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We consider a sequential cascade of molecular first-reaction events towards aterminal reaction centre in which each reaction step is controlled by diffusivemotion of the particles. The model studied here represents a typical reactionsetting encountered in diverse molecular biology systems, in which, e.g., asignal transduction proceeds via a series of consecutive "messengers": thefirst messenger has to find its respective immobile target site triggering alaunch of the second messenger, the second messenger seeks its own target siteand provokes a launch of the third messenger and so on, resembling a relay racein human competitions. For such a molecular relay race taking place in infiniteone-, two- and three-dimensional systems, we find exact expressions for theprobability density function of the time instant of the terminal reactionevent, conditioned on preceding successful reaction events on an ordered arrayof target sites. The obtained expressions pertain to the most generalconditions: number of intermediate stages and the corresponding diffusioncoefficients, the sizes of the target sites, the distances between them, aswell as their reactivities are arbitrary.

A molecular relay race: sequential first-passage events to the terminal reaction centre in a cascade of diffusion controlled processes