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Evaluation of Migration Time and Variance for Accurate Kinetic Studies Based on Affinity Capillary Electrophoresis Nozomu Suzuki, and Kanji Miyabe Anal. Chem., Just Accepted Manuscript • DOI: 10.1021/acs.analchem.7b02598 • Publication Date (Web): 11 Aug 2017 Downloaded from http://pubs.acs.org on August 26, 2017
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Analytical Chemistry
Evaluation of Migration Time and Variance for Accurate Kinetic Studies Based on Affinity Capillary Electrophoresis Nozomu Suzuki* and Kanji Miyabe* Department of Chemistry, College of Science, Rikkyo University, 3-34-1 Nishi-Ikebukuro, Toshima, Tokyo 171-8501, Japan ABSTRACT: 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 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.
Intermolecular interactions play a key role in molecular recognition, sensing, and self-assembly. In living organisms, for instance, signal transduction is the result of intermolecular interactions (i.e., the process of converting external signals into a specific cellular response).1 Understanding the process is essential to find potential drug targets since most of the diseases show abnormal function in the process. The study of molecular interactions also involves characterizing the specificity, stoichiometry, binding strength, and kinetics of binding interactions between the drug and target biocompound. Capillary electrophoresis has been developed over the decades as a qualitative and quantitative analytical tool to study intermolecular interactions. This method is mainly applied to aqueous systems to study biocompounds such as amino acids,2 proteins,3 peptides,4 DNA,5 saccharides,6 and lipids,7 although there are some reports on non-aqueous systems as well.8-10 Recently, the method has been applied to study nano-particles that have potential applications as biocompatible materials.11-14 Following are the advantages of capillary electrophoresis: low sample and reagent consumption; relatively short analysis time; ease of automation; applicability of physiological conditions such as pH, ionic strength, and temperature; and elimination of the need for fluorescent markers and immobilization support.15,16 There are several methods to determine binding constants using capillary electrophoresis.15-17 Affinity capillary electrophoresis (ACE) is the most popular among them.16 Typically, the concentration of one of the two interacting compounds in a buffer is varied and the migration or mobility shift of the other compound is observed. Not only binding constants but also rate constants can be measured using capillary electrophoresis.18-22 Krylov et al. reviewed the kinetic capillary electrophoresis (KCE) technique with various categories of experimental setups.20,21 There are numerical and analytical approaches in KCE, the numerical one being the more generally applicable to any setup. However,
it requires expertise in computational methods to ensure the accuracy of the calculation, and thus, the analytical solution is preferred.20 In this study, the following interaction is considered. S + L ⇄ X
(1)
where S, L, and X represent a solute, ligand, and solute–ligand complex, respectively, and ka [dm3/(mol⋅s)] and kd [1/s] are the association and dissociation rate constants. The association equilibrium constant KA [dm3/mol] can be correlated with the rate constants as KA = ka/kd. Moment analysis (MA) is often used to calculate the height equivalent of the theoretical plate used in high performance liquid chromatography,23-26 but it can also be used to estimate the theoretical plate for capillary electrophoresis. Recently, moment equations were derived based on the moment analysis theory for describing the influence of reaction kinetics and longitudinal diffusion on the elution peak profiles measured by ACE (MA-ACE).27 The equation correlates the rate constants with migration time (tm [s]), variance of the reaction kinetics (σrk2 [s2]), and longitudinal diffusion (σdiff2 [s2]) as follows.
