. The simulated electropherogram is then compared with the experimental one to validate the simulation model and the simulation parameters.  82  §3.7 C o n c l u s i o n s The monotonic transport scheme is used to solve the hyperbolic part of the partial differential equation. With this implementation, numerical dispersion is avoided, and numerical diffusion is greatly reduced. The time and space increments do not need to be extremely small to achieve higher accuracy. At the same time, a circular cell arrangement and manipulation algorithm is implemented to avoid unnecessary calculations and provide an efficient use of the computer's memory. This new C E simulation model can simulate both equilibrium and nonequilibrium C E accurately in a reasonably short time. The simulation can be finished in a few minutes to a few hours depending mainly on the simulation parameters. The high efficiency of this model makes it possible to bridge the gap between theoretical modeling and real C E experiments, and to predict the peak shape and peak position at any given time of a C E process.  83  §3.8 References [I] Galbusera, C , Thachuk, M . , De Lorenzi, E., Chen, D. D. Y., Anal. Chem. 2002, 74, 19031914. [2] Chu, Y . - H . , Avila, L. Z., Gao, J., Whitesides, G. M . , Accounts of Chemical Research 1995, 28, 461-468. [3] Berezovski, M . , Krylov, S. N., J. Am. Chem. Soc. 2002, 124, 13674-13675. [4] Berezovski, M . , Nutiu, R., Li, Y. F., Krylov, S. N., Anal. Chem. 2003, 75, 1382-1386. [5] Bier, M . , Palusinski, O. A., Mosher, R. A., Saville, D. A., Science 1983, 219, 1281-1287. [6] Saville, D. A., Palusinski, O. A., AIChE Journal 1986, 32, 207-214. [7] Palusinski, O. A., Graham, A., Mosher, R. A., Bier, M . , Saville, D. A., AIChE Journal 1986,32, 215-223. [8] Dose, E. V., Guiochon, G. A., Anal. Chem. 1991, 63, 1063-1072. [9] Gas, B., Vacik, J., Zelensky, I., Chromatogr. 1991, 545, 225-237. [10] Ermakov, S., Mazhorova, O., Popov, Y., Informatica 1992, 3, 173-197. [II] Ermakov, S. V . , Bello, M . S., Righetti, P. G., J. Chromatogr. A 1994, 661, 265-278. [12] Dubroakova, E., Gas, B., Vacik, J., Smolkova-Keulemansova, E., J. Chromatogr. 1992, 623, 337-344. [13] Ikuta, N., Hirokawa, T., J. Chromatogr. A 1998, 802, 49-57. [14] Ikuta, N., Sakamoto, H . , Yamada, Y., Hirokawa, T., J. Chromatogr. A 1999, 838, 19-29. [15] Andreev, V. P., Pliss, N. S., Righetti, P. G., Electrophoresis 2002, 23, 889-895. [16] Okhonin, V., Krylova, S. M . , Krylov, S. N.,Anal. Chem. 2004, 76, 1507-1512. [17] Petrov, A., Okhonin, V., Berezovski, M . , Krylov, S. N., J. Am. Chem. Soc. 2005,127, 17104-17110. [18] Bowser, M . T., Chen, D. D. Y., J. Phys. Chem. A 1998, 102, 8063-8071. [19] Strikwerda, J. C , Finite Difference Schemes and Partial Differential Equations, Wadsworth & Brooks/Cole Advanced Books & Software 1989. [20] Press, W. H., Teukolsky, S. A., Vetterling, W. T., Flannery, B. P., Numerical Recipes in C++, Cambridge University Press, New York, N Y , U S A 2002. [21] Peng, X. J., Chen, D. D. Y., J. Chromatogr. A 1997, 767, 205-216. [22] Hawley, J. F., Smarr, L . L., Astrophysical Journal Supplement 1984, 55, 221-246.  84  Chapter 4  Behavior of Interacting Species in Capillary Electrophoresis Described by Mass Transfer Equation  85  Chapter 4 Behavior of Interacting Species in Capillary Electrophoresis Described by Mass Transfer Equation* §4.1 Introduction 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. The efficient ways of interpreting and implementing the mass transfer equation have been demonstrated in Chapter 3. The migration behavior of analytes can be simulated with modern computers in real time or faster. The dynamic complexation of solutes, while migrating at different velocities in a single phase, is a fundamental and crucial assumption for developing practical theories for capillary electrophoresis (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 [2, 3], longitudinal diffusion [4, 5], different path (eddy diffusion) [6], wall adsorption [7], nonhomogeneous electroosmotic flow [8, 9], nonhomogeneous field [10], Joule heating [11], and electromigration dispersion and anomalous electromigration dispersion [12]. Gas and his colleagues have given comprehensive reviews on the effect of these factors [13, 14]. In practice, all of the possible deteriorating phenomena except molecular diffusion can be either eliminated or significantly reduced with carefully designed experiments. However, * A version of this chapter has been published.  Fang, N.; Chen, D.D.Y. Behavior of interacting species in capillary electrophoresis described by mass transfer equation. Analytical Chemistry, 2006, 78(6), 1832-1840.  86  some fundamental and unavoidable factors can also determine the peak shape and peak width even though most of the aforementioned factors are eliminated. As demonstrated in Chapter 2, the migration behavior of individual solutes can produce peak shapes that are characteristic of the relative mobilities of the analyte (/z ), additive (ji ), A  P  and the complex (JUQ) formed in the separation process, and the peak maximums can be determined in each situation. There are six possible orders of the three mobilities: MP > Mc > MA , MP > MA > Mc > Mc > MA > MP, MA > Mc > MP , Mc > MP > MA >  A  N  D  ju > ju > ju . The initial stages of some processes have been demonstrated by Busch et al. A  ?  c  [15], 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 chapter, the mechanisms of all six cases are discussed in more detail, and demonstrate that not only the orders of the three mobilities, but 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 in Chapter 2, 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 chapter, the computer simulation program  SimDCCE,  is used to simulate affinity C E (ACE) experiments with various experimental conditions.  87  The foremost requirement to enable this study is that the simulation program must be accurate and efficient. SimDCCE  is 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. The simulation demonstrates that the analyte plug undergoes interesting processes of expanding and/or shrinking in each case. The well-established linear and nonlinear regression methods can be used to estimate binding constants from A C E experiments [16-18]. 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 [17]. In this chapter, by revealing the mechanisms under various experimental conditions, the deviation of the regression equations in some cases will be demonstrated.  88  §4.2 Mechanisms and Discussion §4.2.1 Simulation Conditions  In a typical A C E 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 free additive concentration ([P]) [19]. The regression methods for calculating binding constants are based on the assumption of the instant establishment of steady state conditions. The steady-state condition is that the concentration of the free additive in the analyte plug is equal to the concentration of the additive present in the B G E ([P]o) [20], that is, [P] = [P]o. However, in many C E processes, [P] is not constant throughout the C E process. In fact, it is often significantly higher or lower than [P]o. 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 C E model is to implement a stable and accurate finite difference scheme to solve the following mass transfer equation efficiently [21-23] dC dt  dC dz  dC 2  z  dz  2  (4.1)  89  in which C is zlj  the concentration of species i at position z and time t, E is the total local z  electric field at position z, //,. is the apparent mobility of the ion i, and D, is the diffusion coefficient of ion i. The simulation program implemented thefirst-orderforward-space scheme, the second-order monotonic transport scheme, and thefirst-orderfully explicit  dC ..  dC ,.  d C ,.  dt  dz  dz  scheme to evaluate the three partial differential terms, - — — , — — and  z  ~-, 2  respectively [24, 25]. The mechanism of A C E 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 B G E 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 4.1. A C E 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 only the most typical examples  90  in each of the six cases are covered in this chapter. The parameters used in all simulation runs are listed in Table 4.1. The diffusion coefficients o f all species are set to 1.0xl0" cm s"'. 6  2  Table 4.1. The experimental conditions for 18 scenarios.  [A] (mM) 0  [P]o (mM)  Length of Capillary (cm)  Length to Detector (cm)  Length of Injection Plug (cm)  Mc  MA  Voltage (kV)  (M- )  (xlO^cn^/Vs)  K 1  A-1  2.0  5.0  64.5  54.3  0.18  1.364  3.699  2.994  10  533  A-2  2.0  50  64.5  54.3  0.18  1.364  3.699  2.994  10  533  A-3 A-4  0.2  5.0  64.5  54.3  0.18  1.364  3.699  2.994  10  533  2.0  5.0  64.5  54.3  0.18  1.364  3.699  1.600  10  533  A-5  2.0  5.0  64.5  54.3  0.18  1.364  3.699  3.500  10  533  B-1  0.1  0.036  47  40  0.18  2.10  2.50  1.80  20  20000  B-2  0.1  0.36  47  40  0.18  2.10  2.50  1.80  20  20000  C-,  5.0  1.5  47  40  0.18  2.10  1.80  2.50  20  500  C-2  5.0  15  47  40  0.18  2.10  1.80  2.50  20  500  D-1  2.0  5.0  47  40  0.18  2.80  1.60  2.10  20  100  D-2  2.0  50  47  40  0.18  2.80  1.60  2.10  20  100  D-3  20  5.0  47  40  0.18  2.80  1.60  2.10  20  100  E-1  2.0  5.0  40  0.18  2.80  2.10  1.60  20  500  E-2  2.0  50  47 47  40  0.18  2.10  1.60  20  500  E-3  0.2  5.0  47  40  0.18  2.80 2.80  2.10  1.60  20  500  E-4  2.0  5.0  47  40  0.18  2.80  2.10  1.60  20  100  E-5  2.0  50  47  40  0.18  2.80  2.10  1.60  20  100  F-1 |  4.0  5.0  47  40  0.18  1.80  2.10  2.40  20  500  91  The computer simulation system SimDCCE, this study of A C E mechanisms. SimDCCE  described in detail in Chapter 3, is used in  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 SimDCCE  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 results. For all 18 scenarios, the time increment (At) is set to 0.001 s, and the space increment (Az) is set to 0.001 cm. 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.  §4.2.2 CASE A:  fi > ¥  v  n> c  //  A  The analyte and additive concentration profiles generated by the simulation program for five scenarios of Case A will be presented to illustrate the mechanism of the electrophoretic migration processes. The conditions in Scenario A - l are obtained from A C E experiments using/>nitrophenol as the analyte and /3-cyclodextrin as the additive, as listed in Table 4.1. The mechanism of this binding interaction in A C E has been discussed in Chapter 2 based on the experimental peak  92  shapes. The change o f the analyte concentration profile, which includes both free and complexed forms, is illustrated in Figure 4.1. During the first 14-second of analyte migration, a high concentration region is first formed at the back o f the analyte plug, because the complex 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 o f this process are illustrated in Figure 4.2.  Figure 4.1. A 3-D line plot demonstrates the change of peak shape (total concentration o f analyte) over the first 14 seconds o f the electrophoretic migration process o f Scenario A - l . Because o f the idle time and the ramp time, only the profiles o f the analyte plug after the first 2 seconds are plotted.  93  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 4.2A). The length of the analyte plug was estimated to be 0.18 cm according to Beckman Coulter P/ACE M D Q injection parameters provided by the manufacturer. This length is used as the length of the injection plugs for all scenarios presented in this chapter. Stage 2: The free additive keeps moving into the analyte plug, and a sweeping effect [26] takes place (Figure 4.2B and 4.2C). 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 4.2C), and the entire analyte plug except the edges now has the same [A ] (4.30 m M or 215% of [A]o) and [P] t  (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 4.2D). 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 4.2E). At the same time, the peak concentration is reduced from 4.30 mM to 0.89 mM (46% of [A]o). 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 B G E ([PJo^ 5 mM).  94  Figure 4.2. The simulated concentration profiles for Scenario A - l . (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.  95  At the end, the concentration profile of the free additive results in a "V" shape (Figure 4.2E), in which the lowest value at the peak position is 4.75 mM. Because [P] is significantly different from [P]o at the peak position, no true steady-state condition is reached at any point during the C E run. The negative additive peaks illustrated in Figure 4.2C-E should not be confused with the large additive trough shown on the right side of Figure 4.2F, which shows a larger portion on the capillary column at a time between Figure 4.2D 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 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 4.1. In Scenario A-2, [P]o is increased by 10 times from 5 mM to 50 mM, and all other conditions are identical to those in Scenario A - l . The capacity factor also increases by approximately 10 times. In other words, most of the analyte will be complexed with the additive not long after the C E process is started. After the sweeping effect (Stage 2), there is  96  not enough free analyte in the analyte plug to produce the front slope (Stage 3). Therefore, the resulting peak (Figure 4.3A) is symmetrical. Because [P] is now nearly equal to [P]o, the steady-state condition is fulfilled. In Scenario A-3, the only condition changed is [A] which is reduced from 2 mM to 0.2 0  mM. Because much less free analyte is present in the analyte band, [P] increases faster and eventually becomes closer to [P]o than in Scenario A - l : the lowest [P] in Figure 4.3B is 4.94 mM, compared with 4.75 mM in Figure 4.2E.  E  54.4 54.6 54.8 55.0 55.2 55.4 55.6  I  54.4 54.6 54.8 55.0 55.2 55.4 55.6  I 54.4 54.6 54.8 55.0 55.2 55.4 55.6  |0.05cm  1.22 cm  |  54.4 54.6 54.8 55.0 55.2 55.4 55.6  On-capillary Position (cm) Figure 4.3. The 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.  97  Although the ratio of [P] to [A] is increased by 10 times in both Scenario A-2 and A-3, 0  0  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. Scenario A-4 and A-5 demonstrate the effects of the differences between the three mobilities. In Scenario A-4, ju is closer to ju than fj. . Because the difference between ju c  A  ?  c  and fj. is smaller, the front slope is not visible (Figure 4.3C). In Scenario A-5, /u is closer k  c  to / / than jU . Because the difference between ju and /u is larger, a larger front slope p  A  c  A  shows up in Figure 4.3D.  §4.2.3 CASE B: / i > fi > // p  A  c  The mechanism of this case is different from Case A. The conditions in Scenario B - l , as listed in Table 4.1, 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 complex (Figure 4.4A). At the same time, a small amount of the additive mixes with the analyte plug at the front (right-hand side) despite /u > ju . One reason for such interaction to occur is diffusion of the additive from high p  A  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  98  Figure 4.4A) and the rear edge. This happens in all other cases, too. However, it is more significant in this scenario, because the difference between /u and ju is smaller (only A  ?  3 x 1 0 ~ c m V ~ V ), compared with 2.3xl0~ cm V~'s~ in Scenario A - l . 5  2  4  2  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 4.4B). There are two plateaus in the migrating plug: one is the free analyte plateau ([A] = [A]o = 0.1 mM), t  and the other is the plateau of the mixture of the free analyte and the complex ([A] = 0.052 t  mM). The small bump on the free analyte plateau is caused by the dissociation of the complex  99  in Peak 1 in Figure 4.4A, 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 4.4C). 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 4.4A travels from the right-hand side of the analyte plug to the left-hand 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 4.4D). Although the final [A ] profiles shown in Figures 4.2E (Scenario A - l ) and 4.4D t  (Scenario B-l) 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]o, while in Case B, [P] inside the analyte plug is greater than [P]o, as shown by the A-shaped traces of [P] in Figure 4.4D. The highest [P] is 4 . 3 x l O M , or 119% of [P] . _5  0  In Scenario B-2, the only change from Scenario B - l is the increase of [P]o by 10 times to 3.6xl0" M , as listed in Table 1. Most of the free analyte is complexed with the additive, 4  and there is not enough free analyte to develop the front slope. As a result, the final analyte plug is not a Gaussian peak, but a peak with a plateau in the middle, as shown in Figure 4.5.  100  0.6 [AJ  o  0.4  ><,  c o CD  i i  0.2  c:  CD O  rz o  "~-\.[PJ \  \  40.0 40.2 40.4 40.6  o  0.52cm  0.0 40.0  40.2  40.4  On-capillary Position (cm)  40.6  Figure 4.5. The simulated concentration profiles for Scenario B - 2 . The black triangle indicates the position where the migration time o f the analyte peak should be measured.  The profile o f free P (dash-dot line) has a similar shape with a height o f 3.626 x 1 0 M - 4  or 101% o f [P]o. Although no Gaussian peak is obtained in Scenario B - 2 , the migration time of the rear edge (left hand side) o f the analyte plateau can be used for the regression methods The reason is that a steady-state condition is reached at the rear edge o f the analyte plug shortly after the C E run begins, and [P] at the rear edge is nearly equal to [P]o for almost the entire process.  §4.2.4 C A S E C : H > H > c  A  H  v  The conditions o f Scenario C - l , as listed in Table 4.1, were created to demonstrate the mechanism in this situation.  101  Stage 1: The analyte plug catches up with the additive in front and forms the complex. Meanwhile, a small amount o f the additive mixes with the analyte plug at the rear edge to form a small peak 1 (Figure 4.6A) for the same reasons as mentioned in Case B . Stage 2: Unlike in Case A , there is no sweeping effect in this case. Because /u > ju , c  A  the complex on both edges migrates ahead of the free analyte (Figure 4.6B). The length of the analyte plug is extended at a relatively fast rate determined by the difference between ju and c  H . A t the same time, the bump resulting from Peak 1 moves toward the outlet within the K  analyte plug.  .2  0.00  3.4  0.05  3.6  3.8  0.10  4.0  0.15  4.2  4.4  0.20  4.6  0.3  40.4  0.4  40.6  40.8  0.5  41.0  41.2  0.6  41.4  41.6  On-capillary Position (cm) F i g u r e 4.6. The simulated concentration profiles for Scenario C - l .  102  Stage 3: As the run goes on, more additive comes into the analyte plug, forming more complex. Figure 4.6C 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]o, 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 4.6D) consists of a large slope and a plateau. The plateau exists because the difference between ju and / / c  A  is not big enough to transform the entire plug into a slope  In Scenario C-2 (Figure 4.7), [P]o is increased by 10 times. As expected, [P] within the analyte plug is now closer to [P]o (-15.12 mM or 101%). The front edge (right hand side) of the peak is saturated with the additive from the start of the electrophoretic migration process, which means a steady state condition is reached for this edge for the entire process.  40.5  41.0  On-capillary Position (cm) Figure 4.7. The simulated concentration profiles for Scenario C-2.  103  The distance traveled by the front edge is the length from the inlet of the capillary to the detector minus the length of the injection plug. Because [P] is much closer to [P] , the mobility difference between Peak 1 and the rest 0  of the analyte plug is smaller. Peak 1 has yet to reach the outlet edge, and remains as a small bump over the top of the plateau with this set of simulation conditions.  §4.2.5 C A S E D:  ju >{i >{i A  c  P  This case is similar to Case A , except that Case A's front slope becomes a rear slope. The conditions in Scenario D - l , as listed in Table 4.1, were created to illustrate the characteristics of this case. Stage 1: After the analyte plug is injected, and the C E process is started, the plug catches up with the additive in front to form the complex (Figure 4.8A). 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 left-hand side of the analyte plug (Figure 4.8B). 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 4.8C). 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 4.8D). The length of the analyte plug increases to 0.38 cm. The concentration of P in the analyte plug is always smaller than [P]o. The [A ] profiles shown in Figures 4.6D (Scenario C-l) and 4.8D (Scenario D-l) have t  similar rear slopes. However, the profiles of [P] are different. In Scenario D - l , [P] is lower  104  than [P] , having a V-shaped profile. In Scenario C - l , [P] is higher than [P] , having a A 0  0  shaped profile. In Scenario D-2, [P] is increased by 10 times. As more complex and less free analyte 0  exist in the analyte plug, the rear slope is much smaller (Figure 4.9A). In Scenario D-3, [A]o is increased by 10 times. It would take much longer time for the additive to sweep through the analyte plug. The concentration of P in the analyte plug increases much slower. Therefore, a longer rear slope is created (Figure 4.9B).  0.55  0.60  0.65  0.70  40.0  40.1  40.2  40.3  40.4  40.5  On-capillary Position (cm)  Figure 4.8. The simulated concentration profiles  for Scenario D - l .  105  40.0  40.2  40.4  '40.0  40.2  40.4  40.6  40.8  41.0 41.2  On-capillary Position (cm) Figure 4.9. The simulated concentration profiles for (A) Scenario D-2 and (B) Scenario D-3.  §4.2.6 C A S E E : / / > ju > ju A  P  c  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 4.10, and the conditions for Scenario E - l are listed in Table 4.1. 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 a small Peak 1 (Figure 4.1 OA) 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 4.10B). The analyte plug is further narrowed until the region containing only  106  0.00 6  0.05  0.10  0.15  0.20  0.15  0.20  0.25  0.30  0.35  0.40  T-  ID  5 •  \  4 • 3 • c  g  j j  2  -t—»  j  03 1 • 0 o c  j j j i  1  0 •  //  \ \  /  ./  \ 0.22cm >l  l-e  o O  1.0  6  1.1  1.2  1.3  1.4  T  5 43 • 2 1 05.8  2  6.0  6.2  6.4  6.6  6.8  7.0  H \ \  \ \ \  \ \  \ \ \  43  44  40  41  42  43  44  On-capillary Position (cm) Figure 4.10.  T h e s i m u l a t e d concentration p r o f i l e s for S c e n a r i o E - l .  the free analyte morphs completely into the plateau of the mixture (Figure 4. IOC). Peak 1 is left outside of the analyte plug, and migrates at a constant speed determined by the capacity factor (k' wK[P] ). Due to diffusion, Peak 1 gets smaller over time, and is hardly visible in 0  Figure 4.10D-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 4.10C-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 4.10D 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]o, 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. 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 migrates further (Figure 4.10E), and eventually, the gap completely disappears. The complex and the additive keep falling behind to create a region containing  108  higher [P] than [P] on the left side of the analyte plug (Figure 4.1 OF). Although the gap with 0  no additive has disappeared, there is still a big additive trough. After traveling 40 cm, the analyte plug is illustrated in Figure 4.10G, 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 4.1 OH. 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 4.10G. By carefully examining Figure 4.10G and 4.10H, one can 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: ju > A  /u . c  This peak splitting phenomenon can be further studied by changing the concentration of the analyte or the additive. In Scenario E-2, [P]o is increased to 50 mM while other conditions are kept the same as in E - l . The final peak for this scenario is displayed in Figure 4.11 A. The left peak is higher and sharper, and the right peak vanishes, because most of the analyte is in the form of complex and is lagging behind. In Scenario E-3, only the [A] is reduced by 10 0  times. The free additive can get to the left side of the analyte band much more easily. The positive additive peak is very small, and the additive trough is narrower. As a result, in Figure 4.1 IB, both the left and the right peaks become narrower.  109  In Scenario E-4, the only change is reducing the binding constant from 500 to 100. Because the free additive can now sweep through the analyte plug more easily, the left peak completely disappears in Figure 4.1 IC. The only peak with a large rear slope looks similar to the one shown in Figure 4.8D (Scenario D-l). Keeping K at 100 M" , [P]o is again increased 1  by 10 times in Scenario E-5 to increase the [C] to [A] ratio. The left peak reappears in Figure 4.11D.  40.0  40.5  41.0  40.0  40.5  41.0  41.5  42.0  42.5  On-capillary Position (cm) Figure 4.11. The simulated concentration profiles for (A) Scenario E-2, (B) Scenario E-3, (C) Scenario E-4, and (D) Scenario E-5.  110  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] invalid. If a much higher additive 0  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 C E 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.  §4.2.7 C A S E F : / / > / / > p. c  p  A  Cases A , C and F share one common property: ju > /u . Following our discussion of c  A  the similarities between the cases, one can expect Case F to show the characteristics of Case A + Case C. Scenario F-1 is simulated with the conditions listed in Table 4.1, and the profiles shown in Figure 4.12 agree with the prediction. Stage 1: The free additive enters the analyte plug to form the complex at the rear edge. At the same time, a small amount of the free additive mixes with the analyte at the front edge of the analyte plug to form a small Peak 1 (Figure 4.12A) for the same reasons as mentioned in Case B. Peak 1 is visible throughout the process.  Ill  5.6  5.8  6.0  6.2  6.4  6.6  40  41  42  43  On-capillary Position (cm) Figure 4.12. The simulated concentration profiles for Scenario F - l .  112  Stage 2: The free additive keeps moving into the analyte plug to form a plateau with a mixture of free and complexed analyte, and a sweeping effect, similar to the one in Case A , takes place to reduce the plug length, as shown in Figure 4.12B. A n additive gap is also formed in the process. Stage 3: The faster migrating complex then moves out of the analyte plug and into the additive gap (Figure 4.12C), and the length of the analyte plug increases. The analyte plug consists of two plateaus which are connected with a slope. Eventually, the additive gap disappears, and an additive valley with two sections, one V-shaped hole and one plateau, is formed (Figure 4.12D). The left plateau on the analyte plug vanishes, while the right plateau becomes wider. Stage 4: The complex at the front edge of the plug continues to move forward to produce another lower plateau where an additive plateau containing higher [P] than [P] is 0  also formed. The analyte plug keeps evolving and extending its length, and the profile reaching the detector is shown in Figure 4.12F. The distance between the analyte plug and Peak 1 gets smaller because the right edge of the analyte plug migrates at a higher speed with higher additive concentration. In this case, the left peak coexists with the negative additive trough, therefore, cannot be used to produce accurate results by the regression methods. The right peak develops after the gap of the free additive is filled. Once again, neither left nor right peaks are suitable for the regression methods. However, similar to Case E, if the additive concentration is high enough to fill the analyte plug quickly, it is possible to use the right peak in the regression methods to calculate approximate binding constants.  113  §4.2.8 Regression Methods Through the discussion of six cases and eighteen scenarios, we can conclude that one relatively accurate data point for the regression methods can be generated when high [P]o and low [A]o are used for the first four cases. To obtain a binding constant with regression method using binding isotherms, one needs to have 5 to 10 data points depending on the range of the isotherm covered by the experiment [27, 28]. High enough [P]o ensures a high capacity factor, and low enough [A] leads to a short time required for the free additive to go through the 0  analyte plug. For the last two cases, because of their complicated mechanisms, even though a high ratio of [P]o to [A]o is achieved, the error in the regression methods can still be fairly large if a wrong peak is chosen. As shown in Figure 4.10, 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. In the next chapter, a better way to determine binding constants is proposed: the enumeration algorithm with computer simulation of A C E , in which [P] in the analyte plug is not assumed constant. Therefore, it can generate accurate results under any given conditions as long as the simulation model is accurate.  114  §4.3 C o n c l u s i o n s 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 A C E 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 finite difference schemes, we are able to simulate many practical scenarios on the C E 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 C E experiments to extract physicochemical parameters, such as binding or dissociation constants. These results re-enforce that the additive concentration has to be much higher than the analyte concentration, if regression methods are to be used to obtain reliable constants based on the measured migration time. If this condition cannot be satisfied for practical reasons, the enumeration method based on properly implemented simulations should be considered. The results shown in the chapter should also provide insights into the phenomena observed in separation systems with much shorter columns, such as microfluidie devices.  115  §4.4 References [I] Giddings, J. C , Unified Separation Science, Wiley-Interscience Publication: New York 1991. [2] Delinger, S. L., Davis, J. M . , Anal. Chem. 1992, 64, 1947-1959. [3] Peng, X., Chen, D. D. Y . , J. Chromatogr. A 1997, 767, 205-216. [4] Jorgenson, J. W., Lukacs, K. D., Anal. Chem. 1981, 53, 1298-1302. [5] Kenndler, E., Schwer, C , Anal. Chem. 1991, 63, 2499-2502. [6] Kasicka, V., Prusik, Z., Gas, B., Stedry, M . , Electrophoresis 1995,16, 2034-2038. [7] Schure, M . R., Lenhoff, A. M . , Anal. Chem. 1993, 65, 3024-3037. [8] Towns, J. K., Regnier, F. E., Anal. Chem. 1992, 64, 2473-2478. [9] Kok, W. T., Anal. Chem. 1993, 65, 1853-1860. [10] Keely, C. A., Vandegoor, T., McManigill, D., Anal. Chem. 1994, 66, 4236-4242. [II] Knox, J. H., Grant, I. H . , Chromatographia 1987, 24, 135-143. [12] Gebauer, P., Bocek, P., Anal. Chem. 1997, 69, 1557-1563. [13] Gas, B., Stedry, M . , Kenndler, E., Electrophoresis 1997, 18, 2123-2133. [14] Gas, B., Kenndler, E., Electrophoresis 2000, 21, 3888-3897. [15] Busch, M . H. A., Kraak, J. C , Poppe, FL, J. Chromatogr. A 1997, 777, 329-353. [16] Rundlett, K. L., Armstrong, D. W., J. Chromatogr. A 1996, 721, 173-186. [17] Bowser, M . T., Chen, D. D. Y., J. Phys. Chem. A 1998,102, 8063-8071. [18] Peng, X. J., Bowser, M . T., Britz-McKibbin, P., Bebault, G. M . , Morris, J. R., Chen, D. D. Y., Electrophoresis 1997, 18, 706-716. [19] Bowser, M . T., Chen, D. D. Y . , Electrophoresis 1997,18, 2928-2934. [20] Galbusera, C , Thachuk, M . , De Lorenzi, E., Chen, D. D. Y . , Anal. Chem. 2002, 74, 1903-1914. [21] Bier, M . , Palusinski, O. A., Mosher, R. A., Saville, D. A . , Science 1983, 219, 12811287. [22] Saville, D. A., Palusinski, O. A., AIChE Journal 1986, 32, 207-214. [23] Palusinski, O. A., Graham, A., Mosher, R. A., Bier, M . , Saville, D. A., AIChE Journal 1986, 32,215-223. [24] Hawley, J. F., Smarr, L . L., Astrophysical Journal Supplement 1984, 55, 221-246. [25] Press, W. FL, Teukolsky, S. A., Vetterling, W. T., Flannery, B. P., Numerical Recipes in C++, Cambridge University Press, New York, N Y , U S A 2002. [26] Quirino, J. P., Terabe, S., Anal. Chem. 1999, 71, 1638-1644. [27] Bowser, M . T., Chen, D. D. Y . , J. Phys. Chem. A 1998,102, 8063-8071. [28] Bowser, M . T., Chen, D. D. Y . , J. Phys. Chem. A 1999,103, 197-202.  116  Chapter 5  Enumeration A l g o r i t h m for Determination of Binding Constants in Capillary Electrophoresis  Chapter 5 Enumeration Algorithm for Determination of Binding Constants in Capillary Electrophoresis* §5.1 Introduction The high throughput evaluation of binding affinities of a large collection of compounds to target biomolecules is one of the bottlenecks in today's drug design, discovery, and evaluation processes. Because capillary electrophoresis (CE) has a well-controlled physical and chemical environment, it can be used to determine binding or dissociation constants of affinity interactions. When C E is used for these studies, it is often referred to as affinity C E or ACE. Linear and nonlinear regression methods have been used to determine binding constants from A C E experimental data as shown in Section 1.3.1 or other references [1-4]. However, because the average mobility of the analyte is used in the regression methods, the errors associated with these methods can be large if the concentration of the additive in the background electrolyte (BGE) is not much higher than the concentration of the injected analyte. In this situation, the mobility of the analyte changes during the course of the electrophoretic migration [3-5]. Moreover, 6 to 10 BGEs with different additive concentrations are normally needed to carry out the experiment for the regression methods, which leads to -18 to 30 C E runs if each experiment is done in triplicates. In addition, the correction factor for higher concentration additives and the mobility of free analyte also need to be measured multiple times. Therefore, a faster and more accurate approach to determining  A version of this chapter has been published. Fang, N.; Chen, D.D.Y. Enumeration algorithm for determination of binding constants in capillary electrophoresis. Analytical Chemistry, 2005, 77(8), 2415-2420.  