Introduction

Mechanical stresses play a critical role during embryogenesis, affecting complex biological tissues such as the skin, the brain, and the interior of organs. There are many studies in the last decades, for example, the pattern of the intestine (Coulombre and Coulombre (1958); Hannezo et al. (2011); Li et al. (2011); Ben Amar and Jia (2013); Shyer et al. (2013)), the brain cortex (Toro and Burnod (2005); Goriely et al. (2015); Ben Amar and Bordner (2017); Tallinen et al. (2016)), and the cir-cumvolutions of the fingerprints (Kücken and Newell (2005); Ciarletta and Ben Amar (2012)) that have been interpreted as the result of compressive stresses generated by growth occurring a few months post fertilization for humans. At a smaller scale, the influence of mechanical stresses superposed to cellular processes (e.g., division, migration) and tissue organization becomes more complex to be identified and quantified. For example, independently of growth, stresses may be generated by different molecular motors, among which myosin II, linked to actin filaments, is the most prevalent during epithelial cell morphogenesis (Vicente-Manzanares et al. (2009)) and cell motility (Cowan and Hyman (2007); Olson and Sahai (2009); Palumbo et al. (2022)). The spatial distribution and dynamics of myosin II greatly influence the morphogenetic process (Levayer and Lecuit (2012); Lv et al. (2022)), as demonstrated for Drosophila (Bertet et al. (2004); Blankenship et al. (2006); Saxena et al. (2014); Shindo and Wallingford (2014)) and also C. elegans embryo (Priess and Hirsh (1986); Gally et al. (2009); Ben Amar et al. (2018)).

The embryonic elongation of C. elegans represents an attractive model of matter reorganization without a mass increase before hatching. It happens after the ventral enclosure and takes around 240 minutes to convert the lima-bean-shaped embryo into a final elongated worm shape: the embryo elongates along the anterior/posterior axis by four times (McKeown et al. (1998)). The short lifetime of the egg before hatching and its transparency makes this system an ideal system to study the forces that exist in the cortical epithelium or its vicinity. However, unlike the embryonic development of Drosophila and Zebrafish, there is neither cell migration, cell division nor a no-table change in embryonic volume (Sulston et al. (1983); Priess and Hirsh (1986)), only a noticeable epidermal elongation drives the whole morphogenetic process of C. elegans in the period post en-closure. There are two experimentally identified driving forces for the elongation: the actomyosin contractility in seam cells of the epidermis, which seems to last during the whole elongation process, and the muscle activity beneath the epidermis that starts after the 1.7 ∼ 1.8-fold stage. The transition is well defined since the muscle participation makes the embryo rather motile impeding any physical experiments such as laser ablation, which could be conducted and achieved in the first period (Vuong-Brender et al. (2017a)). Consequently, the elongation process of C. elegans could be divided into two stages: the early elongation and the late elongation, depending on the muscle activation, and Fig.1(A) shows the whole elongation process (Vuong-Brender et al. (2017a)). Previously, the role of the actomyosin network in the seam cells during the early elongation of C. elegans was investigated (Ben Amar et al. (2018)). Based on the geometry of a hollow cylinder composed of four parts (seam and dorso-ventral cells), a model involving the pre-stress responsible for the enclosure, the active compressive ortho-radial stress joined to the passive stress quantitatively predicts the elongation, but only up to ∼ 70% of the initial length.

Figure 1.

For the late elongation phase, the acto-myosin network and muscles drive together the full elongation during this phase (Vuong-Brender et al. (2017a)). Muscles play an important role since mutants with muscle defects are unable to complete the elongation process, even though the actomyosin network operates normally (Lardennois et al. (2019)). Fig.1(B) shows the schematic image of C. elegans body (Zhang et al. (2011)) with four rows of muscles, two of which are underneath the dorsal epidermis and the other two are under the ventral epidermis. As observed in vivo, C. elegans exhibits systematic rotations accompanying each contraction (Yang (2017); Yang Xinyi et al. (2023)), and deformations such as bending and twisting. Eventually, the C. elegans embryo will complete an elongation from 1.8-fold to 4-fold. We can imagine that once the muscle is activated on one side, it can only contract, and then the contraction forces will be transmitted to the epidermis on this side. So one can wonder how an elongation can occur since striated muscles can only perform cycles of contraction and relaxation, their action will tend to reduce the length of the embryo. Therefore, it is necessary to understand how the embryo elongates during each contraction and how the muscle contractions couple to the acto-myosin activity. This work aims to answer this paradox within the framework of finite elasticity without invoking cell plasticity and stochasticity which cannot be considered driving forces. In addition, several important issues at this stage remain unsettled. First of all, we may also observe a torsion of the embryo but if the muscle activity suggests a bending, it cannot explain alone a substantial torsion. Secondly, a small deviation of the muscle axis (Moerman and Williams (2006)) is responsible for a series of rotations, how to relate these rotations to the muscle activation (Yang (2017)). Since any measurement on a motile embryo at this scale is difficult, it is meaningful to explore the mechanism of late elongation theoretically. Furthermore, muscle contraction is crucial in both biological development and activities and has been studied extensively by researchers (Tan and De Vita (2015); De Vita et al. (2017)), but how it works at small scales remains a challenge.

