Moment Analysis Method for Measurement of Reaction Equilibrium and Rate Constants by Using High-Performance Liquid Chromatography.

The moment analysis method was developed for the determination of association equilibrium constant (KA) and association (ka) and dissociation (kd) rate constants of intermolecular interactions between solute and ligand molecules. They are accurately determined by using moment equations from elution peak profiles because they are measured by using high-performance liquid chromatography (HPLC) under preferable conditions that neither immobilization nor chemical modification (i.e., fluorescence labeling) of solute and ligand molecules is required. To demonstrate the effectiveness of the method, it was applied to the inclusion complex formation system between dibenzo-18-crown-6 (DB18C6) and alkaline earth metal cations, i.e., Mg2+, Ca2+, and Sr2+, as a concrete example. Because the diameter of the three metal cations is smaller than that of the inner cavity of DB18C6, the values of KA, ka, and kd were analytically determined by assuming the stoichiometry of 1:1 between DB18C6 and the metal cation. They reflected the influence of the difference in the size between the inner cavity of DB18C6 and the metal cations on the inclusion complex formation. It seems that the moment analysis method based on HPLC separation is effective for the multifaceted analysis of chemical reactions because some characteristics of the method are different from those of other conventional methods. It must contribute to the dissemination of an opportunity for the study of chemical reactions to many researchers because of the versatility of HPLC.

A study on attempt for determination of permeation kinetics of coumarin at lipid bilayer of liposomes by using capillary electrophoresis with moment analysis theory.

It was tried to develop a moment analysis method for the determination of lipid membrane permeability. The first absolute and second central moments of elution peaks measured by liposome electrokinetic chromatography (LEKC) are analyzed by using moment equations. As a concrete example, elution peak profiles of coumarin in a LEKC system, in which liposomes consisting of 1-palmitoyl-2-oleoyl-sn­glycero-3-phosphocholine (POPC) and phosphatidylserine (PS) are used as a pseudo-stationary phase, were analyzed. It seems that lipid membrane permeability of coumarin across the lipid bilayer of POPC/PS liposomes was measured by the moment analysis method because previous permeability measurements using parallel artificial membrane permeability assay (PAMPA) and Caco-2 cells indicated that coumarin is permeable across lipid bilayer. However, it was also pointed out that the moment analysis method with LEKC is not effective for the determination of lipid membrane permeability and that it provides information about adsorption/desorption kinetics at lipid bilayer of liposomes. Therefore, different moment equations were also developed for the determination of adsorption/desorption rate constants of coumarin from the LEKC data. It was demonstrated that permeation rate constants at lipid bilayer or adsorption/desorption rate constants can be determined from the LEKC data on the basis of moment analysis theory for the mass transfer phenomena of coumarin at the lipid bilayer of POPC/PS liposomes. Mass transfer kinetics of solutes at lipid bilayer should be determined under the conditions that liposomes originally be because they are self-assembling and dynamic systems formed through weak interactions between phospholipid monomers. The moment analysis method using LEKC is effective for the experimental determination of the mass transfer rate constants at the lipid bilayer of liposomes because neither immobilization nor chemical modification of liposomes is necessary when LEKC data are measured. It is expected that the results of this study contribute to the dissemination of an opportunity for the determination of permeation rate constants or adsorption/desorption rate constants at the lipid bilayer of liposomes to many researchers because capillary electrophoresis is widespread.

Moment theory of affinity capillary electrophoresis for analysis of reaction kinetics of intermolecular interactions.

A theoretical basis for affinity capillary electrophoresis (ACE) was systematically developed by applying the moment theory to the study on intermolecular interactions between solute (S) and ligand (L) molecules to form complex (X). Moment equations were developed for both complete filling and partial filling ACE systems, in which SLm or SnL forms as X. They are effective for the determination of equilibrium and kinetic constants of association between S and L and dissociation of X from the first absolute and second central moments of elution peaks measured by ACE. In order to demonstrate their effectiveness, simulative calculations of partial filling ACE behavior was conducted by assuming the stoichiometry of 1:1 between S and L. Because partial filling ACE systems are classified into five categories based on experimental conditions, calculation results for all the five cases are explained. The combination of ACE and moment theory is effective because there are some advantages concerning the accurate analysis of intermolecular interactions. The results of this study provide an opportunity for the study on intermolecular interactions to many researchers because CE is already versatile.

Moment equations for partial filling capillary electrophoresis.

