Gaussian curvature directs the distribution of spontaneous curvature on bilayer membrane necks

Abstract

Formation of membrane necks is crucial for fission and fusion in lipid bilayers. In this work, we seek to answer the following fundamental question: what is the relationship between protein-induced spontaneous mean curvature and the Gaussian curvature at a membrane neck? Using an augmented Helfrich model for lipid bilayers to include membrane-protein interaction, we solve the shape equation on catenoids to find the field of spontaneous curvature that satisfies mechanical equilibrium of membrane necks. In this case, the shape equation reduces to a variable coefficient Helmholtz equation for spontaneous curvature, where the source term is proportional to the Gaussian curvature. We show how this latter quantity is responsible for non-uniform distribution of spontaneous curvature in minimal surfaces. We then explore the energetics of catenoids with different spontaneous curvature boundary conditions and geometric asymmetries to show how heterogeneities in spontaneous curvature distribution can couple with Gaussian curvature to result in membrane necks of different geometries.

Geometric coupling of helicoidal ramps and curvature-inducing proteins in organelle membranes

“…Similarly membrane tubes are present in the ER membrane network [1,[16][17][18], mitochondria [19][20][21], and in the form of cellular nanotubes [22][23][24]. Common to these seemingly disparate membrane structures is the catenoid-like geometry appearing at the neck of buds and the base of tubes [25]. Another striking example is the recently identified helicoidal structure connecting membrane sheets in the peripheral ER of neural and secretory cells [1,26] as well as in the spine apparatus present in dendritic spines [4] (see Fig.…”

“…We previously proposed a minimal model of membrane-protein interactions in catenoid-like necks [25], where our main finding was that at least two distinct curvature-inducing mechanisms are required to constrain a catenoid membrane neck below a critical radius. This has broad implications in ESCRT-mediated budding [9][10][11], yeast [26]).…”

Assembling of Topological Defects at Neck‐Shaped Membrane Parts

“…Furthermore, the budding process may occur due to a local change in the area difference between the outer and inner lipid layer or by a constriction force that pinches the membrane into a budded shape . In addition, the formation of neck‐like membrane structures is an integral part of fusion and fission in cellular membranes …”

“…Local neck structures are well modeled by catenoid geometries in several cases . These geometric shapes exhibit zero mean curvature and negative Gaussian curvature, representing minimal surfaces.…”

“…The Helfrich model for lipid bilayers was used to determine the spontaneous curvature field stabilizing a catenoid‐like membrane neck. They found that the spontaneous curvature field depends on the Gaussian curvature and that the catenoid‐shaped neck has an energy barrier at a critical neck radius, which corresponds to the switch in the sign of the spontaneous curvature . Furthermore, narrow and highly curved fusion pores may be facilitated by accumulation of anisotropic membrane components with orientational ordering.…”