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Determination of binding constants by computer simulation of affinity capillary electrophoresis Fang, Ning 2002

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D E T E R M I N A T I O N OF B I N D I N G C O N S T A N T S B Y C O M P U T E R S I M U L A T I O N OF A F F I N I T Y C A P I L L A R Y E L E C T R O P H O R E S I S by NING FANG B . S c , Xiamen U n i v e r s i t y , 1998 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE THE FACULTY OF GRADUATE STUDIES Depar tment o f C h e m i s t r y We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA September 2002 © N i n g Fang , 2 002 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head o f my department or by h i s or her r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n permission. The U n i v e r s i t y o f B r i t i s h Columb Vancouver, Canada Department of Abstract The aim of this work is to test and improve the mathematical models of affinity capillary electrophoresis (ACE), to implement these models by computer simulation, and to use the simulation programs to determine the medium- and high-affinity binding constants for protein-ligand interactions. The first part of this thesis presents the first mathematical model, the one-cell model, of A C E and the derivations of the equations for both 2:1 (analyte:additive) and 1:1 interactions. The steady state equations were also introduced due to their important roles in analyzing the simulated results. The second part of this thesis presents four simulation programs which were based on the one-cell model and used to simulate the A C E processes, to collect and analyze simulation data, and to estimate the binding constants for the protein-ligand interactions. In this work, we performed several series of experiments and used the simulation programs to determine the binding constants for the 2:1 interaction between flurbiprofen and transthyretin, and the binding constant for the 1:1 interaction between chlorpromazine and quaic riboflavin binding protein (qRfBP). The one-cell model is extremely fast in simulating A C E processes, but it cannot accurately describe the A C E experiments with high concentrations of analyte ([A]) and additive ([P]) or a high ratio of [A] to [P]. Thus, in the final part of this thesis, a new model and its improved implementation algorithm were proposed. n Table of Contents Abstract » Table of Contents "i List of Tables vi List of Figures vii Abbreviations 'x Acknowledgments x Chapter 1 Introduction 1 1.1 General Introduction to Capillary Electrophoresis 1 1.2 CE Set-up 2 1.3 Theory of Capillary Electrophoresis 4 1.3.1 Electrophoretic Mobility, juep 4 1.3.2 Electroosmotic Mobility, fleof 5 1.3.3 Affinity Capillary Electrophoresis (ACE) 9 1.3.4 Regression Methods 11 1.4 Introduction to Computer Simulation in Chemistry 12 Chapter 2 Experiments and Mathematical Description 15 2.1 Introduction 15 2.2 Experimental 16 2.2.1 Materials 16 2.2.2 Equipment IV 2.2.3 1st Series of Experiments 18 iii 2.2.4 2" Series of Experiments 21 2.2.5 3 r d and 4 t h Series of Experiments 22 2.3 Mathematical Model for 2:1 Interactions 26 2.4 Mathematical Derivation for 2:1 Interactions 29 2.4.1 Change in Protein Concentration within the Analyte Plug 29 2.4.2 Effective Mobility of the Analyte Plug 30 2.4.3 Steady State 34 2.5 Mathematical Description of 1:1 Interactions 35 Chapter 3 Computer Implementation - Model One 37 3.1 Introduction 37 3.2 Overview of the Simulation Program 38 3.3 Calculation Module in SimCETonly 43 3.3.1 Procedure of Simulation in SimCETonly 43 3.3.2 Discussions on Steady State 44 3.3.3 Discussions on the Time Increment (At) 48 3.3.4 Discussions on Broadening of Analyte Plug 49 3.4 Calculation Module in SimCE 53 3.4.1 Procedure of Simulation in SimCE 53 3.4.2 Discussions on the Length of the Plug 56 3.4.3 Flow Chart of the Calculation Model 59 3.5 Binding Constants - 2:1 Interactions (Flurbiprofen:TTR) 63 3.6 Binding Constants -1:1 Interactions 74 3.7 Conclusions 77 iv Chapter 4 Computer Implementation - Model Two 79 4.1 Introduction 79 4.2 The New Model 80 4.2.1 Mathematical Description 80 4.2.2 Computer Implementation 82 4.3 Conclusions and Further Development 85 Appendix A Newton-Raphson Method for Solving f(x) = 0 86 A . l Mathematical Description 86 A. 2 Computer Implementation 87 Appendix B Instructions for Using the Simulation Programs 89 B. l Introduction 89 B.2 Naming Convention and Directory Structure 89 B.3 SimCETonly 92 B.4 SimCE 94 B.5 SimCEData •. 96 B.6SimCEGetK 100 References 102 v List of Tables Table 2.1 Experimental Data for the I s Series 20 Table 2.2 Common Parameters for the 1 s t and 2 n d Series 21 Table 2.3 Experimental Data for Series Two 23 Table 2.4 Common Parameters for Series Three and Four 24 Table 2.5 Experimental Data for Series Three 24 Table 2.6 Experimental Data for Series Four 25 Table 3.1 Key Parameters for SimCETonly 43 VI List of Figures Figure 1.1 Schematic representation of a CE 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 The electrical double layer formed at the inner capillary wall in contact with an electrolyte solution 7 Figure 2.1 Electropherogram 19 Figure 2.2 The 2:1 interaction between flurbiprofen (A) and TTR (P) 28 Figure 3.1 Relationships between four simulation programs 39 Figure 3.2 Overview of SimCE 40 Figure 3.3 The formats of output data from SimCETonly and SimCE 42 Figure 3.4 Plots of [P] (free protein) vs. Time and [A] (free analyte) vs. Time 45-46 Figure 3.5 Plots of [A] (free analyte) vs. Time with various time increments 49 Figure 3.6 Broadening of the analyte plug 50 Figure 3.8 A l l pairs of K and jUep A^r which can generate migration times equal to the real (experimental) migration time 55 Figure 3.9 (a) (b) and (c) logK vs. / / A 2 P curves for various length of the plug under different experimental conditions 58-59 Figure 3.10 (a), (b) and (c) The flow chart of the calculation module 60-62 Figure 3.11 Effects of different jUepj 64 Figure 3.12 Effects of different /jep>P 65 Figure 3.13 (a), (b) and (c) The logK vs. (/4 p ,A- /4P,A2P) curves for the 2 n d Series 67-68 Figure 3.14 Shapes of the curves 69 vii Figure 3.15 Finding binding constants from Experiment Set 1 and 2 of the 2" Series 72 Figure 3.16 (a), (b) and (c) Curves for the 3 r d series (chlorpromazine : qRfBP) 74-75 Figure 3.17 Binding constant for chlorpromazine : qRfBP 77 Figure 4.1 Model Two - The capillary is divided into narrow cells 80 Figure 4.2 Model Two - The arrangement of cells in the memory of the PC 84 Figure A. 1 Newton-Raphson method for solving f(\) = 0 86 Figure B . l User Interface of SimCETonly 92 Figure B.2 User Interface of SimCE 95 Figure B.3 User Interface of SimCEData 96 Figure B.4 User Interface of SimCEGetK 101 vm Abbreviations CE: Capillary Electrophoresis A C E : Affinity Capillary Electrophoresis BGE: Background Electrolyte HPLC: High Performance Liquid Chromatography EOF: ElectroOsmotic Flow TTR: TransThyRetin qRfBP: quaic Riboflavin Binding Protein UI: User Interface 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. Mark Thachuk and Dr. Chiara Galbusera for their great job in developing the mathematical model for affinity capillary electrophoresis. I really appreciate the kindness and generosity of Dr. Chiara Galbusera for her experimental data, the detailed explanation, and most importantly, the countless helpful discussions with me. I am especially grateful to David McLaren for his unselfish great jobs to keep our lab rolling smoothly and his valuable help and advice in my research and the writing of this thesis. I want to thank my other lab mates, Wuyi Zha, Jennifer Chu, Junji Kobayashi, Brad Schneider, Cindy Schneider, and Gwen Bebault for sharing their knowledge and experience about CE. 1 also appreciate the generosity of Beckman Instruments, for supplying a second CE instrument and a HPLC instrument. 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! x Chapter 1 Introduction 1.1 General Introduction to Capillary Electrophoresis Electrophoresis, the movement of electrically charged particles through a gas or liquid as a result of an electric field formed between electrodes immersed in a medium, has been a important separation technique in chemical/biochemical research for decades. Capillary Electrophoresis (CE), the separation technique based on the 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 after more than two decade of active research. High speed, high mass sensitivity, high resolution, precision, and the ease of automation are among the advantages usually claimed for CE 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 mainstay of separations of nonvolatile compounds in the modern laboratory, 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 separation techniques can be asked to solve. For instance, separation methods are not typically used in high throughput analyses or for real-time monitoring of reactions. The time-consuming nature of separations has driven many researchers to investigate methods of improving the speed of separation techniques. CE is a relatively rapid separation technique, and usually takes several to tens of minutes. Fused silica capillaries used in CE have a high surface-to-volume ratio, which reduces the influence of the Joule heating generated from the applied field. With typical capillary 1 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. Thus the volumes of buffer and analyte required are also on the order of microliters. Therefore, CE is highly mass sensitive, and ideal for studying small samples, such as DNA/RNA and proteins, where extremely small amounts of analyte are available. Commercial CE instruments are highly automatic. For carrying out a regular experiment, one usually spends minutes to a couple of hours setting up the machine and program, then the instrument could run automatically for hours or days. 1.2 CE Set-up Whether CE 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 which usually provides voltages in the range of 5kV to 30kV. 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. 2 o -I—> S-H Q "3 -O oj Q o Q A o3 - O 3 o3 O ^ m o 1 > ' S o - a • r-H o3 > m O 03 > =3 -i—> OH 3 u O a o a CO CD OH 4) o GO 3 bp 03 o3 GO 3 1.3 Theory of Capillary Electrophoresis 1.3.1 Electrophoretic Mobility, juep There are two forces acting on a charged particle in a homologous electric field. The electrostatic force (Ft,) accelerates the charged particle, and the frictional force (F/) counteracts the electrostatic force and decelerates the charged particle (Figure 1.2): where Q is the net effective charge of the ion (Coulombs), E is the electric field strength ( V • cm" 1), Z is the number of charges on the component, eg is the elemental charge (1.602xl0"1 9 Coulombs), / is the frictional coefficient (g-s" 1 ) and v is the ion velocity (cm-s - 1 ) . In a liquid hydrodynamic medium, the frictional force for a spherical particle is represented by: / = 6-7rrjr where r is the radius of the particle (cm) and r| is the viscosity of the medium (Pa • s ). Fe=QE = Ze0E (1.1) Ff=-f-v (1.2) + + + + © + Figure 1.2 Forces acting on charged particles in a solution 4 During electrophoresis, equilibrium between these two counteractive forces is obtained, at which point the forces are equal. Therefore: (1.3) Fe=~Ff (1 4) The electrophoretic mobility is characteristic of each individual ion, and is determined by the velocity per unit electric field. It can be defined as: (1.5) where fjep is the electrophoretic mobility (cm 2 s"1 - V " ' ). Solving for velocity by substituting Equation 1.4 into Equation 1.5: „ Q . (1.6) 6xr/r This equation implies that the electrophoretic mobility of each species in a solution depends only on its charge and size. 1.3.2 Electroosmotic Mobility, /ie0f 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 CE, the bulk flow of separation buffer through the capillary, is generated due to the ionizable nature of the capillary wall. 5 The fused silica capillary used in CE 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 until pH 9. Therefore, negatively charged Si-O" groups are formed on the wall of the capillary (Figure 1.3). The positively charged particles in the solution are attracted by this layer of negative charges on the capillary wall, forming an electrical double layer. A part of cationic counterions will adsorb to the wall, giving rise to an immobilized compact layer which is called the Stern layer. The charge and potential distribution in the Stern layer are mainly determined by the ionic strength of the solution and the geometrical restrictions of ions. The Stern layer only neutralizes part of the excessive charges at the capillary wall. There are more cations present in the adjacent area because of the residue negative charges, forming a diffuse layer. The electrical double layer at the capillary wall can be illustrated by Figure 1.4. The zeta potential is the potential drop across the diffuse layer. OH o- OH O " inner surface of a capillary o o o o o o o o o capillary wall o JK A A o o o o outer surface of a capillary Figure 1.3 Representation of the silanol groups inside a capillary 6 Stern from the wall Figure 1.4 The electrical double layer formed at the inner capillary wall in contact with an electrolyte solution If an electric field is applied across the capillary, the cations in the diffuse layer begin to migrate towards the cathode. Because these cations are hydrated, they will drag the solvent with them. This movement of solution generates the electroosmotic flow (EOF). The relationship between the velocity of the EOF (veof) and the zeta potential (Q at the capillary/solution interface is given by v =-u .E = - - ^ - E ( L 7 ) where E is the electric field strength, e is the permittivity of the solution, and 7] is the viscosity. The negative sign means that when £ is negative, the electroosmotic flow is towards the cathode, defined as a positive flow. There are several factors that can affect the EOF: 7 1. At higher pH, a greater number of negative charges are present at the capillary wall, resulting in a higher £ potential, thus higher velocity of the EOF. 2. At higher ionic strength, more cations are present in the Stern layer, resulting in a smaller C, potential, thus lower velocity of EOF. 3. At higher temperature, the Stern layer becomes less stable, and the viscosity is reduced, resulting in a higher £ potential and less resistance, thus higher velocity of EOF. Therefore, by various chemical or physical methods, one can control the EOF in order to achieve better separation. Due to the fact that the electroosmotic mobility is often larger than the electrophoretic mobility of most analytes in 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 most slowly. Because the EOF is originated from the wall, it has a flat flow profile, instead of the parabolic flow profile in pressure driven flows such as in chromatography. The peaks in CE are usually much sharper. 8 1.3.3 Affinity Capillary Electrophoresis (ACE) Affinity capillary electrophoresis (ACE) is a powerful tool for studying protein-ligand interactions [1]. 