Universal Proximity Effect in Target Search Kinetics in the Few-Encounter Limit

When does a diffusing particle reach its target for the first time? This first-passage time (FPT) problem is central to the kinetics of molecular reactions in chemistry and molecular biology. Here, we explain the behavior of smooth FPT densities, for which all moments are finite, and demonstrate universal yet generally non-Poissonian long-time asymptotics for a broad variety of transport processes. While Poisson-like asymptotics arise generically in the presence of an effective repulsion in the immediate vicinity of the target, a time-scale separation between direct and reflected indirect trajectories gives rise to a universal proximity effect: Direct paths, heading more or less straight from the point of release to the target, become typical and focused, with a narrow spread of the corresponding first-passage times. Conversely, statistically dominant indirect paths exploring the entire system tend to be massively dissimilar. The initial distance to the target particularly impacts gene regulatory or competitive stochastic processes, for which few binding events often determine the regulatory outcome. The proximity effect is independent of details of the transport, highlighting the robust character of the FPT features uncovered here.

Popular Summary

When a particle attains a certain distance from its origin for the first time, scientists call this event a first passage event. The theory of first passage is an important concept in any field in which a stochastically moving particle reaching a threshold value is of note. A paramount example is gene regulation in living biological cells in which diffusing proteins binding to a specific target on the genome initiate important follow-up reactions. Here we present, for the first time, a full and asymptotically exact analysis of the distribution of first passage times in a finite volume.

We confirm our analytical results by computer simulations of spatially confined systems, and we measure tens of thousands of realizations. In addition to unifying the first-passage universality classes known in the literature, one of our central results is the proximity effect dominating all kinetics in the few-encounters limit: Whenever only a few particles need to arrive at their target, first-passage events in which particles move straight toward their target are decisive (i.e., all other paths are statistically less relevant). We prove that this situation is in fact a universal result for a variety of stochastic processes, even in the presence of external forcing. In other words, the proximity effect is independent of the details of the transport. Our findings shed light on how both the speed and precision of the target search process can be optimized.

We expect that our results will be relevant to fields ranging from biological physics to geophysics to econophysics in which first passage is important.

Figure 1

Schematic of the two limiting forms of long-time FPT behavior, (a) and (b), depending on the effective potential experienced by the searcher in the vicinity of the absorbing target (originating either from an external force or from the geometric spurious drift in the radial direction). (c) Schematics of two molecules (large spheres) searching for a target (small sphere) with two different realizations of direct and indirect trajectories $t_d$ and $t_i$, respectively, depending on the initial distance to the target (see text for details). As soon as a time-scale separation exists between $t_d$ and $t_i$, a robust proximity effect emerges, with any two $t_d$ being typical and very similar while any two $t_i$ tend to be strongly dissimilar. This focusing of $t_d$ in turn enables faster and more precise gene regulation.

Figure 2

Asymptotic collapse and short-time nonuniversality for the rescaled FPT densities $\wp (\theta =t/{\lambda }_0)/\mathcal{C}$ for the various models described in the text. The parameters are $x_0=0.4$ (1D diffusion), $x_a=0.2$, $x_0=0.5$ (2D diffusion), $x_a=0.1$, $x_0=0.4$ (3D diffusion), $x_0=0.3$, $\mathrm{Pe}=3$ (1D biased away from the target), $x_0=0.4$, $\mathrm{Pe}=-3$ (1D biased towards the target), $\nu =-0.25$, $x_a=0.2$, $x_0=0.45$ (noncompact diffusion on fractals), $\nu =0.25$, $x_a=0.2$, $x_0=0.45$ (compact diffusion on fractals), $x_a=0.2$, $x_0=0.5$, $\mathrm{Pe}=3$ (2D radial potential flow away from the target), $x_a=0.2$, $x_0=0.5$, $\mathrm{Pe}=-1.8$ (2D radial potential flow towards the target), and $x_0=-1$, $x_a=2.8$ (overdamped escape from a harmonic potential).

Figure 3

FPT densities for various passive (a) and biased (b) diffusive systems along with mean FPT values (arrows). (a) $x_a=0.006$, $x_0=0.04$ and $\nu =-1/2$ (violet), $\nu =-1/4$ (red), $\nu =-1/4$ (orange), $\nu =1/2$ (yellow), and $\nu =0.7$-strong subdiffusion (blue). The symbols represent results of numerical simulations (see Sec. 2a), solid lines correspond to Eq. (6) (or the Poissonian limit for $\nu <0$), and dashed lines correspond to Eq. (9). Inset: Magnification of the intermediate regime with the asymptotic (10) depicted by dashed lines. Note that the long-time behavior for both $\nu <0$ cases is well within the Poisson-like regime. (b) One-dimensional diffusion under constant bias for $x_0-x_a=0.1$ [orange; $\mathrm{Pe}=-5$ (circles) and $\mathrm{Pe}=5$ (triangles)], 2D diffusion with radial bias for $x_a=0.006$, $x_0=0.04$ [red; $\mathrm{Pe}=0.75$ (circles) and $\mathrm{Pe}=-0.75$ (triangles)], and the OU process [violet; $x_a=3$, $x_0=2.5$ (triangles), $x_a=2.5$, $x_0=2.4$ (circles)]. The symbols represent results of numerical simulations (see Methods), solid lines correspond to Eq. (6) (or the Poissonian limit for $\nu <0$), black dashed lines correspond to Eqs. (9) and (15), respectively, and the blue solid lines correspond to Eq. (10).