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Insights from a New Analytical Electrophoresis Apparatus
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Insights from a New Analytical Electrophoresis Apparatus THOMAS M. LAUE*,X, THERESA M. RIDGEWAY*, JOHN O. WOOLL*, HARVEY K. SHEPARD‡, THOMAS P. MOODY*, TIMOTHY J. WILSON*, JONATHAN B. CHAIRES§, AND DAVID A. STEVENSON* Received February 16, 1996, from the *Department of Biochemistry and Molecular Biology, Rudman Hall, and ‡Department of Physics, University of New Hampshire, Durham, NH 03824-3544, and §Department of Biochemistry, University of Mississippi Medical Center, 2500 North State Street, Jackson, MS 39216. Accepted for publication May 9, 1996X. Abstract 0 Charge is a fundamental property of macromolecules that is inextricably linked to their structure, solubility, stability, and interactions. Progress has been made on the theoretical and structural aspects of these relationships. However, for several reasons, charge is difficult to measure in solution. Consequently, there is a lack of experimental data that, independent of other macro-ion properties, determines the effective charge. To overcome this problem, novel instrumentation and methods are being developed in our laboratory. Described here is an analytical electrophoresis apparatus that permits both the measurement of electrophoretic mobilities and the determination of steady-state electrophoresis concentration distributions. The latter provides a different perspective on the processes that influence macro-ion behavior in an electric field. In addition, the apparatus permits the determination of diffusion coefficients either from boundary spreading during transport or from the decay of a concentration gradient. All of these determinations can be made with a single, 8-µL sample in a variety of solvents, thus providing unique insights into the charge properties of a macro-ion. Presented here is a progress report about this emerging technology, including the description of a prototype apparatus and examples of its use with a DNA oligonucleotide.
Introduction and Background Despite its importance to a variety of molecular properties, the charge on a macro-ion is difficult to measure.1,2 The most direct means of discerning the charge of a macro-ion is by the response of the macro-ion to an applied electric field. However, the parameters (the charge, the field, and the diffusion and frictional coefficients) that describe the behavior of the macro-ion are complicated by their connection with the behavior of the surrounding mobile ionic atmosphere.3-8 The prevailing electrophoretic methods do not attempt to measure these parameters directly. Instead, they are secondary methods that rely on comparisons of the behavior of an unknown to that of better-characterized standards. Even then, there is no independent means of exclusively assessing the charge on a macro-ion in solution. What is measured is the electrophoretic mobility, µ, calculated from the velocity divided by the electric field and equated to the ratio of the charge to the frictional coefficient.2 Unless the frictional coefficient is determined independently, it is not possible to extract the charge from µ. Unfortunately, the effects of an electric field on the frictional coefficient are complicated and difficult to calculate.3-9 In a mobility experiment (Figure 1A), the forces on a particle balance so that a constant-velocity boundary is observed. Pictured in Figure 1A are four forces, though others may have to be included in a more exact model. The four forces are: (1) the electrostatic force, F1 ) QE, where Q is the effective charge (the formal charge less the contributions from counterions) and E is the external electric field; (2) the hydrody X
Abstract published in Advance ACS Abstracts, December 1, 1996.
© 1996, American Chemical Society and American Pharmaceutical Association
Figure 1sDifferent processes are observed in an electrophoretic mobility measurement than in a SSE measurement: (A) mobility measurement (e.g., Figure 2) in which forces are balanced; (B) steady-state measurement in which fluxes are also balanced. The forces and fluxes are described in the text.
namic frictional force, F2 ) -ftv, where ft is the usual Stokes frictional coefficient and v is the constant particle velocity (relative to the laboratory frame of reference); (3) the force due to the motion of the counterions, F3, also called the electrophoretic2 or charged-solvent10 effect; and (4) the relaxation11 or field asymmetry10 effect, F4, due to distortions in the ionic atmosphere. The electrophoretic retardation effect, F3, results from the fact that a macro-ion tends to be surrounded by counterions that flow in the opposite direction to its motion. From the perspective of the macro-ion, it is always moving “upstream” against this flow, so that its velocity relative to its immediate surroundings is higher than that relative to the laboratory reference frame. The field asymmetry effect, F4, is attributed to the dipole moment that is established with the redistribution of the surrounding counterions by the external electric field.10,13 This dipole orients in a direction that (usually) diminishes the external field at the surface of the macro-ion. These effects have been studied for over 70 years,2-11 and good reviews of them are available.1,2,7,10 However, there is much uncertainty in calculations of the contributions of F3 and F4 to the measured mobility. For the moment, though, we will bypass these concerns, but only to argue that mobility measurements and steady-state measurements assess the behavior of macro-ions in an electric field differently. To see this argument, we will assume that the ordinary translational frictional coefficient, ft, can be used for F2. This is justified so long as neither the particle size nor the solution viscosity are affected greatly by the applied field. Of course, ft depends on the size, shape, flexibility, and orientation of the macro-ion. Equally important, the macro-ion velocity will be taken to be that measured relative to the laboratory coordinate system. The issue of the velocity of the particle relative to its immediate surroundings will be considered part of the electrophoretic effect, F3. Likewise, any reduction in the external field at the particle is assumed to be part of the field asymmetry effect, F4. Then, the effective charge (Q) is determined from mobility measurements by the balance of forces: F2 + F3 + F4 ) fEv ) F1 ) QE, and the mobility is µ ) v/E ) Q/fE, where fE is an effective frictional coefficient produced by F2, F3, and F4. In contrast to free boundary motion in mobility experiments, macro-ions in steady-state electrophoresis (SSE) are trapped in a small chamber whose top and bottom are sealed
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Journal of Pharmaceutical Sciences / 1331 Vol. 85, No. 12, December 1996
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Figure 2sIntensity profile of cuvette showing a boundary of DNA (pd(A)20‚pd(T)20, 250 µg/mL in 100 mM KCl, 20 mM Tris, pH 8.0) moving from left to right. The positions of the membranes, M, at either end of the cuvette are noted. As the boundary passes a point, the light intensity increases. Data are shown for intensity profiles acquired at 20, 80, and 150 s after the electric field was applied (E ) 3.0 V/cm).
