Second author

Objects in nature move in random and deceptively complicated ways — each sequential step can go in any direction. Modelling the properties that underlie the first encounter between any 'random walker' — be it an animal, biomolecule or virus — and its target has become a hot topic among physicists. A key parameter is the 'first-passage time' (FPT), which describes the time taken by a walker to first reach its target. But this has proved difficult to apply to real-world conditions — the best estimates have been limited to simple, uniform conditions lacking real-world significance. Olivier Bénichou at the Laboratory of Theoretical Physics of Condensed Matter in Paris and his colleagues have developed a general theory to evaluate the mean FPT in more complex situations (see page 77). Bénichou tells Nature about the work.

Why are first-encounter properties crucial?

Because the random walker — for example, a diffusing biomolecule in a cell — must encounter the target before a reaction or process can occur. For biomolecules, reaction times are often controlled by the time taken for molecules to meet.

Were there real-world motivations for this theoretical work?

Three years ago, we modelled the dynamics between proteins and DNA, which rely on intermittent diffusion strategies. One day I was trying to find my daughter's glasses and it occurred to me that my search strategy — searching slowly in one place and then moving quickly to a new one when the target is not found — was also intermittent. I realized that animals adopt this approach when searching for food. The model we proposed to account for such animal behaviours involved the calculation of mean FPTs of random walks.

How does your theory overcome previous constraints?

Determining FPT in complex environments, which are often multidimensional, is a complicated mathematical task, partly because it depends on the necessary boundary conditions, or confinement, of the system being described. In this analytical theory, we use a mathematical trick to isolate and replace the confinement effect. Then, we relate the mean FPT in confined conditions to properties of random walks in infinite space, which are easier to estimate.

Might this theory prove useful elsewhere?

It could have many uses across disciplines. It should be useful for evaluating the kinetics of reactions between biomolecules in a cell, and for estimating the time a computer virus will take to reach a given node on the Internet.