Cell membranes interact with a myriad of curvature-active proteins that control membrane morphology and are responsible for mechanosensation and mechanotransduction. Some of these proteins, such as those containing BAR domains, are curved and elongated, and hence may adopt different states of orientational order, from isotropic to maximize entropy to nematic as a result of crowding or to adapt to the curvature of the underlying membrane. Here, extending the classical work of Onsager for ordering in hard particle systems and that of [E. S. Nascimento et al., Phys. Rev. E, 2017, 96, 022704], we develop a mean-field density functional theory to predict the orientational order and evaluate the free energy of ensembles of elongated and curved objects on curved membranes. This theory depends on the microscopic properties of the particles and explains how a density-dependent isotropic-to-nematic transition is modified by anisotropic curvature. We also examine the coexistence of isotropic and nematic phases. This theory predicts how ordering depends on geometry but we assume here that the geometry is fixed. It also lays the ground to understand the interplay between membrane reshaping by BAR proteins and molecular order, examined by [Le Roux et al., submitted, 2020].

中文翻译：

---

由密度和曲率驱动的膜上伸长和弯曲的蛋白质有序的理论

细胞膜与多种曲率活性蛋白相互作用，这些蛋白控制膜的形态并负责机械传感和机械转导。这些蛋白质中的某些蛋白质（例如含有BAR结构域的蛋白质）是弯曲和拉长的，因此可能会采用不同的取向顺序状态，即从各向同性到最大熵到向列型（由于拥挤或适应于下层膜的曲率）。在这里，扩展了Onsager在硬质颗粒系统中的经典工作以及[ES Nascimento等人的经典工作。，物理 修订版E，2017，96（022704），我们发展了一种平均场密度泛函理论来预测取向顺序并评估弯曲膜上细长和弯曲物体的集合体的自由能。该理论取决于粒子的微观性质，并解释了各向异性曲率如何改变依赖于密度的各向同性向向列转变。我们还研究了各向同性和向列相的共存。该理论预测了排序如何取决于几何，但是在此我们假设几何是固定的。这也为了解BAR蛋白进行的膜重塑与分子顺序之间的相互作用奠定了基础，该方法已被[Le Roux等，2007年。，已提交，2020年]。