with semipermeable membranes.14,15 An electric field is applied along the chamber so that macro-ions crowd up against one of the membranes. Diffusion produces a macroion flux in the direction opposite to that due to the electric field. When steady state is reached, these two fluxes are matched at each point and a stable concentration gradient is observed. Thus, in SSE the fluxes, as well as the forces, are balanced (Figure 1B).14,15 At any point x in the cuvette, the flux due to electrophoresis, JE, is cv′, where c is the concentration of macroions at x and v′ is their velocity. The flux JE is an effective flux resulting from all of the forces F1, F2, F3, and F4. Because fEv′ ) QE, and hence v′ ) QE/fE, we have JE ) (QE/fE)c. Concentration gradients produce the flux due to diffusion JD ) -∑Dij ∂cj/∂x. Neglecting diffusional cross terms and treating the macro-ion as an ideal, nonassociating component, JD ) -D dc/dx. At steady state, JE + JD ) 0; hence, QEc/fE ) D(dc/dx). The solution to this equation is c ) c0 exp[σ(x - x0)], where c0 is the concentration at a reference point x0 and the exponent, σ ≡ QE/fEDE, includes DE, the diffusion coefficient in the presence of the electric field. Now Q ) σfEDE/ E. Assuming the usual relation fEDE ) kT is still valid, we have Q ) σkT/E, where k is Boltzmann’s constant. Thus, the effective charge can be determined directly from experimental measurements of σ, E, and T, unlike mobility measurements, which also require knowledge of fE or DE.
Experimental Procedures InstrumentsAn apparatus capable of on-line analytical electrophoresis and diffusion measurements has been developed.16 A brief description is provided here. The instrument is an imaging spectrophotometer with a linear photodiode array to measure the light intensities from up to 512 positions along a fused-silica cuvette (Figure 2). The inner dimensions of the cuvette are 2 × 2 × 2 mm, and its top and bottom are open. The open ends of the cuvette are sealed by semipermeable membranes, providing an 8-µL analysis chamber. This arrangement permits the establishment of an electric field along the length of the cuvette, while retaining macro-ions in the field of view. The cuvette is housed in an aluminum water jacket, which also serves as an electrical shield and light mask. Tests reveal that the imaging system has a spatial resolution of >10 µm and the spectrophotometer is useful to 230 nm, with an optical bandpass of ∼5 nm. At 260 nm, the variance of a reading is (0.003 optical density (OD) over the range 0.04-1.0 OD, increasing to (0.09 OD at absorbances approaching 2.0 OD. The linearity of the response is better than (0.015 over the 0.04-1.0-OD range, increasing to (0.04 OD at 2.0 OD. Four measurements can be made with this instrument: (1) the free boundary electrophoretic mobility, µ, measured in an environment
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Figure 3sAbsorbance differences (∆A) between scans taken 10 s apart convert boundaries, like those in Figure 2, into peaks. Shown here are the peaks that result when the first scan of the difference is taken at 20, 80, and 150 s. similar to the descending limb of a Tiselius-type apparatus; (2) the exponent for a steady-state concentration gradient, σ14,15; (3) the diffusion coefficient, DE (in an applied field) using the boundary spreading during a mobility measurement; and (4) the diffusion coefficient, D0 (in zero field) from the relaxation of a gradient.17 These measurements can be made without changing the sample or the arrangement of the apparatus. Moreover, buffer changes are made by electrodialysis, so that measurements for a single sample can be made under a variety of conditions. MaterialssAll buffers and salts were reagent grade and used without additional purification. Equimolar mixtures of singlestranded pd(A)20 (Pharmacia #27-7984-01) and pd(T)20 (Pharmacia #27-7841-01) were heated to 65 °C in 200 mM KCl, 20 mM TrisHCl (pH 8.00) buffer, then cooled to form double-stranded pd(A)20‚pd(T)20. Melting and circular dichroism studies were used to verify that pd(A)20‚pd(T)20 was in a stable duplex form under the conditions of electrophoresis. Capillary zonal electrophoresis was used to test for sample homogeneity. Concentrations were estimated from the A260 measured in the apparatus. Experimental ProceduressAll samples were analyzed at 20 ( 0.1 °C in a fused-silica cell capped by regenerated cellulose membranes (molecular weight cutoff 3500, 800-10000, or 12000-14000). EDTA was excluded from the buffers because it is partially retained by low molecular weight cutoff membranes. The field strengths were calculated with specific conductivities (κ) measured with an Orion Research 811 conductivity meter with a cell cross-sectional area of 0.04 cm2. The specific conductivity is not affected measurably by DNA oligonucleotides at the concentrations used here. Current was supplied by a Keithley 224 programmable current source. Experiments with voltage-sensing electrodes have verified the accuracy of the calculated fields. The buffer flow was adjusted until the pH of the effluents from the cathode and anode electrode chambers were within 0.1 unit of the initial pH. Mobility MeasurementssFor mobility measurements, the velocity of a boundary is monitored (Figures 2, 3, and 4). To determine the velocity, intensity profiles are obtained at regular intervals after the application of power (Figure 2). After conversion to absorbance, successive scans are subtracted, producing a peak (corresponding to a peak in ∆A/∆t with the maximum at the center of the boundary; Figure 3). The slope of the graph of the peak position with time is used to obtain the velocity of the boundary (Figure 4), where the time for each point is the average of the time of the two scans used in the difference. In this way, exact knowledge of the starting position is unnecessary (though it is available as the intercept of this graph). Measurements of µ at several fields (Figure 5) can be obtained rapidly. Alternatively, µ can be determined from the slope of the velocity graphed as a function of E. Moreover, the intercept of such a graph must be zero (within the error of the extrapolation) unless there is a field-independent flow of solvent through the cuvette. Such a flow would distort the gradients and render any analysis meaningless. Steady-State ElectrophoresissA steady-state concentration gradient is shown in Figure 6. Simple theory predicts that the resultant curve is an exponential, c(x) ) c0 exp{σ(x-x0)} (see previous explanation).14,15 Samples were examined at field strengths ranging from 0.04 to 0.3 V/cm.
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Figure 4sSlope of the time dependence of the peak position (xp) is a measure of the velocity. Presented here are data for pd(A)20‚pd(T)20 (195 µM) in 100 mM KCl, 20 mM Tris, pH 8.0 acquired at 3.04 V/cm. These data give v ) 9.09 × 10-4 cm/s and thus yield an estimate of the electrophoretic mobility, µ ) v/E ) 2.99 × 10-4 cm2/V-s.