=
1 + +
=
2 − +
!"" =
2 1 + #$, + $, & +
(2)
(3)
(4)
where CL is the ligand concentration [mol/dm3], vS and vX are the migration velocities of S and X [m/s], respectively, and DL,S and DL,X are the longitudinal diffusion coefficients of S and X [m2/s]. Zm is the length that the solute band travels during the separation process [m] and will be called the migration
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length henceforth. The difference between the migration and effective lengths will be clarified later (see below). For the moment analysis, the information of migration time and variance is defined rigorously. These values can be extracted from the electropherogram as first absolute moment (µ1 [s]) and second central moment (µ2’ [s2]).28
= '１ =
*
(+ )
*
(+ )
,-, = ' ′ =
*
(+ − '１ )
* (+ )
(5)
(6)
where C(t) is the sample concentration measured at time t [mol/dm3] and σtot2 is the variance of the elution peak [s2]. The shape of the peak profile does not have to be assumed to calculate these values. Therefore, it theoretically enables to use various peak shapes (triangular, rectangular, Gaussian, etc.) on the same footing. Furthermore, the two values can be easily calculated on a spreadsheet program, such as Microsoft Excel. Another advantage of this method is the clarity of the factors that affect the resulting rate constants, e.g., longitudinal diffusion and Joule heating, as long as the moment equation has been solved in the past. Here, we discuss in detail the effect of ramp time to increase the applied voltage. Williams and Vigh suggested that the introduction of ramp time is necessary to estimate accurate mobility.29 In this paper, the effects of ramp time on migration time, variance of reaction kinetics, and Joule heating are considered for the analytical equation. This study aims to (1) establish the equation of the analytical solution including the effect of ramp time, (2) describe the phenomena that affect the resulting rate constants, (3) organize the quantification process to estimate rate constants determined by capillary electrophoresis, and (4) clarify the method and best experimental conditions required to achieve high accuracy. In the Theory section, the effect of various factors (reaction kinetics, longitudinal diffusion, sample injection, detector, Joule heating, and ramp time) on migration time and variance are briefly described. Since the effect of ramp time is coupled with reaction kinetics and Joule heating, it is described separately from the other factors (“Consideration of the Effect of Ramp Time on Reaction Kinetics, Diffusion, and Joule Heating”). Most of the equations that appear in the Theory section are necessary to quantify the effect on variance. In the Results and Discussion section, the contribution of various factors to migration time and variance are estimated under typical experimental conditions (“Effect of Reaction Kinetics,” “Effect of Sample Injection,” “Effect of Detector Window,” “Effect of Joule Heating,” and “Consideration of the Effect of Ramp Time on Reaction Kinetics, Diffusion, and Joule Heating”). Titles in this section are the same as those in the Theory section, to emphasize which parts are related. After describing each effect, the quantification procedure of the reaction rate constant of MA-ACE is summarized (Quantification Process). When it is necessary for the process, the equation in the Theory section is referred to. At the end of Results and Discussion, key experimental conditions to increase the accuracy of MA-ACE are discussed (Consideration of Experimental Conditions to Increase Accuracy of MA-ACE).
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THEORY Contributing Factors to Migration Time and Variance. We consider five factors that affect the quantification of rate constants: reaction kinetics,27 longitudinal diffusion, sample injection,30-33 detector,30 and Joule heating.34 The adsorption of chemical species on the surface of the capillary is assumed to be negligible by choosing the appropriate buffer and/or the use of a polymer-coated capillary.35 The total variance can be described by the following equation. ,-, = + !"" + !/0 + 1, + 213,
(7)
where σinj2, σdet2, and σheat2 correspond to the effects of sample injection, detector, and Joule heating on σtot2 [s2]. The units of variance can be converted to [m2] by simply multiplying by the square of solute band velocity at the detector.27,30 In this case, σ’2 [m2] is used instead of σ2 [s2]. It is preferable to consistently use [s2] as the unit of variance because the migration time is described in [s]. However, the equations of variance are often simpler and easier to imagine when described in [m2]. Most of the equations will therefore be described in [m2] in this paper. Since sample injection and detection are independent of the separation process, their effect on effective capillary length (Z [m]) can be described as follows.
= + !/0 + 1,
(8)
where Zinj and Zdet correspond to the contribution of sample injection and detector [m]. By estimating or neglecting Zinj and Zdet, one can obtain the migration time with the following equation.
