the concentrations measured by the two compartments are more different from each other. However, we see that unlike in Eq. \[eq:singleConc\], the relative error increases with the background concentration $\bar{c}$. The reason is that the cell is not measuring a concentration, but rather a *difference* in concentrations, and it is more difficult to measure a small difference on a larger background than on a smaller background [@ellison2016cell]. Eq. \[eq:g\] has been derived more rigorously[@endres2009accuracy], and the problem has been extended to describe rings of receptors [@endres2009accuracy] or detectors distributed over the surface of a circle [@hu2010physical] or a sphere [@endres2008accuracy]. In all cases a term of the form in Eq. \[eq:g\] emerges as the fundamental limit, with the lengthscale $s$ dictated by the particular sensory mechanism and geometry. It is clear that the optimal mechanism would result in an effective compartment size that is roughly half of the cell volume, in which case $s\sim a$.

Do cells reach this limit on gradient sensing? This question has been directly addressed for the amoeba *Dictyostelium discoideum*. Experiments[@van2007biased] have shown that *Dictyostelium* cells exhibit biased movement when exposed to gradients of cyclic adenosine monophosphate as small as $\bar{g} = 10$ nM/mm, on top 