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We apply a recently developed response functions formalism1to 2D bacterial suspensions confined between two surfaces. Individual bacterium performs the usual run-and-tumble motion and responds to a shear flow following the Jeffery equation. The dipolar flow field from swimming provides coupling between the orientation of microswimmers. Under a mean-field approximation, the system can be described by the Smoluchowski equation together with the Stokes equation. We performed linear stability analysis of the system in a slab geometry in 3D and obtained exact solution for the unstable modes which are the left and right polarized laminar flows in the plane. The resulting collective motion is associated with a negative imaginary part of the response function to circular laminar shear flow at low frequencies, which can be traced to adaptive behavior of the cell population when the translational and rotational diffusion constants are sufficiently small. We compare the theoretical results with weak collective oscillations discovered earlier in Yilin Wu’s lab when the cell density exceeds a certain threshold2.
The work is supported in part by the NSFC under grants 11635002 and U1930402, and by the RGC of the HKSAR under grant 12301514.
 Shou-Wen Wang and Lei-Han Tang, Nature Communications 10:5613 (2019).
 Chong Chen, Song Liu, Xia-qing Shi, Hugues Chaté & Yilin Wu, Weak synchronization and large-scale collective oscillation in dense bacterial suspensions, Nature 542, 210-214 (2017).