Vesicle shape transformations driven by confined active filaments

In active matter systems, deformable boundaries provide a mechanism to organize internal active stresses. To study a minimal model of such a system, we perform particle-based simulations of an elastic vesicle containing a collection of polar active filaments. The interplay between the active stress organization due to interparticle interactions and that due to the deformability of the confinement leads to a variety of filament spatiotemporal organizations that have not been observed in bulk systems or under rigid confinement, including highly-aligned rings and caps. In turn, these filament assemblies drive dramatic and tunable transformations of the vesicle shape and its dynamics. We present simple scaling models that reveal the mechanisms underlying these emergent behaviors and yield design principles for engineering active materials with targeted shape dynamics.

In the manuscript entitled "Vesicle shape transformations driven by confined active filaments" the authors study how the shape of an elastic vesicle changes under the forces of contained self-driven filaments. The work seems to be carefully done, with apparently all the necessary attention to detailslike using overlapping beads to avoid a corrugated potential. The manuscript is excellently written, with a well ordered systematic explanation of observed phenomena. The subject of self propelled particles in general, and 'artificial cytoskeleton' are current hot topics of broad interest to physicists and biologists alike. To my knowledge, the presented results are new and exciting. The manuscript thus certainly deserves publication in nature communications.
I have just a few minor comments: -The authors should acknowledge more clearly the fact of violated momentum conservation. While for 2D Systems, the driving force can be generated against the substrate, the 3D Case (especially enclosed in a vesicle) is a bit more difficult. One might expect eg. Vesicles in the polar-prolate state to display net motion. My argument would be that the propulsion force creates a net reaction force. This reaction force gets transmitted via the other contents of the vesicle, to the vesicle membrane, in which case the vesicle would not move. Alternatively, if the filaments are propelled by external forces the vesicle would be propelled. -You write "Vves=4 pi R^3" Dont the vesicle beads also have excluded volume? Thus "Vves=4 pi (Rsigma/2)^3" ? -Please indicate fa=8 also in the Caption of Figure 1 -While I understand your usage of LJ-Units, I prefer to have them explicitly stated in all numbers. Eg. Rves=25sigma. In particular I think providing activity in fa is a bit misleading. In the field People seem to like the Peclet number and Flexure Number as an indicator of activity. Please state them.
-please provide a more accurate equation for the Persistence length of the path (page 3, top left) -The manuscript (almost) limits its study to the case of "strong confinement", or large Peclet numbers. This limitation should be made clear. E.g. I expect that the "independence of fa" found in fig3, would go away if even smaller fa where studied.
-Placement of Citation [64] suggests an explanation for tau propto L/v0, which it is not. Maybe place together with [62,63] -at some points (eg p4, bottom) you use gamma and lambda as symbols for surface tension. Because gamma is also used for friction, I suggest to consistently stick to lambda. -I did not find any data availability statement. I strongly encourage the authors to publish their data in an open access repository.
- Ref [37] indicates that the membrane friction vs monomer friction plays an important role. Can the authors comment?

0.
I understand that state-III translates and IV does not -but does IV spin?

1.
In Figure 3 (and the discussion of it), there is a threshold active force fa ≲ 1. However, the presented data seems to be only for fa ∈ {0, 1, 2}, with the fa = 1 mostly exhibiting the strong force behaviour (or at least non-spherical vesicles). Although it is certainly not the most exciting transition that the authors consider, it is in many ways the baseline and I'd like to see this nailed down better with a finer step size between 0 and 1.

2.
While reading your discussion of the competition between time scales, I kept wanting to think about the ratio as a Péclet number, but I realize that's not quite right. The time τrot isn't a rotational diffusion but is a propulsive timescale, ("timescale equal to the that over which the rod would have moved its own length"). On first reading of the manuscript, the rotational diffusion time scale appears to be ignored; however, on second reading it is including in the guise of persistence length and in the supplementary discussion as τcorr. I had to work through the aspect ratio for filaments in Figure 3 and compare it to aSC = 4.3 in the SI to be sure that it was negligible. I think the authors should include this third timescale in the discussion of τrot and τcoll.