Moment equations were developed for partial filling CE systems, in which solute dissolution phenomena by spherical molecular assemblies or intermolecular interactions take place. Because experimental conditions of partial filling CE are divided into five categories on the basis of the magnitude relationship between the migration velocity of solute molecules and that of molecular assemblies or ligand molecules, the moment equations were systematically developed for each case by using the Einstein equation for diffusion and the random walk model. In order to demonstrate the effectiveness of the moment equations, they were applied to the analysis of partial filling CE behavior, which is correlated with dissolution phenomena of small solute molecules into spherical molecular assemblies as specific examples. Simulation results only in the case that the migration velocity of solute molecules is faster than that of molecular assemblies were represented in this paper. Detailed explanations about the derivation procedure of the moment equations and the simulation results in other cases can be found in the Supporting Information. The moment equations are theoretical bases for applying partial filling CE to the study on solute permeation kinetics at the interface of spherical molecular assemblies and on reaction kinetics of intermolecular interactions.

Kinetic study on solute permeation at the interface of molecular aggregates by partial filling capillary electrophoresis.

Moment analysis method using partial filling CE was developed for the kinetic study on solute permeation at the interface of spherical molecular aggregates. Moment equations for partial filling CE were developed by classifying CE systems into five categories according to the migration velocities of solute and molecular aggregate. The method was applied to the study on the dissolution of electrically neutral solutes into SDS micelles. Elution peaks were measured by partial filling CE while changing the concentration of SDS and the filling ratio of SDS micellar zone to the capillary (ϕM ). Partition equilibrium constants (Kp ) and rate constants of interfacial solute permeation of SDS micelles (kin and kout ) were determined from the first absolute and second central moments of the elution peaks by using the moment equations. Their values were comparable irrespective of ϕM and were almost the same as those previously measured by complete filling CE. The positive correlation of Kp with the hydrophobicity of the solutes was explained in terms of the change in kin and kout . It was demonstrated that the moment analysis method using partial filling CE is effective for studying solute permeation kinetics at the interface of spherical molecular aggregates.

Moment Theory of Chromatography for the Analysis of Reaction Kinetics of Intermolecular Interactions.

Moment theory was applied to the kinetic study of intermolecular interactions. The association equilibrium constant (KA) and association (ka) and dissociation (kd) rate constants of chemical reactions were analytically determined on the basis of the moment theory from elution peak profiles measured by high-performance liquid chromatography (HPLC). The HPLC data were measured under the conditions that neither immobilization nor fluorescence labeling of solute and ligand molecules is required. These are the advantages of the moment analysis method for determining accurate values of KA, ka, and kd. Moment equations were developed on the basis of the Einstein equation for diffusion, the random walk model, and the general rate model of chromatography. The moment analysis method was applied to the inclusion complex formation system between dibenzo-18-crown-6 or dibenzo-15-crown-5 and alkali metal cations. It was demonstrated that the values of KA, ka, and kd can be determined on the assumption that the stoichiometry between crown ethers and cations is 1:1 or 2:1. The influence of the difference in the size between the inner cavity of crown ethers and cations on the association and dissociation of the inclusion complex was considered. The moment analysis method using HPLC is effective for analyzing intermolecular interactions from various perspectives because it is based on the separation technique and has different characteristics from other methods such as spectroscopy. The results of this study contribute to the dissemination of an opportunity for studying intermolecular interactions from equilibrium and kinetic points of view to many researchers because HPLC is widespread.

Simplification of Moment Analysis Procedure for Kinetic Study of Chromatographic Behavior of Core-shell Particles.

The moment analysis method for chromatographic behavior in core-shell columns was simplified. Mass-transfer phenomena other than intra-stationary phase diffusion are analyzed while considering that the packing materials are spherical particles. The manner of intra-stationary phase diffusion is analyzed while assuming a hypothetical flat plate. For most core-shell particles commercially available, the geometry of a spherical thin layer can be supposed as a hypothetical flat plate with a relative error of less than ca. 2% because the thickness of the shell layer is sufficiently smaller than the diameter of whole particle. This supposition makes moment analysis easier because the moment equations for flat plates are simpler than those strictly developed for core-shell particles. Some chromatographic data measured using a core-shell column were analyzed by the simple moment analysis method to confirm its usefulness. It was demonstrated that the method is effective for a preliminary study of mass-transfer kinetics in core-shell columns.

Simple Moment Analysis for a Kinetic Study of the Chromatographic Behavior of Spherical Particles and Silica Monoliths.

