Energetics of DNA twisting

J. Mol. Biol. (1983) 170, 9 8 3 - 1 0 0 7

Energetics of DNA

Twisting

H. Topoisomer Analysis DAVID SHOREt AND ROBERTL. BALDWIN Department of Biochemistry Stanford University Medical Center Stanford, Calif. 94305, U.S.A. (Received 25 March 1983, and in revised form 15 July 1983) A gel electrophoresis method has been developed for resolving small (_-__250bp~:) DNA topoisomers. In this size range only one major topoisomer band is observed, except for ligase closure conditions in which the probabilities are nearly equal for circularization by untwisting and overtwisting the corresponding linear DNA. The two probabilities are nearly equal when ATw is close to 0-5, if the mean helical twist of the linear DNA is n+ATw, where n is an integer and ATw is the fractional twist. We determine ATw of the linear DNA in standard conditions (20°C, no ethidium) by titration experiments in which ATw is varied at the time of ligase closure, either by changing temperature or ethidium concentration, The endpoint (ATw = 0-5) is found when the two topoisomers formed by untwisting and overtwisting are present at equal concentrations. This analysis assumes t h a t the net writhe is zero and the DNA helix is isotropically bendable. The results confirm the analysis of eyclization probabilities given in the preceding paper: ATw = 0 a t the two m a x i m a in the curve of j-factor versus D N A length and ATw = 0-5 a t the minimum. Consequently, we can determine the DNA lengths at which Tw takes on integral values and use them to measure precisely the average helix repeat. F r o m the difference between the ATw values of DNAs with 237 and 247 bp, we obtain an approximate value for the helix repeat of h = 10.4_0.1 bp/turn, in good agreement with earlier values found by the band-shift and nuclease-cutting methods. The twist is integral at 250.8___0.4 bp and from h = 10.4_0.1 we find n = 24; then 250.8/24 gives h = 10.45_+_0.02 bp/turn. The mean linking number (Lk) changes in a stepwise manner as ATw is varied for 250 bp DNAs. This result is expected when the free energy of twisting half a turn becomes large compared to thermal fluctuations. In these experiments, it is possible to obtain the mean Tw value from the mean L/c value only when ATw = 0.5, and consequently the mean Lk value is not simply related to DNA length for 250 bp DNAs except when ATw = 0.5. The relative concentrations of two adjacent topoisomers have been measured as a function of DNA twist for DNAs between 245 and 1361 bp, and the results have been used to analyze the contributions of twisting and writhing in determining t Present address: Medical Research Council Laboratory of Molecular Biology, Cambridge CB2 2QH, England. :~ Abbreviations used: bp, base-pair; kb, 10a base-pairs. 983 0022-2836/83/320983-25 $03.00]0

© 1983 Academic Press Inc. (London) Ltd.

984

D. SHORE AND R. L. BALDWIN the distribution of topoisomers. These experiments are analogous to those for much larger DNAs, which use the breadth of the topoisomer distribution to determine the free energy of supercoiling, as in the work of Depew & Wang (1975) and of Pulleyblank et al. (1975). For the small DNAs, the topoisomer titration data yield an apparent twisting coefficient Kapp that equals Kvw when fluctuations in writhe do not contribute to the topoisomer ratio. Le Bret's (1979) theory for the onset of writhing predicts that about two turns of torsional stress are taken up in twisting before writhing begins. Because the free energy of twisting depends on an inverse function of DNA length, Le Bret's theory can be used to predict that only twisting fluctuations will contribute to Kap~ below a critical DNA length. We find that K~pp/2RT is proportional to N between 245 and 880 bp, as expected if only twisting fluctuations contribute to Kapp in this size range. At about 1000 bp, the plot curves upwards, indicating that fluctuations in writhe begin to contribute to Kapp near l kb. These results are relevant to the problem of analyzing the contributions of twisting and writhing to the free energy of DNA supercoiling.

1. Introduction Our aim here is to develop a second method for analyzing the energetics of DNA twisting, in order to test the conclusions of the preceding paper (Shore & Baldwin, 1983). There we show t h a t the cyelization probability, or j-factor, of a linear DNA oscillates as the length changes: the period is about l0 bpf. The results indicate t h a t cyclization is accompanied by a change in twist (Tw) and t h a t the DNA helix is essentially continuous in nicked circular molecules. We develop first a simplified analysis of the data in which the DNA helix is assumed to be isotropically bendable, 250 bp circles remain planar, and any writhe can be neglected (see below). Then individual circular molecules take on integral values of Tw. I f the mean T w value of the linear DNA is n + A T w , then the change in T w on forming the major circular species (that formed b y minimal untwisting or overtwisting) is either A T w (if A T w is less than 0.5) or 1 - A T w (if A T w is greater than 0.5). Consequently, we develop here a method for measuring ATw. In a closed-circular DNA molecule, the linking number (L/c) is a fixed integral number in the absence of strand scission: Lk is the n u m b e r of times one strand is linked through the other. White (1969) has shown t h a t the L/C value of a closed ribbon is given by the sum of its twist and writhe, Wr (cf. Fuller, 1971; Crick, ]976): L k = T w + Wr.

(1)

T w is the number of times one strand is wrapped around the other (for a more precise definition, see Crick, 1976) and, unlike Lk, T w is a defined q u a n t i t y for a linear DNA. Writhing occurs when a closed DNA circle is bent out of a planar conformation. DNA molecules t h a t are chemically identical but differ in L k are termed topoisomers (topological isomers). When DNA molecules several thousand base-pairs in length are covalently joined into circles, t h e y give rise to a distribution of topoisomers t h a t can be separated by agarose gel electrophoresis. Each band corresponds to a unique topoisomer and successive bands differ in L/C by I. The relative concentrations of the topoisomers in the band pattern can be

See footnote to p. 983.

ENERGETICS OF DNA TWISTING. II.