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Figure 7sDetermination of the diffusion coefficient from the decay of the gradient shown in Figure 6. The slope of a graph of ln(A2/3−A1/3) as a function of time is proportional to π2D/L2. For the data presented here, D is estimated to be 9.6 ± 0.2 × 10-7 cm2/s. Table 1sSSE Properties of pd(A)20‚pd(T)20 in 100 mM XCl, 20 mM Tris pH 8.0 property
K+
CDNA (µg/mL) D0 (cm2/s)
250 9.6 × 10-7 (8.5−10.7) σ (cm-1) 22.3 (21.2−24.4) E (V/cm) 0.100 Q ) σKT/E 5.63 (e- equiv) (5.35−5.90) |Q/40| 0.141 (0.134−0.148) σD0/E (cm2/V-s) 2.14 × 10-4 (1.80−2.48) no. of determinations 4
Na+
Li+
(CH3)4N+
235 7.8 × 10-7 (6.6−8.9) 19.4 (18.7−20.1) 0.122 4.09 (3.89−4.29) 0.102 (0.097−0.107) 1.24 × 10-4 (1.02−1.48) 2
230 7.5 × 10-7 (6.4−8.5) 24.2 (23.5−24.9) 0.137 4.47 (4.35−4.60) 0.112 (0.109−0.115) 1.32 × 10-4 (1.10−1.54) 3
220 8.5 × 10-7 (8.1−8.9) 18.8 (17.8−19.8) 0.122 3.89 (3.69−4.10) 0.0974 (0.092−0.103) 1.31 × 10-4 (1.18−1.44) 2
Table 2sElectrophoretic Mobility of pd(A)20‚pd(T)20 in 100 mM XCl, 20 mM Tris pH 8.0 Figure 5sAlthough generally assumed to be constant, the electrophoretic mobility shows some variation, especially at low E.
Figure 6sA gradient for pd(A)20‚pd(T)20 (195 µM) in 100 mM KCl, 20 mM Tris, pH 8.0 established at a field strength of 0.12 V/cm is shown decaying after the field is shut off. The initial steady-state gradient (dotted line) was formed by application of the field for >24 h. Absorbance profiles acquired at intervals after the field is shut off are roughly sigmoidal due to the rapid collapse of the gradient in the high concentration gradient region near the base of the cell and the slower decay near the top of the cell where the gradient is less steep. Diffusion Coefficient DeterminationssThe diffusion coefficient can in principle be assessed in three ways with this device: (1) by monitoring the decay of a concentration gradient in the absence of an applied field; (2) by monitoring the relaxation following a decrease
property
K+
Na+
Li+
(CH3)4N+
cDNA (µg/mL) 350 215 200 200 µ (cm2/V-s) 3.1 × 10-4 3.4 × 10-4 3.3 × 10-4 4.0 × 10-4 µkT/D0 (e- equiv) 8.1 11.2 11.1 11.9
in the applied field; and (3) by fitting the boundary spreading during transport. At present, only the first method has been characterized properly. The other two methods are currently being explored in our laboratory. The diffusion coefficient in the absence of an electric field can be obtained from the relaxation of a steady-state concentration gradient by a variation of the method of Harned and French17 (Figure 6) in which the time-dependent difference in concentration at “conjugate” positions (i.e., x1 ) m1L and x2 ) m2L, where L is the length of the cuvette and m1 + m2 ) 1) is monitored. It can be shown by separation of variables with a Fourier series expansion in x that for times such that π2D0t/L2 > 1, ln(∆c) = ln B - π2D0/L2 t where ∆c ) cx2 - cx1, B is a constant relating to the initial conditions, and t is the time in seconds. Using only this first-order term yields a very good approximation regardless of which conjugate pair is chosen. A graph using the 1/3 and 2/3 position is presented in Figure 7.
Results Observations on pd(A)20‚pd(T)20sIn Tables 1 and 2 we present representative data for pd(A)20‚pd(T)20 in solvents that were 100 mM in four different cations. For each measurement, the precision is better than (10%. The reproducibility of the measurements with different preparations and different
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Figure 8sThe field dependence of σ/E (V/cm2) is presented for pd(A)20‚pd(T)20 (250 µg/mL) in 100 mM KCl, 20 mM Tris, pH 8.0 acquired at field strengths ranging from 0.04 to 0.27 V/cm.
Figure 9sA graph of the local slope (d ln(A)/dx) as a function of x is one test for sample heterogeneity. Presented here is a graph constructed from the steadystate (t ) 0) data in Figure 7.
cuvettes is good, suggesting that the accuracy of these measurements is satisfactory. Diffusion CoefficientsThe zero-field diffusion coefficient, D0, presented in Table 1, was measured from the decay of gradients, as shown in Figures 6 and 7. Preliminary results suggest that these values are comparable to those obtained from boundary spreading during sedimentation (not shown) and to those predicted from hydrodynamic theory. The fact that D measured in this fashion corresponds to that from other methods indicates that changes in the osmotic potential at the two membranes do not lead to significant bulk fluid flow during the collapse of the gradient. Such a flow might be expected because the change in the osmotic potential at each membrane points in the same direction during the establishment or collapse of a concentration gradient in the cuvette. In the absence of a gradient, or at steady state, there should be no fluid flow due to the osmotic potential. Steady-State ElectrophoresissIn Table 1, the exponent for the steady-state gradient, σ, is presented for each condition, and below it is the field at which the exponent was determined. The field used for the determination must be presented because σ/E depends on E (Figure 8). At first glance, this should not be necessary because σ/E ) Q/fEDE, so that σ/E will be constant and independent of the field so long as Q is independent of the external field and fEDE = kT. With these assumptions, the effective charge can be calculated, Q ) σkT/E (Table 1). Analysis over a wide range of fields (Figure 8) reveals that σ/E deviates systematically at low fields, whereas at fields exceeding ∼0.1 V/cm, σ/E is relatively constant. This observation is consistently made with the different cation types. However, the magnitude and direction of the deviation appears to depend on the concentration of the DNA and of the salt (not shown). A possible source of variation in σ/E is sample heterogeneity. However, a mixture of charges should lead to an increase in σ/E at lower fields where there is a greater contribution to σ from the more highly-charged macro-ions, and just the opposite behavior is observed (Figure 8). Furthermore, the homogeneity of a sample can be tested by examining a graph of the natural logarithm of the absorbance as a function of position in the cell. The slope of such a graph should curve upwards towards the base of the cell if there is charge heterogeneity. However, the slope of the data for pd(A)20‚pd(T)20 (Figure 9) shows no such trend. The slight downward curvature apparent in Figure 9 is suggestive of thermodynamic nonideality. Thus, it appears that sample heterogeneity cannot account for the variation in σ/E. Because σDE/E ) Q/fE ) µ, calculating σD0/E from the values in Table 1 yields an estimate of µ, assuming DE ≈ D0 at the low electric fields applied in SSE.
