[6] Density Gradient Sedimentation Equilibrium

[~] DENSITY GRADIENT SEI)IMENTATION EQUILIBRIUM 111

[6] Density Gradient Sedimentation Equilibrium

By JOHN E. HEARST and CARL W. SCHMID

In 1957, Meselson, Stahl, and Vinograd 1 introduced the remarkable technique of sedimentation equilibriuln in a density gradient for the study of the properties of high molecular weight DNA in solution. Be- cause of the very high molecular weights of DNA, conventional sedimentation equilibrium cannot be used; for even at the lowest centrifugal fields which are practical, the force on these molecules is too large and they become distributed in extremely narrow distributions. Sedimenta- tion equilibrium in a density gradient avoids this problem because the experiment is performed in a solution which is almost precisely buoyant for the DNA, reducing the centrifugal force on the molecules by several orders of magnitude. The band or distribution of DNA in the centrifugal cell is wide enough to be accurately measurable because the force on the DNA molecule is so small. All procedures described in this chapter are equally applicable to proteins, although generally resolution is poorer because of the lower molecular weights. 1~

The centrifugal field also generates an equilibrium density gradient which is caused by the redistribution of the salt and the compression of the solution. Fortuitously, the buoyant density of DNA is a function of base composition, so the technique also provides a method for separating and identifying different varieties of DNA in a mixture. 2 Mesel- son and StahV proved in 1957 that the replication of DNA in Escherichia coli is semieonservative by following the history of density labeled (lSN) DNA which had been transferred to light medium. The density gradient method provided something unique in labeling techniques, for it made possible the physical separation of isotopieally labeled material from unlabeled material.

We present herein the methods for standardizing buoyant density data for DNA. The source of the different density scales is discussed and recommendations are made as to the choice of a standard scale. Factors

1 M. Meselson, F. W. Stahl, and J. Vinograd, Proc. Nal. Acad. Sci. U.S. 43, 581 (1957).

~ J. B. Ifft, "A Laboratory Manual of Analytical Methods of Protein Chemistry" (P. Alexander and H. P, Imndgren, eds.), Vol. 5, p. 151. Pergamon, New York, 1969.

"~ C. L. Sehildkraut, J. Marmnr, and P. Dot3", J. MoI. Biol. 4, 430 (1962). M. Meselson and F. W. Stahl, Proc. Nat. Acad. Sci. U.S. 44, 671 (1958).

1 1 2 M O L E C U L A R ~ V E I G t t T D E T E R M I N A T I O N S [6]

?

5

o0 C:)

+~

+l

+l

¢,D

+1

+

o0

~A

O ~J

0J

o'2

.

[6] DENSITY GRADIENT SEDIMENTATION EQUILIBRIUM 113

TABLE II V VERSUS TEMPER.~.TURE a'b

Salt Empirical relation

Cs formate G = 4.308 (1-0.00334 t°C) X 10 -~° CsC1 G = 0.849 (1-0.00802 t°C) X 10 -~° CsTFA G = 13.83 (1-0.00855/°C) X 10 -~° Cs2SO4 G = 23.01 (1-0.00824 t°C) X 10 -~°

C. W. Schmid and J. E. Hearst, Biopolymers 10, 1901 (1971). b G for CsC1 and Cs~S04 was extensively calibrated in the range 8 ° to 33 °. For CsTFA

and Cs formate the above relations are based on two temperatures 8 ° and 20".

inf luencing reso lu t ion are discussed. F i n a l l y , mos t of the d a t a r equ i red

for the use of th is e legan t tool are compiled. T h e review s t a r t s wi th a discussion of the fac tors governing band

shape or the d i s t r i b u t i o n wi th the in ten t t h a t th is sect ion will p rov ide

the i n tu i t i ve knowledge requ i red for an u nde r s t a nd ing of the com-

p l ica t ions .

M o l e c u l a r W e i g h t a n d the E q u i l i b r i u m C o n c e n t r a t i o n D i s t r i b u t i o n

T h e equ i l i b r ium d i s t r i bu t ion of D N A in the dens i ty g r ad i en t is r e a d i l y ca l cu l a t ed f rom t h e r m o d y n a m i c s . 4-s A t inf ini te d i lu t ion the d i s t r ibu t ion

of a homogeneous D N A will be Gauss i an [Eq. (1 ) ] .