985

fitted to a Gaussian envelope (Depew & Wang, 1975; Pulleyblank et al., 1975), as expected if they are produced by thermal fluctuations at the time of ligase closure (this has been proven directly) and if the free energy varies as the square of a fluctuation in T w and Wr. For small DNAs ( ~ 2 5 0 b p ) , only one or at the most two topoisomers are present at normal concentrations after closure by ligase, as we show here. In the following, we consider the problem of using measurements of such topoisomers to determine A T w of the linear DNA. When an isotropic thin rod is twisted uniformly, the energy required for twisting depends on the square of the turns of twist. This can be applied to DNA molecules (cf. Barkley & Zimm, 1979), homogeneous in length and sequence, for thermal fluctuations in T w by writing: Grw = K r ~ 2,

(2)

where Gr~ is the molar free energy of twisting, T is the number of turns of twist {in the same units as T w or Lk), and Kr~ is a twisting coefficient t h a t is inversely proportional to length. For the relation between Kr~ and the torsional rigidity of the DNA molecule, which is independent of length, see Barkley & Zimm (1979) and Shore & Baldwin (1983). The number of turns of twist, T, required to form a planar covalently closed circle from a linear DNA is: = i+ATw

(i = 0, l, 2 . . . . . - 1 , - 2 . . . . ).

(3)

The relative concentrations of topoisomers produced only by twisting fluctuations are: C~ ~ - = exp [ - G r w ( i ) / R T ] Ci

(4a)

= exp [ - K r w T ~ / R T ],

(45)

where C i is the concentration of topoisomer i and C is the total concentration of the set of topoisomers. When the mean twist of the linear DNA is n ÷ A T w , where n is an integer, then the topoisomer concentrations of the species formed by minimal untwisting ( L k = n) or overtwisting ( L k = n ÷ l) are: C, -~ -- exp [ - K r w ( A T w ) 2 / R T ] Cn+ 1

C

(5a)

= exp [ - K r ~ (l - A T w ) 2 / R T ] ,

(5b)

and their ratio is: Cn+ 1

--

c',

= exp [ - K r w ( 1 - 2 A T w ) / R T ] .

[a2(Wr) = 0].

(5c)

To determine A T w , we note t h a t the topoisomer ratio is 1 when A T w = 0.5 (eqn (5c)). Therefore, we determine when A T w = 0.5 by varying ligase closure

986

D. SHORE

AND

R. L.

BALDWIN

conditions and/or DNA length and finding when the topoisomer ratio is 1. Thus we titrate the Tw of the DNA, by varying temperature or ethidium concentration at the time of ligase closure, to the point where ATw = 0-5 and the topoisomer ratio is 1. From the amount of intercalated ethidium at the endpoint, or knowing the change in DNA twist with temperature, we can compute ATw of the linear DNA in the absence of ethidium, at 20°C. Provided that the DNA helix is isotropically bendable, this procedure remains valid when fluctuations in Wr affect the topoisomer ratio, because they affect the apparent twisting coefficient but do not change the value of ATw = 0-5 at which the topoisomer ratio is unity. Positive and negative fluctuations in Wr are equally likely for a symmetrical DNA helix, and the net Wr value is zero. A bend has been demonstrated in a kinetoplast DNA fragment (Marini et al., 1982), showing that the DNA helix is not symmetric for this fragment. The net writhe need not be zero when such a fragment is bent into a circular form. Such effects are considered as second-order effects here and are neglected in our analysis, whose purpose is to treat the first-order effects. Measurement of the topoisomer ratio as a function of ATw also provides a method for measuring Krw, as equation (5c) indicates. If fluctuations in writhe affect the topoisomer ratio, they will change the value of Krw. We retain the formalism of equation (5c) while recognizing the effect of fluctuations in Wr by changing Krw to Kapp: Cn+ 1

C,

-

exp [-Kapp(1-2ATw)/RT].

[a~(Wr) > 0].

(6)

The procedure for varying ATw is to vary the ethidium concentration at the time of ligase closure, or to vary temperature in the absence of ethidium. The change in ATw with ethidium or with temperature is calibrated by using a large DNA (pBR322, 4362 bp), for which there is a linear relation between mean Tw and mean Lk. To find out whether or not thermal fluctuations in Wr affect the topoisomer ratio, we investigate the length dependence of Kapp. For twisting only: (NKrw/RT) = constant,

(7)

when only the number of base-pairs, N, is varied. For a large DNA, which gives rise to a Gaussian distribution of topoisomers, the variance of Lk is (c£ Depew & Wang, 1975): a2(Lk) = RT/2Ka,p.

(8)

If only fluctuations in Tw are responsible for the topoisomer distribution, then: a2(Lk) -- a2(Tw) = RT/2Kr~.

[aS(Wr) = 0].

(9a)

If, however, fluctuations in Wr do affect the topoisomer distribution, then: a2(Lk) > (RT/2Kr~).

[a2(Wr) > 0].

(9b)

It has been suggested by Vologodskii et al. (1979) that fluctuations in Tw and Wr may be considered to be independent in nicked circles (c£ Benham, 1978) and that

ENERGETICS OF DNA TWISTING. II.

987

therefore a2(Wr) and a2(Tw) add to give a2(Lk). This is controversial (see Discussion) and for our purpose, which is to detect the onset of writhing, equations (9a) and (9b) are sufficient. I t follows from equations (7) to (9) that:

a2(Lk) = N/2(constant)

[a2(Wr) = 0]

(10a)

a2(Lk) > N/2(constant)

[a2(Wr) > 0],

(10b)

where the c o n s t a n t is t h a t given in equation (7). Consequently, if fluctuations in Wr are negligible below a critical length, the plot of (RT/2Kapp) versus N will be linear up to t h a t length and will then bend upwards. These m e a s u r e m e n t s of topoisomer ratios depend on being able to resolve topoisomers of small DNAs, and resolution in gel electrophoresis depends on writhing. We were unable to resolve topoisomers of 250 bp D N A s in agarose gel electrophoresis but, fortunately, we found t h a t we could get good resolution in the t y p e of thin p o l y a c r y l a m i d e gels used for D N A sequencing. These gels are run at high voltages (see Materials and Methods).

