Estimating the torsional rigidity of DNA from supercoiling dataSteven Strogatz Citation: The Journal of Chemical Physics 77, 580 (1982); doi: 10.1063/1.443600 View online: http://dx.doi.org/10.1063/1.443600 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/77/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Polymer induced condensation of DNA supercoils J. Chem. Phys. 129, 185102 (2008); 10.1063/1.2998521 The writhe distribution in DNA plasmids as derived from the free energy of supercoiling J. Chem. Phys. 113, 6950 (2000); 10.1063/1.1310325 Writhing number of supercoiled DNA from its scanning force microscopy imaging J. Vac. Sci. Technol. B 13, 158 (1995); 10.1116/1.587974 Finite element analysis of DNA supercoiling J. Chem. Phys. 98, 1673 (1993); 10.1063/1.464283 Elastic theory of supercoiled DNA J. Chem. Phys. 83, 6017 (1985); 10.1063/1.449637

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580 Letters to the Editor

TABLE I. Equilibrium constants calculated with three differ­ent force fields by direct summation over energy levels, the approximate value le, and experimental values.

T KM HMS CCL 3C Expt. a

275 3.812 3.81 7 3. 818 3.828 3.74, b 3. 75c

298 3.837 3.84 2 3.843 3.854 3.76, d 3.74-3.76, f 3.76. 3.8l,b3.8l1

300 3.839 3.844 3.844 3.855

325 3.861 3.866 3.866 3.878

aAt each temperature, measurements are listed in chrono-logical order of publication.

bReference 6 (273 K). cReference 7 (273 K). dReference 6. -Reference 7. fReference 8 (297 K). 'Reference 9 (296 K). hReference 10 (296 K). iReference 11 (296 K).

tional partition functions, one can break down the vari­ous factors by which the JC value at 298 K must be multiplied to yield the CCL value: from anharmonicity of zero-point energies, 1. 0016; from quantum rotation and rotation-vibration interaction, 0.9957; from the effect of anharmonicity on vibrational excitation, 0.9999. None of the correction factors differ signifi­cantly from unity.

All the experimental values in the 298 K region are lower than the "best" theoretical value 3.84. While the earlier experimental values were about 3.76, the latest value is 3.81. Pyper and his co-workers l1 have placed fairly large error limits on this value (3.81 ± O. 09 and 3.81 ± 0.06). From the point of view of theory, only two factors can now change the values of this equilibrium constant from the "best" CCL values: the use of an-

COMMENTS

other better force field or the conSideration of higher order Born-Oppenheimer corrections. The authors consider it unlikely that either of these factors will shift the calculated value of the equilibrium constant to 3.81 at 298 K.

a)Research supported by U. S. Department of Energy, Office of Basic Energy SCiences, under Contract No. DE-A T03-76ER70l88.

1M. Wolfsberg, J. Chern. Phys. 50, 1484 (1969); M. Wolfs­berg, A. A. Massa, and J. W. Pyper, J. Chern. Phys. 53, 3138 (1970).

2J • R. Hulston, J. Chern. Phys. 50, 1483 (1969). 3M. Wolisberg, J. Chern. Phys. 70, 5322 (1979). In Table I,

A(AZPE) and K(AZPE) for CCL should have been reported O. 3 and 1. 001., respectively.

4J • Bron, C. F. Chang, and M. Wolfsberg, Z. Naturforsch. Teil A 28, 129 (1973).

SR. D. Bardo and M. Wolfsberg, J. Chem. Phys. 62, 4555 (1975).

6L . Friedman and V. J. Shiner, J. Chem. Phys. 44, 4639 (1966).

7J. W. Pyper, R. S. Newbury, and G. W. Barton, J. Chern. Phys. 46, 2253 (1967).

8J. W. Pyper and R. S. Newbury, J. Chern. Phys. 52, 1966 (1970).

9N. A. Jones, R. D. Freisen, and J. W. Pyper, Rev. Sci. Instrum. 41, 1828 (1970).

10J • W. Pyper and L. D. Christensen, J. Chem. Phys. 62, 2596 (1975).

ll J • W. Pyper, R. J. Dunzyk, R. D. Friesen, S. L. Bernasek, C. A. May, A. W. Echeverria, and L. F. Tolrnan, Int. J. Mass Spectrurn. Ion Phys. 23, 209 (1977).

12R. J. Whitehead and N. C. Handy, J. Mol. Spectrosc. 55, 356 (1975).

13 H. C. Allen, Jr. and p. C. Cross, Molecular Vib-Rotors (Wiley, New York, 1963).

14K • Kuchitsu and Y. Morino, J. Chern. Soc. Jpn. 38, 814 (1965).

