SciPost Submission Page
Characteristic features of an active polar filament pushing a load
by Prabhakar Maurya, Shalabh K. Anand, and Sunil P. Singh
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Shalabh Anand · SUNIL PRATAP SINGH |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202410_00008v1 (pdf) |
| Date submitted: | 2024-10-07 15:33 |
| Submitted by: | SINGH, SUNIL PRATAP |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
We present the structural and dynamical behavior of an active polar filament, which is pushing loads of different magnitudes and sizes using overdamped Langevin dynamics simulations. By changing the bending rigidity and the connectivity between the filament and the load, we smoothly vary the filament’s boundary conditions from pivoted to clamped. For the pivot boundary conditions, the active polar filament buckles and displays various fascinating dynamical phases: snake-like motion, rotational motion, bent shape conformations, and helical structures. However, in clamped boundary conditions, the helical phase disappears, and the filament attains either an extended state or a bent state. The transition from the extended-state conformation to the helical state is demarcated using the global helical order parameter in the parameter space of active force and a quantity associated with the boundary condition. We have obtained various power laws relating to the curvature radius of the helical phase, effective diffusivity, and rotational motion of the monomers with the active force. Additionally, we demonstrate that the filament’s effective diffusivity in the helical phase shows a non-monotonic trend with the active force, increasing linearly followed by a sharp reduction at a high strength of the active force.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Strengths
1- Thorough characterization of an active polymer in 3D pushing a load.
2- Reports scaling laws based on the simulation data.
3- Timely study of a system at the interfaces of physics, chemistry, and biology.
Weaknesses
1- A comparison of the 3D simulation data with previous 2D simulation data is missing.
2- The "Model " section should be revised to improve reproducibility.
3- The importance of the linking of the load to the filament is extensively discussed but not well motivated based on biological systems.
Report
The authors use computer simulations based on the overdamped Langevin equation to study the shapes and dynamics of an active polar filament pushing a load in three dimensions. They report snake-like and rotational motion, bent-shape conformations, and helical structures. Simulations are performed for various bending rigidities of the filament, active force strengths, and parameters characterizing the load and its attachment to the filament. To characterize the filament structure, they calculate helical order parameters, filament bending energies, filament sizes, effective filament diffusivities, and the rotational motion of the monomers. They find various power laws and report a non-monotonic behaviour of the effective diffusivity with the active force strength.
The manuscript reports on a very timely study in the field of active matter. The system of active polymers pushing a load provides a novel and synergetic link between physics, chemistry, and biology. Furthermore, the manuscript is written in an accessible way and overall well embedded into the pertinent literature. Therefore, I expect it to eventually be suitable for and of interest to readers of SciPost Physics. However, in my opinion, there are several major and minor issues that - if dealt with by the authors - would improve the quality of the manuscript significantly. Therefore, I ask the authors to consider my issues before I can recommend the manuscript for publication.
Requested changes
Major issues:
1- The authors discuss the system of an active polar filament pushing a load, which they study using computer simulations in 3D. In the literature, data is available for in 2D simulations [Isele-Holder et al., Soft Matter 12, 8495 (2016)]. How do the results in 3D compare to those in 2D?
2- Active-matter systems that contain rigid self-propelled particles are often characterized by the Peclet number, which relates the propulsion velocity with the noise. For studying self-propelled flexible filaments, the flexure number that relates the propulsion force to the bending rigidity appears equally important. I recommend the authors to provide flexure numbers for their system.
3- In Eq. (4), it is unclear to me whether the values of the viscous drag coefficients are the same for all beads i. Is the bead with index i=1 the load? If the friction is implemented bead-wise and does not take into account the elongated structure of the active polymer (such as an anisotropic friction for active rods), I do not understand why a smaller bending rigidity of the bond attaching the load to the filament should increase the friction of the front bead and thus the load. However, the authors write on page 3 "A smaller bending rigidity of the load with the filament provides a higher drag.". A similar statement is found on page 4, where the authors write that "random alignment of the head from the rest of the filament acts as higher load".
4- I am unsure whether I understand Fig. 3 correctly. I do understand that H2 and H4 show finite values for helical order. However, should I not also expect to find values of the order of 1 for straight filaments, such as the conformation shown in Fig. 2(a)? Should I then not expect to find high values of H2 and H4 for small Pe?
5- Are the average bending energy of the filament, shown in Fig. 4, and the curvature of the filament, shown in Fig. 7, related?
