find the value of $\omega$ that maximizes ${\ensuremath{\left\langle v_x \right\rangle}}$ numerically. We show the complete phase diagram in [Fig. ]{}\[fig:rot\]b.

Particle-based model of collective cell migration {#particle-based-model-of-collective-cell-migration .unnumbered}
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We use a minimal model of collective cell migration, describing cells as self-propelled particles connected by springs: $$\begin{aligned}
\label{eq:velocity}
\frac{d}{dt}{ {\mathbf r}}^i &= { {\mathbf p}}^i + \sum_{j\sim i} { {\mathbf F}}^{ij} \\
{ {\mathbf p}}^i &= (\cos\theta^i,\sin\theta^i) \\
\theta^i &= \arctan(\hat{g}_{T,y}/\hat{g}_{T,x}) +\psi^i \\
\frac{d}{dt} \psi^i &= -\tau^{-1} \sin \psi^i+ \sqrt{2 D_\psi} \xi^i(t) \label{eq:psi}\end{aligned}$$ where $\xi^i(t)$ is a Gaussian Langevin noise with zero mean and $\langle \xi^i(t)\xi^j(t') \rangle = \delta^{ij}\delta(t-t')$, with $\delta^{ij}$ the Kronecker delta. In this model, the orientation of an individual cell $\theta^i$ is the cluster’s best estimate of the gradient direction, $\arctan(\hat{g}_{T,y}/\hat{g}_{T,x})$, plus a noise $\psi^i$ which varies from cell to cell. $\tau$ here controls the persistence of this noise, and $D_\psi$ its amplitude; when $D_\psi$ is increased, each individual cell is worse at following the estimate ${\ensuremath{\hat{{ {\mathbf g}}}}_T\xspace}$. The cell-cell forces are $${ {\mathbf F}}^{ij} = -\kappa (|{\ensuremath{d^{ij}}\xspace}-\ell|){\ensuremath{\hat{{ {\mathbf r}}}^{ij}}\xspace}$$ where ${\ensuremath{d^{ij}}\xspace}= |{ {\mathbf r}}^i-{ {\mathbf r}}^j|$ and ${\ensuremath{\hat{{ {\mathbf r}}}^{ij}}\xspace}= ({ {\mathbf r}}^i-{ {\mathbf r}}^j)/{\ensuremath{d^{ij}}\xspace}$. The forces are only between neighboring cells $j\sim i$, where 