We calculate the reaction rate for stochastically gated ligands diffusing in a two-dimensional and a three-dimensional bounded domain with a single small target. Each ligand independently switches between an open and a closed state according to a two-state Markov process; a reaction between ligand and target can only occur when the former is an open state. In the large-time limit the reaction-rate is an exponentially decaying function of time, whose rate of decay is given by the principal eigenvalue of the Laplacian. We calculate the principal eigenvalue using matched asymptotics and determine the leading-order reduction in the reaction rate due to stochastic gating. We also develop a probabilistic interpretation of the reaction rate in terms of the first-passage time density to the target.
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