Absolute vs Convective Instabilities and Front Propagation in Lipid Membrane Tubes

TL;DR

This paper investigates the stability and dynamics of lipid membrane tubes, revealing how dimensionless parameters influence instability types, front propagation, and shape transitions, with implications for understanding axonal pattern formation.

Contribution

It introduces a comprehensive analysis of membrane tube stability using new dimensionless numbers and links nonlinear simulations to pattern formation mechanisms in biological systems.

Findings

Growth rate depends only on the F"oppl--von Kármán number.

The Scriven--Love number determines the nature of instability.

Membrane dynamics can be modeled by extended Fisher--Kolmogorov equations.

Abstract

We analyze the stability of biological membrane tubes, with and without a base flow of lipids. Membrane dynamics are completely specified by two dimensionless numbers: the well-known F\"oppl--von K\'arm\'an number $\Gamma$ and the recently introduced Scriven--Love number $SL$, respectively quantifying the base tension and base flow speed. For unstable tubes, the growth rate of a local perturbation depends only on $\Gamma$, whereas $SL$ governs the absolute or convective nature of the instability. Furthermore, nonlinear simulations of unstable tubes reveal an initially localized disturbance results in propagating fronts, which leave a thin atrophied tube in their wake. Depending on the value of $\Gamma$, the thin tube is connected to the unperturbed regions via oscillatory or monotonic shape transitions -- reminiscent of recent experimental observations on the retraction and atrophy of axons. We elucidate our findings through a weakly nonlinear analysis, which shows membrane dynamics may be approximated by a model of the class of extended Fisher--Kolmogorov equations. Our study sheds light on the pattern selection mechanism in axonal shapes by recognizing the existence of two Lifshitz points, at which the front dynamics undergo steady-to-oscillatory bifurcations.