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Estimating the torsional rigidity of DNA from supercoiling data

Steven Strogatz

Trinity Col/ege, Cambridge CB2 ITQ, England (Received 2 February 1982; accepted 24 March 1982)

Several authors have recently attempted to estimate the torsional rigidity of DNA. 1-5 Many of these esti­mates have been obtained from fluorescence studies of the torsional Brownian motion of DNA. 2,6

A second approach to estimating the torsional rigidity involves measurements of the free energy of DNA super­coiling. 7,8 Barkley and Zimm1 performed a calculation of this sort, in which they assumed "all the free energy of extra supercoiling comes from torsion and none from

bending." Thus if G is the measured free energy

where cP is the net angular twist in a length L. This ex­pression for the torsional energy is strictly correct only if the rate of twist is uniform along the molecule. (See Ref. 9 for a more general discussion-we follow the notation of Barkley and Zimmi for the sake of com­parison. )

.J. Chern. Phys. 77(1), 1 July 1982 0021-9606/82/130580-02$02.10 © 1982 American I nstitute of Physics

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Letters to the Editor 5a1

Barkley and Zimmt did not find "acceptable agree­ment with the observed anisotropy." Typical values inferred from fluorescence studies lie in the range (1. 3-4. 0) x 10-19 erg cm at 293 K, t-4 whereas the super­coiling data appeared to yield values of (0.64-1. 09) x 10-19 erg cm. Barkley and Zimml considered this latter value to be "surprisingly far from the mark. "

It seems that closer agreement may be obtained by modifying the assumptions of Barkley and Zimml to ac­cord with a later finding of Vologodskii et al. 5 Using the same free energy data, 7,8 these workers found that DNA supercoiling is partitioned about equally to writh­ing and twisting. More precisely, if the linking num­bers,to of the circular DNA differs by an amount tl.Lk from that of the relaxed form, then5

tl.Tw"" Wr"" tl.Lk/2 .

Barkley and Zimm,1 however, assumed that a DNA with "T supertwists" and n/2 base pairs has a twist of 2TrT/ (n/2) in the length between adjacent base pairs, and thus a total angular twist of 21fT along the entire mole­cule. Translating to the more precise notation of Fuller, s the total twist btl. Tw is assumed to equal 21fT; and Barkley and Zimm'st T is equivalent to tl.Lk; hence their assumption implies tl.Tw-=tl.Lk. In other words, all of the supercoiling is assumed to be manifested as duplex twisting, and essentially none as writhing.

This twofold overestimate of the twist yields a four­fold underestimate of C. When adjusted accordingly,

ERRATA

the estimate based on supercoiling data is compatible with values inferred from the fluorescence anisotropy decay studies, 1-4.6 thus lending additional credence to the model of Barkley and Zimm. 1 If anything, the cal­culated values appear slightly high; this was to be ex­pected, since by neglecting the contribution of bending energy, we tend to overestimate the torsional rigidity. For if C* is the value obtained by ignoring bending en­ergy, and if C is the "true" value, then

C*cp2/2L = (Ccp2/2L) + U B ,

where UB > 0 is the bending energy. Thus C* > C, i.e., our calculation produces an upper bound on C.

1M. Barkley and B. H. Zimrn, J. Chern. Phys. 70, 2991 (1979).

20. P. Millar, R. J. Robbins, and A. H. Zewail, Proc. Nat!. Acad. Sci. USA 77, 5593 (1980).

3S. A. Allison and J. M. Schurr, Chern. Phys. 41, 35 (1979). 'M. LeBret, Biopolymers 17, 1939 (1978). SA. V. Vologodskii, V. V. Anshelevich, A. V. Lukashin, and

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(1976).

Erratum: Deuterium hyperfine structure measurements and theory for CD3Br [J. Chem. Phys. 76, 1685 (1982)]

S. G. Kukolich and C. D. Cogley

Department o/Chemistry, University 0/ Arizona, Tucson. Arizona 85721

Three typographical errors were detected. Page 1685, paragraph 4, sentence 1 should read: Microwave power from phase locked klystron tubes induces molecular transitions in TMo10 cylindrical copper cavities. The first sen­tence of the figure caption for Fig. 2 should read: Deuterium quadrupole components of the J -= 0- 1, Fl = i- i transition in 79BrCD3 • The first sentence of the figure caption for Fig. 4 should read: Comparison of actual and computer generated spectra for high frequency components of the J=0-1, FI =f-1 transition in 7sBrCD3.

J. Chern. Phys. 77(13), 1 July 1982 0021-9606/82/130581-01 $00.00 © 1982 America!' !!1stit'~te cf P~yti~(

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