Cyclic muscle contractions reinforce the acto-myosin motors and mediate the full elongation of C. elegans embryo

Reviewer #1 (Public Review):

Summary:
The authors have made a novel and important effort to distinguish and include different sources of active deformations for fitting C elegans embryo development: cyclic muscle contractions and actomyosion circumferential stresses. The combination and synchronisation of both contributions are, according to the model, responsible for different elongation rates, and can induce bending and torsion deformations, which are a priori not expected from purely contractile forces. The model can be applied to other growth processes in initially cylindrical shapes.

Strengths:
The model allows us to fit and deduce specific growth patterns, frequencies, and locations of contractions that yield the observed axial elongation during the 240 min of the studied process.

The deformation gradient is decomposed according to muscle and actomyosin activity, which can be distinguished and quantified. An energy-transferring process allows for the retrieval of the necessary permanent deformations that embryo development requires.

Weaknesses:
Despite the completeness of the model, the explanation of the methodology needs to be improved. Parameters and quantities are not always explained in the main text and are introduced on some occasions in an ordered manner. This makes the comprehension and deduction of methodology difficult. There are some minor comments that are listed below. The most important points are:

-How are the authors sure that there is a torsional deformation? Without tracking the muscle fibers, bending with respect to different angles for different Zs may yield a shape similar to the one in Figure 6E. Furthermore, it is unclear why the model yields torsion deformation. If material points of actomyosin rings do not change in reference configuration, no helicoidal growth should be happening.

-The triple decomposition F=F_e*G_i*G_0 seems to complicate the expressions of growth and requires the use of angles alpha and beta due to the initial deformation G_0. Why not use a simpler decomposition F=F_e*G, where G contains all contributions from actomyosin and muscle contractions in a material frame? This would avoid considering angles alpha and beta.

The section "Energy transformation and Elongation" is unclear. Indeed, stresses need to relax, otherwise, the removal of muscle and actin activity would send the embryo back to its initial state. However, the rationale behind the energy transfer is not explained. Authors seem to impose W_c=W_r, and from this deduce the necessary actin contraction after muscle relaxation. Why should energy be maintained when muscle relaxes? Which mechanism physically imposes this energy transfer? Muscle contraction could indeed induce elongation if traction forces at the opposite side of the contracting muscle relax. In fact, an alternative approach for obtaining stress relaxation and axial elongation would be converting part of the elastic deformation F_e to a permanent deformation F_p.

-Self contact is ignored. This may well be a shape generator and responsible for bending deformations. The convoluted shape of the embryo in the confined space deserves at least commenting on this limitation of the model.

Reviewer #2 (Public Review):

Summary:
During C. elegans development, embryos undergo elongation of their body axis in the absence of cell proliferation or growth. This process relies in an essential way on periodic contractions of two pairs of muscles that extend along the embryo's main axis. How contraction can lead to extension along the same direction is unknown.

To address this question, the authors use a continuum description of a multicomponent elastic solid. The various components are the interior of the animal, the muscles, and the epidermis. The different components form separate compartments and are described as hyperelastic solids with different shear moduli. For simplicity, a cylindrical geometry is adopted. The authors consider first the early elongation phase, which is driven by contraction of the epidermis, and then late elongation, where contraction of the muscles injects elastic energy into the system, which is then released by elongation. The authors get elongation that can be successfully fitted to the elongation dynamics of wild-type worms and two mutant strains.

Strengths:
The work proposes a physical mechanism underlying a puzzling biological phenomenon. The framework developed by the authors could be used to explain phenomena in other organisms and could be exploited in the design of soft robots.

Weaknesses:

1. This reviewer considers that the quality of the writing is poor. Because of this the main result of this work, how elongation is achieved by contraction, remains unclear to me. In the opinion of this reviewer, the work is not accessible to a biologist. This is a real pity because the findings are potentially of great interest to developmental biologists and engineers alike.

