Optimization of electric field strength for DNA sequencing in capillary

Anal. Chem. 1003, 65, 2841-2850

2041

Optimization of Electric Field Strength for DNA Sequencing in Capillary Gel Electrophoresis John A. Luckey and Lloyd M.Smith' Department of Chemistry, University of Wisconsin, Madison, Wisconsin 53706

Capillary gel electrophoresis (CGE)has demonstrated the ability to separate DNA sequencing reactions at speeds up to 25 times as great as conventional slab gel electrophoresis. These increased speeds are made possible by the efficient heat dissipation of capillaries, which permits higher electric fields to be employed without deleterious thermal effects. The high electric fields, however, also lead to a reduction in the spacing between bands with a concomitant loss of resolution. The resulting tradeoff between speed and resolution is a very important practical aspect of these high-field separations. This work addresses this question by investigating the band broadening and resolution of DNA fragments as they are separated through a fixed distance of gel at field strengths ranging from 50 to 400 V/cm. It is found that the bandwidths of DNA fragments do decrease with the higher field strengths due to a reduction in the diffusional broadening of bands. However,at sufficiently high electric field strengths, the bands begin to broaden again due to the thermal gradient across the gel. This behavior causes the optimum electric field strength for maximum fragment resolution to depend upon the length of fragments being separated. The relative contributions of diffusion and thermal gradients are discussed and used to predict the ultimate performance of constant field capillary gel electrophoresis.

INTRODUCTION Recent developments in thin gel electrophoresis have been driven by the desire to increase the separation speed of the single-stranded DNA fragments produced in sequencing reactions. The corresponding reduction in electrophoresis time would increase the throughput of automated sequencing instruments and thereby decrease the cost of this aspect of DNA sequencing. One way to decrease the time required for electrophoresis is to increase the electric field driving the separation. This requires, however, that the additional Joule heat generated at these higher fields be effectively dissipated. This may be achieved by decreasing the thickness of the gel and thereby increasing the heat transfer efficiency, an approach which has been recently exploited in the design of several ultrathin gel electrophoresis instruments.14 (1)Luckey,J.A.;Drossman,H.;Kostichka,A. J.;Mead,D.A.;D'Cunha, J.; Norris, T. B.; Smith, L. M. Nucleic Acids Res. 1990, 18, 4417-4421. (2) Swerdlow, H.; Gesteland, R. Nucleic Acids Res. 1990, 18, 14151419. (3) Chen, D.; Harke, H. R.; Dovichi, N. J. Nucleic Acids Res. 1992,20, 4873-4880. (4) Guttman, A.;Cohen, A.S.;Heiger, D. N.; Karger, B. L. Anal. Chem. 1990,62, 137-141. 0003-2700/93/0385-284 1$04.00/0

Thin capillaries (typically 50-pm i.d.1 have been known for many years to be extremely efficient at dissipating Joule heat in electrophoresis.7-9 In open tubular capillary electrophoresis, the resolution of separations is ultimately limited by the longitudinal diffusion of sample bands.' In this limit, the number of theoretical plates, NTH, is

Nm = pV/2D (1) where p is the mobility of the sample constituent, V is the applied voltage, and D is the longitudinal diffusion coefficient. If this equation were strictly applicable to capillary gel electrophoresis,it would implythat higher electric fields would yield better resolution of DNA fragments. In practice, this is not observed.10 Other factors affecting band broadening and spacing decrease the resolution from that obtained under ideal conditions. In recent work, a theoretical model which accurately described the resolution of DNA sequencing reactions as they were electrophoresed through varying lengths of gel was derived.10 This model invoked four sources of band broadening: finite injection volume, finite detection volume, diffusion, and thermal gradient broadening. Using empirically obtained mobilities, predicted resolution curves were generated which matched the observed data quite well. In this paper, we apply this model as well as a mobility model derived elsewhere" to explore the band broadening and resolution of DNA sequencing reactions as they are electrophoresed at various electric field strengths. Since the heat dissipation of capillary gel is so efficient, there is a negligible overall temperature rise inside the gel as the field is increased.11 This allows the effect of the electric field strength to be isolated from the run temperature and more accuratelyevaluated. Theoreticalresolution curves showgood agreement with the observed data a t electric field strengths below 150V/cm and above 300V/cm. Deviationsin the model for the intermediate electric fields are probably due to either a coupling between longitudinal diffusion and thermal gradient broadening or interactions between the migrating DNA and the gel matrix. EXPERIMENTAL SECTION Instrumentation. The fluorescence-basedcapillary gel electrophoresis (CGE) instrument used in this study has been described in detail elsewhere.l'J-lZData points were taken every 500 ms for the 50 V/cm runs, 300 ms for the 75 V/cm run, 200 ms for the 100 and 150 V/cm runs,and 100 ms for the 200,300, and 400 V/cm runs using an IBM AT-compatible computer (5) Brumley, R. L., Jr.; Smith, L. M. Nucleic AcidsRes. 1991,19,41214126. (6) Kostichka, A. J.; Marchbanks,M. L.; Brumley, R. L., Jr.; Drossman, H.; Smith, L. M. Biotechnology 1992, 10, 78-81. (7) Jorgenson, J. W.; Lukacs, K. D. Anal. Chem. 1981,53,1298-1302. (8)Hjerten, S. J. Chromatogr. 1983,270, 1-6. (9) Ewing, A. G.; Walliigford, R. A.;Olefirowicz, T. M. Anal. Chem. 1989,61, 292A-303A. (10) Luckey, J. A.;Norris, T. B.; Smith, L. M. J. Phys. Chem. 1993, 97,3067-3075. (11) Luckey, J. A,; Smith, L. M. Electrophoresis 1993, 14, 492-501. (12) Luckey,J.A.;Drmman,H.;Kostichka,A. J.;Smith,L.M.Methods En~ymol.1993,218, 154-172.