Electrophoretic MobilitysMeasurements of µ were made from the slope of the graphs of the boundary velocity as a function of applied field. In all cases presented in Table 2, µ is independent of E (for large enough E), and there was no evidence of bulk fluid flow. The lone exception is Li+, where there is a slight decrease in µ with decreasing E (not shown). Comparison of the data in Table 2 with that obtained for larger DNA shows that µ for short oligonucleotides is somewhat higher than that for longer DNA. This result is expected for DNA below the limiting length for polyelectrolyte behavior.18-22 The specific cation effect on µ is in accord with earlier results for longer DNA.23 We believe that these values are accurate as well as precise because testing with wellcharacterized proteins gives values of µ that duplicate previous work.16
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Discussion Assuming that the Einstein relation of fD ) kT remains approximately valid for our “dressed” macro-ions (i.e., macroions + counterions) in an electric field, SSE provides a very clean estimate of the effective electric charge. We see from Table 1 that the values of Q obtained from the simple SSE theory, Q ) σkT/E, give a ratio of effective charge to formal charge, |Q/40| (for 20 bp double-stranded DNA), close to previous predictions of 0.12 for single valence salts, neglecting the correction for finite length macroions.18,19 The data in Tables 1 and 2 indicate apparent discrepancies between the results of SSE and electrophoretic mobility. This discrepancy is evident in the seeming disparity between predicted mobility from SSE ()σD0/E) in Table 1 and the measured µ in Table 2. Alternatively, we could state the disparity in terms of an effective charge ()µkT/D0) obtained from the mobilities in Table 2 versus the charges extracted from SSE in Table 1. These apparent discrepancies follow, however, from comparing data obtained at rather different values of the electric field and using values of the macro-ion diffusion coefficient (or frictional coefficient) determined in the absence of an electric field. The question of the actual variation of µ and D with electric field, and the extraction of these values from experiment is of considerable interest and importance. We are currently developing a new apparatus that will enable us to measure σ and µ at the same E field, thus permitting a direct comparison of σD0/E and µ, and also a method for determining DE from electrophoretic mobility data. A different problem, illustrated in Figure 8, is the variation of σ/E with E, because simple theory implies this ratio should be constant. This problem may only be an experimental
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difficulty at low E, because there is no problem when E > 0.1 V/cm. If some real variation of µ with E or of σ/E with E is demonstrated, however, modifications of theory may be required to account for it. ConclusionsElectrophoresis is one of the most widely used techniques in biochemistry and molecular biology. The results presented here demonstrate that much is not understood about the processes that contribute to it. The new apparatus and new methods available hold the promise of providing a deeper understanding.
References and Notes 1. Henry, D. Proc. Royal Soc. London Ser. A 1931, 133, 106-116. 2. Overbeek, J. T. G.; Wiersema, P. H. In Electrophoresis: Theory, Applications & Technique, Vol. II; Bier, M., Ed.; Academic: New York, 1967; pp 1-52. 3. Booth, F. Trans. Faraday Soc. 1948, 44, 955-967. 4. Booth, F. Proc. Royal Soc. London Ser. A 1950, 203, 514-518. 5. Booth, F. Proc. Royal Soc. London Ser. A 1951, 204, 549-556. 6. Henry, D. Trans. Faraday Soc. 1948, 44, 1021-1026. 7. Wiersema, P. H.; Loeb, A. L.; Overbeek, J. T. G. J. Colloid Interface Sci. 1966, 22, 78-99. 8. Onsager, L.; Fuoss, R. M. J. Phys. Chem. 1932, 36, 2689-2821.
9. Gosting, L. J. Am. Chem. Soc. 1952, 74, 1548-1552. 10. Manning, G. Q. Rev. Biophys. 1978, 11, 179-246. 11. Debye, P.; Hu¨ckel, E. In The Collected Works of Peter J.W. Debye, Ox Bow: Connecticut, 1988; pp 264-317. 12. Stigter, D. J. Phys. Chem. 1979, 83, 1663-1676. 13. Stigter, D. J. Phys. Chem. 1978, 82, 1417-1426. 14. Godfrey, J. E. Proc. Natl. Acad. Sci. U.S.A. 1989, 86, 44794484. 15. Laue, T.; Hazard, A.; Ridgeway, T.; Yphantis, D. Anal. Biochem. 1989, 182, 377-382. 16. Ridgeway, T. M.; Hayes, D. B.; Anderson, A. L.; Levasseur, J. H.; Demaine, P. D.; Kenty, B. E.; Laue, T. M. In Biochemical Diagnostic Instrumentation, SPIE Proceedings, Vol. 2136; Bonner, R. F.; Cohn, G. E.; Laue, T. M.; Priezzhev, A. V.; Eds.; SPIE: Bellingham, 1994; pp 263-274. 17. Harned, H.; French, D. Ann. N.Y. Acad. Sci. 1945, 46, 267279. 18. Manning, G. J. Phys. Chem. 1981, 85, 1506-1526. 19. Record, M. T.; Lohman, T. Biopolymers 1978, 17, 159-171. 20. Anderson, C.; Record, M. T. J. Phys. Chem. 1993, 97, 71167126. 21. Olmsted, M.; Anderson, C.; Record, M. Proc. Natl. Acad. Sci. U.S.A. 1989, 86, 7766-7771. 22. Olmstead, M.; Anderson, C.; Record, M. T. Biopolymers 1991, 31, 1593-1604. 23. Ross, P. D.; Scruggs, R. L. Biopolymers 1961, 2, 231-239.
