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.

最小表面作为流体膜的能量最小化剂出现，因此存在于各种生物系统中。内质网和植物类囊体的紧密层状结构由这样的最小表面组成，其中右手和左手螺旋图案嵌入化学计量中，表明全球音高平衡。到目前为止，对最小曲面中螺旋基序的分析处理仅限于小斜率近似，其中基序由调和函数图表示。然而，在大多数生物学和物理相关的方案中，基序间的分离与其音高相当，因此这种近似失败了。在这里，我们提出了构建具有任意分布的螺旋图案的精确最小曲面的方法，表明任何谐波图都可以通过仅利用横向位移来变形为最小表面。我们详细分析了相似和相反手性的图案对，以及具有相似或交替手性的相同图案的无限链。最后，我们研究了具有渐近螺旋结构的螺旋基序集合的面积函数的第二个变体，并表明在这个最小表面的子类中，稳定性要求基序的集合是音高平衡的。