First-Reaction Times at the Terminal Target Site in Molecular Relay Races

In diverse processes encountered in molecular biology, e.g. in intra-cellular signal transduction or in cell-to-cell communication, the search for the terminal reaction centre often necessitates a series of intermediate steps performed in a due order—a cascade of sequential diffusion-controlled reactions involving so-called “messengers”. In a typical situation, a distant cell sends a “messenger” molecule that diffuses in the extracellular medium seeking for a special binding site on the surface of another cell, where it initiates a special kind of a reaction. This process produces the second messenger right at the inner surface of this cell. The second messenger diffuses now within the cell seeking, e.g. the entrance to the nuclear pore. Diffusing through the pore, it penetrates into the nucleus and then searches diffusively for the terminal binding site located on the DNA. In this chapter, we discuss the form of the probability density function (PDF) of the first time instant at which the terminal reaction event in such a reaction cascade takes place. We present a general formalism for the derivation of the PDF and formally exact expressions, which permit to draw some far-reaching conclusions about the behaviour of the cumulants of the PDF and its short- and long-time tails. We also provide some illustrative simple examples—diffusion to a series of multiple ordered targets in unbounded one- and two-dimensional domains—in which the final results have lucid and instructive forms.