remarkable fact that the bacterium’s concentration sensing machinery operates very near the predicted physical limits. If *E. coli* were to use any shorter run times, chemotaxis would be physically impossible. Consequently, the time period for measuring the chemical concentration, $T$ in Eq. \[eq:singleConc\], would be so short that the bacterium would be unable to make an accurate measurement of the chemical concentration.

Single-cell gradient sensing
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Cells are not only able to detect chemical concentrations, they are also able to measure spatial concentration gradients. Many cells, including amoeba, epithelial cells, neutrophils, and neurons, sense gradients by comparing concentrations between compartments in different locations of the cell body [@jilkine2011comparison]. These compartments are typically receptors or groups of receptors on the cell surface, but in a simple model we may treat these compartments as idealized counting volumes as we did before (Fig. \[sensing\]B). The difference in counts between two such compartments provides the cell with an estimate of the gradient. What is the relative error in this estimate?

Consider two compartments of linear size $s$ on either side of a cell with diameter $a$ (Fig. \[sensing\]B). If the compartments are aligned with the gradient $\bar{g}$ of a linear concentration profile, then the 