3.
It is not clear to me how a* is found. It is shown in Figure 4 but the text states that a* = 130 is found from the data in Figure 1, which presumably is only a subset of the data shown in Figure 4 (the small value limit, perhaps?). This is currently unclear.
Minor Comments:

1.
I'm not sure whether or not it would be worth including in the manuscript, but I would be interested to see Figure 4 on a log-log graph.

2.
The first sentence of the abstract reads "... deformable boundaries provide a mechanism to organize internal active stresses and perform work on the external environment." Is the intent of this sentence that deformable boundaries act as the mechanism by which work is done on the external environment? I don't find this very clear and, besides, the manuscript isn't about the vesicles performing work on the external environment. I would like to see this sentence re-worked.

3.
In the introduction, the second half of a sentence reads "confining active particles leads to system-spanning effects" but the issue is that the beginning of the sentence cites "bands or flocks" as examples of phenomena driven by anisotropic interactions. But bands are system-spanning, in at least some sense. I feel this weakens the distinction the authors are trying to make.

4.
In the same sentence, it is written ". . . changing the length and stiffness of active polymers leads to dramatic reorganization of active stresses..." It seems to me that this point could be made stronger by citing the work of the Gompper group that considers active filaments. Similarly, I was surprised that the role of Frank elasticity in active nematics wasn't used to strengthen this argument.

5.
Similarly, I would have liked to see the authors compare and differentiate their work from that of studies of active nematic droplets. It's clear that they aren't the same (vesicle stiffness is not surface tension; self-propelled rods are not active nematic fluids); however, discussing the extent of these similarities and differences might be helpful.

6.
It is written ". . . simulations on infinitely rigid vesicles did not exhibit.. . " These can't really be infinitely rigid. Can this please be said more precisely?

Reviewer #3 (Remarks to the Author):
The authors report on the non-equilibrium shapes, and dynamics, of vesicles that enclose active semiflexible rod. The resulting self-organization of the rods, and the subsequent deformation of the membrane, is interesting. Its an extension to 3D of previous works of very similar nature that were performed in 2D. The change in dimensionality makes this study more relevant to understanding the shapes of biological cells, and artificial vesicles. In its current form I find the work interesting but incomplete in its present form.
2) The method for computing the vesicle shape is not given at all in the main text, it only says "elastic vesicle". I was under the impression it has only bending energy, but then in Fig.6 i find that it has a Young's modulus. This is not clear.
3) The bending modulus used in Fig.1 is not given in the caption, and similarly there is missing information in many later figures. 4) On page 3 we are sent to Fig.6b: What is the source of the feedback between the active filaments and vesicle flexibility that organizes the rotating ring ? Some suggestions and a deeper investigation of this feedback is needed. Is it that the furrow formed by the band "protects" it from running into bands moving in the opposite direction ? 5) The appearance of caps of finite size, which are dynamic, is reminiscent to the spontaneous size of the clusters of active proteins at the tips of protrusions, as shown in: 6) The vesicles in the present study seem not to allow for large deformations, such as narrow protrusions. It seems therefore that there are different classes of shapes that they did not explore here, either using softer vesicles, or larger forces. 7) The snap-shots in Fig.4: please explain which volume fraction or L_rod values have been changed to go from one to the other. 8) In Fig.5b it seems that large f_a inflates the vesicle, so could this increase in volume reduce the volume fraction enough to cause a transition from caps to disordered/uniform phase ? 9) In Fig.5b it seems that increase in f_a decreases the number of caps, and this is not described by eq.2. Can this be rationalized ? 10) The linear dependence on he 3D volume fraction phi in Eq.2 seems strange: intuitively the number of caps should depend on the areal density of rods on the vesicle membrane, so should depend on phi^(2/3) ? it will also fit much better the simulation data. 11) There is no dependence in Eq.2 on the bending modulus of the vesicle. How does it affect the number of caps ? See also the dependence of the size of the protrusions on the bending modulus in the reference given in point 5. 12) The bending modulus of the membrane should be written in Fig.6b. 13) What is the Young's modulus of the vesicle ? Why isnt the threshold of the FvK denoted on Fig.6a ? 14) In Fig.6a there are ordered caps in faceted vesicles, (volume fraction 0.05, low k_ves) which seem to change to a ring at high kappa. Why ? this is not clear. We thank the reviewers for their careful reading of the manuscript and constructive critiques. We have significantly revised the manuscript in response to these comments, and we believe that it now describes our results much more clearly and comprehensively. Especially important critiques were: Reviewer 1 asked us to clarify the description of our simulated dynamics in the main text and the behavior outside of the strong confinement limit, Reviewer 2 felt that the manuscript did not sufficiently explain the broad appeal of our study and in particular what is special about active systems in our context, and Reviewer 3 asked for a more in-depth discussion on the difference between our work and previous studies.
In response to these comments we have extensively rewritten the introduction to (1) more clearly explain the dynamics, its limitations, and the motivation for using a minimal model; (2) to explain the fundamental aspects of our study, including why the effects of internal stresses and boundary conditions are fundamentally different in an active material compared to equilibrium; and (3) we have included references that we overlooked in the previous version as well as a more detailed explanation of the important ways in which our study builds on that previous work. We have also added additional discussion in the results section comparing our findings to those of previous work. Further, we have added discussion and updated figures ( Fig. 3 and Fig. S2) explaining the low-activity behavior when the system falls outside of the strong confinement limit, and information about the dynamics and timescales associated with reaching steady state (new Fig. S4).
We present detailed responses to all critiques along with our revisions to the manuscript point-bypoint below, and we have included a version of the manuscript with tracked changes in the upload. We agree. Although our vesicles do move in the lab frame in many of the steady states, our study focuses on stress organization at a deformable boundary through contact interactions. For this purpose, the momentum conservation in the center of mass frame is irrelevant. For example, when considering microorganisms, it is typical to consider a stroke-averaged force dipole model (i.e., Refs. 57 and 58) or a squirmer model (i.e., Refs. 59 and 60). To capture the details of individual trajectories at walls, the hydrodynamics is important (i.e., Refs. 61 and 62). But when one seeks to understand organization at boundaries at high densities, the relevant phenomenology is robustly captured by self-propelled particle models (i.e., Refs. 63 and 64). It is in this spirit that we study this very simple model. Moreover, we have intentionally not focused on center of mass dynamics in our results, as we agree with the reviewer's point that these are unphysical and unreliable without proper momentum conservation and capturing the physics of a 3D wet system.

