Extended Data Fig. 2: Modelling of droplet–membrane-sheet interaction unveils wetting as a determinant of droplet autophagy.

a, Contact angles of sheets at flat droplet surfaces. There are three distinct wetting states depending on the value of the contact angle. We focus on partial wetting (0° < θ_in < 180°) throughout this study. Sheets in contact with homogeneous, non-phase-separated solutions correspond to dewetting and complete wetting, as described previously⁵. For pinned shapes as depicted here (when the three-phase contact line is located at the rim), the interfacial energy of the partial wetting states is roughly independent of the specific value of the contact angle, because for thin sheets with r_rim ≪ r_sh the area of the rim is negligible with respect to the area of the rest of the sheet. Meanwhile, the value of θ_in becomes important for unpinned sheets (see b–e, g, h). Instability of the open sheet state (which is a pinned shape) towards the closure of the autophagosome is the only state relevant to this study. Therefore the stability diagrams in f and in Figs. 2d, e, 4a, Extended Data Fig. 5a, b are independent of the value of θ_in. b–e, Energy landscapes for autophagosome formation. M_sh = 1/R_sh is the mean curvature of the sheet. M_shr_sh = 2 corresponds to piecemeal autophagy of the droplet, M_shr_sh = −2 to cytosolic autophagy, and the vertical dotted lines correspond to M_sh = 1/R_drop; that is, the state in which droplet and sheet curvatures coincide and the sheet wets the droplet without deforming it. The three coloured lines (in b–e) correspond to different values of r_sh/r_rim (in b, c) or \({\Sigma }_{{\rm{cd}}}{r}_{{\rm{rim}}}^{2}/\kappa \) (in d, e). In all cases, the energy baseline is arbitrary, and the curves have been shifted vertically for clarity to prevent overlapping. In b, c, the sheet grows (blue–yellow–red) at a constant droplet surface tension of \({\Sigma }_{{\rm{cd}}}{r}_{{\rm{rim}}}^{2}/\kappa \)= 0.04. In d, e, the surface tension decreases (blue–yellow–red) at a constant sheet size of r_sh/r_rim = 20. The M_sh = 1/R_drop state is typically stable for small sheet size or high surface tension (blue lines). As the sheet size increases or the surface tension decreases, this state eventually becomes marginally stable (yellow lines; these lines correspond to the values of Σ_cd and r_sh at which the energy barrier disappears and were used to develop the stability diagrams shown in f and in Figs. 2d, e, 4a, Extended Data Fig. 5a, b) and finally unstable (red lines). The instability occurs towards piecemeal autophagy in b, d, which correspond to zero spontaneous curvature contrast (m_cd = 0), or towards cytosolic autophagy in c, e, which correspond to negative contrast (m_cdR_drop = −3). In all cases, the solid lines correspond to θ_in = π/4, and the dashed lines to θ_in = 3π/4. Both lines overlap over a wide region at intermediate values of M_sh, which correspond to the pinned state, and diverge in the unpinned state. Because the instability occurs within the pinned region, it is independent of the value of θ_in. Specific values used are (given in the order blue–yellow–red): b r_sh/r_rim = 5, 12, 20; c r_sh/r_rim = 5, 8.3, 20; d \({\Sigma }_{{\rm{cd}}}{r}_{{\rm{rim}}}^{2}/\kappa \) = 0.2, 0.068, 0.02; and e \({\Sigma }_{{\rm{cd}}}{r}_{{\rm{rim}}}^{2}/\kappa \)= 0.2, 0.054, 0.02. Landscapes can be compared with the stability diagrams b1 and b4 in Extended Data Fig. 5b. The droplet radius is set to R_drop = 50r_rim in all cases. f, Stability diagrams (as shown in Fig. 2d, e) for droplets of sizes R_drop/r_rim = 10, 20, 40, 80 that have a corresponding droplet radius of 0.25, 0.5, 1 or 2 μm (for sheets with r_rim = 25 nm, as in Fig. 2). For each droplet size, we also display a horizontal dotted line for r_sh = 2R_drop that shows the sheet size required for complete droplet sequestration, which increases strongly with droplet size. These lines demonstrate that complete sequestration of large droplets is unlikely, owing to the very large sheet required. Meanwhile, the minimal surface tension necessary for complete droplet engulfment is smaller for larger droplets. g, h, Unpinning of the sheet rim from the droplet surface. Four sequences demonstrating morphological changes of sheets developing at droplets. Two contact angles (45° in g, 135° in h) and both directions of bending, either enclosing the droplet (top panels) or the cytosol (bottom panels) are portrayed in these scenarios. All shapes are calculated including the correct volume, area and contact angle constraints, for R_drop/r_sh = 0.82. The two sides of the sheet (green) cannot be resolved owing to their close proximity. Thus, the sheets are depicted as a single line. Vertical lines indicate transitions between pinned and unpinned shapes.