C= Coexp - ~ (1)

where Co is the concen t ra t ion a t band center, C is the concen t ra t ion

a t a d i s t ance ~ f rom band center, and cr is the s t a n d a r d dev ia t i on of

the G a u s s i a n band. I n ca l cu l a t i ng a mo lecu l a r weight f rom the d i s t r i bu t ion in the band

an effective dens i t y g r a d i e n t m u s t be used. 4 Th i s g rad ien t can be m e a s -

u red e x p e r i m e n t a l l y b y observ ing the spac ing be tween [ ~ N ] D N A and

[ ~ N ] D N A ? ~1 T h e p a r a m e t e r G or (1 + r')/3o~ is ca l cu la t ed f rom Eq. (2) and has been t a b u l a t e d in T a b l e I for severa l sa l ts a t 20 ° and in

T a b l e I I as a func t ion of t empe ra tu r e .

4j. E. Hearst and J. Vinograd, Proc. Nat. Acad. Sci. U.S. 47, 999 (1961). 5j. E. Hearst and J. Vinograd, Proc. Not. Acad. Sci. U.S. 47, 1005 (1961). ~J. E. Hearst, J. B. Ifft, and J. Vinograd, Proc. Nat. Acad. Sci. U.S. 47, 1015 (1961). 7E. F. C,'~'tssa and H. Eisenberg, J. Phys. Ct~em. (}5, 427 (1961).

G. Cohen and H. Eisenberg, Biopolymers 6, 1077 (1968). ~C. W. Schmid and J. E. Hearst, Biopolymers 10, 1901 (1971). ~' J. Vinograd and J, E. Hearst, Forlschr. Chem. Org. Nalurst. 20, 372 (1962). :~ H. Eiscnberg, Biopolymers 5, 681 (1967).

114 MOLECULAR WEIGHT DETERMINATIONS [6]

G - I + F~_ Amps,o (2) ~ff mAr~2ro

Am is the change in mass per nucleotide upon isotopic substitution of 15N for 14N, p~,o is the buoyant density, m is the mass DNA per Cs nueleotide, Ar is the distance between the two peaks. ~ is the angular velocity, and ro is the average position of the bands relative to the center of the rotor. The quantity, r', is the thermodynamic net hydra- tion and the effective gradient is related to flerr by Eq. (3).

tOP) -~2r° (3)

An apparent molecular weight of dry Cs DNA, M~,App, is calculated with Eq. (4)

1 =G(co4ro2'~ M3,A,-~-~ \R--~p~,0] ¢~A,, (4)

where R is the gas constant and T is the absolute temperature. The true molecular weight, M3, must be determined experimentally by extrapolation of the apparent molecular weight to zero concentration2 ~ There are two methods to analyze the bands. The concentration distribution may be numerically integrated to find the apparent molecular weight which is then extrapolated to zero concentration by plotting log MA,p against the average concentration, (C). ~ Equations (5)-(7) define the numerical integrations and operations required for this procedure. Figure 1 presents an example of such an extrapolation.

The variance:

f '° C$2d~ (~) = ~ (5 )

'_° Cd~

The average concentration

C2d~ (6) ( c ) - f,o

Cd~ J - The extrapolation

, p~,oRT In M A , , = m ( 5 ~ o 2 = I n M 3 - - B ' (C} ( 7 )

An alternate method is to measure the band half-width, aApp, directly

~C. W. Schmid and J. E. Hearst, J. Mol. Biol. 44, 143 (1969).


0.8

0.7

7 o.

:~ 0.6 0

ff

0 . 5 - -

0.4--

0.0

I I I I

I I I I 0.1 0.2 0.3 0.4

< 6 " > , O D . ~ 6 5 nrn , I cm

Fie,. 1. Molecular weight of D3 DNA. • 25,000 rpm, O 35,000 rpm. Extrapola- lion to infinite dilution using the moments analysis (Eq. 7).

at the point where the concentration equals 0.606 of that at the maximum concentration. 12 Equation (8) describes the form of this extrapolation, and Fig. 2 presents a representative plot.

p,,oRT In 21IA,, = In (~A,,)2G~o4r02 -- In M3 -- BCo (8)

I n the absence of dens i ty he te rogene i ty , the first of these me thods p ro -

v ides a n u m b e r ave rage mo lecu l a r weighff and the second prov ides a

L I I I I I I

0.7

r,-.

'~ 0.6 -

o

O.b -

0.4 I I I I , I 0.0 O.I 0.2 0.5 0.4 0,5 0.6

C O , O D Z 6 5 nm , I crn

FIC. 2. Molecular weigh|, of D3 DNA. • 25,000 rpm, 0 35,000 rpm. Extrapola- tion to infinite dihdion using a direct measure of the band width (Eq. 8).