2. Materials and Methods (a) Source of DNA molecules The set of molecules from 237 to 254 bp were all derived from a 242 bp fragment by small internal additions or deletions. The construction of these molecules is described in the accompanying paper (Shore & Baldwin, 1983). All other molecules have been described by Shore et al. (1981). The DNA molecules used in this study were all EcoRI restriction fragments and were labeled with 32p at their 5' ends as described in Shore et al. (1981). (b) Enzymes and chemicals Ethidium bromide was Calbiochem (grade B) and was used without further purification. Ethidium bromide concentrations were determined by measuring the optical density at 480 nm of a freshly prepared stock solution (1/~g/ml). Sources of other chemicals and enzymes have been given elsewhere (Shore et al., 1981; Shore & Baldwin, 1983). (c) Ligase closure conditions The conditions for ligase closure are the same as those given by Shore et al. (1981): 20°C, l0 mM-Tris. HC1 (pH 7-5), 50 mM-NaCl, l0 mM-MgCl2, 5 mM-dithiothreitol, 0.25 mM-rATP, 0"2/~g DNA/ml, approx. 0-2/~g phage T4 DNA ligase/ml. Ethidium bromide was added, or the temperature was varied, as described. (d) Gel electrophoresis of small circles Resolution of small DNA circles (237 to 366 bp), differing by only one in linking number, was achieved by electrophoresis at high voltage through thin polyacrylamide gels. Gels were 5% (w/v) acrylamide (acrylamide/N, N'-methylene-bisacrylamide, 19 : 1, w/w), 90 mMTris (pH 8-1), 90 mM-boric acid, 2 mM-EDTA, and had dimensions of 40 cm x 19 c m x 0.038 cm. Electrophoresis was typically carried out at 1000 to 1300 V for 4 h. Gels were blotted onto 3MM paper and the DNA was visualized by autoradiography on Cronex film. Exposure times were adjusted to assure linear response from the film. Quantitation was obtained by scanning the autoradiograph with a Transydyne densitometer and measuring peak areas with a Hewlett-Packard digitizer. Larger DNA circles were separated by agarose gel electrophoresis. Gels were made from 1.2 to 2.5%

988

D. SHORE AND R. L. BALDWIN

agarose (concentration was dependent upon DNA size) and were run in the borate buffer described above or in 10 mM-Tris (pH 8-0), 40 mM-sodium acetate, 2 mM-EDTA. For the 2302 and 4362 bp DNAs, gels contained from 10 to 40 ng ethidium bromide/ml to allow resolution of all topoisomers. Agarose gel electrophoresis was typically carried out at from 3 to 5 V/cm for 12 to 18 h in a horizontal gel apparatus with buffer recirculation. Agarose gels were dried, autoradiographed and quantitated as described above. 3. Results

(a) Small DNAs produce only one major topoisomer species The set of 12 DNA molecules, derived from a common sequence, all containing EcoRI cohesive ends, and ranging in size from 237 to 254 bp, were converted to covalently joined monomer circles using T4 DNA ligase. The products of the ligation reactions were electrophoresed through polyacrylamide gels under conditions t h a t allow the resolution of adjacent topoisomers differing in linking number by one. In each case, the closed circular products were of a single linking number species. Figure 1 shows the results of such an experiment for seven of these fragments. The fragments examined differ in length over an interval of approximately one and one-half turns of the DNA helix. One might expect t h a t most of the fragments, because t h e y will have non-integral values of Tw, would both untwist and overtwist to form at least two different closed circular topoisomers. The fact t h a t a unique topoisomer is always observed indicates t h a t the energy required to change the total twist to an integral value is large compared to ksT (where ]cB is Boltzmann's constant) in this size range. Although no quantitative information is derived from this experiment, it is interesting to note t h a t there is a sharp discontinuity in the pattern of DNA length-dependent mobility between 245 and 249 bp. We demonstrate below t h a t position of exactly half-integral total helical twist occurs in this region. Molecules slightly smaller than this length untwist in order to reach an integral twist, whereas molecules with a larger value of twist will overtwist for joining. As a consequence of forming different linking number circles, the two sets of fragments will differ in their degree of supercoiling under the electrophoresis conditions and thus the mobility is dependent on length in a complex manner. (b) Measurement of the fractional twist of small DNAs Ethidium bromide unwinds the DNA helix upon intercalation, lowering the total twist and changing the distribution of topoisomers upon ligation. We consider two cases. The first case is t h a t of a large DNA (e.g. pBR322 DNA, 4362 bp), for which the free energy of twisting the entire DNA by half a turn is small compared to ksT and there are several topoisomer species in the band pattern. In this case, the average linking number ( L k ) changes continuously as the DNA is unwound by ethidium. Measurement of the change in ( L k ) gives the change in total twist produced by binding ethidium.

--A(Lk) = Tw-Two,

(1 l)

where T w o is the initial twist in the absence of ethidium. (Note t h a t ( W r ) = 0: positive and negative values occur with equal probability.) Measurement of

ENERGETICS OF DNA TWISTING. II.

989

~

254 bp

~

250 bp

245 bp 244 bp

~

242 bp

~

...,,,,,,,,,... ~

" 9 1 " - - " " - - 249 bp

237 bp

~'~

Fro. I. Autoradiogram of closed-circular DNAs separated by polyaerylamide gel electrophoresis as described in Materials and Methods. Lanes contain (from left to right) 237, 242, 244, 245, 249, 250 and 254 bp DNAs. Unless otherwise noted, all ligations reported here were performed under standard conditions (Shore et al., 1981): 20°C in a solution containing l0 mM-Tris. HCI (pH 7.5), l0 mM-MgCI2, 50 m,~l-NaCI, 5 mM-dithiothreitol, and 0.25 mM-rATP.

