Anal. Chem. 1998, 70, 3886-3891
Hydrodynamic Analysis of Macromolecular Conformation. A Comparative Study of Flow Field Flow Fractionation and Analytical Ultracentrifugation Thilo Pauck and Helmut Co 1 lfen*
Max-Planck-Institut fu¨ r Kolloid- und Grenzfla¨ chenforschung, Abt. Kolloidchemie, Kantstrasse 55, 14513 Teltow-Seehof, Germany
A study of the retention and separation of several proteins with various geometries from spheres to elongated conformations and xanthan gum in its rodlike conformation by asymmetrical flow field flow fractionation (aF-FFF) is presented. The obtained data are compared with those from analytical ultracentrifugation (AUC) as a hydrodynamic reference method. The data are discussed within the framework of hydrodynamics and thermodynamics. The results clearly show the failure of FFF to yield the exact geometries of particles differing from spherical shape due to a wrong determined diffusion coefficient for asymmetrical particles. The deviation from the AUC results becomes more pronounced with increasing asymmetry. However, FFF can show a higher resolution than AUC if the samples exhibit a density distribution which is demonstrated for ferritin. Hydrodynamic analysis of macromolecular conformation1 is still important, even in the days of high-resolution techniques like nuclear magnetic resonance (NMR) or X-ray diffraction (XRD). This can be attributed to the fact that hydrodynamic techniques usually work in diluted solutions, whereas NMR requires high concentration, and single crystals are needed for X-ray techniques. Very often, dilute solution conformation or concentration-dependent conformation changes are of interest, which exclude the use of NMR. On the other hand, many polymeric samples cannot be crystallized or it is doubtful that the conformation in the solid state is similar to that in the solution, so X-ray techniques cannot be applied. Furthermore, especially in the field of biochemistry, sometimes only very small sample amounts are available, so sensitive techniques are needed, which can deliver the macromolecular conformation even in such cases. Analytical ultracentrifugation (AUC) is such a technique which is directly able to deliver the particle shape if the hydration of the molecule is known. However, this is a parameter which is unknown or difficult to determine, so usually combinations of analytical techniques like AUC, viscometry, light-scattering, fluorescence depolarization, or electrical birefringence1 are used to derive hydration-independent shape information.2-4 (1) Harding, S. E. Biophys. Chem. 1995, 55, 69-93. (2) Scheraga, H. A.; Mandelkern, L. J. Am. Chem. Soc. 1953, 79, 179-84.
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As AUC is a time-consuming and expensive method, it is highly desirable to find other methods capable of delivering the same hydrodynamic information. In that respect, field flow fractionation (FFF), and especially flow FFF, should be one method of choice because the translational diffusion coefficient distribution can directly be obtained. The diffusion coefficientsrelated to the sedimentation coefficient by the Svedberg equation5sis a shapedependent quantity. This was the motivation for a comparative study of AUC and asymmetrical flow (aF)-FFF on samples with various geometries, because both techniques are capable of determining the translational frictional ratio, which contains the particle shape information. Up to now, the only reported comparative study between AUC and FFF concerns the separation capabilities of AUC and sedimentation FFF for latex mixtures.6 The high efficiency in separation and determination of correct diffusion coefficients of aF-FFF has been demonstrated for spherical latexes.7,8 However, it was of special interest to determine whether the same is true for asymmetrical particles. In this context, we define asymmetrical particles as those with a nonspherical geometry. According to a theoretical work of Gajdos and Brenner, it is expected that the determined diffusion coefficient will be too high.9 Early results on the analysis of aggregated latex particles by sedimentation FFF confirmed this behavior,10-12 although a defined aggregation of latex particles can be questioned from the viewpoint of colloid chemistry. However, all experiments performed so far have only been derived using FFF as analytical method without comparison with other, established methods. Moreover, no samples of well-defined shape have been analyzed, and latex coagulates have been characterized only by electron microscopy, which is known to exhibit artifacts due to drying a solution of interest on the grid. (3) Squire, P. G. Biochim. Biophys. Acta 1970, 221, 425-29. (4) Jeffrey, P. D.; Nichol, L. W.; Turner, D. R.; Winzor, D. J. J. Phys. Chem. 1977, 81, 776-81. (5) Svedberg, T.; Pedersen, K. O. Die Ultrazentrifuge; Steinkopff Verlag: Dresden, Leipzig, 1940. (6) Li, J.; Caldwell, K. D.; Ma¨chtle, W. M. J. Chromatogr. 