30% of the DNA in E. coli bacteria is covered by proteins. Such high degree of crowding affect the dynamics of generic biological processes (e.g. gene regulation, DNA repair, protein diffusion etc.) in ways that are not yet fully understood. In this
paper, we theoretically address the diffusion constant of a tracer particle in a one dimensional system surrounded by impenetrable crowder particles. While the tracer particle always stays on the lattice, crowder particles may unbind to a surrounding bulk and rebind at another or the same location. In this scenario we determine how the long time diffusion constant ${cal D}$ (after many unbinding events) depends on (i) the unbinding rate of crowder particles $k_{rm off}$, and (ii) crowder particle line density $rho$, from simulations (Gillespie algorithm) and analytical calculations. For small $k_{rm off}$, we find ${cal D}sim k_{rm off}/rho^2$ when crowder particles are immobile on the line, and ${cal D}sim sqrt{D k_{rm off}}/rho$ when they are diffusing; $D$ is the free particle diffusion constant. For large $k_{rm off}$, we find agreement with mean-field results which do not depend on $k_{rm off}$. From literature values of $k_{rm off}$ and $D$, we show that the small $k_{rm off}$-limit is relevant for in vivo protein diffusion on a crowded DNA. Our results applies to single-molecule tracking experiments.
