Diffusive search for a stochastically-gated target with resetting
Paul C. Bressloff
Abstract
Abstract In this paper, we analyze the mean first passage time (MFPT) for a single Brownian particle to find a stochastically-gated target under the additional condition that the position of the particle is reset to its initial position x 0 at a rate r . The gate switches between an open (absorbing) and closed (reflecting) state according to a two-state Markov chain and can only be detected by the searcher in the open state. One possible example of such a target is a protein switching between different conformational states. As expected, the MFPT with or without resetting is an increasing function of the fraction of time ρ 0 that the gate is closed or reflecting. However, the interplay between stochastic resetting and stochastic gating has non-trivial effects with regards the optimization of the search process under resetting. First, by considering the diffusive search for a gated target at one end of an interval, we show that the ratio Δ( r ) = T ( r )/ T (0), where T ( r ) is the MFPT at a resetting rate r , exhibits a non-monotonic dependence on ρ 0 even though T ( r ) and T (0) increase monotonically with ρ 0 . In particular, the value of Δ( r ) at the optimal resetting rate (when it exists) decreases with ρ 0 up to some critical value, after which it increases and eventually approaches unity. Second, in the case of a spherical target in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>d</mml:mi> </mml:mrow> </mml:msup> </mml:math> , the dependence of the MFPT on the spatial dimension d is significantly amplified in the presence of stochastic gating.