Mechanics of Multilamellar Membrane Structures
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Multilamellar membrane structures are the common barriers in the cell compartments. One example is the nuclear envelope, which is a unique topological structure formed by lipid membranes in eukaryotic cells. Unlike other membrane structures, the nuclear envelope comprises two concentric membrane shells fused at numerous sites with toroid-shaped pores that impart a geometric genus on the order of thousands. Another example is the endoplasmic reticulum (ER), which is continuous with the nuclear envelope and is extended to the cell periphery. It is the site of molecular mechanisms including protein synthesis and lipid metabolism in the cell. In the vicinity of the nucleus, the stacked sheets of ER are connected through a system of helical pores. The sheets themselves are formed using a double bilayer structure. In addition to these two compartments, the plasma membrane (PM), which is a single bilayer can undergo some shape deformation that brings the apical and basal membranes of the cell together and form another type of multilamellar membarne structure. The two membranes then fuse and form tunnels that are the site of intracellular transport. Although several protein structures are identified to have a role in regulating the geometry of these compartments there is still a lack of consensus on the fundamental forces and physical mechanisms that establish the geometry of multilamellar membrane structures. Here, we use the theory of elasticity and differential geometry to analyze the equilibrium shape and stability of the structure of the nuclear envelope, endoplasmic reticulum, and intracellular transport. For the Nuclear envelope, our results show that modest in- and out-of-plane stresses present in the membranes not only can define the pore geometry, but also provide a mechanism for destabilizing membranes beyond a critical size and set the stage for the formation of new pores. Our results suggest a mechanism wherein nanoscale buckling instabilities can define the global topology of a nuclear envelope-like structure. Similarly for the endoplasmic reticulum, we recruit the equilibrium equation to show that membrane in-plane stress can regulate the key geometric features of this structure including the thickness of the sheets, inter-sheet distance, and diameter of the connecting pores. For the intracellular transport and tunnel opening, our study reveals that hole radius is determined by plasma membrane tension via a commonly used critical length-scale defined by the square root of the ratio of flexural stiffness to in-plane tension. This relationship suggests that the hole diameter increases with a reduction in membrane tension, a finding aligned with the experimental observations but in contrast with the main current model in the literature.