Abstract
This study is concerned with the rotary motions of helical chains that play essential roles in the functions of molecular motors in biological systems. While the standard pictures for the rotary motions of molecular motors may be more or less like the rotations of rigid bodies, this study explores a qualitatively different mechanism for the rotary motions. We take a simple model of a helical chain and highlight its geometric angle shifts induced by internal twist propagation. Such angle shifts, which we call geometric somersaults, can generally arise even under the conditions of zero total angular momentum, and are thereby analogous to the somersault of a falling cat. Helical chirality of the chain and the direction of twist propagation are the decisive factors that determine the direction of the resulting somersaults. As an application, we argue that the geometric somersaults of the helical chain may serve as a prototypical model for the rotary motions of the central shaft of ATP synthase.
Similar content being viewed by others
References
Berg HC (2003) The rotary motor of bacterial flagella. Annu Rev Biochem 72:19–54
Abrahams JP, Leslie AGW, Lutter R et al (1994) Structure at 2.8A resolution of F\(_{1}\)-ATPase from bovine heart mitochondria. Nature 370:621–628
Noji H, Yasuda R, Yoshida M et al (1997) Direct observation of the rotation of F\(_{1}\)-ATPase. Nature 386:299–302
Junge W, Sielaff H, Engelbrecht S (2009) Torque generation and elastic power transmission in the rotary F\(_{\rm O}\)F\(_{1}\)-ATPase. Nature 459:364–370
Wolgemuth CW, Powers TR, Goldstein RE (2000) Twirling and whirling: viscous dynamics of rotating elastic filaments. Phys Rev Lett 84:1623–1626
Jawed MK, Khouri NK, Da F et al (2015) Propulsion and instability of a flexible helical rod rotating in a viscous fluid. Phys Rev Lett 115:168101
Kane TR, Scher MP (1969) A dynamical explanation of the falling cat phenomenon. Int J Solids Struct 5:663–670
Montgomery R (1993) In: Enos MJ (ed.) Dynamics and control of mechanical systems: the falling cat and related problems, volume 1 of Fields Institute Communications (American Mathematical Society, Providence, RI, 1993), pp. 193–218
Dullin HR, Tong W (2015) Twisting somersault. SIAM J Appl Dyn Syst 15:1806–1822
Littlejohn RG, Reinsch M (1997) Gauge fields in the separation of rotations and internal motions in the \(n\)-body problem. Rev Mod Phys 69:213–275
Yanao T, Hino T (2017) Geometric somersaults of a polymer chain through cyclic twisting motions. Phys Rev E 95:012409
Wada H, Netz RR (2009) Hydrodynamics of helical-shaped bacterial motility. Phys Rev E 80:021921
Allen MP, Tildesley DJ (1987) Computer simulation of liquids. Oxford University, Oxford
Guo Z, Thirumalai D (1995) Kinetics of protein folding: Nucleation mechanism, time scales, and pathways. Biopolymers 36:83–102
Kulish O, Wright AD, Terentjev EM (2016) F\(_{1}\) rotary motor of ATP synthase is driven by the torsionally-asymmetric drive shaft. Scientific Reports 6:28180
Acknowledgements
This study has been partially supported by JSPS Grant-in-Aid No. 26800207 and No. 16KT0024.
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Uda, S., Li, M. & Yanao, T. Geometric somersaults of helical chains through twist propagation. Artif Life Robotics 23, 28–33 (2018). https://doi.org/10.1007/s10015-017-0388-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10015-017-0388-8