118  binding constants using A C E is essential for the high throughput screening of thousands of possible compounds in a combinatorial library. Differential equations describing the electrophoretic migration behavior of the analyte in this situation and the approach to obtain their analytical solutions have been demonstrated [6]. However, finding the analytical solution for such differential equations is often complicated. Computer simulation provides a unique opportunity for accurately describing the behavior of analytes in a column when the differential equation is translated into finite difference schemes. If the electrophoretic migration behavior can be accurately predicted by computer simulation, the data from an A C E experiment can be used to estimate binding constants. The accuracy of this approach depends upon the accuracy of the simulation model. On the other hand, the speed of the simulation is also important, due to the fact that thousands of simulation runs are required to determine a single binding constant. Several computer simulation models of C E (or ACE) are presently available [7-11]. However, none of those models are suitable for determining binding constants because of the high demand on computing power and computing time. An efficient algorithm to predict the electrophoretic migration process and to determine the migration time based on peak maximums and peak shapes has been demonstrated in Chapter 2. This algorithm is also known as the One-Cell model because only one cell is monitored in the process of predicting peak maximums. In this chapter, a method for the determination of binding constants by computer simulation of A C E based on the One-Cell model is described.  119  §5.2 Experimental Section §5.2.1 A C E Experiments The 1:1 interaction between p-nitrophenol (analyte, A) and /3-cyclodextrin (additive, P) to form a complex (C) is used as a model system. The experiments were carried out on a Beckman Coulter ProteomeLab PA800 (Beckman Coulter Inc., Fullerton, CA) with a built-in PDA detector. A 64.5 cm long (54.3 cm to detector) x 50 um I.D., fused-silica capillary (Polymicro Technologies, Phoenix, AZ) was used. /3-Cyclodextrin (Sigma, St. Louis, MO) was dissolved in 160 mM borate buffer (pH 9.1) at various concentrations ranging from 1.0 mM to 15.0 mM. />Nitrophenol (Fisher, Fair Lawn, NJ) was dissolved in the same borate buffer at a concentration of 2.0 mM. In each A C E run, a/>nitrophenol solution was injected to form a narrow analyte plug in the inlet of the capillary which was filled with the /3-cyclodextrin solution at various concentrations. The initial concentrations of the analyte (p-nitrophenol) and the additive (/3cyclodextrin) are denoted as [A]o and [P]o, respectively. A 10 kV potential was applied on the capillary, and a temperature of 20 °C was maintained throughout the experiments. Each A C E experiment with different [P]o was repeated three times. Methanol was used as the electroosmotic flow (EOF) marker.  §5.2.2 Mobility of the Analyte To measure the electrophoretic mobility of the free analyte (/J ) , />nitrophenol was epA  injected into the capillary filled with neat borate buffer ([P] = 0), and was driven through by 0  120  a potential of+10 kV. This measurement was carried out before each A C E run, and a total of 24 measurements were taken. Mep.A is normally calculated using the following equation  in which L and LA are the length of the capillary and the length to the detector, respectively, V C  is the potential, t  eo  is the migration time of the EOF marker, and t is the migration time of the m  analyte. To accurately determine the mobility, the idle time (t\, 0.034 min), which is a pause after the instruction is given and before the voltage is applied, and a ramp time (t 0.17 min), u  which is the time required for the voltage to reach the set value, have to be accounted for. The accuracy of eq 5.1 can be improved by adding t\ and t into the equation, and the free analyte r  mobility can be obtained from:  Because the physical condition of the B G E changes after /?-cyclodextrin is added, ju  epA  for each A C E run has to be further corrected by either using neutral EOF markers [12, 13] or using a correction factor which converts the observed net electrophoretic mobility to an ideal state in which the additive concentration approaches 0 [1, 14]. In this chapter, the ju  tpA  input into the simulation program is calculated from the y.  A  values  value measured from an  additive-free solution multiplied by the ratio of the migration times of the EOF markers, which gives the n  A  value under the specific conditions of the specified additive  concentration.  121  §5.3 Enumeration Method §5.3.1 Overview The electrophoretic mobility of the analyte in the presence of the additive (/J*) can be calculated by [14] K  = / > e p , A +/c/"ep,C  in which f  A  and f  c  ( - ) 5  are the fractions of the free analyte and the complex, and ^  e p C  3  is the  electrophoretic mobility of the complex. If a highly concentrated additive solution is used as BGE, f  c  approaches 100%, and ju  epC  can be assumed equal to  which is obtained by  measuring the migration time of the analyte peak. However, this is not always achievable due to limits in solubility or the cost of the additives..Therefore, fj,  c  is often the other unknown  variable in addition to the equilibrium constant (K) [1, 2]. The existence of the two unknowns and the complicated differential equations make the mathematical approach to determine binding constants a challenging task [6]. The approach described in this work utilizes the tremendous power of modern computers to implement an enumeration algorithm for obtaining the binding constant of affinity interactions from less restrictive experimental conditions. The general idea of the enumeration algorithm is that all possible solutions to a given problem are investigated in the process of finding the real solution(s). In our case, all possible combinations of the complex mobility and the binding constant that could result in an analyte migration time are investigated in the effort to find the values that are common in all conditions. This enumeration approach itself is not related to any particular simulation program, but the accuracy of the determined binding constants relies heavily on the accuracy of the  122  simulation model. The simulation program used in this approach must have the ability to not only simulate the electrophoretic migration, but also consider the binding interaction between the analyte and the additive. The simulation program does not need to generate the shape of the entire analyte peak; instead, only the simulated migration times of the peak maximums are required. Chapters 2 and 4 have shown that the shape of an analyte peak is directly related to the relative mobilities of the free analyte, the free additive and the complex. Chapter 2 demonstrates that the peak maximums often reside at the rear or front edge of the initial analyte plug according to the relative mobilities, and that the migrational behavior of peak maximums can be described mathematically by the One-Cell model. In Chapter 3, a more accurate simulation algorithm (the Multi-Cell model) and its implementation are introduced. However, the Multi-Cell model demands a lot more computing power and longer computing time than the One-Cell model. Although both models were implemented in the simulation program SimDCCE,  only the  One-Cell model is used in this chapter because of its fast speed and its ability of predicting the migration times of the peak maximums within the required accuracy. SimDCCE,  written in Microsoft Visual C++ 6.0, is capable of handling the entire  procedure for determining binding constants. Figure 5.1 summarizes the procedure: SimDCCE  is first used to simulate A C E experiments and to generate 3-D surfaces; Then  SimDCCE  is used again to extract 2-D curves from the 3-D data files; Finally, SimDCCE  or  other data analysis software, e.g. SigmaPlot, finds the intersections of the 2-D curves and gives the binding constant and associated standard deviation.  123  Simulation  Figure 5.1. Flowchart of the procedure for determining binding constants.  SimDCCE  requires all experimental conditions: the length of the capillary (L = 64.5 C  cm), the length to the detector (LA = 54.3 cm), the voltage (V=+10 kV), the concentration of the injected analyte ([A] = 2.0 mM), the concentration of the additive present in B G E ([P]o = 0  1.0, 2.0, 3.0, 4.0, 5.0, 10.0, 12.5, 15.0 mM), the migration time of the EOF marker (t  eo  from  each A C E experiment), the corrected electrophoretic mobilities of the free analytes (^ and the electrophoretic mobility of the free additives (ju  P  epA  ),  = 0 for neutral /?-cyclodextrin).  §5.3.2 Step 1: 3 - D Surfaces When the experimental conditions and required values for various constants are sent to the simulation program, migration times are generated and plotted according to the changes of the binding constant and the complex mobility. When the generated migration time  (T ) sim  124  agrees with the experimental migration time (T ), the pair of K and ju exp  c  values are  collected for further consideration. The binding constant and the complex mobility of a given binding interaction can be estimated in certain ranges, which may be wide in the beginning but can be narrowed quickly as one gets more knowledge about this binding interaction. The simulation program has the ability to scan the K vs p  c  plane at user-defined intervals on both axes within the pre-  estimated ranges. Each pair ofK and / /  e p C  given to the simulation program will generate a  simulated migration time. After scanning the entire K vs p  plane, a 3-D surface can be  epC  constructed with K, p  and the simulated migration time (T ) as the x-, y-, and z-axes,  epC  sim  respectively. An example is shown in Figure 5.2, which is plotted using SPSS SigmaPlot. SimDCCE  is used to generate 3-D surfaces. In addition to all experimental conditions,  users have to set the ranges and intervals of K and ju  ep c  Mep A ~ Mep,c >  o  r m  e mobility ratio p  ep  A  /p  c  (or the mobility difference  ) to be scanned by SimDCCE.  Because the  previous studies have shown that the binding constant for the 1:1 interaction between pnitrophenol and /3-cyclodextrin was between 500 and 600 M" , the range and interval of K 1  were set at 200 ~ 800 and 5 M" , respectively. The range of p 1  K  - ju  c  was  (-2.4 ~ -1.0) x 10" cm V"'s"' with an interval of 5 x 1 0 " c m V - y . 4  2  7  2  Because of the highly efficient algorithm, it usually takes only a few minutes on a laptop PC powered by an Intel Centrino™ 1.8 GHz CPU to generate a 3-D surface as shown in Figure 5.2 that contains over 33,000 simulated A C E runs. Therefore, this method is still feasible even if a very large range needs to be tested when the binding constant is completely unknown.  125  Figure 5.2. A 3-D surface which is cut through by a plane to produce a 2-D curve which is then projected onto the bottom plane. (1) The 3D surface, (2) the cutting planes, (3) the intersection between 1 and 2, and (4) the projected intersection curve.  Multiple A C E runs were carried out for each initial additive concentration. All these runs have to be simulated separately. We cannot run just one simulation with the average experimental migration time and the average EOF time of the A C E runs of the same concentration settings. Because the migration times for the EOF markers for individual A C E runs are different, the migration time of the analyte plug can change significantly. Thus, taking averages would give rise to a new source of error.  126  Each simulation run will generate one output data file. The output is customizable, and it must contain at least three columns: the simulated migration time, the binding constant and the complex mobility from which the simulated migration time is generated.  §5.3.3 Step 2: 2-D Curves The 3-D surface described in Figure 5.2 can be intercepted with planes of constant migration times. The planes are selected based on the experimental migration time z  = sim T  = ?exp ±  error  (5.4)  in which error results from the measurements made in the experiment, as well as the simulation program itself. The intersection between these planes and the 3-D surface contains all pairs of K and / i  e p C  which can generate migration times equal to the experimental  migration time with some associated errors. The projection of all those points within the intersection on the K vs //  c  plane gives a 2-D plot (Figure 5.2).  The data extraction function of SimDCCE  is used to extract 2-D curves from the output  data files. By default, the first column of a data file is the simulated migration time. Any row with afirst-columnvalue close to the experimental migration time within user-defined errors will be collected into a new output file, which can be imported into any data analysis programs, such as SigmaPlot, to plot the 2-D curves.  §5.3.4 Step 3: Intersections of the 2-D Curves If another A C E experiment is performed with identical conditions except for the initial concentrations of the analyte or additive, the migration time resulting from the same K and  127  Aep.c values w i l l be different from the previous one, creating a different 3-D surface. A new 2-D curve can be obtained on the basis o f the new experimental migration time. Because the two mobility should have the same values. Therefore, the two 2-D curves have to intersect. The coordinate o f this intersection gives the values for the binding constant and the complex experiments used the same analyte and additive, the binding constant and the complex mobility. If more experiments are performed, the 2-D curves extracted from all 3-D surfaces  800 H  [P] =1mM 0  O  [P] = 2mM 0  [P] = 3 m M 0  700  H  [P] = 4mM 0  [P] = 5mM 0  [P] = 10mM 0  600  [P] = 12.5mM 0  [P] = 15mM 0  500 H  400  4  300  200 5  6  ^epA^ep.C  Figure 5.3. A 2-D graph generated from SimDCCE  with // / / / ep A  e p C  as the x-axis.  [p-Nitrophenol] = 2 m M . Each curve is composed o f data points. Each color o f the curves corresponds to one additive (/j-cyclodextrin) concentration. Three shapes (circle, square, and triangle) in one color correspond to three A C E runs under identical experimental conditions. The red rectangle indicates the intersection o f eight sets o f curves.  128  should all intersect at one point, because there is only one true binding constant for a given binding interaction under specified conditions. Because the conditions of the capillary and its environment could change from one run to another, the calculated ratio of ju  epA  to /j  c  /J.  C  as well as  , instead of /u  epC  ju  ep  A  may vary slightly for any two A C E runs. The  , is used as the x-axis. The free analyte mobilities  obtained from the runs just before an A C E run is used to calibrate the changes caused by slight temperature change, solvent evaporation, or other changes in capillary conditions. Figure 5.3 shows all 2-D curves generated by the simulation programs for the set of experiments with the initial concentration of /?-nitrophenol at 0.2 mM with the x-axis of A'ep.A/-"ep.C •  To obtain the binding constant and complex mobility, only two of these 2-D curves are needed, as demonstrated in Figure 5.4. Each of the curves can be fitted by  in which a, b, and c are three constants defining the curve. The solutions of x and y of any two equations give the values for / / //u epA  spC  and K.  Depending on the information on the binding isotherm carried by the two experiments, however, errors on the obtained values can be different. When the experimental conditions that produced the two curves are significantly different, as depicted in Figures 5.4A-C, the intersections are very clear; however, when the experimental conditions are similar, as shown in Figure 5.4D, the two curves become closer in shape, and the intersection can be less clearly defined.  129  2  3  4  5  6  7  8  2  3  4  5  6  7  8  ^ep.A^ep.C  Figure 5.4. The intersections of two sets of curves. (A) 1 and 15 mM, (B) 2 and 12.5 mM, ( C ) 3 and 10 mM, and (D) 4 and 5 mM.  As discussed in two earlier papers [3,4], the experiments have to cover a significant portion of the binding isotherm for an affinity interaction in order to accurately determine the binding constant and complex mobility. As shown in Figure 5.5, the concentrations used to generate curves in Figure 5.4A-C are at significantly different portions of the binding isotherm, while the concentrations used in Figure 5.4D are too close to provide enough information. It should be noted that the constants could still be obtained in this case, albeit with a larger uncertainty, because the simulation program describes the analyte migration at any given moment, from a plug of pure analyte to that of fully equilibrated with the additive molecules in the B G E .  130  1.6e-4  0.000  0.002  0.004  0.006  0.008  0.010  0.012  0.014  0.016  [P] (M)  Figure 5.5. The binding isotherm.  Table 5.1. Solutions, K(M' ) 1  and ju /jU tpA  , of every two equations representing the curves  epC  shown in Figure 5.3. Additive Concentrations (mM)  Additive Concentrations (mM)  2  3  4  5  10  12.5  15  1  513, 3.37 543,3.15 526, 3.31 529, 3.28 530, 3.27 529, 3.28 525, 3.31  2  550, 3.12 513, 3.37 521, 3.31 525, 3.28 524, 3.29 519, 3.33  3  461, 3.69 497, 3.42 515, 3.31 514, 3.31 508, 3.34  4  550, 3.20 539, 3.25 533, 3.27 524, 3.32  5  534, 3.26 529, 3.28 516, 3.33  10  511,3.31 482, 3.40  12.5  436, 3.50  15  131  Table 5.1 lists the solutions o f every two equations representing the curves. The solutions shown in the shaded cells should not be used because these pairs o f curves are too similar to give clear intersections. The average K and /J  ep A  //^  ep c  in the remaining cells are  524±8 M " and 3.30±0.03, respectively. 