TABLE III MOLECULAI~, WEIGHTS OF HOMOGENEOUS PHAGE D N A ' s a

Phage

Method of analysis T7 T5 T4 ])3

Moments, M X 10 -6 dalton, 24.8 68.7 113 40.6 Na~DNA

Direct measure of a, M X 10 -~ 24.1 80.1 106 36.9 dalton, Na,,DNA

Data from C. W. Schmid and J. E. Hearst, J. Mol. Biol. 44, 143 (1969); Biopolymers 10, 1901 (1971); and unpublished results.

weight average molecular weight? -~ The numerical analysis has the ad- vantage of greater statistical significance and applicability to non- Gaussian bands. I t has the disadvantages of requiring more labor and being more sensitive to small baseline errors. This is the cause for the larger scatter in Fig. 1 than in Fig. 2.

1XIolecular weights for four different phage DNA's have been found by these methods and are presented in Table III.

Density Heterogeneity

The considerations of the last section apply only to a sample that is homogeneous in buoyant density. Fragmented DNA's, in particular, may be heterogeneous in base composition and therefore in buoyant density as well. This density heterogeneity increases the band width and decreases the apparent molecular weight. 13 The most general class of heterogeneity possible is one in which the density and the molecular weight of the DNA may in some way be correlated. Such a system is difficult to deal with experimentally. Most of the density heterogeneity one must deal with is fortunately the result of breakage of very high molecular weight. DNA from higher organisms. For such a case there should not be any correlation between molecular weight and density in the sample, and a simplifying assumption is possible.

In the absence of correlation between density heterogeneity and molecular weight heterogeneity, the variance of the distribution may be assumed to be the sum of a contribution from diffusion and a contribution from density heterogeneity. We may write 14'1~

~'~ N. Suoeka, Proc. Nat. Acad. Sci. U.S. 45, 1480 (1959). 14j. E. Hearst, Ph.D. Thesis, California Institute of Technology, 1961. ~'~C. W. Schmid and J. E. Hearst, Biopolymers, in press (1973).

[6] I)ENSITY GRADIENT SEDIMENTATION EQUILIBRIUM 117

1 Go~4ro2(52) 1 Gf31~2a 2 MA,, RTp,,o = M---~ + ~ <(GC)2> (9)

where <(GC) '~> is the variance of the GC composition, fll~ is related to the lmoyancy gradient by an equation analogous to Eq. (3) and is defined ill the following section, and the parameter , o, is the slope of the plo~ of lmoyant density against GC content for a given salt solution. The derivation assumes a linear dependence of buoyant density versus GC and also assumes tha t all heterogeneity in density arises from GC heterogeneity and therefore there can be no odd bases or glueosylated bases ~' if the analysis is to succeed. From Eq. (9) we can conclude tha t changing the speed of the centrifuge does not aid in separating the effects of diffusion and density heterogeneity, ~7 but changing the salt solution in which the D N A is studied enables one to make this separation. The parameters, G, ft,, and a, are all functions of the salt used. Generally, salts which result in weak gradients such as Cs formate at any arbi t rary speed have a high resolving power with respect to density heterogeneity, and salts with large gradients such as Cs._,SO~ have poor resolving power. Figure 3 shows the distribution of X half-molecules in three different salts. We see tha t the bands are best resolved in Cs formate, next best in CsC1, and worst in Cs2SO~.

Table IV summarizes the data obtained in these three salt solutions on X half molecules. The method should prove to be of great value, in future years, in obtaining molecular weights and evaluating the amount of density heterogeneity in the D N A sample. ~s,~9 The Cs2SO~ estimate of <(GC) °-> is different from tha t for the other two salt solutions.

Distance Fie,. 3. Equilibrium concentration distribution of mechanically sheared halves

of Xei857 DNA in various soh-ents. All traees are at the same magnification. Trace A is cesium formate at 35,000 rpm; trace B is cesium chloride at 35,000 rpm; trace C is cesium sulfate at 30,000 rpm.

16 W. Szybalski, Vol. 12B, p. 330. 17 j. B. Ifft, D. E. Voet, and J. Vinograd, J. Phys. Chem. 65, 1138 (1961). lsC. L. Schildkraut and J. J. Maio, J. MoI. Biol. 46, 305 (1969). 19 H. Yamagishi, J. Mol. Biol. 49, 603 (1970).