6- Many of the figures shown in the manuscript remind me of hysteresis loops. Do the authors expect to find hysteresis if they were to first increase and subsequently decrease Pe in small steps within the same simulation?
Minor issues:
7- In the abstract, the authors mention the effect of "clamped boundary conditions" for attaching the load on the filament conformations. Here, changing the wording such that the meaning of "clampled" is immediately obvious to a reader or adding half a sentence of explanation would be helpful.
8- In the abstract, the authors mention "bent-shape conformations" as one item in a list of four. Because the other three items, snake-like motion, rotational motion, and helical structures all refer to bent filaments, "bent-shape conformations" appears too unspecific.
9- I am not sure whether mentioning nine references [1-9] in line one with a very general sentence is helpful for a reader.
10- In the introduction, the authors mention several examples that appear on the first view only weakly related to self-propelled filaments pushing a load, such as vesicle transport, muscle contraction, and shape changes in neurons. They may want to consider removing those. Instead, discussing the instability of pusher systems in general may be more relevant, such as a long-wavelength instability in active nematics [Thampi et al., EPL 105, 15001 (2014)], and Euler-like buckling for systems of cytoskeltal systems and molecular motors [Vliegenthart et al., Sci. Adv. 6, eaaw9975 (2020)].
11- On page 2, the wording "imposing force along the polymer's conformational specifically unidirectionally along its every tangent-bond vector" should be rephrased.
12- On page 2, I find the statement "leads to the non-monotonic behaviour of the polymer" too vague to be understood by a normal reader. I suggest that the authors add half a sentence specifying what is non-monotonic in the polymer behaviour.
13- On page 2, the authors use "On the other hand", without having used "On the one hand" before. This seems incomplete.
14- On page 2, the authors list various helices present in biological systems that are stabilized by bonds, such as DNA and alpha-helical structures in proteins, when they mention the dynamically stable helical state of active polymers. Are these systems really comparable?
15- On page 3, the authors write "We observe a monotonic shrinkage of the filament with active force.". Do the authors refer to a shrinkage of the contour length of the polymer or of its radius of gyration?
16- On page 3, the authors write "The structure is stable up to a certain bending rigidity, which disappears ...". This reads as if the bending rigidity of the active polymer would disappear, which is probably not what the authors mean to say. The sentence needs to be rephrased.
17- In Eq. (2), I believe the sum should start at i=2.
18- On page 4, the authors provide the size of the load by providing alpha. Intuitively, it appears obvious that a larger load will lead to a higher friction, as one would expect for Stokes friction and 3D hydrodynamics. The authors should define the friction coefficient gamma_i in terms of the load/bead size sigma. This would also clarify "varying its [the load's] connection to the filament as well as its friction and size" to a reader, which the authors write at the beginning of section 3. A priori, the effect of the size of the load in the Langevin approach is not well defined.
19- In the caption of Fig. 2, the authors mention that the front monomer is colored in red. This seems to be a typo, because in the figure one monomer is colored blue and all others are colored red.
20- In the caption of Fig. 3, the authors should refer to Eq. (6).
21- In Eq. (6) or Fig. 3, the <...> brackets indicating the average are missing.
22- In Fig. 3, it is confusing that there is one additional data set for rho=0.05 in subfigure a, such that most colors and symbols in both subfigures do not match.
23- In Fig. 4, it would be easier for a reader if the scaling law ~Pe^{4/3} would be placed next to the guide to the eye.
24- I wonder whether it would be useful to place guides to the eye to indicate scaling behaviour in various other figures where they are still missing as well.
25- The authors discuss the end-to-end distance and the radius of gyration for their active polymers, and the dependence on Pe is qualitatively similar. For passive linear polymers, <Re^2> = 6 <Rg^2>. Can the authors confirm a similar relationship for active polymers?
26- In the discussion of Fig. 6, the authors hypothesize a sharp kink as the reason for the two-step decay shown in subfigure (b). Adding selected simulation snapshots of the filaments to the figure may be helpful to readers.
27- Also the inset of Fig. 7 appears to show a scaling law. I suggest that the authors add a guide to the eye for the scaling exponent.
28- In Fig. 9, several quantities are not dimensionless and should either have units or be converted into dimensionless quantities.
29- Regarding section 3.6, as I mentioned earlier, it is unclear to me how the size of the load enters the model.
30- To me the implications of the manuscript read very weak because they are very vague. I wonder whether the authors may be able to more directly relate their own results to findings in or open questions raised in Refs. [7,8,37,64].
