Geometric Models for Secondary Structures in Proteins

Toda, Magdalena; Athukorallage, Bhagya

doi:10.7546/giq-16-2015-282-300

Cited by 4 publications

(7 citation statements)

References 2 publications

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“…Given a physical model such as the curvature functional F (M ), a basic question one can ask is where it is extremized. That is, it is important to know what surface immersions are extremal for a given functional, because their image surfaces are good candidates for physically relevant objects (see [10,11,5]). To accomplish this for F its first variation is computed, yielding a PDE in the mean curvature H. Solutions to this equation then provide the mean curvature functions corresponding to the surface immersions of interest.…”

Section: The First Variationmentioning

confidence: 99%

“…To further simplify this, recall the Codazzi-Mainardi equation (11), expressed in coordinate form as…”

Section: The Second Variationmentioning

confidence: 99%

“…The Willmore energy has been widely-studied (e.g. [16,17,18,19,11,20,21,22]), though there are still many open questions about its behavior. Indeed, this topic has unified the work of mathematicians, physicists, and biologists in studying elastic phenomena, and has sparked what is now an active area of research.…”

Section: Introductionmentioning

confidence: 99%

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On the variation of curvature functionals in a space form with application to a generalized Willmore energy

2019

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Functionals involving surface curvature are important across a range of scientific disciplines, and their extrema are representative of physically meaningful objects such as atomic lattices and biomembranes.Inspired in particular by the relationship of the Willmore energy to lipid bilayers, we consider a general functional depending on a surface and a symmetric combination of its principal curvatures, provided the surface is immersed in a 3-D space form of constant sectional curvature. We calculate the first and second variations of this functional, extending known results and providing computationally accessible expressions given entirely in terms of the basic geometric information found in the surface fundamental forms. Further, we motivate and introduce the p-Willmore energy functional, applying the stability criteria afforded by our calculations to prove a result about the p-Willmore energy of spheres. 1 2 ON THE VARIATION OF CURVATURE FUNCTIONALSHelfrich model for membrane energy per unit area is given by the functionalwhere H is the membrane mean curvature, K is its Gauss curvature, k, k c are some rigidity constants, and c 0 is a constant known as the "spontaneous curvature". Physically, this high dependence on curvature arises from hydrostatic pressure differences between the fluids internal and external to the membrane.Another noteworthy curvature functional is the bending energy, which quantifies how much (on average) a surface M deviates from being a round sphere. Specifically, the bending energy functional is defined aswhere k 0 is the sectional curvature of the ambient space. This type of energy was first considered by Sophie Germain in 1811 (see [12]) as a model for the bending energy of a thin plate. In particular, she suggested that the bending energy be measured by an integral over the plate surface, taking as integrand some symmetric and even-degree polynomial in the principal curvatures. Note that the functional (2) is one of the simplest models of this kind.Remark. The bending energy also arises in the field of computer vision, where changes in surface curvature are used to simulate natural movement. On the other hand, it is known to these scientists as the surface torsion (see [13]) due to how it measures the change in normal curvature of the surface.From a mathematical perspective, both the Helfrich energy and the bending energy are closely related to the conformally invariant (see [14]) Willmore energy popularized in [15], which is defined as(3) W(M ) :=

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Section: The First Variationmentioning

confidence: 99%

“…To further simplify this, recall the Codazzi-Mainardi equation (11), expressed in coordinate form as…”

Section: The Second Variationmentioning

confidence: 99%

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On the variation of curvature functionals in a space form with application to a generalized Willmore energy

2019

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“…This means that t is the arc length s along the contour curve Γ. Then, in terms of the tangent angle φ we haveṙ = cos φ,ḣ = sin φ (12) and can rewrite equation…”

Section: Euler-lagrange Equationsmentioning

confidence: 99%

“…[1,7,10] and references therein) and in the context of applications, for instance, in structural mechanics, biophysics and mathematical biology (see, e.g. [2,7,8,11,12]).…”

Section: Introductionmentioning

confidence: 99%

Axially Symmetric Willmore Surfaces Determined by Quadratures

Vassilev¹,

Djondjorov²,

Hadzhilazova³

et al. 2016

GIQ

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The work is concerned with a special family of axially symmetric surfaces providing local extrema to the so-called Willmore functional, which assigns to each surface its total squared mean curvature. The components of the position vector of the profile curves of the regarded Willmore surfaces satisfy a system of first-order ordinary differential equations. The solutions of this system are expressed by quadratures in terms of the tangent angle and, in this way, the corresponding Willmore surfaces are determined.

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Application of the moving frame method to deformed Willmore surfaces in space forms

Paragoda

2018

Journal of Geometry and Physics

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Geometric Models for Secondary Structures in Proteins

Cited by 4 publications

References 2 publications

On the variation of curvature functionals in a space form with application to a generalized Willmore energy

On the variation of curvature functionals in a space form with application to a generalized Willmore energy

Axially Symmetric Willmore Surfaces Determined by Quadratures

Application of the moving frame method to deformed Willmore surfaces in space forms

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