Colony and filament morphology of gliding filamentous cyanobacteria, bending and buckling tests.

a, b, Colonies on agar plates (left) and individual filaments in liquid medium (right) of K. animale and O. lutea, respectively. c, Schematic of a microscopic three-point bending test, pushing a filament into the gap between SU-8 pillars using a glass micropipette. d, Schematic of a self-buckling test in a microfluidic chip: A filament glides into a V-shaped obstacle and buckles if its contour length L exceeds the self-buckling threshold Lc.

Single-filament bending measurements using micropipette force sensors.

a, Representative force-deflection measurement for K. animale. Insets: Bottom-view micrographs of the same experiment. b, Box plot of the bending moduli for N = 35 individuals of K. animale and O. lutea, shown as points (each tested 2 − 10 times). Box limits denote the first and third quartile, whiskers the last measurement within the inter-quartile distance away from the respective box limit. A p-value of 6.87 × 10−3 suggests different typical moduli for the two species.

Self-buckling experiments and statistics.

a, Snapshots of K. animale, before and after hitting the obstacle at t = 0. Left, short filament with L < Lc, right, long filament with L > Lc. b, c, Bar plot of the buckling frequency vs. filament length for O. lutea (b, totally N = 388 events) and K. animale (c, totally N = 280 events), together with the logistic regression (dashed curve). d, Box plot of the quantiles of the critical length distribution from the logistic regression p(L). Box limits denote first and third quartile, whiskers the 5th and 95th percentile. e, f, Velocity υ0 immediately before hitting the obstacle vs. filament length L for O. lutea and K. animale, respectively, distinguishing buckling (∼, dark) and non-buckling (−, light). The velocity-dependent median critical length Lc(υ0), as derived from a logistic regression with L and υ0 as independent explanatory variables, is indicated by black lines. Note that axes in b, c, e, f are broken around L = 2 Lc to emphasize the critical region. g, Lc(υ0) (lines) and inter-quartile region (shaded), together with the simple logistic regression from (d), located at the mean velocity υ0 of the population (symbols & error bars).

Kirchhoff theory compared to experimental contours.

a, Time series of a characteristic buckling event of O. lutea, overlaying the contour extracted from the images (green) and the best fit of the solution to Eq. (5) (red). b, Free-gliding friction coefficient fact/v against η from the buckling profile fit. c, d, fact and η vs. filament length L, as determined from the fit, for O. lutea and K. animale, respectively. Light gray indicates L ∈ Lc ± ∆Lc (see Fig. 3d), dark gray L < Lc − ∆Lc, where buckling is not observed. The dashed lines are obtained from Lc of logistic regression. The filament from a is indicated in red.

Details on the microscopic three-point bending tests.

a, Calibration of the force sensing micropipette by the weight of an evaporating water droplet. b, Testing the same filament with three different actuation speeds shows a negligible speed-dependence of the force-deflection curves as compared to the spread between individual measurements. c, Force-deflection curves for increasing (‘pushing’) and decreasing (‘retraction’) deflection, showing the absence of hysteresis. d, Force-deflection measurements for four individual filaments of K. animale.

Collection of buckling events with different shapes of the traps.

Snapshots of K. animale (top row) and O. lutea (bottom row), before and after hitting the obstacle at time t = 0. Left, four short filaments with length below the critical length, right, six long filaments above the critical length.

Active force fact and friction coefficient η vs. filament velocity.

Velocities were measured immediately before hitting the obstacle. Neither fact nor η appears to be correlated with the velocity.

Length distributions of K. animale and O. lutea, together with the buckling probability estimated by the logistic regression.

Colonies were grown in quiescent liquid medium and then gently squeezed in between agar solidified medium and glass. Filaments glide from the colony into the glass-agar contact and form a sub-monolayer. After ∼ 6 h, the layer is then analyzed by our modified U-Net segmentation procedure. Filaments remain intact and alive in this configuration for days, as inferred from their gliding velocity.

Numerical results for the full non-linear Kirchhoff theory.

Eq. (5) is discretized according to the scheme described in Methods and integrated over time, starting from a small-amplitude analytical solution to the linearized equations.

Simulations of self-buckling semi-flexible chains of active overdamped particles. Chains of particles of diameter σ and separation σ/2 are connected by torsional springs to generate an effective bending modulus.

Each particle self-propels along the tangent of the particle chain, defined by the mean orientation of its two bonds. a, Trajectory of active particle chains of 21, 127 and 201 particles, corresponding to filament lengths L of 10σ, 63σ and 100σ, respectively. b, Front-Mid-End angle of particle chains as a function of time, after hitting the obstacle. Active force and torsional spring constant were chosen to obtain, from the analytical expression, Lc ∼ 62.37. The chain hits the obstacle slightly off-center, to induce a non-singular initial condition for self-buckling. For L < Lc (red colors), curvature stagnates and then relaxes very slowly. For L > Lc (green colors), curvature increases further. c, Definition of the Geometry of the particle chains.