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In this manuscript, we introduce a totally asymmetric simple exclusion process on a dynamic lattice with inhomogeneity to model motor-based long-range transport along microtubules (MTs) which exhibit cycles of growth, shortening and regrowth at their growing tips. By mean-field approximation and numerical simulations, we explore phase diagrams for the particle density near the dynamic end of the lattice as well as over the entire lattice. In particular, we find seven different phases for the density over the entire lattice and explore the corresponding phase diagram by analytical analysis. Interestingly, we find that the density may have a linear profile in part of the lattice. When the lattice has a defect, we separate the lattice into two subsystems: a right subsystem with fixed length and a defect in the middle and a left subsystem with dynamic length. When the right subsystem is in maximum-current phase, we calculate an analytical approximation of the density away from the boundaries and the defect site and the density is in association with the reduced hopping rate at the defect site; the density at the left side of the defect site serves as the effective entry rate of the left subsystem. We find that with defects, the linear profiles no longer appear at the left end of the left subsystem and only five possible phases could occur. This is a step-forward to a more comprehensive mathematical description of intracellular long-range transport of organelles along MTs with instability.
Lin et al. (Mon,) studied this question.