0 1993 Amerlcan Chemical Society

2842

ANALYTICAL CHEMISTRY, VOL. 65, NO. 20, OCTOBER 15, 1993

equipped with a 12-bit A/D converter. The same length to detector (40 cm) was employed in all experiments. Sample injectionswere performed by electrokineticinjection (10 sat 150 V/cm) from 6-pL aliquotsof several pooled sequencing reactions to eliminate sample variations. Gel Preparation. All gels used in this study were 4 % total polyacrylamide with 5% bisacrylamide cross-linker containing 8.3 M urea. Fused-silica capillary tubing (375-pmo.d., 50-pm i.d., Polymicro Technologies, Phoenix AZ) was cut into 100-cm segments with detection windows (where the polyimide coating has been removed from the capillary) located 50 cm from the end. The capillaries were cleaned by successively flowing 0.1 M NaOH, deionized water, and acetone through the columns. The inner surface of the capillaries was then derivatizedwith a 0.2% ' solution of y(methoxyacryloxypropy1)trimethoxysilane(Sigma,St. Louis, MO) in 1:lwater, ethanol,which was passed through the columns at room temperature for 30 min. The derivatizing solution was removed by passing acetone and then air through the capillaries. The tubes were then dried under an infrared heat lamp for 45 min. Gel-filled capillaries in TPE buffer (0.75 M Tris base, 10 mM disodium EDTA, brought to pH 7.5 with phosphoric acid) were prepared under high pressure as previously described.1k12 After polymerization, the capillaries were preelectrophoresed in 1X TPE buffer for 4 h at 100 V/cm to remove the mobile reaction products. This step equilibrated the gels with the running buffer, allowing the gels to be run indefinitely without further conductivity changes. Sample Preparation. Fluorescent sequencing reactions were prepared from 10 pg of M13mp18 single-stranded DNA template (UnitedStates Biochemical Corp., Cleveland, OH). The method of generatingthe samples labeled with four fluorophores is described elsewhere.'$ The final volume of the pooled sample was 150 p L , which was sufficient for 25 injections. This sample was stored as a liquid in 5:l formamide/O.Olx TPE at -20 "C in the dark. Prior to injection, a 6-pL aliquot was heated to 90 "C for 4 min and then quickly brought to -20 "C using an ice/ethanol bath to denature the fragments.

5

O0 ,. 1I24 j Y

'

3

0.10-

0.08-

.?I

3

0.060.04-

O'O'L---0.00

300 100

0

400

200

ElezLric Field straqth Man)

Representation of a portion of the measurements obtalned from resolutlon experiments. Peak full widths at half-maximum are plotted versus applied electric field for four representative DNA fragments: (M) base 81, (A)base 180, (0)base 374, and ( 0 )base 570. Error bars reflect the standard devlatlon of the data. Figure 1.

graphically in Figure 1. Figure 1shows a plot of the peak full widths at half-maximum for 4 of the 11 DNA fragments studied. Error bars, which reflect the standard deviation of the data, are small, indicating that the reproducibility of the data is quite good. It is clear from the plots shown that the half-widths of all the DNA fragments are smallest at approximately 200 V/cm. The significance of this minimum will be discussed in a later section. Using the measured half-widths and mobilities, the resolutions of the DNA fragments were calculated from the following equation:

RESULTS

A. Resolution Measurements. Observed Resolution. The underlying data on which this study is based consist of measurements of fragment mobilities and bandwidths for separations through 40 cm of gel a t seven electric field strengths ranging from 50 to 400 V/cm. Three separations were performed for each electric field studied to permit calculations of means and standard deviations of the data. Peak shape measurements were made at 11base positions in the obtained electropherograms (bases 46, 81, 157, 180, 261, 374,421,483,523,570, and 624). These base positions correspond to DNA fragments starting from position 6307 in the M13mp18 sequence published elsewhere (ref 14 contains the M13mp18 sequence) and extending through the multiple cloning site. The base number refers to the length of the DNA fragment in nucleotide units. These positions were selected to be in relatively AT rich regions of the sequence to minimize the effects of GC compressions on the resolution. Furthermore, each position was selected to have an identical base in close proximity (ideally a doublet), so that accurate band separations could be obtained. For each of the 11bases studied, retention time, peak full width at half-maximum (fwhm),and band separation between base x and x + 1were measured, except for bases 46,374,483, and 624 in which base x + 2 was used and bases 523 and 570 in which base x + 3 was used. These values were then tabulated and used to calculate the average retention times, peak half-widths, and observed band separations. The data are given in tabular form in Table I, and a portion is shown (13)Mead, D. A.; McClary,J. A.;Luckey, J. A.;Kostichka,A. J.; Smith, L. M. Biotechniques 1991, 11, 76-87. (14)Yanisch-Perron, C.; Vieira, J.; Messing, J. Gene 1985, 33, 103119.

where AtR is the difference in retention times between the two adjacent peaks, and HW is the full width at halfmaximum. A portion of the results are plotted in Figure 2. B. Theoretical Analysis of Results. From the data presented in Figures 1and 2, it is clear that both peak width and resolution vary with electric field strength and fragment size in a complex manner. To understand the processes affecting the resolution and peak widths of these separations, it is useful to compare the observed behavior to theoretical models. One may begin with eq 22 from ref 10:

R=

LAP (3) 2[112(gT12)1/2+ p1('JT:)1'21 This equation expresses the resolution of two adjacent peaks as a function of the mobilities of both peaks, p1 and pz; the difference between their mobilities; Ap; the length to the detector, L; and their total peak variances UTI2 and UTZ~.If the assumption is made that adjacent peaks are approximately the same width, then eq 3 can be approximated by

R =L & L / ~ ~ ( U T ~ ) ~ / ~ (4) where p is the average of p1 and pz. From this equation it is clear that the resolution depends on the relative mobility difference between adjacent peaks, A p / p , as well as the total peak variance, u+. In general, one expects that both Ap/p and UT^ will be functions of the DNA fragment size and the applied electric field. These dependences can be examined separately as "band-spacing effects" and "band-broadening effects". The former effects influence the spacing of adjacent peaks, while the latter effects determine the thickness of the eluting peaks.


