In various biological processes such as endocytosis and caveolae formation, the cell membrane is locally deformed into curved configurations. Previous theoretical and computational studies to understand membrane morphologies resulting from locally induced curvature are often limited to axisymmetric shapes, which severely restricts the physically admissible morphologies. Under the restriction of axisymmetry, past efforts predict that the cell membrane buds at low resting tensions and stalls at a flat pit at high resting tensions. In this work, we lift the restriction of axisymmetry by employing recent theoretical and numerical advances to understand arbitrarily curved and deforming lipid bilayers. Our non-axisymmetric morphologies reveal membrane morphologies which agree well with axisymmetric studies—however only if the resting tension of the membrane is low. When the resting tension is moderate to high, we show that (i) axisymmetric invaginations are unstable; and (ii) non-axisymmetric ridge-shaped structures are energetically favorable. We further study the dynamical effects resulting from the interplay between intramembrane viscous flow and induced curvature, and find the rate at which the locally induced curvature increases is a key determinant in the formation of ridges. In particular, we show that axisymmetric buds are favored when the induced curvature is rapidly increased, while non-axisymmetric ridges are favored when the curvature is slowly increased: The rate of change of induced curvature affects the intramembrane viscous flow of lipids, which can impede the membrane’s ability to transition into ridges. We conclude that the appearance of non-axisymmetric ridges indicates that axisymmetry cannot be generally assumed when understanding processes involving locally induced curvature. Our results hold potentially relevant implications for biological processes such as endocytosis, and physical phenomena like phase separation in lipid bilayers.
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