The early elongation of the C. elegans embryo has been previously studied, which is characterized by an inner cylinder surrounded by epithelial cells located in the cortical position (Ben Amar et al., 2018). The cortex is composed of three distinct cell types - the seam, dorsal, and ventral cells - that exhibit a unique cytoskeletal organization and actin network configuration. Among these, only the seam cells possess active myosin motors that function in the ortho-radial direction, allowing for the circumferential contraction, and thereby triggering the early elongation process. In this study, we simplify the cylindrical geometry of the body and treat the epidermis as a unified whole with effective activity localized along the circumference, supported by four muscles distributed beneath the epidermis. The muscles do not play a role in the early elongation phase and have, therefore, not been considered previously (Vuong-Brender et al., 2017a; Ben Amar et al., 2018). After this period of approximately a 1.8-fold increase in length, muscles parallel to the main axis and actin bundles organized along the circumference will cooperate to drive further elongation. Table 1 provides the size parameters of C. elegans that will be used hereafter.

We focus on the total deformation of a full cylinder or a thin rod with a length ${\displaystyle L}$ greater than the radius ${\displaystyle R}$ and a central vertical axis along the ${\displaystyle Z}$ direction. Contrary to previous works (Ciarletta et al., 2009; Ben Amar et al., 2018), here we decide to simplify the geometrical aspect due to the mechanical complexity. The biological activity induced by the actomyosin network and muscles is represented by active strains, and the global shape is the result of the coupling between elastic and active strains, modulated by dissipation. Active strains are generated by non-mechanical processes (e.g. biochemical processes): myosin motors and striated muscles derive their energy from ATP hydrolysis, which is converted into mechanical fiber contractions. Due to the significant deformation observed, the central line is distorted and becomes a curve in three-dimensional space, represented by a vector ${\displaystyle {\displaystyle \mathit{r}(\mathit{Z})}}$, as depicted in Figure 3A. Along this curve, perpendicular planar sections of the embryo can be defined, and the deformation in each section can be quantified since the circular geometry is lost with the contractions (Kaczmarski et al., 2022). The geometric mapping is as follows:

where the ${\displaystyle a_i}$ represents the deformation in each direction of the section with ${\displaystyle {\displaystyle a_i\left(0,\mathrm{\Theta },Z\right)=0}}$ so that the ${\displaystyle Z}$-axis is mapped to the centerline ${\displaystyle {\displaystyle \mathit{r}\left(Z\right)}}$. The small quantity ${\displaystyle \epsilon }$ is the ratio between the radius ${\displaystyle R}$ and the length ${\displaystyle L}$ of the cylinder. The axial extension ${\displaystyle \zeta }$ is given by ${\displaystyle {\displaystyle {\mathit{r}}^{\mathrm{\prime }}\left(Z\right)=\zeta {\mathit{d}}_3}}$, where ′ denotes the first derivative with respect to the material coordinate ${\displaystyle Z}$. From the director basis, the Darboux curvature vector reads: ${\displaystyle {\displaystyle \mathit{u}=u_1{\mathit{d}}_1+u_2{\mathit{d}}_2+u_3{\mathit{d}}_3}}$, this vector gives the evolution of the director basis along the filamentary line as: ${\displaystyle {\displaystyle {{\mathit{d}}_i}^{\mathrm{\prime }}\left(Z\right)=\zeta \mathit{u}\times {\mathit{d}}_i}}$, see Figure 3A. Based on the model proposed by Kaczmarski et al., 2022, we define the initial configuration ${\displaystyle {\mathcal{B}}_0}$ with material points ${\displaystyle {\displaystyle \left(R,\mathrm{\Theta },Z\right)}}$, and the mapping function ${\displaystyle {\displaystyle \chi \left(\mathit{X}\right)}}$ connects the initial configuration ${\displaystyle {\mathcal{B}}_0}$ to the current configuration ${\displaystyle \mathcal{B}}$. The geometric deformation gradient is ${\displaystyle {\displaystyle \mathit{F}=}}$ Grad ${\displaystyle {\displaystyle \chi ={\mathit{F}}_{\mathit{e}}\mathit{G}}}$ (Rodriguez et al., 1994; Nardinocchi and Teresi, 2007), where ${\displaystyle {\displaystyle \mathit{G}}}$ is the active strain generated by the actomyosin or the muscles, and ${\displaystyle {\displaystyle {\mathit{F}}_e}}$ is the elastic strain tensor.

Figure 3

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Because of the two stages through which the C. elegans elongates, we need to evaluate the influence of the C. elegans actin network during the early elongation before studying the deformation at the late stage. Thus, the deformation gradient can be decomposed into: ${\displaystyle {\displaystyle \mathit{F}={\mathit{F}}_e{\mathit{G}}_1{\mathit{G}}_0}}$ (Goriely and Ben Amar, 2007) where ${\displaystyle {\displaystyle {\mathit{G}}_0}}$ refers to the pre-strain of the early period and ${\displaystyle {\displaystyle {\mathit{G}}_1}}$ is the muscle supplementary active strain in the late period. Actin is distributed in a circular pattern in the outer epidermis, see Figure 3B, so the finite strain ${\displaystyle {\mathbf{G}}_0}$ is defined as ${\displaystyle {\displaystyle {\mathit{G}}_0=Diag\left(1,g_0(t),1\right)}}$, where ${\displaystyle {\displaystyle 0<g_0<1}}$ is the time-dependent decreasing eigenvalue, operating in the actin zone, and is equal to unity in the other parts: ${\displaystyle g_0(0)=1}$. In the case without pre-strain, ${\displaystyle {\displaystyle {\mathit{G}}_0=\mathit{I}}}$. The deformation gradient follows the description of Equation 1: ${\displaystyle {\displaystyle \mathit{F}=F_{ij}{\mathit{d}}_i\otimes {\mathit{e}}_j,where}}$ and ${\displaystyle {\displaystyle j\in \left\{R,\mathrm{\Theta },Z\right\}}}$.