2845

Table I. 75 V/cm

50 V/cm

base

100 V/cm

150 V/cm

200 V/cm

300 V/cm

400 V/cm

Measured Retention Time (min)

65.8 f 1.9 75.8 f 2.0 102.6 f 2.5 111.4f 2.6 143.2 f 3.2 188.9f 4.2 205.8f 4.1 229.5 f 5.2 242.1 f 5.7 258.7 f 6.2 276.1 f 6.5

86.4 f 4.2 99.5 f 4.7 134.8 f 6.0 146.3 f 6.4 188.6 f 7.7 250.7 f 9.3 274.8 f 9.9 310.2 f 10.4 329.0 f 10.4 353.7 f 10.3 379.1 f 11.3

130.2 f 4.6 150.0f 5.3 203.6 f 7.1 221.3 f 7.6 286.7 f 9.3 384.8 f 11.0 423.3 f 11.9 480.3 f 12.1 512.2 f 11.2 555.6 f 9.0 601.6 f 9.2

46 81 157 180 261 374 421 483 523 570 624

43.7 f 1.4 50.2 f 1.5 67.6 f 1.9 73.3 f 2.0 93.1 f 2.5 120.8 f 3.1 130.9 f 3.3 145.4 f 3.7 153.0 f 3.7 162.5 f 3.6 171.8 f 3.8

31.9f 1.1 36.7f 1.3 49.3 f 1.7 53.3 f 1.9 67.2 f 2.2 85.4 f 2.5 91.9 f 2.6 101.2f 3.0 105.9 f 3.0 111.8 f 3.1 117.4 f 3.2

22.1 f 0.6 25.4 f 0.7 33.9 f 0.9 36.5 f 1.0 45.5 f 1.3 47.1 f 1.6 61.1 f 1.7 66.7 f 1.8 69.5 f 1.8 73.0 f 1.9 76.2 f 2.1

16.3 f 0.3 18.8f 0.3 24.8 f 0.4 26.6 f 0.4 32.9 f 0.5 41.0 f 0.5 43.7 f 0.6 47.5 f 0.6 49.4 f 0.6 51.6 f 0.6 53.5f 0.6

Measured Band Separation (cm)

0.401 f 0.004 0.172 f 0.003 0.167 f 0.006 0.162 f 0.003 0.132f 0.003 0.205f 0.007 0.130f 0.007 0.149f 0.006 0.255 f 0.040 0.199 f 0.007 0.170 f 0.001

46 81 157 180 261 374 421 483 523 570 624

11.5 f 1.1 10.3 f 1.0 7.8 f 0.9 7.2 f 0.9 6.1 f 0.1 5.2 f 0.4 5.9 f 0.3 4.6 f 0.0 5.5 f 0.3 4.6 f 0.2 5.4 f 0.5

14.8 f 0.4 13.0 f 2.0 9.8 f 1.1 8.6 f 0.4 7.1 f 0.1 5.9 f 0.2 5.9 f 0.4 5.0 f 0.3 6.1 f 0.8 5.0 f 0.3 5.8f 0.1

46 81 157 180 261 374 421 483 523 570 624

0.401 0.005 0.164 f 0.004 0.172 f 0.005 0.159 f 0.006 0.128f0.003 0.198 f 0.005 0.126 f 0.005 0.141f 0.009 0.224 f 0.010 0.177f 0.006 0.153f 0.008

0.398 f 0.002 0.171 f 0.003 0.169 f 0.003 0.156 f 0.002 0.124f0.006 0.186 f 0.007 0.111 f 0.005 0.127 f 0.003 0.205 f 0.008 0.163 f 0.007 0.147 f 0.005

0.385 f 0.010 0.170 f 0.005 0.160 f 0.003 0.149 f 0.003 0.108f0.004 0.165 f 0.006 0.103 f 0.003 0.114 f 0.009 0.178 f 0.005 0.129 f 0.004 0.112 f 0.003 Measured Full Width at Half-Maximum ( X W cm) 9.3 f 0.7 11.4 f 0.4 10.8 f 1.1 9.0 f 0.4 7.2 f 1.4 9.1 f 1.5 7.3 f 0.3 6.1 f 0.6 7.1 f 0.7 5.8f 0.6 6.5 f 0.7 6.2 f 0.5 5.5f 1.2 5.7 f 0.3 5.5 f 0.4 4.9f 0.5 4.6 f 0.5 4.7 f 0.2 5.0f 1.1 4.6 f 0.1 5.0 f 0.4 5.0 f 0.4 4.5f 1.3 4.3 f 0.2 4.2f 0.1 4.6 f 0.2 4.6 f 0.2 3.4f 0.2 4.0 f 0.2 3.8 f 0.2 4.1 f 0.3 5.0 f 0.2 5.0 f 0.1

1.5-

1 .o-

0.5-

.

0

0

100

200

400

300

meztxic field straqth Man) Figure 2. Observed resolution of four representative DNA fragments: (m) base 81, (A)base 180, (0)base 374, and (0)base 570 versus the electric field strength used durlngthe separation, Error bars reflect the standard deviatlon of the data.

