Quantifying gliding forces of filamentous cyanobacteria by self-buckling

Reviewer #1 (Public Review):

The paper "Quantifying gliding forces of filamentous cyanobacteria by self-buckling" combines experiments on freely gliding cyanobacteria, buckling experiments using two-dimensional V-shaped corners, and micropipette force measurements with theoretical models to study gliding forces in these organisms. The aim is to quantify these forces and use the results to perhaps discriminate between competing mechanisms by which these cells move. A large data set of possible collision events are analyzed, bucking events evaluated, and critical buckling lengths estimated. A line elasticity model is used to analyze the onset of buckling and estimate the effective (viscous type) friction/drag that controls the dynamics of the rotation that ensues post-buckling. This value of the friction/drag is compared to a second estimate obtained by consideration of the active forces and speeds in freely gliding filaments. The authors find that these two independent estimates of friction/drag correlate with each other and are comparable in magnitude. The experiments are conducted carefully, the device fabrication is novel, the data set is interesting, and the analysis is solid. The authors conclude that the experiments are consistent with the propulsion being generated by adhesion forces rather than slime extrusion. While consistent with the data, this conclusion is inferred.

Summary:

The paper addresses important questions on the mechanisms driving the gliding motility of filamentous cyanobacteria. The authors aim to understand these by estimating the elastic properties of the filaments, and by comparing the resistance to gliding under a) freely gliding conditions, and b) in post-buckled rotational states. Experiments are used to estimate the propulsion force density on freely gliding filaments (assuming over-damped conditions). Experiments are combined with a theoretical model based on Euler beam theory to extract friction (viscous) coefficients for filaments that buckle and begin to rotate about the pinned end. The main results are estimates for the bending stiffness of the bacteria, the propulsive tangential force density, the buckling threshold in terms of the length, and estimates of the resistive friction (viscous drag) providing the dissipation in the system and balancing the active force. It is found that experiments on the two bacterial species yield nearly identical values of 𝑓 (albeit with rather large variations). The authors conclude that the experiments are consistent with the propulsion being generated by adhesion forces rather than slime extrusion.

Strengths of the paper:

The strengths of the paper lie in the novel experimental setup and measurements that allow for the estimation of the propulsive force density, critical buckling length, and effective viscous drag forces for movement of the filament along its contour - the axial (parallel) drag coefficient, and the normal (perpendicular) drag coefficient (I assume this is the case, since the post-buckling analysis assumes the bent filament rotates at a constant frequency). These direct measurements are important for serious analysis and discrimination between motility mechanisms.

Weaknesses:

There are aspects of the analysis and discussion that may be improved. I suggest that the authors take the following comments into consideration while revising their manuscript.

The conclusion that adhesion via focal adhesions is the cause for propulsion rather than slime protrusion is consistent with the experimental results that the frictional drag correlates with propulsion force. At the same time, it is hard to rule out other factors that may result in this (friction) viscous drag - (active) force relationship while still being consistent with slime production. More detailed analysis aiming to discriminate between adhesion vs slime protrusion may be outside the scope of the study, but the authors may still want to elaborate on their inference. It would help if there was a detailed discussion on the differences in terms of the active force term for the focal adhesion-based motility vs the slime motility.

Can the authors comment on possible mechanisms (perhaps from the literature) that indicate how isotropic friction may be generated in settings where focal adhesions drive motility? A key aspect here would probably be estimating the extent of this adhesion patch and comparing it to a characteristic contact area. Can lubrication theory be used to estimate characteristic areas of contact (knowing the radius of the filament, and assuming a height above the substrate)? If the focal adhesions typically cover areas smaller than this lubrication area, it may suggest the possibility that bacteria essentially present a flat surface insofar as adhesion is concerned, leading to a transversely isotropic response in terms of the drag. Of course, we will still require the effective propulsive force to act along the tangent.

I am not sure why the authors mention that the power of the gliding apparatus is not rate-limiting. The only way to verify this would be to put these in highly viscous fluids where the drag of the external fluid comes into the picture as well (if focal adhesions are on the substrate-facing side, and the upper side is subject to ambient fluid drag). Also, the friction referred to here has the form of a viscous drag (no memory effect, and thus not viscoelastic or gel-like), and it is not clear if forces generated by adhesion involve other forms of drag such as chemical friction via temporary bonds forming and breaking. In quasi-static settings and under certain conditions such as the separation of chemical and elastic time scales, bond friction may yield overall force proportional to local sliding velocities.

