Effects of Na' on the Persistence Length and Excluded Volume of T7 Bacteriophage D N A * E. S. SOBEL+ and J.A. HARPST* Department of Biochemistry, Case Western Reserve University, Cleveland, Ohio 44106-4935
SYNOPSIS
Total intensity, Rayleigh light scattering has been used to measure the rms radius, second virial coefficient, persistence length, and excluded volume of homogeneous T7 bacteriophage DNA as a function of Na+ concentration (0.005 to 3.0 M). All parameters decrease sharply
as "a+] increases, and tend to level off at high Na'. The variation of persistence length with [Na+] is consistent with predictions from counterion condensation theory.
INTRODUCTION Understanding the flexibility of DNA has been the objective of numerous studies, almost since discovery of the double-helical structure. In recent years electrostatic contributions to th e persistence length, a direct measure of DNA flexibility, have undergone active experimental and theoretical investigat i ~ n . ' - ' ~I n this article, we report our results on homogeneous, native T 7 DNA (molecular weight, M = 26.43 X lo6)', from low-angle, total-intensity (Rayleigh) light scattering. A major advantage of this work is t hat the large size of T 7 DNA allows the wormlike coil equations of Sharp and B l ~ o m f i e l dt o~ be ~ fit directly to the data in order to obtain the persistence length a and the excluded volume parameter We have thus avoided some of the limitations in interpretation of earlier lightscattering studies on low molecular weight DNA, 4 ~ 1 1 ~ 1which 7 necessitated making estimates of t from other theoretical consideration^.^^"*^^-^^^^^^^^ t.25926
Biopolymers, Vol. 31, 1559-1564 (1991) 0 1991 John Wiley & Sons, Inc. CCC 0006-3525/91/131559-06$04.00
* We are honored to dedicate this work to Dr. Bruno H. Zimm on the occasion of his retirement from the University of California, San Diego. Present address: Department of Medicine, The University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-7280. To whom correspondence should be addressed.
'
*
EXPERIMENTAL T 7 DNA was isolated from coliphage T 7 by standard techniques2s and purified by phenol e x t r a c t i ~ n . ~ ~ T h e coliphage T 7 was obtained from bacteriophage stocks shown to be the wild type by S t ~ d i e r , ~and ' the DNA was later shown by sequencing24to have a molecular weight of 26.43 X Low-angle total-intensity light-scattering measurements were made at 20 k 2°C with a substantially redesigned version of a n earlier i n ~ t r u m e n t . ~ ~ Modifications included a 3 mW helium-neon laser light source and a light-scattering cell, especially designed with a closed, flow-through filtration system. This instrumentation made possible reproducible and continuous measurements over a n angular range of 4'-135". Calibration was performed by previously described methods, 31 except that it was based on the established molecular weightn4of T7 DNA (26.43 X lo6).Further details will be presented elsewhere. T 7 DNA solutions were prepared at eight different ionic strengths from 0.008 to 3M, with Na' as the only cation. T h e basic solvent, BEP, was a sodium phosphate-sodium EDTA (ethylenediaminetetraacetate ) buffer ( 0.00100M NanEDTA, 0.000187M NaHnPO4* H2 0 , and 0.00141 M NanHPO,, p H 6.92). The ionic strength of B EP was 0.008M a n d total N a t concentration = 0.005M. 1559
1560
SOBEL AND HARPST
1.2E-06
I
I
I
I
0.3
0.4
1E-06
8E-07
0
6E-07
B 4E-07
2E-07
a
33734
0
'3.1
0.2
0.5
n
S i r k 2 + 4000c Figure 1. Reciprocal-intensityplot (4"-60") of lightscattering data for T7 DNA in 0.005M Na'. RB is the Rayleigh ratio34at angle 8 . Data are shown ( 0 ) at six concentrations from 9.9 to 58.4 pg/mL; some points at low angles are omitted for clarity. The lines are linear least-squareextrapolations to zero concentration (0)and to zero angle ( * ) from the 4" to 9" region. Above 9" the curves at constant concentration are point-to-point fits
to the data. Table I "a+]
(M) 0.005 0.01 0.02 0.10 0.2Od 0.50 1.00
3.00
