The subdiffusive target problem: Survival probability

TL;DR

This paper analyzes the long-term survival probability of a spherical target in subdiffusive environments, revealing dimension-dependent decay behaviors and implications for enzyme-DNA interactions in crowded cells.

Contribution

It provides exact asymptotic formulas for survival probabilities of targets in subdiffusive media across different dimensions, extending understanding of subdiffusive target problems.

Findings

Survival probability decays as a power law in 1D and 2D with logarithmic corrections.

In 3D, the survival probability remains finite, independent of trap diffusion type.

In a sea of traps, survival probability decays as a stretched exponential in all dimensions.

Abstract

The asymptotic survival probability of a spherical target in the presence of a single subdiffusive trap or surrounded by a sea of subdiffusive traps in a continuous Euclidean medium is calculated. In one and two dimensions the survival probability of the target in the presence of a single trap decays to zero as a power law and as a power law with logarithmic correction, respectively. The target is thus reached with certainty, but it takes the trap an infinite time on average to do so. In three dimensions a single trap may never reach the target and so the survival probability is finite and, in fact, does not depend on whether the traps move diffusively or subdiffusively. When the target is surrounded by a sea of traps, on the other hand, its survival probability decays as a stretched exponential in all dimensions (with a logarithmic correction in the exponent for $d=2$). A trap will therefore reach the target with certainty, and will do so in a finite time. These results may be directly related to enzyme binding kinetics on DNA in the crowded cellular environment.