1  Although two sets of experiments with two different additive concentrations are often enough to determine the binding constant and complex mobility, more experiments could provide added assurance and help to estimate errors in the obtained constants. Figure 5.6 shows three curves generated from the experimental migration times obtained when the additive concentrations are 1, 4, and 15 m M . Three different sets o f solutions can be obtained from the three equations representing the curves.  Figure 5.6. The intersection o f three sets of curves: 1, 4, and 15 m M . Three different sets o f solutions (K and p  ep  A  />  representing the curves. The average K and /J  e p > c  ep A  />  ) can be obtained from the three equations were 525 M " and 3.31, respectively. 1  e p C  132  §5.3.5 Complete Procedure Using  SimDCCE  The 3-step procedure of finding the K value from a set of A C E experiments can be done completely within SimDCCE. The screenshot of SimDCCE to do the first two steps (the generation of 3-D surfaces and extraction of 2-D curves) is shown in Figure 5.7. In addition to the experimental conditions, one must also set the columns in the 3-D surface file and the extraction criteria for the 2-D curve.  W  QUI®  BCD5_1.par - SimDCCE2 File  View  Heb  Use enumeration algorithm to estimate binding interaction properties, - • r  Property  E  3D Surface and 2D C u m B Column 1 in 3D Surface Hie Content Simulate B Extration Conditions Method Base '•.Error ••/-Error Mm  ^  Q  RI 0  B Column 3 in 3D Surface file Content K (Bindin B Extration Condttions B Column 4 in 3D Surface file 19 Column 5 in 3D Surface file B Sort by Columns 2D Curve  2D Curve  SjExperimentalCo... ^-Simulation Para..  5f  View  MutiCdl Model | Q OneCdl Model |  CPU Normal Usage  START  Base an 1385 S8  0 Max 0 B Column 2 in 3D Surface file Content Uep,a/U... H Extration Conditions  B  Choose Simulation Model  View 2D Data  3D Data  v  MobilityRatioJJa/llc  K/logK  1799 3400CO 1786 140000 1773 690000  2 2.05 21  200 200  1762 040000  2 15 2.2 2.25  1721 490000  23 235  200 200 200 200 200  24 2 45  200 200  25  1688 740000 1681 54000(1  255 ?fi  200 200 200  •  \  Single R u n : Multi-Cell Model  rows  200  1712 690000 1704 290000 1696 340000  44 4  get 2D Curve  0 Display 50  SimulatedPeaks  1751 040000 1740.640000 1730 790000  *J  \Enumerat»on Algorithm  j  Enumeration A l g o r i t h m for Estimating Binding Constants Mi approach utilizes the power of modern computers to implement an enumeration algorithm for estimating binding constants cf affinity interactions from less restrictive experimental conditions in affinity C E The general idea cf the enumeration algorithm is that all possible solutions to a given problem are investigated m the process cf finding the real solutionis) In this particular case, all possible combinations of the complex mobility and the binding constant that could result in an experimental analyte migration time are  Find K From Intersections  /(  load.  | Ready  Figure 5.7. The screenshot of SimDCCE for Steps 1 and 2. Region 1: Set the parameters for the 3-D surface and the extraction criteria for the 2-D curve. Region 2: Choose between the two models, start the process to generate a 3-D surface, and extract the corresponding 2-D curve. Region 3: Browse the generated 3-D and 2-D data files.  133  IE BCD5_1.par - SimDCCE2 File  View  HeJp  y ?) .  L: Settings  Value  Property Content  Simulate  H Extration Condrtions E  Column 2 in 3D Surface Rle Content  i  Uep,a/U...  E) Ext ration Conditions •  y  Ranges •  El  Curves  jd Add Curves  Kfflindin  470  < Y< 550  Disable range settings and s h o w all solutions  Column 3 in 3D Surface File Content  <x< 10  0 1  8CD1_1  BCD1.2  BCD1_3  BCD1_1  ExtrationConditions  E) Column 4 in 3D Surface Rle  I M Remove Row  BCD1_2 BCD1_3  El  BCC2_1  522,8  BCD2_2  3.308 503.3  3.308, 503.3  3.074, 538,9  BCD2_3  3.655,474  3.656, 473.9  3.355,507.4  •  Remove All  2D Curve Generate 2D Curve . YES  mM  Column X  Column 2  Column Y  Column 3  Column Z  N/A  Parameter a  0 00633  Parameter b  -0010087  Parameter c  0.00503  Chi~2  384.942  Refresh  v  3.308, 5i 3.074, 5  BCD3_1  3 159 519 3  3.051. 541,9  3.19. 513.4  3.029. 5  BCD3_2  3.227, 511.6  3.113,534  3.335, 493.5  3.152, 5  3.142, 530,5  3.41, 484.4  BCD3_3  3.26, 508.1  3.261. 508.1  BCCH_1  3396, 494.9  3.396, 494.9  BCD4_2  3.317, 502,4  .3.317, 502.4  BCD4_3  .3.308, 503.3  .3.308 , 503.3  3.215, 5 3.443, 4t  3.194, 512.8  BCD5_2 .<!  ^.Simulator! Para... |  3.129, 522.8  BCDS.l  dilution Condttions  JjFxperimenlalCo...  BCD2. 3.308, 51  —  0 Column 5 in 3D Surface Rle Sort by Columns  F:2:_: 3.129,522.8  M 4 • H\  ^  3.137 5 >  i  Single Run: Muiti-CcM Model  X  Enumerate-sigo't--  \ Find K From Intersections ^  Load.  Ready  Figure 5.8. The screenshot of SimDCCE  for Steps 3.  Region 1: Generate the parameters of the regression function for a 2-D curve. Region 2: Find the solution between every two curves. Region 3: Calculate the two unknown variables and the associated standard deviations.  The screenshot of SimDCCE  for Step 3 is shown in Figure 5.8.  After a 2-D data file is generated, SimDCCE  can perform the Levenberg-Marquardt  nonlinear regression [15] on the data points to get the three parameters in eq 5.5. Then all the regression equations for the corresponding 2-D curves can be loaded into a table (Region 2 of Figure 5.8), and SimDCCE  solves for the intersection between every two curves. The  acceptable ranges of the two unknown variables can be set to filter out the "wild" intersections as demonstrated in Figure 5.4D. Finally, the averages of the two variables and the associated standard deviations are calculated and shown in Region 3 of Figure 5.8.  134  The average K value shown as Y in Region 3 of Figure 5.8 is 508 with a standard deviation of 16. This value is not exactly the same as that (525) shown in Figure 5.6. It is not surprising because the value obtained in Figure 5.6 is the solution from only three curves.  §5.3.6 Enumeration Method vs Regression Methods For the purpose of comparison, the binding constant for the 1:1 interaction between pnitrophenol and /3-cyclodextrin was also determined using the regression methods. The binding constant obtained from nonlinear regression on the same set of data is 533±7 M" , and from the three linear regression methods, x-reciprocal, y-reciprocal and 1  double reciprocal [1], the values are 526±6, 547±10 and 534±6 M" , respectively. These 1  values are slightly different due to different variances within these regression methods [3]. All of these values agree with the one generated by the enumeration method. There are three advantages of the enumeration method over the regression methods. First, in theory, the enumeration method is based on the implemented simulation model of A C E and takes into account every moment of the analyte migration. The migration rate could change due to the changes in local conditions, such as the concentrations of the free analyte and free additive, before equilibrium is reached. On the other hand, all four regression methods are based on the average analyte mobility, assuming the analyte migrates with a constant velocity throughout the capillary. If [A]o is not much smaller than [P]o or the length of the injection plug is not infinitely narrow, the migration rate of the analyte plug could differ significantly from the average velocity, especially at the beginning of the A C E process when the equilibrium is not reached. Therefore, the binding constant and the complex mobility obtained from the enumeration  135  method should be more accurate and less dependent on experimental conditions as long as the conditions are given to the simulation program. Second, using the enumeration method, it is possible to obtain a binding constant with only two or three sets of the concentrations of the analyte and the additive as demonstrated in Figures 5.4 and 5.6. While using the regression methods, ~6 to 10 different concentrations of the additive are required. The enumeration method has the potential to be used for high throughput screening of combinatorial libraries. Finally, the enumeration method is applicable to other types of interactions. The regression methods generally cannot be used for higher order binding interactions except for highly cooperative or noncooperative binding [16]. On the other hand, as long as an accurate simulation programs can be written, which in turn depends on the accuracy of the equations describing the analyte behavior, the enumeration method can be used. For a 1:2 interaction, one could in many cases use low concentrations of a higher order species so that the interaction is limited to a 1:1 ratio to obtain first order binding constant, then use higher concentrations of that species for the interaction to obtain the second binding constant.  136  §5.4 C o n c l u s i o n s The enumeration algorithm for determining binding constants itself is sound and solid. As long as the computer-simulation model is accurate, the algorithm can generate accurate results. Although the One-Cell model for A C E simulation can only provide the peak positions, because of its single-cell-based calculation, the simulation speed is extremely fast. Thousands of A C E runs can be simulated within a few minutes, making the enumeration algorithm feasible for the determination of binding constants while simplifying the experimental procedures of A C E . Accurate results can be obtained regardless of the ratio of the initial additive and analyte concentrations. To minimize the adverse effects of changeful capillary conditions, each set of experiments should be done continuously, and the experimental conditions must be controlled. The simulation programs written in-house are combined with commercial data analysis programs to estimate binding constants by plotting 3-D surfaces, extracting 2-D curves, and finding the intersections among the 2-D curves. Using the K vs /u  sp A  ///  e p c  plane can help to  eliminate the effect of the differences in capillary conditions during the A C E experiment. Computer-aided research has become an increasingly important part of chemistry today. The idea of using the enumeration algorithm in determining physical and chemical properties by computer simulation has the potential to solve other chemistry problems when a direct mathematical approach is too difficult or not accurate enough when certain assumptions have to be used. The enumeration approach is equally applicable to binding studies using techniques such as NMR, chromatography, and optical methods. It can also be used in all processes that can be described by rectangular hyperbolas, such as Michaelis-Menten kinetics.  137  §5.5 References [I] Peng, X. J., Bowser, M . T., Britz-McKibbin, P., Bebault, G. M . , Morris, J. R., Chen, D. D. Y., Electrophoresis 1997,18, 706-716. [2] Britz-McKibbin, P., Chen, D. D. Y., J. Chromatogr. A 1997, 781, 23-34. [3] Bowser, M . T., Chen, D. D. Y., J. Phys. Chem. A 1998, 102, 8063-8071. [4] Bowser, M . T., Chen, D. D. Y., J. Phys. Chem. A 1999,103, 197-202. [5] Bowser, M . T., Sternberg, E. D., Chen, D. D. Y., Electrophoresis 1997, 18, 82-91. [6] Galbusera, C , Thachuk, M . , De Lorenzi, E . , Chen, D. D. Y . , Anal. Chem. 2002, 74, 1903-1914. [7] Dose, E . V., Guiochon, G. A., Anal. Chem. 1991, 63, 1063-1072. [8] Hopkins, D. L., McGuffm, V . L., Anal. Chem. 1998, 70, 1066-1075. [9] Ikuta, N., Hirokawa, T., J. Chromatogr. A 1998, 802, 49-57. [10] Ikuta, N., Sakamoto, H . , Yamada, Y., Hirokawa, T., J. Chromatogr. A 1999, 838, 1929. [II] Andreev, V . P., Pliss, N . S., Righetti, P. G., Electrophoresis 2002, 23, 889-895. [12] Shibukawa, A., Lloyd, D. K., Wainer, I. W., Chromatographia 1993, 35, 419-429. [13] Rundlett, K. L., Armstrong, D. W., J. Chromatogr. A 1996, 721, 173-186. [14] Bowser, M . T., Chen, D. D. Y., Electrophoresis 1997,18, 2928-2934. [15] Press, W. H . , Teukolsky, S. A., Vetterling, W. T., Flannery, B. P., Numerical Recipes in C++, Cambridge University Press, New York, N Y , U S A 2002. [16] Bowser, M . T., Kranack, A. R., Chen, D. D. Y.,Anal. Chem. 1998, 70, 1076-1084.  138  Chapter 6  Systematic Optimization of Exhaustive Electrokinetic Injection Combined with Micellar Sweeping in Capillary Electrophoresis  Chapter 6 Systematic Optimization of Exhaustive Electrokinetic injection Combined with Micellar Sweeping in Capillary Electrophoresis* §6.1 Introduction Capillary electrophoresis (CE) has superior separation power, capable of rapid analysis of molecules of similar structure. However, due to the short optical path-length and small sample volume, the sensitivity of C E is often insufficient for samples at low concentrations. Efforts to improve concentration sensitivity usually fall into three categories: (1) Increase the detection path-length by modifying the capillary shape [1-3]; (2) Use more sensitive detection methods, such as laser induced fluorescence (LIF) and linear photodiode array detectors [4]; (3) Introduce a large volume of sample into the capillary, but still obtain a narrow sample zone at the detector by stacking or sweeping based on the differences in the velocity of analyte molecules at different locations of the capillary. Online concentration techniques have been increasingly employed, thanks to their simple experimental setup, two to three orders of magnitude improvement in concentration sensitivity, and the ability to concentrate cationic, anionic and neutral molecules. Combined with sweeping in micellar electrokinetic chromatography (MEKC), online concentration can achieve even greater improvements in sensitivity [5, 6]. A number of reviews on online concentration have been published in the past few years [7-10].  * Portions of this chapter have been accepted or submitted for publication: (1) Fang, N.; Meng, P.J.; Hong, Z.; Chen, D.D.Y. Systematic optimization of exhaustive electrokinetic injection combined with micellar sweeping in capillary electrophoresis, The Analyst, 2006, submitted. (2) Meng, P.J.; Fang, N.; Wang, M . ; Liu, H.; Chen, D.D.Y. Analysis of amphetamine, methamphetamine and methylenedioxymethamphetamine by micellar electrokinetic chromatography using cation-selective exhaustive injection. Electrophoresis, 2006, in press.  140  In field-amplified sample stacking (FASS) [11, 12], the sample, dissolved in a lowconductivity buffer (LCB) or water, is injected hydrodynamically or electrokinetically a certain length into the capillary column filled with a high-conductivity buffer (HCB). Upon application of a voltage of normal or reverse polarity, the sample molecules migrate at a higher velocity in the L C B than in the HCB, which results in stacking at the L C B - H C B boundary. Chien and Helmer derived equations for the electroosmotic-laminar flow in a column with two buffers [13]. Burgi and Chien proposed that the optimal condition for sample stacking with electrokinetic injection is to prepare the sample in a buffer concentration that is about 10 times less than that used for electrophoretic separation [12]. If electrokinetic sample injection is used, a short water plug is often injected before the sample injection to improve reproducibility [14]. The effects of the injection parameters, including the length of the water plug, the sample injection time and voltage, have been studied [15, 16]. Quirino and Terabe reported a million-fold sensitivity increase with cation-selective exhaustive injection-sweeping-MEKC (CSEI-sweeping-MEKC) [6]. Three buffers of different conductivities are used in their method. The sample is prepared in a L C B (water). The capillary column is first flushed with a medium-conductivity buffer (MCB), followed by an injection of a plug of the HCB. The cationic analytes are introduced into the column electrokinetically at the normal polarity, that is, the cathode is on the detector side. The high electric field strength in the sample zone stacks the analytes at the L C B - H C B interface, and the low electric field strength in the HCB zone slows down the cationic analytes. Finally, SDS micelles under a reversed electric field sweep the long sample zone into a sharp peak. Theoretically, the sample zone can be shortened by a factor of l+k\ where k' is the capacity  141  factor [6, 17]. In a similar fashion, anions can be concentrated online with anion-selective exhaustive injection-sweeping-MEKC (ASEI-sweeping-MEKC) [18]. Electrokinetic injection is usually less reproducible than hydrodynamic injection in both the injected sample amount and the migration time due to the mobility, matrix, and instrument biases [19]. Therefore, one or two internal standards are often required [20, 21]. Quirino and Terabe reported poor reproducibility in peak height and area for the analysis of laudanosine and naphthylamine using CSEI-sweeping-MEKC without the use of internal standards [6]. Meng et al. reported a 1000-fold improvement in sensitivity for the analysis of amphetamine and its derivatives using CSEI-sweeping-MEKC over normal M E K C ; however, the reported run-to-run reproducibility with the presence of an internal standard was relatively poor: relative standard deviations (RSD) of the ratios of peak areas of amphetamines to that of the internal standard were more than 10% for low concentrations (0.1 ng/ml) [22]. In the present work, we investigated the causes of the poor reproducibility of peak area and peak height. A systematic optimization method is introduced to improve the detection limit and robustness of the exhaustive electrokinetic injection, followed by sweeping-MEKC. We demonstrated that when the conductivity of the sample itself is very small at a very low concentration, the reproducibility can be greatly improved by adding a small amount of the background electrolyte into the sample solution to slightly increase its conductivity. In addition, several other factors contributing to the sensitivity and reproducibility can be optimized, including M C B and HCB conductivities, the fraction of the capillary filled by the HCB, the electrokinetic injection time, and the micelle concentration. Because of our interest in developing a method to analyze the presence of illicit drugs in various biological matrices, amphetamine and its derivatives are used as analytes in this study.  