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 ^ A P ( 1 8 ) where A is the analyte, P is the additive (such as proteins, surfactants, etc.), and AP is the analyte-additive complex. The equilibrium constant (AT) for Equation 1.8 can be written as: K = - ^ - (1.9) [A][P] where [AP] is the concentration of the analyte-additive complex, [A] is the concentration of the analyte and [P] is the concentration of the additive. The capacity factor (k') can be described as the ratio of the amount of complex to the amount of free analyte, which is very similar to the definition of the capacity factor in chromatography (defined as the ratio of the amount of analyte in the stationary phase compared with the amount of analyte in the mobile phase). k<=HAc_ = [dfl = K[p] (1.10) nc [A] where YlAC is the amount of complex and Ylc is the amount of free analyte. If the fraction of free analyte (J) is defined as [J4]/[J4]O ([A]o is the original analyte concentration), when [P] is much greater than [A]o, substituting [A]o by gives: f = TT< ( U 1 ) \ + k' The electrophoretic mobility of the free analyte is represented by /nep,A- The electrophoretic mobility of complex is represented by /Jep,AP- Then the net electrophoretic mobility of the analyte, which is represented by fi* , can be written as: MeP=MP,A+(l-f)MeP,AP 1 w ~\+k'Mep-A + l+k<Mep<AP d - 1 2 ) 1 , K[P] l + K[P]MEP'A + l + K[P]MEP'AP The Equation (1.12) actually represents the case of an ideal state, in which additives do not affect the physicochemical properties (such as viscosity) of the buffer. But in reality, as shown in Equation (1.6), the electrophoretic mobility is inversely proportional to the viscosity of the solution which will change when the concentration of the additive is changed to give different k' values (Equation (1.10)). So i f we don't ignore the changes in viscosity, a correction factor (v) has to be introduced into Equation (1.12): a 1 K[P] VMcp ~\7k[P]Mep>A + 1UK[P]ME"AP ( U 3 ) The correction factor (v) is defined as 77/77°, where 77 is the viscosity of the buffer at different concentrations of additive 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 analyte and EOF marker, one can obtain the effective electrophoretic mobility (ju^p) of analyte by using these equations: 10 MeP=MA-Meof (114) ..A -^ detector -^ total • -. c\ tA - V Meaf = r / t 0 t a ' ( L 1 6 ) where jUA is the apparent mobility, detector is m e length of the capillary from the injection end to the detector (cm), Ltota] is the total length of the capillary (cm), tA is the migration time of the analyte (s), tuof is the migration time of the EOF marker (s) and V is the voltage applied through the whole capillary (volt). 1.3.4 Regression Methods Equation (1.13) can be rearranged to: , A x , (Wep -MePJ = ] + K [ p ] (MeP.AP ~ Mep,A) (1-17) There are very similar methods for studying complexation in many research areas by transforming derived equations very similar to Equation (1.17). Equation (1.17) can be rearranged to 3 different forms: 1 1 1 1 + — (1.18) VK - Mep.A (Mep.AP ~ Mep,A ) " K iP] Mep,AP ~ Mep.A [P] [P] 1 — 7 ^ ' + (1-19) VMep ~ Mep,A Mep,AP ~ Mep.A (Mep,AP ~ Mep,A ) ' K 11 [p] = ~ K • « ~ ^ a ) + K • (Mep.AP ~ M e p , A ) (1.20) These 3 equations can be referred to as double-reciprocal (Equation (1.18)), ^-reciprocal (Equation (1.19)), and x-reciprocal (Equation (1.20)) plots, respectively. A l l three equations are derived from the same equation. Therefore, theoretically they should generate identical results. However, this is rarely the case due to the different error propagation associated with the variables during the 3 different transformations. A thorough study of this has been published [2, 3]. 1.4 Introduction to Computer Simulation in Chemistry The first electronic digital computer, the Electronic Numerical Integrator and Computer (ENIAC) was developed in 1946. Since then, computers have undergone incredible improvements and have evolved into extremely powerful tools used for a vast number of applications. Computers are playing larger and larger roles in almost every field of modern chemistry. Almost all computing tasks are done by some kind of computer. Besides these obvious applications of computers, computers are involved in many other fields. Almost all instruments are controlled by some kind of computer. Chemists rely more and more on computers to collect, store and analyze huge volumes of data, to draw structures of very complex molecules, to write research papers, and to play music while working. " A simulation is the imitation of the operation of a real-world process or system over time. Simulation involves the generation of an artificial history of a system, and the observation of that artificial history to draw inferences concerning the operating 12 characteristics of the real system." (Banks, J., J.S. Carson, and B.L. Nelson, Discrete-Event System Simulation) [4] The behavior of a system as it evolves over time is studied by developing a simulation model, which usually takes the form of a set of assumptions concerning the operation of the system. Using computers' tremendous computing power to simulate chemical/physical systems is a fairly new field. Here are two examples: 1. The chemical kinetics simulation project, originated at IBM's Almaden Research Center in San Jose, California, is "a multi-faceted effort aimed at understanding, predicting and controlling chemical reactions used in processing of inorganic and polymeric thin films. Modelling and experiment are intertwined to develop new insights to the behavior of these materials under a wide range of conditions, and to use these insights to enhance discovery as well as applications." [5] 2. The use of computer simulations in investigations of protein-protein interactions has been reported recently [6]. There are several advantages of computer simulation, as well as some disadvantages. Some typical advantages of computer simulation are: 1. Simplify the problems. By considering only one condition at a time, one can easily find out the influences of each factor. 2. Once the simulation programs are finished, one can change the conditions and repeat the simulation easily. 3. Simulate processes under extreme conditions which are generally unreachable in the real world. 13 Some disadvantages are: 1. Simulation results may be difficult to interpret. Because in many cases, the outputs are essentially random variables, and/or the assumptions and some simplification are made, it may be hard to determine how "real" the system is. 2. Simulation modeling and analysis can be time consuming and expensive. Usually a huge amount of computing is inevitable. 14 Chapter 2 Experiments and Mathematical Description 2.1 Introduction Before the computer-simulated system of affinity capillary electrophoresis (ACE) can actually be designed and implemented, one must choose some particular system to start with, do some experiments, get as much information from the experiments as one can, and build up a mathematical model. Computer simulation of CE has been demonstrated by several groups [7-10]. These efforts mainly focused on simply simulating the electrophoretic migration behaviors. By considering many factors such as diffusion and electric field, these efforts obtained good results which fitted the experimental data quite well. The aim of this project is to simulate A C E by a computer simulation system. The equilibria of protein-ligand binding are the core of A C E . If the binding constants and the electrophoretic mobility are known, the migration time and/or peak shapes can be simulated. On the other hand, the binding constants can be obtained by using migration time and other conditions as inputs. Protein-ligand binding studies play an important role in many aspects of protein biochemistry and enzymology. The binding of proteins to small molecules generally occurs due to weak hydrophilic and hydrophobic interactions at the protein surfaces. However, when a receptor-like area that can fit only substances with specific chemical functionalities and particular geometries is present, a stronger and more specific binding may occur, leading to medium- to high-affinity interactions. A series of equations can be derived to describe the equilibrium and migration behaviors of A C E . As the equilibria become more and more complicated (higher order 15 equilibria), the complexity of equations rises exponentially. In this project, 2:1 (analyte/additive) and 1:1 stoichiometries are investigated. For the studies of 2:1 stoichiometry, the drug flurbiprofen was used as the analyte and was injected into a capillary which was filled with low concentration of the protein transthyretin (TTR). For the studies of 1:1 stoichiometry, the protein used to fill the whole capillary was quaic riboflavin binding protein (qRfBP) and the analyte injected was chlorpromazine or riboflavin. When the electric field was applied, the migration of the drug and the protein, as well as the complex formed, presented an interesting phenomenon that can be described by a mathematical model. There were four series of experiments done in total. The first and second series were designed for studying the 2:1 interaction between flurbiprofen and transthyretin (TTR). The third series was designed for studying the 1:1 interaction between chlorpromazine and qRfBP. The last series was designed for studying the 1:1 interaction between riboflavin and qRfBP. 2.2 Experimental* 2.2.1 Materials Transthyretin was purified from human plasma as described by Malpeli et al [11]. The homogeneity of the protein preparation was confirmed by SDS PAGE, N-terminal amino acid • sequence and mass spectrometry, and the final product was lyophilised. * Most of the experiments upon which the computer simulations described in this thesis were performed by Dr. Chiara Galbusera. The purpose of this project was to re-investigate the experimental data in a different way. Dr. Galbusera very kindly provided me with the data and the descriptions of her experiments. 16 A stock solution (7.2 |4.M) was prepared by dissolving known amounts of TTR in 50 m M sodium phosphate buffer (pH 7.4), filtered through a 0.45 |im Millipore membrane and appropriately diluted with the same buffer. Flurbiprofen was purchased from Sigma (St. Louis, MO, USA). Sample preparation was carried out by dissolving known amounts of flurbiprofen in a suitable volume of 10 mM sodium phosphate buffer to reach the desired concentration. A l l buffer solutions were prepared fresh daily using doubly distilled water. Phosphate buffer solutions were prepared by mixing analytical grade dibasic sodium hydrogen phosphate and sodium dihydrogen phosphate (Sigma St. Louis, M O , USA) solutions to give a pH of 7.4. A l l solutions were filtered through a 0.2 u\m Millipore (Bedford, M A , USA) membrane filter and degassed by sonication prior to use. The preparation of quaic riboflavin binding protein (qRfBP) was described in detail by Lorenzi, et al [12]. 2.2.2 Equipment CE experiments were performed using a Beckman P/ACE System 5000 (Beckman Instruments Inc., Mississauga, ON, Canada) with a built-in U V detector. The capillary tubing (Polymicro Technologies, Phoenix, AZ) was of uncoated fused silica with an internal diameter of 50 |im, a total length of 47 cm for 2:1 interaction or 60.2 cm for 1:1 interactions, and a length from inlet to detector of 40 cm for 2:1 interaction or 50 cm for 1:1 interactions. 17 2.2.3 1st Series of Experiments The first series of CE experiments (2:1 stoichiometry between flurbiprofen and TTR) were carried out following this procedure: The capillary was rinsed before each run first with NaOH (SDS for the analyses with protein additive [13]), and then with H 2 0 (1000 mbar, 2 min each), and finally conditioned with the background electrolyte (1000 mbar, 3 min). The mobility of the analyte (flurbiprofen) at various concentrations (100, 50 or 25 uM) in buffer was evaluated at different concentrations of additive (TTR) (0.00, 0.36, 0.72, 1.08, 1.44, 1.80, 2.52, 2.88, 3.24, 3.60 uM). The migration time for every experiment was measured from the electropherograms. An example of the electropherograms is shown in Figure 2.1. (Note that the horizontal axis is a mobility scale as opposed to the standard time scale [14].) Electrophoresis was carried out at 20 kV and 25°C; U V signal channels were set at 254 nm for flurbiprofen. The measurements of the ligand mobility in neat buffer (jJep,A) were repeated before carrying out a new protein concentration level. The concentrations of TTR solutions used as additives in the B G E were determined by using a calibration curve and measuring the absorbance at 280 nm. The change in buffer viscosity upon addition of TTR was measured by filling the capillary with solutions containing different concentrations of TTR and measuring the time necessary for a water marker to be pushed through the capillary past the detector when applying a negative pressure to the outlet end [15]. The sample volume introduced into the capillary by applying a pressure of 0.5 psi for 3 sec was calculated to be 3.6 nl. 18 19 Data was collected and processed using System Gold software (Beckman) and a Pentium II PC computer. The data obtained from the first series of experiments described in the previous section is shown in the Table 2.1. Other required parameters are listed in Table 2.2. Given the time of EOF, the electroosmotic mobility (jUeof) can be obtained using Equation (1.16). Table 2.1 Experimental Data for the I s Series P H i S p i l i i H W i i Sot 1 Sol 2 Set 3 [A|«=25u.\1 ' , |A | ,F50JIM | \ | « = I 0 0 ( I M -1.8859X10" 4 -1.8890x10"* -1.8854x10^ IPIo(uM) T, v , (sec) T«„/(M.T) 1 . yi (sec) T („/(soc) !,,/,(sec) rn</(scc) 0 301.062 187.692 301.578 187.776 301.83 188.01 0.36 309.152 190.415 311.213 191.032 312.724 191.829 0.72 322.14 194.849 327.268 196.604 327.256 196.601 1.08 335.9 199.704 349.714 204.117 341.907 201.422 1.44 339.42 200.78 357.756 206.629 362.563 208.085 1.80 356.264 206.299 380.576 213.818 386.441 215.763 2.16 368.175 210.161 397.325 218.779 398.885 219.486 2.52 383.027 214.808 398.263 221.66 413.243 223.096 2.88 395.905 218.817 429.111 227.862 421.803 225.008 3.24 411.86 223.436 446.508 232.298 458.367 234.879 3.60 426.555 227.588 473.226 239.19 505.195 246.142 [P]o : The concentration of protein filling the entire capillary [A] 0 : The initial concentration of analyte (ligand) in the injection plug T^p '• The experimental analyte migration time Teof: The time of electroosmotic flow (EOF) 20 Table 2.2 Common Parameters for the I s and 2" Series The voltage applied across the capillary (E) (V) 20000 The length of capillary (cm) 47.0 The length of capillary to the detector (cm) 40.0 The injection length (cm) 0.18 The mobility of TTR (xlO"4 cm2/V-sec) -1.75±0.1 The injection length is a key parameter in the computer simulation. It is directly related to the validity of the assumptions made for the mathematical models and the computer simulation. In all experiments, the volume introduced into the capillary by applying a pressure of 0.5 psi for 3 seconds was 3.6 nl, and the plug length was 0.18 cm. This calculation is based on Beckman P/ACE injection parameters provided by the manufacturer. The mobility of free analyte (ligand) (jueptA), free protein (TTR) (juepP) and the complex (juepA^p) change when the experimental conditions change, because the mobility is directly related to the viscosity and the electric field according to Equation (1.7), and also related to the actual conditions of the capillary wall. Thus the measurements of the ligand mobility in neat buffer (when [P]o = 0) were repeated before carrying out a new protein concentration level in order to get a suitable adjusted fi A parameter (Table 2.1). 2.2.4 2nd Series of Experiments The second series was also for studying the 2:1 interaction between flurbiprofen and TTR, too. This series of experiments followed procedures similar to the first series of experiments. The only difference was in the setting of the concentrations of flurbiprofen and 21 TTR. In this series, the product of the initial concentrations of flurbiprofen and TTR ( [ P ] o x [ A ] 0 ) was kept constant at 3.60E-11, 1.08E-10 and 1.80E-10 for three sets of experiments, respectively. The measurements of the ligand mobility in neat buffer ( / 4 p , A ) were repeated before carrying out any single experiment, not just before a new set of experiments. (The reason for keeping [P]0 x [A] 0 constant is presented in Chapter 3.) Table 2.4 shows the concentrations of flurbiprofen and TTR, the corresponding EOF time, migration time and the measured ligand mobility in neat buffer (jJep,A)-2.2.5 3 r d and 4th Series of Experiments In the last two series of experiments, the concentrations of qRfBP were kept constant while the concentration of chlorpromazine (for Series Three) or riboflavin (for Series Four) varied in wide ranges. The measurements of the ligand mobility in neat buffer (jiepj) were also repeated before carrying out any single experiment. Each juepj was measured three times and the average value was taken from those measurements. Tables 2.4, 2.5, and 2.6 show all useful data related to the 3 r d and 4 t h Series of Experiments. There are two effective mobilities of qRfBP {pep,p) listed in Table 2.4. They correspond to two isomers of qRfBP. 