For readers from a non-fluids background, some additional discussion of the drag forces, and the forms of friction would help. For a freely gliding filament if 𝑓 is the force density (per unit length), then steady gliding with a viscous frictional drag would suggest (as mentioned in the paper) 𝑓 ∼ 𝑣! 𝐿 𝜂∥. The critical buckling length is then dependent on 𝑓 and on 𝐵 the bending modulus. Here the effective drag is defined per length. I can see from this that if the active force is fixed, and the viscous component resulting from the frictional mechanism is fixed, the critical buckling length will not depend on the velocity (unless I am missing something in their argument), since the velocity is not a primitive variable, and is itself an emergent quantity.

Reviewer #2 (Public Review):

In the presented manuscript, the authors first use structured microfluidic devices with gliding filamentous cyanobacteria inside in combination with micropipette force measurements to measure the bending rigidity of the filaments.

Next, they use triangular structures to trap the bacteria with the front against an obstacle. Depending on the length and rigidity, the filaments buckle under the propulsive force of the cells. The authors use theoretical expressions for the buckling threshold to infer propulsive force, given the measured length and stiffnesses. They find nearly identical values for both species, 𝑓 ∼ (1.0 {plus minus} 0.6) nN∕µm, nearly independent of the velocity.

Finally, they measure the shape of the filament dynamically to infer friction coefficients via Kirchhoff theory. This last part seems a bit inconsistent with the previous inference of propulsive force. Before, they assumed the same propulsive force for all bacteria and showed only a very weak correlation between buckling and propulsive velocity. In this section, they report a strong correlation with velocity, and report propulsive forces that vary over two orders of magnitude. I might be misunderstanding something, but I think this discrepancy should have been discussed or explained.

From a theoretical perspective, not many new results are presented. The authors repeat the well-known calculation for filaments buckling under propulsive load and arrive at the literature result of buckling when the dimensionless number (f L^3/B) is larger than 30.6 as previously derived by Sekimoto et al in 1995 [1] (see [2] for a clamped boundary condition and simulations). Other theoretical predictions for pushed semi-flexible filaments [1-4] are not discussed or compared with the experiments.
Finally, the Authors use molecular dynamics type simulations similar to [2-4] to reproduce the buckling dynamics from the experiments. Unfortunately, no systematic comparison is performed.

[1] K. Sekimoto, N. Mori, K. Tawada and Y. Toyoshima, Phys. Rev. Lett., 1995, 75, 172-175
[2] R. Chelakkot, A. Gopinath, L. Mahadevan and M. F. Hagan, J. R. Soc., Interface, 2014, 11, 20130884.
[3] R. E. Isele-Holder, J. Elgeti and G. Gompper, Soft Matter, 2015, 11, 7181-7190.
[4] R. E. Isele-Holder, J. Jager, G. Saggiorato, J. Elgeti and G. Gompper, Soft Matter, 2016, 12, 8495

Reviewer #3 (Public Review):

Summary:

This paper presents novel and innovative force measurements of the biophysics of gliding cyanobacteria filaments. These measurements allow for estimates of the resistive force between the cell and substrate and provide potential insight into the motility mechanism of these cells, which remains unknown.

Strengths:

The authors used well-designed microfabricated devices to measure the bending modulus of these cells and to determine the critical length at which the cells buckle. I especially appreciated the way the authors constructed an array of pillars and used it to do 3-point bending measurements and the arrangement the authors used to direct cells into a V-shaped corner in order to examine at what length the cells buckled at. By examining the gliding speed of the cells before buckling events, the authors were able to determine how strongly the buckling length depends on the gliding speed, which could be an indicator of how the force exerted by the cells depends on cell length; however, the authors did not comment on this directly.

Weaknesses:

There were two minor weaknesses in the paper.

First, the authors investigate the buckling of these gliding cells using an Euler beam model. A similar mathematical analysis was used to estimate the bending modulus and gliding force for Myxobacteria (C.W. Wolgemuth, Biophys. J. 89: 945-950 (2005)). A similar mathematical model was also examined in G. De Canio, E. Lauga, and R.E Goldstein, J. Roy. Soc. Interface, 14: 20170491 (2017). The authors should have cited these previous works and pointed out any differences between what they did and what was done before.

The second weakness is that the authors claim that their results favor a focal adhesion-based mechanism for cyanobacterial gliding motility. This is based on their result that friction and adhesion forces correlate strongly. They then conjecture that this is due to more intimate contact with the surface, with more contacts producing more force and pulling the filaments closer to the substrate, which produces more friction. They then claim that a slime-extrusion mechanism would necessarily involve more force and lower friction. Is it necessarily true that this latter statement is correct? (I admit that it could be, but is it a requirement?)

Related to this, the authors use a model with isotropic friction. They claim that this is justified because they are able to fit the cell shapes well with this assumption. How would assuming a non-isotropic drag coefficient affect the shapes? It may be that it does equally well, in which case, the quality of the fits would not be informative about whether or not the drag was isotropic or not.