1990, 577, 361-76. (7) Giddings, J. C.; Yang, F. J.; Myers, M. N. Anal. Chem. 1976, 48, 1126-32. (8) Tank, C.; Antonietti, M. Macromol. Chem. Phys. 1996, 197, 2943-59. (9) Gajdos, L. J.; Brenner, H. Sep. Sci. Technol. 1978, 13, 215-40. (10) Berg, H. C.; Purcell, E. M. Proc. Natl. Acad. Sci. U.S.A. 1967, 58, 862-9. (11) Barman, B. N.; Giddings, J. C. Polym. Mater. Sci. Eng. 1990, 62, 186-90. (12) Blau, P.; Zollars, R. L. J. Colloid Interface Sci. 1996, 183, 476-83. S0003-2700(98)00228-5 CCC: $15.00
© 1998 American Chemical Society Published on Web 08/08/1998
Therefore, we used protein and polysaccharide samples of which the solution conformation is precisely known from numerous references. To exclude any error resulting from the specific sample under consideration, identical solutions were investigated by FFF and AUC at the same time to maintain comparability of the results. THEORY Asymmetrical Flow FFF. Field flow fractionation is a family of separation techniques based upon a common separation mechanism. In all subgroups of FFF, the colloidal or macromolecular material under consideration is placed in a thin channel setup between two parallel plates. A force is applied perpendicular to the channel which will cause the sample to migrate to one of the channel walls due to a physical interaction. This channel wall is called the accumulation wall. In aF-FFF, the molecules and particles are driven toward the accumulation wall by a second solvent flow. Due to self-diffusion of the particles, a steady state will build up, showing an exponential concentration c(x) distribution as stated in ref 13:
(
c(x) ) c0 exp -
)
V(x) kBT
(1)
where V(x) is the potential energy of a particle at the position x in the channel, c0 the concentration at the wall x ) 0, kB Boltzmann’s constant, and T the absolute temperature. V(x) is assumed to be linear in x, so that V(x) ) -Fx, with F the force exerted on the particles. The mean thickness of the solute layer l is related to the above equation by
l)
kBT D ) F u0
(3)
vsample 〈v〉
)
t0 tret
(4)
where vsample is the migration velocity of the component of interest, t0 the void timesmeaning the time a hypothetical nonretained component would emergesand tret the retention time of the (13) Wahlund, K. G.; Giddings, J. C. Anal. Chem. 1987, 59, 1332-39. (14) Litzen, A.; Wahlund, K.-G. Anal. Chem. 1991, 63, 1001-7.
t0V˙ crossδ2 6DV0
(5)
V˙ cross is the integral cross-flow velocity, and V0 is the void volume of the FFF channel. As the diffusion coefficient is a temperature- and solventdependent quantity, the calculated data were corrected to standard conditions (293.15 K, water as solvent) via16
( )( )
293.15 ηT,b D T η20,w T,b
D20,w )
(6)
The index b denotes the value for the buffer solution at the temperature T, with η the solvent viscosity. The indexes 20 and w refer to standard conditions at 20 °C in water. The viscosity of the buffer was calculated using the increment system of Laue et al.17 However, it has to be kept in mind that the concentration dependence of the diffusion coefficient, which is especially pronounced for asymmetrical particles, is neglected. But as loading concentrations in FFF are usually very low (∼20 µg), this assumption might be justified. Analytical Ultracentrifugation. A fundamental quantity in AUC is the sedimentation coefficient describing the velocity of the sedimenting species. It can be determined from the time dependence of the moving boundary according to
s)
Under the assumption of a uniform cross flow along the channel, the mean longitudinal solvent velocity 〈v〉 has been derived by Litzen and Wahlund for a trapezoidal channel geometry.14 In this equation, the parameters of the experimental setup enter.8 In chromatography, the displacement velocity is usually expressed in terms of the dimensionless retention ratio R:
R)
tret )
(2)
with u0 the cross-flow velocity and D the diffusion coefficient. We do not derive the retention equations for asymmetrical flow FFF in detail here. The interested reader is kindly refered to ref 13. Usually l is expressed in the dimensionless form of the retention parameter λ, with δ being the channel thickness.