TABLE IV MOLECULAR WEIGHT AND COMPOSITION HETEROGENEITY OF

SHEARED LAMBDA DNA ~

MApp X 10 -e ~ B2a~G X 10 +4 Salt dalton b ~sa X 10 -s c RTp~.o (GC2)112 d

Cs formate 1.17 2.166 4. 254 0. 0435 CsC1 1.88 1.167 2. 544 0. 0435 Cs2S04 7.08 0. 218 0. 246 0. 0616

a C. W. Schmid and J. E. Hearst, Biopolymers, in press (1973). b The apparent molecular weight of Cs,DNA is defined by Eq. (3). For Cs2S04 a

small virial correction was performed by Eq. IIa; for CsC1 and Cs formate no correction was used as the Mapp is essentially a measure of density heterogeneity (Fig. 2).

c CsC1 is taken as the standard [C. L. Schildkraut, J. Marmur, and P: Dory, J. Mol. Biol. 4, 430 (1962)], and Cs fol~nate and Cs2SO4 are calibrated to it as explained in the text. For Cs~S04/~Ba varies by a factor of approximately 2 over the range 25% to 70% GC (C. W. Schmid and J. E. Hearst, Biopolymers, in press (1973); W. Szybalski, Vol. 12B, p. 330).

d Calculated with Eq. (3) assuming for X halves Ms = 21 X 10 +6 daltons. For X NaDNA MW = 32 X 106 , and for CsDNA MW = 42 X 106 •

W e bel ieve this d i s a g r e e m e n t comes f rom the s t rong non l inea r de pe nd -

ence of b u o y a n t d e n s i t y on base compos i t ion in Cs2S04 solut ions. 16

Th i s t e s t sy s t em also suffers f rom a ce r ta in co r re la t ion be tween de ns i t y and m o l e c u l a r we igh t in X halves , s° bu t the resul ts a re encouraging .

T h e B u o y a n t D e n s i t y

T h e b u o y a n t dens i t y of a D N A sample m a y be m e a s u r e d wi th p re - cision. U n d e r f a v o r a b l e c i r cums tances b u o y a n t de ns i t y differences of

0.0005 g / m l are s igni f icant . Since the b u o y a n t de ns i t y depends on such

th ings as t he base compos i t ion , ~,ls the ex ten t of d e n s i t y labe l ing , 3,-~1 and

the presence of unusua l bases , 16 i t is poss ib le to measu re these q u a n t i - t ies f rom a b u o y a n t d e n s i t y de t e rmina t i on .

B y fa r the mos t s a t i s f a c t o r y w a y of d e t e r m i n i n g b u o y a n t de ns i t y is to de t e rmine band pos i t ion r e l a t i v e to a s t a n d a r d b a n d y T h e s t a n d a r d mos t f r equen t l y used is E. coli D N A , and if i t has a dens i ty too nea r t h a t of the D N A of u n k n o w n dens i ty , a s e c o n d a r y s t a n d a r d which has been s t a n d a r d i z e d to E. coli should be used. T h e de ns i t y difference between the s t a n d a r d D N A and the u n k n o w n D N A is then ca l cu l a t ed b y

Eq . (10) and the c a l i b r a t i o n d a t a in T a b l e V.

~W. Doerfler and D. S. Hogness, J. Mol. Biol. 33, 635 (1968). 22 R. L. Baldwin, P. Barrand, A. Fritsch, D. A. Goldthwait, and F. Jacob, J. Mol.

Biol. 17, 343 (1966).


50 2 pl -- p0 -- 9 ~ B (rl 2 -- r02) (10)

where rl and ro are the positions of the two bands relative to the center of rotation, ~ is the angular velocity in radian/second, 0, - po are the two buoyant densities which are t)anded in the salne solution, and fl~ is related to the buoyancy gradient by an equation similar to Eq. (3).

The values of fl~ in Table V arc among the more commonly used values in the literature. The values for CsC1 and Cs formate were measured in this laboratory} and we believe them to be the most precise available. The value for Cs2S0~ is 13% lower than the value determined in this laboratory, but since almost all buoyant densities for D N A in Cs~SO~ in the literature are based on this number, 16 it seems foolish at this point to recommend a change, although such a change may become desirable in the future.

While density differences can be determined with great accuracy, absolute buoyant densities are far less accurate. For example, both 1.7100 g/ml 2 and 1.7035 g /mP 6 have been used as the standard density of E. coli DNA. By accident, the first of these numbers is probably close to the 1 arm pressure buoyant, density of E. coli D N A ~2 and therefore serves as a good standard for buoyant densities at 1 atm. The second number is the buoyant density of E. coli D N A which is banded in the center of a cell at an angular velocity of 45,000 rpm where the pressure is about 160 arm. ~G We favor the use of the first number, that is 1.7100, as the standard density of E. coli since 1 atm numbers are independent of liquid column height, angular velocity, and position of the band. The

TABLE V BUOYANCY GRADIENT CONSTANTS

Salt Cs~S04 ~ CsCP Cs formate c

1~fib g'sec------~ M 10 +1° 14.6 9.35 4.21 em 5

This value is calculated from a buoyancy gradient (W. Szybalski, Vol. 12B, p. 330), assuming that value refers to ro = 6.60 cm.

