of a background concentration of $\bar{c} = 7$ nM. Bias is typically quantified in terms of the chemotactic index (CI), which is the cosine of the angle between the gradient direction and the direction of a cell’s actual motion. By relating the error in gradient sensing (a term of the form in Eq. \[eq:g\] with $s = a$) to the error in this angle, Endres and Wingreen [@endres2008accuracy] obtained an expression for the optimal CI, which they then fit to the experimental data with one free parameter, the integration time $T$. The inferred value of $T = 3.2$ s serves as the physical lower bound on the response time required to perform chemotaxis. Actual response times of *Dictyostelium* cells, as measured by the time from the addition of a chemoattractant to the peak activity of an observable signaling pathway associated with cell motility [@postma2003uniform; @parent2004making], are about $5$$-$$10$ s. Taken together, these results imply that *Dictyostelium* operates remarkably close to the physical limit to sensory precision set by the physics of molecule counting.

Relative changes vs. absolute molecule numbers
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The precision of gradient sensing is often reported in terms of percent concentration change across a cell body. For example, 