Diffusive escape through a narrow opening: new insights into a classic problem

Abstract

We study the mean first exit time (T) of a particle diffusing in a circular or a spherical micro-domain with an impenetrable confining boundary containing a small escape window (EW) of an angular size ε. Focusing on the effects of an energy/entropy barrier at the EW, and of the long-range interactions (LRIs) with the boundary on the diffusive search for the EW, we develop a self-consistent approximation to derive for T a general expression, akin to the celebrated Collins-Kimball relation in chemical kinetics and accounting for both rate-controlling factors in an explicit way. Our analysis reveals that the barrier-induced contribution to T is the dominant one in the limit ε → 0, implying that the narrow escape problem is not "diffusion-limited" but rather "barrier-limited". We present the small-ε expansion for T, in which the coefficients in front of the leading terms are expressed via some integrals and derivatives of the LRI potential. Considering a triangular-well potential as an example, we show that T is non-monotonic with respect to the extent of the attractive LRI, being minimal for the ones having an intermediate extent, neither too concentrated on the boundary nor penetrating too deeply into the bulk. Our analytical predictions are in good agreement with the numerical simulations.

Full distribution of first exit times in the narrow escape problem

“…Equations (2)-(5) completely determine the explicit form of H(r, θ; t). As we already mentioned above, this result is obtained by resorting to a self-consistent closure scheme, developed earlier for the calculation of the mean FRT in certain reaction-diffusion problems [4,46,47]. This approximation consists in replacing the actual mixed boundary condition (1) by an inhomogeneous Neumann condition and in the derivation of an appropriate closure relation, which ensures that the mixed boundary condition (1) holds on average.…”

“…Indeed, in cellular environments, this setting represents one of the most interesting applications of the NEP, as particles such as small signalling molecules or proteins are usually released at some fixed point inside the cell [43] and need to diffusively locate specific targets such as channel or receptor proteins embedded in the cell membrane [44,45]. Our analytical approach is based on a self-consistent closure scheme [4,46,47] that yields analytical results in excellent agreement with numerical solutions. We demonstrate that in these settings the PDF exhibits typically four distinct temporal regimes delimited by three relevant time scales, for which we also present explicit expressions.…”

“…(ii) The concept of the search-controlled versus barrier-controlled NEP proposed in [4,30] holds indeed for the mean FRT, see equation (C.6). In contrast, we observe that two other important characteristic time-scales-t mp and t c -are unaffected by the reactivity and solely depend on the diffusion coefficient and the geometry.…”

“…A striking example of the NEP with very severe escape restrictions is the dire strait problem [20]. Effects of short-ranged (contact) and long-ranged interactions with the confining boundary, which are quite ubiquitous and give rise to 'intermittent motion' characterised by alternating phases of bulk and surface diffusion, were shown to allow for an optimisation of the search times [4,[21][22][23][24][25][26]. It was also studied how molecular crowding in cellular environments impacts the search dynamics [13,27,28].…”

Optimal potentials for diffusive search strategies

“…For example, a nonzero stochastic resetting rate is not always advantageous for a search in bounded spaces [52]. In the so-called narrow escape problem [53][54][55] it has been shown that the dependence of a mean escape time is a nonmonotonic function of the range of interaction with the spherical boundary [56]. Note that impenetrable walls are equivalent to infinite potential barriers, so they introduce constraints on the potentials used in the optimization.…”