Anal. Chem. 2000, 72, 4846-4852
Column Efficiency and Separation of DNA Fragments Using Slalom Chromatography: Hydrodynamic Study and Fractal Considerations Yves Claude Guillaume,*,† Eric Peyrin,| Mireille Thomassin,‡ Anne Ravel,| Catherine Grosset,| Annick Villet,| Jean-Franc¸ ois Robert,§ and Christiane Guinchard‡
Equipe de Chimie Therapeutique, Laboratoire de Chimie Analytique, Laboratoire de Chimie Organique, Faculte´ de Medecine Pharmacie, Place Saint Jacques, 25030 Besanc¸ on Cedex, France, and Laboratoire de Chimie Analytique, Faculte de Pharmacie, Domaine de la Merci, 38700 La Tronche Cedex, France
Fractal geometry has provided a mathematical formalism for describing complex and dynamical structures.1 It has been applied successfully in a variety of areas such as astronomy,2 economics,3,4 and biology.5,6 Because of its success in such a variety of areas, it
is natural to develop fractal applications in chromatography. The fractal concept was used to study the effect of surface irregularity on the accessibility to silylating reactions.7 Grafting of surfaces is indeed a key process in the preparation of chromatographic materials,8-11 given the ability to fine-tune the type of surface absorbate interactions by a suitable choice of the derivatizing agent.12 In recent years, to study the surface properties of chromatographic materials, in general, and reversed- phase materials, in particular, photophysical probes have been successfully used. The structure of the derivatizing layer has been an issue of much debate regarding the questions of whether silanols are evenly distributed on the surface of silica13 or whether they are heterogeneously clustered.14 Lochmuller and his colleagues have used the intermolecular complexation process between ground-state and exited-state pyrene (py) to investigate this problem.15,16 The effect of surface irregularity on parameters such as surface concentration was studied by Farin and Avnir.17 Another application in chromatography was proposed to study the solute retention on immobilized human serum albumin (HSA). This study proposes a mathematical model to provide a more realistic understanding of the molecular processes that take place in the sucrose dependence of dansylamino acid binding on the HSA site II cavity.18 For the separation of biological macromolecules such as DNA, the conventional HPLC modes are usually based on an equilibrium phenomenon between the mobile and stationary phases. In the case of exchange ion chromatography, the retention mechanism is dependent on the electrostatic interaction between the phosphate groups of the DNA molecules and the cationic
* To whom correspondence is to be sent: (tel.) 33 3 81 66 55 46; (e-mail)
[email protected]. † Equipe de Chimie Therapeutique, Faculte ´ de Medecine Pharmacie. ‡ Laboratoire de Chimie Analytique, Faculte ´ de Medecine Pharmacie. § Laboratoire de Chimie Organique, Faculte ´ de Medecine Pharmacie. | Laboratoire de Chimie Analytique, Faculte de Pharmacie. (1) Mandelbrot, B. B. The Fractal Geometry of Nature; Freeman, Academic: New York, 1991. (2) Martinez, V. J. Science (Washington, D.C.) 1999, 5413 (284), 445. (3) Mandelbrot, B. B. Fractals and Scaling in Finance: Discontinuity, Concentration, Risk; Springer-Verlag: New York, 1997. (4) Mandelbrot, B. B. Multifractals and 1/f Noise: Wild-.self-Affinity in Physics; Springer Verlag: New York, 1999. (5) Dewey, T. G. J. Chem. Phys. 1993, 98, 2250. (6) Fedorov, B. A.; Fedorov, B. B.. Schmidt, P. W. J. Chem. Phys. 1993, 99, 4076.
