First-passage dynamics of obstructed tracer particle diffusion in one-dimensional systems

The standard setup for single-file diffusion is diffusing particles in one dimension which cannot overtake each other, where the dynamics of a tracer (tagged) particle is of main interest. In this article we generalise this system and investigate first-passage properties of a tracer particle when flanked by crowder particles which may, besides diffuse, unbind (rebind) from (to) the one-dimensional lattice with rates $k_{\rm off}$ ($k_{\rm on}$). The tracer particle is restricted to diffuse with rate $k_D$ on the lattice. Such a model is relevant for the understanding of gene regulation where regulatory proteins are searching for specific binding sites ona crowded DNA. We quantify the first-passage time distribution, $f(t)$ ($t$ is time), numerically using the Gillespie algorithm, and estimate it analytically. In terms of our key parameter, the unbinding rate $k_{\rm off}$, we study the bridging of two known regimes: (i) when unbinding is frequent the particles may effectively pass each other and we recover the standard single particle result $f(t)\sim t^{-3/2}$ with a renormalized diffusion constant, (ii) when unbinding is rare we recover well-known single-file diffusion result $f(t)\sim t^{-7/4}$. The intermediate cases display rich dynamics, with the characteristic $f(t)$-peak and the long-time power-law slope both being sensitive to $k_{\rm off}$.