Torsional and Bending Rigidity of the Double Helix from Data on Small DNA Rings

Abstract

We have calculated the variance of equilibrium distribution of a circular wormlike polymer chain over the writhing number, ?(Wr)²?, as a function of the number of Kuhn statistical segments, n, For large n these data splice well with our earlier results obtained for a circular freely jointed polymer chain. Assuming that ?(ΔLk)²? = ?(ΔTw)²? + ?(Wr)²? we have compared our results with experimental data on the chain length dependence of the ?(ΔLk) ²? value recently obtained by Horowitz and Wang for small DNA rings. This comparison has shown an excellent agreement between theory and experiment and yielded a reliable estimate of the torsional and bending rigidity parameters. Namely, the torsional rigidity constant is C = 3.0·10^?19 erg cm, and the bending rigidity as expressed in terms of the DNA persistence length is a = 500 A. The obtained value of C agrees well with earlier estimates by Shore and Baldwin as well as by Horowitz and Wang whereas the a value is in accord with the data of Hagerman. We have found the data of Shore and Baldwin on the chain length dependence of the ?(ΔLk) ²? value to be entirely inconsistent with our theoretical results.     

Effect of salt-dependent stiffness on the conformation of a stressed DNA loop containing initially coplanar bends

Closed DNA loops containing one or more bent regions are important structures that occur in the regulation of gene expression. We analyze the response of structures of this type to a change in applied rotation (change in linking deficiency, delta Lk). Our results apply to a closed loop formed from an elastic rod that is intrinsically bent in Nb discrete, 20 degrees steps up to a maximum of 240 degrees, the bent regions being initially coplanar with the plane of the relaxed DNA loop. We determine the effect of changing the intrinsic elastic resistance of the DNA loop to bending and torsional deformations. This relative resistance is expressed by Poisson's ratio v, which depends upon the ratio of bending stiffness to torsional rigidity. Poisson's ratio is primarily a function of salt type and concentration. We find that the tertiary structure of DNA loops changes with delta Lk, but that the geometric response can be either of two quite different types, depending upon the precise (Nb, v) pair. For combinations of Nb and v that are above a critical curve (the Fickel curve), the response to increasing delta Lk is nonmonotonic (NMT region): the distance between the loop closure point and its diametric opposite first decreases, then increases, as delta Lk increases. For combinations of Nb and v that are below the Fickel curve (NMT region), the corresponding diameter never increases, but always decreases with increasing delta Lk. In addition to these results, we define and implement a new measure of tertiary structure in closed DNA: the absolute writhe, AWr.     

Monte Carlo simulations of supercoiling free energies for unknotted and trefoil knotted DNAs.

A new Monte Carlo (MC) algorithm is proposed for simulating inextensible circular chains with finite twisting and bending rigidity. This new algorithm samples the relevant Riemann volume elements in a uniform manner, when the constraining potential vanishes. Simulations are performed for filaments comprising 170 subunits, each containing approximately 28 bp, which corresponds to a DNA length of 4770 bp. The bending rigidity is chosen to yield a persistence length, P = 500 A, and the intersubunit potential is taken to be a hard-cylinder potential with diameter d = 50 A. This value of d yields the same second virial coefficient as the electrostatic potential obtained by numerical solution of the Poisson-Boltzmann equation for 150 mM salt. Simulations are performed for unknotted circles and also for trefoil knotted circles using two different values of the torsional rigidity, C = (2.0 and 3.0) x 10(-19) dyne cm2. In the case of unknotted circles, the simulated supercoiling free energy varies practically quadratically with linking difference delta l. The simulated twist energy parameter ET, its slope dET/dT, and the mean reduced writhe <w>/delta l for C = 3 x 10(-19) dyne cm2 all agree well with recent simulations for unknotted circles using the polygon-folding algorithm with identical P, d, and C. The simulated ET vs. delta l data for C = 2.0 x 10(-19) dyne cm2 agree rather well with recent experimental data for p30 delta DNA (4752 bp), for which the torsional rigidity, C = 2.07 x 10(-19) dyne cm2, was independently measured. The experimental data for p30 delta are enormously more likely to have arisen from C = 2.0 x 10(-19) than from C = 3.0 x 10(-19) dyne cm2. Serious problems with the reported experimental assessments of ET for pBR322 and their comparison with simulated data are noted. In the case of a trefoil knotted DNA, the simulated value, (ET)tre, exceeds that of the unknotted DNA, (ET)unk, by approximately equal to 1.40-fold at magnitude of delta l = 1.0, but declines to a plateau about 1.09-fold larger than (ET)unk when magnitude of delta l > or = 15. Although the predicted ratio, (ET)tre/(ET)unk approximately equal to 1.40, agrees fairly well with recent experimental measurements on a 5600-bp DNA, the individual measured ET values, like some of those reported for pBR322, are so large that they cannot be simulated using P = 500 A, d = 50 A, and any previous experimental estimate of C.     

