### Pattern Formation

The outer coating of many organisms, including insect eggs, plant pollen, mite carapaces, and fungal spores, exhibits a wide range of intricate patterns.  These patterns have counter-intuitive features:  Although a single species reproduces the same pattern robustly, closely-related species may have wildly different patterns.  For example, in pollen (pictured below), the pattern may be used to uniquely identify a species.  We use physical approaches to try to understand the development of such patterns.
 (a) Scanning electron micrographs (SEMs) of plant pollen surfaces (top row, courtesy of A. Radja) are compared to the ordered states of a Brazovskii model on a sphere (bottom row), characterized by angular momentum modes $(\ell_0,m)$ (b) In the left panel, the undulating plasma membrane in a cross-section of pollen grain (a tunnelling electron microscope slice) is indicated by arrow.  In the right panel, the pollen outer surface is shown via SEM.
In pollen grains, the pattern template is likely formed via an undulated plasma membrane.  We modeled these undulations on the surface of a sphere, using a Brazovskii model. Our general approach interprets the pattern formation as a first-order phase transition to a patterned phase with some characteristic angular momentum mode $\ell_0$ on the surface of a sphere, corresponding to a pattern wavelength $\lambda_0 = 2 \pi R/\ell_0$ [1]. Our theory yields predictions for the free energy difference $\Delta \Phi$ between an unpatterned, smooth state and a patterned state characterized by a scalar field $\Psi \equiv \Psi(\theta,\phi)$ describing, for instance, the amount of deposited material on the spherical pollen grain at longitude $\phi$ and colatitude $\theta$.  This scalar field is written as a linear combination of spherical harmonics $Y_{\ell}^m \equiv Y_{\ell}^m(\theta,\phi)$ with a particular $\ell_0$ and some choice of azimuthal quantum numbers $m$: $\Psi(\theta,\phi)=\sum_{m} \Psi_{\ell_0}^m Y_{\ell_0}^m$.  The computed free energy $\Delta \Phi$ (illustrated below), takes into account thermal fluctuations to the following Brazovskii Hamiltonian, perturbatively in the interaction $\mathcal{H}_{\mathrm{int}}$, which includes terms that are cubic and quadratic in the field $\Psi$: $\mathcal{H}[\Psi] = \frac{1}{2} \sum_{\ell,m} [K(\ell-\ell_0)^2+R^2 \tau]|\Psi_{\ell}^m|^2+ \mathcal{H}_{\mathrm{int}}.$
The pattern phases observed in pollen grains are similar to those found in phase-separated lipid vesicles with lipid compositions similar to those found in biological membranes. We are currently working on applying the theory to understand phase transitions in membranes and to see if these patterned phases have some biological relevance.

References
[1] M. O. Lavrentovich, E. M. Horsley, A. Radja, A. M. Sweeney, and R. D. Kamien First-order patterning transitions on a sphere as a route to cell morphology Proceedings of the National Academy of Sciences of the USA  113 5189 (2016)