Recommendation
Ask for major revision
Strengths
1- the paper addresses an interesting scenario, namely the dynamics of active polymers with a heavy head.
2- the numerical results are extensive and more then sufficient to cover the relevant ranges of the Peclet number
3- this contribution is timely since it addresses a setup of current interest in the comunity of active matter
Weaknesses
1- the paper is mainly numerical: the authors do not really attempt to rationalize the numerical data by means of some model or scaling function.
2- the authors do not discuss the relevance of the presented results in relation to the introduction where they mention several interesting scenarios. In which bilogical or technologicla scenario are the phenomena presented here relevant?
Report
The authors report on detailed numerical simulations of active polymers with a load attached to their heads.
The manuscript is, in general, well written, the results are clearly presented and support the conclusions.
I would recommend for publication once the following remarks are addressed
- the results are typically presented in units of the microscopic Peclte number Pe = fl/k_BT. However, in a recent contribution (Molecular Physics, e2384462, 2024), it has been shown that the data can be better understood by looking at a macroscopic Peclet number Pe = f R_G/k_BT. I wonder if this can be the case here as well. For example, I wonder if the reduction in R_E shown in Fig.5a can explain the reduction in the effective diffusion shown in Fig.8.
Minor:
- Check the spelling of Peclet: it is not P'eclet
- In the introduction the authors say "The former one leads to the non-monotonic behavior of the polymer". I do not understand which "non-monotonic behavior" the authors refer to
- similarly for "which offers the appearance of dynamical structures and avoids self-trapping into the spiral form at two dimensions". I do not understand the meaning...
Recommendation
Ask for minor revision
Author: SUNIL PRATAP SINGH on 2025-02-12 [id 5214]
(in reply to Report 2 on 2024-11-13)
Comment: The authors report on detailed numerical simulations of active polymers with a load attached to their heads. The manuscript is, in general, well written, the results are clearly presented and support the conclusions. I would recommend for publication once the following remarks are addressed
Response: We thank the referee for the valuable suggestions and encouraging remarks.
Comment: the results are typically presented in units of the microscopic Peclet number $Pe = fl/k_BT$. However, in a recent contribution (Molecular Physics, e2384462, 2024), it has been shown that the data can be better understood by looking at a macroscopic Peclet number $Pe = f R_G/k_BT$. I wonder if this can be the case here as well. For example, I wonder if the reduction in $R_E$ shown in Fig.5a can explain the reduction in the effective diffusion shown in Fig.8.
Response: We thank the reviewer for this suggestion. Indeed, the diffusivity scales, when presented as a function of $Pe \times R_g$ in the limit of small $Pe$ and the helical phase for large $Pe$. However, in the intermediate regime, due to the structural transition, $R_g$ undergoes a crossover; therefore, the scaling does not hold. To avoid confusion, we have retained the original scaling. A discussion of this context has been added to the main manuscript, and the relevant citations in the revised manuscript have been included.
Comment: Check the spelling of Peclet: it is not P'eclet.
Response: We thank the referee for raising this point. We have reviewed the literature on P\'eclet numbers and found that both spellings are used. We have adopted the nomenclature "Péclet", which is consistent with the following references: Phys. Rev. Res. 6, L032002 2024), Rep. Prog. Phys. 78 (2015) 056601 and THE JOURNAL OF CHEMICAL PHYSICS 146, 154903 (2017) etc.
We respectfully request approval to continue using this spelling.
Comment : In the introduction the authors say "The former one leads to the non-monotonic behavior of the polymer". I do not understand which "non-monotonic behavior" the authors refer to
Response: We thank the reviewer for pointing out this. This sentence is clarified in the revised manuscript.
Comment: similarly for "which offers the appearance of dynamical structures and avoids self-trapping into the spiral form at two dimensions". I do not understand the meaning...
Response: We have also modified this sentence for clarity.
Attachment:
Strengths
1 - including Brownian filament (rather than a passive one)
2 - providing summary statistics of the simulations ($R_g, R_e, H_2, D_p$) for quantitative description of results
3 - analysis of apparent diffusion coefficient
Weaknesses
1 - fixing $l_p$ as a control parameter
2 - insufficient details given on the numerical simulation
3 - disregarding hydrodynamic interactions
Report
# General comments
In this interesting manuscript authors consider a numerical simulation of an active Brownian polymer. Authors describe many critical details of the simulations (numerical scheme, timestep and number of repetitions used). Assuming numbers in the inset of figure 7 indicate typical lengths of the filament used in most simulations, I am unsure whether the Brownian part of the 'active Brownian' polymer really plays a significant role here. It appears that the filament in question has persistence length between 2 and 10 times larger than the length of the filament (this appears consistent with figure 2 showing largely 'smooth' shapes).