Statistical mechanics of DNA topoisomers. The helical worm-like chain

Recent experimental data of Shore & Baldwin (1983b) and of Horowitz & Wang (1984) for the apparent twisting coefficient K, which determines the breadth of the Gaussian distribution of DNA topoisomers with different linking numbers N, show that the product of K and nbp (the number of base-pairs) is nearly a constant for nbp approximately greater than 2000, but that it increases sharply with decreasing nbp for nbp approximately less than 2000. The main purpose of the present paper is to explain theoretically such behavior of K as a function of nbp. Thus the statistical mechanics of DNA topoisomers in general is developed on the basis of a twisted worm-like chain, i.e. a special case of the helical worm-like chain. The previous treatments of the N-dependent ring-closure probability, i.e. the distribution of N, which are valid only for small chain length L, are extended to the range of larger L. The variance of N is then shown to be exactly the sum of those of the writhe Wr and the twist Tw. For small values of L, the distribution of Wr is not Gaussian, and its variance or moment (Wr2) increases rather steeply with increasing L. With these and known Monte Carlo results for freely jointed chains, an empirical interpolation formula for (Wr2) is also constructed to be valid for all values of L. It predicts that (Wr2)/L increases monotonically, with increasing L to its coil-limiting value. On the other hand, the distribution of N is actually Gaussian in the practical range of N for all values of L. The conditional distribution of Wr with N fixed is also evaluated. Finally, K is expressed in terms of the torsional constant C, the stiffness parameter lambda-1 (which is equal to the Kuhn segment length and twice the persistence length for this special case), and (Wr2). The derived equation predicts that nbpK decreases monotonically to its coil-limiting value with increasing nbp. This decrease arises from the fluctuation in Wr and its neglect leads to an underestimate of C by 7 to 10%, even for short DNA with nbp approximately equal to 200. From an analysis of the experimental data of the two groups, the estimates of C = 3.1 to 3.2 X 10(-19) erg cm and lambda-1 = 1000 to 1200 A are obtained.     

Variance of writhe for wormlike DNA rings with excluded volume

We have calculated the variance of the equilibrium distribution of a circular wormlike polymer chain over the writhing number, less than (Wr)2 greater than, with allowance for the excluded volume effects. Within this model the less than (Wr)2 greater than value is a function of the number of Kuhn statistical segments, n, and the chain diameter, d measured in Kuhn statistical lengths, b. Simulated DNA chains varied from 200 to 10,000 base pairs and the d value varied from 0.02 to 0.2. Theory predicts a considerable ionic strength dependence of the DNA superhelix energy as a consequence of the change in the DNA diameter. A comparison with the available experimental data has yielded an estimate of the DNA torsional rigidity, the Kuhn statistical length, and the effective diameter of the double helix under conditions of the complete screening of the DNA electrostatic potential.     

Effect of nucleosome distortion on the linking deficiency in relaxed minichromosomes

The wrapping of closed circular DNA on a protein surface, followed by relaxation with a topoisomerase and removal of proteins, produces a characteristic DNA linking deficiency, delta Lk. We show that the magnitude of delta Lk depends upon the surface shape, and we calculate changes in delta Lk caused by particular distortions of the protein wrapping surface. If the DNA remains attached to the surface during distortion, the DNA winding number, phi, is not altered. The change in delta Lk is then equal to the change in the surface linking number, SLk, which is a straightforward measure of the wrapping of the DNA around the surface. For left-handed wrapping, as in a nucleosome, SLk = -n, the number of times that the DNA axis winds around the axis of the protein complex. We calculate values of SLk for the helical wrapping of a constant length of DNA on protein surfaces having the shapes of cylinders and of ellipsoids and hyperboloids of revolution. If the equatorial radius of the protein is fixed, change in shape from a cylinder to a hyperboloid increases SLk, while the corresponding change to an ellipsoid reduces SLk. We apply the general results to the interpretation of experiments in which minichromosomes are relaxed with topoisomerase at various temperatures and delta Lk is determined. The result is that a distortion of the nucleosome core by at most 5% (the change in the radius at the axial extremity relative to the equator) is sufficient to explain the observed delta Lk changes.     

