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

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 mathematical 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 Lyapunov function to show that the dynamical equations for the flagellar lengths are well-behaved and rule out the possibility of oscillations in the model. We prove global asymptotic stability in the case of two flagella and show that this result generalizes to arbitrary flagellar number. Our analysis reveals a defect in the original active disassembly model, which is manifested as corners in the phase space that yield unphysical solutions. To address this defect, we construct a hybrid system. Although this hybrid system leads to behaviors such as time delays and the accumulation of trajectories in the corners of phase space, we show that these scenarios do not typically arise in applications.