Fig. 3. Membrane deformation through membrane protein hydrophobic mismatch provides a theoretical understanding of the experimental results.

a–e Hydrophobic mismatch as an energy source for membrane-mediated membrane protein interactions (Supplementary Fig. 7, Supplementary Note 1). a Schematic of the inclusion-induced membrane deformation with l_protein, protein hydrophobic thickness, l_bilayer, bilayer hydrophobic thickness, u, hydrophobic mismatch, and d, edge-to-edge distance between proteins (hydrophobic (red) and hydrophilic (blue) protein surfaces). b Perspective schematic representation of 2D membrane profiles (one leaflet) when two cylindrical proteins approach. The space occupied by proteins is not considered part of the deformation field, u_xy, and filled with u₀ for illustration: positive, (e.g., C14, left) and negative (e.g., C20, right) hydrophobic mismatch. Bottom to top: d ~7 nm, ~4 nm, and ~1 nm. c Schematic representation of the 2D deformation energy density, dG_def, for d ~7 nm, ~4 nm and ~1 nm. d 2D elastic mismatch-dependent energy potential as a function of d for the four investigated bilayers. e Compression and bending components of the total 2D elastic mismatch-dependent energy potential, repulsive from ~7 nm to ~3.5 nm separation, and attractive at separation shorter ~3.5 nm. f, g Changes of 2D membrane configurations in association-dissociation events (Supplementary Fig. 9): (f) The five basic local-configurations in microscopic array assembly. g Representative rearrangements of local configurations associated with one-bond (rearrangement 1) and two-bond (rearrangement 2) interactions. h–j Membrane protein automata (Supplementary Fig. 10, Supplementary Note 2, and Supplementary Movie 8): (h) Simulated clusters in membranes of no (left), intermediate (middle), and large (right) hydrophobic mismatch. i Association energy (ΔG_asso), and (j) energy difference between states 1B and 2B (ΔG_diff), as functions of the deformation energy scale factor ψ/ψ_norm, representing the hydrophobic mismatch square (u₀²), with ψ_norm = {1.00 2.06 3.22 4.10} (see “Methods”).