Mitigating memory effects during undulatory locomotion on hysteretic materials

Essential revisions:

1) The paper presents a very detailed and comprehensive study, but one that is scattered over too many disconnected pieces, making the narrative difficult to follow. There is a Main Text, very detailed and compact figures with long captions, a huge array of appendices and supplementary information, and appended videos. The effect is to diffuse the information rather widely, making reading difficult. There is overlap and repetition. We have not previously encountered a relatively short paper with 9 appendices. The authors must reformat the paper to make use of figure supplements in order that the logic and results are more clearly spelled out.

Moreover, the authors cite the supplementary material as if it were a separate publication. It is thus cited differently than Appendix material! We found this quite confusing.

Some minor details of method and snake husbandry might be usefully combined into one Materials and methods appendix with short subsections, while other appendices and supplemental material should be put back into the Main Text. We consider the mechanical RFT model to be a main contribution of the paper and thus should be mostly found in the Main Text. The Introduction and conclusions need to provide a clearer roadmap to all the interlocking parts of the argument and results.

We thank the reviewers for drawing our attention to these weaknesses in our paper. Please find below an itemized list of changes we made to aid reader comprehension.

– As per the reviewers’ suggestion we combined non-essential animal experimental information (collection and IACUC protocols, animal handling procedures) into a single appendix. Information about the animal experiment setup (bed size, cameras, and tracking procedure) is moved to the main text, subsection “Kinematics and ability vary among species.”. Description of C. occipitalis husbandry and the materials used in the construction of the fluidized bed used in the specialist experiments is in the supplemental information as this information is not pertinent to the results nor of broad interest.

– We kept an appendix detailing the calculation of the waveform parameters θ and ξ since this was carried out differently for specialists and nonspecialists, but the details of the procedure are not necessary to understand the results. Notes on this calculation that were in the supplemental were condensed and combined with the Appendix.

– Description of the granular drag experimental setup is moved from the Appendices to the main text, subsection “Granular drag measurements.”.

– Most of the details pertaining to RFT have been moved from the Appendix and supplemental to the main text. The only RFT information remaining in the supplemental is an explanation of the calculation of mechanical cost-of-transport and bodylengths per undulation cycle. We briefly mention these measurements in the main text as they are commonly used in the animal literature such that some readers may be interested in the prediction. However, as these results do not have any bearing on the rest of the study they are presented in the supplemental information. We added a supplemental table of the fourier-fit coefficients used to model the parallel forces.

– The Appendices on the measurements of body lifting and laser line reconstruction of the granular surface have been moved to supplemental. Both experiments and their results support observations and modeling decisions in the main text, however, they are not necessary to understanding the paper or results and we feel it would distract the reader to include this detail in the main text.

– We removed all references to the supplemental material as a citation and replaced them with figure supplements.

The Introduction and conclusions need to provide a clearer roadmap to all the interlocking parts of the argument and results.

We added some more detail on the differences between low-Re swimming, high-Re swimming, and fluids versus granular materials to the Introduction to better set up the main ideas in the paper (changes are further detailed below in response to subsequent critiques). We expanded the overview of the paper contents at the end of the Introduction to better orient the reader to the logical flow. Throughout the paper we edited and in places expanded the transitions between sections to better link them and provide more guidance to the reader.

2) There is also a distinct lack of detail presented on the mathematical/physical modeling. A good example of this lack of detail is found in the section on the drag model. We found no detail in the paper about the form of the "static stress" σ0.

In addition to providing those details in the paper, we would suggest that the authors provide (at least) some heuristic arguments explaining the typical scale of surface stresses they obtain (say, 0.1 N/cm2). Surely this can be explained in terms of grain size, coefficient of static friction, gravity, etc. Without this, the reader has learned nothing from the fact that an unspecified model is consistent with the data.

We thank the reviewers for this suggestion. We note that the original text did explain the method for obtaining σ_o, however, our explanation was clearly lacking clarity. As such we substantially reworked this section to be more explicit. We included the model in equation form, and added an estimation of the static stress using hydrostatic pressure, included as follows,

“The animal behavior indicated that friction between the body and the GM and local, dissipative interactions within the GM were the dominant forces in the system. […] σ_o was calculated by subtracting ρv_d² from σ_n measured at the three lowest speeds collected at a given z and averaging the result.”

