Non-Markovian effects in the first-passage dynamics of obstructed tracer particle diffusion in one-dimensional systems.

Abstract

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 generalize this system and investigate first-passage properties of a tracer particle when flanked by identical crowder particles which may, besides diffuse, unbind (rebind) from (to) the one-dimensional lattice with rates k(off) (k(on)). The tracer particle is restricted to diffuse with rate k(D) on the lattice and the density of crowders is constant (on average). The unbinding rate k(off) is our key parameter and it allows us to systematically study the non-trivial transition between the completely Markovian case (k(off) ≫ k(D)) to the non-Markovian case (k(off) ≪ k(D)) governed by strong memory effects. This has relevance for several quasi one-dimensional systems. One example is gene regulation where regulatory proteins are searching for specific binding sites on a crowded DNA. We quantify the first-passage time distribution, f(t) (t is time), numerically using the Gillespie algorithm, and estimate f(t) analytically. In terms of k(off) (keeping k(D) fixed), we study the transition between the two known regimes: (i) when k(off) ≫ k(D) the particles may effectively pass each other and we recover the single particle result f(t) ∼ t(-3/2), with a reduced diffusion constant; (ii) when k(off) ≪ k(D) unbinding is rare and we obtain the single-file result f(t) ∼ t(-7/4). The intermediate region displays rich dynamics where both the characteristic f(t) - peak and the long-time power-law slope are sensitive to k(off).