22 o & E-i GO •c Q -4—> C e •a <u X W m C N <u © < =t -2.070 -1.992 -1.995 -1.995 -1.987 -1.926 -1.958 -1.937 -1.966 -1.953 \ SET 3 _ 1 p o "o S CD 1-\Cfl 236.63 260.04 275.66 293.58 269.68 293.38 271.23 242.16 288.78 308.80 \ SET 3 , II ••< X S'cJ S (Pi 1- w. 494.14 564.46 671.30 795.64 647.00 605.63 533.96 504.86 785.28 939.60 \ 21 100.0 300.0 150.0 100.0 60.0 50.0 42.9 37.5 33.3 30.0 \ o 0.36 0.72 1.08 1.80 2.16 2.52 2.88 3.24 3.60 \ + < a V =L -1.981 -2.043 -2.025 -2.020 -2.023 -2.023 -2.006 -1.995 -1.994 -1.833 -1.835 1 .NOP -10 o O <i> CD h-212.28 248.66 239.94 248.58 282.26 279.64 281.08 286.32 302.76 354.36 358.42 SET 2 II < X a>;0. H .w 384.15 547.68 500.22 542.84 747.38 734.08 738.56 777.52 939.54 1331.12 1395.00 57 =?! 100.0 500.0 250.0 167.0 125.0 100.0 83.3 71.4 62.5 55.6 50.0 o 0.36 0.72 1.08 1.44 1.80 2.16 2.52 2.88 3.24 3.60 * < . 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Thus in this and the next section, only the 2:1 interactions are discussed. The 1:1 interactions will be discussed solely in Section 2.5. The equilibrium for 2:1 interactions is represented by: 2A+P=^ A 2P (2.1) where A is the analyte (flurbiprofen), P is the protein (TTR), and A 2 P is the complex formed. Previous work has shown the two binding sites to be identical in nature, with the same affinity for the two molecules of flurbiprofen [16]. The mobility of the analyte is given by Mep = fA^ep.A fA2pMep,A2P (2-2) where/A is the fraction of free analyte, fA^P is the fraction of analyte in a complexed form, juepj is the mobility of free analyte and £lep,A2p is m e mobility of complexed analyte. The fractions of free analyte and complex are defined by: L = i ~ f V (2-4) where [A2P] is the concentration of the complex, and [A]o is the initial concentration of the injected analyte in the sample plug. The different mobilities of the free analyte (A), the protein (P), and the complex (A2P) play important roles in the experiment. Under the experimental conditions, the mobility of the additive is higher than the mobility of the free analyte, which itself has a higher mobility 26 than the complex. Figure 2.2(a) shows the situation after the injection and before the start of the CE process. The additive in the capillary, positioned ahead of the analyte, migrates more quickly and can't be reached by the analyte. However, the additive that enters the capillary from behind the analyte plug can migrate into the analyte plug, as demonstrated in Figure 2.2(b). Thus, the amount of the additive in the analyte plug increases during the run. At the beginning of the run, the analyte to additive ratio is very high due to the low starting concentration of the additive and the high analyte concentration. As the run progresses, more protein molecules come into the analyte plug, so the concentration of protein increases in the analyte plug, as demonstrated in Figure 2.2(c), and eventually the amount of protein going into the plug will be equal to that leaving the plug. [17] Several assumptions for the development of a mathematical model have to be made. (1) The width of the analyte plug remains constant. This assumption is valid when the injection length is narrow enough. The detailed discussions justifying this assumption will be presented in the next chapter. (2) Within the analyte plug, rapid (instant) equilibrium is established between analyte and additive. This assumption is valid for interactions with medium- to high-affinity binding constants. (3) The electric field is constant throughout the whole capillary, and the mobilities (jueP,A>Juepj.,r>Mep,p>JuEOF) are independent of space and time. In a typical A C E experimental set-up, the concentrations of analyte and additive are very small (on the order of 10"6 M), comparing with the concentration of the background electrolyte (on the order of 10"3 M). 27 The first assumption shows that all analyte, no matter which form it is, is in the analyte plug. The second assumption shows that there is no difference in mobility on the left and right edge of the analyte plug. Therefore the whole analyte plug is treated as one cell in both the mathematical model and the computer simulation. a) Ligand (A):0 Protein (P): o jlp>|i.A Injection Detection » i t • i n i 1 1 1 1 i I I i • i t i • 11 • I.I.I.I o o o o o o o o o o © o b) Complex (A 2 P):§ 1 1 1 1 S 000 1 1 1 1 i i t I I I I 11 I I • : • • i 3 |lp>M.A>UA2P c) [A2P]>[P]c o O o o o O o Figure 2.2 The 2:1 interaction between flurbiprofen (A) and TTR (P) Note: This figure has been published in "Affinity Capillary Electrophoresis Using a Low-Concentration Additive with the Consideration of Relative Mobilities", Anal. Chem., 2002. 74: p. 1903-1914, reproduced with permission from ACS. 28 2.4 Mathematical Derivation for 2:1 Interactions* 2.4.1 Change in Protein Concentration within the Analyte Plug The time increment is defined as dt. During this very short time interval dt, the analyte plug migrates along the capillary for a distance dx, which can be expressed as: dx = (jUAp+ Meof)-E-dt = lla -E-dt (2.5) where jUAp and juA is the effective and apparent mobility of all species of the analyte, respectively, which are defined in Equation (1.14-1.16). Since all analyte, no matter which form it is, resides in the analyte plug. juA is also the apparent mobility of the analyte plug. From now on, in order to avoid confusion, juA will be called//, andjU2 , where the subscript 1 means this is the effective mobility of the analyte plug in the beginning of the current time interval, while the subscript 2 indicates the effective mobility at the end of the current time interval. The mobility of the analyte plug keeps changing during the migration process. These two variables, //, and ju2, are used to track the changes. At the same time interval dt, the distance that the protein (P) has moved can be expressed as: ^y = (Mep,p+Meof)-E-dt (2.6) where ju p is the effective mobility of protein. The additive entering the capillary from behind the injected sample can migrate into the analyte plug. The increase of the concentration of protein in the plug, [P]ln, is: * The discussions in this section are all related to 2:1 interactions. The discussions on 1:1 interactions are presented in Section 2.5. 29 [ p ] = (dy - dx) • [P\ = - / / , ) • £ • < & • [P]0 ' " a a where [P] 0 is the concentration of the protein in the inlet vial. At the same time some additive migrates out of the analyte plug. The decrease of the concentration of protein in the plug, [P]oul, is: _ (dy - dx) • [P\ _ Qiv, -^)Edt- [P\ a a where the subscript 1 in [P], has the same meaning as that used in fix. So the net effect of the coming-in and moving-out process is: (Men P ~M,)E dt A[P] = [P]in-[Plut =P'P ([P\ - [ /» ] , ) (2.9) a 2.4.2 Effective Mobility of the Analyte Plug The equilibrium is presented as follows: 2A + P <—• A 2 P Start: [A], [P]i+W] [^P}\ End: [A]2 [P]2 [A2P\i [A] is the concentration of free analyte, [P] is the concentration of free protein, and [A2P] is the concentration of the complex. The subscripts (1 and 2) have the same meanings as those used in //, and fi2. The equations for calculating the concentration of each species are: 30 [4+2x[4P]=[4o = > [ A n = [ A l ~ [ A (2.10) [A2P]2=[A]a~[A]l + 8 (2.11) [P]2=[P\+A[P]-S (2.12) [ ^ = [ ^ , - 2 ^ (2.13) where 8 is the change of the concentration after a small time interval dt, and [A]0 is the concentration of the injected analyte, that is, the initial concentration of free analyte in the analyte plug with the assumption that the width of the analyte plug remains constant. The equilibrium constant K is given by: K _ [A2P]2 _ ([Al-jAD/2 + S [A]22-[P]2 ([Al-2S)2([P]l+A[P]-S) If the time increment dt is chosen to be small enough, the change of the concentration is very small compared to the value of the concentration itself, that is: [A]» 8. So Equation (2.14) can be simplified as: ^ ( M 0 - M l ) / 2 ^ [A]U[Pl + MP]-S) Rearranging Equation (2.15), we can get: K[A]l<[P]L+A[P])-[A]* [ A ] L 5 = z 2 \ + K[A]2 By combining Equation (2.16) with Equation (2.12), we have: (2.16) 31 where A[P] can be obtained by Equation (2.9). [A]2 can be calculated in a similar way by combining Equation (2.16) with Equation (2.13). Then [A]2 can be put into Equation (2.2) to calculate ju.2, the effective mobility of the analyte plug at the end of the current time interval, as well as in the beginning of the next time interval. a [A]2 [A]0-[A]2 M l = "'2 =[A]0Mep'A [A]0 Mep'AP ( 2 1 8 ) And the distance that the analyte plug moves during the next time interval can be obtained by: dL = (ju2+jUeof)Edt (2.19) where Lmal is the total length of the capillary, across which the electric field E is applied. The current position of the analyte plug is the sum of dL for each time interval until the analyte plug reaches the detector. In order to simplify Equation (2.14) to Equation (2.15), the concentration of analyte is considered to be much larger than the change in concentration ( [ A ] » 5 ). But this approximation may cause serious problem when the concentration of analyte is not much higher than the concentration of protein, and/or the time increment dt is too large, allowing a large amount of protein to enter into the analyte plug. In these cases, with this approximation the calculation cycles will eventually generate a "0" for ju* and be forced into infinite loops 32 because the analyte plug will never reach the detector and the signal for ending the simulation will never be detected. It is possible to have Equation (2.14) solved without making this approximation. If it is only a 1:1 interaction (forming complex AP) or a 1:2 interaction (forming complex A P 2 ) , a quadratic equation with one unknown can be obtained and solved with ease. But i f it is a 2:1 (forming complex A 2 P) or higher interaction, the equation becomes a lot more complicated. If the approximation is not made for simplifying Equation (2.14), the following equation can be obtained for 2:1 interactions: - 4KS3 + 4K([P\ + A[P] + {A\ )S2 + {-AK[P\[A\ -AKA[P][A\-K[Afx -\)S (2.20) + K[A\] ([/>], + A[P]) - [ ^ ] o ~ [ ^ ] l = 0 Equation (2.20) can be solved using the Newton-Raphson algorithm. The general Newton-Raphson algorithm is presented in Appendix A. It is very important to choose a proper initial approximation. If an improper initial approximation is used, the Newton-Raphson algorithm may not be able to give the desired results. There are two situations in which this algorithm fails to find the correct root. One is that no root is found after the predefined maximum number of iterations has been exceeded. The other is that an improper root out of three roots for a cubic equation is found. To avoid both situations, the result calculated from the simplified equation (Equation 2.16) can be used as a good initial approximation. 33 2.4.3 Steady State When the concentration of protein (TTR) in the analyte plug, as well as the mobility of the entire plug, becomes constant, a steady state condition is reached. There are two possibilities under the steady state condition. The first one is that the amount of protein going into the plug is equal to that leaving the plug. The second one is that the mobility of the analyte plug is equal to the mobility of the free protein. In these experiments, because jUepP > jUepA > jJep^P, only the first case occurs. According to the work by Galbusera et al [17], there are 3 different scenarios, and each of them has a corresponding steady state equation. Because the steady state is part of the discussions in the following sections, all three scenarios and equations are listed as below: (1) The equilibrium concentrations of protein and analyte are equal to the starting concentrations. ( 2 - 2 1 ) [ , 8 ] (2) The concentration of protein at the equilibrium cannot be simply approximated as the starting concentration ([P]o), if the concentration of protein is not sufficiently high and the binding constant has a high value. K [A 2 P] [A]?([P] 0 -[A 2 P]) 2K[P] 0[A] 0 l + K[A]f l + 2K[P] ( 1 [A] 0 ^ep.A-^ep T , ^ r A 1 2 , w r o l f A l ^ep,A "^ep.A^) (2.22) (3) The concentration of analyte at the equilibrium cannot be simply approximated as the starting concentration ([A]o). 34 K = [A 2P] ([A] 0 -[A 2 P]) 2 [P], 1 - V l + 8K[A] 0 [P] 4K[A] 0 [P ] 0 i " c p , A - < = 1 + 10 'ep.A ~ Mep,A2P ) (2.23) V J These three scenarios do not describe the real process very accurately; instead they are the approximations to the real process in order to simplify this complicated process. With these three equations (2.21-2.23), the binding constant and the mobility of the complex can be calculated using a non-linear regression of // A - / /* vs. [P]. In the computer simulation, because the concentrations of both additive (protein) and analyte can be changed step by step with ease, a more accurate description of the real process has been implemented. 2.5 Mathematical Description of 1:1 Interactions 1:1 interactions are similar to 2:1 interactions in many aspects. The differences between them are the focus in this section. The equilibria and basic equations for 1:1 interactions have been presented in Section 1.3.3. The equation for calculating the change of protein concentration in the analyte plug is nearly the same as Equation (2.9), except that the absolute value is used. I a I In the 1:1 interactions, the protein (qRfBP) migrates slower than the analyte (chlorpromazine or riboflavin), that is, the analyte plug actually catches up to the protein in (2.24) 35 the front. (juepP-ju{) is negative in this case, so the absolute value has to be used in order to keep A[P] positive. Although the CE process is different from that of the 2:1 interaction, the net effect is exactly the same as that in the situation where the protein moves faster into the analyte plug from behind. The effective mobility of the analyte plug is calculated in a manner similar to that in Section 2.4.2. A + P « • AP Start: [A]i [P]i+A[P] [AP\ =[A\-[A\X End: [A]2=[A\-8 [P]2 =[P\+A[P]-S [AP]2 =[A]0-[A], + S K _ [AP)2 _ [A]0-[A]l+S [A]2[P]2 ([A]X-S)([P]]+A[P]-S) KS2 -(K[A]X + K([P] , +A[P]) + l)S + K[AUP]\ -([A]U -[A]X) = 0 (2.26) By solving this quadratic equation, the change in the concentration (S ) can be obtained. Then [A]2 andju2 can be calculated. n = i ^ k u + M o n (221) Just as in 2:1 interactions, there are steady states for 1:1 interactions. The counterparts of Equations (2.21-2.23) in 1:1 interactions are: MEP,A - < =K[PUKP,a-MEP,AP) (2.28)[18] ^-< = 1fLCl°1 (M^-^ap) (2-29) = T T ^ T ( ^ - V e p , A P ) (2-30) 1 + A L ^ J O 36 Chapter 3 Computer Implementation - Model One 3.1 Introduction The first model implemented is completely based on the assumptions, theories and equations discussed in Chapter 2. The purpose of the computer simulation (Model One) is to simulate the affinity capillary electrophoresis (ACE) system described in Section 2.2. The mobility of protein (P), TTR for the 2:1 interaction or qRfBP for the 1:1 interactions, and the mobility of analyte (A), flurbiprofen for the 2:1 interaction or chlorpromazine/riboflavin for the 1:1 interactions, at various concentrations were measured by experiments. There are two unknowns remaining, one is the binding constant, the other is the mobility of the complex (A 2 P or AP). One relatively simpler goal of the simulation program is to produce a simulated migration time when the binding constant and the mobility of the complex are given to the programs. Then the simulated migration time is compared to the experimental migration time to see how accurate the simulation programs are. The other goal, which is much more complicated but also more useful, is to estimate binding constants and the mobility of the complex from the experimental migration time. In Model One, the most important assumption, which is worth emphasizing again, is that the length of the analyte plug remains constant during the entire CE process. With this assumption, the complexity of the simulation is reduced tremendously. However, this assumption also brings a few limitations to the simulation programs: 1. Though the migration time can be simulated quite well, it is impossible to describe the peak shapes that may be important in understanding some CE processes. 