Using a finite elasticity model and assuming the embryonic body shape is cylindrical, we can evaluate the geometric bending deformation and the energy released during each muscular contraction on one side since after each contraction, the muscles relax and then muscles on the opposite side undergo a new contraction. This cyclic process leads to a tiny elongation of the cylinder along its symmetry axis at each contraction. Each of them is correlated with the rotational movements of the embryo (Yang (2017)). By repeating these pairs of contractions more than two hundred times, a cumulative extension is achieved, but it must be reduced by friction mechanisms, also evaluated by the model. Furthermore, the mechanical model explains the existence of a torque operating at the position of the head or tail by the coupling of muscle contraction with the orthoradial acto-myosin forces. Finally, the small deviation between the muscles and the central axis experimentally detected (Moerman and Williams (2006)) induces cyclic rotations and possibly torsion leading to fluid viscous flow inside the egg. Quantifying all these processes allows evaluating the physical quantities of the embryo such as the shear modulus of each component, the osmolarity of the interstitial fluid and the active forces exerted by the acto-myosin network and the muscles which are sparsely known in the embryonic stages.

Our mechanical model accounts for the dynamical deformations induced by the internal stimuli of a layered soft cylinder, enabling us to make accurate quantitative predictions about the active networks of the embryo. Furthermore, our model takes into consideration the dissipation that occurs in the late period just before the egg hatch. Our results are consistent with observations of acto-myosin and muscle activity (Vuong-Brender et al. (2017a); Ben Amar et al. (2018); Lardennois et al. (2019)). The architecture of our paper is illustrated in Fig.2.

Figure 2.

Results

It is widely recognized that C. elegans is a well-established model organism in the field of developmental biology. However, it is less widely known that its internal striated muscles share similarities with skeletal muscles found in vertebrates in terms of both function and structure (Lesanpezeshki et al. (2021)). Being contractile, the role of the four axial muscles (as shown in Fig.1) in the final shaping of the embryo is nearly contra-intuitive. In this section, we aim to highlight the action of these muscles when coupled to the acto-myosin network, using a purely mechanical approach. To achieve a quantitative understanding, our method requires a detailed description of the deformation geometry of the embryo, with the body shape represented by a fully heterogeneous cylinder.

Geometric and strain deformations of the embryo

The early elongation of C. elegans embryo was 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 cells - the seam, dorsal, and ventral cells - which exhibit unique cytoskeleton organization and actin network configurations. Among these, only the seam cells possess active myosin motors that function in the ortho-radial direction, allowing for the contraction of the circumference 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. Muscles play no role in the early elongation stage so were not 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 collaborate to facilitate further elongation. Table. 1 provides the size parameters of C. elegans which will be introduced hereafter.

Table 1.

We focus on the overall deformation of a full cylinder or a thin rod, having a length L greater than the radius R and a central vertical axis along the Z direction. Contrary to previous works (Ciarletta et al. (2009); Ben Amar et al. (2018)), here we decide to simplify the geometrical aspect because of the mechanical complexity. The biological activity induced by the acto-myosin 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 through non-mechanical processes (e.g., biochemical processes): myosin motors and striated muscles receive their energy from ATP hydrolysis, which is converted to mechanical contractions of fibers. Due to the significant deformation observed, the central line is distorted and becomes a curve in three-dimensional space, represented by a vector r (Z), as depicted in Fig.3(A). 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:

Figure 3.

