Modelling the motion of organelles in an elongated cell via the coordination of heterogeneous drift-diffusion and long-range transport.

 Cellular distribution of organelles in living cells is achieved via a variety of transport mechanisms, including directed motion, mediated by molecular motors along microtubules (MTs), and diffusion which is predominantly heterogeneous in space. In this paper, we introduce a model for particle transport in elongated cells that couples poleward drift, long-range bidirectional transport and diffusion with spatial heterogeneity in a three-dimensional space. Using stochastic simulations and analysis of a related population model, we find parameter regions where the three-dimensional model can be reduced to a coupled one-dimensional model or even a one-dimensional scalar model. We explore the efficiency with which individual model components can overcome drift towards one of the cell poles to reach an approximately even distribution. In particular, we find that if lateral movement is well mixed, then increasing the binding ability of particles to MTs is an efficient way to overcome a poleward drift, whereas if lateral motion is not well mixed, then increasing the axial diffusivity away from MTs becomes an efficient way to overcome the poleward drift. Our three-dimensional model provides a new tool that will help to understand the mechanisms by which eukaryotic cells organize their organelles in an elongated cell, and in particular when the one-dimensional models are applicable.