Why would the authors focus on this regime in particular? It takes a great amount of effort to deal with stochastic simulations (anything non-deterministic is notoriously hard to debug) and the authors could benefit from highlighting their expertise there. By showcasing that Brownian contributions are indeed significant this manuscript could rise above the level of many publications which deal with active particles.
I find figure 8 especially interesting, but critical dimensionless number of the ratio of persistence length to total filament length is missing from the investigation presented. Furthermore theoretical description of the role of hydrodynamic interations into the apparent diffusion coefficient for long polymers is well established (see JG Kirkwood, J Riseman - J. Chem. Phys. 1948) and these are never negligible even for very slender filaments.
As outlined by the authors, sufficiently long simulation is required to avoid the ballistic regime in case of a simple active particle. However in case of an elastic particle conformation relaxation time (which is likely longer) is also present. Further details describing the procedure leading to the values $D_p$ presented are thus important and great care has to be given into specifying how MSD was post processed to eliminate finite time and initial condition effects.
The authors obtain a power law relationship $R_\beta \sim Pe^{-\beta}$, a relationship between a length and a dimensionless number. I hope this relationship holds regardless of the filament discretisation into finite number of beads. The question arises what sets the lengthscale at fixed $Pe$? In other words, would $R_\beta$ scale as persistence length or total filament length or some other quantity?
Unfortunately, even though the authors chose to contribute to SciPost - a journal which prides itself in spearheading the open-science practice - they do not disclose the code used in their simulations. Such choice makes this numerical-simulation-based manuscript very difficult and perhaps impossible to reproduce / validate. In such cases it is very easy to accidentally miss some vital details - for example number of monomers used in the simulations is never explicitly mentioned in this publication (even though different values are implied in figure 7). With this in mind the statement about the inclusion of hydrodynamic interactions as a next step for the community falls flat - there are multiple packages capable of including those but no entry point for this extension is given.
I hope this serious oversight is very easy to address and high quality, well documented source code repository will accompany this numerics-focussed manuscript. This would also allow for further improvements of simulation method such as using a better integration scheme than naive Euler method presented.
# Nitpicks:
1. figure 2. I assume authors meant in *blue* color rather than in *red* color.
2. $\Delta t = 10^{-4} \tau$ for $\kappa_b = 1000 k_b T / l^2_0$ seems rather large - comparable with spring relaxation time. Some test and at least further comments why this rather large timestep is fine in case of large values of $Pe$ are expected.
3. In Equation (6), in definition of $H_4$, it is not clear what the authors mean by vector squared. Do they mean norm of the vector squared? Moreover wouldn't this sum be simply almost equal to end-to-end distance?
4. $R_\beta$ appears to be unaffected by the chain length but maybe this is because the persistence length is $1000 l_0$ and in all cases a shorter chain was chosen?
# Acceptance criteria:
I hope this manuscipt will eventually find its way into publication.
In my oppinion this manuscript has a potential to fulfill the 'expectations' criterion 2:
"Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work"
rather than criterion 3 indicated by the authors.
Howerver it fails at meeting 'general acceptance' criterion number 2 (compare also with criterion 6.):
"Provide sufficient details (inside the bulk sections or in appendices) so that arguments and derivations can be reproduced by qualified experts"
The critical problem of reproducibility due to lack of supporting software code has to be addressed.
I recommend a revision.
Recommendation
Ask for major revision
Author: SUNIL PRATAP SINGH on 2025-02-08 [id 5198]
(in reply to Report 1 on 2024-11-02)
Comment: In this interesting manuscript authors consider a numerical simulation of an active Brownian polymer. Authors describe many critical details of the simulations (numerical scheme, timestep and number of repetitions used). Assuming numbers in the inset of figure 7 indicate typical lengths of the filament used in most simulations, I am unsure whether the Brownian part of the 'active Brownian' polymer really plays a significant role here. It appears that the filament in question has persistence length between 2 and 10 times larger than the length of the filament (this appears consistent with figure 2 showing largely 'smooth' shapes).
Why would the authors focus on this regime in particular? It takes a great amount of effort to deal with stochastic simulations (anything non-deterministic is notoriously hard to debug) and the authors could benefit from highlighting their expertise there. By showcasing that Brownian contributions are indeed significant this manuscript could rise above the level of many publications which deal with active particles.
