Appendix 1—figure 3. Cell-cell adhesion.

(A) Adhesive energy contribution in a cyclic process, where a protrusion of source cell $\alpha$ against target cell $\beta$ is followed by the inverse retraction event. Both events involve a third party cell $\gamma$, leading to net energy dissipation after the cyclic process has been completed. Protrusion: (i) Three pre-existing cell-cell contacts between $\beta$ and $\gamma$ are torn apart (red dashed contacts); (ii) three new contacts between $\alpha$ and $\gamma$ are formed; (iii) the contact length between source cell $\alpha$ and target cell $\beta$ increases by one unit of length. This implies ${\displaystyle \mathrm{\Delta }{\mathcal{\mathscr{H}}}_{\text{adh}}({\mathcal{𝒯}}_{\text{pro}})=\ell (3B_{\beta ,\gamma }^{\prime }-3B_{\alpha ,\gamma }-B_{\alpha ,\beta })}$. Retraction: (i) Three pre-existing cell-cell contacts between $\alpha$ and $\gamma$ are torn apart (red dashed contacts); (ii) three new contacts between $\beta$ and $\gamma$ are formed; (iii) the contact length between source cell $\alpha$ and target cell $\beta$ decreases by one unit of length. This implies ${\displaystyle \mathrm{\Delta }{\mathcal{\mathscr{H}}}_{\text{adh}}({\mathcal{𝒯}}_{\text{ret}})=\ell (3B_{\alpha ,\gamma }^{\prime }-3B_{\beta ,\gamma }+B_{\alpha ,\beta })}$. Altogether, this leads to ${\displaystyle \mathrm{\Delta }{\mathcal{\mathscr{H}}}_{\text{adh}}^{(\text{cycl})}=\mathrm{\Delta }{\mathcal{\mathscr{H}}}_{\text{adh}}({\mathcal{𝒯}}_{\text{pro}})+\mathrm{\Delta }{\mathcal{\mathscr{H}}}_{\text{adh}}({\mathcal{𝒯}}_{\text{ ret}})=\ell (3(\mathrm{\Delta }B{)}_{\alpha ,\gamma }+3(\mathrm{\Delta }B{)}_{\beta ,\gamma })\ge 0}$, i.e. a (non-negative) dissipative contribution, whose magnitude depends on the dissipation matrix ${\displaystyle (\mathrm{\Delta }B{)}_{\alpha ,\beta }=B_{\alpha ,\beta }^{\prime }-B_{\alpha ,\beta }\ge 0}$. (B) Shear viscosity due to cell-cell adhesion. Consider two rows of adhesive cells sliding past each other as indicated in the figure (left row of cells moves up by one grid site; colors indicate different cells). The associated adhesion energy change (per cell) reads ${\displaystyle \mathrm{\Delta }{\mathcal{\mathscr{H}}}_{\text{adh}}/n_c=2(B^{\prime }-B)\ge 0}$, where $n_c$ denotes the number of cells sliding past each other, and where we assumed cells of like type, i.e. ${\displaystyle B_{\alpha ,\beta }\equiv B}$ and ${\displaystyle B_{\alpha ,\beta }^{\prime }\equiv B^{\prime }}$ ($\alpha \ne \beta$). The condition ${\displaystyle B^{\prime }>B}$, Equation S15e, thus implies positive friction associated with cellular shear flows, whose magnitude is proportional to the number of cells sliding past each other. Note that this shear viscosity vanishes for ${\displaystyle B^{\prime }=B}$, i.e. for zero dissipation matrix.