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)