Considering a filamentary structure Figure 3(B) with different fiber directions ${\displaystyle {\displaystyle \mathit{m}}}$, these directions are specified by two angles ${\displaystyle \alpha }$ and ${\displaystyle \beta }$, as outlined in Holzapfel, 2000: ${\displaystyle {\displaystyle \mathit{m}=\sin \alpha \sin \beta {\mathit{e}}_{\mathit{R}}+\sin \alpha \cos \beta {\mathit{e}}_{\mathbf{\Theta }}+\cos \alpha {\mathit{e}}_{\mathit{Z}},\alpha ,\beta \in \left[-\pi /2,\pi /2\right]}}$. For muscle fibers, ${\displaystyle {\displaystyle {\alpha }_m=0}}$ and ${\displaystyle {\displaystyle {\beta }_m=0}}$, while for hoop fibers in the actin network, ${\displaystyle {\alpha }_a=\pi /2}$ and ${\displaystyle {\beta }_a=0}$. When the muscles are activated and bend the embryo, the actin fibers are tilted with a tilt ${\displaystyle \alpha }$, such that ${\displaystyle {\displaystyle -\pi /2<{\alpha }_a<\pi /2}}$, ${\displaystyle {\alpha }_a\ne 0}$, and ${\displaystyle {\beta }_a=0}$. Each active strain is represented by a tensor ${\displaystyle {\displaystyle {\mathit{G}}_i=\mathit{I}+\epsilon g_i{\mathit{m}}_{\mathit{i}}\mathbf{\otimes }{\mathit{m}}_{\mathit{i}}}}$, where ${\displaystyle g_i}$ represents the activity (${\displaystyle g_m}$ for muscles and ${\displaystyle g_a}$ for actomyosin), and since both are contractile, their incremental activities are negative. Additional calculation details are provided in section Methods and materials.

To relate the deformations to the active forces induced by the muscles and the actomyosin network, we assume that the embryo can be represented by the simplest nonlinear hyperelastic model, called Neo-Hookean, with a strain energy density given by ${\displaystyle {\displaystyle W\left({\mathit{F}}_e\right)=\frac{\mu }{2}\left(tr\left({\mathit{F}}_e{{\mathit{F}}_e}^T\right)-3\right)}}$. In cylindrical coordinates, the total energy of the system, and the associated energy density is:

where ${\displaystyle J=det{\mathbf{F}}_e}$, ${\displaystyle p}$ is a Lagrange multiplier that ensures the incompressibility of the sample, a physical property assumed in living matter. If the cylinder contains several layers with different shear modulus μ and different active strains, the integral over ${\displaystyle S}$ covers each layer. To minimize the energy over each section for a given active force, we take advantage of the small value of ${\displaystyle \epsilon }$ and expand the inner variables ${\displaystyle a}$, ${\displaystyle p}$, and the potential ${\displaystyle V}$ to obtain the associated strain-energy density. Since the Euler-Lagrange equations and the boundary conditions are satisfied at each order, we can obtain solutions for the elastic strains at zero order ${\displaystyle {\mathbf{a}}^{\mathbf{(}\mathbf{0}\mathbf{)}}}$ and at first order ${\displaystyle {\mathbf{a}}^{\mathbf{(}\mathbf{1}\mathbf{)}}}$. Finally, these solutions which will represent a combination of bending and torsional deformation, will last for the duration of a muscle contraction as the actomyosin continues its contractile activity. The evaluation of the elastic energy under actomyosin muscle activity of order ${\displaystyle {\epsilon }^4}$ can be compared with the equivalent energy of an extensible elastic rod and any fundamental deformation will be identified (Kirchhoff and Hensel, 1883; Mielke, 1988; Mora and Muller, 2003; Mora and Müller, 2004; Moulton et al., 2020a). The typical quantities of interest are the curvature achieved during a bending event or a torsion as well as the total elongation ${\displaystyle \zeta }$. In the C. elegans embryonic system, these quantities result from the competition between the active strains due to muscles and the myosin motors and the elasticity of the body. Similar active matter can be found in biological systems, from animals to plants, as illustrated in Figure 4A–C, they have a structure that generates internal stress/strain for the growth or daily activities. By combining anatomy and measurement techniques, we can transform the mechanics of the body under study into a soft sample subjected to localized internal active stresses or localized internal active strains and then deduce its overall deformation mechanisms, some examples are presented in Figure 4D–F.

