First-passage times of multiple diffusing particles with reversible target-binding kinetics

We investigate a class of diffusion-controlled reactions that are initiated at the time instance when prescribed number $K$ among $N$ particles independently diffusing in solvent simultaneously bound to target region. In irreversible target-binding setting, bind stay there forever, and reaction is $K$-th fastest first-passage target, whose distribution well-known. turn, reversible binding, which common for most applications, renders theoretical analysis much more challenging drastically changes times. develop renewal-based approach derive an approximate solution probability density time. This approximation turns out be remarkably accurate broad range parameters. also analyze dependence mean or, equivalently, inverse rate, on main parameters such as $K$, $N$, binding/unbinding constants. Some biophysical applications further perspectives briefly discussed.