Variance of Writhe for Wormlike DNA Rings with Excluded Volume

Abstract

We have calculated the variance of the equilibrium distribution of a circular wormlike polymer chain over the writhing number, <((Wr)² )>, with allowance for the excluded volume effects. Within this model the <((Wr)² )> value is a function of the number of Kuhn statistical segments, n, and the chain diameter, d measured in Kuhn statistical lengths, b. Simulated DNA chains varied from 200 to 10,000 base pairs and the d value varied from 0.02 to 0.2. Theory predicts a considerable ionic strength dependence of the DNA superhelix energy as a consequence of the change in the DNA diameter. A comparison with the available experimental data has yielded an estimate of the DNA torsional rigidity, the Kuhn statistical length, and the effective diameter of the double helix under conditions of the complete screening of the DNA electrostatic potential.     

A semiempirical extension of polyelectrolyte theory to the treatment of oligoelectrolytes: Application to oligonucleotide helix-coil transitions

The interaction of counterions with a suitably long, charged oligomer appears susceptible to treatment in the context of polyelectrolyte theory by the introduction of an end-effect parameter that reflects the reduced association of counterions with the terminal regions of the oligo-ion. Use of a physically reasonable value for the end-effect parameter provides excellent agreement between theory and the experimental data of Elson, Scheffler, and Baldwin [J. Mol. Biol. 54 , 401–415 (1970)] on the dependences of melting temperature on salt concentration and chain length for a series of hairpin helices formed by d(TA) oligomers. The differences in behavior expected for hairpin, dimer, and oligomer-polymer helices are discussed. The salt dependence of the end-joining equilibrium investigated for λ DNA by Wang and Davidson [Cold Spring Harbor Symp. Quant. Biol. 33 , 409–415 (1968)] is treated as an oligomer–polymer interconversion. The dependence of equilibrium constant for this reaction on counterion concentration is in good agreement with that predicted by theory for an end-region totalling 24 nucleotides, the known length of the λ ends.     

Effect of anisotropic bending rigidity and finite twisting rigidity on statistical properties of DNA model filaments

The persistence length and effective long-range bending rigidity are derived for a discrete model of an anisotropically bending filament and shown to be independent of the torsional rigidity. The twisting persistence length is found to be independent of the anisotropic bending rigidity. Other statistical properties are briefly discussed, including the dependence of tangent vector projections on contour length. The dependence of a tensor contraction on contour length is derived for an isotropically bending filament with no equilibrium twist.     

Energetics of DNA twisting. II. Topoisomer analysis

A gel electrophoresis method has been developed for resolving small (approximately equal to 250 bp 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 delta Tw is close to 0.5, if the mean helical twist of the linear DNA is n + delta Tw, where n is an integer and delta Tw is the fractional twist. We determine delta Tw of the linear DNA in standard conditions (20 degrees C, no ethidium) by titration experiments in which delta Tw is varied at the time of ligase closure, either by changing temperature or ethidium concentration. The endpoint (delta Tw = 0.5) is found when the two topoisomers formed by untwisting and overtwisting are present at equal concentrations. This analysis assumes that the net writhe is zero and the DNA helix is isotropically bendable. The results confirm the analysis of cyclization probabilities given in the preceding paper: delta Tw = 0 at the two maxima in the curve of j-factor versus DNA length and delta Tw = 0.5 at 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. From the difference between the delta Tw 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 delta Tw 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 Lk value only when delta Tw = 0.5, and consequently the mean Lk value is not simply related to DNA length for 250 bp DNAs except when delta Tw = 0.5.(ABSTRACT TRUNCATED AT 400 WORDS)     

