Functionals linear in curvature and statistics of helical proteins

Abstract

The effective free energy of globular protein chain is considered to be a functional defined on smooth curves in three dimensional Euclidean space. From the requirement of geometrical invariance, together with basic facts on conformation of helical proteins and dynamical characteristics of the protein chains, we are able to determine, in a unique way, the exact form of the free energy functional. Namely, the free energy density should be a linear function of the curvature * E-mail: feoli@unisannio.it; fax: +39089965275 † E-mail: nestr@thsun1.jinr.ru ‡ E-mail: scarpetta@sa.infn.it 1 of curves on which the free energy functional is defined. We briefly discuss the possibility of using the model proposed in Monte Carlo simulations of exhaustive searching the native stable state of the protein chain. The relation of the model proposed to the rigid relativistic particles and strings is also considered.

Epicycloids generating Hamiltonian minimal surfaces in the complex quadric

“…acting on a space C of curves defined in some interval I ⊂ R and so, in principle, without any boundary conditions (see for example [13,14]). The first variation δQ γ : TC γ → R can be computed as usual to get…”

Helices for mathematical modelling of proteins, nucleic acids and polymers

“…This illustrates the novelty of our approach to our question: instead of knowing a precise energy, we know instead that in biological applications helical curves do arise, so we consider which mathematical models are possible to describe these, if we permit our energy to depend on intrinsic properties of the backbone curve, namely curvature, torsion and their derivatives. In this way we significantly broaden the class of energies considered for this problem by Feoli et al [20].…”

“…[16]), various authors including Feoli et al [20] and Banavar and Maritan [8] have not been concerned to overlook this chemical structure in mathematical modelling of helices. Furthermore it is known that proteins with the same biological task in different animal species have very similar geometric shapes but may only have 25% of their constituent amino acids in common [32].…”

“…More generally it is also known that many sequences fold into the same native state structure [10,17,21]. As a first mathematical approximation we therefore assume the protein backbone, or more generally chain being modelled, by analogy with classical physics, would adopt a shape which minimises some energy depending on the intrinsic properties of the curve, namely its curvature, as in [20], but also possibly its torsion and derivatives, as in [45]. A more advanced model for protein folding considering the assembly of various helicical pieces and their energies of interaction appears in Bruscolini [11].…”

“…We note that a more general variational problem was considered in the more abstract setting of exterior differential systems by Griffiths [24] although equations expressed as above do not appear there. In [20], the energy F (κ) = α + βκ was analysed in detail; while in [40] we analysed the Euler-Lagrange equations more generally and considered some special cases including energies depending on curvature and torsion which give rise to conical helices. In the present paper we broaden the analysis with the basic question in mind "Which energies will give rise to helices as solutions of the corresponding Euler-Lagrange equations?…”