Band-Spacing Effects. A theoretical equation defining terms of the fragment size of the DNA in nucleotide units, N, and the electric field, E, can be derived from the equation given in ref 11 for the mobility of DNA fragments in gel electrophoresis. This semiempirical equation, which describes the dependence of electrophoretic mobility on N and E over the range of DNA size and field strength used in this experiment, is presented below.11 Ap/p in

p

= poexp[- 1

1

+ ( UcN ~E)'/~N~/'

(5)

In this equation,mobility is related to fragment size and field

0.384 f 0.007 0.170 f 0.006 0.162 f 0.006 0.152 f 0.004 0.119fO.000 0.175 f 0.001 0.107 f 0.000 0.120 f 0.004 0.193f 0.010 0.145 f 0.002 0.127 f 0.004

0.380 f 0.008 0.164 f 0.007 0.151 f 0.004 0.141 f 0.007 0.099 f 0.002 0.155 f 0.005 0.091 f 0.001 0.103 f 0.002 0.162 f 0.008 0.115 f 0.006 0.095 f 0.003 9.3 f 1.5 8.3 f 0.8 7.0 f 0.8 6.4 f 0.9 5.7 f 0.6 4.9 f 0.6 4.6 f 0.2 4.4 f 0.2 3.9 f 0.1 4.3 f 0.1 4.4 f 0.2

0.365 f 0.005 0.164 f 0.013 0.150 f 0.011 0.140 f 0.004 0.099 f 0.008 0.148 f 0.004 0.083 f 0.003 0.090 f 0.003 0.144 f 0.005 0.076 f 0.006 0.086 f 0.004 11.9 f 0.6 10.4 f 0.3 7.6 f 0.4 8.0 f 0.5 6.5 f 0.3 6.7 f 0.3 6.7 f 1.0 7.3 f 0.3 6.0 f 0.0 8.1 f 0.5 6.7 f 0.7

strength by three constants: al, c, and PO. The constant p0 is the free solution electrophoretic mobility of single-stranded DNA, equal to 1.22 X 10-4 cm2/V.s, as obtained from the y-intercepts of the predicted mobility curves shown in ref 11. This value is in close agreement to the free solution mobilities obtained by Olivera and co-workers for denatured DNA at high ionic strengths (approximately 1.3 X 10-4 cm2/V.s).15 The remaining constants u1 and c are functions of the gel concentration which are calculated from the data in ref 11 to be 2.76 X 10-20 cm/V and 4.25 X 103,respectively. The behavior of eq 5 is illustrated in Figure 3, where calculated mobility curves are superimposed on observed mobilities. For low electric fields and small fragment sizes, mobility decreases exponentially with increasing fragment size in a manner similar to that described by the Ogston sieving model.16 Increasing the electric field results in mobilities that show positive deviations from the zero-field limit for large DNA fragments. As the electric field strength increases, the mobility difference between adjacent DNA fragments decreases. The effect is most apparent for the longer DNA fragments. Eventually, the mobilities of the larger fragments will reach a saturation value, at which point electrophoresis can no longer resolve adjacent fragments.l'J8 Clearly, the closer one operates electrophoresis to these saturating conditions, the more difficult it will be to resolve adjacent DNA bands. To obtain an expression for Aplp, eq 5 is differentiated with respect to fragment size. Dividing the result by the (15)Olivera,B. M.;Baine, P.; Davidson, N. Biopolymers 1964,2,245257. (16)Rodbard, D.; Chrambach, A. R o c . Natl. Acad. Sci. U.S.A. 1970, 65,970-977. (17)Slater, G.W.;Rousseau, J.; Noolandi, J.; Turmel, C.; Ldande, M. Biopolymers 1988,27, 609-524. (18)Noolandi, J. Adu. Electrophor. 1992,5 , 1-57.

2844


- 9 . 42

W/cm°C), and R, is the resistance per unit length of gel (29 Mnlcm). Combining eq 7 and 8 results in a total peak variance expression that depends only on known constants, the electrophoretic mobility, the applied electric field, and the transverse and longitudinal diffusion coefficients.

h

-9,s -10.0

-10.4 -10.6

-10.8 100

200

300 400 500 Fragment Size

600

700

Flgure 3. Logarithmic plot of the measured fragment mobilities (in cm2/s)versus fragment length at five applied electric fields: (M) 50, (A)100, (0)150, (0)200, and (0) 400 V/cm. The curves shown are generated from eq 5 and are in close agreement with the observed moblllties. At zero applled flekl, eq 5 predicts that the fragment mobilities drop to the value specified by the Ogston extrapolation shown. (Data shown are obtained from ref 11.).

mobility equation yields the following expression for the relative mobility difference:

&= 1.1

~cN~/'(~,E)~/~

+

C

2(1 N3/2(a1E)1/4)2 - 1+ p / 2 ( a 1 E ) 1 / 4

(6)

Equation 6 thus expresses the relative mobility difference in terms of fragment size, electric field strength, and known constants. Band-BroadeningEffects. An equation relating the total peak variance, UT^, to electric field strength and fragment size is now needed to complete the analysis of resolution. If the treatment of band broadening used in ref 10 is applied, we can express the total peak variance of a DNA band as the sum of four independent variances: 2 UT

2 = (I.1EInjtIllj)

+

12

w" ; 2 D/IL ; R2EQ2(TW- T J 2I.~L 4

I.1E

96D,

(7)

where the four terms on the right-hand side refer to the finite injection volume, finite detection volume, diffusion, and thermal gradient variances, respectively. In the above equation, pc~~nj, Ec~~nj, and thj refer to the mobility, field strength, and injection time used during the injection process, w is the width of the focused laser on the capillary (0.005 cm), L is the length to detector (40cm),R is the capillary gel radius (0.0025 cm), n is the temperature coefficient for DNA mobility (approximately 0.02 OC-1 lOJQ), Dll is the longitudinal diffusion coefficient for diffusion in the direction of migration, D, is the transverse diffusion coefficient for diffusion of the DNA perpendicular to the field direction, and T , - T, is the temperature difference between the inner wall and center of the gel-filled capillary. The temperature gradient needed in the thermal gradient term can be calculated by solvingthe heat conduction equation under appropriate boundary conditions.10 This yields a thermal gradient which is equal to the following:

Tw-T , = HR2 -= E2 4k,

4sk&

where H is the power per unit volume of the capillary (in W/cm3), k, is the thermal conductivity of the gel (0.0057 (19) Grossman, P. D. In Capillary Electrophoresis: Theory and Practice; Grossman, P. D., Colburn, J. C., Eds.; Academic Press: San Diego, CA, 1992.