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Insights from a New Analytical Electrophoresis Apparatus THOMAS M. LAUE*,X, THERESA M. RIDGEWAY*, JOHN O. WOOLL*, HARVEY K. SHEPARD‡, THOMAS P. MOODY*, TIMOTHY J. WILSON*, JONATHAN B. CHAIRES§, AND DAVID A. STEVENSON* Received February 16, 1996, from the *Department of Biochemistry and Molecular Biology, Rudman Hall, and ‡Department of Physics, University of New Hampshire, Durham, NH 03824-3544, and §Department of Biochemistry, University of Mississippi Medical Center, 2500 North State Street, Jackson, MS 39216. Accepted for publication May 9, 1996X. Abstract 0 Charge is a fundamental property of macromolecules that is inextricably linked to their structure, solubility, stability, and interactions. Progress has been made on the theoretical and structural aspects of these relationships. However, for several reasons, charge is difficult to measure in solution. Consequently, there is a lack of experimental data that, independent of other macro-ion properties, determines the effective charge. To overcome this problem, novel instrumentation and methods are being developed in our laboratory. Described here is an analytical electrophoresis apparatus that permits both the measurement of electrophoretic mobilities and the determination of steady-state electrophoresis concentration distributions. The latter provides a different perspective on the processes that influence macro-ion behavior in an electric field. In addition, the apparatus permits the determination of diffusion coefficients either from boundary spreading during transport or from the decay of a concentration gradient. All of these determinations can be made with a single, 8-µL sample in a variety of solvents, thus providing unique insights into the charge properties of a macro-ion. Presented here is a progress report about this emerging technology, including the description of a prototype apparatus and examples of its use with a DNA oligonucleotide.
Introduction and Background Despite its importance to a variety of molecular properties, the charge on a macro-ion is difficult to measure.1,2 The most direct means of discerning the charge of a macro-ion is by the response of the macro-ion to an applied electric field. However, the parameters (the charge, the field, and the diffusion and frictional coefficients) that describe the behavior of the macro-ion are complicated by their connection with the behavior of the surrounding mobile ionic atmosphere.3-8 The prevailing electrophoretic methods do not attempt to measure these parameters directly. Instead, they are secondary methods that rely on comparisons of the behavior of an unknown to that of better-characterized standards. Even then, there is no independent means of exclusively assessing the charge on a macro-ion in solution. What is measured is the electrophoretic mobility, µ, calculated from the velocity divided by the electric field and equated to the ratio of the charge to the frictional coefficient.2 Unless the frictional coefficient is determined independently, it is not possible to extract the charge from µ. Unfortunately, the effects of an electric field on the frictional coefficient are complicated and difficult to calculate.3-9 In a mobility experiment (Figure 1A), the forces on a particle balance so that a constant-velocity boundary is observed. Pictured in Figure 1A are four forces, though others may have to be included in a more exact model. The four forces are: (1) the electrostatic force, F1 ) QE, where Q is the effective charge (the formal charge less the contributions from counterions) and E is the external electric field; (2) the hydrody X
Abstract published in Advance ACS Abstracts, December 1, 1996.
© 1996, American Chemical Society and American Pharmaceutical Association
Figure 1sDifferent processes are observed in an electrophoretic mobility measurement than in a SSE measurement: (A) mobility measurement (e.g., Figure 2) in which forces are balanced; (B) steady-state measurement in which fluxes are also balanced. The forces and fluxes are described in the text.
namic frictional force, F2 ) -ftv, where ft is the usual Stokes frictional coefficient and v is the constant particle velocity (relative to the laboratory frame of reference); (3) the force due to the motion of the counterions, F3, also called the electrophoretic2 or charged-solvent10 effect; and (4) the relaxation11 or field asymmetry10 effect, F4, due to distortions in the ionic atmosphere. The electrophoretic retardation effect, F3, results from the fact that a macro-ion tends to be surrounded by counterions that flow in the opposite direction to its motion. From the perspective of the macro-ion, it is always moving “upstream” against this flow, so that its velocity relative to its immediate surroundings is higher than that relative to the laboratory reference frame. The field asymmetry effect, F4, is attributed to the dipole moment that is established with the redistribution of the surrounding counterions by the external electric field.10,13 This dipole orients in a direction that (usually) diminishes the external field at the surface of the macro-ion. These effects have been studied for over 70 years,2-11 and good reviews of them are available.1,2,7,10 However, there is much uncertainty in calculations of the contributions of F3 and F4 to the measured mobility. For the moment, though, we will bypass these concerns, but only to argue that mobility measurements and steady-state measurements assess the behavior of macro-ions in an electric field differently. To see this argument, we will assume that the ordinary translational frictional coefficient, ft, can be used for F2. This is justified so long as neither the particle size nor the solution viscosity are affected greatly by the applied field. Of course, ft depends on the size, shape, flexibility, and orientation of the macro-ion. Equally important, the macro-ion velocity will be taken to be that measured relative to the laboratory coordinate system. The issue of the velocity of the particle relative to its immediate surroundings will be considered part of the electrophoretic effect, F3. Likewise, any reduction in the external field at the particle is assumed to be part of the field asymmetry effect, F4. Then, the effective charge (Q) is determined from mobility measurements by the balance of forces: F2 + F3 + F4 ) fEv ) F1 ) QE, and the mobility is µ ) v/E ) Q/fE, where fE is an effective frictional coefficient produced by F2, F3, and F4. In contrast to free boundary motion in mobility experiments, macro-ions in steady-state electrophoresis (SSE) are trapped in a small chamber whose top and bottom are sealed
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Figure 2sIntensity profile of cuvette showing a boundary of DNA (pd(A)20‚pd(T)20, 250 µg/mL in 100 mM KCl, 20 mM Tris, pH 8.0) moving from left to right. The positions of the membranes, M, at either end of the cuvette are noted. As the boundary passes a point, the light intensity increases. Data are shown for intensity profiles acquired at 20, 80, and 150 s after the electric field was applied (E ) 3.0 V/cm).