where the a_i represents the deformation in each direction of the section with a_i (0, 0, Z) = 0 so that the Z-axis maps to the centerline r (Z). The small quantity ε is the ratio between the radius R and the length L of the cylinder. Referring to the model proposed by B. Kaczmarki et al. (Kaczmarski et al. (2022)), we define the initial configuration ℬ₀ with material points (X, Y, Z), and the mapping function χ (X) links the initial configuration ℬ₀ to the current configuration ℬ₀. The geometric deformation gradient is F = Gradχ = F_eG (Nardinocchi and Teresi (2007)), where G represents the active strain generated by the actomyosin or the muscles, and F_e is the elastic strain tensor. 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. So, the deformation gradient can be decomposed into: F = F_eG₁G₀ (Goriely and Ben Amar (2007)) where G₀ refers to the pre-strain of the early period and G₁ is the muscle-actomyos in supplementary active strain in the late period. Actin is distributed in a circular pattern in the outer epidermis, see Fig.3(B), so the finite strain G₀ is defined as G₀ = Diag (1, g₀(t), 1), where 0 < g₀ < 1 is the time-dependent decreasing eigenvalue, operating in the actin zone and is equal to unity in the other parts: g₀(0) = 1. In the case without pre-strain, G₀ = I. The deformation gradient follows the description of Eq.(1) :F = F_ij d_i ⊗ e_j, i ∈ {1, 2, 3} and j ∈ {R, Θ, Z}.

When considering a filamentary structure with different fiber directions m, these directions are specified by two angles α and β, as outlined in (Holzapfel (2002)): m = sin α sin β e_R + sin α cos β e_𝚯 + cos α e_Z, α, β ∈ [−π/2, π/2], where α and β lie in the range [−π/2, π/2]. For muscle fibers, α_m = 0 and β_m = 0, whereas for hoop fibers in the actin network, α_a = π/2 and β_a = 0. When the muscles are activated and bend the embryo, the actin fibers become inclined with a slope α, such that −π/2 < α_a < π/2, α_a ≠ 0, and β_a = 0. Each active strain is represented by a tensor G_i = I + εg_im_i ⊗ m_i, where g_i represents the activity (g_m for the muscles and g_a for acto-myosin), 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 muscles and the acto-myosin 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 . In cylindrical coordinates, the total energy of the system and the auxiliary energy density is:

where J = det F_{e ′} p is a Lagrange multiplier which ensures the incompressibility of the sample, a physical property assumed in living matter. When the cylinder involves several layers with different shear modulus µ and different active strains, the integral over S covers each layer. To minimize the energy over each section for a given active force, we take advantage of the weakness of ε and expand the inner variables a_i, p_i, and the potential V to obtain the auxiliary strain-energy density. Given that the Euler-Lagrange equations and the boundary conditions are satisfied at each order, we can obtain solutions for the elastic strains at zero order a⁽⁰⁾ and at first order a⁽¹⁾. Finally, these solutions which will represent a combination of bending and torsion deformation will last the time of a muscle contraction while the acto-myosin will continue its contractile activity. The evaluation of the elastic energy under acto-myosin-muscle activity at order ε⁴ can be compared with the equivalent energy of an extensible elastic rod and each basic deformation will be identified (Kirchhoff and Hensel (1883); Mielke (1988); Mora and Müller (2002, 2004); Moulton et al. (2020a)). The typical quantities of interest are the curvature reached during a bending event or a torsion as well as the overall elongation ζ. 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 Fig.3(C)-(E), they have a structure that generates internal stress/strain when growing or activity. Combining anatomy and measurement techniques, we can transform the mechanics of the body under study into a soft sample submitted to localized internal active stresses or localized internal active strains and then deduce its overall deformation mechanisms, some examples are presented in Fig.3(F)-(H).

The early elongation induced by the acto-myosin

Experimental measurements during the early elongation stage reveal the embryonic diameter and the active or passive stresses, detected by fracture ablation in the body vary with the elongation fold (Vuong-Brender et al. (2017a); Ben Amar et al. (2018)). While previous studies have extensively examined this initial stage, we have chosen to revisit it within the context of our geometry, aiming to attain complete control over our structural modeling, encompassing the geometry, shear modulus, and activity of the acto-myosin prior to muscle contraction. At the outset of the second stage, owing to the early elongation, the embryo experiences a pre-strain, which we describe in our model as G₀. The geometry and mechanical information are depicted in Fig.4(A).

Figure 4.