Response: Considering this suggestion, we have performed simulations without thermal fluctuations with the initial configuration of the filament as a straight rod. For this configuration, we do not see any change in the structure. Filament moves in a straight line, see Fig.~1-a. Further, we change the head's orientation from a straight line; however, the rest of the monomers are all aligned in a straight line (see Fig.1 b). In this case, the filament acquires the helix shape due to competition between viscous drag and compressive force. Therefore, without thermal fluctuations, the helical structure depends on the initial configuration. If we start with a straight conformation, we don't see the helix (see Fig.~1-a. However, we observe the helix when we start with conformations are perturbed from the straight rod. Thus, weak thermal fluctuations are important for such systems.
Comment: I find figure 8 especially interesting, but critical dimensionless number of the ratio of persistence length to total filament length is missing from the investigation presented. Furthermore theoretical description of the role of hydrodynamic interactions into the apparent diffusion coefficient for long polymers is well established (see JG Kirkwood, J Riseman - J. Chem. Phys. 1948) and these are never negligible even for very slender filaments.
Response: We thank the reviewer for the suggestion. We have varied $l_p/L$ in the manuscript, both in the inset of $Fig.7a$ and in Fig.7b. In Fig.7a, we have varied $l_p/L$ by fixing $l_p$ and varying $L$ in the range of $l_p/L=0.1$ to $0.4$. In Fig.7b, we have varied $l_p/L$ in the range of $0.5$ to $5$ by fixing $L$ and changing $l_p$. The results are consistent throughout the parameter regime of $l_p/L$. However, we believe that presenting the figure in terms of $l_p/L$ can confuse the reader about which parameter is fixed and which is being varied. We have mentioned the ratio of $l_p/L$ in the revised manuscript. We do agree with the reviewer that hydrodynamic interactions are important for such a system as the drags are anisotropic. We have added a few lines on the importance of the hydrodynamic interactions.
%We thank the reviewer for the suggestion. We have mentioned the ratio of $l_p/L$ in the revised manuscript. Furthermore, to show the influence of the $l_p/L$, we have performed a set of new simulations to show that as long as $l_p/L>>1$, the results of our simulations are identical. We do agree with the reviewer that hydrodynamic interactions could become important. We have mentioned this in the discussion.
Comment: As outlined by the authors, sufficiently long simulation is required to avoid the ballistic regime in case of a simple active particle. However in case of an elastic particle conformation relaxation time (which is likely longer) is also present. Further details describing the procedure leading to the values $D_{p}$ presented are thus important and great care has to be given into specifying how MSD was post processed to eliminate finite time and initial condition effects.
Response: Indeed, we run the simulations for a sufficiently long time. For the final results, we ignore the initial simulation points from our calculations. We only consider points much larger than the relaxation time of the filament. Based on the later time trajectories, we compute the MSD. Additionally, we have generated 10 independent simulation runs for each data point to generate good statistics. From this, we estimate the effective diffusion coefficient of the filament in the diffusive regime.
Comment: The authors obtain a power law relationship $R_{\beta} \sim Pe^{-\beta}$, a relationship between a length and a dimensionless number. I hope this relationship holds regardless of the filament discretisation into finite number of beads. The question arises what sets the lengthscale at fixed $Pe$? In other words, would $R_{\beta}$ scale as persistence length or total filament length or some other quantity?
Response: The inset of Fig. 7a shows that $R_{\beta}$ does not depend on $N$. To verify the dependence with respect to the persistence length of the filament, we performed a new set of simulations for different $l_p$. We have included a new plot of the radius of curvature ($R_\beta$) as a function flexure number ($\chi = l_{0}Pe/l_{p}$, a dimensionless quantity) for different persistence lengths. It matches our scaling behavior, as shown previously. We have included this plot as Fig. 7b in the revised manuscript.
Comment: Unfortunately, even though the authors chose to contribute to SciPost - a journal which prides itself in spearheading the open-science practice - they do not disclose the code used in their simulations. Such choice makes this numerical-simulation-based manuscript very difficult and perhaps impossible to reproduce / validate. In such cases it is very easy to accidentally miss some vital details - for example number of monomers used in the simulations is never explicitly mentioned in this publication (even though different values are implied in figure 7). With this in mind the statement about the inclusion of hydrodynamic interactions as a next step for the community falls flat - there are multiple packages capable of including those but no entry point for this extension is given.