A(Lk) as a function of ethidium concentration, (Et), yields a binding isotherm for ethidium under ligation conditions. The resolution of different topoisomer species produced by such a titration for pBR322 DNA (4362 bp) is shown in Figure 2. Several agarose gels, each containing different concentrations of ethidium, were required to resolve the topoisomer species produced by ligation in concentrations of ethidium up to 0.4/~g/ml. Topoisomer bands (19 total) were ordered relative to each other for eight reaction mixtures and the first moment of each distribution was determined. A plot of A(Lk) as a function of ethidium concentration is shown in Figure 3. Binding is linear for small extents of binding (<:0.043 ethidium ion bound/bp DNA). In the linear region, the change in Tw per bp DNA is - 7 . 7 x 10 -3 (X), where (X) is ethidium concentration in/~g/ml. The fraction of the total ethidium bound to DNA is l'5~o; the DNA molarity (in bp) is 3x lO-TM.

990

D. S H O R E A N D R, L. B A L D W I N

pBR322

4362 bp

OC

Elhidium

FIG. 2. Autoradiogram of topoisomers of a 4362 bp closed-circular DNA separated by agarose gel electrophoresis as described in Materials and Methods. Lane 1 contains an unligated linear marker (L). Succeeding lanes contain DNAs ligated under standard conditions (20°C) in the presence of increasing concentrations of ethidium bromide. The position of open-circle (OC), or form II, DNA is marked. All other bands correspond to closed-circular topoisomers, Adjacent bands differ in linking number by one.

ENERGETICS

OF DNA T W I S T I N G . I I .

991

14 12 A

I0

v

8

i

6


•

•

4

2 0-1

0.2

0-3

Ethidium bromide concenfrotion (/J.glml.)

FIG. 3. Change in the average L k value (first moment of the linking number distribution) as a function of the ethidium bromide concentration for ligation for a 4362 bp DNA (conditions: s e e Fig. 1). The DNA concentration is small compared to the ethidium concentration, which may be taken as the free ethidium concentration.

The second case is that of a small DNA ( ~ 250 bp) for which the free energy of twisting half a turn is large compared to kBT. Figure 4 shows an autoradiogram of an ethidium bromide titration of a 247 bp fragment. A plot of A as a function of (Et) is shown in Figure 5. The striking feature of this titration curve is that it is not smooth, but changes in steps. The distribution of Lk does not reflect the amount of bound ethidium in a simple way. The fraction of sites bound by ethidium should be a smooth function of (Et), yet the observed pattern of topoisomers changes abruptly at critical values of Tw. This can be understood as the consequence of a twisting potential that is steep even for changes of only a fraction of a helical turn. Only when the total twist falls midway between two integral values, where the energy required to twist or untwist the helix for joining is approximately equal, will two topoisomers be observed. Then A will accurately reflect the amount of bound ethidium only at the midpoint of a change between two topoisomers. At the midpoint of the change in , we assume that the twist of the linear molecule before ligase closure is exactly half-integral:

Twm = n + 0 . 5

(at midpoint),

(12)

where n is an integer whose value can be estimated by a procedure discussed below. The difference in twist between the value at the midpoint and the value in the absence of ethidium is denoted by:

ATwmo = T w m - Tw o,

(12a)

and is determined directly from the measured binding isotherm for ethidium bromide (see above). We assume that the affinity of these small DNAs for

~-

Lk.( I )

~..

"~ . . . . . . . .

"~ ......

". . . . . .

~Y"~ I

Ethidium

"~"

~

~

Lk(3)

FIo. 4. Autoradiogram of topoisomers of a 247 bp DNA produced by ligation in increasing concentrations of ethidium bromide. Separation of topoisomers was achieved by polyacrylamide gel electrophoresis as described in Materials and Methods. Topoisomers are labeled according to the order of their appearance in increasing ethidium concentrations (0 to 1-4 ~g/ml).

~

( 2 )

Lk

ENERGETICS

OF DNA T W I S T I N G . I I .

I

i

993

I

5

i

0,5 1.0 I-5 Efhidlum bromide concentration (~g/rnt)

Fro. 5. Change in the average linking number as a function of ethidium bromide concentration at ligation for a 247 bp DNA (data of Fig. 4).

intercalated ethidium is the same as for the 4362 bp DNA shown in Figure 3. Since A T w o, the fractional twist in the absence of ethidium, is A T w o = T w o - n , it follows that: ATw o = Two-n

= 0.5-ATwmo.

(12b)

The value of A T w o is a measure of the amount of twisting or untwisting about the helix axis required to bring the ends into alignment for joining in the absence of ethidium. Figure 6 shows a plot of A T w o as a function of DNA length for the set of fragments from 237 to 254 bp. The data reveal the magnitude and the direction of the change in twist required to yield an integral linking number circle for each of the fragments. The two lines shown in Figure 6 are predicted values based on the helix repeat (see below) and on the measured midpoint (in bp) between two integral values of twist. These data correlate precisely with the measured values of the cyclization probability for this set of fragments (Shore & Baldwin, 1983): the amount of twisting or untwisting required for joining determines the curve of cyclization probability versus DNA length, for small changes in N. Thus we confirm the assumption used in calculating the twisting coefficient in that paper; namely, that the distance (in bp) from the maxima in cyclization probability is a direct measure of the fractional twist, A T w . In writing equation (12), we assume that the ratio of adjacent topoisomers is 1 when ( T w - n ) = 0.5. This assumption is justified by finding that the twisting potential is symmetric (Shore & Baldwin, 1983). (c) D e t e r m i n a t i o n of the helical repeat length o f D N A Careful titration with ethidium to the midpoint where the topoisomer ratio is 1 can give the position of the midpoint to ___0-005 helical turns. A titration

D. SHORE AND R. L. BALDWIN

994

-0.5

g

o

<3

0-5 I

235

I

I

~

~

I

T

240

i

f

v

I

. . . .

245

I

,

250

,

,

,

!

