Diffusion as a Ruler: Modeling Kinesin Diffusion as a Lenth Sensor for Intraflagellar Transport

Abstract

An important question in cell biology is how cells know how big to make their organelles. The eukaryotic flagellum is an ideal model for studying size control because its linear geometry makes it essentially one-dimensional, greatly simplifying mathematical modeling. The assembly of flagella is regulated by intraflagellar transport (IFT), in which trains of kinesin motors walk to the tip of the flagellum and deposit the cargo necessary for the flagellum to grow. The competing length control factor is a length-independent decay of the flagellum. In Chlamydomonas reinhardtii flagella, this process results in initial rapid growth followed by convergence to a steady-state length. Curiously, the rate at which motors are recruited to begin transport is indirectly proportional to the length, implying some kind of communication between the base and the tip. We propose a model in which motors unbind after cargo delivery and diffuse back to the base, and are reused in IFT. In this model, the diffusion time of the motors serves as a proxy for length measurement. To explore the viability of this diffusion-based length control, we computationally built this model in three different ways. First, we built an agent-based model in which we used object-oriented programming to explicitly model flagella and motors, including time dynamics. Second, we modeled the number density along the flagellum as a vector, and built a stochastic matrix to simulate time dynamics and determine a steady-state. Third, we used differential equations to directly solve for the steady-state length. In all three, we found that the diffusion model can achieve steady-state length and an inverse relationship between length and recruitment rate. This is remarkable because this is perhaps the simplest explanation of length control, giving it credence in light of evolution.