Response: We thank the reviewer for pointing out this. We have provided all the information regarding simulation details in the revised version of the manuscript. The simulation code is also shared in the repository with the link provided in the revised manuscript.
Comment: I hope this serious oversight is very easy to address and high quality, well documented source code repository will accompany this numerics-focussed manuscript. This would also allow for further improvements of simulation method such as using a better integration scheme than naive Euler method presented.
Response: We have shared our source code repository with the link embedded in the manuscript.
Comment: figure 2. I assume authors meant in *blue* color rather than in *red* color.
Response: We have corrected the text in the revised manuscript.
Comment: $\Delta t = 10^{-4}\tau$ for $\kappa_{b} = 1000 k_{b}T/l_{0}^{2}$ seems rather large - comparable with spring relaxation time. Some test and at least further comments why this rather large timestep is fine in case of large values of $Pe$ are expected.
Response: We thank the reviewer for this point. A minor correction has been made to the revised manuscript. We presented these parameters in the revised manuscript in the unit of the LJ energy $\epsilon$. This point is revised in the parameter section of the manuscript. This makes $\kappa_{B} = 100 \epsilon/\ell_{0}^{2}$ and $k_{s} = 1000\epsilon/\ell_{0}^{2}$. We have also tested simulations for further smaller integration steps. The results look identical to the presented one.
Comment: In Equation (6), in definition of $H4$, it is not clear what the authors mean by vector squared. Do they mean norm of the vector squared? Moreover wouldn't this sum be simply almost equal to end-to-end distance?
Response: We thank the reviewer for pointing out this. We realized that there was a typo in the manuscript. We have corrected this in our revised version of the manuscript. We have defined a new vector ${\bf u}$ as we use it in the definition of $H_2$ and $H_4$.
Comment: $R_{\beta}$ appears to be unaffected by the chain length but maybe this is because the persistence length is $1000l_{0}$ and in all cases a shorter chain was chosen?
Response: We have varied chain length and persistence length of the filament in the revised manuscript. We observe the same power-law behavior for all the cases. A plot is added in the revised manuscript to demonstrate this; see Fig.7b.
Author: SUNIL PRATAP SINGH on 2025-02-14 [id 5222]
(in reply to Report 3 on 2024-11-20)Comment: The authors discuss the system of an active polar filament pushing a load, which they study using computer simulations in 3D. In the literature, data is available for in 2D simulations [Isele-Holder et al., Soft Matter 12, 8495 (2016)]. How do the results in 3D compare to those in 2D?
Response: The primary difference between our work and that of [Isele-Holder et al., Soft Matter 12, 8495 (2016)] lies in dimensionality. While the models are nearly identical in two dimensions, our results differ in many aspects, though retaining some of the features reported in their work. Of course, three dimensions increase the structural complexity. In 2D, the authors of Soft Matter 12, 8495 (2016) identified different dynamical phases, such as elongation, beating, circling, and rotation. In 3D, we observe additional phases, including extended, rotating, and helical states. The helical phase is not possible in two dimensions. Beating is also challenging to observe in three dimensions without torsional rigidity.
As shown in another article (Soft Matter, 2019, 15, 7926-7933), tuning the torsional rigidity can induce a transition from rotational to beating motion. Therefore, the stable helical phase observed here is distinct from the dynamical features and phases reported in the 2D case, apart from the various physical properties presented in this study.
Comment: Active-matter systems that contain rigid self-propelled particles are often characterized by the Peclet number, which relates the propulsion velocity with the noise. For studying self-propelled flexible filaments, the flexure number that relates the propulsion force to the bending rigidity appears equally important. I recommend the authors to provide flexure numbers for their system.
Response: We thank the reviewer for this suggestion. We have now defined the flexure number and presented the curvature radius as a function of the flexure number for various persistence lengths (see Fig.7 b) in the revised manuscript. However, for the rest of the manuscript, we are using a fixed persistence length ($l_p=1000$). Thus, the flexure number will just be a scaling x-axis by a constant number ($Pel_{0}/l_p)$. We have mentioned the persistence length in the simulation parameters paragraph in the revised manuscript.
Comment: In Eq. (4), it is unclear to me whether the values of the viscous drag coefficients are the same for all beads i. Is the bead with index i=1 the load? If the friction is implemented bead-wise and does not take into account the elongated structure of the active polymer (such as an anisotropic friction for active rods), I do not understand why a smaller bending rigidity of the bond attaching the load to the filament should increase the friction of the front bead and thus the load. However, the authors write on page 3 "A smaller bending rigidity of the load with the filament provides a higher drag.". A similar statement is found on page 4, where the authors write that "random alignment of the head from the rest of the filament acts as higher load".
