variance and $\Delta^i$ are uncorrelated Gaussian noises with zero mean and variance ${\ensuremath{\sigma_{\Delta}}}^2$. Stochastic fluctuations in ligand-receptor binding are taken into account in the term $\delta c^i$, where $(\delta c^i / c({\ensuremath{{ {\mathbf r}}}\xspace}^i))^2 = \frac{1}{n_r} \frac{(c^i+K_D)^2}{c^i K_D}$. This is the error in concentration sensing from a single snapshot of $n_r$ receptors with simple ligand-receptor kinetics and dissociation constant $K_D$ ([@kaizu2014berg], Appendix \[app:conc\]). [Eq. ]{}\[eq:M\] assumes that cell-cell variance additively corrupts the measurement of the concentration $c({\ensuremath{{ {\mathbf r}}}\xspace})$ [*after*]{} an adaptation to the overall level of signal across the cluster $\bar{c}$. This is natural if the primary cell-to-cell variation is [*downstream*]{} of adaptation, as found to be a reasonable model in [@wang2012diverse].

To determine gradient sensing accuracy, we perform maximum likelihood estimation (MLE) of ${\ensuremath{{ {\mathbf g}}}\xspace}$ in [Eq. ]{}\[eq:M\], as in past approaches for single cell gradient sensing [@hu2011geometry]. We obtain the MLE estimator ${\ensuremath{\hat{{ {\mathbf g}}}}\xspace}$ numerically ([*Methods*]{}, Appendix \[app:mle\]), and thus the uncertainty ${\ensuremath{\sigma_{\mathbf{g}}}}^2 \equiv {\ensuremath{\left\langle |\hat{{\ensuremath{{ {\mathbf g}}}\xspace}} - {\ensuremath{{ {\mathbf g}}}\xspace}|^2 \right\rangle}}$ ([Fig. ]{}\[fig:ccv\_illus\]b, symbols), where $\langle \cdots \rangle$ is an average over CCV and ligand-receptor binding. For fixed and roughly circular (isotropic) cluster geometry, if the concentration change across the cluster is small, 