The necessity of two diffusion coefficients to specify eq 7 and 9 is a result of electric field effects which not only increase the mobility with larger field strengths as noted in the previous section but also increase the diffusion coefficientsabove those measured in the absence of a field.10 This forced or kinetic diffusion results in a Dll that is several times greater than DO, the diffusion coefficientmeasured in the absence of an applied field.10 Similarly, diffusion perpendicular to the direction of migration, DI, is likely to differ from the zero-field value. Presumably, this behavior reflects a change in DNA conformation, namely, an orientation or stretching of the DNA in the direction of the field. Evidence for this deformation during gel electrophoresis has been reported in experiments using electric birefringence,2b22linear d i c h r o i ~ m , ~and 3*~~ fluorescence visualization26926 of migrating DNA. From eq 9, it is clear that only the longitudinal diffusion and thermal gradient broadening terms will vary with the electric field used during the separation. Furthermore, an expression for mobility is already available, which leaves only the diffusion coefficients Dll and DL to be defined. Longitudinal Diffusion. In the limit of low electric field, the diffusion-broadening term in eq 9 will dominate the total peak variance. Thus, the peak variance should vary linearly with the reciprocal of electric field strength in this limit, provided the ratio of the longitudinal diffusion coefficient to the mobility of DNA is independent of the electric field strength. This latter condition is satisfied since diffusion and mobility both can be defined using the same frictional coefficient:27128 DIl = kTlf

(10)

1.1 = Q X / f (11) where k is the Boltzmann constant, T is the absolute temperature, Q is the total charge on the migrating particle, X is a factor that accounts for the partial screening of charges by counterions in the solvent, and f is the frictional coefficient. Thus

Dll/1.1=k T / QX

(12)

which will be independent of electric field strength provided the screening factor X is not field dependent. The prediction that peak variance should vary linearly with 1/Eat low fields is experimentally verified in the graph shown in Figure 4. Although interpretations made from this plot should not be overemphasized because of the large error associated with the peak variances (standard deviations are twice that of the half-width measurements), the linearity of (20) Chu, B.; Xu, R.; Wang, Z. Biopolymers 1988, 27, 2005-2009. (21) Lanan, M.; Shick, R.; Morris, M. Biopolymers 1991, 31, 10951104. (22) Chu, B.; Wang, Z.; Xu, R.; Lalande, M. Biopolymers 1990, 29, 737-750. (23) Jonsson, M.; Akerman, B.; Norden, B.Biopolymers 1988,27,381414. (24) Akerman, B.; Jonsson, M.; Norden, B. J . Chem. SOC.,Chem. Commun. 1985,422-423. (25) Schwartz, D. C.; Koval, M. Nature 1989,338,520-522. (26) Smith, S. B.; Aldridge, P. K., Callis, J. B. Science 1989,243,203206. (27) Ornstein, L. Ann. N.Y. Acad. Sci. 1964,121, 321-349. (28) Cussler, E. L. Diffusion: Mass Transfer in Fluid S y s t e m ; Cambridge Univesity Press: Cambridge, UK, 1984.


2845

-

3 . 0 ~ 1 0 ~

N-

2.5-

&

Y

B

2.01.51 .o-

0.5-

0.04, 0

I

I

I

I

5

10

15

20x1O3

l/Electric Field

I

Table 11. Comparison of Observed and Calculated q l / p Ratios base 46 81 157 180 261 374 421 483 523 570 624

obs 1.81 X 1.62 X 7.97 x 5.85 X 3.15 X 1.83 X 1.90 x 4.32 X 2.90 X 2.01 x 2.22 x

10-9 103 10-4 10-4 10-4 10-4 10-4 106 10-4 10-4 10-4

(V) calc 2.12 x 1.24 X 6.75 X 5.99 x 4.37 x 3.29 X 3.01 X 2.72 X 2.57 X 2.42 X 2.28 X

103 103 10-4 10-4 1w 10-4 10-4 10-4 10-4 10-4 10-4

10

15x1 0 "

l/Fragmnt Size

Figure 4. Measured peak variances of four DNA fragments versus the reclprocal of the applied electric field. According to eq 9, at low electric fields (large 114, dlffusionai broadenlngIs predlctedto dominate the peak variance. This results in a peak variance that varies linearly with 1/€In this region. The observeddata clearly support this predlction. The slopes of these lines can be used to obtain ratios of the longitudinal diffusion coefflclent to mobility as descrlbed in the text.

Diilu

5

(m

7% diff -15 31 18 -2.3 -28 -44 -37 -84 13 -17 -2.6

the plots at low electric field is excellent. Diffusion to mobility ratios ( D l ~ / pobtained ) from the slopes of these lines are shown in Table 11. Since an expression for mobility is known, knowledge of these diffusion to mobility ratios is sufficient to define Dll. Alternatively, diffusion to mobility ratios may be obtained by measuring diffusion coefficients at zero field and dividing these values by the extrapolated values for the mobility at zero field obtained from eq 5.lO These diffusion to mobility ratios are also shown in Table I1 for comparison. Except for an anomalous value for the 483-nucleotidefragment,the ratios are reasonably similar with an overall relative difference of 15% ' , Since the experimentally obtained ratios rely on the measured peak variances, which are prone to errors, the diffusion to mobility ratios obtained from the zero-field diffusion coefficients and eq 5 were used below. To obtain the fragment size dependence of Dl~lp,an expression relating it to the size of the DNA fragments is needed. This relationship is obtained from Figure 5, in which Dil/p is plotted versus reciprocal fragment size in nucleotide units. The data for these plots were obtained from ref 10. The resultant plot is fairly linear considering the large relative error associated with the values for the longest fragments. An empirical equation that fits these data well is given below:

D l l / ~= A1 + Bi/N (13) where A1 and B1 are 7.789 X 1od and 9.380 X le2V, respectively. Transverse Diffusion. For the transverse diffusion coefficient, D I,a theoretical relationship to independently

Figure 5. Ratio of statlc diffusion to predicted mobility at zero-fleld strength ( P / p 0 )versus reclprocai fragment length. The line shown isthe best linear fittothedatatakingintoaccountthestandarddeviations of the data.