(7) Farin, D.; Avnir, D. J. Chromatrogr. 1987, 406, 317. (8) Riedo, F.; Gzencz, M.; Liardon, O.; Kovatz, E. Sz. Helv. Chim. Acta 1978, 61, 1912. (9) Korosi. G.; Kovatz, Sz. E. Colloids Surf. 1981, 2, 315. (10) Le Ha, N.; Ungvarai, J.; Kovatz, Sz. E. Anal. Chem. 1982, 54, 2410. (11) Gobet, J.; Kovatz, Sz. E. Adsorpt. Sci. Technol. 1984, 1, 111. (12) Unger, K. K. Porous silica; J. Chromatogr. Library, Vol. 166; Elsevier: Amsterdam, 1979. (13) Unger, K. K.; Roumeliotis, P. J. Chromatogr. 1978, 149, 211. (14) Lochmuller, H.; Wilder, D. R. J. Chromatogr. Sci. 1979, 17, 574. (15) Lochmuller, H.; Colborn, A. S.; Hunicutt, M. L.; Harrius, J. M. J. Am. Chem. Soc. 1984, 106, 4077. (16) Lochmuller, C. H.; Colborn, A. S.; Hunnicutt, M. L.; Harris, J. M. Anal. Chem. 1983, 55, 1344. (17) Farin, D.; Avnir, D. J. Phys. Chem. 1987, 91, 5517. (18) Peyrin, E.; Guillaume, Y. C. Anal. Chem. 1999, 71, 1496.
Novel equations (Guillaume Y. C.; et al. Anal. Chem. 2000, 72, 853) were developed to describe the large double-stranded DNA molecule retention in slalom chromatography (SC). These equations were applied for the first time to model both the “apparent selectivity” and the resolution between two eluted DNA fragments on a chromatogram. A study of the column efficiency corroborated the fact that slalom chromatography is not based on an adsorption or equilibrium phenomenon, but can be attributed to a hydrodynamic phenomenon. Using a combination of the dynamics of DNA fragment progression in the column and fractal considerations, it was shown that the apparent selectivity depends both on the DNA fragment sizes and mobile-phase flow rate and therefore a balance between two hydrodynamic regimes. A chromatographic response function was also used to obtain the most efficient separation conditions for a mixture of DNA fragments in a minimum analysis time. The chromatographic data confirmed that in SC the flow rate can increase or maintain the separation efficiency with an associated decrease in the analysis time. This constitutes an attractive outcome in relation to the classical chromatographic separation.
4846 Analytical Chemistry, Vol. 72, No. 20, October 15, 2000
10.1021/ac000281t CCC: $19.00
© 2000 American Chemical Society Published on Web 09/21/2000
groups of the stationary phase. For hydrophobic-interaction chromatography, an additional contribution to the macromolecule retention is encountered via the hydrophobic effect between the stationary phases and DNA bases. Finally, gel permeation chromatography is dominated by the capability of the DNA molecules to penetrate into the stationary-phase pores in relation to their size. Alternative chromatographic procedures are available for the separation of flexible biological or synthetic polymers. These two techniques, slalom chromatography (SC) and hydrodynamic chromatography (HDC), are based on the use of the laminar flow, which occurs in the interstitial spaces created between the particles packed in the column.19-30 The separation process depends on the flow rate and the particle size of the column packing and not on the pore size or chemical nature. HDC has been principally developed and applied to the separation of synthetic polymers such as the polystyrenes.22-24 The elution order in the HDC is the same as that in gel permeation chromatography due to the exclusion for the large polymers from the low-velocity regions near the particle wall.22 The separation is only valid when the polymer is in a random coil form.22 Separation in SC has been reported for double-stranded DNA molecules.24-29 The elution order for the DNA molecule is the opposite of that expected for a HDC mechanismsthe larger strands are eluted after the smaller ones. When the DNA chain is applied to the chromatographic system, it frequently turns