We develop an elastic–isotropic rod model for superhelical DNA structures where the helical angle is varying as a function of the arc-length. Our motivation for a variable helical angle comes from some experiments and simulations on DNA braids where complex superhelical structures have been observed. The helical solutions are minimizers of a free energy consisting of elastic, entropic and electrostatic terms. These minimizers are obtained within a variational framework where the end-points of the helices are allowed to be variable so that the length of the superhelix is computed as part of the solution. Considering variable curvature solutions brings up the possibility of finding more complex DNA structures because for two (or more) interwound helices there is a geometrical lock-up helical angle which puts a limit on the length of a superhelix. We perform calculations with different ionic concentrations and study the effects of lock up for braided structures. We also extend the variable curvature model to study the formation of plectonemes in the presence of multivalent salts where the supercoiling radius can be regarded as a constant prescribed by the balance of attractive and repulsive forces in DNA–DNA interactions, and provide analytical solutions in terms of elliptic functions for the supercoil parameters.
MarkoJF. Supercoiled and braided DNA under tension. Phys Rev Lett E1997; 55: 1758–1772.
2.
ForthSDeufelCSheininMYDanielsBSethnaJPWangD. Abrupt buckling transition observed during the plectoneme formation of individual DNA molecules. Phys Rev Lett2008; 100(14): 148301.
3.
LipfertJKerssenmakersJWJJagerTDekkerNH. Magnetic torque tweezers: measuring torsional stiffness in DNA and RecA–DNA filaments. Nat Meth2010; doi:10.1038/nmeth.1520.
4.
BrutzerHLuzziettiNKlaueDSeidelR. Energetics at the DNA Supercoiling Transition. Biophys J2010; 98: 1267–1276.
5.
MosconiFAllemandJFBensimonDCroquetteV. Measurement of the torque on a single stretched and twisted DNA using magnetic tweezers. Phys Rev Lett2009; 102(7): 078301.
6.
StrickTRAllemandJFBensimonDBensimonACroquetteV. The elasticity of a single supercoiled DNA molecule. Science1996; 271: 1835–1837.
7.
StrickTRDessingesMNCharvinG. Stretching of macromolecules and proteins. Rep Prog Phys2003; 66: 1–45.
8.
BestemanKHageSDekkerNHLemaySG. Role of tension and twist in single-molecule DNA condensation. Phys Rev Lett2007; 98(5): 058103.
9.
MarkoJF. Torque and dynamics of linking number relaxation in stretched supercoiled DNA. Phys Rev E2007; 76(2): 021926.
10.
MarkoJFSiggiaED. Statistical-mechanics of supercoiled DNA. Phys Rev E1995; 52: 2912–2938.
NizetteMGorielyA. Towards a classification of Euler–Kirchhoff filaments. J Math Phys1999; 40: 2830–2866.
15.
FraserWBStumpDM. The equilibrium of the convergence point in two-strand yarn plying. Int J Solid Struct1998; 35: 295–298.
16.
ColemanBDSwigonD. Theory of supercoiled elastic rings with self-contact and its application to DNA plasmids. J Elasticity2000; 60: 173–221.
17.
ThompsonJMTvan der HeijdenGHMNeukirchS. Supercoiling of DNA plasmids: mechanics of the generalized ply. Proc R Soc Lond A2002; 458: 959–985.
18.
van der HeijdenGHMThompsonJMTNeukirchS. A variational approach to loaded ply structures. J Vib Control2003; 9: 175–185.
19.
ColemanBDSwigonD. Theory of self-contact in DNA molecules modeled as elastic rods. Nuovi Prog Fis Mat Ered Dario Graffi2002; 177: 281–295.
20.
MaffeoCRobertSBrutzerH. DNA–DNA interactions in tight supercoils are described by a small effective charge density. Phys Rev Lett2010; 105(15): 158101.
21.
ArgudoDPurohitPK. The dependence of DNA supercoiling on solution electrostatics. Acta Biomater2012; doi:10.1016/j.actbio.2012.01.030.
22.
ClauvelinNAudolyBNeukirchS. Elasticity and electrostatics of plectonemic DNA. Biophys J2009; 96: 3716–3723.
23.
ClauvelinNAudolyBNeukirchS. Mechanical response of plectonemic DNA: an analytical solution. Macromolecules2008; 41: 4479–4483.
24.
NeukirchSMarkoJF. Analytical description of extension, torque, and supercoiling radius of a stretched twisted DNA. Phys Rev Lett2011; 106(13): 138104.
25.
GelfandIMFominSV. Calculus of Variations. Mineola, NY: Dover Publications, 2000.
26.
BolzaO. Lectures on the Calculus of Variations. New York: Dover Publications, 1961.
27.
ArgudoDPurohitPK. Competition between supercoils and toroids in single molecule DNA condensation. Biophys J2012; 103: 118–128.
28.
HuntNGHearstJE. Elastic model of DNA supercoiling in the infinite-length limit. J Chem Phys1991; 95: 9329–9336.
29.
StasiakAMaddocksJH. Mathematics. Best packing in proteins and DNA. Nature2000; 406: 251–253.
PurohitPK. Shape and energetics of DNA plectonemes. In: GarikipatiKArrudaEM (eds.), IUTAM Symposium on Cellular, Molecular and Tissue Mechanics (IUTAM Book Series, vol. 16(3)). New York: Springer, 2010, pp. 123–138.
41.
EnneperAMullerF. Ellliptische Functionen. Theorie und Geschichte. Halle a.S., 1890.
42.
GreenhillAG. The Application of Elliptic Functions. New York: Macmillan and Co., 1892.
43.
NeukirchSStarostinEL. Writhe formulas and antipodal points in plectonemic DNA configurations. Phys Rev E2008; 78(4): 041912.
44.
FullerFB. Writhing number of a space curve. Proc Natl Acad Sci USA1971; 68: 815–819.