37 2. The length of plug in the experiments has to be very small and has to be measured accurately. Otherwise, this assumption is invalid. (This will be discussed in detail in Section 3.3.) Thus the computer program cannot be used to simulate the phenomenon of large volume sample stacking in capillary zone electrophoresis (CZE). [19] These shortcomings can be overcome by dividing the whole capillary into infinitesimal cells, then calculating the changes of concentrations of each species in each cell. In this way, a well-defined peak shape can be obtained. This is the second model which will be discussed in Chapter 4. This research is focused more on the 2:1 interactions. Unless otherwise noted, all discussions in this chapter are related to the 2:1 interactions. 3.2 Overview of the Simulation Program The programs were written in C/C++ using Microsoft Visual C++ 6.0 Professional Edition, and run on a desktop PC with an Intel Pentium III 566 CPU. There are 4 programs, SimCETonly, SimCE, SimCEData and SimCEGetK. Each of them has different functions. The relationships between these programs are shown in Figure 3.1. SimCETonly* is used to calculate the simulated migration time of the analyte plug when the binding constant (K) and the mobility of the complex (MeP,A2p) a r e given. When K and * The name SimCETonly comes from ".Simulation of CE to get Zkmuhued only". 38 Mep,A2p a r e unknowns, SimCE is used to scan the desired ranges of K and /x A / > , then SimCEData is used to filter and analyse the data files generated by SimCE, and finally SimCEGetK is used to estimate the K and jucp A P by handling the data files from SimCEData. Figure 3.1 Relationships between four simulation programs SimCE and SimCETonly are the most important of the 4 programs and worth a closer look. SimCE and SimCETonly have similar structure. As shown in Figure 3.2, they both consist of three modules, the User Interface (UI), the Calculation Module, and the Data Output Module. The most significant difference between them is that there is a control loop in SimCE, shown in the grey boxes in Figure 3.2. 39 (" START ) User Interface Initiation: Acquire information from the Ul module NO YES Start the loop with current K c u =K m m current n P n A 9 P n i =\i P a r a m e t e r s c a n be set in U l : T h e length o f the ana ly te p lug (dL) T h e length to the de tec to r (L d e t ector) T h e length of t he w h o l e cap i l la ry ( L w h o l e ) T h e t ime interval for o n e s tep (AT) T h e initial c o n c e n t r a t i o n s ( [A] 0 a n d [P] 0 ) T h e range of K, [ K m i n , K m a x ] T h e range of the mobi l i ty of the c o m p l e x , [M-eD.A2P.min' Mep^P.maxl ep,A2P,min etc . B e f o r e en te r ing the ca l cu la t i on m o d u l e , al l p a r a m e t e r s h a v e to be ver i f ied . In the n e s t e d l o o p s , s c a n the w h o l e K v e r s u s \iep A 2 p p l a n e to f ind out w h i c h pa i r s of K a n d j i A 2 p c a n g e n e r a t e the best-f i t m igra t ion t ime c o m p a r i n g to the e x p e r i m e n t a l v a l u e . Calculation Module Data Output Module Increase K c u r and H e p,A2P,cur °y specified steps, respectively T h e hear t of the p r o g r a m . F o r de ta i l ed desc r i p t i on , c h e c k out the f low char t o f the C a l c u l a t i o n M o d u l e . B a s e d o n the r e q u i r e m e n t s , the d e s i r e d in format ion c a n be output in a fo rmat ted w a y . T h e in terva ls b e t w e e n two c o n s e c u t i v e s t e p s d e t e r m i n e h o w f requent ly the v a l u e s of K a n d | ^ e p A 2 p a r © p i c k e d , a n d h o w long it take's to f in ish the run. YES <K_. OR max NO ( END ) Figure 3.2 Overview of SimCE 40 The U l Module is where the user sets up the parameters, such as the electric field, the initial concentrations of the protein and the analyte, the EOF, the length of the plug, etc. Users can also save/load the parameters for different experimental settings. Due to different functions and parameters needed, each program has its own user interface. (Please refer to Appendix B for the screenshots of the user interfaces and the instructions on how to use these programs.) The Calculation Module is where the calculation is actually done. This module is the core of all the simulation programs, and will be discussed in detail in Section 3.3. When the calculation is done, the Data Output Module will take control to output the results. Several powerful data analysis programs, such as Igor, Origin and SigmaPlot, are available to do complicated analysis, and to plot sophisticated graphs. So the results are output in a formatted way by which the data analysis programs can easily take the results as import data. SimCEData can also take the output from SimCE and provides necessary data filtration and analysis functions which are essential to the ultimate goal of the simulation programs - to estimate the binding constant and the mobility of the complex. The formats of the output data from SimCE and SimCETonly are shown in Figure 3.3. Basically, the results are divided into columns and rows. Each column represents one type of desired information, such as the current apparent mobility of the analyte plug, the concentration of free analyte in the plug, and the current migration time. In SimCETonly, each row is a set of information for one step in the simulation program, while in SimCE, each row shows the simulated migration time with a pair of given binding constant and the mobility of the complex (A 2P). A l l conditions (parameters) for this simulation run are shown at the top of the data files generated by SimCE. 41 Users cannot directly control the format of data files generated by SimCE and SimCETonly, although the codes can be modified in order to have different output. But the output of SimCEData can be controlled by users in the User Interface. (For detailed operations, please refer to Appendix B.) (A) Output data from SimCETonly Time 0.000000 0.050000 0.100000 0.150000 0.200000 0.2 50000 0.3 00000 Length 0.000000 0.0052 58 0.010516 0.015774 0.021032 0.02 62 89 0.031547 [P] 0.Q00000e+000 6.3 08670e-011 1.2 61890e-010 1.893070e-010 2 .524406e-010 3 . 15589Be-010 3. 787547e-010 [ A ] . . . . . . . . . U . 1.000000e-004 . 2. 9.99943 6&-005 2. 9.9S8873e-005 2, 9.9S8309e-005 2 . 9.997745e-00S , . 2. 9.997182e-005 -: . 2 . 9.996618e-005 2. 471232e-004 471224e-004 47121Se-004 471207e-004 471198e-004 471190e-004 471181e-004 (B) Output data from SimCE -2 .023200e-004 = Ua -1.7S0000e-004 = U P -2.5000006-004 = Uap (Hin) -1.9000006-004 = Uap (Max) l.OOOOOOe-00 6 = Uap (Step) . 2 .72 64006+002 '.= TimeEOF 2.000000e-005. = [A]0 1.8000006-00 6 = . [n • 2.Q00000e+004 = Voltage . 4.0000006+001 = . The Length .to the Detector. . . . 4.700000e+001 .= Time Interval 5.000000e-002 = The Length of the Plug 1.800000e-001 = The Length of the. Uhole Capillary 8.0000006+000 = logK (Hin) 1.200000e+001 '• = logK (Max) 5.000000e-002 = logK (Step) Simulated Time Ua2 p . logK -V (Ua-Uap) 6 61.400010. 661.600010. 661.800010. 6 62.050010. 662.300010 662.550010 662.900010 663.250010 5p0000e-004 500000e-004 5000006-004 500000e-004 5000006-004 500000e-004 500000e-004 500000e-004 .000000. .050000 . 100000. .150000 .200000 .2 50000 .3 00000 .350000 4.7680ple-4. 7680.01e-4.7680016-768001e-768001e-7 68001e-768001e-7.6800 ie-005 005 005 . 005 005.. 005 005 005 Figure 3.3 The formats of output data from SimCETonly and SimCE Note: " U " was used in place of the symbol |a. The units of time, length, concentration ([A] and [P]) and mobility (Ua2p, Ua-Uap) are second, centimeter, M , and cm 2 IV - sec, respectively. 42 3.3 Calculation Module in SimCETonly 3.3.1 Procedure of Simulation in SimCETonly The underlying idea for SimCETonly is relatively simple. The binding constant (K) and the mobility of the complex (juep^p) are obtained from literature or the method presented later in this thesis. Since K and juep AiP are known, we are able to calculate 8 in Equation (2.16), and JU2 in Equation (2.18). Thus the distance (dL) that the analyte plug moves in one step can be obtained from Equation (2.19). The current position of the analyte plug is the sum of dL for each step until the analyte plug reaches the detector. Once the analyte plug reaches the detector, the time elapsed during the run is the simulated migration time. Before entering the calculation module, users have to set up all the parameters correctly in the User Interface (Ul). The key parameters are listed in Table 3.1. The conditions for a typical experiment are also listed in the table as an example. The value of K and ju A p come from the method presented in the Section 3.4. Table 3.1 Key Parameters for SimCETonly The concentration of protein in the reservoir ([P]o) 3 . 6 x l ( T 6 M The initial concentration of analyte in the plug ([A]o) 1 .0xlO" 5 M The mobility of free analyte (ju A) -0.0002019cm 2/K-sec The mobility of free protein (ju p) - 0.000175 cm 2 IV- sec The time of EOF 338.0 sec The injection length 0.18 cm The time increment (At) 0.05 sec The binding constant (K) 2.0x10'" The mobility of the complex (jUep A^p) -0.0002189cm2 / F-sec 43 Once these parameters are set in the U l of SimCETonly, the simulation starts. It may take from seconds to minutes to finish one run and get the simulated migration time, depending on the power of the computer and how the parameters are set. If the time increment is set to be smaller (O.Olsec) or the length of the injection plug is smaller, the time needed to finish a run becomes longer. The simulated migration time for such conditions, shown in Table 3.1, is 1367.7 seconds. And the experimental value is 1370.0 seconds. The difference is 2.3 seconds - a error less than 0.2%, which is quite small and well within relative standard deviation values for migration times obtained experimentally in typical CE runs. 3.3.2 Discussions on Steady State The output data from SimCETonly provides information about the migration of the analyte plug that cannot be obtained by any other simulation programs published to date. As mentioned in Section 2.4.3, when the concentration of protein (TTR) in the analyte plug and the mobility of the entire plug become constant, a steady state condition is reached. It takes a while for a system to reach the steady state. The faster a given system reaches the steady state, the better the results will be by doing non-linear regression with the steady state equations (Equation (2.21-2.23)). That is because after the steady state has been reached, the mobility of the analyte plug is constant and defined in Equations (2.21-2.23), but before the steady state, the mobility of the analyte plug isn't defined in Equations (2.21-2.23) and is continually changing. 44 The output from SimCETonly provides the concentrations of the free protein and the free analyte in the analyte plug at every step. Plots of the concentrations of the free protein and the free analyte in the analyte plug vs. time show the complete snap-shot images of how the concentrations of protein and analyte change during the migration process. In Figure 3.4 A and B, the data from five different experimental conditions are plotted. The time for reaching the steady state under each condition varies significantly. 200 .; 400 ; 600 /•:; . ;.300.. Migration Time (sec) Figure 3.4 (A) Plots of [P] (free protein) vs. Time and [A] (free analyte) vs. Time The conditions for the three experiments are listed in Row 5 ([A]o/[P]o=33.3), Row 6 ([A]n/[P](F23.1) and Row 10 ([A]0/[P]o=8.3) of Set 3 of the 2 n d Series in Table 2.3. 45 '.. Migration T ime:(sec) Figure 3.4 (B) Plots of [P] (free protein) vs. Time and [A] (free analyte) vs. Time The conditions for these two experiments are listed in Row 3 ([A]o/[P]o=208.3) and Row 4 ([A]0/[P]o=92.6) of Set 3 of the 2 n d Series in Table 2.3. In Figure 3.4 (A), the concentration of analyte is relatively low and the concentration of protein is relatively high, that is, the ratio of [A]o to [P]o ([A]o/[P]o) is low. In these cases, the steady state was reached relatively fast. When [A]o/[P]o=8.3, the steady state was reached after about 100 seconds, which was less than one ninth of the total migration time (939.6 seconds). But as [A]o/[P]o increased, the time for reaching the steady state increased, too. When [A]o/[P]o-33.3, the time for reaching steady state was about 300 seconds, which was now nearly half of the total migration time (647.0 seconds). The conditions with even higher [A]o/[P]o are presented in Figure 3.4 (B). When [A]o/[P]o-92.6, it took the entire migration 46 time to reach the steady state. When [A]o/[P]o was as high as 208.3, the system was far from the steady state. Also from Figure 3.4, both the concentration of analyte and the concentration of protein were changing during the CE migration process, so all three steady state equations (Equation (2.21)-(2.23)) don't describe the real situation properly. When the ratio of [A]o to [P]o ([A]0/[P]o) is relatively low, the fraction of time spends in the non-steady state is relatively small, and the analyte plug migrates at the same speed during most of the migration process. If [A]o/[P]o is lower than 8.3 (the smallest [A]o/[P]o in Figure 3.4), almost all analyte in the analyte plug is consumed by the protein quickly, and the concentration of protein in the analyte plug quickly reaches the same value as that in the background buffer. Thus Equation (2.23) will become more and more valid to describe the process. On the contrary, i f [A]o/[P]o becomes higher and higher, Equation (2.22) will become more and more valid, because the concentration of free analyte in the analyte plug is hardly changed by the small amount of protein coming into the analyte plug. The computer simulation can take care of not only both extreme conditions, but also any combination of [A]o and [P]o theoretically. In the computer simulation, the concentration changes in both analyte and protein are monitored step by step according to the equations in Section 2.4 and 2.5. Thus the computer simulation provides more accurate results than the steady state equations. 47 3.3.3 Discussions on the Time Increment (At) In Equations (2.5-2.8), dx, dy and dt all refer to infinitesimal amounts of distance and time, respectively. But in computer simulation, there is no way to represent the infinitesimal values; therefore finite, very small and pre-defined values are used. The time increment (At) defines how large a step is. The time increment is an important factor in the computer simulation of the CE process. As the time increment decreases, the computational time increases inversely, and the accuracy and precision are improved. The computational time is a major concern in choosing the time increment, especially for SimCE. SimCE scans the K vs. juep AiP plane, and runs a simulation for each point (corresponding to a pair of K and f i e p A^p) on the plane. There are usually hundreds or thousands of points on the plane. If the time increment is too small, it may take several hours to finish the simulation for just one experiment. In this project, all simulations were done with a time increment of 0.05 seconds. No improvement in accuracy or precision was found when the time increment was set to be smaller than 0.05 seconds. Figure 3.5 shows the curves of [A] (free analyte) vs. migration time for one experiment with various time increments. A l l these curves overlap. This figure clearly shows that the different time increments did not change the simulated migration time noticeably. 