A simple procedure of moment analysis was proposed for a kinetic study of the rate processes in the columns packed with full-porous spherical particles and silica monoliths. Previous chromatographic data measured in reversed-phase HPLC systems using Mightysil and Chromolith columns were analyzed by a simple moment analysis. The surface of the packing materials is chemically modified with octadecyl alkyl ligands. A mixture of methanol and water (80/20, v/v) and alkylbenzene homologous series (C6H5CnH2n+1, n = 0 - 7) were used as the mobile-phase solvent and sample probes, respectively. More detailed information about the experimental conditions is provided in Supporting Information. The values of the intra-stationary phase diffusivity (De) and the surface diffusion coefficient (Ds), derived by the simple moment analysis, were almost the same as those by the conventional moment analysis. The simple moment analysis is effective for quantitative studies of mass transfer in chromatographic systems. The previous chromatographic data were also analyzed by assuming external porosity (εe) as typical values, i.e., 0.40 for spherical particles and 0.70 for silica monoliths. The resulting values of De and Ds were of the same order of magnitude as those derived by using εe experimentally measured. Even if εe is assumed to be typical values, the simple moment analysis is effective for preliminary studies of the mass-transfer kinetics in the columns.

Moment analysis of peak broadening in affinity capillary electrophoresis and electrokinetic chromatography.

A moment analysis method is effective for the kinetic study of intermolecular interaction and solute permeation at the interface of spherical molecular aggregates. Association and dissociation rate constants of intermolecular interactions or the rate constants of interfacial solute permeation can be determined on the basis of the moment theory from the first absolute and second central moments of elution peaks measured by affinity capillary electrophoresis (ACE) or electrokinetic chromatography (EKC). In this study, it was discussed how the experimental conditions concerning the concentration of ligand molecule or molecular aggregate should be optimized in ACE or EKC experiments in order to determine the rate constants as accurately as possible. At first, peak broadening due to axial diffusion, reaction kinetics of intermolecular interaction, and mass transfer kinetics of interfacial solute permeation was quantitatively evaluated under hypothetical ACE or EKC conditions, which were chosen on the basis of our previous studies. Second, it was confirmed that some rate constants were determined in the previous studies from ACE and EKC data measured under appropriate experimental conditions. Then, a procedure was considered for determining more accurate analytical results of the objective rate constants from ACE or EKC peaks experimentally measured.

Moment theory for the analytical determination of rate constants for solute permeation at the interface of spherical molecular aggregates.

Moment equations were developed on the basis of the Einstein equation for diffusion and the random walk model to analytically determine the rate constant for the interfacial solute permeation from a bulk solvent into molecular aggregates (kin ) and the inverse rate constant from the molecular aggregates to the bulk solvent (kout ). The moment equations were in good agreement with those derived in a different manner. To demonstrate their effectiveness in one concrete example, the moment equations were used to analytically determine the values of kin and kout of three electrically neutral solutes, i.e. resorcinol, phenol, and nitrobenzene, from the first absolute (µ1A ) and second central (µ2C ) moments of their elution peaks, as measured by electrokinetic chromatography (EKC), in which the sodium dodecyl sulfate (SDS) micelles were used as a pseudostationary phase. The values of kin and kout should be determined with no chemical modifications and no physical action with the molecular aggregates because they are dynamic systems formed through weak interactions between the components. The moment analysis of the elution peak profiles measured by EKC is effective to unambiguously determine kin , kout , and the partition equilibrium constant (kin /kout ) under appropriate experimental conditions.

Moment analysis for mass transfer kinetics at the interface of spherical molecular aggregates.

New moment equations were developed on the basis of the principle of relativity for explaining some characteristics of elution peaks measured by electrokinetic chromatography (EKC) using spherical molecular aggregates. Basic equations representing mass balance and mass transfer kinetics in EKC system in a Galilean coordinate system S were transformed to those in another coordinate system S', which imaginarily moved with respect to S. Moment equations for EKC peaks in S' in the time domain were derived from the analytical solution of the modified basic equations in the Laplace domain. Moment equations for EKC peaks in S were derived from those in S' by the inverse Galilean transformation. The moment equations were used to the re-analysis of EKC data previously measured. The values of permeation rate constants of thymol at the interface of sodium dodecylsulfate micelles were fairly in agreement with those determined in a previous study. The moment equations were also used to the numerical simulation of elution peaks in EKC systems. The influence of some experimental parameters on elution peak profiles was quantitatively analyzed. The moment equations are useful for determining the rate constants of interfacial solute permeation from elution peak profiles measured by EKC.

Moment analysis for reaction kinetics of intermolecular interactions.