λ ) l/δ
sample. For the detailed calculation of the void time, the reader is referred to refs 8 and 14. Assuming an undistorted parabolic flow profile, R is related to λ.15 By combining the Stokes law with λ and eq 4, one obtains
ln(rbnd/rm) ω2t
(7)
with rbnd being the radial position of the boundary at time t, rm the radial position of the meniscus, and ω the angular velocity. From the movement of the second moment of the boundary, one obtains the weight-average value sw. Correction to standard conditions is done according to16
s20,w )
(
)( )
1 - vjF20,w ηT,b s 1 - vjFT,b η20,w T,b
(8)
Here, vj is the partial specific volume of the solute and FT,b the solvent density at temperature T, which was again calculated using the increment system.17 The sedimentation coefficient is a concentration-dependent quantity and must thus be extrapolated to infinite dilution. For the correct extrapolation, the Gralen relations, (15) Wahlund, K.-G.; Winegarner, H. S.; Caldwell, K. D.; Giddings, J. C. Anal. Chem. 1986, 58, 573-8. (16) Tanford, C. Physical Chemistry of Macromolecules; J. Wiley: New York, 1961; Chapter 6. (17) Laue, T. M.; Shah, B. D.; Ridgeway, T. M.; Pelletier, S. L. In Analytical Ultracentrifugation in Biochemistry and Polymer Science; Harding, S. E., Rowe, A. J., Horton, J. C., Eds.; Royal Society of Chemistry: Cambridge, 1992; Chapter 7.
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s20,w ) s020,w(1 - ks)c
(9)
s020,w s20,w ) c 1 + ks
(10)
can be applied, with the latter for more pronouncely concentrationdependentsmore asymmetricalssystems. In these equations, 0 s20,w is the infinite dilution sedimentation coefficient, ks the Gralen coefficient, and c the concentration of the solute. Additional information which can be derived from sedimentation velocity experiments is the sedimentation coefficient distribution, which is proportional to the molar mass distribution by an equation of the form s ) kMa with M the molar mass and k and a constants.18 One advantageous method for the calculation of this distribution is the time derivative method of Stafford:19
g(s*)t )
(
)(
)( )
∂(c(r,t)/c0) ω2t2 r ∂t ln(rm/r) rm
2
(11)
d(ln c) 2RT 2 (1 - vjF)ω dr2
c(r) - cm Φcm(r2 - rm2) + 2Φ
∫ r(c(r) - c r
rm
(13) m) dr
with
Φ)
(14)
with R being the gas constant. Here, it is important to note that s, D, and M are weight-average quantities if polydisperse systems are investigated.23 Hydrodynamic Analysis. One modeling strategy to handle macromolecular conformation in solution is the simple ellipsoid of revolution approach.1 For a comprehensive description of this type of modeling, the reader can consult ref 24. The simplest model for particle conformation in solution is the rigid sphere, and the earliest calculations were based on spherical particles in terms of their frictional properties. Analytical ultracentrifugation allows the measurement of translational frictional ratios f/f0, where f is the frictional coefficient and f0 is the corresponding coefficient of a spherical particle of the same mass and dry volume, from the sedimentation coefficient, s20,w:5
(
)( )
1/3
(15)
0 In terms of the translational diffusion coefficient D20,w the following relation is used:
( )
kBT 4πNA f ) f0 6πη0 3vjM
1/3
1 D020,w
(16)
(12)
In the case of nonideal systems, Mw is an apparent value referred to as Mw,app. Alternatively, the obtained gradients can be evaluated using the M* function, which is defined as21
M*(r) )
RTs D(1 - vjF)
M(1 - vjF0) 4πNA f ) f0 NA6πη0s020,w 3vjM
Very smooth distributions are obtained due to the pairwise subtraction of data files, which cancels out systematic errors and minimizes statistical noise. Sedimentation equilibrium experiments classically deliver the weight-average molar mass by analysis of the equilibrium concentration gradient:20
Mw )
M)
(1 - vjF)ω2 2RT
with the MSTAR program.22 This function is especially useful for unknown polydisperse or nonideal systems and delivers the apparent weight-average molar mass without further assumptions and avoiding any model-dependent fitting. A comparison of obtained diffusion and sedimentation coefficients is possible with the Svedberg equation:5 (18) Lechner, M. D.; Ma¨chtle, W. Prog. Colloid Polym. Sci. 1995, 99, 120-24. (19) Stafford, W. F. Anal. Biochem. 1992, 203, 295-301. (20) Creeth, J. M.; Pain, R. H. Prog. Biophys. Mol. Biol. 1967, 17, 217-87. (21) Creeth, J. M.; Harding, S. E. J. Biochem. Biophys. Methods 1982, 7, 25-34. (22) Co¨lfen, H.; Harding, S. E. Eur. Biophys. J. 1997, 25, 333-46.
