Figure 2. Mechanics of binding.

(A) A schematic of the model. A protein filament (red) may bend and bind to a cylindrical membrane (green) at an angle with respect to the long axis, $\theta$, and the equilibrium conformation may involve deformations of both protein and membrane. (B) Plot of the estimated free energy change due to filament binding against the binding angle, $\theta$, where the cell radius $R_{\mathrm{cell}}=0.5\mu \text{m}$, the pressure difference across the membrane ${\displaystyle p=20\mathrm{a}\mathrm{t}\mathrm{m}}$, the estimated turgor pressure of B. subtilis (Whatmore and Reed, 1990), and the other model parameters are given in Supplementary file 1. The estimate is similar for ${\displaystyle p=0.2\mathrm{a}\mathrm{t}\mathrm{m}}$, but exhibits a more shallow potential well for $R_{\mathrm{cell}}=1.5\mu \text{m}$ (red curve). (C) An approximate phase diagram of protein-membrane binding. The dashed lines delineate regimes, as explained further in Appendix 1.

Figure 2—figure supplement 1. Energetic modeling of MreB binding and deformation of vesicles.

(a) A protein filament is modeled as a curved elastic rod that attaches to the inner membrane in an energetically favorable manner. $U$ denotes the interaction region. In the Monge gauge, a height function $h(x,y)$ parameterizes the membrane shape ($h_0$ in the undeformed state) and the domain of $h$ ($h_0$) is denoted $\mathrm{\Omega }$. The interaction region restricts the value of $h$ over the region $U$. (b) Example solutions for the membrane shape for $p=0$ and ${\displaystyle R_{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}=0.5\mu \text{m}}$, ${\displaystyle 1.5\mu \text{m}}$, and $\mathrm{\infty }$ (left to right) on a finite domain, for the parameter values summarized in Supplementary file 1. (c) (Left) The binding of an MreB filament to a spherical vesicle induces vesicle curvature, a schematic of which is shown. (Right top) A numerical solution for the membrane shape over a finite domain where the undeformed configuration is a sphere with a principal curvature opposite that of an MreB filament. Here $p=0$, the sphere radius $R_{\mathrm{sphere}}=1\mu \text{m}$, and the remaining parameter values are summarized in Supplementary file 1. (Right bottom) MreB filaments binding to vesicles in vitro, with a schematic shown. The image is reproduced here from Salje et al. (2011) under a CC BY 3.0 license (https://creativecommons.org/licenses/by/3.0/).