Moment equations were developed on the basis of the principle of relativity for analyzing elution peak profiles measured by ACE to analytically determine the association (ka ) and dissociation (kd ) rate constants of intermolecular interactions. Basic equations representing the mass balance, mass transfer rate, and reaction kinetics in ACE system in a Galilean coordinate system S were transformed to those in another coordinate system S', which imaginarily moved with respect to S. Moment equations for ACE peaks in S' in the time domain were derived from the analytical solution of the modified basic equations in the Laplace domain. Moment equations for ACE peaks in S were derived from those in S' by the inverse Galilean transformation. The moment equations were used for analyzing some ACE data previously published to determine ka and kd values. It was demonstrated that the moment equations were effective for extracting the information about affinity kinetics of intermolecular interactions from the elution peak profiles measured by ACE. The moment equations were also used to discuss the influence of mass transfer and reaction kinetics on ACE peak profiles. Some results of the numerical calculations are also indicated in Supporting Information.

Kinetic Study of Chiral Intermolecular Interactions by Moment Analysis Based on Affinity Capillary Electrophoresis.

Capillary electrophoresis is a method for analyzing intermolecular interactions that does not require immobilization of molecules to a solid surface or introduction of a luminescent moiety. Recently, an advanced method, moment analysis based on affinity capillary electrophoresis (MA-ACE), was developed. This method can determine not only the equilibrium constant but also the rate constants of an intermolecular interaction. Through MA-ACE, it became possible to theoretically predict an increase in the variance of an observed peak caused by intermolecular interaction. In this study, we confirm the prediction and determine the kinetic constants by using MA-ACE to analyze an intermolecular interaction between cyclodextrin and phenoxypropionic acid. A numerical calculation is performed to confirm that the derived rate constants by MA-ACE are appropriate.

Numerical correction for asymmetrical peak profiles for moment analysis of chromatographic behavior.

A numerical correction method is proposed for the moment analysis of some properties of chromatographic columns, even when the profiles of their elution peaks exhibit some asymmetry. Information on the retention equilibrium and the mass transfer kinetics of a C18-silica gel packing material is derived by the moment analysis from the experimental data obtained under three different sets of conditions: (1) the tailing peaks eluted from the column used; they are not corrected by the numerical method; (2) the same peaks are corrected by the numerical method, which provides the information on the column radial heterogeneity and on the efficiency at its center; and (3) symmetrical peaks having nearly Gaussian profiles eluted from another column packed with the same material; these peaks are not corrected by the numerical method. The analytical results obtained under the three different conditions are compared. The results demonstrate that the numerical correction method allows the determination of the information on the chromatographic behavior of columns from asymmetrical peak profiles, i.e., column efficiency at radial center, order of the polynomial functions for representing the radial distributions of mobile phase flow velocity and column efficiency, ratio of the flow velocity near the wall to that at the column center, and ratio of the column efficiency near the wall to that at the column center.

Moment Analysis of Mass Transfer Kinetics in Micellar Electrokinetic Chromatography Systems.

Moment equations were developed for quantitatively studying the separation characteristics of micellar electrokinetic chromatography (MEKC). They explain how the first absolute and second central moments of elution peaks are correlated with some fundamental parameters of the partition equilibrium and mass transfer kinetics in MEKC systems. In order to discuss the influence of the mass transfer kinetics on peak broadening, the moment equations were used to analyze the separation behavior in MEKC systems. Separation conditions were chosen on the basis of practical MEKC experiments previously conducted. It was quantitatively clarified that both the solute permeation at the interfacial boundary of surfactant micelles and axial diffusion of solute molecules in a capillary had a predominant contribution to the spreading of the elution peaks in MEKC systems. This is a preliminary study for the analytical determination of rate constants concerning solute permeation at the interface of surfactant micelles from elution peak profiles measured by MEKC. In addition, it was also indicated that the experimental conditions of MEKC systems could be controlled so that the interfacial solute permeation would have a predominant role for the band broadening. For example, the contribution of the interfacial permeation was about 33 times larger than that of the axial diffusion of solute molecules under the MEKC conditions in a previous study. This means that the rate constants could appropriately be determined for the interfacial solute permeation.

Moment Analysis Theory for Size Exclusion Capillary Electrochromatography with Chemical Reaction of Intermolecular Interaction.

