an additive constant.

In the limit of $n_r \to \infty$ (neglecting concentration noise), our model becomes a simple linear regression, and the log likelihood can be maximized analytically by finding [$\hat{{ {\mathbf g}}}$]{}such that $\partial_{{\ensuremath{{ {\mathbf g}}}\xspace}} \ln {\ensuremath{\mathcal{L}}\xspace}({\ensuremath{{ {\mathbf g}}}\xspace}; {\ensuremath{\{M^{i}\}\xspace}})|_{{\ensuremath{\hat{{ {\mathbf g}}}}\xspace}} = 0$ (Appendix \[app:mle\]). The result is $\hat{{\ensuremath{{ {\mathbf g}}}\xspace}} = \mathcal{A}^{-1}\cdot \sum_i M^i {\ensuremath{\boldsymbol\delta { {\mathbf r}}^i}\xspace}$, where $\mathcal{A}_{\alpha\beta} \equiv \sum_i\delta r^i_\alpha \delta r^i_\beta$. This estimator is simplest in the limit of roughly circular (isotropic) clusters, where $\sum_i (\delta x^i)^2 \approx \sum_i (\delta y^i)^2 \gg \sum_i \delta x^i \delta y^i$. In this case, $\hat{{\ensuremath{{ {\mathbf g}}}\xspace}} = (\chi^{-1}) \sum_i M^i{\ensuremath{\boldsymbol\delta { {\mathbf r}}^i}\xspace}$ where $\chi = \frac{1}{2}\sum_i |{\ensuremath{\boldsymbol\delta { {\mathbf r}}^i}\xspace}|^2$.

Cluster rotation dynamics {#cluster-rotation-dynamics .unnumbered}
-------------------------

How much speed does a cluster lose by rotating? One possibility is to assume the power expended in generating motility is constant. Consider a circular cluster propelling itself over a surface, with the cells having velocity ${ {\mathbf v}}({\ensuremath{{ {\mathbf r}}}\xspace})$; we expect that the frictional force per unit area between the cluster and substrate will be ${ {\mathbf f}}_{\textrm{drag}} = -\xi { {\mathbf v}}$, where $\xi$ is a friction coefficient with the substrate. If all of the 