Out-of-equilibrium mechanochemistry and self-organization of fluid membranes interacting with curved proteins
New Journal of Physics
The function of biological membranes is controlled by the interaction of the fluid lipid bilayer with various proteins, some of which induce or react to curvature. These proteins can preferentially bind or diffuse towards curved regions of the membrane, induce or stabilize membrane curvature and sequester membrane area into protein-rich curved domains. The resulting tight interplay between mechanics and chemistry is thought to control organelle morphogenesis and dynamics, including traffic,
... luding traffic, membrane mechanotransduction, or membrane area regulation and tension buffering. Despite all these processes are fundamentally dynamical, previous work has largely focused on equilibrium and a self-consistent theoretical treatment of the dynamics of curvature sensing and generation has been lacking. Here, we develop a general theoretical and computational framework based on a nonlinear Onsager's formalism of irreversible thermodynamics for the dynamics of curved proteins and membranes. We develop variants of the model, one of which accounts for membrane curving by asymmetric crowding of bulky off-membrane protein domains. As illustrated by a selection of test cases, the resulting governing equations and numerical simulations provide a foundation to understand the dynamics of curvature sensing, curvature generation, and more generally membrane curvature mechano-chemistry. To quantitatively understand these phenomena, various biophysical studies have exposed artificial lipid membranes to purified proteins in controlled conditions  . At high concentration, curved proteins can induce severe membrane curvature when incubated with liposomes , can stabilize membrane tubes [10, 16] , and can dynamically trigger protein-rich tubular protrusions out of tense vesicles    . At lower concentrations, proteins sense curvature and preferentially adsorb or migrate to favorably curved membranes, as probed in assays involving polydisperse vesicle suspensions , vesicles with membrane tethers [18, 21] or supported lipid bilayers on wavy substrates  . Since protein-rich curved domains sequester apparent membrane area from the adjacent planar membrane, their formation perform work against membrane tension, and thus can be hindered if tension is large enough. This kind of mechano-chemical coupling, tested in vitro by exposing aspirated vesicles to BAR proteins  , has physiological implications during the mechano-protection of stressed cells by the release of membrane area through disassembly of caveolae  , or in the regulation of clathrin-mediated endocytosis by membrane tension  . A number of theoretical and computational studies at various scales have been developed to understand the interaction between curved proteins and membranes. At the nanoscale, all-atom molecular dynamics have described curvature generation by single domains  and curvature maintenance by multiple proteins  . Reaching a micron, coarse-grained molecular dynamics simulations, treating the membrane either molecularly or as a continuum object, have followed the aggregation of multiple proteins to cooperatively form protein-rich curved domains      . Models treating proteins as discrete objects in a continuum membrane have examined membrane-mediated protein-protein interactions [31, 32], or the spontaneous curvature induced by anchored polymers [33, 34] . A fundamental obstacle to develop mean field theories at larger scales based on models where proteins are discrete object, however, is the non-additive nature of membrane-mediated pairwise interactions  . Reaching larger scales, continuum models combining the Helfrich curvature energy [36, 37] with thermodynamic models of mixtures [38, 39] have been quite successful in recapitulating and interpreting quantitative in vitro measurements, see [40, 41] for two recent reviews. These models suggest that, rather than two different mechanisms, curvature sensing and generation are two manifestations of the same mechanochemical coupling. They have provided a background to understand the emergence of heterogeneous proteinrich curved domains using linear stability analysis [42, 43] , or curvature sorting of proteins in equilibrium and at fixed shape between tubes and vesicles [18,    or on wavy surfaces  . Also in equilibrium, proteinmembrane interactions allowing for shape changes were studied in  . With a few exceptions under rather restrictive conditions [49, 50] , previous theories of the interaction between curved proteins and membranes have focused on equilibrium. Yet, cellular functions are fundamentally out-of-equilibrium. Here, we develop a nonlinear and self-consistent continuum theory to study the two-way chemo-mechanical coupling between membranes and curved proteins out-of-equilibrium, and perform numerical simulations. We follow a nonlinear Onsager's variational formalism in which the dynamics emerge from a competition between energy release rate and dissipation. Ingredients in the free energy include the curvature energy of the membrane with a protein-induced spontaneous curvature, the entropy of mixing of proteins, and protein-protein interactions. As it evolves, the system dissipates energy through the drag between proteins and the membrane, and through lipid shear viscosity as the membrane changes shape. See  for a related theoretical work. The setup and ingredients of the theory are given in sections 2 and 3. The resulting governing equations for protein transport and for membrane dynamics are presented in section 4. We particularize this general theory to axisymmetry and present a direct numerical approach to approximate the theory in section 5. In section 6, we present a selection of numerical calculations showing the ability of the theory to describe curvature sensing, generation, and more generally the intimate chemo-mechanical coupling of the membrane-protein system. Finally, we introduce two variants of this model in section 7, one accounting for protein's bending elasticity and the other addressing membrane bending by crowding of bulky off-membrane protein domains [10, 17] , and collect our conclusions in section 8.