New moment equations were developed for size exclusion capillary electrochromatography (SECEC), in which intermolecular chemical reactions simultaneously took place. They explain how the first absolute and second central moments of elution peaks are correlated with some fundamental equilibrium and kinetic parameters of mass transfer and chemical reaction in SECEC column. In order to demonstrate the effectiveness of the moment equations, they were used to predict chromatographic behavior under hypothetical SECEC conditions. It was quantitatively studied how the association and dissociation rate constants of intermolecular interaction affected the position and spreading of elution peaks. It was indicated that both the intermolecular reaction kinetics and axial dispersion of solute molecules in a capillary column had a predominant contribution to the band broadening.

Evaluation of Migration Time and Variance for Accurate Kinetic Studies Based on Affinity Capillary Electrophoresis.

Affinity capillary electrophoresis (ACE) has been widely applied to evaluate binding constants of various systems. Recently, moment equations were derived based on the moment analysis (MA) theory for describing the influence of reaction kinetics and longitudinal diffusion on the elution peak profiles measured by ACE (MA-ACE). The equations enable one to obtain not only the binding constants but also the reaction rate constants from the migration time and variance of elution peaks. However, it is necessary to consider other factors (e.g., sample injection, detector window, Joule heating, and ramp time of the voltage increase) to improve the accuracy of MA-ACE. The variance of these effects was quantified under typical experimental conditions. Such quantification clarified the process to obtain the rate constants. The best experimental conditions to achieve high accuracy were discussed.

Kinetic Study of Interaction between Solute Molecule and Surfactant Micelle.

We developed moment analysis of affinity kinetics by chromatographic capillary electrophoresis (MKCCE) method for the kinetic study of intermolecular interactions. Association and dissociation rate constants of the interaction in a micellar electrokinetic chromatography (MEKC) system between thymol and sodium dodecylsulfate micelle were determined by the MKCCE method. It is a method based on the moment theory for the kinetic study of intermolecular interactions under the conditions that neither immobilization nor chemical modification of molecules is required. In CCE mode, experimental conditions are controlled so that the migration of solute-micelle complex is stopped and only solute molecules migrate in a capillary. Mass transfer behavior of solute molecules in the CCE system is analogous to that in a chromatographic system. However, because it was difficult in practice to really perform CE experiments under the CCE conditions, CE data were measured with changing experimental conditions, i.e., applied pressure, under the conditions that the migration velocity of solute-micelle complex was around zero. The rate constants could be analytically determined from the CE data. In the MKCCE method, it is not necessary to fit elution curves numerically calculated to those experimentally measured for the determination of the rate constants. Regarding the interaction between thymol and SDS micelles, association equilibrium constant and association and dissociation rate constants were determined as 6.35 × 10(3) dm(3) mol(-1), 5.6 × 10(4) dm(3) mol(-1) s(-1), and 8.7 s(-1), respectively. It was demonstrated that the MKCCE method was effective for the kinetic study of intermolecular interactions.

Kinetic Study of the Intermolecular Interaction between 2-Phenoxypropionic Acid and ß-Bromo-cyclodextrin Affixed on the Stationary Phase by Liquid Chromatography.

The intermolecular interaction between 2-phenoxypropionic acid and ß-bromo-cyclodextrin affixed on the stationary phase surface in a chiral HPLC system was studied by the moment analysis method. At first, pulse response and peak parking experiments were conducted to measure some parameters concerning the column geometry, adsorption equilibrium, and mass-transfer kinetics. Then, the first absolute moment (µ1) and second central moment (µ2') of the elution peaks were analyzed by the moment equations, which were developed by assuming that the reaction kinetics between the solute molecules and the functional ligands can be represented by the Langmuir-type rate equation. Finally, the flow-rate dependence of HETP calculated from µ1 and µ2' was analyzed by using the values of the parameters to determine the association and dissociation rate constants of the intermolecular interaction. It was demonstrated that the combination of the chromatographic experiments and moment analysis is one of the effective strategies for the kinetic study of intermolecular interactions.

Moment equations for chromatography based on Langmuir type reaction kinetics.

Moment equations were derived for chromatography, in which the reaction kinetics between solute molecules and functional ligands on the stationary phase was represented by the Langmuir type rate equation. A set of basic equations of the general rate model of chromatography representing the mass balance, mass transfer rate, and reaction kinetics in the column were analytically solved in the Laplace domain. The moment equations for the first absolute moment and the second central moment in the real time domain were derived from the analytical solution in the Laplace domain. The moment equations were used for predicting the chromatographic behavior under hypothetical HPLC conditions. The influence of the parameters relating to the adsorption equilibrium and to the reaction kinetics on the chromatographic behavior was quantitatively evaluated. It is expected that the moment equations are effective for a detailed analysis of the influence of the mass transfer rates and of the Langmuir type reaction kinetics on the column efficiency.