Response: Thanks for pointing this out. The viscous drag $\gamma^h$ is different for the head monomer if the size is taken differently from the others, as in Figure 10, section 3.6. In most cases, we have taken the size of the head monomer to be the same as others. However, the bending rigidity of the head monomer is different from others; therefore, the head can be randomly aligned from the rest of the filament. This leads to an increase in the effective drag on the filament due to random alignment from the orientations of the filaments, which pulls the filament with the active force in different directions. The smaller the bending rigidity, the greater the randomness in the bond alignment, causing the filament to experience more compressive forces.
The referred sentences are modified for clarification.
Comment: I am unsure whether I understand Fig. 3 correctly. I do understand that H2 and H4 show finite values for helical order. However, should I not also expect to find values of the order of 1 for straight filaments, such as the conformation shown in Fig. 2(a)? Should I then not expect to find high values of H2 and H4 for small Pe?
Response: We thank the reviewer for pointing out this. We have corrected the definitions of $H2$ and $H4$ in the revised manuscript.
Comment: Are the average bending energy of the filament, shown in Fig. 4, and the curvature of the filament, shown in Fig. 7, related?
Response: The curvature of the filament is related to the total bending energy. In general, the relation between curvature and energy goes something $U_b\sim \frac{1}{R^2}$ for the case of $R>>\lambda_p$, where $\lambda_p$ is the pitch of the helices Ref[58]. We have plotted the bending energy and curvature radius to establish such a relationship. In our simulations, we do not obtain perfect helices, and the pitch length is variable; therefore, such a relationship is not achieved.
Comment: Many of the figures shown in the manuscript remind me of hysteresis loops. Do the authors expect to find hysteresis if they were to first increase and subsequently decrease Pe in small steps within the same simulation?
Response: Thanks for pointing this out to us. Indeed, the results look like it has hysteresis. However, the deviation from one curve to another curve occurs at the point of the structural transition.
More importantly, simulations for each point are performed with the initial configurations starting from rod-like conformations, and a large portion of the simulations (initial data) are ignored to achieve the steady-state configurations. The final results correspond to the steady-state data. Hence, in this approach, the hysteresis will not be observed.
Comment: In the abstract, the authors mention the effect of "clamped boundary conditions" for attaching the load on the filament conformations. Here, changing the wording such that the meaning of "clampled" is immediately obvious to a reader or adding half a sentence of explanation would be helpful.
Response: Thanks for the suggestion. We have included a sentence for clarification.
Comment: In the abstract, the authors mention "bent-shape conformations" as one item in a list of four. Because the other three items, snake-like motion, rotational motion, and helical structures all refer to bent filaments, "bent-shape conformations" appears too unspecific.
Response: We have revised this sentence.
Comment: I am not sure whether mentioning nine references [1-9] in line one with a very general sentence is helpful for a reader.
Response: We have removed the citation of references from this generic line.
Comment: In the introduction, the authors mention several examples that appear on the first view only weakly related to self-propelled filaments pushing a load, such as vesicle transport, muscle contraction, and shape changes in neurons. They may want to consider removing those. Instead, discussing the instability of pusher systems in general may be more relevant, such as a long-wavelength instability in active nematics [Thampi et al., EPL 105, 15001 (2014)], and Euler-like buckling for systems of cytoskeltal systems and molecular motors [Vliegenthart et al., Sci. Adv. 6, eaaw9975 (2020)].
Response: We thank the referee for this insightful point. In response, we have included a discussion in the introduction regarding the collective dynamics of self-propelled systems, referencing the suggested works. However, previous examples list the single filament buckling; therefore, we have kept that in the introduction.
Comment: On page 2, the wording "imposing force along the polymer's conformational specifically unidirectionally along its every tangent-bond vector" should be rephrased.
Response: The statement is modified for clarity.
Comment: On page 2, I find the statement "leads to the non-monotonic behaviour of the polymer" too vague to be understood by a normal reader. I suggest that the authors add half a sentence specifying what is non-monotonic in the polymer behaviour.
Response: The statement is modified.
Comment: On page 2, the authors use "On the other hand", without having used "On the one hand" before. This seems incomplete.
Response: The statement is modified.