Figure 4

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Experimental measurements during the early elongation stage reveal the embryonic diameter and the active or passive stresses, estimated by the opening of fractures realized in the body by laser ablation (Vuong-Brender et al., 2017a; Ben Amar et al., 2018). These quantities vary with the elongation. While previous studies have extensively investigated this initial stage, we have chosen to revisit it in the context of our geometry, with the goal of achieving complete control over our structural modeling, which includes the geometry, shear modulus, and activity of the actomyosin prior to muscle contraction. At the beginning of the second stage, the embryo experiences a pre-strain due to the early elongation, which we describe in our model as ${\displaystyle {\mathbf{G}}_0}$. The geometry and mechanical information are depicted in Figure 5A.

Figure 5

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Here, we must first accurately determine ${\displaystyle {\mathbf{G}}_0}$ by analyzing the experimental data (Vuong-Brender et al., 2017a; Ben Amar et al., 2018). The cylinder can be divided into three distinct sections: the outer layer is the actin cortex, the thin ring where actin bundles concentrate and work, located within a radius range between ${\displaystyle {R_2}^{\mathrm{\prime }}}$ and ${\displaystyle R_3}$ (as shown in Figure 5A); the middle layer (${\displaystyle {\displaystyle R_2<R<{R_2}^{\mathrm{\prime }}}}$), which is the epidermis but without actin; and the inner part (${\displaystyle {\displaystyle 0<R<R_2}}$), where the muscle is located, along with some internal organs, tissues, and fluids. So, we treat the outer and middle layers as incompressible, but the inner part as a compressible material, except for the muscles. The initial deformation gradient: ${\displaystyle {\mathbf{F}}_0=}$ Diag ${\displaystyle (r^{\mathrm{\prime }}(R),r(R)/R,\lambda )}$, and ${\displaystyle {\mathbf{G}}_0=}$ Diag ${\displaystyle (1,g_0,1)}$ with ${\displaystyle {\displaystyle 0<g_0<1}}$ in the actin layer, but with ${\displaystyle g_0=1}$ in the part without actin. ${\displaystyle {\mathbf{G}}_0}$ represents the circumferential strain exerted by actin during the early elongation and is a slowly varying function of time. By applying the principles of radius continuity, radial stress continuity, and incorporating the zero traction condition on the face of the cylinder, we can determine that ${\displaystyle g_0=0.88}$ when the elongation ${\displaystyle \lambda =1.8}$, at the end of the early elongation. As illustrated in Figure 5B, the results of our model and the experimental data are in good agreement, demonstrating the consistency of the geometric and elastic modeling together with the choice of a pre-strain represented by ${\displaystyle {\mathbf{G}}_0}$ which gives a good prediction of the early elongation. More details can be found in the Appendix 3.

From the first stage elongation represented by the blue dots and the blue dashed line in Figure 5C we can extract the time evolution of the contractile pre-strain ${\displaystyle g_0(t_i)}$ derived from our elastic model. To explain ${\displaystyle g_0(t)}$ quantitatively, we propose a phenomenological dynamical approach for the population of active myosin motors. This equation takes into account the competition between the recruitment of new myosin proteins from the epidermal cytoskeleton, which is necessary to elongate the embryo, and the detachment of these myosins from the actin cables, which is damped by the compressive radial stress. It reads:

where ${\displaystyle X_g=1-g_0(t)}$, ${\displaystyle p_1}$ is the ratio of the freely available myosin population to the attached ones divided by the time of recruitment (given in minutes), while ${\displaystyle p_2}$ is the inverse of the debonding time of the myosin motors from the cable: ${\displaystyle p_2=6}$ min⁻¹. The debonding time increases (or decreases) when the actin cable is under radial compressive (or tensile) stress, see Appendix 3, Equation 35. ${\displaystyle {\tau }_v}$ is the viscoelastic time estimated from laser ablation fracture operated in the epidermis (Vuong-Brender et al., 2017b): ${\displaystyle t_v=6s}$ and ${\displaystyle {\tau }_p}$ is the time required to activate the myosin motors ${\displaystyle t_p=1200s}$ (Howard, and Clark,, 2002. This equation is similar to the model derived by Serra et al., 2021) for the viscous stress that occurs during gastrulation.

Note that only ${\displaystyle p_1}$ and ${\displaystyle p_3}$ need be obtained by comparing ${\displaystyle g_0(t)}$ deduced from Equation 3 with the values derived from our elastic model. The result of Equation 3 with ${\displaystyle p_1=0.6}$ and ${\displaystyle p_3=0.75}$ are shown in Figure 5C with a rather good agreement.

The experimental regulation of muscle contraction in C. elegans (Yang, 2017) suggests a cyclic process in which two muscles on one side of the embryo contract quite at the same time and then stop, while on the opposite side, the two muscles begin to contract, but with a delay. Let us first consider that only the muscles are active (see the schematic Figure 6A for the structure, then (B) and (C) for the bending). In this case, due to the geometry, bending to the left occurs when the left muscles are activated and then to the right for the symmetrical right muscles.