Solvents of higher ionic strength were prepared by adding appropriate amounts of NaCl and NaOH to obtain the desired total Na+ concentration at pH 6.92. Above 0.1M N a + , the ionic strength is within 1%of the Na+ concentration. For each light-scattering series at a given ionic strength, five or six DNA solutions were prepared separately and extensively dialyzed against the appropriate buffer. At each ionic strength, two sets of measurements a t concentrations from 10 to 60 pg/ mL were made with DNA from different stock preparations. The one exception was in 0.2M Na+, for which results have been included from seven sets of measurements, one in BEP and the others in a similar buffer, BPES," with a higher phosphate concentration. Clarification of solutions was accomplished by filtration3' through MF-Millipore filters (Type GSWP, 0.22-pm pore size) directly into the flow-through cell, to minimize turbulence and handling. Appropriate sedimentation and viscosity measurements, performed as described previo ~ s l y ,indicated ~ ~ , ~ ~the DNA was intact, homogeneous, and unaffected by filtration. Scattering data were initially displayed and evaluated with standard reciprocal-intensity plots, one of which is shown in Figure 1 for T7 DNA in 0.005 M Na+. Extrapolations to zero concentration ( c = 0) and zero scattering angle ( 0 = 0) were done with a linear least-squares executed by computer. Difficulties with the extrapolations to zero angle have been discussed p r e v i o ~ s l y . ~For ~~~~'~~~~ display purposes linear extrapolations from the 4' to 9" angular range have been made (Figure I ) and also used to establish the lines at 0 = 0, from which the values of second virial coefficients A' (Table 1) were obtained.34 These values of A2 were the same
Summary of Light-Scattering Results on T7 DNA
y'z
(R (nm)
A* x lo4 (mole cm3/g2)
a (nm)
c
700 -+ 30" 670 k 20 590 3- 20 550 k 10 530 k 20 490k 7 484 r+ 8 460 Y 6
18 k 4 ' 13 zk 2 9 +-I 4 f 1 3.1 2 0.5 2.1 k 0.8 1.8 k 0.7 1.6 f 0.8
78 k 2b 74 f 2 68 k 2 55 k 2 48 k 2 48 f 1 45 k 2 38 ? 1
0.14 f O.Olb 0.12 0.01 0.08 f 0.01 0.05 0.01 0.071 2 .005 0.054 f .005 0.049 k .007 0.053 f .005
-
* *
Averages of least-squares fit to Debye equation (see text) weighted by standard deviations from individual P-' (0) curves. Averages of least-squares fit of P-' (8) curve to equations of Sharp and Bloomfield weighted by standard deviations from individual P-' (8) curves. Weighted averages of linear least-squares extrapolation to 0 = 0 over 4"-9" angular region. Weighted average of six determinations in BPES and one in low phosphate buffer BEP (see text). a
EFFECTS OF Na'
1561
glement is expected to begin. Values of c * obtained ) "a+] = 0.005M from the rms radius ( ( R 2 ) 1 ' 2 at (Table I ) are 128 pg/mL from a suggestion by d e G e n n e ~ , 23 ~ ~ pg/mL ,~~ from the more stringent requirement for hexagonally close-packed s p h e r e with ~ ~ the ~ ~ above ~ ~ radius, or 16 pg/mL for the same spheres, arranged at the corners of a cube and just touching each other. For polyelectrolytes in monovalent salt, c* = 49 pg/mL from an expression by Odijk.2,20,39 Since the concentrations of our measurements bracket or fall below all these estimates of c * and the data are linear over a sixfold concentration range, we conclude our extrapolations to c = 0 are valid and provide reliable values of the scattering functions, P-' ( 0 ) . The linear angular range is limited to values of P-' (19)< 1.33,3,22p33 which restricts the linear region in these experiments to < 9" in 0.2M and < 6" in 0.005M Na+. At the lowest ionic strengths we have too few data points for a reliable linear extrapolation. Eisenberg and his collaborators have pointed out that the useful range of angles can be e~tended".'~ by applying the Debye e q ~ a t i o n , ~ ' 0.0
0.34
2
+
P ( 8 ) = 7(e-" x - 1 1 , Figure 2. Scattering curves at zero concentration for native T7 DNA at different ionic strengths. The ordinate is K c / R, = [ P - ' ( 0 ) 1/ M , where K is an optical constant.34 Data points ( 0 )are shown between 10' and 70'. Solid lines are the best-fitting theoretical curves, described in the text. Na+ molarity is indicated by the numbers next to each curve. The data at 3.OM Na+ are scaled as shown; each line at successively lower ionic strengths is displaced upward by 1 X
within experimental error as those derived from true extrapolations to 8 = 0, discussed in the following section.
