Author Response
eLife assessment
In this study, the authors offer a theoretical explanation for the emergence of nematic bundles in the actin cortex, carrying implications for the assembly of actomyosin stress fibers. As such, the study is a valuable contribution to the field actomyosin organization in the actin cortex. While the theoretical work is solid, experimental evidence in support of the model assumptions remains incomplete. The presentation could be improved to enhance accessibility for readers without a strong background in hydrodynamic and nematic theories.
To address the weaknesses identified in this assessment, we plan to expand the description of the theoretical model to make it more accessible to a broader spectrum of readers. We will discuss in more detail the relation between the different mathematical terms and physical processes at the molecular scale, as well as the experimental evidence supporting the model assumptions. We will also discuss more explicitly how our results are relevant to different systems exhibiting actomyosin nematic bundles beyond stress fibers.
Public Reviews:
Reviewer #1 (Public Review):
Summary:
In this article, Mirza et al developed a continuum active gel model of actomyosin cytoskeleton that account for nematic order and density variations in actomyosin. Using this model, they identify the requirements for the formation of dense nematic structures. In particular, they show that self-organization into nematic bundles requires both flow-induced alignment and active tension anisotropy in the system. By varying model parameters that control active tension and nematic alignment, the authors show that their model reproduces a rich variety of actomyosin structures, including tactoids, fibres, asters as well as crystalline networks. Additionally, discrete simulations are employed to calculate the activity parameters in the continuum model, providing a microscopic perspective on the conditions driving the formation of fibrillar patterns.
Strengths:
The strength of the work lies in its delineation of the parameter ranges that generate distinct types of nematic organization within actomyosin networks. The authors pinpoint the physical mechanisms behind the formation of fibrillar patterns, which may offer valuable insights into stress fiber assembly. Another strength of the work is connecting activity parameters in the continuum theory with microscopic simulations.
We thank the referee for these comments.
Weaknesses:
This paper is a very difficult read for nonspecialists, especially if you are not well-versed in continuum hydrodynamic theories. Efforts should be made to connect various elements of theory with biological mechanisms, which is mostly lacking in this paper. The comparison with experiments is predominantly qualitative.
We agree with the referee that the manuscript will benefit from a better description of the theoretical model and the results in relation with specific molecular and cellular mechanisms. We will further emphasize how a number of experimental observations in the literature support our model assumptions and can be explained by our results. A quantitative comparison is difficult for several reasons. First, many of the parameters in our theory have not been measured, and in fact estimates in the literature often rely on comparison with hydrodynamic models such as ours. Second, the effective physical properties of actomyosin gels can vary wildly between cells, which may explain the diversity of forms, dynamics and functions. For these reasons, we chose to delineate regimes leading to qualitatively different emerging architectures and dynamics. In the revised manuscript, we will make this point clearer and will further study the literature to seek quantitative comparison.
It is unclear if the theory is suited for in vitro or in vivo actomyosin systems. The justification for various model assumptions, especially concerning their applicability to actomyosin networks, requires a more thorough examination.
We thank the referee for this comment. Our theory is applicable to actomyosin in living cells. To our knowledge, reconstituted actomyosin gels currently lack the ability to sustain the dynamical steady-states involved in the proposed self-organization mechanism, which balance actin flows with turnover. In addition to actomyosin gels in living cells, in vitro systems based on encapsulated cell extracts can also sustain such dynamical steady states [e.g. https://doi.org/10.1038/s41567-018-0413-4], and therefore our theory may be applicable to these systems as well. Of course, with advancements in the field of reconstituted systems, this may change in the near future. We will explicitly discuss this point in the revised manuscript.
The classification of different structures demands further justification. For example, the rationale behind categorizing structures as sarcomeric remains unclear when nematic order is perpendicular to the axis of the bands. Sarcomeres traditionally exhibit a specific ordering of actin filaments with alternating polarity patterns.
We agree and will avoid the term “sarcomeric”.
Similarly, the criteria for distinguishing between contractile and extensile structures need clarification, as one would expect extensile structures to be under tension contrary to the authors' claim.
We plan to clarify this point by representing in a main figure the stress profiles across dense nematic structures (currently in Supp Fig 2), along with a more detailed description. In short, depending on the parameter regime, the competition between active and viscous stresses in the actin gel determine whether the emergent structures are extensile or contractile. In our system tension is positive in all directions at all times. However, in “contractile” structures, tension is larger along the bundle, whereas in “extensile” structures, tension is larger perpendicular to the bundle. This is consistent with the common expression for active stress of incompressible nematic systems [see e.g. https://doi.org/10.1038/s41467-018-05666-8], that takes the form –zQ, where z is positive for an extensile system, showing that in this case active tension is negative along the nematic direction. This point, also been raised by another referee, will be clarified and connected to existing literature.
Additionally, its unclear if the model's predictions for fiber dynamics align with observations in cells, as stress fibers exhibit a high degree of dynamism and tend to coalesce with neighboring fibers during their assembly phase.