255

DNA length (base-pairs}

FIo. 6. The fractional twist (the difference between the total twist and the closest integer) as a function of DNA length for the series of DNAs between 237 and 254 bp. The lines are theoretical lines based on the precise measurement of the position of one-half fractional twist and the value of the helix repeat (see the text). experiment was performed on a mixture of two small fragments differing in length by l0 bp (237 and 247 bp). By covalently cyclizing both fragments in the same reaction mixture, identical conditions of ionic strength, t e m p e r a t u r e and ethidium concentration are assured. The resolution of the four resulting topoisomer species is achieved by polyacrylamide gel electrophoresis, as shown in Figure 7. Quantitation of such a gel is shown in Figure 8. According to equation (12b), the twist of the shorter (237 bp) fragment can be expressed as: Two(1 ) = n+O'5-ATwmo(1),

(13a)

and Figure 8 gives ATwmo = -0.173-t-0.005. The integral part of the initial twist of the larger fragment (247 bp) will be ( n + 1) and: Two(2 ) = ( n + l) + 0 . 5 - ATwmo(2).

(13b)

From Figure 8, ATwmo -- -0"136 +_0"005. The helix repeat (h), or the n u m b e r of base-pairs per turn of the helix, is given simply by the ratio of the n u m b e r of base-pairs to the number of helical turns: h = 10/[1 -- ATwmo(2) + ATwmo(1)].

(14)

The value we obtain from this experiment is h = 1 0 / [ 1 + 0 . 1 3 6 - - 0 . 1 7 3 ] = 10-4±0.1 bp per turn. We m a y also calculate h more precisely if we known n, the integral part of the total twist (see Discussion). (d) Measurement of the twisting coe~cient of DNA from studies of

ethidium binding The titration curves shown in Figure 8 provide q u a n t i t a t i v e information on the relative probabilities of untwisting or overtwisting to reach an integral twist. We

Ethidium

m

FIO. 7. Autoradiogram of topoisomers of 237 and 247 bp closed-circular DNAs produced by ligation in increasing concentrations of ethidium bromide. DNAs in each lane were cyclized in the same reaction mixture, containing from 0 to 120 ng ethidium bromide/ml. Topoisomers were separated by polyacrylamide gel electrophoresis and are labeled as in Fig. 4.

237 bk Lk ( I

247 bk Lk ( I

237bpLk ( 2 )

247 bp Lk ( 2

D. S H O R E A N D R. L. B A L D W I N

996

I

I

0-12

0.I4

I .....

l

I

I

I

I-0

A ..j V

<3 i 0.5

0.t0

0.I6

0.t8

0-2(

- ( T w - Tw O) (turns)

F](}. 8. Quantitationof ethidium titration of 237 bp (O) and 247 bp (O) closed-circulartopoisomers. The change in linking number is plotted against the change in helical twist produced by ethidium binding. interpret these data by equation (6) as outlined in the Introduction. A plot of the data for the 247 bp fragment is shown in Figure 9. The slope gives the value of Kapp/RT (see Fig. 12). The effect of fluctuations in the number of bound ethidium ions is discussed below.

(e) Measurement of the twisting coe~cient from the temperature dependence of the topoisomer distribution We have measured the topoisomer distribution of a 245 bp circle as a function of temperature from 20 to 0°C. By lowering the temperature at ligation for this fragment, we increase its total twist sufficiently to initiate the formation of a new topoisomer species. The titration is analogous to that caused by untwisting the 247 bp fragment with ethidium, except that in this case we proceed from linking number n - 1 to n. The temperature dependence of the helix pitch was determined by measuring the change in the topoisomer distribution of pBR322 supercoils in an analogous manner to the ethidium bromide measurements (data not shown). We obtain a value of - 2-72 x 10- s turns per deg.C per bp. The temperature dependence of the topoisomer distribution of the 245 bp fragment was followed by polyacrylamide gel electrophoresis, and is shown in Figure 10, together with the titration curve taken from quantitation of the autoradiogram. The temperature midpoint of the change in (Lk) is 10.6°C and the change in Tw in going from 20 ° to 10-6°C is (9-4°)(2-72 x 10-5)(245) = 0-062 turns. Thus the DNA length at which Tw (20°C) is half integral is 245+(0.062)(10.4)= 245.64 bp. From the ethidium titration of the 247 bp fragment (Fig. 8), we get ATwmo =--0.136 turns, so that the DNA length at which Tw (20°C, no ethidium) is half integral is 247-(0-136)(10.4) = 245.59 bp, in good agreement with the temperature titration

ENERGETICS

OF DNA TWISTING.

II.

997

po

;o ~- o

-bO I

O.l

t

-0-05

0

0,05

O-iO

i-2 ATw

Fro. 9. Plot of the ethidium titration of topoisomers of 247 bp fragment according to eqn (6). The slope of the line is equal to (K.pp/RT). (See the text.)

of the 245bp fragment. The plot of ( 1 - 2 A T w ) versus T ln(C.+l/C.) for the temperature shift titration of the 245 bp DNA is shown in Figure 10(c). The value of (K.pp/RT) obtained from this plot is 16.7, which agrees within experimental error with the value obtained by ethidium titration of the 247 bp fragment: (K~pp/RT) = 17-2. (f) Dependence on DNA length of (RT/2Kap p) We have also measured the value of (Kapp/RT) for DNAs of 366, 880 and 1361 bp by the methods described above. The 366 bp fragment was titrated with ethidium bromide and topoisomer separation was achieved by polyacrylamide gel electrophoresis. Topoisomer distributions of the 880 and 1361 bp fragments were both shifted by increasing the temperature at ligation; see Figure ll for temperature titration of the 880 bp DNA. The 880 bp fragment was also titrated by ethidium. Topoisomers of the 880 and 1361 bp fragments were separated on 2.5% and 2.0% agarose gels, respectively. The same value of (Kapp/RT) was obtained for the 880 bp fragment by ethidium titration and by temperature titration. Fragments of 2302 and 4362 bp were analyzed by the technique of Depew & Wang (1975), which fits a Gaussian curve to the mass distribution of topoisomers (data not shown). Data from the experiments described above are shown in Figure 12, where the value RT/2Kapp is plotted as a function of DNA length. Values of (NK~pp/RT) are given in the Figure legend. Two length regimes clearly emerge. Fragments below 1 kb define a straight line with an intercept at the origin, in agreement with equation (10a): twisting fluctuations will have an inverse length dependence with a zero intercept. The point at which writhing fluctuations begin to contribute (cf. Vologodskii et al., 1979) is defined by the abrupt increase in slope at about 1 kb.