with semipermeable membranes.14,15 An electric field is applied along the chamber so that macro-ions crowd up against one of the membranes. Diffusion produces a macroion flux in the direction opposite to that due to the electric field. When steady state is reached, these two fluxes are matched at each point and a stable concentration gradient is observed. Thus, in SSE the fluxes, as well as the forces, are balanced (Figure 1B).14,15 At any point x in the cuvette, the flux due to electrophoresis, JE, is cv′, where c is the concentration of macroions at x and v′ is their velocity. The flux JE is an effective flux resulting from all of the forces F1, F2, F3, and F4. Because fEv′ ) QE, and hence v′ ) QE/fE, we have JE ) (QE/fE)c. Concentration gradients produce the flux due to diffusion JD ) -∑Dij ∂cj/∂x. Neglecting diffusional cross terms and treating the macro-ion as an ideal, nonassociating component, JD ) -D dc/dx. At steady state, JE + JD ) 0; hence, QEc/fE ) D(dc/dx). The solution to this equation is c ) c0 exp[σ(x - x0)], where c0 is the concentration at a reference point x0 and the exponent, σ ≡ QE/fEDE, includes DE, the diffusion coefficient in the presence of the electric field. Now Q ) σfEDE/ E. Assuming the usual relation fEDE ) kT is still valid, we have Q ) σkT/E, where k is Boltzmann’s constant. Thus, the effective charge can be determined directly from experimental measurements of σ, E, and T, unlike mobility measurements, which also require knowledge of fE or DE.
Experimental Procedures InstrumentsAn apparatus capable of on-line analytical electrophoresis and diffusion measurements has been developed.16 A brief description is provided here. The instrument is an imaging spectrophotometer with a linear photodiode array to measure the light intensities from up to 512 positions along a fused-silica cuvette (Figure 2). The inner dimensions of the cuvette are 2 × 2 × 2 mm, and its top and bottom are open. The open ends of the cuvette are sealed by semipermeable membranes, providing an 8-µL analysis chamber. This arrangement permits the establishment of an electric field along the length of the cuvette, while retaining macro-ions in the field of view. The cuvette is housed in an aluminum water jacket, which also serves as an electrical shield and light mask. Tests reveal that the imaging system has a spatial resolution of >10 µm and the spectrophotometer is useful to 230 nm, with an optical bandpass of ∼5 nm. At 260 nm, the variance of a reading is (0.003 optical density (OD) over the range 0.04-1.0 OD, increasing to (0.09 OD at absorbances approaching 2.0 OD. The linearity of the response is better than (0.015 over the 0.04-1.0-OD range, increasing to (0.04 OD at 2.0 OD. Four measurements can be made with this instrument: (1) the free boundary electrophoretic mobility, µ, measured in an environment
1332 / Journal of Pharmaceutical Sciences Vol. 85, No. 12, December 1996
Figure 3sAbsorbance differences (∆A) between scans taken 10 s apart convert boundaries, like those in Figure 2, into peaks. Shown here are the peaks that result when the first scan of the difference is taken at 20, 80, and 150 s. similar to the descending limb of a Tiselius-type apparatus; (2) the exponent for a steady-state concentration gradient, σ14,15; (3) the diffusion coefficient, DE (in an applied field) using the boundary spreading during a mobility measurement; and (4) the diffusion coefficient, D0 (in zero field) from the relaxation of a gradient.17 These measurements can be made without changing the sample or the arrangement of the apparatus. Moreover, buffer changes are made by electrodialysis, so that measurements for a single sample can be made under a variety of conditions. MaterialssAll buffers and salts were reagent grade and used without additional purification. Equimolar mixtures of singlestranded pd(A)20 (Pharmacia #27-7984-01) and pd(T)20 (Pharmacia #27-7841-01) were heated to 65 °C in 200 mM KCl, 20 mM TrisHCl (pH 8.00) buffer, then cooled to form double-stranded pd(A)20‚pd(T)20. Melting and circular dichroism studies were used to verify that pd(A)20‚pd(T)20 was in a stable duplex form under the conditions of electrophoresis. Capillary zonal electrophoresis was used to test for sample homogeneity. Concentrations were estimated from the A260 measured in the apparatus. Experimental ProceduressAll samples were analyzed at 20 ( 0.1 °C in a fused-silica cell capped by regenerated cellulose membranes (molecular weight cutoff 3500, 800-10000, or 12000-14000). EDTA was excluded from the buffers because it is partially retained by low molecular weight cutoff membranes. The field strengths were calculated with specific conductivities (κ) measured with an Orion Research 811 conductivity meter with a cell cross-sectional area of 0.04 cm2. The specific conductivity is not affected measurably by DNA oligonucleotides at the concentrations used here. Current was supplied by a Keithley 224 programmable current source. Experiments with voltage-sensing electrodes have verified the accuracy of the calculated fields. The buffer flow was adjusted until the pH of the effluents from the cathode and anode electrode chambers were within 0.1 unit of the initial pH. Mobility MeasurementssFor mobility measurements, the velocity of a boundary is monitored (Figures 2, 3, and 4). To determine the velocity, intensity profiles are obtained at regular intervals after the application of power (Figure 2). After conversion to absorbance, successive scans are subtracted, producing a peak (corresponding to a peak in ∆A/∆t with the maximum at the center of the boundary; Figure 3). The slope of the graph of the peak position with time is used to obtain the velocity of the boundary (Figure 4), where the time for each point is the average of the time of the two scans used in the difference. In this way, exact knowledge of the starting position is unnecessary (though it is available as the intercept of this graph). Measurements of µ at several fields (Figure 5) can be obtained rapidly. Alternatively, µ can be determined from the slope of the velocity graphed as a function of E. Moreover, the intercept of such a graph must be zero (within the error of the extrapolation) unless there is a field-independent flow of solvent through the cuvette. Such a flow would distort the gradients and render any analysis meaningless. Steady-State ElectrophoresissA steady-state concentration gradient is shown in Figure 6. Simple theory predicts that the resultant curve is an exponential, c(x) ) c0 exp{σ(x-x0)} (see previous explanation).14,15 Samples were examined at field strengths ranging from 0.04 to 0.3 V/cm.
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Figure 4sSlope of the time dependence of the peak position (xp) is a measure of the velocity. Presented here are data for pd(A)20‚pd(T)20 (195 µM) in 100 mM KCl, 20 mM Tris, pH 8.0 acquired at 3.04 V/cm. These data give v ) 9.09 × 10-4 cm/s and thus yield an estimate of the electrophoretic mobility, µ ) v/E ) 2.99 × 10-4 cm2/V-s.
