Exact minimal parking garages: pitch balance of helical motifs in optimally packed lamellar structures

  Luiz C. B. da Silva  ,  Efi Efrati  
Department of Physics of Complex Systems, Weizmann Institute of Science

Minimal surfaces arise as the ground state of a variety of manmade and naturally occurring membranes and lamellar structures. They minimize area among all surfaces with a given boundary and should naturally appear when surface tension is the dominant effect. Despite that, minimal surfaces can be also relevant in systems dominated by membrane bending energy as they form a special class of critical points of the Helfrich free energy. Investigations of the interior of thylakoids and of the endoplasmic reticulum revealed that their lamellar structure is composed of helical motifs [1,2] and, in addition, right- and left-handed motifs were observed in both systems in arrangements that suggested global pitch balance [3,4]. So far the analytical treatment of such helical motifs in minimal surfaces has been limited to the small-slope approximation [5]. However, the resulting surfaces are not exactly minimal, which is exceptionally apparent in the immediate vicinity of the helical structures rendering the small slope approximation irrelevant in the biologically and physically relevant regimes. With the purpose of bridging this gap, we provide a recipe for constructing exact and analytically tractable minimal surfaces in which the appropriate distribution of helical motifs are embedded. This is based on a representation of minimal surfaces known as Enneper immersion [6], from which follows that any approximated minimal surface can be deformed into an exact minimal one through an explicit, yet non-local operation by only exploiting lateral displacements. We analyze the fundamental structure and interaction between two helical motifs of opposite and similar handedness. Finally, we apply the devised method to build a minimal surface with an arbitrary array of N helical motifs and show that area minimization, i.e., positive second variation, leads to global pitch balance.

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