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.

Keywords: Brazovskii; pattern formation; phase transitions; pollen.

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.

Fig. S1.

(A) The ratio $\mathrm{\Delta }L$, calculated in Eq. S53, of the two loop contributions to the correlation function in the ordered state (Eq. S44) plotted for various values of ${\ell }_0$, $m_{\mathrm{1,2}}$. Note that this factor is small for all directions except when $m_1=-m_2$. (B) The plotted ratio shows that the scattering function $A_{{\ell }_0}$ decays rapidly with increasing distance $\left|\mathrm{\Delta }m\right|$ away from the special direction $m_1=-m_2$.

Fig. 2.

Plots of metastable, ordered states (Eq. 6) with ${\ell }_0=15$ with identical energies, within our approximation, chosen by changing the phases $e^{i{\theta }_m\pi /4}$ of the directions $c_m$ of the spherical harmonic modes. The bright yellow and dark purple regions indicate, respectively, regions of greater and lesser values of the ordered state $\overline{\mathrm{\Psi }}$. For these plots, we have chosen three nonzero $c_m$s with $m=\mathrm{4,5,7}$ and phases determined by the triple $\left({\theta }_4,{\theta }_5,{\theta }_7\right)$ shown above each plot. In the Bottom row, we highlight a region of the pattern as we vary one of the phases. Note that even though these states all have the same value of ${\ell }_0$ and choice of ms, changing the relative phases can substantially alter the resulting pattern.

Fig. 3.

The free energy difference $\mathrm{\Delta }\mathrm{\Phi }$ between ordered states and the disordered phase as a function of the reduced temperature $\tau <0$ for ${\ell }_0=12$, $R=10$, and ${\lambda }_4=0.01$. The plot legend shows the chosen combination of ms. The cubic term coefficient is zero except for the $\left|m\right|=\mathrm{0,10}$ case, for which ${\lambda }_3=0.015$. When ${\lambda }_3=0$, single $\left|m\right|\approx {\ell }_0/2$ modes are favored for these modest values of ${\ell }_0$. At higher values of ${\ell }_0$, we find that linear combinations are more favorable, instead. The presence of a cubic term favors the formation of phases with hexagonal patterns. As we decrease the temperature (increasing $-\tau$), the ordered states become more favorable. There are a wide variety of metastable ordered states.