On the energy density of helical proteins

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.

Forceless Sadowsky strips are spherical

“…Nor does it include the narrow limit (w → 0) of the functional for annular strips derived in [6], nor, seemingly, the functional for narrow residually-stressed strips derived in [7]. It would be interesting to find all functionals of the form (2) (or, more generally, functionals with l = l(κ, η, κ , η , ...) [31]) with unconstrained spherical solutions, analogous to all functionals with forceless helical solutions having been characterised in [2].…”

“…The case l = (Aκ+Bη)κ gives the isotropic Kirchhoff rod having both bending and torsional stiffness [19], while l = (A + Bη 2 )κ 2 describes a thin strip whose material frame is locked to the Frenet frame and which therefore bends only about a single principal axis [20]. The linear function l = A + Bκ + Cτ , meanwhile, which gives rise to generalised (Lancret) helices (having constant η), has been proposed for protein chains [2]. Functionals U as in Eq.…”