In the present work, we focus on the self-organization of a periodic patch of actomyosin gel. However, in adherent cells boundary conditions play an essential role, e.g. with inflow at the cell edge as a result of polymerization and exclusion at the nucleus. In ongoing work, we are studying with the present model the dynamics of assembly and reconfiguration of dense nematic structures in domains with boundary conditions mimicking in adherent cells, as suggested by the referee. We would like to note, however, that the prominent stress fibers in cells adhered to stiff substrates, so abundantly reported in the literature, are not the only instance of dense nematic actin bundles, and may not be representative of physiologically relevant situations. In the present manuscript, we emphasize the relation of the predicted organizations with those found in different in vivo contexts not related to stress fibers, such as the aligned patterns of bundles in insects (trachea, scales in butterfly wings), in hydra, or in reproductive organs of C elegans; the highly dynamical network of bundles observed in C elegans early embryos; or the labyrinth patters of micro-ridges in the apical surface of epidermal cells in fish. We will further emphasize these points in the revised manuscript.
Finally, it seems that the microscopic model is unable to recapitulate the density patterns predicted by the continuum theory, raising questions about the suitability of the simulation model.
We thank the referee for raising this question, which needs further clarification. The goal of the microscopic model is not to reproduce the self-organized patterns predicted by the active gel theory. The microscopic model lacks essential ingredients, notably a realistic description of hydrodynamics and turnover. Our goal with the agent-based simulations is to extract the relation between nematic order and active stresses for a small homogeneous sample of the network. This small domain is meant to represent the homogeneous active gel prior to pattern formation, and it allows us to substantiate key assumptions of the continuum model leading to pattern formation, notably the dependence of isotropic and deviatoric components of the active stress on density and nematic order (Eq. 7) and the active generalized stress promoting ordering.
We should mention that reproducing the range of out-of-equilibrium mesoscale architectures predicted by our active gel model with agent-based simulations seems at present not possible, or at least significantly beyond the state-of-the-art. We note for instance that parameter regimes in which agent-based simulations of actin gels display extended contractile steady-states are non-generic, as these simulations often lead to irreversible clumping (as do many reconstituted contractile systems), see e.g. https://doi.org/10.1038/ncomms10323 or https://doi.org/10.1371/journal.pcbi.1005277. Very few references report sustained actin flows or the organization of a few bundles
(https://doi.org/10.1371/journal.pcbi.1009506). While agent-based cytoskeletal simulations are very attractive because they directly connect with molecular mechanisms, active gel continuum models are better suited to describe out-ofequilibrium emergent hydrodynamics at a mesoscale. We believe that these two complementary modeling frameworks are rather disconnected in the literature, and for this reason, we have attempted substantiate our continuum modeling with discrete simulations. In the revised manuscript, we will better frame the relationship between them.
Reviewer #2 (Public Review):
Summary:
The article by Waleed et al discusses the self organization of actin cytoskeleton using the theory of active nematics. Linear stability analysis of the governing equations and computer simulations show that the system is unstable to density fluctuations and self organized structures can emerge. While the context is interesting, I am not sure whether the physics is new. Hence I have reservations about recommending this article.
We thank the referee for these comments. In the revised manuscript, we will highlight the novelty of the paper in terms of the theoretical model, the mechanism of patterning of dense nematic structures, the nature and dynamics of the resulting architectures, their relation with the experimental record, and the connection with microscopic models.
We will emphasize the fact that nematic architectures in the actin cytoskeleton are characterized by a co-localization of order and density (and strong variations in each of these fields), that recent work shows that isotropic and nematic organizations coexist and are part of a single heterogeneous network, that the emergence and maintenance of nematic order requires active contraction, and that the assembly and maintenance of dense nematic bundles involves convergent flows. None of these key features can be described by the common incompressible models of active nematics. To address this, we develop here a compressible and density dependent model for an active nematic gel. We will carefully justify that the proposed model is meaningful for actomyosin gels, and we will highlight the commonalities and differences with previous models of active nematics.
Strengths:
(i) Analytical calculations complemented with simulations (ii) Theory for cytoskeletal network
Weaknesses:
Not placed in the context or literature on active nematics.
We agree with the referee that the manuscript requires a better contextualization of the work in relation with the very active field of active nematics. In the revised manuscript, we will clearly describe the relation of our model with existing ones.
Reviewer #3 (Public Review):
The manuscript "Theory of active self-organization of dense nematic structures in the actin cytoskeleton" analysis self-organized pattern formation within a two-dimensional nematic liquid crystal theory and uses microscopic simulations to test the plausibility of some of the conclusions drawn from that analysis. After performing an analytic linear stability analysis that indicates the possibility of patterning instabilities, the authors perform fully non-linear numerical simulations and identify the emergence of stripelike patterning when anisotropic active stresses are present. Following a range of qualitative numerical observations on how parameter changes affect these patterns, the authors identify, besides isotropic and nematic stress, also active self-alignment as an important ingredient to form the observed patterns. Finally, microscopic simulations are used to test the plausibility of some of the conclusions drawn from continuum simulations.
