Pollen Cell Wall Patterns Form from Modulated Phases.

The ornately geometric walls of pollen grains have inspired scientists for decades. We show that the evolved diversity of these patterns is entirely recapitulated by a biophysical model in which an initially uniform polysaccharide layer in the extracellular space, mechanically coupled to the cell membrane, phase separates to a spatially modulated state. Experiments reveal this process occurring in living cells. We observe that in ∼10% of extant species, this phase separation reaches equilibrium during development such that individual pollen grains are identical and perfectly reproducible. About 90% of species undergo an arrest of this process prior to equilibrium such that individual grains are similar but inexact copies. Equilibrium patterns have appeared multiple times during the evolution of seed plants, but selection does not favor these states. This framework for pattern development provides a route to rationalizing the surface textures of other secreted structures, such as cell walls and insect cuticle.

Aspects of nucleation on curved and flat surfaces.

We investigate the energetics of droplets sourced by the thermal fluctuations in a system undergoing a first-order transition. In particular, we confine our studies to two dimensions with explicit calculations in the plane and on the sphere. Using an isoperimetric inequality from the differential geometry literature and a theorem on the inequality's saturation, we show how geometry informs the critical droplet size and shape. This inequality establishes a "mean field" result for nucleated droplets. We then study the effects of fluctuations on the interfaces of droplets in two dimensions, treating the droplet interface as a fluctuating line. We emphasize that care is needed in deriving the line curvature energy from the Landau-Ginzburg energy functional and in interpreting the scalings of the nucleation rate with the size of the droplet. We end with a comparison of nucleation in the plane and on a sphere.

First-order patterning transitions on a sphere as a route to cell morphology.

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.