  The mean time for a diffusing ligand to activate a target protein located on
the surface of a microdomain can regulate cellular signaling. When the ligand
switches between various states induced by chemical interactions or
conformational changes, while target activation occurs in only one state, this
activation time is affected. We investigate this dynamics using new equations
for the sojourn times spent in each state. For two states, we obtain exact
solutions in dimension one, and asymptotic ones confirmed by Brownian
simulations in dimension 3. We find that the activation time is quite sensitive
to changes of the switching rates, which can be used to modulate signaling.
Interestingly, our analysis reveals that activation can be fast although the
ligand spends most of the time 'hidden' in the non-activating state. Finally,
we obtain a new formula for the narrow escape time in the presence of
switching.
