Global asymptotic stability of the active disassembly model of flagellar length control

TL;DR

This paper proves the global asymptotic stability of an active disassembly model for flagellar length control, extending the analysis to multiple flagella and exploring implications for sensory neurons.

Contribution

It provides a rigorous mathematical proof of stability for the flagellar length control model and generalizes it to multiple flagella, offering insights into cellular size regulation mechanisms.

Findings

Proved global asymptotic stability of the model for two flagella.

Extended stability analysis to systems with up to twenty flagella.

Ruled out oscillations in the model, confirming robustness.

Abstract

Organelle size control is a fundamental question in biology that demonstrates the fascinating ability of cells to maintain homeostasis within their highly variable environments. Theoretical models describing cellular dynamics have the potential to help elucidate the principles underlying size control. Here, we perform a detailed study of the active disassembly model proposed in [Fai et al, Length regulation of multiple flagella that self-assemble from a shared pool of components, eLife, 8, (2019): e42599]. We construct a hybrid system which is shown to be well-behaved throughout the domain. We rule out the possibility of oscillations arising in the model and prove global asymptotic stability in the case of two flagella by the construction of a suitable Lyapunov function. Finally, we generalize the model to the case of arbitrary flagellar number in order to study olfactory sensory neurons, which have up to twenty cilia per cell. We show that our theoretical results may be extended to this case and explore the implications of this universal mechanism of size control.