On mathematical model of DNA mobility

V. S. Novoselov

Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, no. 3 (2014), pp. 36-45

Résumé

The superhelical DNA molecule is a linearly extendet elastic polimer made from repeating nucleotides. It is start from of a two-chain interwound structure DNA molecules. The DNA double helix is stabilited of the hydrogen bonds between the nitrous bases (nucleobases) align perpendicular to the axis of the molecule. The backbone of the DNA molecule strand is made from alternating phosphate and sugar residues. In a prosesses named replication and transcription the enzymes break the hydrogen bonds and originate the so-called open states. With due regart for a quantum mechanical effect the forces of the self-cotact-atoms into the superhelikal molecule is completaty unstated. In part because, the mechanical methods inability to account explicitle for interactions spiraling sulvent. As enzymes unwinds the DNA double helix, they inducte the forced rotation round tangents to the spiral. This suggest the double helix modeled as two linear chans of penduls (the nucleobases) connected by springs (the sugar- phocphate backbones). It is known such the coupled-pendulum system is then an example of the sine-Gordon model. There the problem of the angular mobility nucleobases redices to the soliton solutions of the sine-Gordon (SG) equation. From a more biology viewpoint, solton may bewiewed, as solitary excitation arising from large-amplitude, the refore nonlinear effect produce, which can migrate as particle. The principal features of the solitonic exctations are stable dynamic and static mode in extended system. In this article an direct analytical construction one-solution (kink) and oscillating-soliton (bion) solutions of SG equation is given. The soliton solutions SG equation applicability on the description of the DNA mobility in a processes replication and transcription is discussed. On the basis of the sine-Gordon model we show that when the probability density of the excitations and of the kinetic equation coefficients are considered. The comparison of energy levet are realized for the different conditions DNA. Bibliogr. 23.

Détail

Keywords: DNA mobility, replication, kink, bion.
Mots-clés : transcription, soliton

Bibliographie

[1] M. S. Giljarov (gl. red.), Encyclopedic Dictionary of Biology, redkol: A. A. Baev, G. G. Vinberg, G. A. Zavarzin i dr., Sov. jenciklopedija, M., 1989, 863 pp.

[2] Bresler S. E., Molecular Biology, Nauka, L., 1973, 578 pp.

[3] Vol'kenshtejn M. V., Molecular Biophysics, Nauka, M., 1975, 616 pp.

[4] Vol'kenshtejn M. V., Biophysics, Nauka, M., 1981, 576 pp.

[5] Chosh A., Bansal M., “A glossury of DNA structures from A to Z”, Acta Crystallogr. D, 59:4 (2003), 620–626 | DOI

[6] Nechiporenko Ju. D., Gurskij G. V., “Thermodynamic binding model DNA ligands”, Biofizika, 48 (2003), 773–796

[7] Nechiporenko Ju. D., Polozov R. V., Nechiporenko D. Ju., Il'icheva I. A., Vorob'ev E. A., Gorohovskij G. V., Gurskij G. V., “In Mathematical models of gene regulation: mechanical perturbations of DNA structure”, ICE-2006 (mathematics, computer, education), v. 2, Nauch. issled. centr, M.–Izhevsk, 2006, 393–403

[8] Benham C., Mielke S., “DNA mechanics”, Annu Rew. Biomed Eng., 7 (2005), 21–53 | DOI

[9] Novoselov V. S., Analytical mechanics of systems with variable masses, Izd-vo Leningr. un-ta, L., 1969, 240 pp.

[10] Englander S. W., Kallenbach N. R., Heeger A. J., Krumhansl J. A., Litwin S., “Nature of the open state in long polynucleotide double helices: possibility of solution excitations”, Proc. Natl. Acad. Sci. USA, 77:12 (1980), 7222–7226 | DOI

[11] Reznichenko G. Ju., Mathematical models in biology and ecology, Izhevsk. in-t komp'juternyh issledovanij, M.–Izhevsk, 2003, 184 pp.

[12] Jakushevich L. V., How to build a mathematical model of the unwinding of the DNA double helix?

[13] Jakushevich L. V., Krasnobaeva L. A., Shapovalov A. V., Kinetro N. D., “One and two-soliton solutions of cinus-Gordon equation in the annex to the DNA”, Biofizika, 50:3 (2005), 450–455

[14] Shustorovich E. M., Chemical bond, Nauka, M., 1973, 232 pp.

[15] Tahtadzhan L. A., Faddeev L. D., Hamiltonian approach to the theory of solitons, Nauka, M., 1986, 528 pp. | MR

[16] Zaharov V. E., “The inverse scattering”: Kunin I. L., The theory of elastic media with microstructure, Nauka, M., 1975, 226–273

[17] Dodd R., Eilbeck J., Gibbon J., Morris H., Solitons and nonlinear wave equations, Per. s angl. V. I. Macieva, V. P. Gurarija, ed. A. B. Shabat, Mir, M., 1988, 694 pp.

[18] Mahan'kov P. G., “Physics of elementary particles and atomic nuclei”, Fizika jelementarnyh chastic i atomnogo jadra, 4:1 (1983), 123–180

[19] Novoselov V. S., Korolev V. S., Analytical Mechanics controlled system, studies allowance, Izd-vo S.-Peterb. un-ta, St. Petersburg, 2005, 298 pp.

[20] Novoselov V. S., Statistical Dynamics, proc. allowance, Izd-vo S.-Peterb. un-ta, St. Petersburg, 2009, 393 pp.

[21] Tihonov A. N., “Systems of differential equations containing small parameters in the derivatives”, Matem. sb., 31(73):3 (1952), 575–586 | MR | Zbl

[22] Vasil'eva A. B., Butuzov V. F., Asymptotic expansion of solutions singularly perturbed equations, Nauka, M., 1973, 272 pp. | MR | Zbl

[23] Dobrynina V. I., Biological Chemistry, Medicina, M., 1976, 504 pp.