around the spherical obstacles; the larger the fragments the more difficulty it is to travel across the interstitial spaces created inside the column. These two techniques, HDC and SC, cannot be explained in terms of an equilibrium constant between the mobile and the stationary phase and are based on a hydrodynamic process. In a recent paper,31 Guillaume et al. developed a novel theory to describe this separation hydrodynamic process by slalom chromatography. This report proposes a reanalysis of this mathematical model using fractal considerations in terms of column efficiency and both apparent selectivity and resolution between two adjacent peaks on the chromatogram. MODEL Apparent Selectivity in Slalom ChromatographysFractal Considerations. In a recent paper,31 a novel mathematical model was developed to describe the fractionation of large doublestranded DNA in “slalom chromatography”. This fractionation is based on a new hydrodynamic process that is determined by the progression of the mobile phase flow through the interstitial (19) Dimarzio, L. A.; Guttmann, C. M. Macromolecules 1970, 2, 131. (20) Small, H. J. J. Colloid. Interface Sci. 1974, 48, 147. (21) Hoangland, D. A.; Prudhomme, R. K. Macromolecules 1989, 22, 775. (22) Stegeman, G.; Kjraak, J. C.; Poppe, H. J. Chromatogr. 1991, 550, 721. (23) Stegeman, G.; Kjraak, J. C.; Poppe, H.; Tijssen, R. J. Chromatogr., A 1993, 657, 283. (24) Venema, E.; Kraak, J. C.; Poppe, H.; Tijssen, R. J. Chromatogr., A 1996, 740, 159. (25) Hirabayashi, J.; Kasai, K. In Molecular Interactions in Bioseparations; Ngo, T. T., Ed.; Plenum Press: New York, 1993; Chapter 5, p 69. (26) Boyes, E.; Walker, D. G.; Mcgeer, P. L. Anal. Biochem. 1988, 170, 127. (27) Hirabayashi, J.; Ito, N.; Naguchi, K.; Kasai, K. Biochemistry 1990, 29, 9515. (28) Kasai, K. I. J. Chromatogr. 1993, 618, 203. (29) Peyrin, E.; Guillaume, Y. C.; Grosset, C.; Ravel, A.; Villet, A.; Alary, J.; Favier, A. J. Chromatogr., A. In press. (30) Wang, Q. H. Anal. Chem. 1997, 69, 361. (31) Peyrin, E.; Guillaume, Y. C; Villet, A.; Favier, A. Anal. Chem. 2000, 72, 853.
Figure 1. Representation of the progression of the DNA chains (arrows) through the closed-column packing particles.
spaces created between the highly packed particles inside the column (Figure 1). The separation was treated as the result of a slowing down of the large double-stranded DNA fragments in relation to their size, with the flow direction changing around the particles. Following this description, this separation mode seems to be close to a new type of electrophoretic technique such as the capillary electrophoresis using liquid crystals as replaceable media.32 The differences are constituted by the use of the electric field for the DNA migration and the creation of very narrow interstitial spaces, but both systems are based on the DNA separation in a very compact and dense media. The major advantages of the slalom chromatography mode consists of the rapidity and the simplicity of the experimental procedure. For example, three fragments of approximatively 4, 9, and 23 kb can easily be separated in less than 2 min with a conventional chromatographic system using a gel permeation column.28 However, the resolution is inferior to that of capillary gel electrophoresis.28 It is well-known that the steady-state extension, R, of a single molecule in a hydrodynamic flow is characterized by a dumbbell model consisting of two beads connected by a spring representing the entropic elasticity of a wormlike chain. When a force F is applied across the ends of the chain, the extension, R, is given by33
FA 1 R ) 1kT 4 L
(
-2
)
-
1 R + 4 L
(1)
where A is the persistence length, L the total curvilinear length, (32) Rill, R. L.; Liu, Y.; Van Wincle, D. H.; Locke, B. R. J. Chromatogr., A 1998, 817, 287.
