Anal. Chem. 2009, 81, 343–348
General Approach for Certain Quantitative Calculations for Instance of the Variance of Reversible Adsorption to the Capillary Wall in CE ´ kos Ve´gva´ri,§ Ferenc Kila Viktor Farkas,†,‡ Melinda Rezeli,†,‡ A ´ r,‡,| and Stellan Hjerte ´ n*,† Department of Biochemistry and Organic Chemistry, BMC, Uppsala University, SE-751 23 Uppsala, Sweden, Department of Analytical and Environmental Chemistry, and Institute of Bioanalysis, University of Pe´cs, H-7624 Pe´cs, Hungary, and Clinical Protein Science, Department of Electrical Measurements, Lund University, SE-221 00 Lund, Sweden Miniaturization of analytical separation methods offers several advantages, including short run times, high resolution, and high recovery of the sample constituents. To optimize these parameters, the reversible adsorption (to minimize loss in resolution), as well as the irreversible adsorption (to minimize loss of analytes) must be quantified. However, no useful equation is available for the calculation of the variance of reversible adsorption. Therefore, we have taken another approach to quantify the reversible interaction. The method is unique and important since no equation for calculation of this variance is required. Instead, two experiments are required, which are run under such conditions that the variance of a certain parameter has the same numerical value in the two experiments (one with and without EOF), except for the variance of reversible adsorption. The approach is universal in the sense that it can be used for many different mathematical concepts and be modified to also cover certain functions other than a sum of parameters. We have also introduced a simple expression for the irreversible adsorption, which shows that the hydrophobic interaction from only two methyl groups in the coating gives rise to as much as 40-50% loss of protein, and the width of the zones in the capillary with this coating was 8-15% larger compared to the zone width in the polyacrylamide-coated capillaries. The reproducibility in migration time, peak area, and peak width in two consecutive runs in capillaries with two methyl groups in the coating was very low, but in EOF-free polyacrylamide-coated capillaries extremely high, indicating that the reversible and irreversible adsorption of proteins to this coating is negligible. The scanning detector, frequently used in free zone electrophoresis in the 1960s-1970s, gives true separation parameters and is, therefore, much preferable to the stationary detector used in most CE experiments, because this detector gives apparent separation parameters. Electroosmosis generates a plug flow, which merely displaces electrophoretic zones; i.e., this flow cannot, in practice, affect the shape of the zone nor its width when the radius of the separation channel is much larger than the width of the Helmholtz double layer. Accordingly, the resolution is the same in an EOFgenerating and in an EOF-free capillary if the electrophoretic * To whom correspondence should be addressed. E-mail: Stellan.Hjerten@ biorg.uu.se. Tel.: +46 18 471 4461. Fax: +46 18 558431. E-mail: ferenc.kilar@ aok.pte.hu. Tel.: +36-72-536273. Fax: +36-72-536254. † Uppsala University. ‡ Department of Analytical and Environmental Chemistry, University of Pe´cs. § Lund University. | Institute of Bioanalysis, University of Pe´cs. 10.1021/ac8010457 CCC: $40.75 2009 American Chemical Society Published on Web 11/26/2008
migration times and distances are the same in the two capillaries, provided that the variances for reversible and irreversible adsorption of the analyte onto the wall of the two capillaries are zero. However, all published coatings for generation of EOF seem to have hydrophobic groups that (may) cause interactions with the analytes, particularly with proteins. Capillaries with such coatings are used, for instance, to transfer proteins into a mass spectrometer for qualitative and quantitative analysis. The great variations in reported quantitative protein determinations of the same proteins in different laboratories may be caused by irreversible adsorption of the proteins onto the wall of the capillary or by the fact that the formula previously derived for such determinations is not correct when the stationary detector is used to record the electropherogram. All published equations for the reversible interaction contain constants that, in practice, cannot be determined. Therefore, we have developed an approach that is not based on any equation, but yet gives a high accuracy in the determination of the variance of the reversible adsorption. This unique way to solve a problem is to some extent universal and can, therefore, be employed also in disciplines outside the field of separation science, but it is surrounded by many requirements, which limits its application range. To illustrate the method, we chose to coat one capillary with acrylamide, which we have earlier shown gives a negligible adsorption of proteins,1 and one with dimethyldiallylammonium chloride (DMDAAC) to get a coating with a high EOF and a moderate protein adsorption, induced by the two methyl groups. If this coating gives both reversible and irreversible adsorption of proteins, one can expect many of the published coatings to give still stronger interaction. THEORY Experimental Determination of the Variance of Reversible Adsorption of an Analyte onto the Capillary Wall. Several equations have been derived for the variance of the reversible adsorption of an analyte onto the capillary wall.2-5 They all have the form 2 σads ) Cadsv2t ) CadsLv
(1)
where Cads is the adsorption coefficient, v is the velocity of the analyte, t is the migration time, and L is the migration distance. (1) Hjerte´n, S.; Mohabbati, S.; Westerlund, D. J. Chromatogr., A 2004, 1053, 181–199.
