Absolute/convective instabilities and front propagation in lipid membrane tubes

Biological lipid membranes make up the cell boundary, and are often found in cylindrical shapes. For example, axonal flows bring lipids from the growth cone of a neuron to its cell body. We investigate the stability of membrane tubes with and without a base flow of lipids. We confirm tubes are stable at low tensions and unstable at high tensions, yet for unstable tubes we find increasing the base flow velocity changes the nature of the instability from absolute to convective. In the former case, an initially local perturbation will grow faster than it is convected downstream, and eventually will invade the entire domain. In the latter case, the perturbation is convected faster than it spreads, and at long times a stationary observer will see no disturbance---though the perturbation continues to grow. Nonlinear simulations reveal an initially localized disturbance results in propagating fronts, which leave a thin tube in their wake. Depending on the base tension, the thin tube is connected to the unperturbed regions via oscillatory or monotonic shapes---reminiscent of experimental observations in axons. Our study sheds light on the pattern selection mechanism in axonal shapes, and we determine the base tension at which the front dynamics undergo steady-to-oscillatory bifurcations.