b Several calculations and measurements of this quantity are in good agreement: J. E. Hearst, J. B. Ifft, and J. Vinograd, Proc. Nat. Acad. Sci. U.S. 47, 1015 (1961); J. Vinograd and J. E. Hearst, Fortschr. Chem. Org. Naturst. 20, 372 (1962); W. Bauer, F. Prindaville, and J. Vinograd, Biopolymers 10, 2615 (1971); C. W. Schmid and J. E. Hearst, Biopolymcrs 10, 1901 (1971).

c C. W. Schmid and J. E. Hearst, Biopolymers 10, 1901 (1971).

~'~ W. Bauer, F. Prindaville, and J. Vinograd, Biopolymers 10, 2615 (1971).


disadvantage of this density scale is that the CsC1 solution which one makes up to band the DNA in the center of the cell must be 0.003-0.006 g/ml lower in density than the stated buoyant density in order to correct for the fact that the band is at high pressure. 22 It would clearly be counterproductive to consider still other standard densities at this point.

The other method for determining buoyant density is to band the DNA in two solutions of different initial density and interpolate the band to the root mean square position of the cell, assuming this is the position where the density of the solution is that of the starting solution. 1° This procedure is far less accurate than using a marker, and it is not recommended unless one is using a new salt solution or a substance other than native DNA where standards are not available.

The GC differences in two DNA's can be calculated directly from their spacing in a gradient by Eq. (11).

AGC = °~2(r12 - - r°2) (11) 2a[3 B

The important parameter, aflB, is tabulated in Table IV. For CsC1 the G C data of Schildkraut, Marmur, and Dory were employed. 2 These data were selected because they are clearly presented and done with care. The ratio of the quantity afl~ to the same quantity in another salt can be obtained from the relative spacing of two DNA bands in the two solutions. This method was used to obtain the values of aflB for Cs formate and Cs~S0415 in Table IV. This method assumes a linear dependence of buoyant density on GC, and, while we trust the results for Cs formate, we believe that the absence of linearity in Cs2S0416 makes the figure for that salt very unreliable.

The buoyant properties of single-stranded DNA have been studied under alkaline and neutral conditions. In neutral CsC1 the buoyant density of single-strand DNA is about 0.02 g/ml denser than native DNA. A larger difference is observed in Cs2S04.1G The buoyant density of single stranded DNA is dependent on its base composition. 23 Under alkaline denaturing conditions the buoyant density is dependent on the GT composition. ~

For preparative and analytical studies, it is often desirable to enhance density differences. Synthetic polynucleotides have been bound to single-strand DNA's to increase the density differences between com-

~S. Riva, I. Barrai, L. Cavalli-Sforza, and A. Falaschi, J. Mol. Biol. 45, 367 (1969). J. Vinograd, J. Morris, N. Davidson, and W. F. Dove, Jr., Proc. Nat. Acad. Scl. U.S. 49, 12 (1963).


plementary strands. ~,26 Intercalating dyes produce a density difference between DNA's of different superhelical densityY 7 Heavy metal (Hg 2+ and Ag ÷) ions enhance density differences arising from differences in base compositionY s-3° By using different salts, base compositional differences may be enhanced (Cs formate) or diminished (Cs2SO~) relative to the differences observed in CsC1 (Fig. 3). Little use has been made of the greater resolution of such salts as Cs formate. The resolution of 3. halves in Cs formate is comparable to the separation in Hg 2+ - Cs~S04. 31

A nonequilibrium method for separating components that would ordinarily be difficult to resolve has been considered. This technique em- ploys high speed centrifugation which results in the formation of very sharp bands, followed by slow speed centrifugation so that the components sediment away from each other22

Finally, since there still is some experimental error in the fl~'s for many salt systems and since they are unknown for other salts and macromolecules (denatured DNA, viruses, etc.) it would be very bene- ficial if the pr imary data were presented in the literature as well as the buoyant density differences. Such data should be presented as the distances between bands multiplied by ~-~ to eliminate angular velocity and the position of the bands in the gradient as parameters. We suggest the symbol A to represent this quant i ty; its definition is contained in Eq. (12), where f is the arithmetic mean of the position of the two bands.

A = ~:~(Ar) (12)

The value of A between calf thymus DNA and E. coli DNA calculated from the data of Schildkraut, Marmur, and Doty 2 is -1 .31 × 107 cm2/ sec 2, where the negative indicates that calf thymus DNA is less dense then the reference E. coli DNA.