142  With the optimized CSEI-sweeping-MEKC method, amphetamine, methamphetamine and methylenedioxy-methamphetamine (MDMA) were analyzed, and the detection limits for all three amine drugs are improved to below 0.01 ng/ml, an over 3000-fold improvement, compared to conventional C E with photodiode array detection [23], with good reproducibility (< 2% RSD).  143  §6.2 Experimental Section Apparatus. The experiments were carried out on a Beckman Coulter ProteomeLab PA800 (Beckman Coulter Inc., Fullerton, CA) with a U V detector. A 60 cm long (50.2 cm to detector) x 75 um I.D., fused-silica capillary (Polymicro Technologies, Phoenix, AZ) was used. A temperature of 25 °C was maintained for all C E experiments. The detection wavelength was 200 nm. Reagents and Solutions. One ml capsules of 1 mg amphetamine (l-phenylpropan-2amine), methamphetamine ((S)-N-methyl-l-phenylpropan-2-amine) and methylenedioxymethamphetamine (MDMA) methanol solutions were purchased from Cerilliant (Round Rock, TX). The three amine drugs have similar structures and characteristics. Sodium dihydrogen phosphate, sodium dodecyl sulfate (SDS), phosphoric acid, methanol, hydrochloric acid, and sodium hydroxide were purchased from Sigma (St. Louis, MO). Water was purified by a NANOpure Infinity Reagent Grade Water System (Apple Scientific Inc., Chesterland, OH). The three amine drugs were mixed and diluted with the purified water to 50 ng/ml.  Amphetamine  Methamphetamine  MDMA  Figure 6.1. Molecular structures of 3 amine drugs studied.  144  Table 6.1. The composition of the buffers.  Phosphate (mM)  Methanol (v/v)  SDS (mM)  MCB  100 mM  20%  0  HCB  200 mM  0  0  100 mM  20%  Micellar Buffer  20, 40  This sample stock solution was further diluted to suitable concentrations and adjusted to a certain conductivity for optimization purposes. In all sample solutions, the concentrations of the three amphetamines (in ng/ml) were the same. Buffer solutions were prepared by mixing sodium dihydrogen phosphate, SDS, and methanol stock solutions and diluting with purified water. The high conductivity buffer (HCB), medium conductivity buffer (MCB), and micellar buffer for sweeping were prepared as listed in Table 6.1. All three buffers were adjusted to pH 3.0 with 4M phosphoric acid. All solutions were filtered through 0.45 /mi membrane filters prior to the C E experiments. The composition and pH of the buffers have been studied by Meng et al. [22]. Capillary Electrophoresis Procedure. Prior to use, the capillary column was treated with IM NaOH, methanol, and purified water for 10 minutes each. Then, the capillary was flushed with the M C B for 30 minutes and was conditioned overnight. The conductivity of a sample or buffer solution was measured by flushing the column with that solution for 10 minutes and recording the current under the application of a voltage of 18 kV. The electrophoretic mobilities of the amphetamines in the M C B (or the HCB) were measured by flushing the column with the M C B (or the HCB) for 6 minutes, followed by a pressure injection of an mixture of the three amphetamines (1 /xg/ml) for 3 seconds at 0.5 psi (3447 Pa), and finally applying a voltage of 18 kV at the normal polarity.  145  The EOF mobilities in M C B (//  e0jMCB  ) and HCB (ju  method proposed by Williams and Vigh [24]. The / /  e o M C B  eoHCB  ) were determined using the  and / /  e o H C B  were  1.3xlO~ cm V"'s"' and 1.6 xlO" cm V"'s"', respectively. The average bulk mobility for the 5  2  5  2  H C B - M C B buffer system can be estimated with consideration of both the electroosmotic flow and the laminar flow, and its value is likely between / /  e o M C B  and ju  eoHCB  [13].  Exhaustive Electrokinetic Injection and Sweeping Procedure. This procedure was similar to the CSEI-sweeping-MEKC procedure reported by Quirino and Terabe [6]. At the beginning of each run, the capillary was conditioned with the M C B . The H C B was injected at 0.5 psi for 2 to 6 minutes to occupy a certain length of the column, followed by a short plug of purified water (0.5 psi for 5 s). A sample solution was then placed on the inlet of the capillary and a M C B on the outlet. With the application of a voltage of 18 kV at the normal polarity, the cationic samples were introduced into the capillary through the water plug at high velocities and were then slowed down in the HCB. The electrokinetic injection was run for 3 to 21 minutes. The sample solution and the M C B were then replaced by the micellar buffer at both ends of the capillary. The voltage was then switched to the reverse polarity, thus permitting the entry of micelles from the cathodic vial into the capillary to sweep the stacked analytes to a narrower band. Finally, the separation was performed using M E K C in the reversed-migration mode. The sample and micellar buffer were replaced after each run to ensure reproducibility.  146  §6.3 R e s u l t s a n d D i s c u s s i o n Electric Field Distribution. A capillary column of length L is filled with two buffers ( H C B and M C B ) of the same electrolyte with different conductivity. The M C B is first introduced to fill the entire column. Then the H C B is injected at a low pressure to occupy a fraction x of the column, where 0 <x <1. Assuming a uniform radius of the column and a sharp boundary between two buffers, the H C B fraction x can be determined by [13] ^ C B  X=  7  ) -  7  1  (  6  1  )  ( M C B / HCB ) ~ 1 7  7  in which / is the electric current when a voltage V is applied across the column containing both buffers, and 7 CB and 7 CB are the currents under the same voltage when the column is H  M  completely filled with only the H C B or M C B , respectively. The conductivity ratio of the H C B to the M C B (<j)) can be calculated as the ratio of JHCB to / M C B as the current is directly proportional to the conductivity. The electric field strengths in H C B and M C B can be calculated using V ^  H C B =  77  J I ^  (  6  -  2  A  )  L(x -x(p + <p) E rn = MCB  —  M  j /  i  i\  (6.2b) v /  L(x - xf + (p) Maximum Effective Injection Time. After a sample is introduced by an electric field, it migrates in the H C B and M C B zones at different electrophoretic mobilities of ^/ Mtp MCB > respectively. The two mobilities are measured by separate C E runs.  ep H C B  and  With the assumption of a sharp boundary between the M C B and H C B for the entire electrokinetic injection phase, the time for the sample to travel through the H C B zone (^HCB)  147  can be calculated by dividing the initial length of the HCB zone by the velocity of the sample in the HCB zone:  HCB =  xL  —  xL (x - x0 + <p) 2  =  •^HCB/^ep.HCB  The  l - ) 6  3  /^ep.HCB^  is independent of the electroosmotic flow (EOF) because the EOF is a bulk  fHCB  flow and is the same throughout the entire column. However, the distance that the sample travels while staying in the HCB zone (/'  ' H C B = (^P,HCB XL  + / 0 / / V H C B '  w  h  e  r  e  HCB  V™  ) is related to the EOF:  i s t  h  electroosmotic mobility.  e  The time for the sample to travel in the M C B before it is eluted can be calculated by T -/'  /  = MCB  ,  "cs \p  ,  V/^ep.MCB  +  (6 4) V -^7 U  M:o)-^MCB  The maximum effective injection time (t ) is equal to tncB plus max  time is longer than t  IMCB-  If the injection  , the analyte will start to migrate out of the column at the outlet of the  max  capillary. ^rnax  =  ^HCB  +  ^MCB  L (X - X(f> + <j>) X<p(M  B +MJ  2  =  EPMC  + /"ep.HCB - *(/*cp,HCB + /"eo )  ^ e , H C B < > e p ,MCB  V  P  +  (6-5)  /"eo )  The EOF mobility /u , usually small (less than 10% of ju eo  MCB  and / /  epHCB  ) at low pH,  is often negligible to give _L (x-x<f) + <p) 2  max  J/  "  *^  e p  ,MCB +  J.  (1 -  ^/"ep.HCB/^ep.MCB  .HCB  ,r  r\  '  Please note that eqs 6.5 and 6.6 are derived with the assumption of a stable sharp boundary between two buffer zones; however, in reality, this may not always be true. The  148  estimated maximum effective injection times are compared with the experimental results later in this chapter. In these experiments, three cations were analyzed at the same time. Obviously, the fastest migrating cation (amphetamine) determined the maximum electrokinetic injection time. The fj.  M C B  and ju  H C B  for amphetamine were measured with separate C E runs to be  1.74xl0~ cm V"'s'' and 2.31 x 10~ cm V"'s"', respectively. The / / 4  2  4  2  e p M C B  and / /  e p H C B  forthe  slowest cation (MDMA) were 1.66xlO" cm V"V and 2.23xlO~ cm V"'s"', respectively. 4  2  4  2  Conductivity of Sample Solutions. The samples were prepared in an L C B . The conductivity of the L C B plays an extremely important role in the efficiency and reproducibility of the electrokinetic injection. In field-amplified sample injection, the conductivity of the separation buffer is much higher than the conductivity of the sample solution. Burgi and Chien proposed that the value of 10 is the optimal field enhancement factor to minimize the peak variance [13]. However, the peak variance of the injected sample zone is not relevant in this study because a long sample zone, sometimes a full column of sample, is the result of the exhaustive electrokinetic injection. A wide range of sample conductivities was tested for optimal injection conditions. A series of samples containing the same concentrations of each of the three amine drugs (1 ng/ml) were prepared in the phosphate buffers of different concentration, ranging from 0.2 mM to 6 mM. The capillary was flushed with each sample solution for at least 10 minutes, and then the current was recorded under the application of an 18 kV voltage. The plot of phosphate concentration vs. current shows good linearity for the whole concentration range (Figure 6.2A).  149  o 1 0  i  2  i  4 Current (|aA)  t  i  6  8  Figure 6.2. Current (proportional to sample conductivity) vs. electrokinetic injection efficiency. The current of a sample solution was measured under the application of an 18 kV voltage for a capillary column filled with that sample solution. The M D M A peak area is plotted against the current. The nonlinear regression equation is shown on the graph. Figure A shows a linear relationship between the phosphate concentration and the current.  150  The electrokinetic injection and sweeping were carried out for each sample solution as described in the experimental section. The injection time was 6 minutes for all sample solutions. As shown in Figure 6.2, when the current was in the range of 7.72 fiA (the 6 mM phosphate LCB) to 1.35 juA (the 1 mM phosphate LCB), the M D M A peak area was inversely proportional to the current, which clearly demonstrates that a high field strength established at the injection point can greatly improve the efficiency of the sample introduction. On the other hand, the electrokinetic injection was reproducible in this range. Seven measurements were taken for the sample solution in 1 mM phosphate on the same day with. The average peak area and the standard deviation were 17011 ± 202, or 1.2% RSD. Further decrease in the sample conductivity did not increase the sample injection efficiency, and the peak area was no longer reproducible. The possible reason is that the voltage drop along the capillary is very much related to the conductivity of injected sample. When the sample conductivity is extremely low, minor changes in the sample, even due to the variation of analyte concentration itself, or minor changes in sample matrix during the sample handling or due to ions leaching out from container walls will significantly change the voltage drop at the injection point, thus, changing the amount of injected analyte during the long injection process. The amphetamine and methamphetamine peak areas followed the similar trend and had similar relative standard deviations. In all other experiments in this study, the samples were prepared in 1 mM phosphate buffer to achieve high efficiency and reproducibility. The water plug was not necessary to obtain high reproducibility in our experiments. With ImM phosphate buffer in the sample solution, the field enhancement factor was reduced,  151  and the accumulation of cations at the injection point was no longer serious enough to degrade the injection efficiency. Similar reproducibility was achieved with or without the water plug. However, the water plug was still used to ensure the best results. The current during the electrokinetic injection processes can be used to judge whether a CE run is reproduced. A smooth current curve indicates a good run; while a rough current curve is often caused by irreproducible incidents, such as the introduction of air bubbles or small particles during the injection. Figure 6.3 shows the good and bad currents for two runs with identical conditions. The resulting peak areas from these two runs differed by about 10%, which is rather significant compared to the 1 to 2% relative standard deviation of the good runs.  0  2  4  6  8  Time (minute)  Figure 6.3. The current curves are used as signals for reproducible runs.  152  Conductivity of M C B and H C B . It is obvious that when the fraction of the HCB (x) is kept constant, the higher the H C B - M C B conductivity ratio (0), the longer the maximum sample injection time. However, it is difficult to achieve a value of ^ that is greater than 5 because of the two limiting factors: ( 1 ) The HCB conductivity cannot be too high because the maximum current for the Beckman Coulter P A 8 0 0 C E system is  3 0 0 JUA  and because a very  high current is usually unfavorable for C E experiments; ( 2 ) The M C B conductivity cannot be too low because it has to be high enough, compared to that of the sample solution (LCB), to ensure an enhanced field which is determined by the ratio of the two conductivities. In our experiments, the M C B was a 1 0 0 mM phosphate buffer with 2 0 % v/v methanol, and the HCB was a 2 0 0 mM phosphate buffer without methanol. The value of 0 was 2 . 7 2 . The 2 0 % methanol in the M C B is required for good separation of the three amine drugs. Sweeping Process. After the electrokinetic injection, the vials containing the micellar buffer (20%o methanol) are placed at both the cathode and the anode. In 2 0 % methanol, the SDS cannot form micelles effectively due to its high critical micelle concentration (CMC). Upon the application of a reversed electric field, the SDS enters the H C B zone from the inlet end of the capillary. A lot more micelles can be formed in the H C B zone as there is no methanol, and the sweeping occurs. The sweeping process is efficient because the sample ions and the micelles move in the opposite directions: The negatively charged micelles pick up sample ions and migrate towards the detector. In the meantime, the positively charged sample ions migrate towards the inlet of the column. The boundary of the M C B and HCB zones also helps the sweeping process. When the micelles carry sample ions into the M C B zone, the 2 0 % methanol in the M C B destroys most  153  of the micelles, and the sample cations are released and move back into the H C B zone. The stacking of the sample cations occurs at this boundary. In the electrokinetic injection step, the samples can be introduced to fill both the HCB and M C B zones. In the sweeping step, given long enough time, the sample ions in the M C B zone can enter the H C B zone and be swept by the micelles. However, the sample ions that stay in the M C B zone when the SDS arrives, cannot be swept effectively because the micelles are mostly destroyed in the M C B with 20% methanol. As a result, a small plateau is created in front of the sharp analyte peak. Therefore, the fraction of the H C B zone and the electrokinetic injection time should be optimized to ensure all sample ions injected into the M C B zone can move back into the H C B zone in time for them to be swept by the micelles. After the sweeping process ends, the swept sample ions are separated while traveling in the M C B zone. The length of the M C B zone should be optimized to ensure good resolution of the samples. Fraction of H C B . Two sets of CSEI-sweeping-MEKC experiments were carried out to test the effect of different fractions of the HCB. The pressure injection time of the H C B at 0.5 psi was 2 and 4 minutes for the two sets, respectively. 