Reactive Boundaries and Cover Times for Diffusing Particles under Confinement

From protein-DNA binding and receptor activation to enzyme kinetics and intracellular signaling, diffusive searchers are involved in some of the most fundamental processes in biology. A quantitative theory of these processes requires accounting for physical realities that classical models neglect: targets are often only partially reactive, requiring multiple encounters before binding succeeds; a messenger molecule may need to activate an entire set of targets before a biological response is triggered; and in many cellular contexts, many copies of a molecule search simultaneously. Here, we present closed-form analytical results for one or more diffusing particles searching for one or more partially reactive targets under spherical confinement. For multiple targets, we derive a transition probability matrix between targets and recast the diffusive environment as a Markov chain, applying a generalization of the coupon collector's problem to obtain the mean cover time (MCT) and mean activation time (MAT) in closed form in the small-target limit. The generalized coupon collector formula provides a tractable method for computing the MCT and MAT for a reasonable number of targets (N < 1000), offering a pathway for optimizing the spatial arrangement of reactive target sites under confinement. We also show that the MFPT, MCT, and MAT all exhibit clean scaling laws with the number of simultaneously searching particles. All results are validated against Brownian simulations, as well as equivalent Markov chain simulations, yielding excellent agreement throughout.