48 M i g r a t i o n T i m e ( S e c o n d ) Figure 3.5 Plots of [A] (free analyte) vs. Time with various time increments The conditions for this experiment are shown in Row 5 of Set 3 of the 2 n d Series in Table 2.3. Note: Three different types of lines are used to present the three overlapped curves. 3.3.4 Discussions on Broadening of Analyte Plug In the mathematical model and the computer simulation, an assumption of constant length of the analyte plug is made, which helps to simplify the real process. In this section, a detailed analysis of this assumption is conducted to determine whether this assumption is valid, and if not, how to eliminate this assumption. If the length of the analyte plug is not small enough and the mixture of the analyte and the additive is not fast enough, then the edge of the analyte plug on the inlet side does not 49 move at the same speed as that on the outlet side. The analyte plug will undergo broadening during the migration process. This scenario is described in Figure 3.6. The concentrations of free protein and free analyte on the inlet side are denoted as [P]in and [A] i n , respectively. And the concentrations of free protein and free analyte on the outlet side are denoted as [P]o ut and [A] o ut, respectively. Obviously, [P]jn > [P]0ut and [A]jn < [A]OLit, because the protein continuously comes into the analyte plug from behind to form complex with the analyte on the inlet side of the capillary. There are concentration gradients existing in the analyte plug, which leads to a mobility gradient (juin < juoul) according to Equation 2.18. This broadening-enabled model is very complicated to implement in the computer simulation. And it is possibly impossible to implement when the whole analyte plug is considered as one cell, as in the model described in this chapter, although it is possible i f the capillary is divided into thousands of very narrow cells, and the changes in concentrations for each cell are calculated separately. [Plin • M i in [Plout-IAl ^out lout E O F Pro te i n c o m i n g into the p lug f r o m b e h i n d Analyte plug > ° ° out of the p lug P ro te i n go i ng M o r e c o m p l e x A 2 P a n d pro te in P o n the inlet s i d e M o r e f r ee a n a l y t e A o n the out let s i d e Figure 3.6 Broadening of the analyte plug 50 An approximation is used in order to make things easier. // , , [A] i and [P] 1 are the effective mobility of the entire plug, the concentration of free analyte and the concentration of free protein, respectively, in the very beginning of each step. /i2, [A] 2 and [P]2 are the same factors at the end of the step, and they are calculated in the same way by equations presented in Section 2.4. In the approximation, one step is divided into two separate stages. The first stage is that protein comes into the plug from behind and interacts with the analyte at the inlet edge of the analyte plug. The equilibrium is reached very quickly in a small region on the inlet side of the plug, and the concentrations of free analyte and free protein in that small region become [A] 2 and [P]2, respectively. But in other regions on the outlet side, the concentrations of free analyte and free protein are unchanged at [A]i and [P]i, respectively. Thus the inlet edge of the plug moves at//,, and the outlet edge of the plug moves at// 2, which causes the broadening in the approximation. As time goes by, more and more regions on the inlet side change the concentrations of analyte and free protein to [A] 2 and [P]2. At the end of the current step, the entire plug moves at ju2. From the approximation described above, the change in the length of the analyte plug (Adl) is calculated by: Ml = (jU2 -//,) • E • dt (3.1) The approximation actually amplifies the effect of the broadening theoretically, because the concentrations of free analyte and free protein at the outlet edge are higher than [A]i and [P]i, respectively, due to diffusion. But even with this amplified broadening, the outcomes of the simulation were very close to those without considering the broadening. Use the same conditions listed in Table 3.1 as example. With consideration of these broadening effects, the length of the plug is changed from 0.18cm to 0.1801cm at the end of the run. This 51 difference is too small to affect the migration time noticeably. If the length of the injection plug is very narrow, the effect of broadening may become significant. However, in actual practice, the length of the injection plug cannot be extremely small due to the limitations of the instrument. Thus, the effect of broadening can be omitted in Model One. 52 3.4 Calculation Module in SimCE 3.4.1 Procedure of Simulation in SimCE There are two unknowns, the binding constant (K) and the mobility of the complex A 2 P (MepA2p)- The direct mathematical approach to finding these unknowns is investigated by Galbusera et al [17]. The equations and the derivations presented in that paper are highly complex. In the computer simulation of CE, an alternative approach is proposed. This method takes advantage of the PC's tremendous computing powers, enumerating all possible values of K and f i e p A P , running a simulation for each set of K and Hep<AiP, comparing the simulated migration time to the real migration time, and finally yielding estimates for the values of K and MepA2p- The combined efforts of SimCE, SimCEData and SimCEGetK implement this approach. SimCE is built on the top of SimCETonly. As shown in Figure 3.2, the major difference between SimCE and SimCETonly is the additional loop control in SimCE. In the User Interface (UI) of SimCE (Figure B.2 in Appendix B), users can define the ranges and steps of K and juep AiP. Then SimCE scans the K vs. jUep,A2p plane, getting all points on this plane based on the preset ranges and steps on both axes. Each pair of K and MeP,A2p * s P u t m t 0 the calculation module to generate a simulated migration time. After finishing scanning the entire K vs.juAP plane, we will be able to plot 3-D graphs. Choosing K as x-axis, Mep,A2p a s y-axis, and simulated time as z-axis, 3-D graphs can be plotted using SigmaPlot. An example of this kind of graph is shown in Figure 3.7. 53 (puooas) awn UOIJBJ6!|/\| psieiniuis 09 a O o o II 2. o © < o o o o II m O +3 DO a 3 o u 09 "in © o II u E •~ o S3 s J > o u a a» CD •B a o 5 6b T3 3 4> oo • i—I t/) CD 5b § a o EC s o T3 O CD " > a o CD 09 CD a o oa s o B CD ia H u ^—» o Z 54 The 3-D graph is cut through by a plane perpendicular to the axis of migration time. The migration time has a value equal to the experimental migration time, that is, the T =T plane. By doing this, all pairs of K and a p which can generate simulation experimental 1' 2 migration time equal to the experimental migration time are selected and plotted (Figure 3.8). SimCEData was designed to plot the data similar to this set from a group of experiments efficiently and precisely. Other data analysis programs such as SigmaPlot, Origin and Igor can also be used to generate these types of curves, but users need to do much more work in order to get the curves. 13 12 A 4 1.1 H o 10 4 9 i -0.00026 -0.00025 -0.00024 -0.00023 -0.00022 -0.00021 -0.00020 -0.00019 Mobi lity of the Complex A2 P (cm /V sec) Figure 3.8 A l l pairs of K and jUep A^p which can generate migration times equal to the real (experimental) migration time 55 Thus, if two experiments with different conditions (usually the difference is in the initial concentrations of protein and analyte) are properly chosen, two curves can be generated and an intersection between these two curves is expected. The pair of K and jUep,A2p o n t m s intersection can generate a simulated migration time that is equal to the experimental values for both experiments. This K value is the estimated binding constant. If a series of curves obtained from a series of experiments are put together, under the ideal situation, all these curves should intersect at a common point. But in reality, these curves usually do not intersect at only one point; instead they intersect inside a small region. This is due to a number of factors, including inherent deviation in experimental migration times in CE, the fact that Model One has some assumptions and limitations, and the fact that the length of the analyte plug is not perfectly narrow and may not be identical for each run. SimCEGetK is used to estimate the binding constant from these scattered intersections by looking for the point with the minimal standard deviation or the highest overlapping percentage. (SimCEGetK is discussed later in Section 3.4.4.) 3.4.2 Discussions on the Length of the Plug The length of the injection plug is a very important factor in Model One of the computer simulation. When the length of the analyte plug is very small, and the concentration of protein in the background buffer is relatively high, the interaction and the movement of protein and analyte in the plug are so fast that the influence of various lengths of the plug can be ignored. But if the length of the plug is not small enough to be overlooked, the positions of the calculated curves change significantly with variations in the length of the analyte plug. 56 Three examples are shown in Figure 3.9 (a)-(c). In Figure 3.9 (a), the initial concentrations of analyte and protein for an experiment are lOOfiM and 3.60uM, respectively. The experimental migration time is 505.2 sec. In SimCE, the length of the analyte plug is set to be 0.01cm, 0.03cm, 0.09cm, 0.18cm, 0.30cm and 0.50cm for a series of runs, which will generate a series of K ^s.juepAP curves. In Figure 3.9 (a), the curves for the small lengths of the analyte plug (L=0.01, 0.03, and 0.09cm) are closely overlapping with each other, but when the length of the plug increases (L=0.18, 0.30, 0.50cm), the curves move farther and farther to the left. Thus the measurement of the length of the injection plug is very important in this case. But because the final length of the injection plug cannot be measured directly, and usually is measured (or estimated) based on the injection time, the injection pressure and the characteristics of the instrument, unavoidable errors are introduced into the simulation program. In Figure 3.9 (b), the initial concentration of analyte is 50|iM, and all other conditions remain the same as those in Figure 3.9 (a). The curves for larger length of the analyte plug are closer to the "L=0.01cm" curve. In Figure 3.9 (c), the initial concentration of analyte is 25fiM. A l l curves in Figure 3.9 (c) are very close to one another and it can be deduced that the influence of different lengths of the analyte plug is insignificant in this case. Obviously, for Model One, smaller injection lengths will give better results. But the injection length is limited by the instrument itself, as well as the experimental conditions. In all experiments, the injection length is estimated to be 0.18cm (Section 2.3). 57 12 H 11 CD o 10 H 8 -0 .00026 Common conditions: [P]0= 3.60 uM [A]0= 100 nM AT = 0.05 sec Figure 3.9 (a) o • v L=0.01cm L=0.03cm L=0.09cm L=0.18cm L=0.30cm L=0.50cm -0 .00025 -0 .00024 -0 .00023 -0 .00022 -0.00021 -0 .00020 -0 .00019 Mobility of the Complex A2P (cm 2 V" 1 sec 1) 13 12 11 CO o 10 8 -0 .00026 Common conditions: [P]0= 3.60 uM [A] 0 =50uM AT = 0.05 sec O T V • L = 0 .01cm L = 0 .03cm L = 0 .09cm L = 0 .18cm L = 0 .30cm L = 0 .50cm Figure 3.9 (b) -0 .00025 -0 .00024 -0 .00023 -0 .00022 -0.00021 -0 .00020 -0 .00019 Mobility of the Complex A2P (cm 2V" 1sec~ 1) 58 Figure 3.9 (a) (b) and (c) logK vs. JUAIP curves for various length of the plug under different experimental conditions 3.4.3 Flow Chart of the Calculation Model The flow chart of the calculation model is shown in Figure 3.10. Detailed explanations are provided on the right hand side of each box. 59 K e y V a r i a b l e s : L : the length of the whole capillary Ldetector: the length of the capillary from injection to the detector I: the current position of the analyte plug dl: the distance that the analyte plug moves during the current step ,A2P aO: [A]0; the initial concentration of analyte in the analyte plug pb: [P]b; the concentration of the protein in the inlet vial a l . p i : the [A] and [P] in the analyte plug at the beginning of a given step u l : the mobility of the analyte plug at the beginning of a given step a2. p2: the [A] and [P] in the analyte plug at the end of a given step u2: the mobility of the analyte plug at the end of a given step dnew: the current length of the analyte plug during the current step ConAnew: the current total concentration of A and A^P in the analyte plug during the current step dt: the time interval between two consecutive steps J u m p back to here QSTART ) Initiation Data Output Module Access Point 1 Temp = Edt L • dnew p12=p1+Temp*(up-u1 )*(pb-p1) C o n t i n u e to the i—* next p a g e d n e w is equa l to the initial length of the p lug. C o n A n e w is equa l to the initial concent ra t ion of A , [A] Q . T h e r e a re two a c c e s s points for the Da ta Output Modu le . O n e is here, the other is ou ts ide the Ca lcu la t i on M o d u l e . Th i s p rov ides flexibil ity for data output. Temp is a temporary var iab le . T h e vo l tage E i s 2 0 k V in all expe r imen ts . P\2 = A[P] = [P]aM -[/>]„ The Flow Chart of the Calculation Module (Page 1) Figure 3.10 (a) 60 dp=(K*a1*a1 * p 1 2 - ( C o n A n e w - a 1 )I2)I (1+m_fK*a1*a1) K[A,}\[P\+A[P})-dp = 8-\ + K[A]t Here [A] 0 c h a n g e s after e a c h s tep if b roaden ing is cons ide red . N O tse Newto r R a p h s o n Mgorithrrv> Y E S x=dp U s e dp a s the initial approx imat ion Ca l cu l a te f(x) and f (x) x1 =x-f(x)/f (x) fix) = -4KS3 + 4K([P]t + A[P] + [A], )S2 + (-4K[Pl[Al -4KA[P][Al -K[A$-\)6 + K[A]2]([P]]+A[P])-[A]0-[Al = 0 Y E S Re t r y_number i nc reases by 1. If the root with requi red a c c u r a c y has b e e n found, cont inue on with this root. dp=x1 T o avo id an infinite loop, exit the loop if the m a x i m u m number of i terations has been r e a c h e d . B e c a u s e the initial approx imat ion w a s properly c h o s e n , T h e N e w t o n - R a p h s o n algor i thm never fa i led to f ind the root with required a c c u r a c y for all s imulat ion runs wh ich have been per formed s o far. p2=p12-dp; a2=a1-2*dp; u 2 = u a * a 2 / C o n A n e w + u a 2 p * ( C o n A n e w - a 2 ) / C o n A n e w [A. P]2 = [ A ] o [A]l+s C o d e segmen t for the N e w t o n -R a p h s o n algor i thm [P]2=[P\+A[P]-S [ ^ = [ , 4 ] , - 2 * [A]2 [A]0 ^ [A]0 ep,A2P The Flow Chart of the Calculation Module (Page 2) Figure 3.10(b) 61 1 dl=u2*Temp*dnew l=l+dl N O Y E S ut=ua*a1/ConAnew+ ua2p* (Con A n e w - a 1 )/Con A n e w dl2=ut*Temp*dnew ConAnew=ConAnew*dnew/(d l2 -d l+dnew) dnew=dnew+(di2-dl) a2=a2*dnew/(dl2-dl+dnew) N O The inlet side of the analyte plug moves along the capillary for a distance dl. dl=jUA Edt The current position of the analyte plug is updated, too. The similar equations are used to calculate dl2, which is the distance for which the outlet side of the analyte plug moves along the capillary. The updated total concentration of A and A 2P in the analyte plug and the updated length of the analyte plug can be calculated. The concentration of the free analyte is also changed due to the change in the length of the plug. C o d e segment for deal ing with broadening. If you choose not to cons ider broadening, this code segment will not be run. The current step has finished. The end of the current step means the start of the next step. Thus we put u2 into ul, a2 into al,p2 into pi. ^ 3 / Jump back to the starting point to do the next cycle. This loop executes until the analyte plug reaches the detector. The Flow Chart of the Calculation Module (Page 3) Figure 3.10 (c) 62 3.5 Binding Constants - 2:1 Interactions (Flurbiprofen:TTR) The binding constants for the 2:1 interaction between flurbiprofen (A) and TTR (P) can be estimated using Model One of the A C E computer simulation. The experiments are described in Section 2.2. There are two series of experiments, Series One (Section 2.2.3) and Series Two (Section 2.2.4), providing the data for the