Advertisement

Journal of Mathematical Biology

, Volume 69, Issue 6–7, pp 1801–1813 | Cite as

On the energy density of helical proteins

  • Manuel Barros
  • Angel FerrándezEmail author
Article

Abstract

We solve the problem of determining the energy actions whose moduli space of extremals contains the class of Lancret helices with a prescribed slope. We first see that the energy density should be linear both in the total bending and in the total twisting, such that the ratio between the weights of them is the prescribed slope. This will give an affirmative answer to the conjecture stated in Barros and Ferrández (J Math Phys 50:103529, 2009). Then, we normalize to get the best choice for the helical energy. It allows us to show that the energy, for instance of a protein chain, does not depend on the slope and is invariant under homotopic changes of the cross section which determines the cylinder where the helix is lying. In particular, the energy of a helix is not arbitrary, but it is given as natural multiples of some basic quantity of energy.

Keywords

Energy action Lancret helix Protein chain 

Mathematics Subject Classification (2010)

53C40 53C50 

Notes

Acknowledgments

The authors wish to thank the referees for their constructive comments and suggestions for improvement in the article. They are also indebted to Professor F. Carreras for the excellent graphics accompanying the examples. MB has been partially supported by Spanish MEC-FEDER Grant MTM2010-18099 and J. Andalucía Regional Grant P09-FQM-4496. AF has been partially supported by MINECO (Ministerio de Economa y Competitividad)and FEDER (Fondo Europeo de Desarrollo Regional) project MTM2012-34037, and Fundación Séneca project 04540/GERM/06, Spain. This research is a result of the activity developed within the framework of the Programme in Support of Excellence Groups of the Región de Murcia, Spain, by Fundación Séneca, Regional Agency for Science and Technology (Regional Plan for Science and Technology 2007–2010).

References

  1. Barros M (1997) General helices and a theorem of Lancret. Proc AMS 125:1503–1509CrossRefzbMATHMathSciNetGoogle Scholar
  2. Barros M, Ferrández A (2009) A conformal variational approach for helices in nature. J Math Phys. 50:103529CrossRefMathSciNetGoogle Scholar
  3. Barros M, Ferrández A (2010) Epicycloids generating Hamiltonian minimal surfaces in the complex quadric. J Geom Phys 60:69–73CrossRefGoogle Scholar
  4. Cahill K (2005) Helices in biomolecules. Phys Rev E 72:062901CrossRefGoogle Scholar
  5. Calugareanu G (1961) Sur les classes d’isotopie des noeuds tridimensionnels et leurs invariants. Czechoslovak Math J 11:588–625MathSciNetGoogle Scholar
  6. Crane R (1950) Principles and problems of biological growth. Sci Mon 6:376–389Google Scholar
  7. Feoli A, Nesterenko VV, Scarpetta G (2005) Functionals linear in curvature and statistics of helical proteins. Nucl Phys B 705:577–592CrossRefzbMATHMathSciNetGoogle Scholar
  8. Ferrández A, Guerrero J, Javaloyes MA, Lucas P (2006) Particles with curvature and torsion in three-dimensional pseudo-Riemannian space forms. J Geom Phys 56:1666–1687CrossRefzbMATHMathSciNetGoogle Scholar
  9. Galloway J (2010) Helical imperative: paradigm of growth, form and function. In: Encyclopedia of life sciences. Wiley, ChichesterGoogle Scholar
  10. Levien R (2008) The elastica: a mathematical history. University of California at Berkeley. http://www.eecs.berkeley.edu/Pubs/TechRpts/2008/EECS-2008-103.html
  11. Liu Z, Qin L (2005) Electron diffraction from elliptical nanotubes. Chem Phys Lett 406:106–110CrossRefGoogle Scholar
  12. Loui AH, Somorja RL (1982) Differential geometry of proteins: a structural and dynamical representation of patterns. J Theor Biol 98:189–209CrossRefGoogle Scholar
  13. Loui AH, Somorja RL (1983) Differential geometry of proteins: helical approximations. J Mol Biol 168:143–162CrossRefGoogle Scholar
  14. McCoy JA (2008) Helices for mathematical modelling of proteins, nucleid acids and polymers. J Math Anal Appl 347:255–265CrossRefzbMATHMathSciNetGoogle Scholar
  15. Pauling L, Corey RB, Branson HR (1951) The structure of proteins: two hydrogen-bonded helical configurations of the polypeptide chain. Proc Natl Acad Sci USA 37:205Google Scholar
  16. Pauling L, Corey RB (1953a) Two pleated-sheet configurations of polypeptide chains involving both Cis and Trans amide groups. Proc Natl Acad Sci USA 39:247Google Scholar
  17. Pauling L, Corey RB (1953b) Two rippled-sheet configurations of polypeptide chains, and a note about the pleated sheets. Proc Natl Acad Sci USA 39:253Google Scholar
  18. Thamwattana N, McCoy JA, Hill JM (2008) Energy density functions for protein structures. Q J Mech Appl Math 61(3):431–451CrossRefzbMATHMathSciNetGoogle Scholar
  19. White J (1969) Self-linking and the Gauss integral in higher dimensions. Am J Math 91:693–728CrossRefzbMATHGoogle Scholar
  20. Wu Z, Yung EKN (2006) Axial mode elliptical cross-section helical antenna. Microw Opt Technol Lett 48:2080–2083CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Departamento de Geometria y Topología, Facultad de CienciasUniversidad de GranadaGranadaSpain
  2. 2.Departamento de MatemáticasUniversidad de Murcia MurciaSpain

Personalised recommendations