Scaling of long bone fracture strength with animal mass

Most long bone fractures are the result of bending and/or torsional loading. To allometrically relate bone torsional and bending strength to animal mass (M), we define the bone strength index SB = J/dl where J = midshaft cross section polar moment of inertia, d = diameter, and l = length. In geometrically similar scaling, one would expect SB alpha M2/3. In this study, long bone geometric parameters were measured for 12 species of Artiodactyls. The relationships determined for length and diameter are similar to those reported by previous investigators (l alpha d3/4, l alpha M1/4). For the Artiodactyls studied, we found that SB alpha M0.82. Data previously collected by Biewener on a wide range of mammals (non-Artiodactyls) showed different scaling characteristics (l alpha d0.89, l alpha M0.31). However, our analysis of his data suggests roughly similar scaling of the torsional and bending strength index, SB alpha M0.77. It therefore appears that, in spite of differences in scaling of length and external diameter, the bending and torsional strengths scale similarly across a broad range of animals.     

Topological distributions and the torsional rigidity of DNA. A Monte Carlo study of DNA circles

Distributions of the linking number of circular DNA molecules, defined as the sum of twist and the writhing number, are obtained by Monte Carlo simulations of small, randomly closed DNA circles. We estimate the relative contributions of fluctuations in twist and writhe to the linking number distribution, as functions of DNA size. Published experimental data on topoisomer distributions in circular DNA molecules are interpreted to estimate the torsional rigidity of DNA in solution. We show that ignoring the writhe component of the linking number distribution, even for DNA circles as small as 250 base-pairs, leads to an underestimate for the torsional stiffness of the double helix. The value of the torsional modulus obtained from this analysis, C = 3.4 X 10(-19) erg cm, is from 10 to 40% larger than that estimated by others and more than twice as large as the values obtained from fluorescence depolarization or other time-resolved spectroscopic measurements. We also develop further the theoretical treatment of ring closure probabilities for DNA described in the previous article. It is shown that the torsional part of the ring closure probability, phi 0,1 (tau 0) is a periodic function of DNA length that contributes strongly to the ring closure probability for short chains but makes negligible contributions for chains over 1000 base-pairs in length.     

The influence of salt on the structure and energetics of supercoiled DNA.

We present a detailed computational study of the influence of salt on the configurations, energies, and dynamics of supercoiled DNA. A potential function that includes both elastic and electrostatic energy components is employed. Specifically, the electrostatic term, with salt-dependent coefficients, is modeled after Stigter's pioneering work on the effective diameter of DNA as a function of salt concentration. Because an effective charge per unit length is used, the electrostatic formulation does not require explicit modeling of phosphates and can be used to study long DNAs at any desired resolution of charge. With explicit consideration of the electrostatic energy, an elastic bending constant corresponding to the nonelectrostatic part of the bending contribution to the persistence length is used. We show, for a series of salt concentrations ranging from 0.005 to 1.0 M sodium, how configurations and energies of supercoiled DNA (1000 and 3000 base pairs) change dramatically with the simulated salt environment. At high salt, the DNA adopts highly compact and bent interwound states, with the bending energy dominating over the other components, and the electrostatic energy playing a minor role in comparison to the bending and twisting terms. At low salt, the DNA supercoils are much more open and loosely interwound, and the electrostatic components are dominant. Over the range of three decades of salt examined, the electrostatic energy changes by a factor of 10. The buckling transition between the circle and figure-8 is highly sensitive to salt concentration: this transition is delayed as salt concentration decreases, with a particularly sharp increase below 0.1 M. For example, for a bending-to-twisting force constant ratio of A/C = 1.5, the linking number difference (delta LK) corresponding to equal energies for the circle and figure-8 increases from 2.1 to 3.25 as salt decreases from 1.0 to 0.005 M. We also present in detail a family of three-lobed supercoiled DNA configurations that are predicted by elasticity theory to be stable at low delta Lk. To our knowledge, such three-dimensional structures have not been previously presented in connection with DNA supercoiling. These branched forms have a higher bending energy than the corresponding interwound configurations at the same delta Lk but, especially at low salt, this bending energy difference is relatively small in comparison with the total energy, which is dominated by the electrostatic contributions. Significantly, the electrostatic energies of the three-lobed and (straight) interwound forms are comparable at each salt environment.(ABSTRACT TRUNCATED AT 400 WORDS)     

Cofilin increases the bending flexibility of actin filaments: implications for severing and cell mechanics

