Ultradonut Topology of the Nuclear Envelope

Abstract

The nuclear envelope 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. Despite the intriguing architecture and vital biological functions of the nuclear membranes, how they achieve and maintain such a unique arrangement remains unknown. Here, we used the theory of elasticity and differential geometry to analyze the equilibrium shape and stability of this structure. Our results show that modest inand 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.nuclear envelope | lipid membranes | topology | buckling instability T he cell nucleus is bounded by two lipid bilayers arranged in a unique geometry called the nuclear envelope. These two bilayers are shaped into concentric spheres that are maintained at a remarkably uniform spacing of ∼ 30−50 nm (1), and yet are fused together at thousands of pores (holes) at an average spacing of ∼ 250−500 nm from each other [based on areal density measurements (2-4)]. At these pores, the membranes locally take on a toroid shape with a radius of curvature on the order of ∼ 20 nm ( Fig. 1). Lipid structures such as vesicles, spherocylinders (bacterial membranes), and biconcave discoids (red blood cell membranes) are all closed structures with no holes, and hence possess a zero genus (5) (Fig. 1). A donut, on the other hand, is also a closed structure but has one hole, and as a result has a genus of 1 (Fig. 1). If we fuse two donuts, we get a shape with a genus of 2, and if we fuse thousands of donuts and bend them to form a sphere, we obtain a nucleus-membrane-like structure ( Fig. 1). This structure, therefore, can be thought of as an "ultradonut" with a genus on the order of thousands. How the two membranes assemble in a unique arrangement with such a large number of local fusions is a fundamental question in both physics and biology that is still unresolved (6-10). To seek an answer to this puzzle, we ask three natural questions based on the common notions in the field of membrane physics.First, can membrane curvature-mediated interactions determine optimal pore number and interpore separation? This principle has been successfully used to predict the interactions of membraneembedded proteins and nanoparticles (11-13). However, in the case of nuclear envelope, Fig. 1C (14) shows that the curved shapes of the membranes at the pores do not persist over interpore length scales; The "curvature memory" of the membranes is lost beyond ∼ 100 nm and they remain essentially flat in between the pores. As a result, a pore does not sense the presence of other p...

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