RESULTS AND DISCUSSION For this investigation it is most important to obtain accurate determinations of the intramolecular scattering functions P-' ( B ) at each ionic strength. These curves (Figure 2 ) have been derived from linear extrapolations to c = 0 in the reciprocalintensity plots described above. There has been some concern as to whether light-scattering measurements on a large DNA such as T7 can be made at concentrations low enough to avoid problems due to coil overlap." Several estimates are available for the overlap concentration c*, at which coil entan3925,26,33*34
X
(1)
where
Here no is the refractive index of solvent, A 0 is the wavelength of light in vacuo, and ( R 2 )is the meansquare radius. In our experiments the Debye equation is valid for angles at or below 12" in 0.2M and 9" in 0.005M N a + . Data from these applicable angular ranges are fit to eq. ( 1 ) by standard statistical methods35to provide reliable, limiting values of (R 2 ) (Table I ) . All indicated errors are standard deviations obtained from least-squares analysis. A summary of our results is given in Table I. The parameters shown are from the first systematic study of ionic-strength effects on the properties of T7 DNA, determined by light-scattering measurements that extend into the limiting angular region of the scattering curve. Each entry in the table is an average of the two sets of measurements (seven in 0.2M Na+) ,weighted35by the standard deviations from analyses of individual data sets. The rms radii (Table I ) , obtained from eq. ( 1) ,decrease smoothly as ionic strength increases, and appear to approach
1562
SOBEL AND HARPST
a plateau above 0.5M Na'. The second virial coefficients also decrease in a smooth manner with increasing ionic strength. This behavior is consistent with theoretical expectations and with other experimental data, 4 ~ 1 7including recent results from small DNA fragments.41 Persistence lengths and excluded volume parameters are listed in columns 4 and 5 of Table I. These values were obtained by fitting the light-scattering equations of Sharp and BloomfieldZ5to experimental P-'(O) curves over the angular region of 4"-70". The approach described previously3 was modified to conduct the curve fitting by computer. The program utilized the iterative method of Marquardt to minimize deviations between the theoretical curves and experimental points, weighted by their standard dev i a t i o n ~Each . ~ ~ fit was checked by visual comparison of the curves. This procedure provided unique values of a and E without recourse to ancillary theory or other experimental data. Average scattering curves at c = 0, along with the best-fitting theoretical curves, 25 are given in Figure 2 for the 10°-70" range. The figure shows that for each ionic strength, appropriate values of a and E provide a remarkably good fit. At the lowest ionic strengths, experimental and theoretical curves could not be matched as closely as in higher salt, reflecting the greater technical difficulties experienced and
larger standard deviations obtained in the measurements. Comparison of our persistence lengths with others from the literature is shown in Figure 3. Light-scattering data4," on Col El DNA ( M = 4.35 X lo6), interpreted in different ways, provide three sets of points in the figure. The data were originally analyzed on the basis of excluded-volumetheory, derived for infinite molecular weight polymers, and values of E consistent with4 or obtained from'' viscosity measurements on DNA. These values of c are higher than those in Table I, either because of the theoretical approach used or because the entire effect of salt on viscosity is assumed to be due to changes in excluded v ~ l u m e . ' ~ Consequently, ~~~ persistence lengths from the earlier analyses by Eisenberg's group4*" are consistently lower than ours (Figure 3 ) . Because of this overestimate of the excludedvolume effect for Col El DNA, corrections have been applied by Manning27and by Post.15 They derived persistence lengths from the same data that show a stronger ionic-strength dependence (cf. Figure 3) than the original a n a l y s e ~ . ~As ~ "shown in Figure 3, our results agree well with those of Manning and Post for Na' concentrations at 0.2M Na' and below but differ at 1.0 and 3.0 M Na+ . The differences at high Na' concentrations persist regardless of the theory used. Flow birefringence measurements on T7 DNA have also been made as a function of ionic strength" and analyzed to give persistence lengths. The actual values of a depend upon the anisotropy of polarizability of a single base pair, which has not been well established. Figure 3 includes one set of persistence lengths,12 which are based on an assumed value of the anisotropy that gives a = 47.7 nm in 0.2 M Na'. Under these conditions, there is excellent agreement with our results (Figure 3 ) . The close correspondence of persistence lengths from flow birefringence l2 with those from this work and from the data of Borochov et a1.,4 as corrected by Manning2' and Post, l5 strongly suggests that reliable lightscattering results may be used to calibrate the optical anisotropy factor. A number of other techniques have been used in the last decade to measure persistence lengths of DNA. Values of a from linear dichroism measurements on T7 DNA" are also included in Figure 3. These data fall generally below, and exhibit a different ionic-strength dependence than, the results from flow birefringence or light scattering. Above 0.05M Na+, the persistence length remains nearly constant. Maret and Weill l6 have used magnetic bi-