13% of the capillary column was filled by the H C B with a 2-minute injection; 26% with a 4-minute injection. The same sample solution (1 ng/ml of each of the three amphetamines in a 1 mM phosphate buffer) was used in both sets. In each set, the samples were electrokinetically injected for up to 21 minutes, and both peak areas and heights were measured. The results for the fastest migrating cation (amphetamine) and the slowest migrating cation (MDMA) are shown in Figure 6.4.  154  Figure 6.4. The effect of H C B fraction. The electrokinetic injection time vs. (A) the M D M A peak area, (B) the M D M A peak height, (C) the amphetamine peak area, and (D) the amphetamine peak height. In (C), the star shows a point that is not used in the regression. The explanation is in the text.  As shown in Figure 6.4A and C, both M D M A and amphetamine peak areas increased almost linearly as the electrokinetic injection time increased. However, the better fit of the data is generated by nonlinear regression using y = a(\~ e ~ ), where a and b are regression x  bx  parameters. The increase rate of the peak area slowed down slightly over time due to the lessenhanced field caused by the increased conductivity of the sample solution. The peak areas could increase as long as the injected sample zone did not exceed the length of the capillary after a long electrokinetic injection. Using eq 6.5, the maximum electrokinetic injection time (t ii) for amphetamine was 19.3 minutes with 26% H C B , and 18.7 a  minutes with 13% HCB. For M D M A , the maximum electrokinetic injection times were  155  longer at 21.6 and 20.4 minutes with 26% and 13% HCB, respectively. These calculated maximum injection times were verified experimentally: In Figure 6.4A, with 26% HCB, the M D M A peak area continued to increase all the way up to 21 minutes; while in Figure 6.4C, with 26% HCB, the amphetamine peak area remained approximately the same after 19 minutes. Another observation from Figure 6.4A and C is that under the same injection time, slightly larger peak areas were produced with 26% HCB than 13% H C B due to the fact that a larger field enhancement factor at the injection point and a higher current were obtained with 26% HCB. The plots of peak height vs. electrokinetic injection time (Figure 6.4B and D) show a similar trend, but the linear increase of the peak height stopped sooner than the peak area: With 26%o HCB, the M D M A and amphetamine peak heights rose to the maximum at about 19 and 17 minutes, respectively; with 13% HCB, the M D M A and amphetamine peak heights stopped increasing after about 8 and 7 minutes, respectively. The electropherograms for 13, 19 and 21 minutes of the electrokinetic injection with 26% H C B and 20 mM SDS are shown in Figure 6.5A-C. From these figures, we can clearly see that once the small steps show up in front of the sharp peaks, the peak height remains nearly constant. This has been explained by the sweeping mechanism. When the small plateaus connected all three peaks at the bottom, the peak areas could not be measured accurately. Therefore, with 13% HCB, only the peak areas with up to 13minute injection were shown in Figure 6.4A and C.  156  Effect of SDS Concentration. The micelle concentration also plays an important role in determining the resolution and sensitivity. Two SDS concentrations, 20 and 40 mM, were tested. All other conditions were the same. The resulting peaks are shown in Figure 6.5. The peak areas were about the same using both SDS concentrations because the SDS buffer was used only for the sweeping, and not for the injection. The three amine drugs were nicely separated in both SDS concentrations. As the SDS concentration increased, the peaks of three amine drugs became sharper, but they were positioned more closely. Higher SDS concentration could improve sensitivity, and lower SDS concentration was good for resolution.  Electrokinetic Injection Time 13 min 19 min 21 min  20 mM CD  i—  !Q t_  SDS Concentration  njCD O  cCD  .o  40 mM  s  o CO  <  Time Figure 6.5. Electropherograms of the mixture of the three amine drugs. (1) M D M A , (2) methamphetamine, (3) amphetamine. The three panels in one row used the same SDS concentration and the three panels in one column share the same electrokinetic injection time. The x- and y-axes of all the panels are on the same scale. The range of x-axis (migration time) in all six panels is 56 seconds.  157  In the next section, we will demonstrate that the peak area is a better choice for calibration than the peak height. Wide spacing of the peaks (high resolution) can be helpful in measuring the peak area for samples of high concentrations when the peaks become broader. The 20 mM SDS buffer was chosen for constructing the calibration curves. Calibration Curves. The exhaustive electrokinetic injection and sweeping procedure described in the experimental section was followed strictly. The sample concentrations were in the range of 0.01-10 ng/ml. The optimal conditions were chosen as follows: the samples in 1 mM phosphate buffer, 4-minute HCB injection at 0.5 psi, 13-minute electrokinetic injection, and 20 mM SDS. The plot of M D M A concentration vs. peak area (Figure 6.6A) shows a good linearity for the range of 0.01-10 ng/ml. Figure 6.6B is an enlargement for the concentrations below 1 ng/ml. The plot of M D M A concentration vs. peak height (Figure 6.6C) is no longer a straight line, but a curve that can be well fitted by a logarithmic equation. The data points for the low concentrations (< 1 ng/ml) still follow a linear relationship (Figure 6.6D). The L O D for M D M A was 6 pg/ml (ppt), calculated as three times the standard deviation of the noise divided by the slope of the straight line in Figure 6.6D. For methamphetamine and amphetamine, the plot of concentration vs. peak area (Figure 6.6E) and the plot of concentration vs. peak height (Figure 6.6F and G) are similar to the M D M A calibration curves. The LODs for methamphetamine and amphetamine were 8 and 7 pg/ml (ppt), respectively. The reported LODs of M D M A , methamphetamine, and amphetamine using C E - D A D was 80, 20, and 30 ng/ml, respectively [23]. Thus, the concentration sensitivity improvement  158  over C E - D A D was over 10000-fold, 2500-fold, and 4000-fold for M D M A , methamphetamine, and amphetamine, respectively. The peak area is a better choice for calibration than the peak height due to its better reproducibility and linear relationship with concentration. The peak area was affected only by the electrokinetic injection process, while the peak height was affected by both the injection and the sweeping processes. On the other hand, the peak height rose to a maximum as the concentration increased, which made the determination of higher concentrations using the peak height less accurate. As the sample concentration increased, the sample peaks got broader, and the micellar sweeping effect reached a saturation point. When the sample concentrations were 20 ng/ml or higher, the 3 peaks overlapped, making it impossible to accurately measure peak areas. This problem can be easily solved by reducing the injection time.  159  0  2  ,  4  6  8  10  0  2  4  6  8  10  Concentration (ng/ml)  Figure 6.6. The calibration curves for the three amine drugs. (A) The plot of M D M A concentration vs. peak area for the range of 0.01 to 10 ng/ml. The linear regression equation is shown besides the regression line. (B) An enlargement for the range of 0.01 to 0.8 ng/ml. The dash line fits the points in this small range, and the solid line represents the regression equation for the range of 0.01 to 10 ng/ml. The two lines are slightly different. (C) The plot of M D M A concentration vs. peak height. A logarithmic equation is used to fit the data. (D) Linear regression on the M D M A concentrations lower than 1 ng/ml. (E) The methamphetamine and amphetamine calibration curves based on peak area. (F) The methamphetamine and amphetamine calibration curves based on peak height. (G) Linear regression of the methamphetamine and amphetamine concentrations lower than 1 ng/ml.  160  §6.4 C o n c l u s i o n s A systematic optimization of exhaustive electrokinetic injection and sweeping processes for the detection of three amine drugs was carried out. In general, there are five key factors: the conductivity of the sample solution, the conductivities of the HCB and M C B , the fraction of the HCB, the electrokinetic injection time, and the micelle concentration. An internal standard is usually necessary for C E analysis with electrokinetic injection because the injection reproducibility often suffers from the matrix bias and instrument bias [19]. In this chapter, we demonstrate that high reproducibility is achievable by controlling the conductivity of the sample solution. When a sample is dissolved in water, the conductivity of the sample is intimately related to the sample characteristics, as well as the changes in the sample solution during the injection. By dissolving the sample in a solution with a low conductivity, the injection process is no longer dictated by the sample, making the injection process, as well as the whole analytical procedure, much more reproducible. The calibration curves can be constructed without an internal standard, and a several thousand-fold improvement in sensitivity was achieved. The results presented show that CSEI-sweepingM E K C has the potential to be applied to the successful detection of amphetamines in human hairs and body fluids.  161  §6.5 References [I] Moring, S. E., Reel, R. T., Vansoest, R. E. J., Anal. Chem. 1993, 65, 3454-3459. [2] Xue, Y. J., Yeung, E. S., Anal. Chem. 1994, 66, 3575-3580. [3] Wang, T. S., Aiken, J. H., Huie, C. W., Hartwick, R. A., Anal. Chem. 1991, 63, 13721376. [4] Culbertson, C. T., Jorgenson, J. W., Anal. Chem. 1998, 70, 2629-2638. [5] Quirino, J. P., Terabe, S., Science 1998, 282, 465-468. [6] Quirino, J. P., Terabe, S., Anal. Chem. 2000, 72, 1023-1030. [7] Quirino, J. P., Terabe, S., J. Chromatogr. A 2000, 902, 119-135. [8] Osbourn, D. M . , Weiss, D. J., Lunte, C. E., Electrophoresis 2000, 21, 2768-2779. [9] Breadmore, M . C , Haddad, P. R., Electrophoresis 2001, 22, 2464-2489. [10] Lin, C. H . , Kaneta, T., Electrophoresis 2004, 25, 4058-4073. [II] Mikkers, F. E . P., Everaerts, F. M . , Verheggen, T., J. Chromatogr. 1979,169, 11-20. [12] Burgi, D. S., Chien, R. L., Anal. Chem. 1991, 63, 2042-2047. [13] Chien, R. L., Helmer, J. C , Chem. 1991,55, 1354-1361. [14] Wey, A. B., Thormann, W., Chromatographia 1999, 49, S12-S20. [15] Liu, S. PL, Li, Q. F., Chen, X. G., Hu, Z. D., Electrophoresis 2002, 23, 3392-3397. [16] Sun, S. W., Tseng, H. M . , J. Pharm. Biomed. Anal. 2004, 36, 43-48. [17] Quirino, J. P., Terabe, S., Anal. Chem. 1999, 71, 1638-1644. [18] Kim, J. B., Otsuka, K., Terabe, S., J. Chromatogr. A 2001, 932, 129-137. [19] Krivacsy, Z., Gelencser, A., Hlavay, J., Kiss, G., Sarvari, Z., J. Chromatogr. A 1999, 534,21-44. [20] Dose, E. V., Guiochon, G. A., Anal. Chem. 1991, 63, 1154-1158. [21] Leube, J., Roeckel, O., Anal. Chem. 1994, 66, 1090-1096. [22] Meng, P. J., Fang, N., Wang, M . , Liu, H., Chen, D. D. Y . , Electrophoresis 2006, accepted. [23] Boatto, G., Faedda, M . V., Pau, A., Asproni, B., Menconi, S., Cerri, R., J. Pharm. Biomed. Anal. 2002, 29, 1073-1080. [24] Williams, B. A., Vigh, C.,Anal. Chem. 1996, 68, 1174-1180.  162  Concluding Remarks Two computer simulation models of affinity capillary electrophoresis (ACE) are presented in this thesis. The One-Cell model (Chapter 2) describes the migration of peak maximums and, therefore, is extremely fast. The Multi-Cell model (SimDCCE, Chapter 3) is an efficient implementation of a robust finite difference scheme for solving the mass transfer equation. A C E experiments usually have well-controlled physical and chemical environments: constant temperature and pH, stable current, and rapidly reached equilibrium. Under these conditions, the key assumption that the local electric field strength is constant throughout the capillary is valid, and the calculations required for an accurate simulation run can be reduced because only the analyte plug and its two boundaries are tracked. With accurate simulation, the enumeration method (Chapter 5) can be used to estimate equilibrium or kinetic constants even when the ratio of the additive and analyte concentrations is not large enough for the system to reach the steady state conditions required for regression methods. The next step in developing simulation models of A C E is to simulate multiple-additive, multiple-analyte, and multiple-stoichiometry equilibria. The regression methods for determining binding constants generally cannot be used for higher order binding interactions except for highly cooperative or noncooperative binding, and the methods' accuracy is not assured when multiple additives or analytes are present in the system. In such cases, the enumeration method has important advantages in accuracy and speed. The exhaustive electrokinetic injection and sweeping-MEKC experiments described in Chapter 6 are much more complicated than A C E experiments, as a number of additional  163  factors must be considered: one sample solution and two buffers with different conductivities, a small water plug at the injection point, constantly changing current, the sweeping effect to reduce the long initial sample plug to a very short one, the formation of micelles from SDS, and the effect of organic contents in the buffers. Many other C E experiments share a similar level of complexity. To simulate these complex systems, the local electric field strength cannot be assumed constant, protons and hydroxide ions must be included in the calculations, the effects of organic contents on the equilibrium and kinetic constants and solution viscosity should be considered, and the simulation program should be capable of handling multiple-step separation procedures. A new simulation system, to be called Virtual Capillary Electrophoresis (VCE), could be built to fulfill all these requirements. The ultimate goal of V C E would be to simulate all C E experiments in a way similar to carrying out experiments on a real instrument. One use of V C E could be to study complex phenomena, such as dynamic complexation, sweeping/stacking of sample analytes, pH depletion, and peak splitting. In Chapter 4,1 have demonstrated that SimDCCE is capable of studying A C E in a comprehensive way, and similar studies of complex systems could be done with V C E . More importantly, V C E may be used to find the optimum conditions for a target system before any real experiments are carried out, which may speed up method development process and save time and resources. SimDCCE and V C E may be developed in parallel with different emphases: speed and accuracy for SimDCCE and functionality and versatility for V C E . The exhaustive electrokinetic injection and sweeping M E K C method presented in Chapter 6 has shown great potential in analyzing trace amounts in biological matrices. This method has been used successfully in quantifying amphetamine and its derivative at as low as  164  6 pg/ml (6 ppt) and tyrosine and nitro-tyrosine at about 10 ng/ml (10 ppb). These two applications are joint projects with forensic chemistry and anesthesia research groups, respectively. More applications can be expected as opportunities arise. Many online concentration techniques have been developed in the past decade. However, due to their high complexity, most of these methods have not been fully understood and systematically optimized. As mentioned earlier, V C E may be able to study these methods comprehensively and suggest optimum conditions for improved sensitivity and reproducibility.  165  Appendix  A. SimDCCE Guide B. Business Plan  Appendix A SimDCCE Guide §A.l Introduction to User Interface The computer simulation program SimDCCE,  written in Microsoft Visual C++ version 6,  runs on Microsoft Windows operating systems. The user interfaces (UI) were designed in accordance with all typical Windows applications. As such, the basics of operation that apply to other Windows applications also apply to SimDCCE.  These basics will not be covered in  this guide.  ^  NP2BCD5.par - SimDCCE2  D;  GS B  41 .  I Settings  A smiaoon run has been started. You cannot start another run untif tfws one is either finished or stopped.  Property B  B  B  Value RESUME  Instrument Length of Capillary (cm)  64 5  Length to Detector (cm)  54.3  Voltage (V)  10059  Idle Time (s)  2.04  Ramp Time (s)  10.2  Temperature (C)  20  y-axis 4  (1-800)  A L L Y It jenaoled me x and/or y axes are automatically adjusted to fit the entire x anavor y data ranges REFRESH E V E R Y * SECONDS Specifies the  (concentration)  [  Adjust Automatically  Zoom x  [ADJUST  STOP  Adjust Automatically I  Y Max (M) =  MIGRATION TIME interval  Chenicalx The Analyte (A)  p-nitrophenol  Refresh every  The Additive (P)  b-cyclodextnn  Show Curves  Reaction Type  A:P = 1 1  j i . j n Sample iM.  0002  PI in Sample (M)  0  [A] in BGE (M)  0  Fin  0 005  BGE(M)  Mofaitties  Ki  0 1  Deiay  </ (P free] |  |  [Complex]]  tt Signal II  [tt  Signal III  0  | « / |Atotal) |  millisec | J< [P lolalj |  for refreshing the peaks DELAY i MILLISECOIICS Specifies the real-time delay after every  Time (s) = 17 238  | On-Capi!lary Position (cm)  1 108  (cra2^Ts)  Free Analyte (A)  -0 0002357  Free Additive (P)  0  Compleit (C)  0.0002994  Electroosmotic Row (EOF)  0  Length of Capillary (cm) Specifies the length of the capillary  i Experimental Conditions  \ . Simulation Parameters j  ^  ell  1  4 » H ~ \ s i n g l e Run: Multi-Cell M o d e l / ~ Enumeration Algorithm  /  Find K From Intersections  X  Load.  