Figure 6

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To be more quantitative, we assume that the left side muscles are activated during a short period with an active constant strain value ${\displaystyle g_m}$ in the region ${\displaystyle M_1}$ and ${\displaystyle M_2}$, as shown in Figure 6A, B; if the muscles are perfectly vertical, ${\displaystyle {\alpha }_m={\beta }_m=0}$ in the initial configuration. In fact, the two muscles on the same side are not rigorously in phase, and one may have a small delay. For simplicity, we assume that they contract simultaneously. During the entire initial period when the muscles are not activated, the actin fibers are distributed in a horizontal loop on the outer surface of the epidermis, but as soon as the muscle starts to contract, the actomyosin network will be reoriented (Lardennois et al., 2019). The fibers are then distributed in an oblique pattern, which eventually causes the twisting of the embryo, see the schematic diagram in Figure 6D–F. When this region is activated with a constant strain value, ${\displaystyle g_a}$, the angle of the actin fibers will change according to the amplitude of the bending caused by the muscle contraction. In this situation, the angle of the actin fibers may change from ${\displaystyle {\alpha }_a\in \left[0,\frac{\pi }{2}\right]}$, but ${\displaystyle {\beta }_a}$ is not changed and ${\displaystyle {\beta }_a=0}$.

Throughout the entire process, the muscle and actomyosin activities are assumed to operate simultaneously. Our modeling allows us to evaluate the bending and torsion generated independently by muscles and actin bundles, culminating in a complete deformation under coupling. Furthermore, the angle of the actomyosin fibers varies during muscle contraction. We maintain a constant activation of the actomyosin network and gradually increase the muscle activation.

As a result, the structure will be bent, causing a change in the angle of the actin fibers, ultimately yielding a deformation map that is presented in Figure 7A–C. As myosin activation increased, we observed a consistent torsional deformation (Figure 7E) that agrees with the patterns seen in the video (Figure 7D). However, significant torsional deformations are not always present. In fact, other sources can induce torsion as a lack of symmetry of the muscle axis. We will now discuss torsion due to muscle contraction and the angle of deviation from the axis. In Figure 7F, G, we demonstrate that the curvature provided by the model increases with muscle activation and that the torsion is not simply related to the activation amplitude, as it also depends on the value of the angle ${\displaystyle {\alpha }_a}$, reaching a maximum at about ${\displaystyle \pi /4}$. Detailed calculations are given in section Methods and materials and Appendix 4.

Figure 7

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During the late elongation process, the four internal muscle bands contract cyclically in pairs (Yang, 2017; Williams and Waterston, 1994). Each contraction of a pair increases the energy of the system under investigation, which is then rapidly released to the body. This energy exchange causes the torsional bending energy to be converted into elongation energy, leading to an increase in length during the relaxation phase, as shown in Figure 1 of the Appendix 5. With all the deformations obtained, Equation 2 can be used to calculate the accumulated energy ${\displaystyle W_c}$ produced by both the muscle and the actomyosin activities during a contraction. Subsequently, when the muscles on one side relax, the worm body returns to its original shape but with a tiny elongation corresponding to the transferred elastic energy. In this new state, the actin network assumes a ‘loop’ configuration with a strain of ${\displaystyle \epsilon g_{a1}}$ once relaxation is complete. If all the accumulated energy from the bending-torsion deformation is used to elongate the worm body, the accumulated energy ${\displaystyle W_c}$ and the energy ${\displaystyle W_r}$ after muscle relaxation are equivalent. The activation of actin fibers ${\displaystyle g_{a1}}$ after muscle relaxation can be calculated and determined by our model.

Once the first phase is well characterized, we can quantify the total energy resulting from both muscle and actomyosin after each contraction. We then used our model to predict elongation in wild-type C. elegans, unc-112(RNAi) mutant and spc-1(RNAi) pak-1(tm403) mutant and further compared the results with experimental observations. The wild-type length increases from about 90 to 210 during the muscle-activated phase (Lardennois et al., 2019), the phase lasts about 140 min, and the average time interval between two contractions is about 40 s (Yang, 2017), so the number of contractions can be estimated to be about 210 times. Unfortunately, due to the difficulty of performing quantitative experiments on an embryo that is constantly moving in the egg shell between bending, torsion, and even rotations around its central axis, one can hypothesize this scenario: at each step i between the state ${\displaystyle A_{i,0}}$ to ${\displaystyle A_{i,1}}$ and then ${\displaystyle A_{i,2}}$, the whole mechanical muscle-myosin energy is transferred to the elongated step ${\displaystyle A_{i,2}}$, after resulting in a small ${\displaystyle \delta {\zeta }_i}$.

Considering the experimental results shown in Figure 8B, we determine the optimal values for the activation parameters: ${\displaystyle g_m=-0.15}$ and ${\displaystyle g_a=-0.01}$ assuming that all the energy accumulated during the muscle activation is transferred to the elongation process (${\displaystyle W_r=W_c}$). The elementary elongation ${\displaystyle \delta {\zeta }_i}$ is gradually increased over time, which is shown as the black line in Figure 8A. At the beginning, ${\displaystyle \delta {\zeta }_i}$ is about ${\displaystyle 0.5\mu m}$, but at the end of this process, ${\displaystyle \delta {\zeta }_i}$ is about ${\displaystyle 1.5\mu m}$, indicating that the worm will grow up to ${\displaystyle 290\mu m}$. The result is significantly larger than our actual size ${\displaystyle 210\mu m}$. As the elongation progresses, we assume that there is an energy transfer between bending-torsion-contraction and elongation but it may not be fully effective, meaning that a significant part of the energy is lost. From the experimental data, we estimate that the energy loss gradually increases, from full conversion at the beginning to only 40% of the accumulated energy used for elongation at the end of the process (${\displaystyle W_r=0.4W_c}$). For ${\displaystyle \delta {\zeta }_i}$ it induces a first increase and then a decrease, which is shown as the blue line in Figure 8A and is responsible for the slowdown around 200 min. This possibility, which may be not the only possible one, leads to the estimated elongation being in good agreement with the experimental data (see the blue-dashed curve in Figure 8B). Indeed it is possible that the C. elegans elongation requires other transformations that cost energy. As the embryo gradually elongates, energy dissipation, and the biomechanical energy required to reorganize the actin bundles may be two factors contributing to the increased energy loss underlying the hypothesis.