7
s
A A
A A A
Na [MI Figure 3. Graph of persistence length vs log [ Na+] from several investigations. Corrections to total [ Naf ] were applied where necessary. Data shown are as follows: This work ( 0 ) ;Manning's27( * ) and Post's'' ( 0 )corrections of data of Borochov et al.;4Kam et al. (A);"Rizzo and .12 Schellman (A);lo Cairney and Harrington (0)
EFFECTS OF Na’
refringence to determine a for erythrocyte DNA (average M = 4.2 X lo6) as a function of ionic strength. Their results appear to agree with the linear dichroism data” and differ similarly from ours (Figure 3). However, Maret and Weill applied no excluded-volume corrections, l6 and there may be additional complications from sample polydispersity. Transient electric birefringence ( TEB ) has been employed to determine persistence lengths for short DNA fragments at low ionic strength^.'-^,^^*^^,^^ The show a threefold decrease in a to 50 nm, as [ Na+] increases from to 10-3M, with little further decrease to 0.004M Na’. Although the conditions of these measurements differ markedly from those for light scattering and the investigators recognize problems in interpreting TEB data, 14~19221 their inconsistency with the results in Figure 3 continues to be tr0ub1ing.l~ It was noted above that the values for t, obtained from the original analysis of light-scattering data on Col El DNA,4*11 are consistently greater than ours (Table I ) , although the two estimates show nearly parallel decreases of t with increasing “a+]. Manning’s 27 and Post’s l5 recalculations of the Col El data indicated a marked diminution of the excluded-volume effect. To estimate the equivalent values oft, we have used the original rms radii,4 the redetermined persistence lengths, and the equation 3,4*25
(R2)= L2
( E
+
L’(f-l) 2)(t 3)
+
where L is the contour length of the molecule, L‘ = L/2a = number of Kuhn statistical segments, and other parameters are defined above. These calculated values of c are about one-half those in Table I at 0.2 M Na+ and below. A tentative but tantalizing conclusion from this comparison is that there is a molecular-weight dependence of the excluded-volume parameter between M = 4 X lo6and 25 X lo6. This observation, along with related comments by Eisenberg’s group 4~17and by Hagerman, 21 strongly suggests that further investigations in this molecular-weight range by light scattering and other techniques would be valuable for improving our knowledge of excluded-volume effects. Understanding the flexibility of DNA, quantitated by the persistence length, requires the application of polyelectrolyte theory. Recent theoretical developments have been discussed by several
1663
authors 13,18.20,21.43 and are outside the scope of this article. We simply remark here that our light-scattering values of a vs ionic strength agree closely with predictions of counterion condensation theory, as presented by Manning.13We believe the differences between our data and other theoretical predictions and experimental measurements can be resolved by further work with the available techniques in an effort to provide overlapping results and to apply appropriate theory. The authors thank Dr. A. M. Jamieson for expert assistance with various aspects of this work. This investigation was supported in part by research grant GM-29828 from the U.S. Public Health Service and by the Department of Biochemistry,Case Western Reserve University.
REFERENCES 1. Frontali, C., Dore, E., Ferrauto, A., Gratton, E., Bettini, A., Pozzan, M. R. & Valdevit, E. (1979) Biopolymers 1 8 , 1353-1373. 2. Odijk, T. (1979) Biopolymers 18,3111-3113. 3. Harpst, J. A. (1980) Biophys. Chem. 11,295-302. 4. Borochov, N., Eisenberg, H. & Kam, Z. (1981) Biopolymers 2 0 , 231-235. 5. LeBret, M. (1981 ) CR Acad. Sci. Paris 292 (Series 11) , 291-294. 6. Schurr, J. M. & Allison, S. A. ( 1981) Biopolymers 20, 251-268. 7. Elias, J. G. & Eden, D. (1981) Macromolecules 1 4 , 410-419. 8. Hagerman, P. J. & Zimm, B. H. (1981) Biopolymers 20,1481-1502. 9. Hagerman, P. J. (1981) Biopolymers 20,1503-1535. 10. Rizzo, V. & Schellman, J. (1981) Biopolymers 2 0 , 2143-2163. 11. Kam, Z., Borochov, N. & Eisenberg, H. (1981) Biopolymers 2 0 , 2671-2690. 12. Cairney, K. L. & Harrington, R. E. (1982) Bwpolymers 21,923-934. 13. Manning, G . S. (1983) Biopolymers 22,689-729. 14. Hagerman, P. J. (1983) Biopolymers 22,811-814. 15. Post, C. B. (1983) Biopolymers 22, 1087-1096. 16. Maret, G. & Weill, G. (1983) Biopolymers 22, 27272744. 17. Borochov, N. & Eisenberg, H. (1984) Bwpolymers 23, 1757-1769. 18. Stigter, D. (1985) Macromolecules 18, 1619-1627. 19. Lewis, R. J., Pecora, R. & Eden, D. (1986) Macromolecules 19,134-139. 20. Schurr, J. M. & Schmitz, K. S. ( 1986) Ann. Reu. Phys. Chem. 37,271-305.
