  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.