The A p p r o a c h to Equil ibrium

The approach to equilibrium is a transport process in a centrifugal field and may be used to measure the sedimentation coefficient. The ap-

W. C. Summers, Biochim. Biophys. Acta 182, 269 (1969). -~ H. Kubinski, Anal. Biochem. 35, 298 (1970). ~ R. Radloff, W. Bauer, and J. Vinograd, Proc. Nat. Acad. Sci. U.S. 57, 1514 (1967). ~ G. Corneo, E. Ginelli, and E. Polli, J. Mol. Biol. 48, 319 (1970). "0' R. H. Jensen and N. Davidson, Biopolymers 4, 17 (1966). ~ U. S. Nandi, J. C. Wang, and N. Davidson, Biochemistry 4, 1687 (1965). 31 j. C. Wang, U. S. Nandi, D. S. Hotness, and N. Davidson, Biochemistry 4, 1697

(1965). = R. Anet and D. R. Strayer, Biochem. Biophys. Res. Commun. 34, 328 (1969).


proach to equilibrium has all the thermodynamic complications ~'~ men- tioned for sedimentation equilibrium in addition to the problem of an accurate description of the salt distribution. To simplify the description of the salt distribution, sedimentation may be studied in either a pre- formed gradient 34 or a centrifugally generated equilibrium gradient25 I t is most convenient in analytical ultraccntrifugation to employ an in- ternally generated equilibrium density gradient. This may be accomplished by low speed centrifugation and by waiting about 24 hours before taking data. ~*,35

A complete description of the nonequilibrium concentration distribution is very difficult. To avoid this problem the sedimentation coefficient, DM:, can be related to moments of the concentration distribution as shown in Eqs. (13) and (14)Y,35

1.4 i5i.2 I I ] @1 I I I I ] I I I

^

6 0

v ID i

^

N

0.9 oO

v '..-.-' 0.8

£

°°' 0.7

0.6

0.5 0 4 8 12 f6 20 24 28 52 56 40 44 48

T ime /52 rain

FIG. 4. Sedimentation velocity of phage a DNA in a buoyant density gradient. The moments are conveniently expressed in arbitrary chart dimensions. The initial point was taken 40 hours after the run was started. The average concentration in- creased through the range 0.11~).21 OD.~5 .... ~ c,,, during the time sequence followed.

~J. E. Hearst, Biopolymers 3, 1 (1965). u R. L. Baldwin and E. M. Shooter, "Ultracentrifugal Analysis in Theory and

Experiment" (J. W. Williams, ed.), p. 143. Academic Press, New York, 1963. M. Meselson and G. M. Nazarian, "Ultracentrifugal Analysis in Theory and Experiment" (J. W. Williams, ed.), p. 131. Academic Press, New York, 1963.

[6] DENSITY GRADIENT ~EI)IMENTATION EQUILIBRIU~I 123

2Dr (~(t)2) _ (~(~)2) _- [(~(0)o.) _ (~(~)2)] exp a2 (13)

[RTo~,o]D (14) DM3 = [ Go~4ro2j a~

The proportionali ty constant between M:, and a"- is identical to the quant i ty appearing at eqnilibrimn [Eq. (4)]. Use of G from Tables I or I I defines M:~. for dry Cs DNA. l) is the diffusion constant for the temperature and solvent employed and (8(t)'-') is the time-dependent variance of the band.

Transpor t data demonstrating the use of Eq. (2) are presented in Fig. 4. The slope of this plot, 2D/d-' is listed in Table VI together with the necessary corrections to a solvent of the viscosity and density of water at 20°. 3~,37 The value is also corrected to the molecular weight and partial specific volume of Na DNA. D M 3 / R T which equals M3/] is multiplied by 0.759 to correct to Na DNA and by (1 - ~p), 0.444, s,3s to form the sedimentation coefficient.

This method of measuring the sedimentation coefficient eliminates the problems of speed effects on high molecular weight DNA's and avoids problems of convection that are experienced in the absence of density stabilization. The sedimentation coefficients found by this procedure are in reasonable agreement with S2o,w's found by conventional boundary sedimentation (Table VI) . The moment analysis, however, is lengthy, and it is not likely that this method will receive much use except in special circumstances.

The Buoyant Medium

When banding DNA it is prudent to check the solvent density before starting the run. This is conveniently (tone by measuring the refractive index of the solution. Table VII contains the parameters which relate the index of refraction to the solution density by the equation 0 -~5°c = a n~ °c - b . The a's, b's, and approximate buoyant densities of DNA are tabulated for numerous salt solutions. The relations summarized in Table VII and those below refer to a salt dissolved in water.