Here, we need to first accurately determine G₀ 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 R₂^′ and R₃ (as shown in Fig.4(A)); the middle layer (R₂ < R < R₂^′), which is the epidermis but without actin; and the inner part (0 < R < R₂), 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 the muscles. The initial deformation gradient: F₀ =Diag(r^′(R), r(R)/R, λ), and G₀ = Diag(1, g₀, 1) with 0 < g₀ < 1 in the actin layer but with g₀ = 1 in the part without actin. G₀ represents the circumferential strain exerted by actin during the early elongation and is a slowly varying function of time. r(R) is the radius after early elongation. 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 g₀ = 0.88 when the elongation λ = 1.8, i.e., at the completion of the early elongation. As illustrated in Fig.4(B), the results of our model and the experimental data are in good agreement, this demonstrates the consistency of the geometric and elastic modeling together with the choice of a pre-strain represented by G₀ which gives a good prediction of the early elongation. More details are displayed in Appendix 3.

From the first stage elongation represented by the blue dots and the blue dashed line in Fig.4(C) we can extract the time evolution of the contractile pre-strain g₀(t_i) derived from our elastic model. To explain quantitatively g₀(t), we suggest a phenomenological dynamical approach for the population of active myosin motors. This equation considers the competition between the recruitment of new myosin proteins from the epidermis cytoskeleton necessary to extend the embryo and the debonding of these myosins from the actin cables, which is damped by the compressive radial stress. It reads:

with X_g = 1 − g₀(t), p₁ is the ratio between the free available myosin population and the attached ones divided by the time of recruitment (given in minutes), while p₂ is the inverse of the debonding time of the myosin motors from the cable: p₂ = 6 min⁻¹. The debonding time increases (or decreases) when the actin cable is in radial compressive (or tensile) stress, see Appendix 3, Eq.(35). τ_υ is the visco-elastic time estimated from laser ablation fracture operated in the epidermis (Vuong-Brender et al. (2017b)): t_υ = 6s and τ_p is the time required for the activation of the myosin motors t_p = 1200s (Howard and Clark (2002)). This equation is similar to the model derived by Serra et al (Serra et al. (2021)) for the viscous stress occurring in gastrulation.

Notice that only p₁ and p₃ must be obtained by comparison of g₀(t) deduced from Eq.(3) with the values deduced from our elastic model. The result of Eq.(3) with p₁ = 0.6 and p₃ = 0.75 are shown in Fig.4(C) with a rather good agreement.

Shape of the embryo under muscles and acto-myosin contraction

The regulation of the muscle contraction in C. elegans detected experimentally (Yang (2017)) indicates a cyclic process, where two muscles contract on one side of the embryo quite simultaneously and then stop, while on the opposite side, the two muscles start their contractions, but only after a delay. Let us consider first that only muscles are active (see the schematic Fig.5(A) for the structure, then (B) and (C) for the bending). In this case, due to the geometry, only a bending deformation occurs on the left for active muscles localized on left and then on the right for the symmetric muscles on right.

Figure 5.

To be more quantitative, we assume that the left side muscles are activated during a short period with an active constant strain value g_m in the region M₁ and M₂, as shown in Fig.5(A)-(B); if the muscles are perfectly vertical, α_m = β_m = 0 in the initial configuration. In fact, the two muscles on the same side are not always fully in phase and one may present of small delay. For simplification, we assume them are perfectly synchronous. During the full initial period where muscles are not activated, the actin fibers are distributed in a horizontal loop on the outer surface of the epidermis, but once the muscle starts to contract, the acto-myosin network will be re-orientated (Lardennois et al. (2019)). The fibers will be then distributed in a sloping pattern causing eventually the twisting of the embryo, see the schematic diagram shown in Fig.5(D)-(F). When this region is activated with a constant strain value, g_a, the angle of the actin fibers will change following the amplitude of the bending by the muscle contraction. In this situation, the angle of the actin fibers may change from but β_a is not modified and β_a = 0.

Throughout the entire process, the muscle and acto-myosin activities are assumed to work almost 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 acto-myosin fibers varies during muscle contractions. We maintain a constant activation of the acto-myosin network and gradually increase muscle activation.

