Immersed surfaces and membranes transformations
Abstract
Physical and biological observation methods provide a variety of bilayer membranes’ shapes and their transformations. Besides, the topological and geometrical methods allow us to deduce a classification of all possible membrane surfaces. This doublesided approach leads to a deeper insight into membranes properties. Our goal is to apply an appropriate mathematical technique for classifying vesicles (closed surfaces in mathematical terminology) and for their transformation ways. The problem turned out to be an intricate one, and to our knowledge no mathematical techniques have been applied to its solution. We find that all vesicles can be decomposed in a small number of universality classes generated by a few ‘bricks’: a torus, a screwed torus, and the real projective plane. We consider several ways of transforming membrane surfaces, bearing in mind that they possess an additional extremal property. Our method exploits different constructions of minimal surfaces in S^{3}. We estimate energetic barrier for transformation of minimal membrane surfaces using the socalled doubling procedure. This problem is far from being a pure theoretical exercise. For instance, almost all cells’ biological functions, or tumor progression, are accompanied by apparently singular cell membrane transformations.
 Publication:

Physica Scripta
 Pub Date:
 June 2015
 DOI:
 10.1088/00318949/90/7/074003
 Bibcode:
 2015PhyS...90g4003K