mean concentrations at each compartment are $\bar{c}_1$ and $\bar{c}_2 = \bar{c}_1 + a\bar{g}$. The mean molecule counts in the two compartments are roughly $\bar{n}_1 = \bar{c}_1s^3$ and $\bar{n}_2 = \bar{c}_2s^3$, and the difference is $\Delta\bar{n} = \bar{n}_2 - \bar{n}_1 = a\bar{g}s^3$. The variance in this difference is $\sigma_{\Delta n}^2 = \sigma_{n_1}^2 + \sigma_{n_2}^2 \sim \bar{n}_1^2/(s\bar{c}_1DT) + \bar{n}_2^2/(s\bar{c}_2DT)$, where the first step assumes the two compartments are independent, and the second step uses Eq. \[eq:singleConc\] for the variance in each compartment’s measurement. For shallow gradients, where the limits on sensing are generally probed, we have $a\bar{g} \ll \bar{c}_1$, and therefore we may assume $\bar{c_1} \approx \bar{c_2} \approx \bar{c}$, where $\bar{c}$ is the mean concentration at the center of the cell. Thus $\sigma_{\Delta n}^2 \sim 2(\bar{c}s^3)^2/(s\bar{c}DT)$, and the relative error in the cell’s estimate of the gradient is then $$\label{eq:g}
\frac{\sigma_g}{\bar{g}} = \frac{\sigma_{\Delta n}}{\Delta \bar{n}} \sim \sqrt{\frac{\bar{c}}{s(a\bar{g})^2DT}},$$ where the factor of $2$ is neglected in this simple scaling estimate. As in Eq. \[eq:singleConc\], we see that the relative error decreases with $s$, since the molecule counts in each compartment are larger, and also with $D$ and $T$, since more independent measurements can be made. Additionally, the relative error decreases with $a\bar{g}$, since 