For CsC1 a more exact relation between refractive index and density is o 25°c = 1.1584 - 10.2219 n~ °c + 7,5806 (n~°c) ~- which covers the density range 1.1 g/ml < o < 1.9 g/ml?" For cesium sulfate a more exact relation is 4°

~" R. Brunner and J. Vinograd, Biochim. Biophys. Acta 108, 18 (1965). "~ P. A. Lyons and J. F. Rile3,, J. Amer. Cl~em. Soc. 76, 5216 (1954). ~J. E. Hearst, J. Mol. Biol. 4, 415 (1962). 3,~j. B. Ifft, W. l~. Martin, III, and K. Kinzie, Biopolyrners 9, 597 (1970). 4°D. B. Ludlurn and R. C. Warner, J. Biol. Chem. 240, 2961 (1965).

1 2 4 M O L E C U L A R W E I G H T D E T E R M I N A T I O N S [6]

,b T

X

A ~ x

0

o 'N

N

ff

Z

d d d d ÷1 ÷1 ÷1 ÷1

,d ,d d d

d ~ d ~

d d d d

d d d d

S

09

?

.£

8

, . ~ ¢D

[6] DENSITY GRADIENT SEDIMENTATION EQUILIBRIUM 1-05

p = 0.9954 -t- 11.106607 -- no) - - 2 6 . 4 4 6 0 0 ~ - ~7o) 2

w h e r e no is t h e r e f r a c t i v e i ndex of w a t e r .

S e v e r a l o b s e r v a t i o n s on the a p p l i c a b i l i t y of d i f f e ren t sa l t s a re s u m -

m a r i z e d be low. Cs a c e t a t e , Cs f o r m a t e , and m i x t u r e s of these two sa l t s

p r o v i d e s h a l l o w d e n s i t y g r a d i e n t s w i t h v e r y good r e s o l v i n g power . E q u i -

l i b r a t i o n in t he se s a l t so lu t ions is s low and the i n i t i a l d e n s i t y n m s t be

a d j u s t e d v e r y close to t he buoyan t , d e n s i t y of t he D N A . T h e s e sa l t s a re

i d e a l l y su i t ed for v e r y h igh m o l e c u l a r w e i g h t D N A . I n Cs a c e t a t e so lu-

t i ons i t is diff icul t to o b t a i n d i sce rn ib l e bands w i t h a n y t h i n g o t h e r t h a n

v e r y h igh m o l e c u l a r w e i g h t D N A . I n Cs=,SO~ and Cs~Se04 the s teep

d e n s i t y g r a d i e n t p r o d u c e s poor r e so lu t ion , r a p i d e q u i l i b r a t i o n , and a

m e d i u m for b a n d i n g low m o l e c u l a r w e i g h t s amples . C s B r a n d C s I a re

n e a r l y s a t u r a t e d a t t he dens i t i e s r e q u i r e d for b a n d i n g D N A and a re of

l i t t l e use.

T h e t r i f l u o r o a e e t a t e ion is a d e n a t u r a n t for D N A and e i t h e r p o t a s s i u m

or ce s ium t r i f l u o r o a e e t a t e se rves as a 1)uoyant so lven t for D N A / 1 So-

TABLE VII APPROXIMATE DNA BUOYANT DENSITIES AND P~EFRACTIVE INDEX

EQUATIONS FOR SEVERAL SALTS

Salt p~.0 a - b

Cs acetate 1. 959 h 10.1428 b 12. 5548 b 1. 946~ 10.7527 ~ 13. 4247 ~

50-50 acetate-formate 1. 864 a - - - - Cs formate 1. 767 ~ 13.7363" 17. 4286 ~ CsC1 1. 710(F - - I - - I Cs trifluoroacetate 1. 576 d - - - - CsBr 1.63 a 9.9667" 12.2876 ~ CsI 1.55" 8. 8757 ~ 10.8381" Cs2 oxalate 1. 508 b 10. 5904 b 13. 2769 b Cs2Se04 1. 479 ~ 12. 0919" 15.1717" Cs2SO4 1. 4260 ~ - - s - - J K trifluoroacetate 1.52 ~ - - - - Li silicotungstanate 1.138 ~ - - - -

T4 DNA [J. E. Hearst and J. Vinograd, Proc. Nat. Acad. Sci. U.S. 47, 1005 (1961); J. Vinograd and J. E. Hearst, Fortschr. Chem. Org. Nalurst. 20, 372 (1962); J. E. Hearst, Ph.D. Thesis, California Institute of Technology, 1961].

b Calf thymus DNA [L. Zolotor and R. Engler, Biochim. Biophys. Acta 52, 145 (1967)]; coefficients a and b refer to p23O and n ~°.

c 50% GC DNA (W. Szybalski, Vol. 12B, p. 330). 50% GC DNA [M. J. Tunis and J. E. Hearst, Biopolymcrs 6, 1345 (1968)].

e T7 DNA at 25°C [M. J. Tunis and J. E. Hearst, Biopolymers 6, 1325 (1968)]. I See text.