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The derivations are based on certain assumptions about the adsorption mechanism. Whether these assumptions are always fulfilled in CE experiments is not easy to decide. Since the constant Cads cannot be determined by any known theory, we 2 need another way to determine σads . Assume that we do two electrophoresis experiments in a CE instrument with a scanner or a whole-tube detector under the same conditions, including the same migration time and field strength, one in the presence and one in the absence of electroosmosis. The total experimentally determined variances (σ2tot,ep+eo and σ2tot,ep), which can both be calculated from the width of the peaks, can then be written as a sum of variances: 2 2 2 2 2 2 σtot, ep+eo ) (σinj + σdiff + σsedD + σ∆κ,∆pH + σsed, h + 2 2 2 + σads + σint )ep+eo (2a) σcurv
2 2 2 2 2 2 σtot, ep ) (σinj + σdiff + σsedD + σ∆κ,∆pH + σsed, h + 2 2 2 + σads + σint )ep (2b) σcurv 2 2 2 2 2 2 2 2 where σinj , σdiff , σsedD , σ∆κ,∆pH , σsed, h, σcurv, σads, and σint are the variances for injection, diffusion, sedimentation with diffusion, ionic strength and pH differences, horizontal sedimentation, capillary curvature, wall interaction, and interaction with the buffer, respectively. See the definitions and detailed descriptions of these variances in ref 1. 2 2 The variances σtot, ep+eo and σtot, ep can be determined from the electropherograms recorded in the presence and the absence of 2 EOF, respectively. Observe, however, that σads, ep+eo cannot be calculated from eq 2a since all the other variances cannot be calculated or only inaccurately. Therefore, recalling the above assumed experimental conditions (with a scanning detector), a subtraction of eq 2b from eq 2a gives
2 2 2 2 σads, ep+eo - σads, ep ) σtot, ep+eo - σtot, ep
(3)
2 2 where σads, ep+eo and σads, ep are the variances for the reversible adsorption of the analyte onto the capillary wall in the experiments performed in the presence and in the absence of EOF, respectively; σ2tot, ep+eo is the measured total variance in an electrophoretic 2 experiment performed in the presence of EOF; σtot, ep is the measured total variance in an analogous electrophoretic experiment performed in the absence of EOF. Equation 3 permits calculation of the difference in the experimentally determined variances for reversible adsorption of an 2 analyte in the presence of EOF (σads, ep+eo) and in its absence 2 (σads, ep) and is valid when a scanning detector6,7 (but not a stationary detector) is used to record the electropherograms.
(2) Giddings, J. C. Dynamics of Chromatography, Part I, Principles and Theory; Marcel Dekker, Inc.: New York, 1965. (3) Wieme, J. R. Chromatography, A Laboratory Handbook of Chromatographic and Electrophoretic Methods; Heftmann, E., Ed.; Van Nostrand Reinhold: New York, 1975. (4) Gasˇ, B.; Sˇtı`dry´, M.; Rizzi, A. M.; Kenndler, E. Electrophoresis 1995, 16, 958–967. (5) Poppe, H. Adv.Chromatogr. 1998, 38, 233–300. (6) Hjerte´n, S. Chromatogr. Rev. 1967, 9, 122–219. (7) Hjert´en, S. Protides of the Biological Fluids, Proceedings of the 7th Colloquium, Bruges 1959; Peeters, H., Ed.; Elsevier: Amsterdam, 1960.