Analytical Chemistry, Vol. 72, No. 20, October 15, 2000
4847
k the Boltzmann constant, and T the absolute temperature. By a limited development of eq 1 to the second order in relation to R/L, the following equation is obtained:
R ) L
-1 x1 + 4FA 3kT
(2) R)
In addition, it has been shown by Perkins et al.34 that the actual bead radius, rb, for a chain in an elongation flow is equal to RL0.54, where R is a constant. Using the rb and L values of a single phage DNA molecule calculated from the measurements of its elongation,35 R can be determined. Knowing the rb value, the value of F can be obtained with the well-known Stockes relation
F ) 6πηrbν
tR tNR
(
RRT ) 1 - K
)
R dp
For high DNA fragment size (>Dc) (region II)
dp - KRA dp - KRB
(8)
In the extreme case where R , dp, eq 8 can be rewritten as
R)1+
(
)
RB K (R - RA) 1 + K dp B dp
R K′′(dp/R) (e - 1) dP
R)
(9)
(5)
)
RA eK′′(dp/RA) - 1 RB eK′′(dp/RB) - 1
(10)
If A belongs to region I and B has sufficient size to belong to region II, combining eqs 5-7 gives
-1
R)
dp - KRA K′RB(eK′′(dp/RB) - 1)
(11)
-1
(6)
In these equations, K, K′, K′′ are characteristics of the structural arrangement of the spherical particles inside the column and are named directional change factors. Dc is the critical value of the DNA size and corresponds to the change in the DNA fragment hydrodynamical progression mechanism in the column between the two regions I and II. Dc depends on both the particule arrangement of the column used and the mobile-phase flow rate, F. Therefore, a critical value of F, noted Fc, can be defined. At a very a low mobile-phase velocity, the F value is very weak (eq 3) and Rf to its random coil (eqs 1 and 2). Thus, the DNA chain is in a random coil configuration in such way that RRTf1 (eq 4). As the mobile phase velocity increases, both F and R increase (eqs 2 and 3), and RRT is related to the steady-state extension as (33) Bustamante, C.; Marko, J. F.; Siggia, E. D.; Smith, S. B. Science (Washington, D.C.) 1994, 265, 1599. (34) Perkins, T. T.; Smith, D. E.; Larson, R. G.; Chu, S. Science (Washington, D.C.) 1995, 268, 83. (35) Perkins, T. T.; Smith, D. E.; Chu, S. Science (Washington, D.C.) 1997, 276, 2016. (36) Hirabayashi, J.; Kasai, K. J. Chromatogr. 1996, 722, 135.
4848
R)
If the two fragments A and B belong to region II, combining eqs 6 and 7 gives
For low DNA fragment size ( Fc (region II). The apparent separation factor R between two adjacent DNA fragments on the chromatogram is defined by the following equation
Analytical Chemistry, Vol. 72, No. 20, October 15, 2000
Fractal geometry (FG) was used in this manuscript to investigate the effect of the DNA fragment size, noted D, on the apparent selectivity, R. A series of n DNA fragments was introduced. This series was composed of n terms from the smallest fragment size (level 0, size D0) to the highest fragment size (level n - 1). Do corresponds, obviously, to the size of the unretained fragment. At level L, the DNA fragment size has a DL size. The parameter
qL )
DL DL - 1
(12)
where qL > 1 and 1 e L e n - 1 was introduced to characterize the DNA fragment size in the series. Assuming a self-similar fractal nature of the DNA fragment series implies that qL must be independent of L: qL ) q ) constant. Thus, at level L, the DNA fragment size can be expressed as
DL ) DoqL Thus, the sum of all the DNA fragment sizes is
(13)
n-1
DT )
∑D
k
(14)
k)0
DT )
Doqn - Do q-1
(15)
Equations 12 and 14 reflect the fractal nature of the DNA fragment series. The corresponding apparent selectivity between two adjacent DNA fragments in the fractal series was defined as
RL )
RRTDL RRTDL-1
(16)
where RRTDL-1 and RRTDL represent the relative retention time of the DNA fragments with a respective size equal to DL-1 and DL. The variation curve RL vs L is an image of the variation in the apparent selectivity vs the DNA fragment size. As L increases, the apparent selectivity is determined between two adjacent fragments in the fractal series, i.e, between two DNA fragments for which both their sizes and their size differences increase. Resolution in Slalom Chromatography. The resolution of a pair of DNA molecules can be measured from the chromatogram using retention time tR and peak width ω using the following equation
Rs ) 2
tRB - tRA ωA + ω B