1664
SOBEL AND HARPST
21. Hagerman, P. J. (1988) Ann. Rev. Biophys. Biophys. Chem. 17,265-286. 22. Harpst, J. A. & Dawson, J. R. (1989) Biophys. J. 65, 1237-1249. 23. Reed, C. & Reed, W. (1990) J. Chem. Phys. 92,69166926. 24. Dunn, J. & Studier, F. W. (1983) J.Mol. Biol. 166, 477-535. 25. Sharp, P. & Bloomfield, V. A. (1968) Biopolymers 6, 1201-1211. 26. Bloomfield, V. A., Crothers, D. M. and Tinoco, I., Jr. (1974) Physical Chemistry of Nucleic Acids, Harper & Row, New York, pp. 151-258. 27. Manning, G. S. (1981) Biopolymers 20, 1751-1755. 28. Bachrach, M. & Friedman, A. (1971 ) Appl. Microbiol. 22, 706-715. 29. Mandell, J. D. & Hershey. A. D. (1960) Anal. Bwchem. 1,66-77. 30. Studier, F. W. (1979) Virology 96, 70-84. 31. Harpst, J. A., Krasna, A. I. & Zimm, B. H. (1968) Biopolymers 6,585-594. 32. Harpst, J. A., Krasna, A. I. & Zimm, B. H. (1968) Biopolymers 6, 595-603.
33. Zimm, B. H. (1948) J. Chem. Phys. 16,1099-1116. 34. Tanford, C. ( 1961) Physical Chemistry of Macromolecules, John Wiley & Sons, New York, pp. 275-316. 35. Bevington, P. ( 1969) Data Reduction and Error Analysis, McGraw-Hill, New York, pp. 204-242. 36. deGennes, P. G. (1976) Macromolecules 9,587-593. 37. Yu, T. L., Reihanian, H., Southwick,J. G. & Jamieson, A.M. (1980) J. Macromol. Sci. Phys. B 1 8 ( 4 ) , 777791. 38. Weissberg, S. G., Simha, R. & Rothman, S. (1951) J. Res. Nat. Bur. Std. 47, 298-314. 39. Odijk, T. (1979) Macromolecules 12,688-693. 40. Debye, P. (1947) J. Phys. Colloid Chem. 61,18-32. 41. Nicolai, T. & Mandel, M. (1989) Macromolecules 22, 438-444. 42. Douthart, R. J. & Bloomfield, V. A. (1968) Biopolymers 6, 1297-1309. 43. Schmitz, K. S. (1990) An Introduction to Dynamic Light Scattering by Macromolecules, Academic Press, New York, pp. 225-250.
Received July 23, 1991 Accepted August 7,1991
SYNOPSIS
Total intensity, Rayleigh light scattering has been used to measure the rms radius, second virial coefficient, persistence length, and excluded volume of homogeneous T7 bacteriophage DNA as a function of Na+ concentration (0.005 to 3.0 M). All parameters decrease sharply
as "a+] increases, and tend to level off at high Na'. The variation of persistence length with [Na+] is consistent with predictions from counterion condensation theory.
INTRODUCTION Understanding the flexibility of DNA has been the objective of numerous studies, almost since discovery of the double-helical structure. In recent years electrostatic contributions to th e persistence length, a direct measure of DNA flexibility, have undergone active experimental and theoretical investigat i ~ n . ' - ' ~I n this article, we report our results on homogeneous, native T 7 DNA (molecular weight, M = 26.43 X lo6)', from low-angle, total-intensity (Rayleigh) light scattering. A major advantage of this work is t hat the large size of T 7 DNA allows the wormlike coil equations of Sharp and B l ~ o m f i e l dt o~ be ~ fit directly to the data in order to obtain the persistence length a and the excluded volume parameter We have thus avoided some of the limitations in interpretation of earlier lightscattering studies on low molecular weight DNA, 4 ~ 1 1 ~ 1which 7 necessitated making estimates of t from other theoretical consideration^.^^"*^^-^^^^^^^^ t.25926
Biopolymers, Vol. 31, 1559-1564 (1991) 0 1991 John Wiley & Sons, Inc. CCC 0006-3525/91/131559-06$04.00
* We are honored to dedicate this work to Dr. Bruno H. Zimm on the occasion of his retirement from the University of California, San Diego. Present address: Department of Medicine, The University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-7280. To whom correspondence should be addressed.