As a result, the bending of the model will be increased, causing a change in the angle of the actin fibers, ultimately yielding a deformation map that is presented in Fig.6(A)-(C). Upon increasing myosin activation, we observed a consistent torsional deformation (Fig.6(E)) that agrees with the patterns seen in the video (Fig.6(D)). However, significant torsional deformations are not always present. In fact, other sources can induce torsion as the default of symmetry of the muscle axis, which we will discuss later. In Fig.6(F)-(G), we demonstrate that the curvature provided by the model increases as muscle activation increases and that the torsion is not simply related to the activation amplitude since it also depends on the value of the angle α_a, reaching a maximum at approximately π/4. Detailed calculations are shown in Section Methods and Materials and Appendix 4.

Figure 6.

Energy transformation and Elongation

During the late elongation process, the four internal muscle bands cyclically contract 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 torsion-bending energy to convert into elongation energy, leading to a length increase during the relaxation phase, as shown in Fig.1 of the Appendix 5. With all deformations obtained, Eq.(2) can be used to calculate the accumulated energy W_c produced by both the muscles and the acto-myosin activities during a contraction. Subsequently, when the muscles on one side relax, the worm body reverts to its original shape but with a tiny elongation corresponding to the transferred elastic energy. This new state involves the actin network adopting a ‘loop’ configuration with a strain of εg_a1 once relaxation is complete. If all accumulated energy from the bending-torsion deformation goes towards elongating the worm body, the accumulated energy W_c and energy W_r following muscle relaxation are equivalent. The activation of actin fibers g_a1 after muscle relaxation can be calculated and determined by our model.

Once the first stage is well characterized, we can quantify the total energy resulting from both muscles and acto-myosin after each contraction. Then, we used our model to predict elongation in the 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 length of the wild type elongates from approximately 90 µm to 210 µm during the muscle-activated phase (Lardennois et al. (2019)), the phase lasts about 140 minutes, and the average time interval between two contractions is about 40 seconds (Yang (2017)), so the number of contractions can be estimated to be around 210 times. Unfortunately, due to the difficulty to realize quantitative experiments on an embryo always in agitation in the egg shell between bending, torsion, and also rotations around its central axis, one can hypothesize several scenarios: at each step i between state A_i,0 to A_i,1 and then A_i,2, the whole mechanical muscle-myosin energy is transferred to the elongated step A_i,2, leading to a small δζ_i.

Considering the experimental results shown in Fig.7(B), we determine the optimal values for the activation parameters: g_m = −0.15 and g_a = −0.01 assuming that all the accumulated energy during muscle activation is transferred to elongation(W_r = W_c). The elementary elongation δζ_i will be gradually increased with time, which is shown as the black line in Fig.7(A). At the beginning, δζ_i is about 0.5µm, but at the end of this process, δζ_i is about 1.5µm, indicating that the worm will elongate up to 290µm. The result is significantly higher than our actual size 210µm. When the elongation proceeds, we assume a transfer of energy between bending-torsion-contraction and elongation but it may be not fully effective, which means a significant part of the energy is lost. From the experimental data, we evaluate 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 (W_r = 0.4W_c). It induces for δζ_i a first increase and then a decrease, which is shown as the blue line in Fig.7(A) and is responsible for the slowdown around 200 mins. This option, which may be not the only possible one, leads to the estimated elongation having a good agreement with experimental data (see the blue-dashed curve in Fig.7(B)). Indeed it is possible that the C. elegans elongation requires other transformations which will cost energy. As the embryo gradually elongates, energy dissipation and the biomechanical energy required to reorganize the actin bundles may be two factors that contribute to the increased energy loss that underlies the hypothesis.

Figure 7.

Moreover, it was reported in (Norman and Moerman (2002)) that the knockdown of unc-112(RNAi), known for impairing muscle contractions, results in the arrest of elongation of embryos at the twofold stage, indicating that muscles have no activation, g_m = 0 in our model, and no accumulated energy can be converted into elongation. Another mutation concerning 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 the pre-stretch caused by myosin will not be maintained and will decrease. These aforementioned findings are fully consistent with a variety of experimental observations and are shown in Fig.7(B).

Embryo rotations and dissipation

The main manifestation of the muscle activity, independently of the elongation, is probably the constant rotations 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 approximately ±45^◦ and the deviation β_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 geometrical factor about a_g = sin(6π/180) cos(π/4). Then a simple estimation of the muscle activity in terms of torque reads Λ_m ∼ µ_mπR³s_mp_ma_g(εg_m) where s_m is the surface of the muscle pair on the left (or right) compared to the section of the cylinder: s_m = 0.025 and p_m is the distance of the muscles from the central axis of the embryo: p_m = 0.75 while g_m = 0.15 according to the analysis of the elongation. So the muscles on one side contribute to a torque Λ_m along the axis of symmetry given by Λ_m = 4.657µ_mπR³ · 10⁻⁵.