The paper is well written, figures are mostly clear and the theoretical analysis presented in both, main text and supplement, is rigorous. Mechano-chemical coupling has emerged in recent years as a crucial element of cell cortex and tissue organization and it is plausible to think that both, isotropic and anisotropic active stresses, are present within such effectively compressible structures. Even though not yet stated this way by the authors, I would argue that combining these two is of the key ingredients that distinguishes this theoretical paper from similar ones. The diversity of patterning processes experimentally observed is nicely elaborated on in the introduction of the paper, though other closely related previous work could also have been included in these references (see below for examples).
We thank the referee for these comments and for the suggestion to emphasize the interplay of isotropic and anisotropic active tension, which is possible only in a compressible gel. We thank the suggestions of the referee to better connect with existing literature.
To introduce the continuum model, the authors exclusively cite their own, unpublished pre-print, even though the final equations take the same form as previously derived and used by other groups working in the field of active hydrodynamics (a certainly incomplete list: Marenduzzo et al (PRL, 2007), Salbreux et al (PRL, 2009, cited elsewhere in the paper), Jülicher et al (Rep Prog Phys, 2018), Giomi (PRX, 2015),...). To make better contact with the broad active liquid crystal community and to delineate the present work more compellingly from existing results, it would be helpful to include a more comprehensive discussion of the background of the existing theoretical understanding on active nematics. In fact, I found it often agrees nicely with the observations made in the present work, an opportunity to consolidate the results that is sometimes currently missed out on. For example, it is known that self-organised active isotropic fluids form in 2D hexagonal and pulsatory patterns (Kumar et al, PRL, 2014), as well as contractile patches (Mietke et al, PRL 2019), just as shown and discussed in Fig. 2. It is also known that extensile nematics, \kappa<0 here, draw in material laterally of the nematic axis and expel it along the nematic axis (the other way around for \kappa>0, see e.g. Doostmohammadi et al, Nat Comm, 2018 "Active Nematics" for a review that makes this point), consistent with all relative nematic director/flow orientations shown in Figs. 2 and 3 of the present work.
We thank the referee for these suggestions. Indeed, in the original submission we had outsourced much of the justification of the model and the relevant literature to a related pre-print, but this is not reasonable. In the revised manuscript, we will discuss our model in the context of the state-of-the-art, emphasizing connections with existing results.
The results of numerical simulations are well-presented. Large parts of the discussion of numerical observations - specifically around Fig. 3 - are qualitative and it is not clear why the analysis is restricted to \kappa<0. Some of the observations resonate with recent discussions in the field, for example the observation of effectively extensile dynamics in a contractile system is interesting and reminiscent of ambiguities about extensile/contractile properties discussed in recent preprints
(https://arxiv.org/abs/2309.04224). It is convincingly concluded that, besides nematic stress on top of isotropic one, active self-alignment is a key ingredient to produce the observed patterns.
We thank the referee for these comments. We will expand the description of the results around Figure 3. We are reluctant to extend the detailed analysis of emergent architectures and dynamics to the case \kappa > 0 as it leads to architectures not observed, to our knowledge, in actin networks. We will expand the characterization of emergent contractile/extensile networks by describing the distribution of the different components of the stress tensor across the bundles and will place our results in the context of related recent work.
I compliment the authors for trying to gain further mechanistic insights into this conclusion with microscopic filament simulations that are diligently performed. It is rightfully stated that these simulations only provide plausibility tests and, within this scope, I would say the authors are successful. At the same time, it leaves open questions that could have been discussed more carefully. For example, I wonder what can be said about the regime \kappa>0 (which is dropped ad-hoc from Fig. 3 onward) microscopically, in which the continuum theory does also predict the formation of stripe patterns - besides the short comment at the very end? How does the spatial inhomogeneous organization the continuum theory predicts fit in the presented, microscopic picture and vice versa?
We thank the referee for this compliment. We think that the point raised by the referee is very interesting. It is reasonable to expect that the sign of \kappa will not be a constant but rather depend on S and \rho. Indeed, for a sparse network with low order, the progressive bundling by crosslinkers acting on nearby filaments is likely to produce a large active stress perpendicular to the nematic direction, whereas in a dense and highly ordered region, myosin motors are more likely to effectively contract along the nematic direction whereas there is little room for additional lateral contraction by additional bundling. In the revised manuscript, we envision to further deepen in this issue in two ways. First, we plan to perform additional agent-based simulations in a regime leading to kappa > 0. Second, we will modify the active gel model such that kappa < 0 for low density/order, so that a fibrillar pattern is assembled, and kappa > 0 for high density/order, so that the emergent fibers are highly contractile.
Overall, the paper represents a valuable contribution to the field of active matter and, if strengthened further, might provide a fruitful basis to develop new hypothesis about the dynamic self-organisation of dense filamentous bundles in biological systems.
