and decoupling. These approximations are exact for the rigid cluster rotation in Sec. \[sec:rotate\], but only approximate for this self-propelled particle model. As a consequence, at small $D_\psi$, simulated clusters have chemotactic indices slightly exceeding our predictions ([Fig. ]{}\[fig:Dpsi\]b). This error likely arises from emergent couplings between cluster shape and $\Delta_i$ – clusters may spread perpendicular to ${\ensuremath{\hat{{ {\mathbf g}}}}_T\xspace}$, weakening the decoupling approximation ([Fig. ]{}\[fig:Dpsi\]a). The approximation of cluster isotropy can be removed (Appendix \[app:timeaverage\]), and does not resolve the violation of the bound ([Fig. ]{}\[fig:Dpsi\]b). Despite this potential error source, the model captures CI variation over a broad range of parameters (Appendix \[app:params\]).]{}

Discussion
==========

Our study results in several predictions and suggestions for experiments that investigate collective chemotaxis. For example, we predict that, when CCV is large, gradient sensing error is insensitive to background concentration ([Fig. ]{}\[fig:ccv\_illus\]c). This is consistent with recent measurements on developing organoids that show that the up-gradient bias is not strongly dependent on mean concentration [@ellison2016cell], though in contrast with results on lymphocyte clusters [@malet2015collective]. Furthermore, if CCV limits collective chemotaxis, clusters gradient sensing [*in vivo*]{} should have tightly regulated expression of proteins relevant to the signal response. Interestingly, measurements of zebrafish posterior 