reasonable tradeoff between speed and angular velocity for a cluster. If the power available for cell motility is a small amount of the cell’s energy budget [@purcell1977life; @flamholz2014quantified; @katsu2009substantial], other tradeoffs may be more important and additional modeling will be necessary.

We consider a circular cluster traveling towards its best estimate of the gradient with speed $v(\Omega)$ given by [Eq. ]{}\[eq:tradeoff\], and traveling in the direction of the estimator ${\ensuremath{\hat{{ {\mathbf g}}}}_T\xspace}$. We can then determine when the cluster maximizes its mean velocity in the direction of the increasing gradient, which we choose to be $x$, ${\ensuremath{\left\langle v_x \right\rangle}}$, as a function of $\Omega$. This average is $${\ensuremath{\left\langle v_x \right\rangle}} = \sqrt{v_\textrm{max}^2 - \frac{\gamma_r}{\gamma_t} \Omega^2} \times {\ensuremath{\left\langle \frac{{\ensuremath{\hat{{ {\mathbf g}}}}_{T,x}\xspace}}{|{\ensuremath{\hat{{ {\mathbf g}}}}_T\xspace}|} \right\rangle}}$$ We know from our results above and in Appendix \[app:timeaverage\] that, for a fixed configuration, ${\ensuremath{\hat{{ {\mathbf g}}}}_T\xspace}$ has a Gaussian distribution with mean ${ {\mathbf g}} = g \hat{{ {\mathbf x}}}$ and variance given by [Eq. ]{}\[eq:snrt\]. The average of $\frac{{\ensuremath{\hat{{ {\mathbf g}}}}_{T,x}\xspace}}{|{\ensuremath{\hat{{ {\mathbf g}}}}_T\xspace}|}$ depends only on the signal-to-noise ratio ${\ensuremath{\textrm{SNR}}}_T \equiv \frac{1}{2}(g^2/\sigma_{{ {\mathbf g}},T}^2)$, with ${\ensuremath{\left\langle \frac{{\ensuremath{\hat{{ {\mathbf g}}}}_{T,x}\xspace}}{|{\ensuremath{\hat{{ {\mathbf g}}}}_T\xspace}|} \right\rangle}} = C({\ensuremath{\textrm{SNR}}}_T^{-1/2})$.

Given the angular velocity $\Omega$, we can work out the 