Delayed Feedback Model of Axonal Length Sensing

A fundamental question in cell biology is how the sizes of cells and organelles are regulated at various stages of development. Size homeostasis is particularly challenging for neurons, whose axons can extend from hundreds of microns to meters (in humans). Recently, a molecular-motor-based mechanism for axonal length sensing has been proposed, in which axonal length is encoded by the frequency of an oscillating retrograde signal. In this article, we develop a mathematical model of this length-sensing mechanism in which advection-diffusion equations for bidirectional motor transport are coupled to a chemical signaling network. We show that chemical oscillations emerge due to delayed negative feedback via a Hopf bifurcation, resulting in a frequency that is a monotonically decreasing function of axonal length. Knockdown of either kinesin or dynein causes an increase in the oscillation frequency, suggesting that the length-sensing mechanism would produce longer axons, which is consistent with experimental findings. One major prediction of the model is that fluctuations in the transport of molecular motors lead to a reduction in the reliability of the frequency-encoding mechanism for long axons.