for diffusive rearrangements, we expect $\tau_r \sim R^2/D_{\textrm{eff}}$, and for persistently rotating clusters, $\tau_{\textrm{r}} \sim R/v_{\textrm{cell}}$ (Appendix \[app:mechanisms\]).

We have assumed that the CCV is time-independent over our scale of interest – consistent with the long memory found in [@sigal2006variability]. If $\Delta$ changes faster than the cluster re-arranges, our results can be straightforwardly modified. Generalizations of [Eq. ]{}\[eq:snrt\] and [Eq. ]{}\[eq:snrt\_simp\] to this case are provided in Appendix \[app:timeaverage\].

![image](rotate_omega3){width="180mm"}

Tradeoffs in collective accuracy and motility: cluster rotation {#sec:rotate}
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Our central result ([Eq. ]{}\[eq:snrt\]) shows that clusters can improve their chemotactic accuracy by changing cell positions. The simplest mechanism to do this is cluster rotation, which occurs in border cell clusters [@combedazou2016myosin] and transiently in leukocyte clusters [@malet2015collective]. When should a cluster actively rotate in order to increase its accuracy? Rotation creates an important tradeoff: more work must be put into rotating, and therefore less into crawling up the gradient [^2]

For constant work of motility, the maximum speed of a cluster of radius $R$ that rotates with angular speed $\Omega$ is ([*Methods*]{}) $$v(\Omega) = v_\textrm{max}\sqrt{1 - \frac{1}{2}\Omega^2\tau_{\textrm{rot}}^2} \label{eq:rot_tradeoff_main}$$ where $\tau_\textrm{rot} = R/v_\textrm{max}$ and $v_\textrm{max}$ the maximum cluster speed absent rotation. The cluster cannot rotate faster than $\Omega_\textrm{max} = 