Miroslav MAKSIMOVIĆ, Ljubica VELIMIROVIĆ, Marija NAJDANOVIĆ

Infinitesimal bending of DNA helices

The mathematics of DNA molecules is often studied as a double helix model. This paper focuses on the modelling of infinitesimal bending of DNA helices. The paper checks the flexibility of DNA molecule, i.e. the flexibility of double helix in infinitesimal bending theory. Actually, first, we find infinitesimal bending field of an arbitrary curve on the helicoid, that leaves bent curves on the helicoid, and show that helix bending on the helicoid is not possible. Finally, we deal with the infinitesimal bending of helicoid, using PDEs. Visualization of infinitesimal bending was done in Mathematica.

PDF

___

[1] Aleksandrov AD. O beskonechno malyh izgibaniyah neregulyarnyh poverhnostei. Matematiceskii Sbornik 1936; 1 (43) 3: 307-322 (in Russian).
[2] Benham CJ, Harvey S, Olson WK, Sumners DWL, Swigon D. Mathematics of DNA Structure, Function and Interactions. The IMA Volumes in Mathematics and its Applications 2009; 150: 293-320.
[3] Catalan E. Sur les surfaces réglées dont l’aire est un minimum. Journal de Mathématiques Pures et Appliquées 1842; 7: 203-211.
[4] Efimov N. Kachestvennye voprosy teorii deformacii poverhnostei. Uspekhi Matematicheskikh Nauk 1948; 3.2: 47-158 (in Russian).
[5] Gözütok U, Çoban HA, Sağiroğlu Y. Ruled surfaces obtained by bending of curves. Turkish Journal of Mathematics 2020; 44 (1): 300-306.
[6] Hagerman K, Hagerman P. Helix rigidity of DNA: the meroduplex as an experimental paradigm. Journal of Molecular Biology 1996; 260 (2): 207-223.
[7] Ivanova Karatopraklieva I, Sabitov IKh. Surface deformation. Journal of Mathematical Sciences New York 1995; 70 (3): 1685-1716.
[8] Ivanova Karatopraklieva I, Sabitov IKh. Bending of surfaces II. Journal of Mathematical Sciences New York 1995; 74 (3): 997-1043.
[9] Mills J, Hagerman P. Origin of the intrinsic rigidity od DNA. Nucle Acids Research 2004; 32 (13): 4055-4059.
[10] Najdanović M, Velimirović Lj. Infinitesimal bending of curves on the ruled surfaces. The University Thought - Publication in Natural Sciences 2018; 8 (1): 46-51.
[11] Rýparová L, Mikeš J. Infinitesimal rotary transformation. Filomat 2019 - The 20th Geometrical Seminar 2018; 33 (4): 1153-1157.
[12] Steinhaus H. Mathematical Snapshots. 3rd ed. New York, NY, USA: Dover, 1999, pp. 231-232.
[13] Velimirović LS. Change of geometric magnitudes under infinitesimal bending. Facta Universitates 2001; 3 (11): 135-148.
[14] Velimirović LS. Infinitesimal bending. Niš, Serbia: University of Niš Faculty Sciences and Mathematics, 2009.
[15] Velimirović LS, Ćirić M, Zlatanović M. Bending of spherical curves. In: Proceedings of 25th National and 2nd International Scientific Conference MoNGeometrija; Belgrade, Serbia; 2010. pp. 657-667.
[16] Velimirović LS, Ćirić M, Cvetković M. Change of the Willmore energy under infinitesimal bending of membranes. Computers and Mathematics with Applications 2010; 59 (12): 3679-3686.
[17] Velimirović LS, Ćirić M, Velimirović N. On the Willmore energy of shells under infinitesimal deformations. Computers and Mathematics with Applications 2011; 61 (11): 3181-3190.
[18] Velimirović LS, Cvetković M, Ćirić M, Velimirović N. Analysis of Gaudi surfaces at small deformations. Applied Mathematics and Computation 2012; 218: 6999-7004.
[19] Vologodskii A, Du Q, Frank-Kamenetskii M. Bending of short DNA helices. Artificial DNA: PNA & XNA 2013; 4 (1): 1-3.
[20] Vologodskii A, Frank-Kamenetskii M. Strong bending of DNA double helix. Nucle Acids Research 2013; 41 (14): 6785-6792.
[21] Watson JD, Crick FHC. Molecular structure of nucleic acids: a structure for deoxyribose nucleic acid. Nature 1953; 171 (4356): 737-738.