3888 Analytical Chemistry, Vol. 70, No. 18, September 15, 1998
with NA being Avogadro’s number. To get quantitative shape information, the Perrin function could be applied.25 But as the “hydrated” or “swollen” specific volume enters this equation, which is a quantity that is notoriously difficult to measure with any meaningful precision, this has not been done in this study, and only the frictional ratios will be compared.1 MATERIALS AND METHODS Asymmetrical Flow FFF. The asymmetrical flow FFF setup used in this work was developed in-house and is described in detail elsewhere.8,26 All experimental parameters are mentioned in the elugrams. From eq 14, it becomes clear that, in the case of a diffusion coefficient distribution, the weight-average diffusion coefficient Dw has to be considered, which was derived by peak analysis. Analytical Ultracentrifugation. For all ultracentrifuge experiments, a Beckman Optima XL-I (Beckman, Spinco Division, Palo Alto, CA) analytical ultracentrifuge equipped with integrated scanning absorption and Rayleigh interference optics (RI) was used. Sedimentation velocity and sedimentation equilibrium experiments were performed at 293.15 K using either titanium double-sector cells or six-channel KEL-F centerpieces for sedimentation velocity and sedimentation equilibrium, respectively.27 (23) Fujita, H. Foundations of ultracentrifugal analysis; John Wiley and Sons: New York, 1975; p 229. (24) Harding, S. E.; Rowe, A. J. Int. J. Biol. Macromol. 1982, 4, 160-4. (25) Perrin, F. J. Phys. Radium 1936, 7, 1-11. (26) Tank, C.; Ph.D. Thesis, TU-Berlin, 1996. (27) Yphantis, D. A. Ann. N.Y. Acad. Sci. 1960, 88, 586-601.
Table 1. Experimental Conditions for AUC Experiments sedimentation velocity sample BSA ferritin apoferritin fibrinogen xanthan gum
rotor speed wavelength (rpm) (nm) 50 000 25 000 30 000 40 000 30 000
280 280 280 280 RI
sedimentation equilibrium rotor speed (rpm)
wavelength (nm)
10 000 3 700 5 000 6 000 3 000
280 280 280 280 RI
The applied rotor speeds and wavelengths used are given in Table 1. Several experiments with different polymer concentrations ranging from 0.1 to 1.5 g/L have been performed to enable accurate extrapolation to infinite dilution. Reagents and Samples. The protein samples (ferritin, apoferritin, BSA, fibrinogen) were used without further purification as purchased from Sigma. All proteins except fibrinogen were measured in purified water. To prevent aggregation and filament formation in the case of fibrinogen, a solution at pH 7.3 containing 0.5 M potassium chloride, 1 mM magnesium chloride, and 0.1 mM EDTA was used. The addition of any surfactant, although commonly used in flow FFF to reduce interaction with the membrane material, was avoided to prevent conformational changes and complex formation.28 Xanthan gum (Kelco Int., Keltrol, grade RD) was eluted in a phosphate chloride buffer at pH 6.5 and an ionic strength of I ) 0.3. The polymer was dissolved in deionized water under gentle stirring and dialyzed into the buffer overnight.29 RESULTS AND DISCUSSION Figure 1 displays the elution curves obtained by field flow fractionation of the various samples investigated in this study. Especially the ferritin sample exhibits the high resolution of the applied experimental setup. Besides the monomer, also the dimer and even the trimer are well resolved. Comparing this with the AUC results of the same sample depicted in Figure 2, it becomes obvious that a sample with a density distribution due to an inhomogeneous loading with iron oxide is better analyzed by a method sensitive only to differences in the particle size. The density inhomogenity causes a strong smearing of the different peaks in the AUC, as can be clearly seen from the AUC equations, where the partial specific volume contributes to the buoyancy force in the AUC cell. Table 2 summarizes all data for the proteins and the xanthan sample as determined from the ultracentrifuge experiments. The hydrodynamic values are corrected over a concentration series to infinite dilution, indicated by the superscript 0 in the case of AUC. For a better comparison, the corrected diffusion coefficients from FFF and AUC are plotted versus the molecular weight in a double logarithmic manner in Figures 3 and 4. Employing the scaling relation D ) AM-e, spheres should result in a limiting exponent of e ) 1/3 and rods with e ) 0.85, as cited by Harding.1 (28) Goddard, E. D. Colloids Surf. 1986, 19, 301-29. (29) Dhami, R.; Harding, S. E.; Jones, T.; Hughes, T.; Mitchell, J. R.; To, K. Carbohydr. Polym. 1995, 17, 93-9.