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Figure 7sDetermination of the diffusion coefficient from the decay of the gradient shown in Figure 6. The slope of a graph of ln(A2/3−A1/3) as a function of time is proportional to π2D/L2. For the data presented here, D is estimated to be 9.6 ± 0.2 × 10-7 cm2/s. Table 1sSSE Properties of pd(A)20‚pd(T)20 in 100 mM XCl, 20 mM Tris pH 8.0 property
K+
CDNA (µg/mL) D0 (cm2/s)
250 9.6 × 10-7 (8.5−10.7) σ (cm-1) 22.3 (21.2−24.4) E (V/cm) 0.100 Q ) σKT/E 5.63 (e- equiv) (5.35−5.90) |Q/40| 0.141 (0.134−0.148) σD0/E (cm2/V-s) 2.14 × 10-4 (1.80−2.48) no. of determinations 4
Na+
Li+
(CH3)4N+
235 7.8 × 10-7 (6.6−8.9) 19.4 (18.7−20.1) 0.122 4.09 (3.89−4.29) 0.102 (0.097−0.107) 1.24 × 10-4 (1.02−1.48) 2
230 7.5 × 10-7 (6.4−8.5) 24.2 (23.5−24.9) 0.137 4.47 (4.35−4.60) 0.112 (0.109−0.115) 1.32 × 10-4 (1.10−1.54) 3
220 8.5 × 10-7 (8.1−8.9) 18.8 (17.8−19.8) 0.122 3.89 (3.69−4.10) 0.0974 (0.092−0.103) 1.31 × 10-4 (1.18−1.44) 2
Table 2sElectrophoretic Mobility of pd(A)20‚pd(T)20 in 100 mM XCl, 20 mM Tris pH 8.0 Figure 5sAlthough generally assumed to be constant, the electrophoretic mobility shows some variation, especially at low E.
Figure 6sA gradient for pd(A)20‚pd(T)20 (195 µM) in 100 mM KCl, 20 mM Tris, pH 8.0 established at a field strength of 0.12 V/cm is shown decaying after the field is shut off. The initial steady-state gradient (dotted line) was formed by application of the field for >24 h. Absorbance profiles acquired at intervals after the field is shut off are roughly sigmoidal due to the rapid collapse of the gradient in the high concentration gradient region near the base of the cell and the slower decay near the top of the cell where the gradient is less steep. Diffusion Coefficient DeterminationssThe diffusion coefficient can in principle be assessed in three ways with this device: (1) by monitoring the decay of a concentration gradient in the absence of an applied field; (2) by monitoring the relaxation following a decrease
property
K+
Na+
Li+
(CH3)4N+
cDNA (µg/mL) 350 215 200 200 µ (cm2/V-s) 3.1 × 10-4 3.4 × 10-4 3.3 × 10-4 4.0 × 10-4 µkT/D0 (e- equiv) 8.1 11.2 11.1 11.9
in the applied field; and (3) by fitting the boundary spreading during transport. At present, only the first method has been characterized properly. The other two methods are currently being explored in our laboratory. The diffusion coefficient in the absence of an electric field can be obtained from the relaxation of a steady-state concentration gradient by a variation of the method of Harned and French17 (Figure 6) in which the time-dependent difference in concentration at “conjugate” positions (i.e., x1 ) m1L and x2 ) m2L, where L is the length of the cuvette and m1 + m2 ) 1) is monitored. It can be shown by separation of variables with a Fourier series expansion in x that for times such that π2D0t/L2 > 1, ln(∆c) = ln B - π2D0/L2 t where ∆c ) cx2 - cx1, B is a constant relating to the initial conditions, and t is the time in seconds. Using only this first-order term yields a very good approximation regardless of which conjugate pair is chosen. A graph using the 1/3 and 2/3 position is presented in Figure 7.
Results Observations on pd(A)20‚pd(T)20sIn Tables 1 and 2 we present representative data for pd(A)20‚pd(T)20 in solvents that were 100 mM in four different cations. For each measurement, the precision is better than (10%. The reproducibility of the measurements with different preparations and different
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Figure 8sThe field dependence of σ/E (V/cm2) is presented for pd(A)20‚pd(T)20 (250 µg/mL) in 100 mM KCl, 20 mM Tris, pH 8.0 acquired at field strengths ranging from 0.04 to 0.27 V/cm.
Figure 9sA graph of the local slope (d ln(A)/dx) as a function of x is one test for sample heterogeneity. Presented here is a graph constructed from the steadystate (t ) 0) data in Figure 7.
cuvettes is good, suggesting that the accuracy of these measurements is satisfactory. Diffusion CoefficientsThe zero-field diffusion coefficient, D0, presented in Table 1, was measured from the decay of gradients, as shown in Figures 6 and 7. Preliminary results suggest that these values are comparable to those obtained from boundary spreading during sedimentation (not shown) and to those predicted from hydrodynamic theory. The fact that D measured in this fashion corresponds to that from other methods indicates that changes in the osmotic potential at the two membranes do not lead to significant bulk fluid flow during the collapse of the gradient. Such a flow might be expected because the change in the osmotic potential at each membrane points in the same direction during the establishment or collapse of a concentration gradient in the cuvette. In the absence of a gradient, or at steady state, there should be no fluid flow due to the osmotic potential. Steady-State ElectrophoresissIn Table 1, the exponent for the steady-state gradient, σ, is presented for each condition, and below it is the field at which the exponent was determined. The field used for the determination must be presented because σ/E depends on E (Figure 8). At first glance, this should not be necessary because σ/E ) Q/fEDE, so that σ/E will be constant and independent of the field so long as Q is independent of the external field and fEDE = kT. With these assumptions, the effective charge can be calculated, Q ) σkT/E (Table 1). Analysis over a wide range of fields (Figure 8) reveals that σ/E deviates systematically at low fields, whereas at fields exceeding ∼0.1 V/cm, σ/E is relatively constant. This observation is consistently made with the different cation types. However, the magnitude and direction of the deviation appears to depend on the concentration of the DNA and of the salt (not shown). A possible source of variation in σ/E is sample heterogeneity. However, a mixture of charges should lead to an increase in σ/E at lower fields where there is a greater contribution to σ from the more highly-charged macro-ions, and just the opposite behavior is observed (Figure 8). Furthermore, the homogeneity of a sample can be tested by examining a graph of the natural logarithm of the absorbance as a function of position in the cell. The slope of such a graph should curve upwards towards the base of the cell if there is charge heterogeneity. However, the slope of the data for pd(A)20‚pd(T)20 (Figure 9) shows no such trend. The slight downward curvature apparent in Figure 9 is suggestive of thermodynamic nonideality. Thus, it appears that sample heterogeneity cannot account for the variation in σ/E. Because σDE/E ) Q/fE ) µ, calculating σD0/E from the values in Table 1 yields an estimate of µ, assuming DE ≈ D0 at the low electric fields applied in SSE.