2. The authors assume that the embryo is elastic throughout all stages of development. Is this assumption appropriate? In my opinion, the authors need to critically discuss this assumption and provide justification. Would this still be true for the adult? If so could the adult relax back to the state prior to elongation? The embryo should be able to do that, if the contractility of the epidermis were sufficiently reduced, right?

3. The authors impose strains rather than stress. Since they want to understand the final deformation, I find this surprising. Maybe imposing strain or stress is equivalent, but then you should discuss this.

4. Does your mechanism need 4 muscle strands or would 2 be sufficient?

5. It is sometimes hard to understand, whether the authors are talking about the model or the worm.

Author Response

Reviewer 1 (Public Review)

Summary: The authors have made a novel and important effort to distinguish and include different sources of active deformations for fitting C elegans embryo development: cyclic muscle contrac- tions and actomyosion circumferential stresses. The combination and synchronisation of both contributions are, according to the model, responsible for different elongation rates, and can in- duce bending and torsion deformations, which are a priori not expected from purely contractile forces. The model can be applied to other growth processes in initially cylindrical shapes.

Strengths: The model allows us to fit and deduce specific growth patterns, frequencies, and lo- cations of contractions that yield the observed axial elongation during the 240 min of the studied process.

The deformation gradient is decomposed according to muscle and actomyosin activity, which can be distinguished and quantified. An energy-transferring process allows for the retrieval of the nec- essary permanent deformations that embryo development requires.

Weaknesses: Despite the completeness of the model, the explanation of the methodology needs to be improved. Parameters and quantities are not always explained in the main text and are intro- duced on some occasions in an ordered manner. This makes the comprehension and deduction of methodology difficult. There are some minor comments that are listed below. The most important points are:

How are the authors sure that there is a torsional deformation? Without tracking the muscle fibers, bending with respect to different angles for different Zs may yield a shape similar to the one in Figure 6E. Furthermore, it is unclear why the model yields torsion deformation. If material points of actomyosin rings do not change in reference configuration, no helicoidal growth should be happening.

Our torsional deformations were obtained computationally, and the results are plotted in Figure 6 according to our formalism. In our approach, the torsional deformation results from the interaction between the vertical muscles and the circumferential actin network: the muscles bend the cylinder and the bending modifies the direction of the actin fibers, as demonstrated in the experiment.

-The triple decomposition 𝐹 = 𝐹𝑒 ⋅ 𝐺𝑖 ⋅ 𝐺0 seems to complicate the expressions of growth and requires the use of angles alpha and beta due to the initial deformation 𝐺0. Why not use a simpler decomposition 𝐹 = 𝐹𝑒 ⋅ 𝐺, where 𝐺 contains all contributions from actomyosin and muscle contrac- tions in a material frame? This would avoid considering angles alpha and beta.

𝐺0 represents the active strain during the early elongation stage and 𝐺𝑖 during the late elongation stage respectively. Such a decomposition which is not mandatory, allows a better un- derstanding. In addition, due to the late elongation stage, both muscle and actin networks must be considered, and their orientation changes with deformation. Therefore, it is clearer and simpler to express the active strain in terms of alpha and beta angles.

The section "Energy transformation and Elongation" is unclear. Indeed, stresses need to relax, oth- erwise, the removal of muscle and actin activity would send the embryo back to its initial state. How- ever, the rationale behind the energy transfer is not explained. Authors seem to impose 𝑊𝑐 = 𝑊𝑟, and from this deduce the necessary actin contraction after muscle relaxation. Why should energy be maintained when muscle relaxes? Which mechanism physically imposes this energy transfer? Muscle contraction could indeed induce elongation if traction forces at the opposite side of the contracting muscle relax. In fact, an alternative approach for obtaining stress relaxation and axial elongation would be converting part of the elastic deformation 𝐹𝑒 to a permanent deformation 𝐹𝑝.