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bound ethidium Each step in the topoisomer titration curve of average linking number versus ethidium concentration (Fig. 5, 247 bp) becomes progressively broader. The same behavior is observed in the ethidium titrations of all 12 DNA samples, and the shape of the first transition has been examined in detail for the 237 and 247 bp

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1002

D. SHORE AND R. L. BALDWIN

DNAs (see Fig. 8). The effect results from the broadening of the topoisomer distribution by a superimposed distribution in the number of ethidium molecules bound: cf. Le Bret (1979). The effect is easy to understand and is not difficult to treat quantitatively if one has an accurate distribution function for the bound ethidium at low extents of ethidium binding (J. A. Schellman, personal communication). It is intrinsically interesting, because most physical measurements of ligand binding give only average values of the number of ligands bound, without giving information about the distribution of bound ligand. We will not treat the problem here, but only point out its consequences for our work. It is obvious from Figure 5 that the broadening of a step in a topoisomer titration curve by the distribution of bound ethidium can be large compared to thermal broadening, which we want to measure in order to get (Kap,/RT). It is also obvious that we cannot use the second or third step in the titration curve for this purpose without a quantitative theory for the broadening effect of bound ethidium. We use the first step in the curve when it takes place at low extents of bound ethidium: (Et)< 0.15#g/ml. This corresponds to 3% of the maximal amount of ethidium that can be bound (N/2, where N is the number of bp: Bauer & Vinograd, 1970). The justification for using the first step in the curve, with this restriction, is that it gives the same value for (Kapp/RT) when ATw is varied either by ethidium binding or by changing temperature. This has been tested for 880 bp and for 247 to 245 bp fragments as discussed above.

4. Discussion (a) Resolution of small circle topoisomers We describe here a polyacrylamide gel electrophoresis technique for the resolution of topoisomer species of small DNA circles ( < 500 bp) that has allowed us to determine the fractional twist of the corresponding linear DNAs and the twisting energetics of the nicked circles. The basic problems faced in the separation of small circles is the virtual absence of writhe in these molecules under standard conditions of temperature and ionic strength. Agarose get electrophoresis does not provide resolution of open from closed circles of 250 bp DNA, except when ethidium bromide is added to the gel, in which case separation from open circles is achieved, but topoisomers comigrate. I f electrophoresis in thin potyacrylamide gels at high voltages is used instead, then migration of small DNA circles is extremely sensitive to small differences in conformation. Relative mobility as a function of linking number is similar to that of larger DNAs, in that a minimum in mobility is observed as the linking number is changed (see Fig. 4). Such behavior suggests that molecules ligated in the absence of ethidium are positively supercoiled under electrophoresis conditions. As the linking number is lowered by ligation at increasing values of (Et), first relaxed and then negatively supercoiled molecules are produced. Gels are typically run at high voltages (1000 to 1300 V across 40 cm) and the glass plates are commonly held at 50 to 60°C. These conditions may enhance conformational differences between topoisomers and may also change the sieving properties of the polyacrylamide gel.

ENERGETICS OF DNA TWISTING. II.

1003

(b) Measurement of fractional twist in small linear DNAs We have studied the topoisomers present after ligation as a function of ethidium bromide concentration at ligation for a series of DNAs between 237 and 254 bp. Typically, only a single topoisomer species is observed initially and the first transition to a new species occurs abruptly at a critical (JEt) value, which differs for each fragment. An identical experiment performed with pBR322 DNA (4362 bp) shows t h a t the average shift in linking number is a smooth function of (Et) as expected, and reveals the ethidium binding isotherm for the ligation conditions. The step-function nature of the ethidium binding curve for small DNAs can be understood as a direct consequence of the high energy of twisting at such short DNA lengths. When concentrations of two adjacent topoisomers are equal, we may assume as a consequence of the symmetry of the twisting potential (Shore & Baldwin, 1983), that the fractional twist of the linear DNA is 0.5. Knowing the ethidium bromide concentration or temperature at such a titration midpoint, we are able to calculate the fractional twist in the absence of ethidium or at a standard reference temperature. The discovery that thermal fluctuations in supercoiling give rise to an entire set of topoisomers with different linking numbers, either when a nicked DNA circle is sealed with Escherichia eoli DNA ligase (Depew & Wang, 1975) or when a covalently closed circle is opened and resealed with a type I topoisomerase (Pulleyblank et al., 1975), showed that a new tool was avilable for analyzing the helical parameters of DNA. Wang (1979a,b) used this to measure the pitch of the DNA helix for inserts in plasmid DNAs, and Peck & Wang (1981) studied the pitch of the helix as a function of DNA base composition and sequence. Their method is based on determining changes in , the center of the linking number distribution for DNAs large enough ( > 2 kb) to give Gaussian band patterns. Consequently, it is important to understand the behavior of in response to changes in helical twist. We find here that, for DNAs less than 1 kb, is not a direct measure of the helical twist before closure. As shown in Figure 5, changes in a stepwise manner as Tw is varied and the value of provides a measure of the twist only at the midpoint of a step. This helps to clarify the meaning of for larger DNAs. For DNAs larger than 2 kb, where the free energy of twisting by half a turn is less than kT, the non-integral residual of measures ATw, the fractional twist of the corresponding linear DNA. For DNAs less than 1 kb, can be used to measure ATw but this must be done by a titration experiment in which Tw is varied until the relative amounts of two adjacent topoisomers are equal. (c) The helix repeat The exact value of h, the helix repeat (bp per turn), has been measured by a careful titration with ethidium bromide of two fragments differing in length by exactly 10 bp. By measuring the ethidium titration curves for both the 237 and 247 bp DNAs in the same experiment, we determine the difference in fractional twist between the two fragments to high precision (_0.01 turn). We assume that