Figure 1. Elution profiles of all samples under consideration. Experimental details are mentioned in the different plots. Focusing time in all cases was 35 s, with an inject flow velocity of 0.1 mL/min.
Figure 2. Sedimentation coefficient distribution g(s*) of ferritin as obtained from a sedimentation velocity experiment. For details, see Table 1.
Figure 3 depicts the diffusion data as obtained from AUC. The globular proteins show the expected scaling relation, and the exponent of e ) 0.35 is in good agreement with the theory. Nearly the same value is obtained from the FFF data (e ) 0.34). Regression over the asymmetrical particles BSA dimer, fibrinogen, Analytical Chemistry, Vol. 70, No. 18, September 15, 1998
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Table 2. Thermodynamic and Hydrodynamic Data As Obtained by AUC and FFFa AUC s020,w sample
(S)
Mw (g/mol)
BSA 4.7 64 800 apoferritin 17.9 450 000 ferritin 26.2 622 000 fibrinogen 7.67 313 600 xanthan 12.97 5 500 000 gum
aF-FFF D020,w (10-7 D020,w (10-7 f/f0 f/f0 cm2 s-1) cm2 s-1) (FFF) (AUC) 6.89 3.84 2.91 2.03 0.13
8.53 4.19 4.10 2.39 0.68
1.1
1.2
1.8 2.2
2.1 14.2
a The frictional ratios are given for comparison of the asymmetrical samples.
Figure 3. Double-logarithmic plot of the corrected diffusion coefficient versus the molecular weight as obtained from AUC data. O, Globular particles; b, asymmetrical particles.
Figure 4. Double-logarithmic plot of the corrected diffusion coefficient versus the molecular weight as obtained from FFF data. O, Globular particles; b, asymmetrical particles.
and xanthan yields e ) 0.92 from the AUC data. This is in good agreement with the value of 0.85 for the scaling of the diffusion coefficients of rodlike particles reported by Harding.1 From Figure 4 for the FFF results, it becomes clear that the asymmetrical particles do not exhibit a good correlation of D20,w vs Mw but show a highly nonlinear behavior which becomes more pronounced with increasing axial ratio of the material of interest. To be sure that the determined diffusion coefficient distribution from FFF in case of xanthan is not artificially determined, e.g., by selective adsorption of high-molecular-weight material on the membrane, the polydispersity was estimated by AUC employing 3890 Analytical Chemistry, Vol. 70, No. 18, September 15, 1998
the method of Lechner.30 We obtained a value of Mw/Mn ) 2.2 by transforming the measured value for Mz/Mw according to ref 23, which is inicated by an error bar in Figure 4 for xanthan. This procedure was necessary, as a direct determination of Mn from sedimentation equilibrium requires very high quality data and an estimate for the number-average Mn,app at the meniscus.20 From Figure 4, it becomes clear that even the smallest species do not show the expected scaling behavior. In the case of the globular particles, the obtained values are of the same order of magnitude as those from AUC (Table 1). The determined hydrodynamic radii coincide within the experimental error. Comparing the diffusion coefficients and the translational frictional ratios in case of fibrinogen and xanthan, the difference is evident. The higher diffusion coefficients would result in too small hydrodynamic radii, meaning that asymmetrical particles are eluted much faster than the equivalent sphere. In the classical work of Berg and Purcell, the finite size of the particles is assumed to be the major contribution to the deviation between theory and experiment.10 Rigorous calculations by Gajdos and Brenner show that, in the case of asymmetrical particles, the movement in a laminar flow for sufficiently small particles is independent of their shape, and the asymmetry contributes only to a smearing of the determined signal due to an orientational averaging over transversal intrinsic diffusivities of the particle.9 Experimental work showed that the deviation between experiment and theory increases with increasing particle size, a point also observed in this study.31 Similar results have recently been published by Blau