'
*
EXPERIMENTAL T 7 DNA was isolated from coliphage T 7 by standard techniques2s and purified by phenol e x t r a c t i ~ n . ~ ~ T h e coliphage T 7 was obtained from bacteriophage stocks shown to be the wild type by S t ~ d i e r , ~and ' the DNA was later shown by sequencing24to have a molecular weight of 26.43 X Low-angle total-intensity light-scattering measurements were made at 20 k 2°C with a substantially redesigned version of a n earlier i n ~ t r u m e n t . ~ ~ Modifications included a 3 mW helium-neon laser light source and a light-scattering cell, especially designed with a closed, flow-through filtration system. This instrumentation made possible reproducible and continuous measurements over a n angular range of 4'-135". Calibration was performed by previously described methods, 31 except that it was based on the established molecular weightn4of T7 DNA (26.43 X lo6).Further details will be presented elsewhere. T 7 DNA solutions were prepared at eight different ionic strengths from 0.008 to 3M, with Na' as the only cation. T h e basic solvent, BEP, was a sodium phosphate-sodium EDTA (ethylenediaminetetraacetate ) buffer ( 0.00100M NanEDTA, 0.000187M NaHnPO4* H2 0 , and 0.00141 M NanHPO,, p H 6.92). The ionic strength of B EP was 0.008M a n d total N a t concentration = 0.005M. 1559
1560
SOBEL AND HARPST
1.2E-06
I
I
I
I
0.3
0.4
1E-06
8E-07
0
6E-07
B 4E-07
2E-07
a
33734
0
'3.1
0.2
0.5
n
S i r k 2 + 4000c Figure 1. Reciprocal-intensityplot (4"-60") of lightscattering data for T7 DNA in 0.005M Na'. RB is the Rayleigh ratio34at angle 8 . Data are shown ( 0 ) at six concentrations from 9.9 to 58.4 pg/mL; some points at low angles are omitted for clarity. The lines are linear least-squareextrapolations to zero concentration (0)and to zero angle ( * ) from the 4" to 9" region. Above 9" the curves at constant concentration are point-to-point fits
to the data. Table I "a+]
(M) 0.005 0.01 0.02 0.10 0.2Od 0.50 1.00
3.00
Solvents of higher ionic strength were prepared by adding appropriate amounts of NaCl and NaOH to obtain the desired total Na+ concentration at pH 6.92. Above 0.1M N a + , the ionic strength is within 1%of the Na+ concentration. For each light-scattering series at a given ionic strength, five or six DNA solutions were prepared separately and extensively dialyzed against the appropriate buffer. At each ionic strength, two sets of measurements a t concentrations from 10 to 60 pg/ mL were made with DNA from different stock preparations. The one exception was in 0.2M Na+, for which results have been included from seven sets of measurements, one in BEP and the others in a similar buffer, BPES," with a higher phosphate concentration. Clarification of solutions was accomplished by filtration3' through MF-Millipore filters (Type GSWP, 0.22-pm pore size) directly into the flow-through cell, to minimize turbulence and handling. Appropriate sedimentation and viscosity measurements, performed as described previo ~ s l y ,indicated ~ ~ , ~ ~the DNA was intact, homogeneous, and unaffected by filtration. Scattering data were initially displayed and evaluated with standard reciprocal-intensity plots, one of which is shown in Figure 1 for T7 DNA in 0.005 M Na+. Extrapolations to zero concentration ( c = 0) and zero scattering angle ( 0 = 0) were done with a linear least-squares executed by computer. Difficulties with the extrapolations to zero angle have been discussed p r e v i o ~ s l y . ~For ~~~~'~~~~ display purposes linear extrapolations from the 4' to 9" angular range have been made (Figure I ) and also used to establish the lines at 0 = 0, from which the values of second virial coefficients A' (Table 1) were obtained.34 These values of A2 were the same
Summary of Light-Scattering Results on T7 DNA
y'z
(R (nm)
A* x lo4 (mole cm3/g2)
a (nm)
c
700 -+ 30" 670 k 20 590 3- 20 550 k 10 530 k 20 490k 7 484 r+ 8 460 Y 6
18 k 4 ' 13 zk 2 9 +-I 4 f 1 3.1 2 0.5 2.1 k 0.8 1.8 k 0.7 1.6 f 0.8
78 k 2b 74 f 2 68 k 2 55 k 2 48 k 2 48 f 1 45 k 2 38 ? 1
0.14 f O.Olb 0.12 0.01 0.08 f 0.01 0.05 0.01 0.071 2 .005 0.054 f .005 0.049 k .007 0.053 f .005
-
* *
Averages of least-squares fit to Debye equation (see text) weighted by standard deviations from individual P-' (0) curves. Averages of least-squares fit of P-' (8) curve to equations of Sharp and Bloomfield weighted by standard deviations from individual P-' (8) curves. Weighted averages of linear least-squares extrapolation to 0 = 0 over 4"-9" angular region. Weighted average of six determinations in BPES and one in low phosphate buffer BEP (see text). a
EFFECTS OF Na'
1561
glement is expected to begin. Values of c * obtained ) "a+] = 0.005M from the rms radius ( ( R 2 ) 1 ' 2 at (Table I ) are 128 pg/mL from a suggestion by d e G e n n e ~ , 23 ~ ~ pg/mL ,~~ from the more stringent requirement for hexagonally close-packed s p h e r e with ~ ~ the ~ ~ above ~ ~ radius, or 16 pg/mL for the same spheres, arranged at the corners of a cube and just touching each other. For polyelectrolytes in monovalent salt, c* = 49 pg/mL from an expression by Odijk.2,20,39 Since the concentrations of our measurements bracket or fall below all these estimates of c * and the data are linear over a sixfold concentration range, we conclude our extrapolations to c = 0 are valid and provide reliable values of the scattering functions, P-' ( 0 ) . The linear angular range is limited to values of P-' (19)< 1.33,3,22p33 which restricts the linear region in these experiments to < 9" in 0.2M and < 6" in 0.005M Na+. At the lowest ionic strengths we have too few data points for a reliable linear extrapolation. Eisenberg and his collaborators have pointed out that the useful range of angles can be e~tended".'~ by applying the Debye e q ~ a t i o n , ~ ' 0.0
