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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 timereaches an immobile target located on the surface of a boundedthree-dimensional domain (e.g., a hemoglobin molecule or the cellular nucleus)is a decisive characteristic time-scale in diverse biophysical and biochemicalprocesses, as well as in intermediate stages of various inter- andintra-cellular signal transduction pathways. Adam and Delbr\"uck put forth thereduction-of-dimensionality concept, according to which a ligand first bindsnon-specifically to any point of the surface on which the target is placed andthen diffuses along this surface until it locates the target. In this work, weanalyse the efficiency of such a scenario and confront it with the efficiencyof a direct search process, in which the target is approached directly from thebulk and not aided by surface diffusion. We consider two situations: (i) asingle ligand is launched from a fixed or a random position and searches forthe target, and (ii) the case of "amplified" signals when $N$ ligands starteither from the same point or from random positions, and the search terminateswhen the fastest of them arrives to the target. For such settings, we go beyondthe conventional analyses, which compare only the mean values of thecorresponding FPTs. Instead, we calculate the full probability density functionof FPTs for both scenarios and study its integral characteristic -- the"survival" probability of a target up to time $t$. On this basis, we examinehow the efficiencies of both scenarios are controlled by a variety ofparameters and single out realistic conditions in which thereduction-of-dimensionality scenario outperforms the direct search.

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