Abstract
We consider a model of a biomembrane with attached proteins. The membrane is represented by a near spherical continuous surface and attached proteins are described as discrete rigid structures which attach to the membrane at a finite number of points. The resulting surface minimises a quadratic elastic energy (obtained by a perturbation of the Canham-Helfrich energy) subject to the point constraints which are imposed by the attachment of the proteins. We calculate the derivative of the energy with respect to protein configurations. The proteins are constrained to move tangentially by translation and by rotation in the axis normal to a reference point. Previous studies have typically restricted themselves to a nearly flat membrane and circular inclusions. A numerically accessible representation of this derivative is derived and employed in some numerical experiments.
Highlights
The morphology of cell membranes and a variety of functions are well-known to be regulated by the interplay between surface proteins and the curvature of the membrane
Biological membranes are composed of a lipid bilayer, this layer is believed to act like a fluid in the lateral direction and elastically in the normal direction
With the explicit formula we find, the algorithm to construct the gradient would require solving 1 linear system and evaluating 3N functionals, where these functionals are relatively cheap to evaluate compared to a linear solve for a fourth order PDE
Summary
The morphology of cell membranes and a variety of functions are well-known to be regulated by the interplay between surface proteins and the curvature of the membrane. Biological membranes are composed of a lipid bilayer, this layer is believed to act like a fluid in the lateral direction and elastically in the normal direction This means that in principle, any proteins which may be embedded into or attached to the surface of the membrane may move freely. We further note the work of [5] which considers point constraints in a Kirchoff plate, this bears a striking similarity to the biological problems of optimising the locations of constraints with respect to the an elastic membrane energy It is widely accepted in the literature that the near stationary state of lipid membranes are minimisers of the Canham-Helfrich energy [6, 25], M κ 2. With the explicit formula we find, the algorithm to construct the gradient would require solving 1 linear system and evaluating 3N functionals, where these functionals are relatively cheap to evaluate compared to a linear solve for a fourth order PDE
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