0.34
2
+
P ( 8 ) = 7(e-" x - 1 1 , Figure 2. Scattering curves at zero concentration for native T7 DNA at different ionic strengths. The ordinate is K c / R, = [ P - ' ( 0 ) 1/ M , where K is an optical constant.34 Data points ( 0 )are shown between 10' and 70'. Solid lines are the best-fitting theoretical curves, described in the text. Na+ molarity is indicated by the numbers next to each curve. The data at 3.OM Na+ are scaled as shown; each line at successively lower ionic strengths is displaced upward by 1 X
within experimental error as those derived from true extrapolations to 8 = 0, discussed in the following section.
RESULTS AND DISCUSSION For this investigation it is most important to obtain accurate determinations of the intramolecular scattering functions P-' ( B ) at each ionic strength. These curves (Figure 2 ) have been derived from linear extrapolations to c = 0 in the reciprocalintensity plots described above. There has been some concern as to whether light-scattering measurements on a large DNA such as T7 can be made at concentrations low enough to avoid problems due to coil overlap." Several estimates are available for the overlap concentration c*, at which coil entan3925,26,33*34
X
(1)
where
Here no is the refractive index of solvent, A 0 is the wavelength of light in vacuo, and ( R 2 )is the meansquare radius. In our experiments the Debye equation is valid for angles at or below 12" in 0.2M and 9" in 0.005M N a + . Data from these applicable angular ranges are fit to eq. ( 1 ) by standard statistical methods35to provide reliable, limiting values of (R 2 ) (Table I ) . All indicated errors are standard deviations obtained from least-squares analysis. A summary of our results is given in Table I. The parameters shown are from the first systematic study of ionic-strength effects on the properties of T7 DNA, determined by light-scattering measurements that extend into the limiting angular region of the scattering curve. Each entry in the table is an average of the two sets of measurements (seven in 0.2M Na+) ,weighted35by the standard deviations from analyses of individual data sets. The rms radii (Table I ) , obtained from eq. ( 1) ,decrease smoothly as ionic strength increases, and appear to approach
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SOBEL AND HARPST
a plateau above 0.5M Na'. The second virial coefficients also decrease in a smooth manner with increasing ionic strength. This behavior is consistent with theoretical expectations and with other experimental data, 4 ~ 1 7including recent results from small DNA fragments.41 Persistence lengths and excluded volume parameters are listed in columns 4 and 5 of Table I. These values were obtained by fitting the light-scattering equations of Sharp and BloomfieldZ5to experimental P-'(O) curves over the angular region of 4"-70". The approach described previously3 was modified to conduct the curve fitting by computer. The program utilized the iterative method of Marquardt to minimize deviations between the theoretical curves and experimental points, weighted by their standard dev i a t i o n ~Each . ~ ~ fit was checked by visual comparison of the curves. This procedure provided unique values of a and E without recourse to ancillary theory or other experimental data. Average scattering curves at c = 0, along with the best-fitting theoretical curves, 25 are given in Figure 2 for the 10°-70" range. The figure shows that for each ionic strength, appropriate values of a and E provide a remarkably good fit. At the lowest ionic strengths, experimental and theoretical curves could not be matched as closely as in higher salt, reflecting the greater technical difficulties experienced and
larger standard deviations obtained in the measurements. Comparison of our persistence lengths with others from the literature is shown in Figure 3. Light-scattering data4," on Col El DNA ( M = 4.35 X lo6), interpreted in different ways, provide three sets of points in the figure. The data were originally analyzed on the basis of excluded-volumetheory, derived for infinite molecular weight polymers, and values of E consistent with4 or obtained from'' viscosity measurements on DNA. These values of c are higher than those in Table I, either because of the theoretical approach used or because the entire effect of salt on viscosity is assumed to be due to changes in excluded v ~ l u m e . ' ~ Consequently, ~~~ persistence lengths from the earlier analyses by Eisenberg's group4*" are consistently lower than ours (Figure 3 ) . Because of this overestimate of the excludedvolume effect for Col El