We determined the flexural (bending) rigidities of actin and cofilactin filaments from a cosine correlation function analysis of their thermally driven, two-dimensional fluctuations in shape. The persistence length of actin filaments is 9.8 μm, corresponding to a flexural rigidity of 0.040 pN μm². Cofilin binding lowers the persistence length ∼5-fold to a value of 2.2 μm and the filament flexural rigidity to 0.0091 pN μm². That cofilin-decorated filaments are more flexible than native filaments despite an increased mass indicates that cofilin binding weakens and redistributes stabilizing subunit interactions of filaments. We favor a mechanism in which the increased flexibility of cofilin-decorated filaments results from the linked dissociation of filament-stabilizing ions and reorganization of actin subdomain 2 and as a consequence promotes severing due to a mechanical asymmetry. Knowledge of the effects of cofilin on actin filament bending mechanics, together with our previous analysis of torsional stiffness, provide a quantitative measure of the mechanical changes in actin filaments associated with cofilin binding, and suggest that the overall mechanical and force-producing properties of cells can be modulated by cofilin activity.     

Bone modeling response to voluntary exercise in the hindlimb of mice

The functional adaptation of juvenile mammalian limb bone to mechanical loading is necessary to maintain bone strength. Diaphyseal size and shape are modified during growth through the process of bone modeling. Although bone modeling is a well-documented response to increased mechanical stress on growing diaphyseal bone, the effect of proximodistal location on bone modeling remains unclear. Distal limb elements in cursorial mammals are longer and thinner, most likely to conserve energy during locomotion because they require less energy to move. Therefore, distal elements are hypothesized to experience greater mechanical loading during locomotion and may be expected to exhibit a greater modeling response to exercise. In this study, histomorphometric comparisons are made between femora and tibiae of mice treated with voluntary exercise and a control group (N = 20). We find that femora of exercised mice exhibit both greater bone growth rates and growth areas than do controls (P < 0.05). The femora of exercised mice also have significantly greater cortical area, bending rigidity, and torsional rigidity (P < 0.05), although bending and torsional rigidity are comparable when standardized by bone length. Histomorphometric and cross-section geometric properties of the tibial midshaft of exercised and control mice did not differ significantly, although tibial length was significantly greater in exercised mice (P < 0.05). Femora of exercised mice were able to adapt to increased mechanical loading through increases in compressive, bending, and torsional rigidity. No such adaptations were found in the tibia. It is unclear if this is a biomechanical adaptation to greater stress in proximal elements or if distal elements are ontogenetically constrained in a tradeoff of bone strength of distal elements for bioenergetic efficiency during locomotion.     

Gapped DNA and cyclization of short DNA fragments

We use the cyclization of small DNA molecules, approximately 200 bp in length, to study conformational properties of DNA fragments with single-stranded gaps. The approach is extremely sensitive to DNA conformational properties and, being complemented by computations, allows a very accurate determination of the fragment's conformational parameters. Sequence-specific nicking endonucleases are used to create the 4-nt-long gap. We determined the bending rigidity of the single-stranded region in the gapped DNA. We found that the gap of 4 nt in length makes all torsional orientations of DNA ends equally probable. Our results also show that the gap has isotropic bending rigidity. This makes it very attractive to use gapped DNA in the cyclization experiments to determine DNA conformational properties, since the gap eliminates oscillations of the cyclization efficiency with the DNA length. As a result, the number of measurements is greatly reduced in the approach, and the analysis of the data is greatly simplified. We have verified our approach on DNA fragments containing well-characterized intrinsic bends caused by A-tracts. The obtained experimental results and theoretical analysis demonstrate that gapped-DNA cyclization is an exceedingly sensitive and accurate approach for the determination of DNA bending.     

Torsional rigidities of weakly strained DNAs

Measurements on unstrained linear and weakly strained large (> or =340 bp) circular DNAs yield torsional rigidities in the range C = 170-230 fJ fm. However, larger values, in the range C = 270-420 fJ fm, are typically obtained from measurements on sufficiently small (< or =247 bp) circular DNAs, and values in the range C = 300-450 fJ fm are obtained from experiments on linear DNAs under tension. A new method is proposed to estimate torsional rigidities of weakly supercoiled circular DNAs. Monte Carlo simulations of the supercoiling free energies of solution DNAs, and also of the structures of surface-confined supercoiled plasmids, were performed using different trial values of C. The results are compared with experimental measurements of the twist energy parameter, E(T), that governs the supercoiling free energy, and also with atomic force microscopy images of surface-confined plasmids. The results clearly demonstrate that C-values in the range 170-230 fJ fm are compatible with experimental observations, whereas values in the range C > or = 269 fJ fm, are incompatible with those same measurements. These results strongly suggest that the secondary structure of DNA is altered by either sufficient coherent bending strain or sufficient tension so as to enhance its torsional rigidity.     