(17)
where the subscripts A and B refer to the first and second eluting peaks, respectively. Assuming that the two peaks have a similar bandwidth, i.e., ω ) ωA ) ωB, it can be expressed in relation to the column plate number N determined on the B peak by the well-known equation
tR2 B
NB ) 16
ω2
(18)
Thus
1 1 1 ) N ω 4 x B tR B
(19)
Rewriting eq 17 using 19
Rs )
tRB - tRA 1 NB x 4 tRB
(20)
Using the apparent selectivity equation between A and B, eq 20 can be rewritten as
Rs )
R-1 1 N 4x B R
(21)
It is important to note here that for two DNA fragments belonging to region I and in the extreme case where R , dp, eq 21 can be rewritten as
(
)
RB K (RB - RA) 1 + K dp dp 1 Rs ) xNB RB 4 K 1 + (RB - RA) 1 + K dp dp
(
)
(22)
EXPERIMENTAL SECTION Apparatus. The HPLC system consisted of a Merck Hitachi pump L7100 (Nogent sur Marne, France), an Interchim Rheodyne injection valve model 7125 (Montluc¸ on, France) fitted with a 20µL sample loop, and a Merck L 4500 diode array detector. A GF250 porous silica column Zorbax for gel permeation chromatography (particle size, 4 µm, column size, 250 mm × 4.6 mm, exclusion size for globular protein, 3 × 105 Da) and a C1Kromasil column (particle size, 5 µm, column size, 150 mm × 4.6 mm) supplied by Interchim were used with controlled temperature at 25 °C in a TM no. 701 Interchim Crococil oven. Reagents. Lambda DNA (48.50 kb) and restriction enzyme KpnI were supplied by New England Biolabs (Gagny, France). Ethanol, EDTA, sodium hydrogen phosphate and sodium dihydrogen phosphate were purchased from Prolabo (Paris, France). Water was obtained from an Elgastat option water purification system (Odil, Talant, France) fitted with a reverse osmosis cartridge. Digestion of λ DNA. Restriction enzyme KpnI was used for the cleavage of the λ DNA into three fragments of different sizes 29.95, 17.05, and 1.50 kb. The λ DNA (2 µg) was treated with 3U of KpnI in 15 µL of the reaction mixture at 37 °C for 3 h, precipitated by ethanol, dissolved in 20 µL of water, and stored at -20 °C until use. Chromatographic Conditions. The mobile phase consisted of a sodium phosphate salt 0.01 M EDTA, 0.001 M mixture at pH ) 6.8. Twenty microliters of DNA solution was injected, and the retention times were measured for different flow rates varying from 0.1 to 1.5 mL/min. The retention time tNR corresponding to the void fraction was obtained using the 1.5 kb fragment which was not retained. RESULTS AND DISCUSSION Model Validation. The asymmetry factors of all peaks calculated from measurements made at 50% of the total peak height were in the range 1.00 e As e 1.10. The corresponding peak widths were determined. The retention time values for the 17.05, 29.95, and 48.50 kb fragments (tR) and for the 1.50 kb fragment which corresponded to the void volume marker (tNR) were obtained at various flow rates and at the maxima of the chromatographic peaks. All the experiments were repeated three times Apparent Selectivity Model. Whatever the flow rate variation, all the fragments were arranged on a chromatogram in the same order. Considering only the retained fragments, the 17.05 kb fragment eluted first and the 48.50 kb fragment eluted with the highest retention time:
48.50 kb > 29.95 kb > 17.05 kb Analytical Chemistry, Vol. 72, No. 20, October 15, 2000
4849
From the tR and tNR values, the experimental R values were calculated for the different chromatographic conditions for two adjacent peaks on the chromatogram, i.e., the pair {48.5, 29.95 kb} and {29.95, 17.05 kb}. The variation coefficients of the R values were less than 4% in most cases, indicating a high reproducibility and good stability for the chromatographic system. With a weighted nonlinear regression (WNLIN) which was used in earlier chromatographic studies,37-39 the data reported by Hirabayashi et al.27 for four different columns of dp ) 9 µm at a flow rate equal to 0.3 and 0.6 mL/min and our data for the Zorbax column of dp ) 4 µm and the Kromasil column of dp ) 5 µm obtained at various flow rates were fitted to eqs 8, 10, and 11. The steady-state extension, R, for the different DNA fragments was calculated from eqs 1 and 3 using, by approximation, the linear velocity of the mobile phase as a v value. After the WNLIN procedure, the calculated directional change K, K′, and