Fig. 3.

Mechanical model of undulating and pearled tubule formation. (A) Unduloid-like shapes solution for Helfrich energy minimization at different values of nondimensional parameter, $\alpha$. For α ∼0.75, the membrane takes on a cylindrical shape (purple line); for α > 0.75, the unduloid becomes a sphere similar to a string of pearls (gray line). (B) Schematic depiction of membrane tubule formation due to the compressive stresses applied by liquid–liquid phase separation on the membrane. On a flat membrane, the density of protein segments decreases with increasing distance from the membrane surface, such that x is greater than x’. Therefore, if the membrane remains flat, there will be an increasing number of unsatisfied potential protein–protein interactions as distance from the surface increases. These unsatisfied interactions create a driving force for membrane bending, which increases the density of protein segments at a distance from the membrane (x’ < x), leading to more overlap among the proteins and stronger protein–protein interactions. (C) Schematic of the axisymmetric simulations depicting the simulation domain and the boundary conditions. The yellow region represents the bare membrane, and the green region is the area coated by the proteins. The dashed lines indicate the cap of the tubule, assumed to have a constant curvature. The inset shows the spontaneous curvature distribution along the tubule region used to model the membrane shape. (D) Undulating tubules minimize the membrane bending energy as the spontaneous curvature increases for uniform bending rigidity of the membrane (κ = 80 k_BT). (E) Percentage of change in the tubule diameter ((D−D_{κ = 25 kBT})/D_{κ = 25 kBT}) as a function of the bending rigidity for three different values of spontaneous curvature. The dashed lines show a square root dependence on the bending modulus by fitting to the curve ($A\sqrt{\kappa }+B$), where for the gray line, A = 5.4, B = −26.44; for the pink line, A = 2.71, B = −12.9; and for the blue line, A = 1.53 and B = −7.4. (F) Pearled tubules minimize the bending energy of the membrane for heterogeneous membrane rigidity (κ_ratio= κ_{protein-domain}/κ_{bare membrane}), C₀ = 3.5 μm⁻¹. (G) Percentage of change in the tubule diameter ((D − D_{κ = 25 kBT})/D_{κ = 25 kBT}) as a function of the bending rigidity for three different values of spontaneous curvature for κ_ratio = 20. The dashed lines are the fitted curve ($A\sqrt{\kappa }+B$), where for the gray line, A = 10.98, B = −51.31; for the pink line, A = 4.22, B = −19.58; and for the blue line, A = 3.1 and B = −13.