Shape transition from elliptical to cylindrical membrane tubes induced by chiral crescent-shaped protein rods

Proteins often form chiral assembly structures on a biomembrane. However, the role of the chirality in the interaction with an achiral membrane is poorly understood. Here, we report how chirality of crescent-shaped protein rods changes their assembly and tubulation using meshless membrane simulations. The achiral rods deformed the membrane tube into an elliptical shape by stabilizing the edges of the ellipse. In contrast, the chiral rods formed a helical assembly that generated a cylindrical membrane tube with a constant radius in addition to the elliptical tube. This helical assembly could be further stabilized by the direct side-to-side attraction between the protein rods. The chirality also promotes the tubulation from a flat membrane. These results agree with experimental findings of the constant radius of membrane tubules induced by the Bin/Amphiphysin/Rvs (BAR) superfamily proteins.


S1. EFFECTS OF HOOKS IN ACHIRAL RODS
To clarify the effects of the hook-shaped structure of the rods, we compared the results with the achiral protein rods with and without the hook particles [the rods shown in Fig. 1(a) and Fig. S1(a)]. Both of them exhibit the rod assembly and elliptical tube formation as shown in Fig. S2. The rod assemblies along the azimuthal (θ) and longitudinal (z) directions occur at larger rod curvatures C rod for the rods with the hooks. This is caused by the excluded volume interaction of the hook particles; the rods without the hooks can form a more compact assembly on the elliptical tube [compare Fig. S2

S2. DEPENDENCE ON ATTRACTION SITES AND HOOK POSITION IN CHIRAL RODS
To confirm that our conclusions are robust to changes in the chiral protein structures, we examined the helical tube formation by four different types of the protein structures as shown in Figs. S1(b)-(e). The first three models have different segment pairs for the side-to-side attractive interaction, and in the last model, hook parti- The magenta and green lines represent the qθ and qz modes for the rods without the hooks, respectively. The black and gray lines represent the qθ and qz modes for the rods with the hooks, respectively.
cles are connected with the second and third segments. All four models exhibit the helical cylinder formation as shown in Fig. S3. The pair of the third and fourth segments (and also the second and fourth segments) has a contact in the helical assembly of the original model so that the attraction strength ε for the assembly is close to that of the original model. In contrast, a greater value of ε is required for the attraction between the third segments, since they have no contact in the original rod assembly. The helical assembly can also be formed when the hook position is shifted to the inner position [hook . Thus, the protein behavior does not qualitatively change under the above structure modifications. We conclude the helical assembly on a membrane is a general phenomenon for the chiral rods.

MOVIE CAPTIONS
Movie 1: Formation of the helical cylinder of chiral protein rods without direct attraction between the rods at C rod r rod = 3.2, N = 4800, and R cyl /r rod = 1.18. An equilibrium conformation at C rod r rod = 2.8 is used as an initial state.
Movie 2: Formation of the helical cylinder of chiral protein rods at ε/k B T = 2, C rod r rod = 2, N = 9600, and R cyl /r rod = 1.31. An equilibrium conformation at ε/k B T = 0.5 and C rod r rod = 1.5 is used as an initial state.
Movie 3: Tubulation from flat tensionless membrane induced by chiral protein rods at high rod density φ rod = 0.2 and C rod r rod = 2.5. An equilibrium confor- Radial distribution of the membrane particles of the tubules along the z axis that is taken along the eigenvector of the greatest eigenvalue of the gyration tensor for the middle region of each tubule. The center of the cylindrical coordinate for the i-th particle is set to the center of the particles in the sliced region for −0.2r rod < z − z i < 0.2r rod . The green crosses and rod dots represent the data corresponding to the upper and lower tubules shown in (a), respectively. The mean radius calculated from 12 tubules is shown in Fig. 3. mation at C rod r rod = 0 is used as an initial state.
Movie 4: Tubulation from flat tensionless membrane induced by chiral protein rods at low rod density φ rod = 0.05 and C rod r rod = 3. An equilibrium conformation at C rod r rod = 0 is used as an initial state.