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

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.

arXiv comment16 pages, 4 figures

References in corpus (5)

• The Geometry of Soft Materials: A Primer
• Emergent superfluid crystals, frustration, and topologically defected states in multimode cavity QED
• Metastability in Fluctuation Driven First-order Transitions: Nucleation of Lamellar Phases
• Order and disorder in columnar joints
• Fast and accurate evaluation of Wigner 3j, 6j, and 9j symbols using prime factorisation and multi-word integer arithmetic

Cited by in corpus (11)

• Pollen Patterns Form from Modulated Phases
• Orientational Phase Transitions and the Assembly of Viral Capsids
• A rationale for mesoscopic domain formation in biomembranes
• Modeling mechanochemical pattern formation in elastic sheets of biological matter
• On the mechanism behind the inverse melting in systems with competing interactions
• Phase diagram of SALR fluids on spherical surfaces
• Aspects of nucleation on curved and flat surfaces
• Crystalline membrane morphology beyond polyhedra
• Model plasma membrane indicates the inner leaflet is poised to initiate compositional heterogeneities
• Phase Diagrams of Multicomponent Lipid Vesicles: Effects of Spherical Topology and Finite Size
• Invariant Theory and Orientational Phase Transitions