following discussions. First of all, the proper x- and y-axes have to be chosen in order to represent the correct relationship between the binding constant and the mobility of complex. The y-axis is logK, instead of K. The only reason is that logIC is simpler to be entered in the user interfaces of the simulation programs. In Figure 3.8, the x-axis is the mobility of the complex. But from Table 2.3, 2.5 and 2.6, the mobility of free analyte (jJe/lA) varies for any single experiment due to the changes in the conditions of the capillary and background buffer. Various values of jUEP,A will lead to various values of fJep,A2P- Thus i f more than one Figure 3.8-like curves are put together into one graph., the mobility of the complex is no longer suitable for the x-axis. Instead, the difference between the mobility of free analyte and the mobility of the complex (jieP^4-JUep,A2p) is used as the x-axis. In this way, the effects of the different capillary and buffer conditions are offset. The accurate measurement of the mobility of the analyte (jUepj) is crucial in the effort to acquire binding constants. Figure 3.11 was made in order to show how various //eA^'s affect the results. The conditions of the experiment for the curves shown in Figure 3.11 are listed in Row 9 in Set 1 of Series Two (Table 2.3). Five different values of fJEP,A were used in SimCE and generated five curves. This figure clearly demonstrates the importance of measuring fJepA carefully before any individual experiment. /uep,A was measured three times 63 for each experiment and the average of those three values was used. The differences between the three measured values were usually not bigger than the differences between the curves shown in Figure 3.11. This is where the errors associated with the final results come from. 12.5 12.0 11.5 11.0 ^ 10.5 CD O 10.0 9.5 9.0 H 8.5 0 u.ep,A = -0.00019943 cm /V-sec -0.00019893 crri /V-sec ^ep,A ^ep,A u e p A = -0.00020043 cm ' /V-sec -0.00019993 crrT/V-sec -0.00019843 cm 2 /V -sec 2 Row 9 in Set 1 of the 2"u Series (Table 2.3) — i 1 1 1 — 1e-5 2e-5 3e-5 4e-5 l ^ e p , A - ^ e p , A 2 p ( C m 2 / V - S e C ) 5e-5 6e-5 Figure 3.11 Effects of different /jePrA From Tables 2.2 and 2.4, it is clear that jjePtP, the mobility of TTR (or qRfBP for 1:1 interactions), changes in the same way as juEP,A. That is, /4Ap is not a constant, either. There is no easy way to measure juep,p while the capillary is filled with protein in background buffer. Fortunately, the effects of variations in fJuPiP are much smaller than that of variations in /JEP,A-The experiment with conditions listed on Row 6 of 3 r d Series of experiments in Table 2.5 is 64 used as an example here to investigate this effect. Figure 3.12 shows that no matter which /iepPis chosen (-1.55X10"4 or -1.43X10"4 cm2/V-sec), the resulting curves are essentially the same. Thus, a single value of juep,p is used to simulate all experiments related to this protein. 7.5 7.0 6.5 CO o 6.0 H 5.5 5.0 -\ 4 .5 1e -5 2 e - 5 O | L l e p p = -1.43x10" 4 cm2/V-sec (D U l e p p = -1.55x10" 4 cm 2/V-sec ^ ^ ^ ^ 3 e - 5 4 e - 5 5 e - 5 6 e - 5 7 e - 5 ^ e p , A ^ e p , A P ( C m / V ' S e C ) Figure 3.12 Effects of different jjcp>P 8 e - 5 9 e - 5 In Series Two, the product of the concentrations of analyte and protein, [A]o[P]o, was fixed at one of the three values 3.6x10"", 1.08xl0"10 and 1.8xl0"10 M 2 . The reason for doing this was explained mathematically by Galbusera et al [17]. The explanation is directly related the steady state equation, Equation (2.23): 65 u 1-Vl + 8K[A] 0[P] 0 4K[A] 0[P] 0 (M'ep.A " M-ep,A2p) (2.23) When [A]o[P]b is a constant, 1 + 1-Vl+8K[A] 0[P] 0 4K[A] 0[P] 0 is a constant for the whole set of experiments. Thus i f there are more than two sets of experiments with different values of [A]o[P]o and u , e p j A 2 p , the value of the constant 1 + 1-Vl + 8K[A] 0[P] 0 4K[A] 0[P] 0 can be solved from two simultaneous equations. So does the binding constant K. If [A]o[P]o is not fixed, it becomes very difficult to solve the value of K. Three sets of experiments in Series Two (keeping [A]0[P]o constant) underwent the procedure described in Section 3.4.1. The outcomes were three graphs, Figure 3.13 (a), (b) and (c). Series Two, instead of Series One, is first discussed, because in Series One, the measurement of the mobility of analyte (JIVP,A) was performed only once for each set of experiments, which would lead to inaccuracy due to the fact that JUL,PIA changed for every experiment. In Figure 3.13 (a), (b) and (c), all of the curves composed of a bunch of points have shapes of inverse functions, and they are nearly "parallel" to each other. The simplest steady state equation (Equation (2.21)) is used here for studying the shapes of these curves. Although Equation (2.21) does not reflect the real process precisely, it is suitable to be used as an approximation. 66 12 A 11 o 10 Figure 3.13 (a) R o w N o . 4 [ A y [ P ] 0 = 3 0 . 8 3 R o w N o . 6 [ A y [ P ] 0 = 11.11 R o w N o . 7 [A ] 0 / [P ] 0 =7 .72 R o w N o . 8 [ A l o / I P l ^ S . 5 6 R o w N o . 9 [A ] 0 / [P ] 0 =4.34 R o w N o . 1 0 [ A ] 0 / [ P ] 0 = 3 . 4 3 R o w N o . 11 [A ] 0 / [P ] 0 =2 .78 TV ^ m Set 1 of the 2 n d Series (Table 2.3) 58 W o 5 • • • i ' - P S " 1e-5 2 e - 5 3 e - 5 4 e - 5 5 e - 5 ^ e p , A - | ^ e p , A 2 p ( C I T I / V " S e C ) Figure 3.13 (b) o T V • • • R o w N o . 5 [A] 0/[P] 0=86.81 R o w N o . 6 [A] 0 / [P] 0=55.56 R o w N o . 7 [A] Q / [P] 0=38.56 R o w N o . 8 [ A y [ P ] 0 = 2 8 . 3 3 R o w N o . 9 [A] Q / [P] 0=21.70 R o w N o . 10 [A] Q / [P] 0=17.16 R o w N o . 11 [A] 0 / IP] 0=13.89 Set 2 of the 2 n d Series (Table 2.3) •••• mm U v V V 1e-5 2 e - 5 3 e - 5 4 e - 5 l ^ e p , A - ^ e p , A 2 p ( C m 2 / V " S e C ) 5e-i-5 67 12 H 11 o 10 H 9 H o o o o o o o o V V V • Figure 3.13 (c) V V T IB O V O V O V O » O « • O • O • O • O • R o w N o . 4 [A] 0/[P] 0=92.59 o R o w N o . 5 [A] 0/[P] 0=33.33 • R o w N o . 8 [A] 0/[P] 0= 13.02 v R o w N o . 9 [A] 0/[P] 0= 10.28 . R o w No. 10 [ A y t P ^ S . 3 3 Set 3 of 2nd Series (Table 2.3) r v ^ 7 T P 7 O O O 1e-5 2e-5 3e-5 4e-5 l ^ e p , A - ^ e p , A 2 p ( c m 2 / V ' s e c ) 5e-5 6e-5 Figure 3.13 (a), (b) and (c) The logK vs. ( / 4 p , A - / 4 p , A 2 P ) curves for the 2 n d Series Equation (2.21) can be rearranged to Equation (3.2): A M-ep.A "Hep (M-ep.A " Hep,A2P ) " (Pep.A " M-ep ) Further rearrangement of Equation (3.2) gives: 1 = 2K[P] 0[A] 0 (3.2) M"ep,A " M-ep,A2P * M'ep.A " P-ep 2K[P] 0[A] 0 (3.3) The logarithm of both sides of Equation (3.3) is taken in order to plot "logK" along the y-axis. 68 log M-ep.A "M'ep,A2P -1 V Uep,A " Hep J = log(2K[P] 0[A] 0) (3-4) If we setx = ^ e p A -u e p > A 2p, y = log(2K[P] 0[A] 0) and a = f i e p - A - u * , Equation (3.4) can be rewritten as: log f 1 ^ x/a-l •y (3.5) Equation (3.5) is a simple function. The plots of this function with a=l, 2, 3 (Figure 3.14) have a very similar pattern as Figure 3.13. The value of a ( a = f i e p A - u^p ) differs from experiment to experiment, which gives the "parallel" curves of similar shapes shown in Figure 3.13 (a), (b) and (c). Figure 3.14 Shapes of the curves 69 More complicated steady state equations will give this similar pattern. When the concentrations of analyte become much higher than the concentrations of protein, Equation (2.22) takes control of the shapes of the curves. When the concentrations of analyte become equal to or even smaller than the concentrations of protein, Equation (2.23) takes control of the shapes of the curves. Due to the fact that there should be only one binding constant (K) for a specified interaction, ideally all curves for the same set of experiments in Figure 3.13 should almost overlap with some slight offsets. If these curves do nearly overlap, then the model built for the computer simulation can be considered quite accurate. However, under these conditions these curves do not overlap. This indicates that Model One is not the best model for the simulation of these kinds of CE experiments. Although Model One is not the best, under some well-designed circumstances, this model can properly describe the real process and the binding constants can be obtained through computer simulation. The relative positions of the curves are important. The curve for an experiment with higher concentration of protein and lower concentration of analyte, that is, smaller ratio of [A]o to [P]o ([A]o/[P]o), is closer to the upper right corner of the graph. (There are some exceptions in Figure 3.13 - such as the curve for Row 9 of Set 1. This could be the result of inaccurate measurement of juepj, or the result of an abnormal injection which made the length of the injection plug differ from the expected value, i.e., 0.18cm.) From the point of view of the steady state, when [A]o/[P]o is small (less than 10), the steady state will be reached very quickly (a few seconds under extreme conditions), and the steady state equation (Equation 2.23) can describe the process correctly. 70 From the point of view of computer simulation, when [A]o/[P]o is high, a relatively small amount of protein enters the analyte plug, and most protein is consumed by analyte on the inlet side of the analyte plug to form the complex ( A 2 P ) . This leads to a lower concentration of A 2 P and higher mobility on the outlet side of the analyte plug than the calculated values (Section 2.4). This indicates that the mixing of the protein in the plug is not fast enough if the length of the analyte plug is not negligible. The result of this effect is that a smaller-than-real-value binding constant is obtained under such conditions. Therefore, the curves with high [A]o/[P]o reside on the lower left part of the graphs. On the contrary, when [A]n/[P]o is low, the steady state is reached faster, and the problem associated with high [A]0/[P]o can be ignored. Model One now can describe the CE process correctly in such cases, and the binding constants (K) obtained from these curves agree well with the experimentally determined values. By considering the changes in concentrations of both analyte and protein, Model One of the computer simulation can definitely provide a better result than the steady state equations. But due to its own limitations, Model One is not the perfect simulation model for the CE process. A better model is discussed briefly in Chapter 4. In Figure 3.13 (a), (b) and (c), the curves with [A]n/[P]o smaller than 20 overlap with or are very close to one another. According to the discussions above, these curves with [A]o/[P]o smaller than 20 can provide accurate information about the binding constants. Set 1 and Set 2 are used here to obtain the binding constant. Row 10 and 11 in both Set 1 and 2, which have the lowest [A]o/[P]oin Series Two, are put together in Figure 3.15. There is a clear intersection in Figure 3.15, which is the binding constant (in the form of logK). 71 From Figure 3.15, the binding constant (logK) is 10.30±0.1, and jUep,A-/jepA2p is (1 .70±0.02)x lO~ 5 cm 2 /Vsec . These results agree well with the ones presented by Galbusera, et al. [17] 12.5 12.0 11.5 11.0 -CD O 10.5 -10.0 9.5 H 9.0 8.5 Row No.10 in Set 1 Row No.11 in Set 1 Row No.10 in Set 2 Row No.11 in Set 2 1e-5 2e-5 3e-5 4e-5 5e-5 / W W (cm2V-1sec-1) Figure 3.15 Finding binding constants from Experiment Set 1 and 2 of the 2 n d Series The binding constant can be obtained either directly from Figure 3.15 or by using SimCEGetK. In SimCEGetK there are two methods for getting K. Each curve is composed of tens or hundreds of points. One can choose to calculate the standard deviation of the mobility of the complex for the points on different curves at the same K value. The final result of the 72 binding constant is the K value coming up with the smallest standard deviation. In other words, the K value at which the mobility of the complex has the narrowest spread is the final result. Or one can choose to sum up the percentage of overlap area between any two curves. The K value at which a highest sum is obtained is the final result. Recall that in Section 3.3.2, when [A]0/[P]o=6.9, it took about 100 sec out of 400 sec in total for the system to reach the steady state. In Set 2 & 3 of Series Two (keeping [A]o[P]o constant), the minimum of [A]n/[P]o was larger than 8, which meant that even more time (more than one fourth of the total migration time) was spent in the unsteady state This was not good enough for achieving the accurate simulation using this one-cell model. Thus, new series of experiments were designed. In Series Three and Four, the 1:1 interactions between qRfBP and chlorpromazine/riboflavin were studied. This time the concentrations of the analyte and the additive were carefully selected. But there are some limits for the combination of [A]o and [P]o. When [A]o/[P]o was too small, that is [P]o was large, there were some difficulties to detect or measure the peaks, because the peaks were either too small or too broad. Another way to improve the computer simulation of A C E is to use a new model, which is discussed in Chapter 4. 73 3.6 Binding Constants - 1:1 Interactions In this section, we are attempting to find the binding constants for the 1:1 interactions. Two series of experiments were done for two 1:1 interactions, chlorpromazine : qRfBP and riboflavin : qRfBP. In the 2 n d series (for 2:1 interaction), the measurement of the mobility of analyte was performed only once for each experiment. This caused relatively large errors (Figure 3.11). Because the accurate measurement of fJLEP,A was crucial, in the new series, the measurement was performed three times for each experiment, and the average was taken. The 1:1 interaction between chlorpromazine and qRfBP was investigated first. Figures 3.16 (a), (b) and (c) are the graphs for three sets of Series Three, respectively. cr> o 6 .0 A 7.0 A 6 .5 A [A ] 0 = 7 5 u.M [A ] 0 = 6 0 u M [A ] 0 = 5 0 u.M [A ] 0 = 4 0 | i M [A ] 0 = 2 5 u M [ A ] 0 = 15 u M [ A ] 0 = 1 0 u . M [A ] 0 = 2 .5 u M 5 .5 A 5 .0 A Figure 3.16 (a) Set 1 of the 3 r d Series 4 . 5 0 2 e - 5 4 e - 5 6 e - 5 (cmVsec" 1) 8 e - 5 ^ep,A^ep,AP 74 7.0 A 6.5 A 6.0 CO o 5.5 A 5.0 4.5 .8 T * [P]0 = 0.8 u.M AT = 0.05 sec L = 0.18 cm • [A] 0 = 1.0 u M o [A] 0 = 2 . 5 ^ M T [A] 0 = 5.0 u.M V [A] 0 = 2 5 u M • [A] 0 = 40 |xM • [A] 0 = 60 • [A] 0 = 75 u M •°4 • l , v v y "".| • • . vvv. Figure 3.16 (b) - - - ^ • • • o n ' u 2 U Set 2 of the 3 r d Series 2e-5 4e-5 6e-5 > " e A ^ e A ^ ( C m 2 V " 1 s e C " 1 ) 8e-5 7.0 6.5 A 6.0 CO o 5.5 5.0 [P]0 = 0 . 1 8 u M A T = 0.05 s e c L = 0.18 c m o T V [A]0 = 1 HM [A]Q = 5 nM [A]Q=10nM [A]Q = 15 | iM [A]„ = 50 liM [A]0 = 75 nM Figure 3.16 (c) Set 3 of the 3 r d Series 2e-5 4e-5 6e-5 Mep,A-Vep,Ap (cmV1sec-1) 8e-5 Figure 3.16 (a), (b) and (c) Curves for the 3 r d series (chlorpromazine : qRfBP) 75 Three sets of experiments were done. The concentration of protein was fixed for each set. According to Equation (2.30), which is similar to Equation (2.23), curves with the shapes of inverse functions were expected when the concentration of protein was fixed. And this was exactly what was observed. In Figure 3.16 (a), (b) and (c), wide ranges of concentrations of analyte were chosen ([A]o/[P]o was from 1 to 100) so that a better understanding of the relative positions of the curves could be gained. When [A]0/[P]o becomes smaller and smaller, the curves move to the upper-right of the graph and become infinitely closer to a curve which corresponds to extremely low [A]0/[P]o- When [A]o/[P]o becomes bigger, the curves move toward the lower-left of the graph, and the shapes of the curves start to change. As discussed in Section 3.3.2, when [A]o/[P]o becomes higher, Equation (2.30) can no longer solely describe the curve shape, and Equation (2.29) takes more and more control over the curve shape. So the curve shape is now determined by the combined descriptions of Equation (2.29) and (2.30). But Equation (2.29), unlike Equation (2.28) and (2.30), has both [P]0 and [A] 0 components. Thus the curves with high [A]o/[P]o are no longer "parallel" to the ones with low [A]o/[P]o- This scenario can be observed best in Figure 3.16 (c), in which [A]0/[P]o was as high as 400. The binding constant for 1:1 interaction between chlorpromazine and qRfBP can be obtained from the intersection of the curves corresponding to the lowest [A]o/[P]o in Set 1 and Set 2 of Series Three. From Figure 3.17, the binding constant (in the form of logK) is 5.89±0.02, and jUepA-jU.epMP is (4.58 ± 0.05) x 10.