1004

D. SHORE AND R. L. BALDWIN

the two fragments produce circles t h a t differ in linking number by one. The difference in fractional twist between the two fragments allows us to determine precisely the number of base-pairs per one turn of the helix. A simple calculation yields h = 10.4±0-1 bp/turn. The added 10bp sequence is 5' G-G-G-C-T-CA-T-T-C 3' (60°/o G +C). Earlier measurements made by two different methods give values of 10-6(_0-1)bp/turn for natural DNAs of similar G + C content, either in solution (Wang, 1979a,b; Peck & Wang, 1981) or bound to surfaces (Rhodes & Klug, 1980,1981). The band-shift method of Wang (1979a,b) is closely related to the ethidium titration experiment we use here. For B-form fibers at high humidity, or wetted fibers, values of 10-0(_+0-15) and 9.9(+0.14) are obtained by X-ray diffraction methods (Zimmerman & Pheiffer, 1979). We note that the measured position of integral twist at 250.8 bp corresponds to a helix repeat of either 10-0, 10.45 or 10.9 bp/turn with 25, 24 or 23 total helical turns in the DNA molecule. The helix repeat in solution (Peck & Wang, 1981; Rhodes & Klug, 1981) and in crystals (Dickerson & Drew, 1981) depends slightly on sequence. The helix repeat that we measure corresponds to that of the ten added base-pairs in the 247 bp fragment (see above), which is close to the average value for the whole DNA. The measured position of integral twist at 250.8_+0.4 bp allows us to determine very precisely the average value of the helix repeat: h = (250.8/24) = 10.45_ 0.02 bp/turn. (d) The torsional rigidity of DNA By using equation (6) with DNA fragments from 237 to 880 bp, we obtain a value for (NKrw/RT), the length-independent twisting constant, of 4200_+200. For reasons discussed above, we associate Kapp solely with twisting in this size range. The value obtained by the topoisomer titration method is slightly higher than that derived by measuring cyclization probability, or j-factor, as a function of length for fragments from 237 to 254 bp. In that case (NKrw/RT) =3400_400. We do not know the reason for the difference. The value of Krw obtained from j-factor data is based on comparisons of maxima and minima in j, which are determined by extrapolation, not directly (see Shore & Baldwin, 1983). In the topoisomer titration method, the fact that the same value of (Kapp/RT) is obtained by varying ATw either by changing temperature or by binding ethidium suggests that there is no serious systematic error in either method. The effect of fluctuations in bound ethidium is discussed above, and we point out that the transition between two topoisomers must be induced by binding low amounts of ethidium (<0.15 #g/ml) in order to use this method. Our value for the temperature coefficient of the helical twist ( - 2 . 7 x 10-5 turns/deg. C per bp) is smaller than that found by Depew & Wang ( 1 9 7 5 : - 3 . 4 x 1 0 -5 turns/deg. C per bp) but our ionic conditions (10mM-Mg 2+, 50mM-Na +) are different from theirs (2 mM-Mg2+) and the pitch of the helix is known to vary with the ionic environment (Anderson & Bauer, 1978). The torsional rigidity of DNA, calculated by equation (15) of Shore & Baldwin (1983) from (NKr~/RT) = 4200 is C = 2.9x 10-19ergcm. This is significantly higher than the value of 1.3 x 10-19 to 1.4 x 10-19 erg cm found by time-resolved

ENERGETICS OF DNA TWISTING. II.

1005

measurements of the fluorescence depolarization of ethidium intercalated into DNA (Thomas et al., 1980; Millar et al., 1982); see also the spin label results of Hurley et al. {1982). Part of the difference might be caused by conditions: Na + versus Mg2+ . Millar et al. (1982) find, however, that their value of C is independent of Na + concentration above 0.1 M. Consideration of the sources of error in the first method of measuring C given by Shore & Baldwin (1983) and in the second method given here leads to the conclusion that both values of C should be underestimates. If writhing contributes to values of C measured by both methods for DNAs of less than 1 kb, then C will be too low. If fluctuations in bound ethidium contribute significantly to values found by the second method using ethidium bromide titration, then C will be too low. If the temperature coefficient of the helical pitch has been underestimated by us, then the value of C found by the second method, using temperature titration, will be too low. If, in the first method, the two ends of a DNA strand need not be exactly in register across a nick, then less twisting will be required and the value of C will be too low. Consequently, it appears that there is a real difference between the values of C found from the free energy of twisting and from the kinetics of fluorescence depolarization of ethidium intercalated into DNA. Values of C found from fluorescence depolarization data might be affected by the tilt angle of the bound ethidium and by the effects of intercalated ethidium on the local torsional properties of DNA. (e) Relative contributions of twist and writhe to topoisomer distributions

The problem of determining the relative contributions of twist and writhe to the linking number distribution is a problem of long standing: see Benham (1978), Le Bret (1978,1979), Vologodskii et al. (1979), Our results (Fig. 12) indicate that fluctuations in writhe are damped out below a critical length. This is the behavior expected from Le Bret's (1979) theory for the onset of writhing: about two turns of torsional stress will be absorbed by twisting before the planar circle buckles and writhing begins (see also Benham, 1983; Le Bret, 1983). The exact number of turns of twisting that precedes writhing depends on Poisson's ratio for a DNA molecule, which is not known accurately. The free energy of twisting a molecule by a given number of turns is inversely related to DNA length; consequently, there is a critical length below which thermal fluctuations will rarely exceed two turns of twist and so cause writhing. Our data (Fig. 12) indicate that this length is about 1000 bp. One can calculate from ( N K r w / R T ) = 3800 that the free energy of twisting a DNA molecule by half a turn is almost equal to kBT at 1000 bp. The free energy of twisting the molecule by two turns will be 16 times larger. Consequently, it is reasonable to suppose that thermal fluctuations will rarely exceed the energy needed to twist two turns when the length is 1000 bp. To decide this point a priori, it is necessary to know more about the theory of writhing. The probability of writhing is enhanced by the fact that there are many different ways that writhing can occur: cf. Benham (1978,1983); Vologodskii et al. (1979); Le Bret (1978,1979,1983). It has been suggested by Vologodskii et al. (1979) that the variances of T w and