Table 111. Comparison of DI with Static Diffusion Coefficients base D J p obs (V) est D I (cm2/s) Do (cm2/s) 46 81 157 180 261 374 421 483 523 570 624

1.70 X 2.01 x 5.42 X 3.85 X 8.52 X 5.38 X 5.17 X 3.46 X 6.01 X 2.29 X 4.21 X

10-6 10-6 10-6 10-6 10-6 10-6 10-6 10-6 10-6 10-6 10-6

1.72 X 1.77 X 3.63 X 2.39 X 4.17 X 2.03 X 1.79 X 1.08 X 1.78 X 6.42 X 1.12 x

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measured parameters is lacking. One is therefore restricted to obtaining values for D I from the measured peak variances in the high-field limit, where the thermal-broadening term is dominant. Again, an assumption must be made concerning the ratio of the transverse diffusion coefficient to the mobility in order to proceed. Unlike the case for Dll, there is no theoretical basis to assert that D J p will be independent of the applied field. On the contrary, it is likely that the ratio of the transverse diffusion coefficient to the mobility will decrease with increasing field. This is because one expects that diffusion perpendicular to the direction of migration will decrease the more the DNA is stretched or oriented by the action of the field. In the absence of a theoretical framework with which to assign a field dependence for this ratio, we have chosen to assume that the field dependence of this ratio will be negligible compared to the E5 dependence of the remainder of the thermal gradient term. This allows an estimate of D I / p to be made from a linear plot of the total peak variance minus the diffusion variance versus E5. The resultant values for D J p are given in Table 111. No clear trend is apparent for the transverse diffusion to mobility ratios. Accordingly, the value for this ratio for all N was taken as the average for subsequent analysis. D,/p = 4.36 X

lo4 V

(14) As the values for p are known from eq 5, it is now possible to estimate the values of the transverse diffusion coefficients. These results are presented in Table I11and shown graphically in Figure 6 along with the diffusion coefficients measured in the absence of a field. Interestingly, the values for the transverse diffusion do not vary dramatically with fragment size. However, the values do seem reasonable considering the DNA orientation that occurs during gel electrophoresis.

2848


Fragment

j

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Size

Static Diffusion

46

-400.0

-

421 523 624

100

200

300

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400

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Relative magnitudes of the measured static diffusion coefficients and the predicted transverse dlff usion coefficients at 200 V k m . Surprisingly,the magnitudeof the transverse diffusion does not vary greatly over the size range of DNA fragments shown.

C. Predicted Resolution Behavior. At this point eq 6, 9,13, and 14 may be combined with eq 4 to yield an expression for the resolution in terms of known parameters.

.(

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~cN~/~(u,E?/~ c 2(1 N3/2(a,E)'/4)2- 1 N3/2(alE)'/4

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Electric Field (V/cm) Predicted effect of band spacing on resolution for six DNA fragments ranging in size from 46 to 624 nucleotides In length. Flgure 7.

Figure 8.

=

0

)'I2(15)

This expression for resolution consists of five components: band spacing, finite injection volume, finite detection volume, diffusion, and thermal gradient broadening, which are all expressed in terms of fragment size, electric field strength, and known constants. Since both injection and detection variances are independent of the applied field strength, the field dependence of the denominator is entirely determined by longitudinal diffusion and thermal gradient broadening. We thus have only three field-dependent factors in eq 15: band-spacing effects, diffusion, and thermal gradient broadening. To obtain a clear picture of the importance of these effects in determining how the resolution changes with field strengths, these three terms were considered successively. Band Spacing. Since resolution is predicted to be proportional to the relative mobility difference, plots based on eq 6 of the relative mobility difference versus field strength for various fragment sizes should allow a qualitative understanding of these band-spacing effects on resolution. Such plots are shown in Figure 7. The relative mobility curves in Figure 7 clearly show maxima at zero applied field for all DNA fragments, illustrating that the physical spacing between adjacent bands decreases as the applied electric field increases. This change in spacing is most noticeable for the longest DNA fragments. For a fragment that is 610 nucleotides in length, the relative mobility difference drops by roughly a factor of 2 when the field is increased from 50 to 400 V/cm, while it does not decrease observably for a 46-nucleotide fragment. Thus band spacing generally decreases with increasing DNA fragment size and increasing field strength. This is one of the fundamental reasons why longer DNA fragments are more poorly resolved when the field is increased. Band Broadening: Diffusion. If the effect of longitudinal diffusion is included with the band-spacing effects, the theoretical resolution plots shown in Figure 8 are obtained. These calculated curves also include the contributions from

the finite injection and detection volumes. The addition of these band-broadening effects has a dramatic influence on the predicted resolution curves. Even though band-spacing effects decrease resolution at higher fields, as observed in Figure 7, band broadening from diffusion is so much less a t higher fields that the predicted resolution actually increases with field strength. Furthermore, resolution now is not highest for the shortest DNA fragments. Rather, predicted resolutions that include diffusional broadening are highest for DNA fragments that are around 200 nucleotides in length. Resolution decays sharply for fragments smaller than 200 nucleotides in length and more gradually for longer fragments. This is in agreement with the observed data. However, these plots show no significant decrease in resolution as the electric field is increased further to 500 V/cm. This behavior is inconsistent with the observed data, which indicate optimum electric field strengths between 100 and 300 V/cm. Clearly, the thermal gradient term must be included in this analysis to provide a correct description of the behavior. Band Broadening: Thermal Gradient. Adding the thermal gradient-broadening term in the analysis yields theoretical resolution plots which are shown in Figure 9. Comparing these plots to those in Figure 8, it is observed that the addition of a thermal gradient-broadening term leads to a pronounced drop in resolution a t large DNA sizes and at high electric fields. This behavior results in resolutions that peak a t intermediate field strengths. For fragments that are roughly 200 nucleotides in length, the resolution is highest at field strengths that are roughly 250 V/cm. The optimum field strength to resolve longer DNA fragments drops from 250 V/cm for 200-nucleotide fragments to 100 V/cm for 800nucleotide fragments (data not shown). This behavior is in good agreement with the observed results, which show that better resolution of long DNA fragments is obtained at fields that are less than 200 V/cm. In examining Figure 9, it is clear that the agreement of the theoretical resolution plots to the observed data is best at low and high electric fields. The theoretical curves overestimate the observed resolution for intermediate field strengths between 150 and 300 V/cm by approximately 15% compared to an overall agreement within 5 ?6 . This may be due to uncertainty in the field dependence of the thermal gradient variance. It is more likely, however, that the discrepancy indicates the presence of a third field-dependent peak variance. This variance could take the form of a coupling between the diffusional and thermal gradient variances analogous to eddy diffusion in packed-column chromatog-