DNA, corrections have been applied by Manning27and by Post.15 They derived persistence lengths from the same data that show a stronger ionic-strength dependence (cf. Figure 3) than the original a n a l y s e ~ . ~As ~ "shown in Figure 3, our results agree well with those of Manning and Post for Na' concentrations at 0.2M Na' and below but differ at 1.0 and 3.0 M Na+ . The differences at high Na' concentrations persist regardless of the theory used. Flow birefringence measurements on T7 DNA have also been made as a function of ionic strength" and analyzed to give persistence lengths. The actual values of a depend upon the anisotropy of polarizability of a single base pair, which has not been well established. Figure 3 includes one set of persistence lengths,12 which are based on an assumed value of the anisotropy that gives a = 47.7 nm in 0.2 M Na'. Under these conditions, there is excellent agreement with our results (Figure 3 ) . The close correspondence of persistence lengths from flow birefringence l2 with those from this work and from the data of Borochov et a1.,4 as corrected by Manning2' and Post, l5 strongly suggests that reliable lightscattering results may be used to calibrate the optical anisotropy factor. A number of other techniques have been used in the last decade to measure persistence lengths of DNA. Values of a from linear dichroism measurements on T7 DNA" are also included in Figure 3. These data fall generally below, and exhibit a different ionic-strength dependence than, the results from flow birefringence or light scattering. Above 0.05M Na+, the persistence length remains nearly constant. Maret and Weill l6 have used magnetic bi-
7
s
A A
A A A
Na [MI Figure 3. Graph of persistence length vs log [ Na+] from several investigations. Corrections to total [ Naf ] were applied where necessary. Data shown are as follows: This work ( 0 ) ;Manning's27( * ) and Post's'' ( 0 )corrections of data of Borochov et al.;4Kam et al. (A);"Rizzo and .12 Schellman (A);lo Cairney and Harrington (0)
EFFECTS OF Na’
refringence to determine a for erythrocyte DNA (average M = 4.2 X lo6) as a function of ionic strength. Their results appear to agree with the linear dichroism data” and differ similarly from ours (Figure 3). However, Maret and Weill applied no excluded-volume corrections, l6 and there may be additional complications from sample polydispersity. Transient electric birefringence ( TEB ) has been employed to determine persistence lengths for short DNA fragments at low ionic strength^.'-^,^^*^^,^^ The show a threefold decrease in a to 50 nm, as [ Na+] increases from to 10-3M, with little further decrease to 0.004M Na’. Although the conditions of these measurements differ markedly from those for light scattering and the investigators recognize problems in interpreting TEB data, 14~19221 their inconsistency with the results in Figure 3 continues to be tr0ub1ing.l~ It was noted above that the values for t, obtained from the original analysis of light-scattering data on Col El DNA,4*11 are consistently greater than ours (Table I ) , although the two estimates show nearly parallel decreases of t with increasing “a+]. Manning’s 27 and Post’s l5 recalculations of the Col El data indicated a marked diminution of the excluded-volume effect. To estimate the equivalent values oft, we have used the original rms radii,4 the redetermined persistence lengths, and the equation 3,4*25
(R2)= L2
( E
+
L’(f-l) 2)(t 3)
+
where L is the contour length of the molecule, L‘ = L/2a = number of Kuhn statistical segments, and other parameters are defined above. These calculated values of c are about one-half those in Table I at 0.2 M Na+ and below. A tentative but tantalizing conclusion from this comparison is that there is a molecular-weight dependence of the excluded-volume parameter between M = 4 X lo6and 25 X lo6. This observation, along with related comments by Eisenberg’s group 4~17and by Hagerman, 21 strongly suggests that further investigations in this molecular-weight range by light scattering and other techniques would be valuable for improving our knowledge of excluded-volume effects. Understanding the flexibility of DNA, quantitated by the persistence length, requires the application of polyelectrolyte theory. Recent theoretical developments have been discussed by several
1663
authors 13,18.20,21.43 and are outside the scope of this article. We simply remark here that our light-scattering values of a vs ionic strength agree closely with predictions of counterion condensation theory, as presented by Manning.13We believe the differences between our data and other theoretical predictions and experimental measurements can be resolved by further work with the available techniques in an effort to provide overlapping results and to apply appropriate theory. The authors thank Dr. A. M. Jamieson for expert assistance with various aspects of this work. This investigation was supported in part by research grant GM-29828 from the U.S. Public Health Service and by the Department of Biochemistry,Case Western Reserve University.
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Received July 23, 1991 Accepted August 7,1991