K′′ characteristic of the columns was used to estimate the R values with the measured values for the two DNA fragment pairs at the different flow rates. The correlation between all the predicted and experimental R values exhibited slopes equal to 0.98 with r2 > 0.99. This excellent approximation between the predicted and experimental values can be considered adequate to verify the model. Size Dependence on Apparent Selectivity: Fractal Approach. The data reported previously27 were used to determine the RL versus L plots for two DNA fragment fractal series for which Do ) 5 kb and (i) q ) 1.1, n ) 27 (ii) q ) 1.25, n ) 14 (Figure 2). The plots were divided into two regions, one for levels L over Lc and a second for levels below Lc. I. For L , Lc corresponding to D , Dc, the DNA size was small in such way that DL-1 and DL f Do (the DNA fragment size of the unretained peak). Thus, the steady-state extensions, R, of the DL-1 and DL fragments were similar and therefore RLf1 (eq 9). It can be noted that the electrophoretic procedure is able to extend the analysis possibilities to smaller DNA (e5 kb) molecules than in SC due to the smaller spherical obstacles (18 nm).32,40 As L increased, but still remained Dc, the shape of the RRTDL-1 and RRTDL variations with L, i.e, with the DNA fragment size, changed (eq 6) producing a decrease of RL. At a sufficiently high level of L, the RRT of a DNA fragment was proportional to its size, explaining that, for relatively low q values, corresponding to a weak increase in DL in relation to DL-1 (q ) 1.1, for example, Figure 2A), RL was relatively independent of L and f1. This was clearly objectivized, by using a weak q value, (q ) 1.1), which in the two extremities of the two regions I and II (DL-1 and DL , Dc and DL-1 and DL . Dc), the apparent selectivity RL, was relatively independent of the level L value. This reflects the fractal character (37) Peyrin, E.; Guillaume, Y. C. Anal. Chem. 1999, 71, 1496. (38) Peyrin, E.; Guillaume, Y. C.; Guinchard, C. Anal. Chem. 1998, 70, 4235. (39) Guillaume, Y. C.; Peyrin, E. Anal. Chem. 1999, 71, 1326. (40) Rill, R. L.; Locke, B. R.; Liu, Y.; Van Winckle, D. H. Proc. Natl. Acad. Sci. U.S.A. 1998, 95, 1534.
4850
Analytical Chemistry, Vol. 72, No. 20, October 15, 2000
Figure 2. Selectivity RL as defined by eq 16 (A) q ) 1.1 (B) q ) 1.25 for an Asahipak GS-310 column with a packing particle size fixed at 9 µm and a flow rate equal to 0.3 mL/min using experimental data reported by Hirabayashi et al.27
of the apparent selectivity for both the smallest and the largest DNA fragments. Flow Rate Dependence on the Column Efficiency H. The H values were calculated from the peak widths for the 48.50,
Figure 3. Dependence of plate height on flow rate for a DNA fragment size (A) 48.50, (B) 29.95, (C) 17.05 kb. Experimental data obtained using a C1 Kromasil column (see Experimental Section) with a particle diameter equal to 5 µm.
Figure 4. Theoretical flow-rate dependence on selectivity R between the two DNA fragments 17.05 and 29.95 kb for a particle diameter of 4 µm. Experimental data obtained using a Zorbax column (see Experimental Section) with a particle diameter equal to 4 µm.
29.95,and 17.05 kb DNA fragments. The variation coefficient of the H values was less than 3%. Figure 3 shows, for example, and for the Kromasil column of dp ) 5 µm, the liquid velocity dependence on the H values for the three DNA fragments. To interpret the experimental results for H, the expression for H given by Dawkins and Yeadon concerning polymer was used. This experimental H value is given by41
term III in eq 23 was assumed to be also nil. Therefore, following eq 23, H should be a constant in the range of the mobile phase flow rate:
H)A h+
B h +C hν ν
(23)
in which A h, B h , and C h are coefficients depending on several parameters, where the term I(A h ) depends on both the eddy diffusion and molecular weight distribution for solute dispersion in the mobile phase, term II(B h /ν) results from the dispersion due to the molecular diffusion in the longitudinal direction in the mobile phase, and term (III)(C h ν) results from solute dispersion due to mass transfer. The very weak value of the diffusion coefficient in the mobile phase of macromolecules such as large DNA fragments (