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2847

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Flgurr 8. Predicted versus observed resolution curves (y-axls,resolution; x-axis, electric field strength (V/cm)). Predicted curves were generated from eq 15 in the text, assuming no thermal gradient broadening.

raphy,2Qan adsorption/desorption mechanism30that accounts for transient trapping of the DNA on the gel fibers, or some other broadening mechanism. The inclusion of such a term allows a more accurate fit to the observed data in the intermediate range of field strengths (not shown). However, even without this added term, the similarity in shape and magnitude of the resolution plots is encouraging and suggests that the resolution of DNA fragment separations is primarily determined by the broadening effects outlined in this work. Furthermore the estimated value of the transverse diffusion coefficient is reasonable when compared to the diffusion coefficients measured a t zero electric field. Future work will focus on means to analyze and understand the fragment size and electric field dependences of the tranverse diffusion coefficient. An alternative means of viewing the behavior of eq 15 is presented in Figure 10. In this figure, the theoretical resolution obtained from eq 15 is plotted versus fragment size and applied electric field, yielding a resolution surface. Whereas Figure 9 presented slices of the predicted resolution surface for various DNA fragment sizes, Figure 10 displays the complete behavior of our predicted resolution equation. The general shape of this surface reveals a well-defined resolution maximum at roughly 20 nucleotides and 250 V/cm. The resolution drops off sharply from this maximum as the fragment size decreases and shows a more gradual decrease (29) Giddings, J. C.Dynamics of Chromatography; Marcel Dekker: New York, 1965. (30) Liu,J.; Dolnik, V.; Hsieh, Y.-Z.; Novotny, M. Anal. Chem. 1992, 64, 1328-1336.

as the fragment size increases. Furthermore, the field a t which the resolution is maximum decreases steadily as the fragment size increases. The downward slopes that occur as the field increases are due entirely to the thermal gradient term discussed above. Figure 10 clearly shows that the optimum regime for obtaining high-resolution separations for this system is at intermediate applied fields (150-250V/cm) and at intermediate fragment lengths (100-300nucleotides). It also provides an informative picture as to how the resolution changes with both fragment size and applied electric field.

DISCUSSION The observed data in conjunction with the theoretically derived resolution plots clearly demonstrate that increasing the applied electric field does not always increase the resolution that is obtained as would be predicted for a purely diffusion-limited separation. The critical factor that limits the optimum electric field is not diffusion or relative mobility changes, as shown in Figure 8, but the thermal gradient variance, which varies strongly with the electric field strength. In its absence, the resolution would increase with increasing electric field strengths and only be limited by the decreasing mobility difference between adjacent DNA fragments at high fields. The importance of thermal broadening indicates that thin gels, which minimize the effect of the thermal gradient term, offer a clear advantage in resolving power, allowing long reads to be performed at high electric fields. Because of thermal gradient broadening, gels must be operated with the realization that there exists a tradeoff in using very large field strengths. While the sample will elute

2848


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Predicted versus observed resolution curves including all four peak variancesand band-spacingeffects (y-axis, resolution: x-axis, electric field strength). The addition of the thermal gradient-broadeningleads to a pronounced drop in the resolution at high fields. Predicted resolution curves overestimatethe observed resolutions at intermediate field strengths. This is probably evidence for another broadening term that depends on a low power of the electric field strength. Flgure 9.

in much less time at higher fields, the length of read will decreasedramatically,because the increased thermal gradient across the gel greatly increases the band broadening for the larger DNA fragments. However, if the length of read does not have to be much greater that 200 bases, then the use of very high fields is justified. If longer reads are desired, then the field strength must be decreased, involving significant increases in electrophoresis time. This tradeoff between speed and length of read is evident in the data shown in Figure 11A,B. Figure 11A shows fourdye sequencedata which were collected at 400 V/cm. Roughly 650 bases of information are displayed. The primer peak eluted in 15 min while the retention time of the pentet C (blue trace) sequence at 665 bases was 55 min. The pentet T (red trace) at base position 317 is also clearly seen at the beginning of the fourth line of sequence. For comparison, sequence data that were collected at 100 V/cm are shown in Figure 11B. An identical region of data is shown, namely, from the primer peak to the pentet C at 665 bases. In this run, however, the retention time of the primer was 60 min and the pentet C eluted at 289 min. Comparing the relative position of the pentet T in Figure 11A and B, one finds that the pentet T is shifted toward the primer at the lower electric field, appearing at the end of the third line of sequence. This is a result of the larger mobility difference between adjacent peaks for the long DNA fragments at lower fields (refer to Figures 3 and 7). This increase in mobility difference and the reduced thermal gradient at low electric field translate into better resolution for the long DNA fragments and poorer

Fragment Size

8 O z v /

1000

Three-dimensionalrepresentationof thepredicted resdutkn equation (eq 15 in the text). Resolution is plotted versus fragment size and electric field strength. The curve superimposed on the surface denotes the maximum resolution for each fragment size and clearly shows the gradual decrease in the optimum field strengthas thefragment size increases. Flgure 10.