Electrophoretic MobilitysMeasurements of µ were made from the slope of the graphs of the boundary velocity as a function of applied field. In all cases presented in Table 2, µ is independent of E (for large enough E), and there was no evidence of bulk fluid flow. The lone exception is Li+, where there is a slight decrease in µ with decreasing E (not shown). Comparison of the data in Table 2 with that obtained for larger DNA shows that µ for short oligonucleotides is somewhat higher than that for longer DNA. This result is expected for DNA below the limiting length for polyelectrolyte behavior.18-22 The specific cation effect on µ is in accord with earlier results for longer DNA.23 We believe that these values are accurate as well as precise because testing with wellcharacterized proteins gives values of µ that duplicate previous work.16
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Discussion Assuming that the Einstein relation of fD ) kT remains approximately valid for our “dressed” macro-ions (i.e., macroions + counterions) in an electric field, SSE provides a very clean estimate of the effective electric charge. We see from Table 1 that the values of Q obtained from the simple SSE theory, Q ) σkT/E, give a ratio of effective charge to formal charge, |Q/40| (for 20 bp double-stranded DNA), close to previous predictions of 0.12 for single valence salts, neglecting the correction for finite length macroions.18,19 The data in Tables 1 and 2 indicate apparent discrepancies between the results of SSE and electrophoretic mobility. This discrepancy is evident in the seeming disparity between predicted mobility from SSE ()σD0/E) in Table 1 and the measured µ in Table 2. Alternatively, we could state the disparity in terms of an effective charge ()µkT/D0) obtained from the mobilities in Table 2 versus the charges extracted from SSE in Table 1. These apparent discrepancies follow, however, from comparing data obtained at rather different values of the electric field and using values of the macro-ion diffusion coefficient (or frictional coefficient) determined in the absence of an electric field. The question of the actual variation of µ and D with electric field, and the extraction of these values from experiment is of considerable interest and importance. We are currently developing a new apparatus that will enable us to measure σ and µ at the same E field, thus permitting a direct comparison of σD0/E and µ, and also a method for determining DE from electrophoretic mobility data. A different problem, illustrated in Figure 8, is the variation of σ/E with E, because simple theory implies this ratio should be constant. This problem may only be an experimental
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difficulty at low E, because there is no problem when E > 0.1 V/cm. If some real variation of µ with E or of σ/E with E is demonstrated, however, modifications of theory may be required to account for it. ConclusionsElectrophoresis is one of the most widely used techniques in biochemistry and molecular biology. The results presented here demonstrate that much is not understood about the processes that contribute to it. The new apparatus and new methods available hold the promise of providing a deeper understanding.
References and Notes 1. Henry, D. Proc. Royal Soc. London Ser. A 1931, 133, 106-116. 2. Overbeek, J. T. G.; Wiersema, P. H. In Electrophoresis: Theory, Applications & Technique, Vol. II; Bier, M., Ed.; Academic: New York, 1967; pp 1-52. 3. Booth, F. Trans. Faraday Soc. 1948, 44, 955-967. 4. Booth, F. Proc. Royal Soc. London Ser. A 1950, 203, 514-518. 5. Booth, F. Proc. Royal Soc. London Ser. A 1951, 204, 549-556. 6. Henry, D. Trans. Faraday Soc. 1948, 44, 1021-1026. 7. Wiersema, P. H.; Loeb, A. L.; Overbeek, J. T. G. J. Colloid Interface Sci. 1966, 22, 78-99. 8. Onsager, L.; Fuoss, R. M. J. Phys. Chem. 1932, 36, 2689-2821.
9. Gosting, L. J. Am. Chem. Soc. 1952, 74, 1548-1552. 10. Manning, G. Q. Rev. Biophys. 1978, 11, 179-246. 11. Debye, P.; Hu¨ckel, E. In The Collected Works of Peter J.W. Debye, Ox Bow: Connecticut, 1988; pp 264-317. 12. Stigter, D. J. Phys. Chem. 1979, 83, 1663-1676. 13. Stigter, D. J. Phys. Chem. 1978, 82, 1417-1426. 14. Godfrey, J. E. Proc. Natl. Acad. Sci. U.S.A. 1989, 86, 44794484. 15. Laue, T.; Hazard, A.; Ridgeway, T.; Yphantis, D. Anal. Biochem. 1989, 182, 377-382. 16. Ridgeway, T. M.; Hayes, D. B.; Anderson, A. L.; Levasseur, J. H.; Demaine, P. D.; Kenty, B. E.; Laue, T. M. In Biochemical Diagnostic Instrumentation, SPIE Proceedings, Vol. 2136; Bonner, R. F.; Cohn, G. E.; Laue, T. M.; Priezzhev, A. V.; Eds.; SPIE: Bellingham, 1994; pp 263-274. 17. Harned, H.; French, D. Ann. N.Y. Acad. Sci. 1945, 46, 267279. 18. Manning, G. J. Phys. Chem. 1981, 85, 1506-1526. 19. Record, M. T.; Lohman, T. Biopolymers 1978, 17, 159-171. 20. Anderson, C.; Record, M. T. J. Phys. Chem. 1993, 97, 71167126. 21. Olmsted, M.; Anderson, C.; Record, M. Proc. Natl. Acad. Sci. U.S.A. 1989, 86, 7766-7771. 22. Olmstead, M.; Anderson, C.; Record, M. T. Biopolymers 1991, 31, 1593-1604. 23. Ross, P. D.; Scruggs, R. L. Biopolymers 1961, 2, 231-239.
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