Fig. 1: Effective interfacial tension results from cyclic protein attachment and detachment at pattern interfaces.
From: Protein pattern morphology and dynamics emerging from effective interfacial tension
a, Coupling membrane attachment and detachment driven by NTP hydrolysis with fast cytosolic and slow membrane diffusion creates intracellular reaction–diffusion patterns. b, Patterns of diverse McRD systems are described by the movement of the interface (black line) between domains of high- and low-density or domains dominated by proteins A and B. The simulated membrane density of species A is shown in blue in the symmetric PAR system (f) on the surface of an ellipsoid [(a, b) = (1, 0.6)]. c, The straight pattern interface is sustained by cyclic fluxes of attachment and detachment fa,d(0) in an attachment and detachment zone, respectively, which induce a shallow cytosolic gradient counteracting membrane diffusion. d, In phase-separating liquids, the (exchange) chemical potential at the interface shows a gradient from outwards- to inwards-curved interfaces (orange). e, In McRD systems, an area difference between the attachment and detachment zones arises at curved interfaces. The blue shading symbolizes the membrane-density gradient (left). The (local) reactive turnover balance of the total attachment–detachment fluxes Ja(κ) = Jd(κ) can only be fulfilled if the local attachment–detachment fluxes fa,d(κ) change compared with their values fa,d(0) at a straight interface. The resulting curvature-induced cytosolic density gradients (equations (4) and (5)) induce mass transport from outwards- to inwards-bending regions (orange). f, In the symmetric PAR model, anterior (A, blue) and posterior (B, red) PAR proteins detach each other mutually (see ‘PAR system’ in Methods). g,h, The non-equilibrium Gibbs–Thomson relation equation (5) is verified in numerical simulations of a circular interface (inset) of the symmetric PAR system. The scaling for varying interface curvature \((\delta {\eta }_{{\rm{A}}}^{{\rm{stat}}}-\delta {\eta }_{{\rm{B}}}^{{\rm{stat}}}) \approx \sigma \kappa\) and the approximate scaling \(\sigma \approx \sqrt{{D}_{{\mathrm{m}}}^{{\rm{A}}}}+\sqrt{{D}_{{\mathrm{m}}}^{{\rm{B}}}}\) with the membrane diffusion coefficients are shown (κ ≈ 0.14; Supplementary Sections 3.6.4 and 3.6.5). The model equations and simulation parameters for panels b, g and h are given in ‘PAR system’ in Methods.