Comment: On page 2, the authors list various helices present in biological systems that are stabilized by bonds, such as DNA and alpha-helical structures in proteins, when they mention the dynamically stable helical state of active polymers. Are these systems really comparable?
Response: These examples are listed to demonstrate the helical states in a living matter, although these states originate from different physical mechanisms. The sentence is rephrased for clarity in the revised manuscript.
Comment: On page 3, the authors write "We observe a monotonic shrinkage of the filament with active force.". Do the authors refer to a shrinkage of the contour length of the polymer or of its radius of gyration?
Response: This sentence is rephrased.
Comment: On page 3, the authors write "The structure is stable up to a certain bending rigidity, which disappears ...". This reads as if the bending rigidity of the active polymer would disappear, which is probably not what the authors mean to say. The sentence needs to be rephrased.
Response: This sentence is rephrased.
Comment: In Eq. (2), I believe the sum should start at i=2.
Response: The equation is corrected.
Comment: On page 4, the authors provide the size of the load by providing alpha. Intuitively, it appears obvious that a larger load will lead to a higher friction, as one would expect for Stokes friction and 3D hydrodynamics. The authors should define the friction coefficient $\gamma_i$ in terms of the load/bead size sigma. This would also clarify "varying its [the load's] connection to the filament as well as its friction and size" to a reader, which the authors write at the beginning of section 3. A priori, the effect of the size of the load in the Langevin approach is not well defined.
Response: Thanks for pointing out this issue. We have defined the friction coefficient of the load in Equation 4. A few lines about the choice of the $\gamma^h$ are included in the results section 3 in line 4.
Comment: In the caption of Fig. 2, the authors mention that the front monomer is colored in red. This seems to be a typo, because in the figure one monomer is colored blue and all others are colored red.
Response: The sentence is rephrased.
Comment: In the caption of Fig. 3, the authors should refer to Eq. (6).
Response: We have cited Eq.6 in the revised manuscript.
Comment: In Eq. (6) or Fig. 3, the <...> brackets indicating the average are missing.
Response: We have modified the definition of Equation 6.
Comment: In Fig. 3, it is confusing that there is one additional data set for rho=0.05 in subfigure a, such that most colors and symbols in both subfigures do not match.
Response: We have included $\rho=0.05$ in the figure to make it consistent throughout the manuscript.
Comment: In Fig. 4, it would be easier for a reader if the scaling law $~Pe^{4/3}$ would be placed next to the guide to the eye.
Response: The suggestion is included in the revised manuscript.
Comment: I wonder whether it would be useful to place guides to the eye to indicate scaling behaviour in various other figures where they are still missing as well.
Response: The suggestion is included in the revised manuscript.
Comment: The authors discuss the end-to-end distance and the radius of gyration for their active polymers, and the dependence on Pe is qualitatively similar. For passive linear polymers, $<Re^2> = 6 <Rg^2>$. Can the authors confirm a similar relationship for active polymers?
Response: For the passive limit, the relationship between $<R_e> \approx \sqrt{12} <R_g>$ holds in our simulations, which is the case of the rod-like polymer as $l_p/L=5??$. However, in the helical phase, this relation does not hold.
Comment: In the discussion of Fig. 6, the authors hypothesize a sharp kink as the reason for the two-step decay shown in subfigure (b). Adding selected simulation snapshots of the filaments to the figure may be helpful to readers.
Response: We thank the reviewer for the suggestions. Here are a few snapshots. We have included a snapshot in the revised manuscript at $Pe=400$.
Comment: Also the inset of Fig. 7 appears to show a scaling law. I suggest that the authors add a guide to the eye for the scaling exponent.
Response: The suggestion is included in the revised manuscript.
Comment: In Fig. 9, several quantities are not dimensionless and should either have units or be converted into dimensionless quantities.
Response: The length is in units of the bond length $\ell_0$, and time is in units of $\tau$ as mentioned in the model section. Hence, the frequency is units of $\tau^{-1}$. All the parameters presented are dimensionless quantities.
Comment: Regarding section 3.6, as I mentioned earlier, it is unclear to me how the size of the load enters the model.
Response: This discussion is included and highlighted in the revised manuscript on page 5 at the top of the page.
Comment: To me the implications of the manuscript read very weak because they are very vague. I wonder whether the authors may be able to more directly relate their own results to findings in or open questions raised in Refs. [7,8,37,64].
Response: We have rephrased the summary and the open questions presented in the summary of the manuscript. For clarity, the revisions are highlighted in the updated version of the manuscript.
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