Figure 8

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In addition, it has been reported in Norman and Moerman, 2002 that the knockdown of unc-112(RNAi), known the to affect muscle contractions, leads to the arrest of elongation of the embryos at the twofold stage, indicating that the muscles have no activation, ${\displaystyle g_m=0}$ in our model, and no accumulated energy can be converted into elongation. Another mutation affecting the embryos consisting of mutant cells with pak-1(tm403), known to regulate the activity of myosin motors leads to a retraction of the embryo, so that the pre-stretch caused by myosin is not maintained and decreases. The results are fully consistent with a number of experimental observations and are shown in Figure 8B.

The main manifestation of the muscle activity, independent of the elongation, is probably the constant rotation of the embryo despite its confinement in the egg. This can be explained by a small angular deviation of the muscle sarcomeres from the central axis due to their attachment to the inner boundary of the cell epidermis, the so-called ‘dense bodies’ (Moerman and Williams, 2006). Since they cross the horizontal plane at about ±45° and the deviation ${\displaystyle {\beta }_m}$ from the anterior-posterior axis is estimated to be about 6° each active muscle on the left (or on the right) contributes to the torque via a geometric factor of about ${\displaystyle a_g=\sin (6\pi /180)\cos (\pi /4)}$. Then a simple estimate of muscle activity in terms of torque is ${\displaystyle {\mathrm{\Lambda }}_m\sim {\mu }_m\pi R^3{s}_m{p}_m{a}_g(\epsilon g_m)}$ where ${\displaystyle s_m}$ is the area of the left or right muscle pair compared to the section of the cylinder: ${\displaystyle s_m=0.025}$ and ${\displaystyle p_m}$ is the distance of the muscles from the central axis of the embryo: ${\displaystyle p_m=0.75}$ while ${\displaystyle g_m=0.15}$ according to the elongation analysis. So the muscles on one side contribute to a torque ${\displaystyle {\mathrm{\Lambda }}_m}$ along the symmetry axis, given by ${\displaystyle {\mathrm{\Lambda }}_m=4.657{\mu }_m\pi R^3\cdot {10}^{-5}}$.

Let us now consider the dissipative torque, assuming that the rotational dynamics are stopped by friction after a bending event. Two cases can be considered: either the dissipation comes from the viscous flow or from the rubbing of the embryo when it folds. The fluid dissipation results from the rotations in the interstitial fluid inside and along the eggshell as the anterior-posterior axis remains parallel to the eggshell (Bhatnagar et al., 2023). The interstitial fluid, of viscosity ${\displaystyle \eta }$ contains a significant amount of sugars and other molecules necessary for embryo survival and that is more viscous than water (Soesanto and Williams, 1981; Chadwick, 1985; Bouchard et al., 2007; Telis et al., 2007; Hidayanto et al., 2010; Labouesse, 2023). However, values for sucrose or sorbitol at the concentration of 1 mole/liter indicate a viscosity on the order of the viscosity of water, which is 1 mPas. For example at 0.9 mole/liter and temperature of 20°, an aqueous solution of sorbitol has a viscosity of 1.6 mPas, which can be extrapolated to ${\displaystyle \eta =1.9}$ mPas at 1.2 mole/liter (Jiang et al., 2013). The estimate given by an embryo located in the middle of the egg shell, gives a weaker viscous torque once evaluated by ${\displaystyle {\mathrm{\Lambda }}_v=4\pi \eta {\mathrm{\Omega }}_{e}L{R}^2(R_{egg}^2/(R_{egg}^2-R^2))}$ according to a classical result reported by Landau and Lifshitz, 1971; Landau and Lifshitz, 2013.