Reviewer #1 Comments
Modification: We have removed references to the vesicle motion in the results section of the manuscript, and now make this feature/limitation of this work more explicit in the revised manuscript in the second paragraph of the Methods section.
Indeed, both the vesicle and filament monomers have excluded volume. However, this is only a nominal volume; since the vesicle does not maintain a perfectly spherical shape, it might not be useful to be this precise in the available volume for the filaments. Additionally, this only changes the volume by an order O(R 2 ) correction, which is not a significant difference.

Modification:
We have added a sentence to the Methods section clarifying this point: Note that the volume fraction is defined with respect to the nominal volume of the undeformed vesicle, without considering the finite size of the filament and vesicle monomers.
3. Please indicate fa=8 also in the Caption of Figure 1.

Modification:
We have added the value of fa to the Fig. 1  While we agree that the flexure number is a useful parameter, we do not think that it is the natural control parameter for our system. In particular, as we note in the Supplemental Material (Sec. E), we find that the filament persistence length is renormalized by activity so that it scales as 1/Pe 2 , and therefore is not adequately captured by the flexure number. Instead, we define a rigidity parameter χ that scales with the passive filament persistence length, which leads to a more intuitive expression for the critical value of Pe below which the filaments are no longer in the strong confinement limit. We agree that at small values of fa we should see a dependence on the activity, this was stated in the main text of the original manuscript.

Modification:
In response to this comment, we have performed additional simulations and analysis at low values of the activity parameter and included this data in Fig. 3. As expected, there is a weak dependence on activity at low values of fa. We also show an additional snapshot in Fig. S2.

Placement of Citation [64] suggests an explanation for tau propto L/v0, which it is not.
Maybe place together with [62,63].
Thank you for pointing out this misplaced footnote.

At some points (eg p4, bottom) you use gamma and lambda as symbols for surface tension.
Because gamma is also used for friction, I suggest to consistently stick to lambda.

Modification:
We have corrected the inconsistent usage of lambda for the interfacial tension parameter.
9. I did not find any data availability statement. I strongly encourage the authors to publish their data in an open access repository.

