We introduce and analyze two general dynamical models for unidirectional movement of particles along a circular and an open chain of sites. The models include a soft version of the simple exclusion principle, thus allowing to model and analyze the evolution of "traffic jams" of particles along the chain. A unique feature of these two new models is that each site along the chain may have a different size. Although nonlinear, the models are amenable to rigorous analysis. We prove that the dynamics always converges to a steady-state, and that the steady-state densities along the chain and the steady-state output flow rate can be derived from the spectral properties of a suitable matrix, thus eliminating the need to numerically simulate the dynamics until convergence. This spectral representation also allows for powerful sensitivity analysis, i.e. understanding how a change in one of the parameters affects the steady-state. The site sizes and the transition rates from site to site play different roles in the dynamics, and for the purpose of maximizing the steady-state output (or production) rate the site sizes are more important than the transition rates. The new models can be applied to study various natural and artificial systems including networks of ribosome flow during mRNA translation, phosphorelay, the movement of molecular motors along filaments of the cytoskeleton, and pedestrian and vehicular traffic.
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