resolution for the short fragments. However, the cost in achieving this better resolution for the long DNA fragments was a factor of 5 longer separation time. Another method to increase the resolution and length of read in DNA sequencing is to increase the length to the detector while maintaining a constant applied electric field. The effect of increasing the length to the detector is rather


+--Primer

A 22.328.9 min

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2849

2850


improved by operating at a low field. However, most sequencing projects are less interested in resolution optimization than in overall throughput. To address this issue, it is necessary to determine the field strength at which capillary gel electrophoresis is most efficient (Le., provides the most sequence information in the least amount of time). This field strength can be determined by weighing the expected gain in resolution against the added cost in electrophoresis time. A useful parameter that weighs these considerations is the theoretical plate number, NTH,of the separation. The definition of NTHis given below: where L is the length to the detector and aT2 is the total variance of the peak. The larger the value of NTH,the more efficient the separation is. For a constant length to detector, the optimum separation conditions will be those that minimize the peak variance. The electric field that minimizes this variance can be determined by examining Figure 1, which shows plots of the full widths at half-maximum for five DNA fragments. From these plots, it is evident that the peak halfwidths show minima which all center around 200 V/cm. This is a rather interesting result in that it suggests that the use of electric fields lower than 200 V/cm to resolve larger DNA fragments may not be worth the added cost in electrophoresis time. CONCLUSIONS The effect the applied electric field has on the quality of the separation has been addressed in this work. It has been shown that in capillary gel electrophoresis the resolution of the separation is not limited solely by longitudinal broadening, but by a combination of longitudinal diffusion, thermal gradient-broadening, and band-spacing effects. This conclusion supports the previous work of Kambara3l and Grossman,lg who studied resolution in slab and capillary gel electrophoresis, respectively. In this study, diffusional broadening dominates at electric fields less than 200 V/cm, while the thermal gradient effect dominates a t higher fields. Furthermore the field effects on band spacing, although important, were not the primary reason for the decrease in resolution at high fields. Rather, the critical factor is the thermal gradient broadening. This is in contrast to resolution studies in open tubular capillary electrophoresis where the limiting factors were injection and longitudinal diffusion variances.3The importance of the thermal gradientbroadening term in CGE stems from its strong dependence on the electric field strength and the much smaller transverse diffusion coefficients of DNA in a gel matrix, resulting in maximal resolutions that occur a t intermediate field strengths. The results of this work also extend the band-broadening theory derived in a previous paperlo to CGE separations at much higher applied fields. In that study, band broadening was found to be dominated by injection and diffusional effects. Although this appears to conflict with the importance of the (31) Niehikawa, T.; Kambara, H. Electrophoresis 1991, 12, 623-631. (32) Cheng, Y.F.;Wu, S.;Chen, D. Y.; Dovichi, N. J. Anal. Chem. 1990,62,496-503. (33) Delinger, S. L.;Davis, J. M. Anal. Chem. 1992, 64, 1947-1959. (34) Huang, X.;Coleman,W. F.; Zare, R. N. J.Chromatogr. 1989,480, 95-110. (35) Tanaka, T.; Hocker, L.; Benedek, G. J. Chem. Phys. 1973, 59, 5151-5159.

thermal gradient term observed here, the difference is because the field strength used in that study was low enough (150 V/cm) that thermal gradient effects were negligible. Theoretical resolution curves were generated using equations derived in two previous papers. In the absence of a means to measure the diffusion coefficients of DNA perpendicular to the applied field, estimates were made based on the measured peak variances. The magnitude of these transverse diffusion coefficients were found to be between 1 X 10-10 and 5 X 10-10 cm2/s, roughly 10-1000 times smaller than the diffusion coefficients measured in the absence of a field. These values are reasonable considering the fact that the DNA becomes oriented in the direction of the applied field during electrophoresis and is thus expected to diffuse more slowly perpendicular to the field. The relative insensitivity of this transverse diffusion to the length of the DNA fragment is surprising nonetheless. One possible explanation is that this diffusion may be determined by the collective diffusion of the gel matrix35 rather than by the size of the DNA fragment. Finally, the finding that better resolution is obtained for longer DNA fragments a t low electric fields is tempered by the consideration that much longer electrophoresis times are needed. The electric field that affords the best resolution in the least amount of time is determined primarily by the field at which the peak variance is at a minimum. Although knowledge of the exact value of the transverse diffusion coefficient is lacking, an estimate of this optimum field can be made from plots of the peak widths at varying fields. From these plots, it is observed that the peak variances for all DNA fragments studied are minimum at roughly 200 V/cm. This result indicates that for the conditions used in this experiment (Tris/phosphate/EDTA buffer, 50-pm-i.d. capillaries, 4 7% T/5 % C polyacrylamide gel), capillary gel electrophoresis is most efficient at this intermediate field strength. Different buffer compositions, gel thicknesses, and concentrations will likely shift this optimum field, but the underlying competition between diffusion and thermal gradient broadening will always favor an intermediate operating field strength, since neither process can be entirely eliminated in gel electrophoresis. Improvement in the performance of capillary gel electrophoresis clearly must now focus on minimizing the thermal gradient contribution to band broadening. To this end, it is predicted that smaller bore capillaries and less conductive buffers will allow the application of much higher electric fields without sacrificing resolution of very large DNA fragments. ACKNOWLEDGMENT The authors thank Michael Giddings for his help in the preparation of the four color plots shown in Figure 11. These plots were generated using his base-calling program (Nucleic Acids Res., submitted for publication). John Luckey is an NIH predoctoral trainee, Biotechnology Training Grant GMO8349-03. This work was supported by NIH Genome Grant HG00321. RECEIVEDfor review April 12, 1993. Accepted July 14, 1993." @

Abstract published in Advance ACS Abstracts, September 1,1993.

Optimization of electric field strength for DNA sequencing in capillary

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