Modification:
We have added data availability and code availability statements. We have posted the modified LAMMPs source code, all of the analysis scripts, and the data files used in the study on the Open Science Framework site. Because of the large size of the raw simulation trajectories (2 TB) we are making it available upon request rather than directly posting it on the Open Science Framework Site.

Ref [37] indicates that the membrane friction vs monomer friction plays an important role. Can the authors comment?
In Ref.
[37] (now Ref. [45]) they do not study or obtain conclusions about the effects of friction between membrane and filament monomers; rather, they find that friction between the membrane and the substrate for interface that the flexocyte is crawling on plays an important role. Varying membrane-filament friction could indeed be interesting, but given the already very large parameter space that characterizes our system, it is beyond the scope of the present article.

As stated above, I think that it is incumbent on the authors to better argue that their study is of broad interest to the readers of Nature
Communications. This is my principle concern -I enjoyed the reading the manuscript.

I was not convinced by paragraph three of the introduction, any more than I would say the same of any system. Yes, internal effects and boundary conditions determine the behaviour of systems -essentially all systems, not just active systems. That being said, I do certainly agree with the authors that there is something interesting in active systems (which often have selforganized scales) and their confinements (which introduce competitive scales). However, I wasn't convinced by the argument presented here. I think this might need to be re-posed. In some ways, I think this connects to my comment on the publishability: The science is strong but placing it in the most impactful context seems essential if it is to be accepted here.
Thank you for highlighting this. We agree that the original manuscript did not sufficiently explain what is special about active systems in our context, and in particular why a model that accounts for internal stress organization, boundary forces, and feedback between the two is of broad interest. We will break our response to this critique and the corresponding changes to the manuscript into two parts.
A. Why active systems are special. In the original version of the manuscript, we argued that there are two mechanisms for organizing active stress, inter-particle interactions that realign active forces and deformable boundaries that exert passive reaction stresses that further shape and organize active stress. As the reviewer points out, all systems experience effects due to internal stresses and boundaries. However, as also pointed out by the reviewer, in contrast to the systems traditionally studied in materials physics, active systems lack a separation of scales between correlations and observables. Consequently, the effects of boundaries are nonextensive, and can qualitatively change the macroscale behaviors of an active system. Moreover, the processes that lead to macroscale observables are statistical at every relevant scale in an active material. Energy is continuously input at the component-scale, which drives active forces that self-organize into mesoscale active stresses in a manner that depends on the local physical environment; in turn, these active stresses drive macroscale behaviors. Thus, the relationship between organization of internal stresses and macroscale observables fundamentally differs in active systems, in comparison to traditional materials.
Previous works that studied these effects independently have already established unique aspects of active materials. However, currently we know little about what happens when both mechanisms occur simultaneously and can couple with each other. Our study addresses this in the context of a minimal microscopic model.

Modification:
Based on the reviewer's critique, we have now modified the introduction to more clearly explain (while trying to maintain brevity) why the behaviors of internal stress organization and boundaries are fundamentally different in active systems, and why it is important to learn what happens when these effects couple to each other.
B. Even though active systems are special, why should someone outside of this immediate field care about coupling between internal stress organization and boundary forces? The case we make is the following: dynamical organization of active stress is the physical mechanism at play in diverse cellular processes, from cell division to endocytosis and motility. Thus, understanding this phenomenon is of fundamental interest both to cell biology and nonequilibrium statistical mechanics. Moreover, learning to harness and design such capabilities in biomimetic or synthetic systems would lead to transformative new applications, making this study of interest to disciplines across physics, chemistry, and materials science. However, identifying and understanding transferable design principles for the organization of active stresses is not achievable in the context of complex and faithful models of cytoskeletal dynamics. Thus, minimal mechanical models are essential for this exploration. Further, the complexity of the emergent behavior in this intrinsically nonequilibrium class of materials warrants developing models that allow us to ask specific questions. In our case, the question is: what are the physical mechanisms underlying shape transformations in vesicles deformed by active stresses?

Modification:
We have extensively modified the introduction to hopefully better emphasize these points.

In Methods, the simulations of the filaments are described in detail, but the vesicle is not described at all. I see that details are given in the SI but vesicle is on equal footing to the filaments in this study. What is the mesh? Is vesicle rigidity the same as the filaments? Is it implemented in the same way? Why can't filaments go through the pores in the mesh?
We agree that the description of the vesicle was underdeveloped in the main text.