It should be noted that this estimation assumes that the two cylinders: the egg shell and the embryo have the same axis of symmetry and concerns the beginning of the muscle activity where the radius is about ${\displaystyle 8.2\mu m}$, the length is ${\displaystyle 90\mu m}$, the radius of the shell is about ${\displaystyle R_{egg}=15\mu m}$ and the length ${\displaystyle L_{egg}=54\mu m}$, see Figure 7D. The angular velocity ${\displaystyle {\mathrm{\Omega }}_e}$ is more difficult to evaluate but it is about 90° per 2 s deduced from videos. As the embryo approaches the eggshell, the friction increases, and two eccentric cylinders of different radii must be considered with the two symmetry axes separated by a distance ${\displaystyle d}$. The hydrodynamic study in this case is far from being trivial, and seems to have been first initiated by Zhukoski, 1887, who proposed the use of bipolar coordinates for the mathematical treatment. Many subsequent papers were published after, using barious simplified assumptions, and the study was completely revisited by Ballal and Rivlin, 1976. Here, we focus on the limit of a small gap ${\displaystyle \delta }$ between the rotating body shape and the egg and by considering an asymptotic analysis at small ${\displaystyle \delta }$ of the general result derived in Ballal and Rivlin, 1976. Thus, the viscous torque reads ${\displaystyle {\tilde{\mathrm{\Lambda }}}_v=2\sqrt{2}\pi \eta {\mathrm{\Omega }}_e{L}_c{R}_{egg}^2\sqrt{R_{egg}/\delta }\sqrt{(R_{egg}-d)/d}}$, where ${\displaystyle L_c}$ is the zone of contact with the egg and ${\displaystyle d=R_{egg}-\delta -R}$. This approach is an approximation since the embryo is more toroidal than cylindrical (Yang, 2017), but the evaluation of the dissipation is satisfactory for ${\displaystyle \delta =0.5\mu m}$, ${\displaystyle {\mathrm{\Omega }}_e\sim \pi /4s^{-1}}$ and ${\displaystyle \mu ={10}^{5}Pa}$. Going back to the first model of dissipation with the same data, the ratio between the dissipative viscous torque and the active one gives: ${\displaystyle {\mathrm{\Lambda }}_v/{\mathrm{\Lambda }}_m=0.02}$, which is obviously unsatisfactory. Finally, the dissipative energy ${\displaystyle {\mathcal{E}}_{diss}}$ during one bending event leading to an angle of ${\displaystyle \pi /2}$ is ${\displaystyle {\mathcal{E}}_{diss}=1/2{\mathrm{\Lambda }}_m\times (\pi /2{)}^2}$ which represents 4% of the muscle elastic energy during the bending so at the beginning of the muscle activity (Appendix 5, Equation 45), the dissipation is present but is negligible. At the very end of the process, this ratio becomes 60% but as already mentioned, our estimate for the dissipation becomes very approximate, increases a lot due to the embryo confinement, and does not include the numerous biochemical steps necessary to reorganize the active network: actomyosin and muscles.

The finite strain ${\displaystyle {\mathbf{G}}_0}$ maps the initial stress-free state ${\displaystyle {\mathcal{B}}_0}$ to a state ${\displaystyle {\mathcal{B}}_1}$, which describes the early elongation. We then impose an incremental strain field ${\displaystyle {\mathbf{G}}_1}$ that maps the state ${\displaystyle {\mathcal{B}}_1}$ to the state ${\displaystyle {\mathcal{B}}_2}$, which represents late elongation. So, the deformation gradient can be expressed as: ${\displaystyle \mathbf{F}={\mathbf{F}}_e{\mathbf{G}}_1{\mathbf{G}}_0}$ (Goriely and Ben Amar, 2007). The deformation gradient ${\displaystyle \mathbf{F}=F_{ij}{\mathbf{d}}_i\otimes {\mathbf{e}}_j,}{\displaystyle i\in \left\{1,2,3\right\}}$ and ${\displaystyle j}$ labels the reference coordinates ${\displaystyle \left\{R,\mathrm{\Theta },Z\right\}}$ finally reads:

where ${\displaystyle \lambda }$ is the axial extension due to the pre-strained.

We define ${\displaystyle {\mathbf{G}}_0=Diag\left(1,1+\epsilon c,1\right)}$, since ${\displaystyle g_0=0.88}$ can be determined by the early elongation, we write it in the form of ${\displaystyle 1+\epsilon c}$ to simplify the calculation, ${\displaystyle c\approx -0.6}$ (${\displaystyle \epsilon \approx 0.2}$).

Consider fibers oriented along the unit vector ${\displaystyle \mathbf{m}=(\sin \alpha \sin \beta ,\sin \alpha \cos \beta ,\cos \alpha )}$ for the incremental strain ${\displaystyle {\mathbf{G}}_{\mathbf{1}}}$, where ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ characterize the angles with ${\displaystyle {\mathbf{e}}_{\mathbf{z}}}$ and ${\displaystyle {\mathbf{e}}_{\theta }}$ and can be found in Figure 3A. The active filamentary tensor in cylindrical coordinates is then ${\displaystyle \mathbf{G}={\mathbf{G}}_0\left(\mathbf{I}+\epsilon g\mathbf{m}\otimes \mathbf{m}\right)}$, see Holzapfel, 2000 which reads:

The associated strain-energy density can be obtained by expanding the inner variables ${\displaystyle a_i}$ (see Equation 1), ${\displaystyle p_i}$, and the potential ${\displaystyle V}$ (see Equation 2) as:

For each cross-section, the associated Euler-Lagrange equations take the following form:

where ${\displaystyle j=1,2,3,k=0,1}$, in association with boundary conditions at each order ${\displaystyle k=0,1}$ on the outer radius ${\displaystyle R_i}$:

We solve these equations order by order, taking incompressibility into account. At the lowest order, the incompressibility imposes the deformation: ${\displaystyle {\mathbf{a}}^{\left(0\right)}=\left(r\left(R\right),\mathrm{\Theta },0\right)}$ and the Euler-Lagrange equation gives the Lagrange parameter ${\displaystyle p^{(0)}=P_0(R)}$ which finally read:

and the boundary condition ${\displaystyle {\sigma }_{rr}(R=1)=0}$ gives the value of ${\displaystyle P_0=1/\lambda }$. At ${\displaystyle \mathcal{O}\left(\epsilon \right)}$, the Euler-Lagrange equations are again automatically satisfied, and at ${\displaystyle \mathcal{O}\left({\epsilon }^2\right)}$, the crucial question is to get the correct expression for ${\displaystyle a_i^{\left(1\right)}}$ and ${\displaystyle p^{\left(1\right)}}$. Based on the previous subsection, the following form is intuited:

As before, the Euler-Lagrange equations and the incompressibility condition give the two constants ${\displaystyle q_1=\frac{1}{8}\left(-2+{\lambda }^3\right)}$, ${\displaystyle q_2=\frac{1}{8}\left(2+3{\lambda }^3\right)}$ and ${\displaystyle h_1}$, ${\displaystyle h_2}$, ${\displaystyle h_3}$ are a function of ${\displaystyle R}$. With the solutions for active strain ${\displaystyle {\mathbf{a}}^{\left(0\right)}}$ and ${\displaystyle {\mathbf{a}}^{\left(1\right)}}$, full expressions of them are given in the Appendix 4. The second order energy takes the form:

and we can transform it into this form:

where ${\displaystyle A_i}$, ${\displaystyle B_i}$, ${\displaystyle C_i}$, and ${\displaystyle D_i}$ are functions of ${\displaystyle R}$ and ${\displaystyle \mathrm{\Theta }}$.

Comparing with the energy of an extensible elastic rod (Moulton et al., 2020a; Kirchhoff and Hensel, 1883; Mielke, 1988; Mora and Muller, 2003; Mora and Müller, 2004), we recognize the classic extensional stiffness ${\displaystyle K_0}$, bending stiffness ${\displaystyle K_1}$ and ${\displaystyle K_2}$, torsional stiffness ${\displaystyle K_3}$ coefficients:

where ${\displaystyle A_2}$, ${\displaystyle B_2}$, ${\displaystyle C_2}$, and ${\displaystyle D_2}$ are related to the shear modulus μ, so for a uniform material with no variations of shear modulus, ${\displaystyle S_K}$ represents the cross-section of the cylinder. But if not, we need to divide different regions to perform the integration.

We now focus on the intrinsic extension and curvature of the cylindrical object induced by the active strains, which requires the competition of the active forces with the stiffness of the cylindrical beam according to the relationships,

each ${\displaystyle H_i}$ is calculated by:

where ${\displaystyle A_1}$, ${\displaystyle B_1}$, ${\displaystyle C_1}$, and ${\displaystyle D_1}$ are related to the shear modulus μ, the fiber angles ${\displaystyle \alpha }$ and ${\displaystyle \beta }$, and the activation ${\displaystyle g}$. So, the integration region ${\displaystyle S_H}$ is divided into the different parts of the embryo which all contribute to the deformation.

To calculate the intrinsic Frenet curvature and torsion, we use the following equation:

Finally, in the case of the activated left side muscles, we can calculate the intrinsic extension and curvatures using Equation 16, ${\displaystyle {\hat{u}}_{1m}=0}$ and ${\displaystyle {\hat{u}}_{3m}=0}$, so Frenet’s intrinsic curvature and torsion Equation 18, ${\displaystyle {\hat{\kappa }}_m={\hat{u}}_{2m}}$ and ${\displaystyle {\hat{\tau }}_m=0}$. For the activated actin case, we can calculate the intrinsic extension and curvatures by Equation 16, ${\displaystyle {\hat{u}}_{1a}=0}$ and ${\displaystyle {\hat{u}}_{2a}=0}$, and the Frenet’s intrinsic curvature and torsion Equation 18, ${\displaystyle {\hat{\kappa }}_a=0}$ and ${\displaystyle {\hat{\tau }}_a=\frac{{\hat{u}}_{3a}}{{\hat{\zeta }}_a}}$. These quantities have been used to calculate the deformation images, as shown previously in Figure 7.

After obtaining all solutions, we can use these quantities to calculate the accumulated energy ${\displaystyle W_c}$ in the system after one contraction by Equation 2. By defining the activation (${\displaystyle g_m}$ and ${\displaystyle g_a}$) and the energy conversion efficiency, we can obtain the activation (${\displaystyle g_{a1}}$) of the actin bundles after a single contraction and further calculate the elongation.

All calculation results and parameters are presented in the Appendix.

The finite deformation gradient: ${\displaystyle {\mathbf{F}}_0=Diag(r^{\mathrm{\prime }}(R),r(R)/R,\lambda )}$, and ${\displaystyle {\mathbf{G}}_0=Diag(1,g_0,1)}$, ${\displaystyle 0<g_0<1}$ in the actin layer, and ${\displaystyle g_0=1}$ in the part without actin bundles. To determine the finite strain ${\displaystyle {\mathbf{G}}_0}$, we converted the four muscle parts into thin layers attached to the epidermis in equal proportions to ensure the continuity of the model, and divided our model into four parts, see Figure 1. The actin layer(${\displaystyle {\displaystyle R_3<R<{R_2}^{\mathrm{\prime }}}}$), epidermis layer(${\displaystyle {\displaystyle R_2<R<{R_2}^{\mathrm{\prime }}}}$), and muscle layer(${\displaystyle {\displaystyle {R_1}^{\mathrm{\prime }}<R<R_2}}$) are considered as incompressible materials, and the inner part(${\displaystyle {\displaystyle 0<R<{R_1}^{\mathrm{\prime }}}}$) is compressible. The Neo-Hookean energy function is used for the incompressible parts:

Appendix 2—figure 1

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According to the Euler-Lagrange equations, we can obtain the radius of the actin layer(${\displaystyle r_a}$), the epidermis layer(${\displaystyle r_e}$), and the muscle layer(${\displaystyle r_m}$):

where A, E. and M are constants, and the Lagrange multiplier ${\displaystyle q}$ in the actin, epidermis, and muscle layers:

For the compressible part, we take the energy function form (Holzapfel, 2000):

where ${\displaystyle \kappa }$ is a material constant. The radius of the inner part is:

where ${\displaystyle a}$ is a constant.