and Zollars, who studied the retention behavior of asymmetrical latex coagulates by means of sedimentation field flow fractionation.12 Another complication when using FFF for hydrodynamic analysis might be found in the framework of hydrodynamics. In any hydrodynamic method, an extrapolation to infinite dilution of the solute has to be done, which is impossible in FFF as one has always a finite concentration near the accumulation wall which is unknown. The extrapolation avoids errors due to interparticle interactions during the transport process. In recent theoretical work of Beckett and Giddings, they ascribe the deviation to a reduction of the entropy S of the system under consideration in the case of aspherical particles.32 The entropic term enters the retention equations by the exponential concentration profile eq 1 via the Helmholtz free energy A. The change in A is given by
∆A ) ∆V - T∆S
(17)
By defining the entropic change over the number of degrees of freedom of a rodlike particle, Nrod, to the degrees of freedom of an ideal sphere, Nsphere, one obtains
∆S ) K ln
Nrod Nsphere
(18)
As depicted schematically in ref 32, it is possible for a thin rod to approach the accumulation wall with its center of mass (30) Lechner, M. D. In Analytical Ultracentrifugation in Biochemistry and Polymer Science; Harding, S. E., Rowe, A. J., Horton, J. C., Eds.; Royal Society of Chemistry: Cambridge, 1992; Chapter 16. (31) Gajdos, L. J.; Brenner, H. J. Colloid Interface Sci. 1977, 58, 312-56. (32) Beckett, R.; Giddings, J. C. J. Colloid Interface Sci. 1997, 186, 53-9.
much nearer than half of its length 2R, whereas the sphere cannot come closer than its radius R. Thus, not all possible states of rotational diffusion are accessible to the rod anymore. As a result, the entropy decreases. Combining eqs 1 and 17, one obtains
(
c(x) ) c0 exp -
) ( )
∆A(x) ∆S exp kBT kB
(19)
The free energy does not change once the system is in the steady state. Therefore, the middle concentration increase results in a larger distance to the accumulation wall upon a decrease in ∆S. The decrease in entropy of the system if an asymmetrical particle approaches the accumulation wall causes the particles to move at a greater distance from the channel wall to an area of faster longitudinal flow. Thus, it will be eluted earlier, and the determined diffusion coefficient will be too high. CONCLUSION From the above-presented data, it becomes clear that the hydrodynamic radii calculated from FFF measurements are in good agreement with reality as long as particles are spherical. If unknown samples are investigated and an asymmetry cannot be ruled out, the obtained values have to be considered with care. Any hydrodynamic analysis with the measured diffusion coefficients by flow FFF must fail, as the apparent values are far
too high in the case of asymmetrical particles. This is clearly revealed when the diffusion coefficients and the calculated frictional ratios are compared between AUC and FFF. For globular particles, the hydrodynamic radii calculated via the Stokes-Einstein equation show very good agreement. From the considerations of Gajdos and Brenner9 and the independent work of Beckett and Giddings,32 no way to overcome this problem can be found. For scientific work to characterize, e.g., complex colloids, proteins, or polysaccharides, FFF cannot serve as a stand-alone method to evaluate particle sizes and/or conformations unless it is known that the material under consideration is of spherical shape. However, in the case of samples with a density distribution, flow FFF has clear advantages over all fractionating methods, which are sensitive to particle density (e.g., AUC, sedimentation FFF). ACKNOWLEDGMENT The authors thank the Max-Planck-Society for financial support. Dr. Rajesh Dhami, NCMH, Nottingham, UK, is acknowledged for providing the xanthan sample he used in the referenced publication.
Received for review February 27, 1998. Accepted June 11, 1998. AC980228T
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