Maxim O. Lavrentovich, Eric M. Horsley, Asja Radja, Alison M. Sween   and  Randall D. Kamien

Significance

Pollen grains, insect eggshells, and mite carapaces of different species exhibit an amazing variety of surface patterning, despite having similar developmental characteristics and material properties. This pattern formation is robust enough to warrant its use in taxonomic classification. Focusing on pollen, we propose a theory of transitions to spatially modulated phases on spheres to explain both the variability and robustness of the patterns. We find that the sphere geometry allows for a wider variety of patterns compared with planar surfaces. A species may robustly “choose” among the possibilities by locally nucleating a patch of the pattern. We expect our theory to describe a wide variety of pattern-forming processes on spherical geometries.

Abstract

We propose a general theory for surface patterning in many different biological systems, including mite and insect cuticles, pollen grains, fungal spores, and insect eggs. The patterns of interest are often intricate and diverse, yet an individual pattern is robustly reproducible by a single species and a similar set of developmental stages produces a variety of patterns. We argue that the pattern diversity and reproducibility may be explained by interpreting the pattern development as a first-order phase transition to a spatially modulated phase. Brazovskii showed that for such transitions on a flat, infinite sheet, the patterns are uniform striped or hexagonal. Biological objects, however, have finite extent and offer different topologies, such as the spherical surfaces of pollen grains. We consider Brazovskii transitions on spheres and show that the patterns have a richer phenomenology than simple stripes or hexagons. We calculate the free energy difference between the unpatterned state and the many possible patterned phases, taking into account fluctuations and the system’s finite size. The proliferation of variety on a sphere may be understood as a consequence of topology, which forces defects into perfectly ordered phases. The defects are then accommodated in different ways. We also argue that the first-order character of the transition is responsible for the reproducibility and robustness of the pattern formation.

See: http://www.pnas.org/content/113/19/5189.abstract.html?etoc

PNAS May 10 2016; vol.113; no.19: 5189–5194

Fig. 1.

(A) Electron micrographs of pollen grains. The surface coat of the pollen, called exine, exhibits different patterns, ranging from stripes to many different patchy arrangements. Appearing below each micrograph is a corresponding height function representation constructed from our theory with the indicated spherical harmonics. (B, Left) Transmission electron microscopy (TEM) cross-section of an early pollen developmental stage. The surface of the immature cell undulates (yellow arrows) with a length scale consistent with the final patterning of the mature grain shown in a scanning electron microscopy (SEM) image in B, Right.