Characteristic features of an active polar filament pushing a load

Comment: The authors replied that the primary difference between their work and Isele-Holder et al., Soft Matter 12, 8495 (2016) is the dimensionality--and that the models are nearly identical in 2D. However, they neither cite nor discuss this work in their manuscript. Isele-Holder's publication has a very similar title and discusses the results of 3D simulations. Therefore, a reader could think that the manuscript under consideration (on 2D results) adds no new knowledge. I expect the authors to cite the study of Isele-Holder et al. in their manuscript and to discuss their findings in light of these previous results. They should state clearly where their results are in line with the 3D findings and where the different dimensionality leads to qualitatively or quantitatively different results.

Response: We have included the suggested article Soft Matter 12, 8495 (2016) in the references of the revised manuscript. We missed citing this article in the previous version, so we thank the reviewer for pointing this out. While our results share some features with those reported in Soft Matter 12, 8495 (2016), they differ in several key aspects. In three dimensions, the structural and dynamical complexity of the active polymer increases. For example, observing a beating phase in 3D is difficult without considering the filament’s torsional rigidity. Conversely, a helical phase will not appear in 2D.
In summary, the dynamics in 3D and 2D exhibit significant differences. In 2D, Soft Matter 12, 8495 (2016), authors identified distinct dynamical phases, including elongation, beating, circling, and rotation. In contrast, our 3D study reveals the presence of helical phase alongside extended and rotating states.

Beyond these differences, we have introduced a minimal model that incorporates cargo attachment as a control parameter, denoted by $\rho$. This addition underscores the novelty of our manuscript compared to previous studies. We have included a few additional lines in the Introduction to emphasize the primary distinctions. More specifically, we have stated that a 3D filament studied in the manuscript could bring different phases that are absent in the existing 2D results.

Comment: It is great that the authors have introduced the flexure number that also takes into account the elasticity of the self-propelled filament. Does it mean that in the current manuscript, the flexure numbers are one or two orders of magnitude smaller than in Isele-Holder et al. (2016), i.e., that the filaments are much stiffer than in the previous work? How do the different ranges of flexure numbers affect the results?

Response: We agree with the reviewer the calculations are performed for very stiff filaments $\l_p\ge L$. But, we have also considered the variation of the bending rigidity, that is, persistence length. By doing so, we vary the flexure number. Our study does not show any significant variation in response to changes in the flexure number. As shown in Fig. 7b, the radius of curvature as a function of $\chi$ remains consistent across different values of the filament's persistence length $l_p$. We observe the same power-law behavior with a flexure number for all persistence lengths. That concludes that changing the stiffness of the filament does not change our results, specifically in the limit of $\chi <1$.

Comment: 3 - I do not entirely understand how the simulation snapshots that the authors have sent in their rebuttal letter and the one in Fig. 2(e) of the manuscript are related to the local minimum of the tangent-tangent correlation function in Fig. 6(b). I suggest that the authors extend the discussion and indicate in the simulation snapshot the arc lengths s for the local minimum and maximum in the correlation function.

Response: In the snapshot shown in Fig. 2(e), the filament exhibits buckling at a certain distance from the tail end. This buckling results in local minima in the tangent correlation. Intuitively, this can be explained as follows: The front load opposes forward motion under compressive force, causing resistance that induces bending along the filament’s backbone. Figure 2(e) in the manuscript visually represents this buckled conformation. We have clarified this in the revised manuscript.

Comment: at pag.3 the authors say "This article presents a minimal model for the active polar filament in three dimensions (3D) with the active force applied along its backbone" which is not true. This article is presenting a minimal model about the polymer pushing on a cargo. Please rephrase

Response: We thank the reviewer for this suggestion. We have rephrased this sentence in the revised manuscript.

Comment: the authors say "The sensitivity of the structural dynamics to the boundary conditions of the front monomer and its load has already been emphasized for the case of polar semi-flexible and flexible polymers" hence they should state why and how the present work is going beyond.

Response: We thank the reviewer for pointing out this. The boundary condition is important in defining the behavior of an active filament. For example, an active filament with follower forces exhibits beating and rotational motion. We have rephrased this sentence in the revised version and stated the importance of our work specifically in the Introduction section.

Comment: the authors say "Our results show a monotonic decrease in both the radius of gyration and the end-to-end distance of the filament as the active force increases." These results have already been reported and discusses. The authors should focus on the novelty of their work and not give the reader the feeling that they are redoing what is already in the literature.

Response: Indeed, the effect of active forces on a filament's structure is already reported. However, we report here the variation of $R_e$ or $R_g$ with $Pe$ for varying the parameter $\rho$, which is beyond the scope of the previous work. To avoid confusion, we have rephrased the sentence more clearly and highlighted our contributions in the revised manuscript.

Comment : in Fig.5 I suggest to normalize $R_e$ and $R_g$ by their equilibrium values

Response: We thank the reviewer for this suggestion, we updated the plot in the revised manuscript (see the figure attached).

Comment: in Fig.9 I suggest to rescale the MSD by the countour lenght (as indeed suggested also in the main text)

Response: We have rescaled Fig.9a as per suggestion (see the figure attached).

Attachment:

plots.pdf