\sqrt{2}/\tau_\textrm{rot}$.

If a cluster follows its best estimate ${\ensuremath{\hat{{ {\mathbf g}}}}_T\xspace}$ with speed $v(\Omega)$ given by [Eq. ]{}\[eq:rot\_tradeoff\_main\], it can improve its velocity in the gradient direction by rotating when the averaging time $T$ is long compared with $\tau_\textrm{rot}$ ([Fig. ]{}\[fig:rot\]a) [^3] We find that the optimal rotation speed $\Omega$ that maximizes the upgradient speed depends only on the signal-to-noise ratio without rotation, ${\ensuremath{\textrm{SNR}}}_0 \equiv \frac{1}{2}g^2/\sigma_{{\ensuremath{{ {\mathbf g}}}\xspace},0}^2$ and $T/\tau_\textrm{rot}$ ([Fig. ]{}\[fig:rot\]b, [*Methods*]{}). Mammalian cells have speeds in the range of microns/minute and radii of tens of microns, so to benefit from averaging ($T \gtrsim\tau_\textrm{rot}$), $T$ must be longer than tens of minutes. The timescale $\tau_\textrm{rot}$ and the signal-to-noise-ratio ${\ensuremath{\textrm{SNR}}}_0$ both depend on cluster size – larger clusters with more cells are both better gradient sensors and more difficult to drive to large angular speeds. As a consequence, the optimal $\Omega$ is highly cluster-size-dependent: as cluster size decreases, there is a continuous transition to nonzero optimal $\Omega$ if $T$ is sufficiently long ([Fig. ]{}\[fig:rot\]c).

![image](vary_kappa_label){width="170mm"}

Linking chemotactic accuracy and fluidity {#sec:fluidity}
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Clusters with more cell rearrangement are more accurate by [Eq. ]{}\[eq:snrt\_simp\]. To further quantify the consequences of cell rearrrangements, we model a cluster of cells as self-propelled particles 