By considering the boundary condition ${\displaystyle {\sigma }_{rr}=0}$ on the outer border ${\displaystyle R=1}$:

the continuity of the radius:

and the continuity of the radial stresses ${\displaystyle {\sigma }_{rr}}$ in ${\displaystyle {R_2}^{\mathrm{\prime }}}$ and ${\displaystyle R_2}$ :

Using the above equations, we can obtain expressions for the constants ${\displaystyle E}$, ${\displaystyle M}$, ${\displaystyle C_a}$, ${\displaystyle C_e}$, and ${\displaystyle C_m}$.

We replace all size and material parameters and ${\displaystyle \kappa =100}$, ${\displaystyle \lambda =1.8}$ in the last condition ${\displaystyle {\sigma }_{mrr}\left({{\mathrm{R}}_1}^{\mathrm{\prime }}\right)={\sigma }_{irr}\left({{\mathrm{R}}_1}^{\mathrm{\prime }}\right)}$:

where ${\displaystyle {\sigma }_{irr}}$:

and obtain the relation with constant ${\displaystyle A}$ and ${\displaystyle g_0}$.

Finally, by prescribing a zero traction condition on the top of the cylinder, the muscle part was noticeably considered inextensible, so there was no stress on the top:

where:

all solutions ${\displaystyle g_0=0.88}$, ${\displaystyle A=0.08}$, and ${\displaystyle r_a(1)=0.73}$ can be determined. The result is also in good agreement with the experimental data (Vuong-Brender et al., 2017a; Ben Amar et al., 2018).

The time required for non-muscle myosin detachment is estimated to be ${\displaystyle {\tau }_0=10s}$ for free actomyosin filaments. If the actin filament is subjected to an external load perpendicular to its axis, detachment can be facilitated, or on the contrary, inhibited, see Howard, and Clark,, 2002, page 169–170. In the present case, the stresses acting in the radial direction of an actin bundle are compressive and thus will retard the detachment. This energy must be compared to the energy of detachment of all myosin motors from the bundle. The corresponding elastic energy associated with the radial deformation for an actin cable of length ${\displaystyle l_a}$ with radius ${\displaystyle r_b=0.05\mu m}$ estimated from Lardennois et al., 2019 and shear modulus ${\displaystyle {\mu }_a=5KPa}$ is given by ${\displaystyle 1/2{\mu }_a(1-g_0(t))\pi l_a{r}_b^2={210}^{-11}{l}_{a}J}$. This result must be compared to the individual energy of detachment times the number of myosin motors on a cable. This number is difficult to determine but an estimate is given by the length of the cable ${\displaystyle l_a}$ divided by the distance between 2 myosin anchoring sites, which is about ${\displaystyle 5nm}$ while the attachment energy per motor is about ${\displaystyle 6k_{b}T}$. These values are for skeletal muscle myosins (Howard, and Clark,, 2002) and must be taken with caution. Nevertheless, the order of magnitude of the detachment energy for a collection of myosin heads from actin cable can be estimated to be ${\displaystyle 4.8l_a{10}^{-12}J}$. Then the ratio between the two quantities is of the order ${\displaystyle 4(1-g_0(t))}$, which explains that the time scale of debonding for a cable under compressive stress in the direction orthogonal of its axis is then:

where ${\displaystyle p_3}$ is a positive constant of order one that cannot be predicted exactly. This time scale justifies the exponential correction in Equation 3 of the manuscript.

The paper discusses the case with pre-strain. Here, are more details about the case without pre-strain (${\displaystyle {\mathbf{G}}_{\mathbf{0}}\mathbf{=}\mathbf{I}}$). Up to the lowest order, the solution of the Euler–Lagrange equations is obviously given by ${\displaystyle {\mathbf{a}}^{\left(0\right)}=\left(R,\mathrm{\Theta },0\right)}$, and ${\displaystyle p^{\left(0\right)}=1}$, so there is no deformation and the zero and first order energies vanish. At order ${\displaystyle \mathcal{O}\left({\epsilon }^2\right)}$ of the elastic energy, the solutions for the active strain components are

where ${\displaystyle f_{1,2}}$ are functions related to the active stress ${\displaystyle g}$ and the fiber angles ${\displaystyle \alpha }$ and ${\displaystyle \beta }$:

The stiffness coefficients:

The deformation coefficients:

The first order solutions of the theory with the case of pre-strain, ${\displaystyle h_{1,2,3}\left(R\right)}$, are related to the activation ${\displaystyle g}$ and the fiber angles ${\displaystyle \alpha }$ and ${\displaystyle \beta }$:

The stiffness coefficients are:

The deformation coefficients are: