Facilitated target search

A facilitated target search combines bulk diffusion with lower-dimensional motion on a substrate, such as sliding on DNA. In the scaling model $\tau =A/\lambda +B\lambda$, positive constants $A,B$ balance relocation and local exploration. Differentiation gives ${\lambda }_{\ast }=\sqrt{A/B}$ and ${\tau }_{\min }=2\sqrt{AB}$. These expressions require an accessible optimum within the physically allowed range of sliding lengths.

For an unbiased one-dimensional walk with step size $a$, diffusion coefficient $D$ fixes step interval $a^2/(2D)$. A small dissociation probability $P$ per step gives about $1/P$ steps before release. Thus lifetime depends on $D$, while the root-mean-square net displacement before release is $a/\sqrt{P}$. Net displacement is not the cumulative distance along all backtracking steps. Dimensional consistency of the measured or assumed diffusivity is necessary before applying the estimate.