Critical bifurcations in deformable membrane necks under inhomogeneous curvature:constriction frustration vs. abscissional elongation

A Aguilar-Maldonado ¹ Francisco Monroy ^2,3,4* Jose Antonio Santiago ^1,3,4*

• ¹ Departamento de Matem´aticas Aplicadas y Sistemas Universidad Autonoma Metropolitana Cuajimalpa Vasco de Quiroga 4871, 05348 Cd. de Mexico, MEXICO, Ciudad de México, Mexico
• ² Physical Chemistry, Complutense University of Madrid, Madrid, Madrid, Spain
• ³ Departamento de Quimica Fisica Universidad Complutense de Madrid Av. Complutense s/n, 28040, Madrid, SPAIN, Madrid, Spain
• ⁴ Research Institute Hospital 12 de Octubre, Madrid, Catalonia, Spain

Catenoid necks, as minimal surfaces with zero mean curvature (K = 0), minimize bending energy and serve as geometric scaffolds for scissional membrane remodeling. We apply the Canham-Helfrich model of flexible membranes to analyze deformable spontaneous curvature (K0), a key regulator of membrane scission events in cellular compartmentalization. To model functional membrane necking, we examine deformed catenoidal shapes with variable mean curvature (δK ̸ = 0) near the minimal-energy catenoid (K = 0), varies along either constrictional or elongational pathways. Using the Euler-Lagrange equilibrium equations, we derive inhomogeneous catenoid solutions, revealing metastable singularities departing from the critical catenoid of maximal area-a tipping point for scission. Using functional second-derivative analysis, we further examine how inhomogeneous K0 affects stability. The transition between frustrated constriction and abscissional elongation is numerically analyzed through conformal solutions to the governing inhomogeneous K0-field.