Molecular characterization of transesterification activity of novel lipase family I.1

Lipase’s thermostability and organic solvent tolerance are two crucial properties that enable it to function as a biocatalyst. The present study examined the characteristics of two recombinant thermostable lipases (Lk2, Lk3) based on transesterification activity. Conversion of C12-C18 methyl ester with paranitrophenol was investigated in various organic solvent. Both lipases exhibited activity on difference carbon chain length (C12 - C18, C18:1, C18:2) of substrates. The activity of Lk2 was higher in each of substrate compared with that of Lk3. Experimental findings showed that the best substrates for Lk2 and Lk3 are C18:1 and C18:2 respectively, in agreement with the computational analysis. The activity of both enzymes prefers on nonpolar solvent. On nonpolar solvent the enzymes are able to keep its native folding shown by the value of radius gyration, solvent–enzyme interaction and orientation of triad catalytic residues. Lk3 appeared to be more thermostable, with maximum activity at 55°C. The presence of Fe³⁺ increased the activity of Lk2 and Lk3. However, the activity of both enzymes were dramatically decreased by the present of Ca²⁺ despite of the enzymes belong to family I.1 lipase known as calcium dependent enzyme. Molecular analysis on His loop of Lk2 and Lk3 on the present of Ca²⁺ showed that there were shifting on the orientation of catalytic triad residues. All the data suggest that Lk2 and Lk3 are novel lipase on the family I.1 and both lipase available as a biocatalyst candidate.     

Anisotropy in sickle hemoglobin fibers from variations in bending and twist

We have studied the variations of twist and bend in sickle hemoglobin fibers. We find that these variations are consistent with an origin in equilibrium thermal fluctuations, which allows us to estimate the bending and torsional rigidities and effective corresponding material moduli. We measure bending by electron microscopy of frozen hydrated fibers and find that the bending persistence length, a measure of the length of fiber required before it starts to be significantly bent due to thermal fluctuations, is 130microm, somewhat shorter than that previously reported using light microscopy. The torsional persistence length, obtained by re-analysis of previously published experiments, is found to be only 2.5microm. Strikingly this means that the corresponding torsional rigidity of the fibers is only 6x10(-27)Jm, much less than their bending rigidity of 5x10(-25)Jm. For (normal) isotropic materials, one would instead expect these to be similar. Thus, we present the first quantitative evidence of a very significant material anisotropy in sickle hemoglobin fibers, as might arise from the difference between axial and lateral contacts within the fiber. We suggest that the relative softness of the fiber with respect to twist deformation contributes to the metastability of HbS fibers: HbS double strands are twisted in the fiber but not in the equilibrium crystalline state. Our measurements inform a theoretical model of the thermodynamic stability of fibers that takes account of both bending and extension/compression of hemoglobin (double) strands within the fiber.     

Torsional rigidity of DNA and length dependence of the free energy of DNA supercoiling

By analyzing the Boltzmann populations of DNA topoisomers that differ only in their linking numbers, the dependence of the free energy delta G tau of DNA supercoiling on the linking number alpha has been determined for DNA rings as small as 200 base-pairs (bp) in length. All experimental data can be fitted by the relation delta G tau = K (alpha-alpha)2, where alpha is a constant for a given DNA at a given set of conditions and K is a DNA length-dependent proportionality constant. For DNA rings with length N larger than 2000 bp, K is inversely proportional to N and the product NK is nearly a constant around 1150 RT X bp. For rings smaller than 2000 bp NK increases steadily with decreasing N; for a 200 bp ring NK is 3900 RT X bp. The increase in NK when N decreases can be interpreted as a result of the decrease in the contribution of the fluctuation in the writhing number to the equilibrium distribution in alpha. Assuming that the writhing contribution approaches zero for DNA rings 200 bp in size, the torsional rigidity of the DNA double helix is calculated to be 2.9 X 10(-19) erg cm. In addition, the large value of K for the small circles allows precise calculation of the helical repeat of DNA. For the 210 bp rings, the repeat is measured to be 10.54 bp.