Construction of exact minimal parking garages: nonlinear helical motifs in optimally packed lamellar structures

TL;DR

This paper develops a method to construct exact minimal surfaces with embedded helical motifs, overcoming previous small-slope approximation limitations, and analyzes stability conditions related to motif arrangements.

Contribution

It introduces a novel approach to deform harmonic graphs into exact minimal surfaces with arbitrary helical motif distributions, including stability analysis.

Findings

Exact minimal surfaces with arbitrary helical motifs can be constructed from harmonic graphs.

Stability of these surfaces requires pitch balance among motifs.

Analysis of motif pairs and chains reveals conditions for minimality and stability.

Abstract

Minimal surfaces arise as energy minimizers for fluid membranes and are thus found in a variety of biological systems. The tight lamellar structures of the endoplasmic reticulum and plant thylakoids are composed of such minimal surfaces in which right and left handed helical motifs are embedded in stoichiometry suggesting global pitch balance. So far, the analytical treatment of helical motifs in minimal surfaces was limited to the small-slope approximation where motifs are represented by the graph of harmonic functions. However, in most biologically and physically relevant regimes the inter-motif separation is comparable with its pitch, and thus this approximation fails. Here, we present a recipe for constructing exact minimal surfaces with an arbitrary distribution of helical motifs, showing that any harmonic graph can be deformed into a minimal surface by exploiting lateral displacements only. We analyze in detail pairs of motifs of the similar and of opposite handedness and also an infinite chain of identical motifs with similar or alternating handedness. Last, we study the second variation of the area functional for collections of helical motifs with asymptotic helicoidal structure and show that in this subclass of minimal surfaces stability requires that the collection of motifs is pitch balanced.