=
4
(9)
where vB is the solute band velocity [m/s]. Effect of Reaction Kinetics and Diffusion. The units of σrk2 and σdiff2 can be transformed from [s2] to [m2] (eqs. 3 and 4) by multiplying σrk2 by vB2 = (vS+vXKACL)2/(1+KACL)2. Zm in eqs. 3 and 4 can be replaced by tm(vS+vXKACL)/(1+KACL) based on eq. 2. The following equation is thus obtained:
′ =
2 −
1 +
′ !"" =
2$, + $,
1 +
(10)
(11)
To determine the diffusion coefficients in eq. 11, an equation proposed by Wilke and Chang can be used.36 $=
7.4 × 10:; ? @AB,3 +.C
(12)
where D is the diffusion constant of a molecule [m2/s], T is the absolute temperature [K], α is the association coefficient, M is the molecular weight, η is the viscosity [N⋅s/m2], and Vb,a is the molar volume at the normal boiling point. From this equation, diffusion coefficients can be estimated without performing an experiment. If the diffusion coefficient of one sample D1 [m2/s] is known from an experiment, that of another sample D2 [m2/s] in the same solvent can be estimated by solely calculating Vb,a for each (define the value as Vb,a1 and Vb,a2, respectively). The unknown D2 can be written as D2 = D1Vb,a1/ Vb,a2.
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Analytical Chemistry
Effect of Sample Injection. The general process and type of flow are described before considering the effect of sample injection. Typically, hydrodynamic injection is used rather than electrokinetic injection, because the injected sample amount depends on the electrical charge of the samples for the latter. After sample injection, separation by applying voltage follows. The voltage is gradually increased up to the separation voltage for the ramp time (trp), and then, constant voltage is applied. The typical analyzing process and type of flow can be described as follows (Fig. 1).
(Step 1) Sample injection by pressure: laminar flow. (Step 2) Running buffer injection by pressure: laminar flow. (Step 3) Separation by applying voltage (0 < t ≤ trp): plug flow. (Step 4) Separation by applying voltage (trp < t ≤ tm): plug flow.
Figure 1. Sample distribution during injection and separation. Effects of longitudinal and radial diffusion are not included for clarity.
For the sample injection process (Steps 1 and 2), the type of flow is laminar because the Reynolds number of the pressuredriven flow is quite small in the case of capillaries. For the separation process (Steps 3 and 4), it becomes plug flow due to electro-osmosis. For hydrodynamic injection, fluid velocity along the longitudinal axis (u) [m/s] can be described as follows.37 ∆F D= H − H 4@G +
(13)
where ∆p is the pressure drop [N/m2], L is the total capillary length [m], r0 is the inner radius of the capillary [m], and r is the radial position [m] in the capillary. The average speed is ∆pr02/(8ηL). The effect of injection on length and variance can be described as follows (see Supporting Information for the derivation).
I/0 =
H+ ∆FJ J + 2∆F 12@G
′!/0 =
H+ K ∆FJ J + ∆FJ ∆F J + ∆F 288@ G
(14)
(15)
where t1 and t2 correspond to the time [s] for Steps 1 and 2. Similarly, ∆p1 and ∆p2 correspond to the pressure at Steps 1 and 2. The unit of σ’inj2 is [m2]. The equation of variance for a rectangular pulse of width Winj [m] is also described in Supporting Information. Effect of Detector Window. The effect of detector window can be described with the following equation.38
1, =
M 1, 2
’ 1, =
M 1, 12
(16)
(17)
Effect of Joule Heating. Grushka et al. described the effect of Joule heating on the height equivalent to a theoretical plate by the following equation.34 O=
’213, P+ C Q K B R Λ 4 =
24$8B > E2ΛCbR02B2. From this equation, the variance of the heat can be described as follows. ’213, =
P+ C Q K B R Λ 4
24$#8B