- 5cm. 2/V • sec . 76 1e-5 2e-5 3e-5 4e-5 5e-5 6e-5 7e-5 8e-5 ^e P ,A-|^ep,Ap(cm /V-sec) Figure 3.17 Binding constant for chlorpromazine : qRfBP 3.7 Conclusions Model One of the computer simulation presented in this chapter accurately describes the migration behaviour of a ligand in a CE system when a very low concentration of protein is used as an additive and when a 2:1 or 1:1 stoichiometry is present. In order to obtain accurate binding constants from Model One, carefully designed experimental settings are very important. There are several important rules and recommendations: 77 1. The accurate measurement of the mobility of the analyte for every experiment is very important. Thus, three or more measurements of /j^j for any single experiment are required. 2. The concentrations of protein and analyte should be set properly. Usually [A] 0/[P]o should be equal to or smaller than 3 in order for Model One to be applicable. 3. The length of the injection (analyte) plug should not be too large. 0.2cm or shorter is recommended. 4. For 2:1 interactions, the experiments performed with constant values at [A]o[P]o allow for finding binding constants from only two sets of experiments with different [A]o[P]o- For the same reason, the experiments for 1:1 interactions should also be performed with constant values of [ P ]Q . 78 Chapter 4 Computer Implementation - Model Two 4.1 Introduction Treating the whole injection plug as one cell is not the best way to describe the process in CE. There are many factors that can either broaden the injection plug or change the concentration distribution of each species or the pattern of the electric field within the capillary. The diffusion coefficient is the most significant factor which was unable to be incorporated into Model One. Diffusion arises from the random motion of molecules and the Gaussian shape of peaks is the direct result of this diffusion phenomenon. In Model One, with the assumptions of rapid equilibrium and constant plug length, the concentrations of all species are considered identical anywhere within the injection plug. But the truth is that the peak is a typical Gaussian peak, that is, it is more concentrated in the center, and decreases rather quickly on both sides. Additionally, because of diffusion, the length of the injection plug is not constant during the entire process. The second important factor is the electric field. The electric field of a specified position in the capillary depends on the local ion concentrations as well as on the global ion concentration profiles through the current /. [7] Although Model One did not consider these factors, the results were still quite good, because the effects of these factors were relatively small, and could be omitted when the injection length was very narrow (0.18cm). But when the injection length becomes larger, or when the peak shape is also of interest, a new model is needed to describe the CE process more accurately. At this stage, 79 only preliminary results have been recorder and the new model has not been studied thoroughly. But even at this stage of research and development, the new model looks promising and should be able to estimate the binding constant more accurately than Model One. 4.2 The New Model 4.2.1 Mathematical Description In this model, the entire capillary is divided into very narrow cells, as shown in Figure 4.1. Between adjacent cells, there is migrational flux of each species. The concentrations of analyte and protein in each cell can be calculated. This physical system can be described by the following equation [7]: dC . dFZL 9 2 C . ZJ Z,l jry ZJ dt dz dz (4.1) where Cz; is the concentration of ion / at position z and time t, Fzj is the migrational flux (whose sign indicates direction) of ion / at position z and time t. Protein O Analyte Figure 4.1 Model Two - The capillary is divided into narrow cells 80 The ion flux is given by FZJ = Ctjft,Et (4.2) where Ez is the total local electric field, and fj,i - . + fi is the apparent mobility of the ion /. At the current stage of development of the simulation programs, the local electric field is assumed to be constant throughout the capillary. The migrational fluxes of protein and analyte are obtained by: Fp = E(jUA2P [A2P]+ JUp [P]free) (4.3) FA=E(2jUA2P[A2P] + jUAU]fre,) (4-4) With a selected time increment At and a selected space increment Az, the first and second derivative terms on the right side of Equation (4.1) are evaluated in the usual manner as: dz 2Az £j d CZI ^ CZ+AZJ — 2CZJ + C.^j dz2 ' (Az)2 Thus the change in the total concentration of analyte in cell j is a U L ] = ~ W + , - 0 + ; ^ (4-5) 2Az (Az) Similarly, the change in the total concentration of protein in cell j is 4?L ] = ~ - Fi'x)+Trtr (kS1- ipL, 1+k21) (4.6) 2Az (Az) 81 4.2.2 Computer Implementation A sound mathematical model is fundamental for the computer simulation. Additionally, the computer implementation must reflect the model correctly and efficiently in order to be successful. In this model, the entire capillary is equally divided into m very narrow cells. The length of each cell (J) can be calculated by dividing the length of the whole capillary by m. The way that the data for each cell is stored and manipulated in the memory of the computer is the key to the efficiency of the simulation programs, especially when the simulation programs are not designed for running on a main frame computer, but on a less powerful PC. A common approach to data storage for all cells is to allocate an array of m elements. Each element in this array stores the data for each cell of the divided capillary. Equation (4.5) and (4.6) are used to calculate the changes in the concentrations of protein and analyte in every cell for one step. Theoretically, this kind of memory manipulation is good for computer simulation of CE for any given set of conditions. But when the capillary is very long or it is divided into very narrow cells, the required computing power to run the simulation is tremendous, and sometimes unaffordable. For the affinity capillary electrophoresis (ACE) technique presented in this thesis, the concentrations in most cells do not change; instead only the concentrations in the cells within or near to the analyte plug change. So there is no need to track the changes in concentration for every cell. Following these lines, a new approach to manage the data storage in the PC's memory was implemented. The whole capillary is still divided into m cells. The cells within and near to the analyte plug are called active cells, in which there is a noticeable amount of analyte. A l l other 82 cells are called inactive cells, in which there is no analyte. Only the concentration changes in active cells need to be tracked and stored. At the beginning of the A C E experiments, the analyte exists only in the analyte plug. Due to the applied electric field, the analyte plug moves towards the detector; due to diffusion, the analyte spreads to the nearby cells. To determine whether a cell is active or not, a user-defined threshold concentration of analyte is used. When the concentration of analyte in one cell is greater than the threshold amount, this cell is activated; when the concentration of analyte in one cell is smaller than the threshold amount, this cell is deactivated. An array with n elements is allocated in the memory, n can be much smaller than m, but it should be large enough to store all the data for active cells. This array is arranged as a circle (Figure 4.2), which makes it possible to reuse elements. Two pointers (pi and p2) are used, 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. In the beginning of a simulation run, pi points at element No.O, and p2 points at element Noy* (j = Lphlo II), which is the last cell within the analyte plug. Any cell outside this range (from j+l to n) has the same default conditions: the concentration of protein is [P]o and the concentration of analyte is 0. In the first step, the changes in the concentrations of protein and analyte for the cell from pl(0) to p2(j) are calculated by Equation (4.5) and (4.6). Some analyte moves into cell No./'+l, so the pointer p2 is moved to y'+l, too. At the same time, the concentration of analyte in the cell No.O decreases. In every following step, the pointer p2 is moved one step forward. But the pointer pi still points at the first cell until the concentration of analyte in the first cell is lower than the 83 predefined threshold value. In other words, when the concentration of analyte in the leftmost cell is small enough to be ignored, the pointer pi is moved forward. As the pointers pi and p2 keep being moved forward, p2 may finally bypass the last physical element in the memory of the PC, No.n of the array. But because the elements are arranged as a circle, logically there is no LAST element. The pointer p2 is moved to No.O again after No.rc. This operation is valid also due to the release of elements on the side of the pointer pi. PI Figure 4.2 Model Two - The arrangement of cells in the memory of the PC An important part of this new model is to determine the time increment At and the space increment Az. 84 Because the pointer p2 is moved forward one cell per step, the time increment At has to be quite small, otherwise, the changes in concentrations are too steep to describe the real process. The space increment Az has to be quite small, too. Obviously, narrower cells can give more accurate results. 4.3 Conclusions and Further Development The new approach to memory arrangement and manipulation is very promising and makes the calculation-intense simulation of long and complicated CE experiments possible on a PC. The new model may produce better results than Model One in estimating the binding constants from the affinity capillary electrophoresis (ACE) experiments. And it is also possible to obtain precise peak shapes using this new model. The relationship between the optimal time increment At and the optimal space increment Az need to be figured out. The programs for this model need to be completed in order to predict the binding constants and generate the peak shapes. 85 Appendix A Newton-Raphson Method for Solving f(x) = 0 [20] A.1 Mathematical Description A function y=f (x) is plotted as Figure A. 1. p is the intercept of the function on the x-axis, and also one of the roots of y—f (x). Let x' be an approximation to p such that f'(x') ^ 0 and \x '—p\ is "small". Consider the first Taylor polynomial for fix) expanded about x', O - x ' ) 2 (A.1) 2 Sincef(p) — 0, this equation, with x — p, gives: 0=f(x') + (j>-x')f'(x')4 (A.2) 2 y. Slope/'to) y=f(x) Slope/'(Po) Figure A. 1 Newton-Raphson method for solving f(x) = 0 [20] 86 y 2 Assuming that since \p—x'| is small, the term involving (p - x') is much smaller and that 0 « / ( * ' ) + ( />"* ' ) / ' (* ' ) (A.3) p~x'-f(x')/f(x') (A.4) This sets the stage for the Newton-Raphson method, which starts with an initial approximationpo and generates the sequence {p„} defined by: Pn=Pn-]-^r^Z, forn>\. (A.5) A.2 Computer Implementation To find a solution tof(x) — 0, given the differentiable function f(x) and an initial approximation po, the following procedure is implemented: INPUT: initial approximation (po); tolerance (TOL); maximum number of iterations (No). OUTPUT: approximate solution for p or a message of failure. Step 1: Set/= 1. Step 2: While / <= N 0 do Steps 3-6. Step 3: Setp=Po-f(po) If' (po). Step4: lf\p-p0\< TOL then OUTPUT (p); (Procedure completed successfully.) STOP. 87 Step 5: Se t /= /+ l . Step 6: Set po = p. (Update po) Step 7: OUTPUT ("Method failed" after N 0 iterations); STOP. (Procedure completed unsuccessfully.) Appendix B Instructions for Using the Simulation Programs B.l Introduction There are four simulation programs, and in the following sections they will be referred to as SimCETonly (generating only one simulated migration time when given the values of binding constant (K) and the mobility of the complex (fJ-ep<AP)), SimCE (scaning the K vs-JuePA1p p ' a n e to get a simulated migration time for any point on that plane), SimCEData (data analysis and filtration of the output from SimCE) and SimCEGetK (estimation of the binding constant (K) based on the output from SimCEData). The simulation programs run on Windows 9x/NT/2000/XP. The user interfaces (Ul) were designed in the form that all other typical applications in Windows follow. Thus, the experience gained from other windows applications by a user should apply to the simulation programs as well. The basics of operation in windows operating systems will not be covered in this section; instead only a couple of important points (naming convention and directory structure) will be discussed in Section B.2 followed by a discussion of the four programs one by one in Sections B.3-B.6. B.2 Naming Convention and Directory Structure There are 4 programs dealing with more than 4 kinds of data and generating hundreds of input and output files, so giving each different kind of data file a meaningful and clear 89 name, as well as saving data files in well-organized directories, are essential to identify data files of different kinds and different experimental conditions. The name of a file in the computer is composed of two parts: the file name and the extension. The extension shows what kind of file it is, and the file name tells us what this file is about. Each file corresponds to one experimental condition. Therefore, experimental conditions are the perfect identifiers for individual files. The key parameters of the experimental conditions are the initial concentrations of protein and analyte, the length of the analyte plug and the time increment. With all these four parameters, an experimental condition is well defined and distinct. So the recommended name of a file is the combination of all four parameters. For example, one experimental condition has the following four parameters: the initial concentrations of protein and analyte are 2.16xlO~ 6Mand5.0xlO~" 5M, respectively, the length of the plug is 0.18 cm, and the time increment is 0.05 sec. The file name for this condition can be A050P216L018T005. Here A, P, L and T stand for the initial concentration of analyte (A), the initial concentration of protein (P), the length of the plug (L) and the time increment (T), respectively. And the numbers following these letters represent the corresponding values of the four key parameters. "050" is used to represent a value of 5.0xlO~ 5Mfor the initial concentration of analyte. The reason for using "050" instead of "50", "5" or any other possible format is that only three different initial concentrations of analyte ( 10.0xl0" 5A/ , 5.0xlO~ 5 M and 2 .5x l0~ 5 M) were used in this series of experiments, and they can be represented best as "100", "050" and "025". 