Modification:
We have added the following paragraph to the Methods section that discusses the vesicle in much more detail: We agree that the dynamics of this system is an important aspect for understanding of the interplay between self-organizing active agents and deformable boundaries. However, given the large parameters set and the complexity of behaviors already present at steady state, we feel that analyzing the transient approach to steady-state falls beyond the scope of the current paper. Therefore, we have limited our focus on understanding the long-lived dynamical steadystates which will help set the foundation for future studies of this system that look at the dynamics of these states.
In general, we find that the dynamical steady states are very long-lived compared to the total simulation time, and reconfiguration events are quite rare.

Modification:
We have added an additional figure to the Supplemental Material (Fig. S4) that better shows this. We have additionally added the following text to SI Sec. F: The first 10% of the simulation is used for initialization, during which the filament positions and orientations are randomized. The final 10% of the simulation is used to analyze the resulting vesicle conformations and filament organizations. Note that the system reaches a long-lived steady state configuration very rapidly, and tends to exist in that state for nearly the entirety of the simulations (see Fig. S4).

I understand that state-III translates and IV does not -but does IV spin?
Yes, we find that all cap-forming states can spin. However, as pointed out by Reviewer 1, our system does not conserve momentum or account for hydrodynamic interactions. As a result, we have decided to de-emphasize the net vesicle motion as it may be confusing to readers.

Modification:
We have added the following text in the Results section: We note that states with net polarity can exhibit center-of-mass motion, but more comprehensive models that account for momentum conservation would be important to study such effects. Therefore, in what follows we focus on the internal motions and shape transformations of the vesicle.
6. In Figure 3 (and the discussion of it), there is a threshold active force fa ≲ 1. However, the presented data seems to be only for fa ∈ {0, 1, 2}, with the fa = 1 mostly exhibiting the strong force behaviour (or at least non-spherical vesicles). Although it is certainly not the most exciting transition that the authors consider, it is in many ways the baseline and I'd like to see this nailed down better with a finer step size between 0 and 1.
This is a good point (Please also see our response to Rev. 1 comment 6).

Modification:
We have performed additional simulations (shown in Fig. 3) that better showcase the weak dependence on activity for small values of fa. We have also placed an additional snapshot in Fig. S2 to make the "ragged caps" more apparent.
7. While reading your discussion of the competition between time scales, I kept wanting to think about the ratio as a Peclet number, but I realize that's not quite right. The time τrot isn't a rotational diffusion but is a propulsive timescale, ("timescale equal to the that over which the rod would have moved its own length"). On first reading of the manuscript, the rotational diffusion time scale appears to be ignored; however, on second reading it is including in the guise of persistence length and in the supplementary discussion as τcorr. I had to work through the aspect ratio for filaments in Figure 3 and compare it to aSC = 4.3 in the SI to be sure that it was negligible. I think the authors should include this third timescale in the discussion of τrot and τcoll.
We see that a lack of explicit mention of this timescale in the text was confusing in the previous version. Based on the observation that cap and ring formation does not occur outside of the strong confinement limit, and to simplify the presentation, we have assumed that the system is in the strong confinement limit during this discussion. This implies that the rates of reorientation and collision are very fast compared to the rate of rotational diffusion. While the assumption of the strong confinement limit was mentioned in the previous version, we agree that we should make its implication on the rotational timescale clearer.

Modification:
We have added the following text just before the timescale analysis: In the following discussion we assume the strong confinement limit; in particular, we assume that the timescale governing the rotational diffusion of the filaments, τcoll, is long compared to the other relevant timescales.
Modification: For additional clarity, we also add the following sentence after our timescale analysis: That is, we assume that the rotational diffusion timescale of the filaments, τcorr, is much larger than both τrot and τcoll. Figure 4 but the text states that a* = 130 is found from the data in Figure 1, which presumably is only a subset of the data shown in Figure  4 (the small value limit, perhaps?). This is currently unclear.

