### **Pollen grains**

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.

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.

**Lipid membranes**

The heterogeneous mixture of lipids found in biological membranes provides an interesting playground for exploring the patterned phases. We look at model membranes consisting of various mixtures of common lipids: DSPC/DOPC/POPC. These mixtures, in the presence of cholesterol, phase separate if the temperature is sufficiently low. The different phases can induce different membrane curvatures, creating the possibility of generating the modulated phases found in pollen. Neutron scattering techniques may be used to identify this phase separation in small (nanoscopic) vesicles [2]. We found that melatonin, a small molecule that associates with one of the lipid phases, can preserve the phase separation:

**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)[2] D. Bolmatov et al

*Deciphering melatonin-stabilized phase separation in phospholipid bilayers*Langmuir**35**12236 (2019)