Fig. 1: Growing E. coli filaments undergo a buckling instability.

A Phase contrast images of a representative E. coli filament growing on 10 μg/ml aztreonam agarose pads. Following cell segmentation and tracking (Methods), the curvature of the filaments is computed along the midline. At a given location (e.g., yellow point) the curvature, ∣κ∣, is computed as the inverse of the radius of the circumference that is tangent to that point (green circle): ∣κ∣ = 1/R. As time progresses and the filament grows, the snapshots reveal a buckling instability where large values of ∣κ∣ are reached by the central region of the filament (see E). B Cartoon of the mechanical toy model. Forces are exerted by a chain of connected springs (black). Nodes (grey circles) are subjected to elastic and bending forces. Due to cellular growth, the elastic energy relies on the time-dependent distance, L(t), between consecutive nodes (Methods). Bending energy depends on the angle θ as determined by neighboring segments. C Snapshots of a simulated growing filament (initial length: 50 μm; doubling time: 50 min). Color scales for each time point indicate the elastic (top) and bending (bottom) energies using pair-wise moving average coordinates (Methods). As time progresses, elastic energy accumulates by the central region. Buckling leads to a relaxation of the elastic energy at the expense of an increase of the bending energy (black triangles). D Ratio of the total energy between soft and stiff filaments (soft: \(\widetilde{\Omega }=0.9\); stiff: \(\widetilde{\Omega }=2\); sample size: n = 100 simulations). As time progresses, the panel shows that stiff filaments that are not able to buckle increase their energy. E Density histograms of the curvature of filaments as a function of the relative position as obtained in experiments (left) and simulations (pair-wise moving average coordinates; right). Sample size: n = 22 (experiments) and n = 100 (simulations). The color scale represents probability. Larger curvatures (buckling) accumulate in the central region of the filament.