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...

Mechanics of nuclear membranes

“…Consistent with the notion that nuclear membrane tension may have an important role in nuclear membrane structure, a computational approach that minimized bending and stretching energies using the Helfrich-Canham model (Torbati et al, 2016) predicted a decrease in the nuclear membrane separation with an increase in membrane tension. When the membrane was assumed to be under compression (negative tension), the bilayer separation increased significantly, eventually undergoing a bucking instability at a critical compressive stress.…”

“…If the areal stretch modulus and the bending rigidity of lipid membranes is known, it is possible to calculate their equilibrium shapes under defined geometric constraints (boundary conditions), and under known applied mechanical stresses. This approach has been used to calculate equilibrium shapes of the nucleus and nuclear membranes (Lim et al, 2007;Noguchi, 2016;Torbati et al, 2016), which is discussed in the next section. This approach has also been extensively used to predict the shapes of a diverse array of other lipid membrane structures, such as the membranes of Tensile and shear forces from the cytoskeleton can act on the ONM, and forces from the lamina and/or chromatin can act on the INM.…”

“…One explanation for the observed membrane bending that leads to pore formation is that this bending may be a type of buckling instability that is induced owing to a build-up of in-plane compressive stresses in the membranes. Computational modeling using the Helfrich-Canham model suggests that the buckling response of the nuclear membrane is strongly dependent on the two-dimensional membrane area that lies in between adjacent nuclear pores (Torbati et al, 2016). The nuclear membrane is predicted to be highly stable against buckling if the membrane length between pores is less than 250 nm, while it is predicted to be highly unstable at lengths above 500 nm (Torbati et al, 2016).…”

“…Computational modeling using the Helfrich-Canham model suggests that the buckling response of the nuclear membrane is strongly dependent on the two-dimensional membrane area that lies in between adjacent nuclear pores (Torbati et al, 2016). The nuclear membrane is predicted to be highly stable against buckling if the membrane length between pores is less than 250 nm, while it is predicted to be highly unstable at lengths above 500 nm (Torbati et al, 2016). Thus, buckling is predicted to occur for membrane patches that are in the range of 250 to 500 nm diameters, which is consistent with experimental measurements of nuclear pore spacing (Belgareh and Doye, 1997;D'Angelo et al, 2006;Dultz and Ellenberg, 2010).…”

Geometry of the nuclear envelope determines its flexural stiffness

“…, 2003 ). We have previously shown that a flat shape near the pore necessitates prescription of tension in a membrane lacking curvature-inducing proteins ( Torbati et al. , 2016 ).…”