On the energy density of helical proteins
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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.
KeywordsEnergy action Lancret helix Protein chain
Mathematics Subject Classification (2010)53C40 53C50
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).
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