Figure A . l . The user interface of SimDCCE is divided into three regions: 1. the settings panel defines the experimental conditions and simulation parameters; 2. the simulation control panel; 3. the display panel shows the animation of the simulated concentration profiles.  167  §A.2 T u t o r i a l f o r S i m u l a t i o n w i t h M u l t i - C e l l M o d e l §A.2.1 Step 1: Set Experimental Conditions SimDCCE  can be used to simulate D C C E experiments with real or hypothetical  experimental conditions. This tutorial uses SimDCCE to simulate a real A C E experiment, consisting of the analyte £>-nitrophenol that interacts with the additive /?-cyclodextrin (/3-CD). The experimental conditions are listed in Table A . 1. Entering these conditions into SimDCCE setting panel yields Figure A . 2 .  Table A . l . The list of the experimental conditions Length of the capillary  L = 64.5 cm  Length to the detector  .Ld = 54.3 cm  Voltage  V=+ 10059 volts  Idle time  2.04 s  Ramp time  10.2 s  [A] in the injected sample plug  [ A ] = 2.0 m M  c  0  [A] in the B G E  [A] = b  0  [P] in the injected sample plug  [P]o = 0  [P] in the B G E  [P]b = 5.0 m M  Migration time of the E O F marker  tco = 946.86 s  Electrophoretic mobility of the free analyte Electrophoretic mobility of the free additive  =-2.411 xlO" c m / V - s 4  M e p A  fj.  2  = 0 for neutral /3-CD  p  Viscosity correction factor  v= 1.023  Length of the injection plug  0.18 cm  Diffusion coefficients  10~ cm V 6  for all species  Migration time of the analyte peak  t = 1385.88 s  Estimated binding constant  K= 533 M "  Estimated complex mobility  m  Mcp,c =-7.31xl0"  5  1  cm /V-s 2  168  Property  •  B  B  Value  hstrunent  Length of Capilary (cm)  64.5  Length to Detector (cm)  543  Voltage (V)  10059  Idle Time (s)  2.04  Ramp Time (s)  102  Temperature iC)  20  Chemicals The Analyte (A)  p-nlrophenol  The Additive (P)  b-cydodextrin  Reaction Type  A:P-1:1  [A] in Sample (M)  0 002  [Pj in Sample (M)  0  [A] in B G E ( M )  0  [P]in B G E ( M )  0 005  M o b i l i t i e s (cm2/Vi) Free Analyte (A)  -0.0002411  Free Additive (P)  0  Complex IC)  -7.31E-005  Electroosmotic Row (EOF)  0  Time of EOF Marker (s)  946%  ..:..  Migration Time of A (s)  B  Viscosity Correction A  1023  Viscosity Correction P  1  Viscosity Correction C  1.023  Diffusion Coefficients ( c m " 2 / s )  -=•::-:  Free Analyte Free AddBve Complex  B  1E-00C  Use Length of Injection Plug (...  0.18  Binding Constant  533  Association Rate {/s/M)  0  Dssociation Rate i/s)  0  Use Equilibrium or Kinetic..  Equilibrium  L e n g t h o f C a p i l l a r y (cm) Specifies the length of the capillary  Figure A.2. Screenshot of the settings panel after all the experimental conditions were set in.  Notes about the experimental conditions: 1. The mobility of the free analyte (//  ep A  ) was measured using separate C E runs,  during which the separation buffer did not contain any additive. 2.  The binding constant (K) and the complex mobility (ju ) were not obtained epc  directly from an experiment. Instead, these values are determined using either the  169  regression methods or the enumeration method. These values are required to run a single simulation to show the mechanism. 3. The length of the injection plug can either be measured with a preliminary experiment, or estimated from the injection parameters provided by the manufacturer of the C E instrument. 4. The interaction between the analyte and the additive in this experiment occurs very quickly, so the system can be assumed to be equilibrated at any given position and moment. Consequently, this experiment is an equilibrium C E experiment in which the equilibrium constant is used in the calculation but the kinetic constants (association and dissociation rates) are not needed. 5. The diffusion coefficients for all species were set at 10~ cm s" in this tutorial. No 6  2  1  real values were obtained from neither experiments nor the literature.  §A.2.2 Step 2: Set Simulation Parameters for the Calculation Module The parameters used in the calculation module are listed as follows: 1. Length of a Cell (Az) and Time Increment (At): The accuracy and speed of the simulation depend on the values of Az and At. The elaborated explanation has been presented in Chapter 3. For this simulation run, Az and At were set at 0.001 cm and 0.001 s, respectively. 2. Maximum Number of Active Cells: This number is usually set between 10,000 and 50,000. If this number is too small, the number of active cells will exceed the size of the allocated memory, and the simulation will fail.  170  3. Simulation Stop Time (s): The simulation will stop at the set simulation stop time. If a complete simulation run is desired, the simulation stop time must exceed the full experimental time. 4. Abandon Off-column Cell? (Yes or No): This setting determines whether cells are deactivated and abandoned after they run out of the capillary column. In most cases, this setting should be "Yes" to ensure the best simulation efficiency. 5. Finite Difference Scheme (FDS): The type of FDS to be used in the calculation can be selected. The user can choose between the "forward-space method" or the "monotonic transport method". 6. Concentration Thresholds: Two concentration thresholds can be set to signal the activation and deactivation of the cells. 7. Types of Concentration Threshold: The concentration threshold can be set to the concentration of one of the following species: total analyte (free + complexed), total additive (free + complexed), free analyte, free additive, and complex. If none of the above are needed, select N/A. 8. Operators and Values of Concentration Threshold: The concentration threshold can be set on either the inlet or outlet side of the active segment. On each side, the threshold condition is defined using the combination of one operator, two values (VI and V2), and the concentration of the species defined by the type ([Type]). The threshold condition can be one of the following: [Type] >V1; [Type] <V1; V I < [Type] <V2; [Type] is NOT between VI and V2; The change of [Type] from the concentration in the B G E is greater than V2.  171  Property E  Value  3D Surface and 2D Curve  B One Ceil Model B  Multi-Cell Model Length of CeH (cm)  0.001  Time Increment (s)  0001  Max Number of Active C _  20000  Slmulation Stopping Time... Abandon Off-column Cells'' YES Finite Difference Scheme  Monotonic Transport  El first Unknown Variable Q  Second Unknown Variable  B  Concentration Threshold 1 Type  [AHC]  Inlet - Operator  >-V1  Inlet-VI  IE-009  Inlet-V2  0  Outlet - Operator  >-V1  Outlet-VI Outlet-V2  E 0  El Concentration Threshold 2 Type  IP]  Inlet - Operator  N/A  lr»et-V1  IE-009  lntet-V2  0  Outlet - Operator  Off (V2)from [BGE]  Outlet-VI  1E40S  Outlet-V2  IE-009  B  Output Options  3  Bectropherogram  S  Signal Multipliers  S Snapshots  NO. 01 1.0.1.0.0.0.0.0.0 N/A. 1. NO  Length of Cell (cm) Specifies the length of one cell  Figure A.3. Screenshot of the settings panel after the simulation parameters used in the calculation module were set.  §A.2.3 Step 3: Set Simulation Parameters for Outputs There are five types of outputs for SimDCCE  simulations: electropherograms, automatic  snapshots, manual snapshots, animations, and 3D meshes. Electropherograms are generated in a similar fashion as the real C E instrument: the concentrations of all species in the detector cell, which is designated at the same position as the real detector window, are presented as U V or LIF signals.  172  Snapshots are taken at a moment to show the concentration distribution of all species throughout the capillary length. Snapshots can be taken either automatically by SimDCCE  at a  user-defined time interval or manually by the user. A series of snapshots can be put together to clearly illustrate the mechanism of the binding interaction and electrophoretic migration process as detailed in Chapter 4. When continuous snapshots are shown at a very short time interval, an animation is created. SimDCCE  is the first simulation program of C E to demonstrate the electrophoretic  migration processes in real-time or faster. 3D meshes are created to demonstrate the change of peak shape (Figure A.4). This peak shape change can also be demonstrated with a contour plot (Figure A.5).  173  0.00  0.05  0.10  0.15  0.20  0.25  0.30  0.35  On-capillary Position (cm) Figure A . 5 . Contour plot representation o f the peak shape changes during the first 14 minutes of a simulation.  The five types o f SimDCCE  outputs are controlled by different parameters. The  parameters controlling electropherograms and automatic snapshots are listed below: 1. Peak Detection (Yes or No): " N o " , i f a single simulation run is conducted. "Yes", i f the enumeration method is used with the M u l t i - C e l l model. For this tutorial, Peak Detection should be set to " N o " . 2. Electropherogram - "Want it? " (Yes or No): " Y e s " denotes that an electropherogram is desired. 3.  Electropherogram - Time Interval: A n electropherogram is essentially a collection of signals recorded at the specified time interval.  174  4.  Signal Multipliers: The absorption at a given moment in a real, non-simulated electropherogram is the sum of U V or LIF signals of all the analyte species that exist at the detector in that moment. SimDCCE uses the signal multipliers to simulate the ability of each individual species to absorb U V light or emit LIF. There are three sets of such multipliers, each corresponding to one of the three generated signals.  5. Snapshot - Format: Choose the format of the generated snapshot data files. If snapshots are not required, select N/A. 6.  Snapshot - Time Interval: This parameter determines how often a snapshot is taken. It works together with the Simulation Stop Time to determine the total number of snapshots to be generated. Be advised not to set a small time interval and a large simulation stop time together; otherwise, a huge number of data files may be generated and the simulation may slow down significantly.  7. Snapshot - Single File (Yes or No): The snapshot data files can also be output to a single file.  a Peak " Detection a  Electropherogram  a  Signal Multipliers  Want it?  YES  Time Interval fs)  0.1  Set 3-A  01 1.0.1.0.0.0.0.0.0 1 0 1 0 0 0 0  Set3-P  0  Set3-C  0  Set 1 -AFree An Sell -PFreeAd Set 1 - C Complex Set2-A Set2P Set2-C  a  NO YES.  Snapshots Format  N/A, 1. NO N/A  Time Interval (s)  1  Single file''  NO  Figure A.6. The parameters for generating electropherograms and automatic snapshots.  175  Thus far, the Settings Panel, as it is related to the simulation of a single D C C E experiment using the Multi-Cell model, has been explained. The other three types of outputs manual snapshots, animations, and 3-D meshes - can only be recorded or displayed after the simulation is started, and will be explained in Step 4.  §A.2.4 Step 4: Start the Simulation Run SimDCCE  has a variety of functions: the "Enumeration Algorithm" and "Find K From  Intersections" functions are used for the enumeration algorithm as explained in Chapter 5, while the "Load and Plot Data Files" page is covered later in this appendix. A screenshot of the control panel and display panel is shown in Figure A.7. If this window is different, click on the "Single Run: Multi-Cell Model" tag on the bottom of the display panel.  ^  ADJUST AUTOMATICALLY: If enabled, the x and/or y axes are automatically adjusted to fit the entire x and/or y data ranges  START  REFRESH EVERY | SECONDS Specifies the MIGRATION TIME interval for refreshing the peaks  y-axis (concentration) 1/ Adjust Automatically  y Adjust Automatically Zoom x  (1-800)  Refresh every  Delay  millisec  0  Show Curves: SC. \r m  | | *  IP free; | [ jj  *  I [ K Signal II | |  DELAYS MILLISECONDS Specifies the realtime delay after every showing cf the peaks.  Y Max (M) =  s  0 1  [Complex! ]  [ V I A total]]  | jj [P I  A  SHOW CURVES Specifies what curves to show. SAVE SNAPSHOT. A snapshot can be saved while the simulation is running or has been paused  |  v  Time Elapsed (s) j On-Capillary Position (cm)  0.0  Detector  Peak Height  pq  (M)  00  •K  < > » \ S i n g l e Run:  ;  '_i:i- > l l  "odel <  •  - ;- :—  ;  /  Find K From Intersections  /  Load and Plot Data Files  /  Figure A . 7 . Screenshot of the control panel and the display panel.  176  There are many buttons and edit boxes on the control panel. Brief explanations are provided in the textbox on the right side of the window, as well as below. 1. X-Axis Adjust Automatically: If this option is active, SimDCCE  will automatically  adjust the range of the x-axis so that only the active segment of the entire capillary is shown. The active segment is the collection of active cells determined by the concentration threshold conditions. If this option is inactive, the full capillary length is shown on the display panel. 2. X-Axis Zoom (T-800x): When the "Adjust Automatically" option is inactive, the X Axis Zoom becomes available. With this setting, the user can zoom in on the x-axis and select the portion of the capillary to be displayed using the horizontal scroll bar. This setting can allow the user to observe the change of the peak shapes and the migration of the peaks towards the detector simultaneously. 3. Y-Axis Adjust Automatically: Similar to its x-axis counterpart, activating this option enables SimDCCE  to automatically adjust the range of the y-axis to show the full  peak heights. 4. Y-Axis Y Max (Maximum Concentration): When the "Adjust Automatically" option is inactive, this setting becomes available to allow the user to define the maximum value for the y-axis. 5. Refresh Rate: This setting specifies the time interval between two consecutive snapshots. Faster refresh rate can make the animation smoother, but will also slow down the simulation speed. One tenth of a second is often sufficient for both animation and simulation speed.  177  6.  Delay: During the simulation run, the user may want to capture snapshots at desired moments. The delay can be used to slow down the animation to facilitate the capture of a precise moment, while keeping the same time interval for taking the snapshots.  7.  Show Curves: Eight curves with different colors can be shown simultaneously. Activate the button(s) representing the curves to be displayed. Signals I, II, and III correspond to the three sets of Signal Multipliers.  Any of the above settings can be adjusted while the simulation is running, paused, or stopped. It is now time to start the simulation run! Click on the big "START" button. Colourful curves will subsequently evolve on the display panel. The "START" button turns into the "PAUSE" button, and the "STOP" button becomes available. Experiment with different settings to find your preferred "view".  II  PAUSE  ADJUST  STOP  AUTOMATICALLY  'f  e n a b l e d , the x a n d / o r y a x e s a r e automatically  y-axis (concentration)  adjusted to fit the entire x a n d / o r y data r a n g e s  if  if  Adjust Automatically  Zoom x Refresh every  0  1  Adjust Automatically  the M I G R A T I O N TIME  millisec  Delay fo  Show Curves  interval f o r r e f r e s h i n g the p e a k s DELAY*  jf  [P free; ]  Si Signal II  m\M  REFRESH EVERY * SECONDS Specifies  Y Max (M) =  (1-800)  0.289  | # JC  [Complex]]  [ ^  [A total] |  | jj  [F total; |  Signal III  On-Capillary Position (cm)  MILLISECONDS S p e c i f i e s t h e real-time  Time (s) = 16 340 0 6S 7  5 000e-003  [X] (M)  0 0 <  Figure A.8. Screenshot after the simulation is started.  178  The simulation can be paused or resumed at any time. A snapshot can be taken when the simulation is running or paused, but not when the simulation is stopped. Click on the "Save Snapshot" button (the button with an image of Gaussian curves and a floppy disk) to capture a snapshot. A window pops up asking the user for a location in which to save the snapshot data file, which assumes an ".sss" extension. The "3D Mesh Generation" button indicates that 3-D meshes can be generated. When the button appears as fll, no 3-D meshes will be generated. When the button appears as E2, the generation of 3-D meshes is active. If the user clicks on the active button, a window asking for required settings appears (see Figure A.9). Please pay attention to the IMPORTANT NOTICES placed at the lower half of the popup window. The simulation may have to be restarted if the simulation has run longer than the specified time span.  3D Mesh Settings Yes. Please generate a 3D mesh data file Data Filename (<Parameter  filename>.