Modification:
We have now clarified in the caption how we estimated a*.
Specifically, the data in Fig. 4 consists of all data points from Fig. 1 in which the system forms rings or caps. To estimate a*, we fit a line to the data points with ncap <= 7 (we did not include data points with more caps because the identification and counting becomes unreliable).
9. I'm not sure whether or not it would be worth including in the manuscript, but I would be interested to see Figure 4 on a log-log graph.
We show here Fig. 4 on a log-log scale (removing the points corresponding to rings which have no caps, as they are not representable on a log scale). We have decided not to include this in the manuscript, as we think it doesn't show the trend any better than that linear scale plot, and it does not allow us to include the ring state.
10. The first sentence of the abstract reads ". . . deformable boundaries provide a mechanism to organize internal active stresses and perform work on the external environment." Is the intent of this sentence that deformable boundaries act as the mechanism by which work is done on the external environment? I don't find this very clear and, besides, the manuscript isn't about the vesicles performing work on the external environment. I would like to see this sentence reworked.
We agree that the intent of this sentence was unclear. Our original thought was that, in biological contexts, the active material is often contained within a deformable container, and thus the container acts as the mediator of interactions between active material and the external environment; however, as pointed out by the reviewer, this is tangential to the focus of this paper.

Modification:
We have removed the portion of this sentence that mentions performing work so that it more clearly emphasizes the role of the deformable boundary on the organization of active stresses, which is the main focus of this work.
11. In the introduction, the second half of a sentence reads "confining active particles leads to system-spanning effects" but the issue is that the beginning of the sentence cites "bands or flocks" as examples of phenomena driven by anisotropic interactions. But bands are systemspanning, in at least some sense. I feel this weakens the distinction the authors are trying to make.
The point of this sentence is not to say that confinement and anisotropic interactions are mutually exclusive in their ability to create system-spanning effects. In fact, it is exactly because both can serve as mechanisms of generating large-scale behaviors that we are interested in combining them in this work.

Modification:
We have modified the text in this paragraph to clarify our meaning.
12. In the same sentence, it is written ". . . changing the length and stiffness of active polymers leads to dramatic reorganization of active stresses. . . " It seems to me that this point could be made stronger by citing the work of the Gompper group that considers active filaments. Similarly, I was surprised that the role of Frank elasticity in active nematics wasn't used to strengthen this argument.

Modification:
We have added additional references from Gompper and co-workers here.
While we agree that Frank elasticity is very relevant in this context for bulk simulations of active filaments, it's not clear if there are well-defined Frank elastic constants for our system, due to the fact that it is dilute and undergoes a number of structural transformations. Hence, we have not explicitly mentioned it in this work in order to avoid confusion.
13. Similarly, I would have liked to see the authors compare and differentiate their work from that of studies of active nematic droplets. It's clear that they aren't the same (vesicle stiffness is not surface tension; self-propelled rods are not active nematic fluids); however, discussing the extent of these similarities and differences might be helpful.
This is a good point, we did not do a good job of emphasizing the novel features of our simulations in comparison to the previous studies on active droplets. We are only aware of a few studies of active droplets, which are now cited as Refs. 47 -50. These works provide great motivation for our study, but as pointed out by the reviewer the physics is quite different. Most importantly, the theoretical formulations used in these works necessarily require assumptions about the types of organization the active particles can undergo as well as the form of the hydrodynamic boundary conditions. In contrast, the filament organization and interaction with the membrane is an emergent property in our computational model, which as we show leads to filament organizations that could not be considered in these previous works.

Modification:
We now summarize these differences in the introduction: More closely related to our work are simulation studies of droplets containing active material that show tantalizingly life-like behaviors such as motility and division [47][48][49][50]. These elegant studies highlight the importance of understanding the types of emergent behaviors that arise when active matter and deformable boundaries are combined. However, the continuum hydrodynamic theories employed in these works require key assumptions about the nature of particle organization and particlemembrane interactions.
14. It is written ". . .simulations on infinitely rigid vesicles did not exhibit. . . " These can't really be infinitely rigid. Can this please be said more precisely?
We did not make this point sufficiently clear in the original version of the manuscript. Our infinitely rigid vesicle is indeed infinitely rigid; we implement it by fixing the positions of the vesicle monomers in their initial spherical configuration so that the active filaments are unable to deform the vesicle.

Modification:
We have added a sentence in the Supplemental Material (Sec. F) that clarifies this: Infinitely rigid vesicles are implemented by not integrating the equations of motion for the vesicle monomers so that they are always in their initial (spherical) configuration.

Modification:
We have added the value of fa to the Fig. 1 caption.