Search efficiency in the Adam-Delbr\"uck reduction-of-dimensionality scenario versus direct diffusive search

TL;DR

This paper compares the efficiency of the Adam-Delbruck reduction-of-dimensionality search scenario with direct diffusive search, analyzing full FPT distributions and survival probabilities for single and multiple ligand cases.

Contribution

It provides a comprehensive analysis of both search strategies by calculating FPT distributions and survival probabilities, highlighting conditions where reduction-of-dimensionality outperforms direct search.

Findings

Reduction-of-dimensionality can be more efficient under certain parameter regimes.

Full FPT distributions offer deeper insights than mean FPT comparisons.

Multiple ligand scenarios enhance search efficiency, especially with fastest-first arrival.

Abstract

The time instant -- the first-passage time (FPT) -- when a diffusive particle (e.g., a ligand such as oxygen or a signalling protein) for the first time reaches an immobile target located on the surface of a bounded three-dimensional domain (e.g., a hemoglobin molecule or the cellular nucleus) is a decisive characteristic time-scale in diverse biophysical and biochemical processes, as well as in intermediate stages of various inter- and intra-cellular signal transduction pathways. Adam and Delbr\"uck put forth the reduction-of-dimensionality concept, according to which a ligand first binds non-specifically to any point of the surface on which the target is placed and then diffuses along this surface until it locates the target. In this work, we analyse the efficiency of such a scenario and confront it with the efficiency of a direct search process, in which the target is approached directly from the bulk and not aided by surface diffusion. We consider two situations: (i) a single ligand is launched from a fixed or a random position and searches for the target, and (ii) the case of "amplified" signals when $N$ ligands start either from the same point or from random positions, and the search terminates when the fastest of them arrives to the target. For such settings, we go beyond the conventional analyses, which compare only the mean values of the corresponding FPTs. Instead, we calculate the full probability density function of FPTs for both scenarios and study its integral characteristic -- the "survival" probability of a target up to time $t$. On this basis, we examine how the efficiencies of both scenarios are controlled by a variety of parameters and single out realistic conditions in which the reduction-of-dimensionality scenario outperforms the direct search.