$g R_\textrm{cluster} \ll 1$, ${\ensuremath{\sigma_{\mathbf{g}}}}^2$ can be approximated by assuming $\delta c^i$ is constant across the cluster, resulting in $${\ensuremath{\left\langle |\hat{{\ensuremath{{ {\mathbf g}}}\xspace}} - {\ensuremath{{ {\mathbf g}}}\xspace}|^2 \right\rangle}} \approx \frac{2}{\chi} \left({\ensuremath{\sigma_{\Delta}}}^2 + \frac{1}{n_r} \frac{(\bar{c}+K_D)^2}{\bar{c} K_D} \right) \label{eq:combined_error}$$ Here, $\chi = \frac{1}{2}\sum_i |{\ensuremath{\boldsymbol\delta { {\mathbf r}}^i}\xspace}|^2$ is a shape parameter, and ${\ensuremath{\boldsymbol\delta { {\mathbf r}}^i}\xspace}= { {\mathbf r}}^i - { {\mathbf r}}_{\textrm{cm}}$ is cell position relative to cluster center of mass. Evaluating this expression reveals that it is an excellent approximation to the numerically-obtained uncertainty (dashed lines, [Fig. ]{}\[fig:ccv\_illus\]b).

The approximate expression for the uncertainty, Eq. \[eq:combined\_error\], allows us to quantify the relative contribution of receptor-ligand fluctuations and CCV to the gradient sensing error. For background concentrations $\bar{c}$ near the receptor-ligand equilibrium constant $K_D$ and for typical receptor numbers in eukaryotic cells ($n_r \sim 10^5$ [@hesselgesser1998identification; @macdonald2008heterogeneity]), $\delta c / \bar{c}$ can be smaller than 0.01. Protein concentrations, on the other hand, often vary between cells to 10%-60% of their mean [@niepel2009non] – hence we estimate ${\ensuremath{\sigma_{\Delta}}}\approx 0.1-0.6$. Thus, we expect CCV to dominate gradient sensing error and that the error from concentration sensing and receptor binding can be neglected completely if ${\ensuremath{\sigma_{\Delta}}}> 0.1$ ([Fig. ]{}\[fig:ccv\_illus\]b). [Eq. ]{}\[eq:combined\_error\] also reveals that 