strongly on cluster rearrangement mechanism. Finally, our results show that noise in cell motility can be beneficial for collective sensing.

Methods
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Maximum likelihood estimation of gradient direction in the presence of cell-cell variation and ligand-receptor noise {#maximum-likelihood-estimation-of-gradient-direction-in-the-presence-of-cell-cell-variation-and-ligand-receptor-noise .unnumbered}
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We compute the maximum likelihood estimate of gradient direction given the measured signal at cell $i$, $M^i$, given by [Eq. ]{}\[eq:M\]. If the cluster of cells is in a shallow linear gradient, with concentration $c_0$ at the cluster’s center of mass ${\ensuremath{{ {\mathbf r}}}\xspace}_{\textrm{cm}} = N^{-1} \sum_i {\ensuremath{{ {\mathbf r}}}\xspace}^i$, then $c({\ensuremath{{ {\mathbf r}}}\xspace}) = c_0 \left[1 + { {\mathbf g}} \cdot ({\ensuremath{{ {\mathbf r}}}\xspace}-{\ensuremath{{ {\mathbf r}}}\xspace}_{\textrm{cm}}) \right]$ and thus $\bar{c} = c_0$. $M_i$ is then $M^i = 1 + {\ensuremath{{ {\mathbf g}}}\xspace}\cdot{\ensuremath{\boldsymbol\delta { {\mathbf r}}^i}\xspace}+ (\delta c^i/c_0) \eta^i + \Delta^i$ with ${\ensuremath{\boldsymbol\delta { {\mathbf r}}^i}\xspace}= {\ensuremath{{ {\mathbf r}}}\xspace}-{\ensuremath{{ {\mathbf r}}}\xspace}_{\textrm{cm}}$ and $(\delta c^i/c^i)^2 = \frac{1}{n_r} \frac{(c^i+K_D)^2}{c^i K_D}$, i.e. $(\delta c^i/c_0)^2 = \frac{1}{n_r} (1+{\ensuremath{{ {\mathbf g}}}\xspace}\cdot{\ensuremath{\boldsymbol\delta { {\mathbf r}}^i}\xspace}+K_D/c_0)^2\frac{1+{\ensuremath{{ {\mathbf g}}}\xspace}\cdot{\ensuremath{\boldsymbol\delta { {\mathbf r}}^i}\xspace}}{K_D/c_0}$.

$\Delta^i$ are uncorrelated between cells, with a Gaussian distribution of zero mean and standard deviation [$\sigma_{\Delta}$]{}, i.e. ${\ensuremath{\left\langle \Delta^i\Delta^j \right\rangle}} = {\ensuremath{\sigma_{\Delta}}}^2 \delta^{ij}$ with $\delta^{ij}$ the Kronecker delta function. As $\eta^i$ and $\Delta^i$ are both Gaussian, 