precision of concentration sensing were first introduced by Berg and Purcell almost 40 years ago[@berg1977physics]. Berg and Purcell began by considering an idealized cell that acts as a perfect counting instrument. Their simplest model assumed that the cell is a sphere in which molecules can freely diffuse in and out (Fig. \[sensing\]A). The concentration of these molecules is uniform in space, and the cell derives all its information about the concentration by counting each molecule inside its spherical body. The expected count is $\bar{n} = \bar{c}V$ where $\bar{c}$ is the mean concentration and $V$ is the cell volume. However, since molecules arrive and leave via diffusion, there will be fluctuations around this expected value. Diffusion is a Poisson process, meaning that the variance in this count $\sigma_n^2$ equals the mean $\bar{n}$. Therefore the relative error in the cell’s concentration estimate is $\sigma_c/\bar{c} = \sigma_n/\bar{n} = 1/\sqrt{\bar{c} V}$.

![image](fig2.png){width=".75\textwidth"}

The cell can improve upon the relative error in its concentration estimate by time-averaging over multiple measurements. However, consecutive measurements are only statistically independent if they are separated by a sufficient amount of time such that the molecules inside the cell volume are refreshed. The amount of time required is characterized by 