Let us consider now the dissipative torque, assuming that the dynamics of rotation is stopped by friction after one bending event. Two cases can be considered: either the dissipation comes from 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 egg shell since the anterior-posterior axis remains parallel to the egg-shell (Bhatnagar et al. (2023)) The interstitial fluid, of viscosity η contains a significant amount of sugar and other molecules which are required for embryo survival and then is more viscous than water (Labouesse (2023)). However, values for sucrose or sorbitol at the concentration of 1 mole/liter indicates a viscosity of order a few times the viscosity of water, which is 1 mPas. For example at 0.9 mole/liter and temperature of 20^◦, an aqueous solution with sorbitol has a viscosity of 1.6 mPas, which can be extrapolated to η = 1.9 mPas at 1.2 mole/liter (Jiang et al. (2013)). The estimation given by an embryo located in the middle of the egg-shell, gives a weaker viscous torque once evaluated by according to a classical result reported by Landau et al. (Landau and Lifshitz (1971, 2013)). It has to be mentioned 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 8.2 µm, the length is 90 µm,the radius of the shell about R_egg = 15µm and the length L_egg = 54µm, see Fig.6(D). The angular velocity Ω_e is more difficult to evaluate but it is about 90^◦ per two seconds deduced from videos. When the embryo approaches the egg shell, the friction increases, and two eccentric cylinders of different radii have to be considered with the two axes of symmetry separated by a distance d. The hydrodynamic study in this case is far from being trivial, and seems to have been initiated first by Zhukoski (Zhukoski (1887)) who suggested the use of bipolar coordinates for the mathematical treatment. Many following contributions established with different simplified assumptions have been published after and the study was fully revisited by Ballal and Rivlin (Ballal and Rivlin (1976)). Here we focus on the limit of a small gap δ between the rotating body-shape and the egg and by considering an asymptotic analysis at small δ of the general result derived in (Ballal and Rivlin (1976)). Thus, the viscous torque reads is the zone of contact with the egg and d = R_egg − δ − R. This approach is an approximation since the embryo has more of a torus shape than a cylinder (Yang Xinyi et al. (2023)) but the evaluation of the dissipation is satisfactory for δ = 0.5µm, Ω_e ∼ π/4 s⁻¹ and µ = 10⁵P a. Coming back to the first model of dissipation with the same data, the ratio between the dissipative viscous torque and the active one gives: Λ_υ/Λ_m = 0.02, which is obviously unsatisfactory. Finally, the dissipative energy ? during one bending event leading to an angle of π/2 is ξ_diss = 1/2Λ_m ×(π/2)² which represents 4% of the muscle elastic energy during the bending so at the beginning of the muscle activity (Appendix 5, Eq.(45)), the dissipation exits but is negligible. At the very end of the process, this ratio becomes 60% but as yet mentioned our estimation for the dissipation becomes very approximate, increases a lot due to the embryo confinement, and does not involve the numerous biochemistry steps necessary to reorganize the active network: acto-myosin and muscles.

Discussion

Since the discovery of the muscle activity before the egg hatch of the C. elegans embryo, it has become critical to explain the role of mechanical forces generated by muscle contraction on the behavioral and functional aspects of the epidermis. We provide a mechanical model in which the C. elegans is simplified as a cylinder, and the muscle bands and actin that drive its elongation are modeled as active structures in a realistic position. We determine the fiber orientation using experimental observations and then calculate the deformation by tensorial analysis involving the strains generated by the active components. Although a special focus is made on late elongation, its quantitative treatment cannot avoid the influence of the first stage of elongation due to the acto-myosin network, which is responsible for a pre-strain of the embryo. In a finite elasticity formalism, the deformations induced by muscles in a second step are coupled to the level of strains of the initial elongation period. For that, we need to revisit the theory of the acto-myosin contraction and previous results (Ciarletta et al. (2009); Vuong-Brender et al. (2017a); Ben Amar et al. (2018)) to unify the full treatment. In particular, a model for the recruitment of active myosin motors under forcing is presented which recovers the experimental results of the first elongation stage.