In this section, we do assume that all the energy accumulated by the muscle contrac- tions will be converted into the energy necessary for elongation, and as our estimate in the article shows, 𝑊𝑐 is indeed greater than 𝑊𝑟, indicating that a significant fraction of 𝑊𝑐 is converted into dissipation and friction, but also into the reorganization of the actin cables. Indeed, elongation of the cylinder induces a significant reduction in the experimentally observed and also in the actin cable density. However, this reduction in cable density is not observed experimentally. Thus, elon- gation requires a reorganization of the actin network, which is part of the energy consumption and which explains the existence of a permanent deformation 𝐹𝑝.

Self contact is ignored. This may well be a shape generator and responsible for bending deforma- tions. The convoluted shape of the embryo in the confined space deserves at least commenting on this limitation of the model.

Thank you for your suggestion. We have considered the effect of contact between C. elegans and the eggshell in the energy dissipation section but we also agree that the self-contact of the worm in confinement will be important. Here, we focus mainly on active filaments: actomyosin and muscle, and we restrict ourselves to a cylindrical shell that is far from the embryo.

Reviewer 2 (Public Review)

Summary

During C. elegans development, embryos undergo elongation of their body axis in the absence of cell proliferation or growth. This process relies in an essential way on periodic contractions of two pairs of muscles that extend along the embryo’s main axis. How contraction can lead to extension along the same direction is unknown.

To address this question, the authors use a continuum description of a multicomponent elastic solid. The various components are the interior of the animal, the muscles, and the epidermis. The different components form separate compartments and are described as hyperelastic solids with different shear moduli. For simplicity, a cylindrical geometry is adopted. The authors consider first the early elongation phase, which is driven by contraction of the epidermis, and then late elongation, where contraction of the muscles injects elastic energy into the system, which is then released by elongation. The authors get elongation that can be successfully fitted to the elongation dynamics of wild-type worms and two mutant strains.

Strengths

The work proposes a physical mechanism underlying a puzzling biological phenomenon. The framework developed by the authors could be used to explain phenomena in other organisms and could be exploited in the design of soft robots.

Weaknesses

1. This reviewer considers that the quality of the writing is poor. Because of this the main result of this work, how elongation is achieved by contraction, remains unclear to me. In the opinion of this reviewer, the work is not accessible to a biologist. This is a real pity because the findings are potentially of great interest to developmental biologists and engineers alike.

We regret that, despite a general introduction and a number of figures, the work does not seem accessible to biologists.

1. The authors assume that the embryo is elastic throughout all stages of development. Is this assumption appropriate? In my opinion, the authors need to critically discuss this assumption and provide justification. Would this still be true for the adult? If so could the adult relax back to the state prior to elongation? The embryo should be able to do that, if the contractility of the epidermis were sufficiently reduced, right?

Soft tissues are elastic, the modeling of soft tissues, even with large deformations, is now well established. The difference between a worm embryo and an adult is first of all the quality of the tissues, their low degree of heterogeneity, the weakness of the muscles and the absence of bones. As for the question of complete relaxation of the stresses, the fact that different components are attached to each other limits complete relaxation. We keep our fingerprints and cortical undula- tions, although they originate from an elastic instability that occurs in fetal life. It never disappears.

The authors impose strains rather than stress. Since they want to understand the final deformation, I find this surprising. Maybe imposing strain or stress is equivalent, but then you should discuss this.

Perhaps, the referee has in mind the question of active strain versus active stress and is concerned about the representation of biological forces such as those produced by actomyosin or muscle. In fact, both exist in morphoelasticity and are, of course, related. Usually, the choice is dictated by the simplicity of deriving quantitative results for comparison with experiments.

1. Does your mechanism need 4 muscle strands or would 2 be sufficient?

First, the 4 muscle strands are consistent with real C. elegans structures, and second, although we assume that two muscles on the same side contract simultaneously, their size and position affect the deformation results. Also, the time period we consider is just before the worm hatches. After that, the worm has to slide on the ground. So efficient muscles are needed.

1. It is sometimes hard to understand, whether the authors are talking about the model or the worm.

It will be corrected in the new version.