does edge current affect the previously established design principle?

To answer this question, the orientation dynamics of bacteria with the heading angle $\theta$ is considered at the vicinity of the sharp areas of the boundaries (“*tip*”). Given the CCW fluid flow near boundary reorients the bacteria at the *tip*, the effective torque for reorientation $\bm{\tau_e} = - \partial U_e/\partial \theta$ has geometry-dependent potential $U_e = -2 \gamma_e \sin\theta\cos\Psi$ with $\gamma_e$ representing the ratio of fluid flow to bacterial swimming ($\gamma_e > 0$). This reorientation maintains the CCW rotation of the edge current along boundary in both FMV pairing (Fig. \[fig5\](b), left) and AFMV pairing (Fig. \[fig5\](b), right). In addition, the result of the bi-particular collision near the *tip* between bacteria coming from different circular parts[@beppu] also affects the vortex pairing. Bacterial collision is ruled by a polar alignment as a source of geometry-dependence[@vicsek; @supplement]. By considering the most probable configurations (general case detailed in [@supplement]), the orientation near the *tip* is decided by the potential of FMV pairing $U_p^{FMV} = 2\gamma_p\sin\Psi$ at $\theta = 0$ (Fig. \[fig5\](b), left) or of AFMV pairing $U_p^{AFMV} = 2\gamma_p\cos\Psi$ at $\theta = \pi/2$ (Fig. \[fig5\](b), right), with the strength of the polar alignment $\gamma_p>0$. 