41 M. J. Tunis and J. E. Hearst, Biopolymers 6, 1325 (1968).


dium iodide serves as a inexpensive salt with better resolution than cesium chloride. 4-~

Finally, several attempts have been made to calculate and measure the composition density gradient of many salts, la,39,4° While this is useful for the study of new materials in density gradients, analytically this quantity should not be used for the determination of 1)uoyant density differences between two bands or for the calculation of molecular weights using band widths unless corrections are made.

Experimental Procedures

All DNA studies utilize the ultraviolet optical system of the analytical ultracentrifuge. In recent years this means that the experimenter has flexibility with respect to the wavelength of light being utilized because most instruments are now equipped with a monoehromator. The most common wavelength used for DNA work is 265 nm because of the ab- sorption maximum of nucleic acids near this wavelength and because the high pressure mercury light source has a strong emission band at that wavelength. Most instruments arc also equipped with a photoelectric scanner which eliminates the need for photography, and with double- sector centerpieces, makes double-beam optics possible so that baselines are indeed flat even in very strong salt gradients. A major time-saving device is the multiplexer which makes it possible to run as many as six cells at a time and record the cells independently. I t is sometimes necessary to use a - 1 o wedge as the top window of the cell to compensate for the refractive index gradient generated by the salt solution. The gradient, if sufficiently high, bends the light beam so far that partial obstruction of the light beam occurs.

I t is advisable to avoid the use of aluminum or aluminum-filled Epon centerpieces because of the corrosive effects of salt solutions. The most satisfactory centerpieces are fabricated from titanium, ~3 charcoal-filled Epon, or Kel-F.

To avoid leaks, it is advisable to clean all cell parts before assembly and to lubricate the screw ring and upper Bakelite gasket. The cell should be tightened with a torque of 120-125 inch-pounds. About 30 seconds should be allowed for the flow of the centerpieces and gaskets during tightening. Scratched windows should be avoided. For DNA work, syringe needles must not be used to fill the cells because the DNA is broken by the shear in passing through the needles. This laboratory fills the partially assembled cells from the top using glass pipettes.

42 R. Anet and D. R. Strayer, Biochem. Biophys. Res. Commun. 37, 52 (1969). ~'~3. E. Hearst and H. B. Gray, Jr., Anal. Biochem. 24, 70 (1968).


Knowledge of the density of the homogeneous solution is necessary. Densities are most conveniently determined by weighing a known volume of solution in a 0.3-ml calibrated inicropipette. No at tempt is made to control the temperature of the sample during the measurement, which is performed at room temperature. The resulting densities are accurate to +_0.001 g/ml. Data for tile refractive index vs. density of salt solution greatly simplify the setting up of runs. Tim data for many salts can be obtained from the International Critical Tables, 44 and when not, available are readily measured. Certain of these data are given in Table VII.

I t is usually desirable to buffer the solution. The addition of buffer is readily accomplished during mixing of the concentrated salt solution with water and maerospeeies. Changes in pH up to 0.5 pH unit have been observed for buffers diluted into 7 molal CsC1 solutions. Buffers in low concentration do not, interfere with refractive index measurements. For Cs salts, additive volumes may be assumed in preparing solutions of de- sired density from concentrated stock solutions and water or buffer solutions. The additive mixing relation is discussed in more detail by Vinograd and tIearst . 1°

It, is necessary to estimate the amount of DNA or protein to add to the cell in order to end the run at equilibrium with a measurable band. I f Co is the polymer concentration at band center and C~ is the initial concentration uniformly distributed over the cell a helpful relation- ship is ~°

Co = 0 .40 - L Ci

where L is the length of the liquid column and ~ is the standard deviation of tim equilibrium band. Generally DNA increases in concentration by a factor of 10-20 in a full centrifuge cell so it is common to start a run with a DNA concentration about cw~6o = 0.03. " J a J 1 c m

A c k n o w l e d g m e n t

We wish to acknowledge that this work was supported in part by Grant GM 11180 of the U.S. Public Health Service and that C. W. S. was supported by a predoetoral fellowship No. 1-FO1-GM 46,314-01.

*4"Inlernational CriHral Tables." McGraw-Hill, New York, 1933.

[6] Density Gradient Sedimentation Equilibrium

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