100(i

D. SHORE AND R. L. BALDWIN

Wr should add to give a2(Lk), because fluctuations in Tw and Wr are independent processes in nicked circles (cf. Benham, 1978). I f this is correct, it would mean t h a t a2(Wr) can be determined from a2(Lk) when the value of Krw is known (see eqn (10a)). There are reasons to suppose, however, t h a t fluctuations in Tw and Wr are not independent. C. J. Benham pointed out to us t h a t if thermal fluctuations first cause some n u m b e r of turns of twist before writhing begins (Le Bret, 1979), this is unlike the behavior expected for independent processes. B . H . Zimm pointed out to us t h a t the nicked circles t h a t are covalently closed by DNA ligase are a special subset t h a t already have the conformations of covalently closed circles, again implying t h a t Tw and Wr are not independent in this subset. Until more is known about the relationship between Tw and Wr, it seems better to use d a t a such as those in Figure 12 to determine the onset of writhing t h a n to calculate the actual magnitude of a2(Wr). (f) Other related work At the time our papers were submitted for publication, we sent copies to scientists working on related problems and received a n u m b e r of interesting replies. Before receiving our paper, Shimada and Y a m a k a w a sent us their t h e o r y for the effect of DNA twist on cyclization probability and topoisomer distributions (Shimada & Yamakawa, 1984). The theory is developed for a helical wormlike coil model and analytical solutions are obtained. I t was remarkable to us to find t h a t t h e y had predicted the behavior of the j-factor t h a t we observed. Horowitz & Wang (1983) sent d a t a as a function of DNA length for the q u a n t i t y we denote as (NKapp/RT). Their results, which are obtained by a technique similar in principle to ours, agree quantitatively for 200 bp DNAs but t h e y observe a decrease in (NKapp/RT) between 200 and 1000 bp, in contrast to our results (see the legend to Fig. 12). The reason for the difference is not known. Their results agree with ours in showing t h a t (NKapp/RT) rises sharply below 2000 bp. T h e y avoid use of the Tw value of the linear DNA in developing their equations, and a reference linking n u m b e r plays the role in their theory t h a t ATw, the fractional twist of the linear DNA, has in our equations. Our work was helped by discussions with Drs C. J. Benham, P. J. Hagerman, J. A. Schellman, J. Widom and B. H. Zimm. We thank Drs C. J. Benham, M. Le Bret, H. Yamakawa and J. C. Wang for copies of their unpublished manuscripts. We also thank Dr J. C. Wang for his comments on the manuscript. D.S. received predoctoral training support from an NIH training grant. The research was supported by NIH grant GM 2 R01 19988-22. REFERENCES Anderson, P. & Bauer, W. (1978). Biochemistry, 17, 594-601. Barkley, M. D. & Zimm, B. H. (1979). J. Chem. Phys. 70, 2991-3007. Bauer, W. & Vinograd, J. (1970). J. Mol. Biol. 47, 419-435. Benham, C. J. (1978). J. Mol. Biol. 123, 361-370. Benham, C. J. (1983). Biopolymers, in the press. Crick, F. H. C. (1976). Proc. Nat. Acad. Sci., U.S.A. 73, 2639-2643.

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Depew, P~. E. & Wang, J. C. (1975). Proc. Nat. Acad. Sci., U.S.A. 72, 4275-4279. Dickerson, R. E. & Drew, H. R. (1981). J. Mol. Biol. 151,535-556. Fuller, F. B. (1971). Proc. Nat. Acad. Sci., U.S.A. 68, 815-819. Horowitz, D. S. & Wang, J. C. (1983). J. Mol. Biol. In the press. Hurley, I., Osei-Gyimah, P., Archer, S., Scholes, C. P. & Lerman, L.S. (1982). Biochemistry, 21, 4999-5009. Le Bret, M. (1978). Biopolymers, 17, 1939-1955. Le Bret, M. (1979). Biopolymers, 18, 1709-1725. Le Bret, M. (1983). Biopolymers. In the press. Marini, J. C., Levene, S. D., Crothers, D. M. & Englund, P. T. (1982). Proc. Nat. Acad. Sci., U.S.A. 79, 7664-7668. Millar, D. P., Robbins, R. J. & Zewail, A. H. (1982). J. Chem. Phys. 76, 2080-2094. Peck, L. J. & Wang, J. C. (1981). Nature (London), 292, 375-378. Pulleyblank, D. F., Shure, M., Tang, D., Vinograd, J. & Vosberg, H.-P. (1975). Proc. Nat. Acad. Sci., U.S.A. 72, 4280-4284. Rhodes, D. & Klug, A. (1980). Nature (London), 286, 573-578. Rhodes, D. & Klug, A. (1981). Nature (London), 292, 378-380. Shimada, J. & Yamakawa, H. (1984). Macromolecules. In the press. Shore, D. & Baldwin, R. L. (1983). J. Mol. Biol. 170, 957-981. Shore, D., Langowski, J. & Baldwin, R. L. (1981). Proc. Nat. Acad. Sci., U.S.A. 78, 48334837. Thomas, J. C., Allison, S. A., Appelof, C. J. & Schurr, J. M. (1980). Biophys. Chem. 12, 177-188. Vologodskii, A. V., Anshelevich, V. V., Lukashin, A. V. & Frank-Kamenetskii, M.D. (1979). Nature (London), 280, 294-298. Wang, J. C. (1979a). Proc. Nat. Acad. Sci., U.8.A. 76, 200-203. Wang, J. C. (1979b). Cold Spring Harbor Symp. Quant. Biol. 43, 29-33. White, J. H. (1969). Amer. J. Math. 90, 693-728. Zimmerman, S. B. & Pheiffer, B. H. (1979). Proc. Nat. Acad. Sci., U.S.A. 76, 2703-2707.

Edited by M. Gellert