TY - JOUR
T1 - Construction of exact minimal parking garages: nonlinear helical motifs in optimally packed lamellar structures
AU - da Silva, Luiz C. B
AU - Efrati, Efi
PY - 2021/2
Y1 - 2021/2
N2 - 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 comprised 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 analyse 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.
AB - 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 comprised 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 analyse 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.
UR - http://www.scopus.com/inward/record.url?scp=85102893267&partnerID=8YFLogxK
U2 - https://doi.org/10.1098/rspa.2020.0891
DO - https://doi.org/10.1098/rspa.2020.0891
M3 - مقالة
SN - 1364-5021
VL - 477
JO - Proceedings of the Royal Society. A, Mathematical, physical, and engineering sciences
JF - Proceedings of the Royal Society. A, Mathematical, physical, and engineering sciences
IS - 2246
M1 - 20200891
ER -