The elongation process of C. elegans during the late period is much more complex than the early elongation stage which is caused only by actin contraction. During the late elongation, the worm is distorted by the combined action of muscle and acto-myosin, resulting in an energy-accumulating process. Bending deformation is a phenomenon resulting from unilateral muscular contraction, and during the late elongation, significant torsional deformation is observed, indicating that the bending process induces a reorientation of the actin fibers. It is worth noting that the embryo is always rotating in tandem with the muscle activity making difficult any experimental measurements. However, our model can predict that if the muscles are not perfectly vertical, torque exits that causes rotation and eventually torsion. The accumulated energy is then partly turned into energy for the ongoing action of actin, allowing the embryo to elongate when the muscle relaxes. Both sides of the C. elegans muscles contract in a sequential cycle, repeating the energy conversion process, and eventually completing the elongation process. However, the energy exchange between bending and elongation is limited, among other factors, by the viscous dissipation induced by rotation, which is also evaluated in this study. Not investigated in detail here is the necessary re-organization of the active networks (acto-myosin and muscles) due to this tremendous shape transformation of the embryo. In parallel to elongation, the cuticle is built around the body (Page and Johnstone (2007)). This very thin and stiff membrane ensures protection and locomotion post-hatching. Clearly, these processes will perturb muscle activity. These two aspects which intervene in the final stage of the worm confinement play a very important role at the frontier across scales between genetics, biochemistry and mechanics.

Finally, the framework presented here not only provides a theoretical explanation for embryonic elongation in C. elegans, it can also be used to model other biological behaviors, such as plant tropism (Moulton et al. (2020b)) and elephant trunk elongated (Schulz et al. (2022b,a)) and bending. Our ideas could potentially be used in the emerging field of soft robotics, like octopus legs-inspired robots (Kang et al. (2012); Nakajima et al. (2013); Calisti et al. (2015)), which is soft and its deformation induced by muscles activation. We can reliably predict deformation by knowing the position of activation and the magnitude of forces in the model. Furthermore, residual stresses can be incorporated into our model to fulfill design objectives.

Methods and Materials

The model has been presented in a series of articles by A. Goriely and collaborators (Goriely and Ta- bor (2013); Moulton et al. (2013, 2020b); Kaczmarski et al. (2022); Goriely et al. (2022)). As one imagines, it is far from triviality and most of the literature on this subject concerns either full cylinders or cylindrical shells in torsion around the axis of symmetry, the case of bending is more overlooked. However, the geometry of the muscles in C. elegans leads automatically to a bending process that cannot be discarded. We take advantage of these previous works and apply the methodology to our model.

The deformation gradient

The axial extension ζ is obtained by r^′ (Z) = ζ d₃, where ^′ denotes the first derivative with respect to the material coordinate Z. From the director basis, the Darboux curvature vector reads: u = u₁d₁ + u₂d₂ + u₃d₃, this vector gives the evolution of the director basis along the filamentary line as: d_i^′ (Z) = ζ u × d_i, see Fig.3(A).

The finite strain G₀ can map the initial stress-free state ℬ₀ to a state ℬ₁, which reflects early elongation process. After in the residually stressed ℬ₁, we impose an incremental strain field G₁ maps the body to ℬ₂, which represents late elongation process. So, the deformation gradient can be expressed as: F = F_eG₁G₀ (Goriely and Ben Amar (2007)). The deformation gradient F = F_ij d_i ⊗ e_j, i ∈ {1, 2, 3} and j labels the reference coordinates {R, Θ, Z} finally reads:

where λ is the axial extension due to the pre-strained.

We define G₀ = Diag (1, 1 + εc, 1), since g₀ = 0.88 can be determined through the early elongation, we write it in the form of 1 + εc to simplify the calculation, c ≈ −0.6 (ε ≈ 0.2).

Let us consider fibers oriented along the unit vector m = (sin α sin β, sin α cos β, cos α) for the incremental strain G₁, where α and β characterise the angles with e_z and e and can be found in Fig.3(A). The active filamentary tensor in cylindrical coordinates is then G = G₀ (I + εg m ⊗ m), see (Holzapfel (2002)) which reads:

The energy function

The auxiliary strain-energy density can be obtained by expanding the inner variables a_i p_i and the potential V (see Eq.(2)) as:

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

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

We solve these equations order by order and consider incompressibility. At the lowest order, the incompressibility imposes the deformation: a⁽⁰⁾ = (r (R), Θ, 0) and the Euler-Lagrange equation gives the Lagrange parameter p⁽⁰⁾ = P₀(R) which finally read:

and the boundary condition σ_rr(R = 1) = 0 gives the value of P₀ = 1/λ. At 𝒪 (ε), the Euler-Lagrange equations are again automatically satisfied, and at 𝒪(ε²), the crucial question is to get the correct expression for and p(1). Based on the previous subsection, the following form is instuited.

