Bending instability of rod-shaped bacteria

As can be intuited from everyday experience, a thin-walled tube (e.g., a drinking straw) subject to bending reaches a critical curvature at which an instability occurs, localizing the deformation into a narrow region. This instability has been extensively studied since the seminal work of Brazier nearly a century ago. However, the scenario of pressurized tubes has received much less attention. Motivated by rod-shaped bacteria such as E. coli, whose cell walls are much thinner than their radii and are subject to a substantial internal pressure, we study, theoretically, how such instability is affected by this internal pressure. We find that while the bending rigidity of the cell wall has almost no effect on the critical curvature, the internal (turgor) pressure significantly postpones the onset of the instability.