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