Searching for partially reactive sites: Analytical results for spherical targets

How do single or multiple (sub)diffusing particles search for a target with a partially reactive boundary? A finite reaction rate which is typical for many chemical or biochemical reactions is introduced as the possibility for a particle to find a target but not to "recognize" it. The search is not finished until the target is found and recognized. For a single searching particle, the short- and long-time regimes are investigated, with a special focus on comparison between perfectly and partially reactive boundaries. For multiple searching particles, explicit formulas for the probability density of the search time are given for subdiffusion in one and three dimensions. The dependence of the mean search time on the density of particles and the reaction rate is analyzed. Unexpectedly, in the high density limit, the particles undergoing slower subdiffusive motion find a target faster.