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When the adsorption onto the EOF-free coated capillary is 2 negligible (σads, ep ) 0), the variance for the adsorption onto the 2 EOF-generating capillary wall (σads,ep+eo ) can be calculated by the expression (see eq 3) 2 2 2 σads, ep+eo ) σtot, ep+eo - σtot, ep
(4a)
In the case σ2ads,ep< σ2ads,ep+eo, but not negligible we get 2 2 2 σads,ep+eo > σtot,ep+eo - σtot,ep
(4b)
In the Appendix, we make a generalization (eq 14) of the above eq 3. Equations 17 and 18 are examples of an analogous approach. Determination of the Numerical Value of the Adsorption Coefficient Cads in Eq 1. The numerical value of the constant 2 Cads () Cads,ep+eo) in eq 1 can be calculated, since σads,ep+eo can experimentally be obtained from eq 4a. This value of Cads,ep+eo can 2 then be used for the calculation of σads,ep+eo for other values of the parameters v, F, and L in eq 1, provided that the EOFgenerating coating is the same in the experiments. However, in many experiments Cads increases with an increase in v. This method might be the only approach permitting experimental determination of the adsorption coefficient Cads,ep+eo. This constant is a function of several parameters,2-5 including the distribution coefficient. This informative coefficient (or related parameters) can now be calculated when the numerical value of Cads,ep+eo is known. A prerequisite is that none of the analytes in the sample is irreversibly adsorbed to the capillary wall (or the coating), since the adsorbed molecules affect the surface properties of the wall coating and, therefore, also the distribution coefficient. 2 2 Determination of the Variances σtot, ep+eo and σtot, ep in Eq 3 When the Stationary Detector Is Used To Record the Electropherograms. The experimental determination of these parameters is very simple when a scanning6,7 detector is employed (see previous section) because tep ) tep+eo. For the stationary detector, the determinations are much more complicated and time-consuming because this requirement of the migration time is not fulfilled without a series of precautions, because all analytes have its own, specific migration time. The question is then: How should the migration distance of each analyte be chosen to attain the same migration time in the absence and presence of EOF. Lep uepFeptep ) Lep+eo uep+eoFep+eotep+eo
(5a)
where L is the migration distance, u is the mobility, F is the field strength, and t is the migration time. Characteristic of the experimental conditions for the scanning detector is that Fep ) Fep+eo and tep ) tep+eo (see previous section). Accordingly Lep uep ) Lep+eo uep+eo
(5b)
for this detector. The stationary detector can, accordingly, also 2 be used for experimental determination of the variance σtot, ep+eo 2 2 2 and σtot, ep and thereby σads, ep+eo - σads, ep (eq 3), provided that the
experimental conditions are such that eq 5b obtains. Therefore, the ratio between the migration distances of a given analyte in the experiment without EOF and the experiment with EOF must be equal to the ratio of the mobilities of this analyte in these two experiments. Observe that the migration direction in the absence of EOF in our experiment is opposite to that in the presence of EOF. A series of experiments must be done to fulfill all the requirements: (a) determine uep and uep+eo first for one particular analyte in the sample and then for the other analytes; (b) cut the fused-silica tubing in accordance with eq 5b. Observe that Lep and Lep+eo are shorter than the total length of the capillaries; (c) measure the total length of the two pieces of the tubing; these lengths must be known to apply the voltage satisfying the above requirement Fep ) Fep+eo; (d) control that the EOF does not change in this series of runs; (e) repeat these procedures for each analyte. The characteristic feature of the separation patterns obtained by scanning is that the migration time (tep and tep+eo) between each scanning can be chosen freely.6 In most experiments, this time is constant. The voltage is switched off during the scanning. However, the scanning time should be as short as possible in order to minimize diffusion (although the effect of diffusion will disappear by the subtraction; see eq 3). One can therefore 2 2 calculate σads,ep+eo - σads, ep for each analyte in the sample from one electropherogram (one run), when the recording is done with a scanning detector. Observe that the migration times are the same for all the analytes in the electropherogram recorded by a scanner, whereas the migration distances vary. With the use of the scanning detector, the migration times and the field strengths in the experiments in the absence and presence of EOF will be equal (for each individual analyte), i.e., tep ) tep+eo and Fep ) Fep+eo. Loss of Analyte Caused by Irreversible or Extremely Slow Reversible Adsorption onto the Naked or Coated Wall of the Separation Channel. The amount of material in an electrophoretic zone is determined by eq 6 when a stationary detector is used to record the separation pattern (to be published): Q ) πR2vA
(6)
where R is the inner radius of the capillary, v is the migration velocity of the analyte, and A is the peak area. Observe that Q is proportional not only to peak area but also to the migration velocity8 and the radius squared, when the electropherograms are recorded by a stationary detector. For quantitative calculations, the radius of the capillary should be determined. For a comparison of the amount of an analyte adsorbed to the capillary wall in the absence and the presence of EOF, we use the parameter qrel, defined as
qrel )
Qep - Qep+eo × 100% Qep
(7)
where qrel is the difference between the amount of the analyte in the zone in the EOF-free and the amount of the analyte in the zone in the EOF-generating capillary, expressed in percentage of the amount of analyte in the zone in the EOF-free capillary. Using eq 6, eq 7 can be written
qrel )
vepAep - vep+eoAep+eo × 100% vepAep
(8)
For the special case that the migration velocity of an analyte is the same in the absence and in the presence of EOF, as it is for lysozyme in the experiment described herein, we get
qrel )
Aep - Aep+eo × 100% Aep
(9)
In this special case |uep+eo|F - |uep|F ) |uep|F, i.e., |uep+eo| ) 2|uep|
(10)
Accordingly, the electroosmotic velocity (veo) is twice the electrophoretic velocity (vep) when the analyte migrates in different directions in the EOF-free and in the EOF-generating capillary, but with the same net velocity. Increase in Zone Width Caused by Rapid, Reversible Adsorption of the Analyte onto the Naked or Coated Wall of the Separation Channel. The difference between the width of a zone in the presence and the absence of EOF is 2σads,ep+eo - 2σads,ep and expressed in percentage of the total zone width in the absence of EOF:
∆σads,ep+eo,rel )
2σads,ep+eo - 2σads,ep × 100% 2σtot,ep
(11a)
When σads,ep ) 0 or
