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

TL;DR

This paper models a sequential cascade of diffusion-controlled molecular reactions, deriving exact probability densities for the timing of the final event in various spatial dimensions, applicable to biological signaling pathways.

Contribution

It provides the first exact analytical expressions for the timing of the terminal reaction in a general multi-stage diffusion-controlled cascade across different dimensions.

Findings

Exact probability density functions derived for the terminal reaction time.

Results applicable to 1D, 2D, and 3D systems with arbitrary parameters.

Model captures complex biological signaling cascades as molecular relay races.

Abstract

We consider a sequential cascade of molecular first-reaction events towards a terminal reaction centre in which each reaction step is controlled by diffusive motion of the particles. The model studied here represents a typical reaction setting encountered in diverse molecular biology systems, in which, e.g., a signal transduction proceeds via a series of consecutive "messengers": the first messenger has to find its respective immobile target site triggering a launch of the second messenger, the second messenger seeks its own target site and provokes a launch of the third messenger and so on, resembling a relay race in human competitions. For such a molecular relay race taking place in infinite one-, two- and three-dimensional systems, we find exact expressions for the probability density function of the time instant of the terminal reaction event, conditioned on preceding successful reaction events on an ordered array of target sites. The obtained expressions pertain to the most general conditions: number of intermediate stages and the corresponding diffusion coefficients, the sizes of the target sites, the distances between them, as well as their reactivities are arbitrary.