the diffusion time, $\tau \sim V^{2/3}/D \sim a^2/D$, where $D$ is the diffusion constant and $a$ is the cell diameter. In a time period $T$ the cell makes $\nu = T/\tau$ independent measurements, and the variance is reduced by the factor $1/\nu$. This gives the long-standing lower limit $$\label{eq:singleConc}
\frac{ \sigma_c }{\bar{c}} = \frac{ \sigma_n}{\bar{n}} \sim \frac{1}{\sqrt{a\bar{c}DT}}$$ for the cell’s relative error in estimating a uniform concentration. The relative error decreases with $a$ and $\bar{c}$, since the molecule count is larger, and also with $D$ and $T$, since more independent measurements can be made. Berg and Purcell derived this limit more rigorously[@berg1977physics], and the problem has been revisited more recently to account for binding kinetics, spatiotemporal correlations, and spatial confinement [@bialek2005physical; @kaizu2014berg; @bicknell2015limits]. In all cases a term of the form in Eq. \[eq:singleConc\] emerges as the fundamental limit for three-dimensional diffusion.

Do cells reach this limit? Berg and Purcell themselves asked this question in the context of several single-celled organisms, including the *Escheria coli* bacterium [@berg1977physics]. Motility of *E. coli* has two distinct phases: the run phase in which a cell swims in a fixed direction, and the tumble phase in which the cell erratically rotates in order to 