90 In some cases, not all four key parameters are needed to identify a specified experimental condition, or more parameters may be needed. If so, use whatever identifiers are necessary to distinguish experimental conditions. With the file name set, we can now consider the extensions needed to distinguish the parameter files and data files generated by each program. The parameter files are used in SimCE and SimCETonly to save all parameters for any individual experimental condition in order to avoid re-inputting these parameters. Because the numbers of parameters saved in the parameter files for SimCE and SimCETonly are different, two extensions are used, ".par" is used for the parameter files for SimCE, while ".pal" is used for SimCETonly. The output data files from SimCE are formatted in the manner shown in Section 3.2, and have an extension ".dat". The output data files from SimCETonly have an extension ".dal". The output data file from SimCE (*.dat) are analyzed by SimCEData to generate new output data files which have an extension ".out". Thus far all the data files are named in a meaningful and organized way. But since there are hundreds of data files generated by these simulation programs, the data files have to be put into well organized directories in order to let users find any data file for any given experimental condition easily. However, there are generally no strict rules for organizing data files. Usually the data files corresponding to a series of experiments should be saved in a directory with a unique directory name. 91 B.3 SimCETonly This program is used to generated a simulated migration time when a pair of K and MeP a,p a r e given- The screenshot of the user interface is shown in Figure B. 1. • SimCE Program 1 - Model One - for obtaining Tsim only File Operations: New Open Save J Save As File name (without extension): A050P180L018T005 U (cn^/V-sec)- 1 - j - Initial Concentration (M) —i Ua= j -0.0001889 • [A]0= J5e-005 Time of EOF = (213.818 Up= j -0000175 [P]0= jl.Be-006 Voltage (V)= 120000 I Unknows: Uap= j-0.0002059 K= [ieToiO r Length to the detector (cm): L(detector) = p0 Directory: f~ Considering Broadening P More accurate calculation Length of a cell (cm): | C:\My DocumentsSNingVSi j method (eg. Newton Algorithm) j Length of whole capillary(cm): L(whole) = Time Increment (sec): dT= J"-'® dL= FTiT Start Exit Interaction Type: r 1:1 (AP) 2:1 (A2P) r CPU Time: C Critical C High <* Normal C Idle Figure B. 1 User Interface of SimCETonly A l l the required parameters need to be set in the proper edit boxes. The values of these parameters can be in general format (such as 999.99) or scientific format (such as 1 .OE-5). Besides the parameters related to the experimental conditions, there are other options users have to set correctly. (1) "Considering Broadening": If this box is checked, the broadening of the analyte plug will be considered in the calculation module as discussed in Section 3.3.2. By default, the box is unchecked. 92 (2) "More accurate calculation method (e.g. Newton Algorithm)": If this box is checked, the calculate module follows the more accurate and intense algorithm to find the roots of the quadratic or cubic equations, that is, the assumption of [A]» 5 is not made. If there is a cubic equation, then the Newton-Raphson Algorithm is used. (See Section 2.5.2). By default, this box is checked. (3) "Interaction Type": This program can deal with 2:1 or 1:1 interactions between analyte (A) and protein (P). Users must make sure to select the correct interaction type; otherwise strange results may be obtained. (4) "CPU Time": This setting is not related to chemistry. It is for you to set how much CPU power will be put into the calculation. If you set it to "Critical" or "High", then most power of the PC will be used to do the calculation, and as a result, you may have difficulty in doing other things with your computer. If it is set to "Idle", then only when the CPU has nothing else to do will it carry out the calculations. This "Idle" setting is useful when you run the simulation programs and at the same time you want to do other things such as word processing or data analysis with your computer. The default setting is "Normal". In most cases, just keep this setting. When all the parameters are set, you can save these settings into a parameter file. In the "File name (without extension)" edit box, enter the desired file name of the parameter file. In the "Directory" edit box, enter the directory in which the file will be saved. You can also click on the button next to the "Directory" edit box to open a directory selection dialog, and select the desired destination of this parameter file. After the file name and directory are set, click on the "Save" button to save the parameters into a parameter file. The other way to 93 save parameters is to click on the "Save As . . . " button. In the popup dialog, navigate to the directory you want to save the file, and then you either enter a new file name or choose an existing parameter file to overwrite. Once you save the parameters into a file, you can open the saved parameter file the next time you want to run the simulation under the same experimental conditions. To open a parameter file, click on the "Open" button, and then choose the file in the typical open file dialog. If you want to reset all the parameters to default values, just click on the "New" button. Most default values of the parameters are zero. A useful trick for entering experimental conditions: You may have already saved the parameters for one experimental condition. When you want to enter a new experimental condition which is similar to the old one, you can just open the saved parameter file, change parameters wherever necessary, and "Save as" a new parameter file with a new file name corresponding to the new experimental conditions. Finally, click on the "Start" button to start the simulation run. Once it starts, the "Start" button changes to a "Pause" button which can be used to pause the run, and the "Exit" button becomes a "Stop" button which can be used to terminate the run. B.4 SimCE The user interface of SimCE is very similar to that of SimCETonly. You need to deal with the parameter files in the same way as you do with SimCETonly. In this section, we will focus on the differences. 94 One significant difference is that there are range settings for the binding constant (K) and the mobility of the complex (ft A P ) on the right-bottom corner of the user interface. The minimum, maximum and step values for both K and ja M P have to be set properly. As shown in the screenshot (Figure B.2), the minimum of jU^A^p is -0.00025cm2 IV-sec, the maximum is -0.00019cm2 IV-sec, and the increment for each step is lx l0" 6 cm 2 /F-sec. For those familiar with computer programming, these settings do the same thing as the "FOR i=start_value TO endvalue STEP increment" statement in BASIC (a common programming language). For the binding constant, logK is used instead of K. File Operations: New Open | Save As... Save Output Directory: C:\My Documents\Ning\ . . j Filename: 050130005018 V Detailed Output r U (cm2A/*sec) Ua= j-0.000188903 Up = •0.000175 CPU Time <~ Critical C High <• Normal r* Idle Interaction Type: r 1 A:P r 1 & 2: HAP) 2(AP2) 1 (A2P) Start Exit f" Consider Broadening J7 M ore Accurate Algorithm (e.g. Newton Algorithm) Initial Concentration (M) [A]0 m j5e-005 [P]b = 1.8e-006 Time of eof • 213.818 Voltage|V)= 20000 Length of the Whole Capillary (cm): L(whole) = Length to the D etector (cm): L(detector) • Time Increment (sec): dT = Length of the Plug (cm): L = 47 40 0.05 0.18 Uap= 1-0.00025 [ bgK-0.00019 5e-007 12 0.05 Figure B.2 User Interface of SimCE 95 B.5 SimCEData SimCEData does the data analysis and filtration tasks on the output data files generated by SimCE. You can use any data analysis software, such as SigmaPlot or Origin, to finish the same tasks that SimCEData can do. However, SimCEData provides a direct, convenient and task-oriented approach. The Screenshot of this program is shown in Figure B.3. Mr SimCEData Choose Data Files: -Data View: 025216005018.dat 02507200501E i.dat 02510800501E i.dat 02514400501E i.dat 02518000501E i.dat 02503600501E i.dat V Constant column width for all rows Add.. Removel FtemoveAII I 17 Show only 50 rows (faster than show all rows) Column 1 j Column 2 Column 3 364.250005 •2.500000e-004 8.000000 364.350005 -2.500000e-004 8.050000 — 364.450005 -2.500000e-004 8.100000 364.550005 •2.500000e-004 8.150000 364.700005 -2.500000e-004 8.200000 364.850005 -2.500000e-004 8.250000 365.050005 -2.500000e-004 8.300000 365.200005 -2.500000e-004 8.350000 365.450005 -2.500000e-004 8.400000 M • J C F ; ccnnnc i Knnnnn„ nn.i o acnnnn Sorting and Outputting Settings-Apply settings in this box to all data files r Sort by Columns: \7 1. J Column 1 T 2. j • j [Ascendant * \ j « J Ascendant Output Columns Columns: I Column 1 ——~3 V Output this column This is the No. (l ~Z. column in the output file(s). Output all columns Output no columns Range Settings Columns: | Column 1 No range specified C Range <=> 0.15 " 3 r +/- 1 Base: J3G8175 Save output into this directory: ••—— JE:\Programming\A2P_0.18cm_set2 Browse! CPU Time: C Critical C High <? Normal C Idle Start Exit Figure B.3 User Interface of SimCEData 96 The user interface of SimCEData is more complicated than the previous two programs. A l l components on the user interface will be discussed one by one. (1) Selecting and viewing data files: The top half region is for selecting and viewing data files generated by SimCE. Data files can be added into the list box* on the left by clicking on the "Add" button and then selecting files from the popup "Open File" dialog. You can select multiple files in the "Open File" dialog.^ Once a data file is added to the list box, the name of the data file shows up in the list box. At this point if you select and highlight an entry in the list box by clicking on it, then the contents of that data file will be read and displayed in the "Data View" box on the right. By default, only the first 50 rows of the data file are read in order to speed up the file-reading process. You can change this setting by clearing the checkbox before "Show only 50 rows". This way you will be able to view all the data in the data files. But please be patient when you want to view all rows, because a data file usually contains thousands of rows, and takes a few seconds to read. Another setting for data selection and viewing is the "constant column width for all columns". If you are sure that any individual column in your data files has a constant column width, you can clear this checkbox to speed up the file-reading process a little. Generally, it is a good idea to keep this box checked to avoid any unpredicted changes to the width of a column. * A list box displays a list of items, filenames in this case, that the user can view and select. * To select multiple files in the "Open File" dialog, you can hold down the "Ctrl" key and left-click on a file to add this file to your selection, or you can select a file (e.g. "filel") and then hold down the "Shift" key and left-click on a second file (e.g. "file2") to select multiple files between "file 1" and "file2". 97 (2) Sorting and output settings: These settings sit on the lower left region of the user interface. The data file can be sorted according to the primary and the secondary sorting columns. In the "Sort by columns" box, check the box before the primary and/or the secondary sorting column to activate these settings. Then choose the columns that are going to be sorted and the order of sorting (ASCENDANT or DESCENDANT). If not all columns in the data files generated by SimCE are needed in the new output data files, you can choose whatever columns you want to output. You can also control the positions of the outputted columns in the new output data files. If you want to output most of the columns, you first click on the "Output all columns" and then de-select the columns you don't want. On the other hand, if you want only a few columns, you may first click on the "Output no columns" and then select the columns you want. It is very likely that you want all data files having the same settings on sorting and output control. If so, you can just set up one data file, and click on the "Apply all settings in this box to all data files" button to make all data files have the same settings. (3) Range settings: These are the most important settings in SimCEData. Any column in any data file can have its own range settings. As discussed in Section 3.4, SimCE scans the K - M e p , A 2 p p l a n e to get simulated migration times for every point on that plane. Then SimCEData picks only the 98 pairs of K and /J-epAiP that can give the simulated migration time that is close enough to the real value. In other words, SimCEData will copy into the new output files only the rows with simulated migration times that fall into the range which is centered on the real experimental migration time. There are 3 ways to define a range. The first one is to specify the minimal and maximum values of the range. The other two methods are to define the errors allowed based on the real (base) value: [Basex(I-err%),Basex.(] + err%)\ or [Base-err,Base + err] where err is a positive real number. These methods do not make big differences to the output. The user is free to choose any one of them which best fits his/her taste. When everything is set, click on the "START" button to start the analysis. Once done, you have data files with extension ".out" in your specified directory. At this point, the data in the ".out" files can be plotted by commercial data analysis software. As mentioned in the Section 3.2, the data files are well formatted in order to make data transfer easy. The data files can be opened with any text editor, such as Notepad. In the text editor, you select all rows and columns, which is normally achieved by a hot key CTRL-A, then copy and paste them into the data tables of the data analysis software. In my personal opinion, the best text editor to do such a job is Ultraedit. Ultraedit is an excellent text editor with its powerful column manipulation capability and can be found at www.ultraedit.com. 99 B.6 S imCEGetK This program is used to estimate K and HepAP based on the output files from SimCEData. As discussed in Section 3.4, at the current stage of research and development of the CE simulation system, this program can only be used as a reference. You should always plot the output data from SimCEData in SigmaPlot, Origin or Igor. The file list box on the left is the place for you to select data files. Click on the "Add. . . " button to add data files, use the "Remove" button to remove a single data file from the list box, or use the "Remove A l l " button to remove all data files from the list box. At the top-right corner, you can tell SimCEGetK which columns are K and jUep,A2p m the data files, respectively. A l l the data files are required to have the same column sequence. Recall that in SimCE the ranges and steps of K and jUep A i P need to be set in order to determine how frequently the values of K and fj. A^p are taken for calculating the simulated migration time from the K - juepAP plane. In SimCEGetK, the step width for K is also required. It is best to have this value the same as that in SimCE, although it can be larger. Two methods have been implemented to estimate the K values so far. Two buttons corresponding to each of these two methods sit at the left-bottom region. Click on one of them to estimate the K and fi A P that are shown in the "Results" box. Only one pair of K and juep A^P is shown in the "Results" box. But if you want to check the detailed results, you can check the "Output results" box and specify the directory for saving the output result files. If you choose the "Standard Deviation (SD)" method, the output result file will be named as "finalbySD.out". If you choose the "Percentage of Overlapping", the output result will be named as "fmalbyPO.out". 100 SimCEGetK File List RO'vV'3; R0W13.out R0W25.out R0W26.out R0W12.out ROW38.out Add. . . | Remove j Remove All j Find K by Percentage of Overlapping Find K by Stardard Deviation Exit Column information: — Mobility of the complex: Binding constants (logK) Column 1 Bjj jCoTurnn 2 ~ ^ j Step Step of mobility of complex jle-OOG Step of logK: 0.02 |- Output Directory: — — P" Output results C:''.My DocumentsSNing\SimCESDat a\A2 ... | Results:-Binding constant (logK) = j* Mobility of complex -^ W s e c ) Figure B.4 User Interface of SimCEGetK Then you can open the output result files, "finalbySD.out" or "fmalbyPO.out", with any text editor. The output result file has its own format. In "finalbySD.out", for each logK value, the standard deviation, and the minimum and maximum of }X .p are the output. 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