(A) A single filament shown at different time points of its movement under an agar pad. Timelapse images were captured at 2 second intervals using fluorescence microscopy. The scale-bar shown on the first image applies to all subsequent ones. The extreme points of the trajectory across the time lapse are marked with blue and orange asterisks on each image. (B) Top: X- and Y-coordinates of filament’s centre throughout the recorded time-lapse. The points show observations, while the line shows a spline fit to this data. Bottom: The distance (gray) between filament centre and one of the extreme ends of its trajectory, shown with blue asterisk on panel A, and filament speed (orange) throughout the time-lapse. The speed is calculated from the spline fitted to the x- and y-coordinates shown in the top panel. Gray backdrop regions indicate time-points with speed below a set threshold, indicating reversal events. (C) Top: Mean filament speed from 65 different filaments observed under agar, plotted against filament length. Bottom: Distribution of dwell times, as calculated from independent reversal events. For the same analyses for observations on glass, see ((Fig. S3).

(A) A cartoon of the biophysical model (see Methods). Cells are modelled as beads, connected by springs, with preferred rest length l0. Cells self-propel with a propulsion force fa. It is assumed that cells regulate the direction of fa, and this regulation is modelled by a function ω, which includes a self-regulatory element, implemented as a confining potential Uω (shown as blue lines in the panels above cells). The value ωi of each cell is represented with an orange dot and its affected by random fluctuations, a mechanical feedback from neighbouring cells (Vω, red arrows), and an external signal fext, present at the ends of the track L (green arrow) (see Eq. (5)). (B) Example of reversal process: the snapshots show a coordinated filament approaching the border (top); after reaching it, the closest cells reverse their propulsion under the action of fext (centre). This prompts the rest of the cells to reverse and the filament coordinates again to travel in the opposite direction (bottom). (C) Simulated trajectories (gray lines) and absolute speed value (orange lines) as a function of time, presented as in (Fig. 1B, for Nf =70 units in a track of length L/l0 =400. The different panels display a typical trajectory for a well behaved filament (top, Kω = 20, ωmax = 30), a filament with little cell-to-cell coupling (centre, Kω = 1, ωmax = 30) or with little memory (bottom, Kω = 30, ωmax = 1). The gray bands highlight reversal events and their duration. (D) Contour plots of the synchronisation S (top) and of the reversal efficiency M (bottom) as a function of the cell-to-cell coupling Kω and of the cell memory ωmax. Black symbols highlight the systems showcased in panels (C).

(A) Top: Individual images from a TIRF microscopy time-lapse, obtained with excitation using 473nm laser. The left and right panel show images with emission filters centered at 625nm and 525nm respectively. Note that these image shows a thin (≈ 200 nm) section of the cell membrane. Bottom: The position (top) and speed (bottom) of individual cells during a TIRF time-lapse movie (see the associated Movie S 8). Different cells’ trajectories and speed are shown in different shades of gray. The inset on the upper panel shows the normalised distance travelled by each cell, revealing high coordination in their movement. (B) Top: The same TIRF image as shown on the right side of top panel A, but displaying the trajectories of some of the membrane bound protein complexes (orange lines) and the angle of this trajectory with the long axis of the filament; θ. Bottom: The distribution of the angle θ (top) and the relation between rotational speed of the protein complexes and the linear speed of the filament (bottom). The black and orange lines in the bottom panel show the diagonal and the correlation fit (see legend) between linear and angular speed. Both angle and speed measurements are collected from 11 filaments and 391 trajectories.

(A) A filament forming a plectoneme during movement under agar, shown at different time points as indicated on each panel. The scale-bar shown on the first image applies to all subsequent ones. The orange and blue asterisks indicate the end-points of the filament’s trajectory. See also the corresponding supplementary Movie S 11. (B) Top: The x- and y-coordinates of the two ends of the filament over time. Data points for the x- and y-coordinates are shown in gray and black respectively. Lines are spline fits to the data and their orange and blue colors indicate the filament end that stays close to the corresponding trajectory end-point shown on the images in panel A. Bottom: Speed and distance of each end, respective to the end-point of the trajectory close to them. Speed is shown as blue and orange lines, calculated from the respective position data shown in the top panel. Distance is shown in orange and blue data points, indicating the filament end that stays close to the corresponding trajectory end-point shown on the images in panel A, while gray lines show spline fits. The shaded areas indicate the reversals. Note that the second reversal involves only the end that stays close to the blue asterisk on panel A.

A. Reversal frequency against track length, determined from extreme boundaries of filament movement during observation time. Data collected from 65 filaments observed as moving under agar. B. Reversal frequency against filament length. C. Reversal frequency against filament length normalised by track length. D. Reversal frequency against filament mean speed.

A. Single frame from a movie, showing filaments on glass (this image is associated with Movie S 2). Two selected filaments are highlighted with their trajectories, colored in blue and orange. B. Trajectory (blue and orange) and speeds (gray) of filaments shown in panel A. Trajectory colors are matching to panel A. Trajectories are only shown in terms of either the x- or y-axis for simplicity, with the axis the the most significant movement shown.

Left panel. Distribution of dwell times of filaments, observed while moving on glass. Data is collated from 42 filaments. Right panel. Mean filament speed on glass. Each data point is from a single filament, where mean speed is calculated from filaments’ movement across observation time.

Distribution of dwell times of leading (head) and trailing (tail) ends of filaments. Data is collated from 632 reversal events across 65 filaments. Only those reversals, where dwell times estimated from tracking filament ends differed less than a threshold from those estimated from filament centre, are included.

Simulated distribution of dwell times. The parameters values are as in Fig. 2C: (Left) Kω = 20, ωmax = 10; (Centre) Kω = 1, ωmax = 30; (Right) Kω = 30, ωmax = 1.

Number of reversals nr against the number of expected reversals as observed in experiments under agar (left) or in simulations (right). One outlier filament in the experimental data involved reversals prior to track ends, but in a consistent location in the track. Simulated data have been obtained with Kω = 20 and ωmax = 10, 20 and several different values of L.

(A) TIRF image of F. draycotensis filaments illuminated with light of wavelength 473nm and imaged under an emission filter centred on 625nm. (B) Same filaments under an emission filter centred on 525nm, showing regularly placed fluorescent protein complexes. (C) TIRF image of a bundle of filaments stained using Concanavalin-A and illuminated with light of wavelength 473 nm. Top and middle panels show images with emission filters centred on 625 and 525nm, respectively. Surrounding sheath of exopolysaccharide is visible (false colored in yellow). The third panel shows the composite of these two emission channels.

Scanning electron microscopy (SEM) images of the F. Draycotensis cyanobacteria filaments. (A) cyanobacteria filaments without further treatment, Mag.: × 40 k, EHT: 8 kV; (B) filaments following plasma cleaning inside SEM chamber, Mag.: × 25 k, EHT: 8kV. Performed on a Zeiss Gemini, imaged on silicon wafer

(A). Two time points in the movement sequence of a filament, the position and speed data for which is shown in panel B. The location of two protein complexes are shown with arrows, indicating that the complex further up the filament starts its movement earlier than the one at the tip of the filament. See also the associated Movie S10. (B) The position (top) and speed (bottom) of individual cells. Different cells’ trajectories and speed are shown in different shades of gray.

Distribution of filament lengths that exhibit buckling or plectonemes, shown for filaments observed on a glass slide (n = 50), or sandwiched under agar (n = 11)