*.asc)  0  Position of Capillary Time Span: jo  14  second  Curves: [  [*  IP free] ]  ( X Signal II ]  [J  [Complexl)  |^  [Atotal] J  [  it |F total; ]  | it Signal III j  IMPORTANT NOTICE 1 Be careful of setting the Time Span and the Position of Capillary to be monitored Too long time or distance will produce a huge data file 2 The refreshing interval, set in the "Single Run Multi-Cell Model" window, is used to determine how frequently the data would be collected for the 3D mesh OK  Cancel  Figure A.9. Screenshot for the window asking for 3-D mesh settings.  179  §A.3 Load and Plot Data Files in S i m D C C E The generated data files, including electropherograms, snapshots, 3-D meshes, 3-D surfaces, and 2-D curves, can be loaded and plotted within SimDCCE.  Click on the "Load and  Plot Data File" tag to switch to this function. The workplace under this tag is divided into three parts: a setting panel for curves at the top-left, a data browser at the top-right, and a display panel underneath (Figure A. 10).  §A.3.1 Browse Data Files Data files are composed of a number of columns. A proper name is given to each column, followed by a list of numbers. A data file has been loaded and shown in the data browser in Figure A. 10.  Property El Y Plot Range B Curve 1 Enabled Data File X Column No Y Column No Lines between Dots Curve Color B X Data Range (Read On B Y Data Range (Read On  f  u  Value 0.1  E \My Dc<uments\StmCE\Paper4_ACEMgchanisrn\Dat;|[^ E Display 50  True  CelINo 1552 1553 1554 1555 1556 1557 1558  E \My Documents\SimCE\ 2 5 True  • ::::::  1 552. 2 578 0.0.00420176  1 LOAD Curves  jjp  2 PLOT 100% Curve 1  1 552  Position(cm) 1 552 1 553 1 554 1 555 1 556 1 557 1 558  VIEW [FreeA] 1 222317e-009 4 246427e-009 1 030856e-008 2 153159e-008 4 159738e-003 7 701655e-008 1 392135e-007  [Complex *• 3 2£7475e 1 131672e 2 747224a 5 738143e 1 108561e 2 052458e 3 70993 l e v >  v  2 14072  2 578  0 00426641  « < • » \  Single Run: Multi-Cell Model  /  Enumeration Algorithm  ^ Find K From Intersections  \ L o a d and Hot Data Files /  Figure A.10. Screenshot showing the "Load and Plot Data File" workplace, with a loaded data file.  180  SimDCCE  requires the name and location of the data file be entered directly in the edit  box. The user can also click on the OS. button, and then select the appropriate data file. Once the file is selected, simply click on the "View" button to display the data. In order to speed up the loading process, only the first 50 rows of the data file will be loaded. To view the full data file, clear the checkbox before "Display X X rows". The user can also change the number of rows to be loaded.  §A.3.2 Curve Settings The full list of settings available for plotting the curves is shown in Figure A. 11. A total of five curves can be plotted at the same time. By default, each curve is plotted with its own x and y scales and ranges. However, the user has the option to use the same scales and ranges for all curves. The scales and ranges can be adjusted to one of three settings: 1. use individual settings for each curve; 2. use the same scale, but a different start point; 3. use the same scale and start point. The settings available for each curve are identical. 1. Enabled (True or False): A given curve is enabled and will be plotted if "True" is selected. 2. Data File: Each curve can use a different data file. The user defines the location and name of the data file under this setting. 3. X and Y Column Numbers: There are usually more than two columns of data existing in the data file. The user must specify which columns are used as the x and y data.  181  4.  Lines between Dots (True or False): By selecting "False", a single point for each data point is plotted. By selecting "True", consecutive data points will be connected with a line.  5. Curve Color: Each curve can be plotted with different color. 6. X / Y Data Ranges (Read Only): SimDCCE  automatically maintains these values  when a data file is loaded. These values represent the real data ranges. 7. Auto Adjust X / Y (True or False): If "True" is selected, SimDCCE  will automatically  adjust the x/y scale and range to fit all the data points. If "False" is selected, the user must define the x/y scale and range. 8. X / Y Plot Ranges: The ranges of x- and y-axes on the plot. When the Auto Adjust X / Y options is "True", the X / Y Plot Ranges are the same as the X / Y Data Ranges (Read Only), as in Figure A. 11, 9. X / Y Transform Ranges: This option allows the plot to be shrunk, expanded, or translated. The settings can be saved to a Curve Setting File by clicking on LSJ, and then providing the file name and location. The saved setting file can be loaded later with the button  to  plot the same set of curves again.  182  Property B  B  Value  Common Settings X Axis  Use individual settings  B  0.1  X Plot Range Xmin  0  Xmax  1  Y Axis  Use individual settings  B  0.1  Y Plot Range Yrren  0  Ymax  1  Curve 1 Enabled  True  Data Rle  E \ M y DocumentsVSmCE-  XCdumn No  2 3  Y Column No Lines between Dots  True  Curve Color  •  B  •  X Data Range {Read Only)  1.552  XMax  2 578  Y Data Range (Read Onty) Y Max  Auto Adjust X B  B  X Plot Range  B  0.000139352 0  : Tiue 1 552.2 578  XMin  1552  XMax  2578  X Transform  1.0  X Scale  1  X Translation  0  AutoAdjust Y B  OOOOcS  1552.2578  XMm  YMin  B  -  Y Plot Range  True 0.0.00139352  YMin  0  Y Max  000139352  Y Transform  1.0  Y Scale  1  Y Translation  0  Curve 2  B  Curve 3  B  Curve 4  B  Curve 5  Figure A.11. Screenshot s h o w i n g the c u r v e settings.  § A . 3 . 3 Plot Curves  A f t e r the c u r v e s are p r o p e r l y set, c l i c k o n I '**  1 L 0 W O m e  * j t o l o a d the data into the  m e m o r y , p r o m p t i n g a w i n d o w s h o w i n g the l o a d i n g i n f o r m a t i o n to appear. I f a l l c u r v e s are  loaded successfully, c l i c k i n g o n  j»> 2 PLOT 100* w i l l  p l o t the c u r v e s i n their f u l l scales a n d  ranges.  183  1 552  Curve 1 ys  1 57752  2 578  1.57752  2 578  0 001414%  1.552  <*  0 00507692  1 03448e-020 J <  V  Figure A.12. Two curves are plotted at the same time using different colors. The origin of the scale and range is determined by the drop box above the curves.  Figure A. 12 shows two curves plotted at the same time. The x and y ranges of the graph are shown on the top and left of the display panel. Each curve is plotted with its own x and y scales and ranges. By selecting the curve from the drop-down combo box, the user can choose the origin of the scales and ranges are shown on the graph (Figure A.12). The x and y values of the point under the mouse cursor are displayed in the boxes positioned in the middle of the x and y axes, respectively. In Figure A.12, the mouse cursors pointing to the same spot result in the same x value, but two different y values, because the two curves share the same x-axis, but have different y-axes.  184  §A.3.4 Zoom In and Out The user can use the zoom function to enlarge a given graphical region. This is done as follows: 1. Move the mouse cursor to one of the four corners of the desired region. 2.  Press and hold the left mouse button.  3. Move the mouse cursor to the diagonally opposing corner. 4.  Release the left mouse button.  Figure A. 13 demonstrates the zoom-in action and the enlarged graph. The horizontal and vertical scroll bars can be used to move the visible potion of the entire graph. The user can switch back to the full scale by clicking on  9 2 PLOT 100%  or double-clicking on the graph.  0 00141496  0 000729249  Curve 1  1 56558 0 001^496 j \  /  0 00131474  1  0.000794654  I  vj  1 1663  1 71045  " " " ^ ^  —  ....  JI, g—p—i  >  Figure A . 13. Snapshot of the Zoom function.  185  Appendix B Business Plan This is a proposal for the A P R U Enterprise Business Plan Competition - "Extra Chapter Challenge". More information regarding this competition can be found at http://www.fas.nus.edu.sg/APRU/ECC/callForPapers.htm. "This is a business plan competition open to doctoral student in any discipline at Association of Pacific Rim University member universities. The aim is to create value from graduate student research by encouraging commercial enterprise, technology transfer models, social enterprises, non-governmental organizations, and other entities that create economic and social value beyond the dissertation."  186  Ning Fang <.  ™  f S2 SoleIn  Department of Chemistry U n i v e r s i t  y  o f  t: (604) 822-2847 ^ Supervisor: Prof. David D.Y. Chen, Department of Chemistry, U B C 2  0  3  6  M  a  i  n  Columbia Vancouver, B C , Canada V6T 1Z1  B r i t i s h  Subject: Proposal for the Doctoral Thesis Business Plan Competition  Unified Separation Science and High-Efficiency Computer Simulation Background  In the past two decades, great efforts have been taken to unify the numerous separation techniques under one set of theory. In the early 1990s, Giddings first proposed "a theoretical consideration of displacement and transport exerts a unifying influence on separation science, bringing diverse methods under a common descriptive umbrella" [1]. In the late 1990s, Bowser et al. went one step further towards unified separation science by comparing fundamental equations in the major areas of separation science, clarifying the meaning of the capacity factor, and redefining the separation factors [2]. Those developments have been focused on macroscopic (often average) properties of separation systems: the average analyte migration rate, the steady state, resolution, sensitivity, precision, etc. My research project is to study microscopic/instantaneous behaviors of separation systems, which are essential to understand complex phenomena, such as dynamic complexation, sweeping/stacking of sample analytes, and pH depletion. Computer simulation is one of the best ways of visualizing the mechanism behind any separation system. The driving force for chemical separation in chromatography is chemical equilibrium, and that in electrophoresis and centrifugation is physical field. C E uses the driving forces of both chemical equilibrium and physical field in its separation process, and therefore has the advantage of being a starting point for the development of a unified theory for separation science. I have successfully developed a computer simulation system of capillary electrophoresis [3]. The simulation model is based on the differential mass transfer equation, the governing principle of analyte migration in all separation techniques. The simulation program is highly efficient, and it is the first to demonstrate the mechanism of affinity interactions in C E in real time or faster. Various C E modes, such as the affinity C E (ACE) method, the Hummel-Dreyer  187  (HD) method, vacancy affinity C E (VACE) method, vacancy peak (VP) method, and CE frontal analysis (CE-FA) have been simulated.  l|NP2KD5.par-SimDCCr:2 File  View  Help  • c? y 51 Settings  A s#nulation run has been started.  Property  or stopped.  Value  B Instrument H Length of Capillary (cm) Length to Detector (cm) Voltage (V) Idle lime (s) Ramp Time (s) Temperature (C)  RESUME 64.5 54.3 10059 2.04 10.2 20  STOP  y-axis (concentration) |  iff Adjust Automatically Zoom x  (1-800)  Adjust Automatically  V Max (VI)  B Chemical The Analyte (A) The Additive (P) Reaction Type [A] in Sample (M) [P] in Sample (M) [A]m BGE(M) [P] in BGE (M) B Mobilities (cm2yV=) Free Analyte (A) Free Additive (P) Complex IC) Bectroosmotic Row (EOF)  p-nitropheno! b-cyclode>tnn A P - 1:1 0.002 0 0  Delay 0  Refresh every JQ.1  millisec  Show Curves «y [Pfree] | | tf [Complex] | # Signal III  : cos -0.0002357 0 0.0002994 0  =I  0.380  ^jAtotai;]  St  il  ADJUST A U T O M A T I C A I I V If enabled, the x and/or y axes are automatically adjusted to fit the entire x ancVor y data ranges. REFRESH EVERY # SECONDS: Specifies the MIGRATION TIME interval for refreshing the peaks. DELAY t MILLISECONDS: Specifies the real-time delay after every  Time (s)= 17 238 I On-Capillary Position (cm)  1.108  5 028e-003  Length of Capillary (cm) Specifies the length of the capillary ^Experimental Conditions Ready  Simulation Parameters  M < > » \ S i n g l e Run: Multi-Cell Model X Enumeration Algorithm  /  Find rC From Intersections  /  Load?  Figure 1. SimDCCE: the simulation program of dynamic complexation capillary electrophoresis. The user interface of SimDCCE is divided into three regions: 1. the settings panel defines the experimental conditions and simulation parameters; 2. the simulation control panel; 3. the display panel shows the animation of the simulated concentration profiles.  Future Work  The ultimate goal is to develop a computer program to simulate all separation techniques. Certainly, it needs years of hard work to accomplish such an ambitious goal. However, it is clear that the following objectives should and can be done in the near future. 1. Build a framework to accommodate all separation techniques. The framework provides a skeletal support used as the basis for individual separation techniques being constructed as separated or connected modules. Simulation of dynamic complexation capillary electrophoresis (SimDCCE) will be the first module in the framework. The framework must be strictly based on the unified separation science and capture the  188  2.  3. 4. 5.  similarities among all separation techniques. This approach should make it much easier to add new simulation modules in the future. Develop the simulation of chromatography. Chromatography is usually simpler than capillary electrophoresis because the separation in chromatography is based on the distribution of the analyte between two immiscible phases (the stationary phase and the mobile phase), and because the stationary phase has a mobility of zero. Design a more user-friendly interface. The simulation program has to be accessible to professionals in the fields of chemistry, biochemistry and pharmaceutics. Find a better way to input more complex experimental conditions. Add the support of transportation and interaction of H and OFT to the simulation of CE. This would enable us to study many important phenomena related to pH boundaries and pH depletion. +  6. Refine the calculation modules to further improve simulation efficiency. The final product of this stage will be a new simulation program with better user interface, better flexibility of adding or deleting any simulation module, more simulated separation techniques, more functionality, and better performance.  Market Values and Audiences  The simulation program can be used for educational and research purposes. 1. The simulation program is a study tool to help undergraduate and graduate students to understand separation science. It is often difficult to let students have hands-on experiences on expensive analytical instruments. The simulation provides unique and valuable opportunities for students to experiment each technique with different conditions. SimDCCE has already been used in a graduate level analytical chemistry course to help students understand the mechanisms of complex affinity interactions in CE. (Please contact Prof. David Chen at the Chemistry Department, University of British Columbia for more details.) The simulation program can be distributed with a textbook. 2. The high efficiency of this simulation model makes it possible to bridge the gap between theoretical modelling and real C E experiments, and to predict the peak shape and peak position at any given time of a C E process. The simulation results provide guidance for scientists who are interested in using the migration times obtained from C E experiments to extract physicochemical parameters, such as binding or dissociation constants [4]. The results should also provide insights into the phenomena observed in separation systems with much shorter columns, such as microfluidic devices. Therefore, analytical and/or pharmaceutical labs and companies, which are interested in drug and receptor binding, detailed mechanisms of separation systems, and development of analytical methods, can use the simulation program to facilitate their research.  189  People and Budget  A designer and programmer (myself), 4 months of work to build the framework and add the simulation module for chromatography. -$8000 C A D . A programmer (preferably a senior undergraduate or co-op computer science student), 2 months of work to develop a new user interface, -$3000 C A D . Several voluntary testers.  Feasibility  The most difficult part of this project, the development of an efficient implementation of numerical method to calculate the analyte migration and affinity interactions, has been finished, and the results have been published [3-5]. The further development should be a lot easier.  References  Giddings, J.C., Unified Separation Science. 1991: Wiley-Interscience Publication: New York. Bowser, M.T., et al., Redefining the separation factor: A potential pathway to a unified separation science. Electrophoresis, 1997. 18(15): p. 2928-2934. Fang, N. and D.D.Y. Chen, General approach to high-efficiency simulation of affinity capillary electrophoresis. Analytical Chemistry, 2005. 77(3): p. 840-847. Fang, N. and D.D.Y. Chen, Enumeration algorithm for determination of binding constants in capillary electrophoresis. Analytical Chemistry, 2005. 77(8): p. 24152420. Fang, N., E. Ting, and D.D.Y. Chen, Determination of shapes and maximums of analyte peaks based on solute mobilities in capillary electrophoresis. Analytical Chemistry, 2004. 7 6 ( 6 ) : p. 1708-1714.  190  

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