3) The issue of the drag anisotropy is a fascinating one, particularly in comparison to the behavior of long slender objects in low Reynolds number Newtonian fluids. But here we were slightly confused about the results, and think the presentation needs to be reworked. Consider the following: in normal slender body hydrodynamics there are drag coefficients for motion normal (perpendicular) and tangential (parallel) to the filament, call them ζpara and ζperp, and one writes the total drag force F (per unit length) on a filament with velocity v as

F = (ζparatt + ζperpnn)dotv

where ζperp approximately 2 ζpara = 4 pi eta /(log(L/d)+const), where η is the viscosity, L the total length, and d the filament diameter, and the constant is order unity. The coefficients zeta (which are basically the Cn and Ct coefficients in subsection “Drag anisotropy is not strongly dependent on speed or depth”) thus do not depend on the angle of motion. So, we think the real question here is whether the experimental results in this paper are consistent with such constant zetas. Therefore, instead of (or in addition to) plotting the ratio σn/ σt as a function of βd in Figure 6A, consider plotting versus tan(βd) and/or a plot of (σn/ σt)/tan(βd) vs βd to see if a straight line emerges, which would be an indication of fluid-like behavior (or deviation from it). Also, the quantity Kfluid mentioned in the figure appears to be undefined.

We thank the reviewers for pointing out this lack of clarity and the suggestion of plotting the ratio in relation to tan(β_d) to highlight the difference between drag in GM versus viscous fluids. We updated the first paragraph in subsection “Drag anisotropy is not strongly dependent on speed or depth.” to point out that it is known in subsurface drag that the prefactors are functions of drag angle.

We included the following paragraph in subsection “Drag anisotropy is not strongly dependent on speed or depth.”, which references the plot of $\frac{\frac{{\sigma }_n}{{\sigma }_t}}{tan({\beta }_d)}$ which has been added to Figure 6B.

“In viscous fluids the constants C_n and C_t are independent of drag angle such that $\frac{\frac{{\sigma }_n}{{\sigma }_t}}{tan({\beta }_d)}$ is constant. Consistent with results for subsurface drag, Maladen et al., 2009, 2011b, we find that at the surface $\frac{\frac{{\sigma }_n}{{\sigma }_t}}{tan({\beta }_d)}$ is a nonlinear function of β_d (Figure 6B).”

4) Continuing with this point, the authors need to write down precisely a mathematical statement of resistive force theory for these systems. Is it like what is written above except for the definitions of the zetas (Cs), or something else?

We expanded subsection “Surface resistive force theory model.“ to include the mathematical definition of RFT used in our model. We moved most of the description of the calculation from the Appendix and supplemental to the main text with the exceptions described in response to revision 1.

Indeed, the formulation of granular RFT is as written above with functions f_n and f_t in place of the constants. In our case, we note that we used a fourier fit to the data to obtain f_t (as noted in the main text). We chose to do this to allow us to carry out the RFT calculation without having a model for the form of the tangential forces as the models previously used in subsurface drag did not sufficiently capture the shape of the curve.

5) The section comprising paragraph four-six in subsection “Surface resistive force theory model” is similarly in need of revision. It continually refers to supplemental material [21] and simply says whether the model works or doesn’t but does not convey any real physical insight to the reader without turning to that material.

We thank the reviewers for pointing out this lack of clarity. To address this, we moved the figure comparing RFT-calculated movement speeds with those measured in experiment from the supplemental to the main text (Figure 6 in the revised manuscript). We expanded our Discussion of this comparison in the main text to better present the important findings.

6) The authors may wish to note that another medium that has memory is a fluid moving at high Reynolds number. Recent work from the lab at Courant has shown that hysteresis arises in model experiments of flocks due to the finite lifetime of vortices shed by leading swimmers.

This is an excellent point. We have added the following to the Introduction to include the time-dependent flows occurring at high-Re:

“At the other end of the spectrum, animals move in materials whose state depends on the history of interaction. […] However, unlike fluids, they can bear internal stress. At the free surface, gravity is often insufficient to return mounds of material created by a locomotor to the undisturbed state, exemplified by the tracks left behind by a passing animal.”

7) The authors frequently make reference and comparisons to subsurface locomotion and to low-Re swimmers, without really explaining what Reynolds number is. They seem to assume the reader is simply familiar with these cases, without explanation. It might be useful to briefly bring together these comparisons in the Introduction, so that later reference to them seem less disconnected.

We thank the reviewers for pointing out this oversight. We have added a definition of Reynolds number to the Introduction as follows:

“At one end of the spectrum, where deformations are short-lived Cohen and Boyle, 2010, small fluid swimmers like the nematode Caenorhabditis elegans Wen et al., 2012, spermatazoa Gray, 1953, and bacteria in water Rodenborn et al., 2013, are propelled by the viscous force of the material resisting the animal’s +body shape changes. […] Notably, in both viscous and frictional fluids, the material surrounding the submerged swimmers continuously re-flows around the body of the animal such that the animal is always surrounded by material.”

In addition to the added the paragraph discussing high-Re swimming (copied above in response to comment 6) this provides a better explanation for motion in viscosity versus inertia-dominated systems.