Curvature-mediated feedback leads to turbulence of growing interfaces

The cytoplasmic membrane of cells is a complex object which dynamically undergoes shape transitions during many cellular processes. One important aspect is the generation and maintenance of curvature, which defines the morphology of cells. Recently, it has been shown that cytoskeletal motor proteins induce membrane deformations via highly cooperative interactions between motor proteins and lipids which depend on membrane morphology [1]. Motivated by this work, we have studied the dynamics of a growing interface driven by the local density of membrane proteins. A key feature of our model is a coupling between interface morphology and attachment kinetics of proteins. For the deterministic case, we find that morphological coupling results in chaotic dynamics, reminiscent of the Kuramoto-Shivashinsky equation. The emergence of chaos is quantitatively confirmed by numerically determining the spectrum of Lyapunov exponents. We find that our model is similar to the KPZ model with non-uniform growth velocity along the interface. Our results further show that the growth kinetics do not fall into the KPZ universality class.

[1] Reconstitution reveals how myosin-VI self-organises to generate a dynamic mechanism of membrane sculpting, B.Rogez, L.Würthner, et al., Nat. Commun. (2019)