As before, the Euler-Lagrange equations and incompressibility condition give the two constants and h₁, h₂, h₃ are a function of R. With the solutions for active strain a⁽⁰⁾ and a⁽¹⁾, 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 A_i, B_i, C_i and D_i are functions of R and Θ.

The method for determining deformation

Comparing with the energy of an extensible elastic rod (Moulton et al. (2020a); Kirchhoff and Hensel (1883); Mielke (1988); Mora and Müller (2002, 2004)), we recognize the classic extensional stiffness K₀, bending stiffness K₁ and K₂, torsional stiffness K₃ coefficients:

where A₂, B₂, C₂ and D₂ are related to the shear modulus µ, so for a uniform material with no variations of shear modulus, 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 curvatures of the cylindrical object induced by the active strains, this requires the competition of the active forces with the stiffness of the cylindrical bar following the relationships,

where H_i calculated by:

where A₁, B₁, C₁ and D₁ are related to the shear modulus µ,the fiber angles α and β, and the activation g. So, the integration region 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, for the case of activated left side muscles, we can calculate intrinsic extension and curvatures by Eq.16), û_1m = 0 and û_3m = 0, so the intrinsic Frenet curvature and torsion Eq.(18), and . For the activated actin case, we can calculate intrinsic extension and curvatures by Eq.(16), û_1a = 0 and û_2a = 0, and the intrinsic Frenet curvature and torsion Eq.(18), and . These quantities have been used to calculate the deformation images, as shown previously in Fig.6.

After obtaining all solutions, we can use these quantities to calculate the accumulated energy W_c in the system after one contraction by Eq.(2). By defining the activation (g_m and g_a) and the conversion efficiency of energy, we can obtain the activation (g_a1) of the actin bundles after once contraction and further calculate the elongation.

All calculation results and parameters are presented in the Appendix.

Acknowledgements

It is a pleasure to acknowledge fruitful discussions with Michel Labouesse and Kelly Molnar about experimental observations concerning the embryonic C.elegans elongation. We also thank Alain Goriely for his help with the technical aspects of nonlinear elasticity. Both authors acknowledge the support of the contract EpiMorph (ANR-2018-CE13-0008). Anna Dai acknowledges the support of the CSC (China Scholarship Council), file No. 201906250173.

Appendix 1

Size and material parameters

Appendix 1–table 1.

Appendix 2

Analytical model of the early elongation

The finite deformation gradient: F₀ = Diag(r^′(R), r(R)/R, λ), and G₀ = Diag(1, g₀, 1), 0 < g₀ < 1 in the actin layer, and g₀ = 1 in the part without actin budles. To determine the finite strain G₀, we converted the four muscle parts into thin layers attached to the epidermis in equal proportions to ensure the continuity of the model, and divide our model into four parts, see Fig.1. The actin layer(R₃ < R < R₂^′), epidermis layer(R₂ < R < R₂′) and muscle layer(R₁^′ < R < R₂) are considered as incompressible materials, and the inner part(0 < R < R₁′) is compressible. The Neo-Hookean energy function is used for the incompressible parts:

Appendix 2–figure 1.

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

where A, E and M are constant, and the Lagrange multiplier p in the actin, epidermis and muscle layer:

For the compressible part, we take the energy function form (Holzapfel (2002)):

where κ is a material constant. The radius of the inner part:

where a is a constant.

By considering the boundary condition σ_rr = 0 on the outer border R = 1:

the continuity of the radius:

and the continuity of the radial stresses σ_rr in R₂^′ and R₂ :

According to the above equations, we can obtain expressions for constants E, M, C_a, C_e and C_m.

We substitute all size, material parameters and κ = 100, λ = 1.8 into the last condition σ_mrr (R₁′)= σ_irr (R₁′):

where σ_irr:

and obtain the relationship with constant A and g₀.

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

where:

all solutions g₀ = 0.88, A = 0.08 and 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)).

Appendix 3

Time-scale for myosin detachment

The time required for non-muscle myosin detachment is estimated to be τ₀ = 10s for free acto-myosin filaments. If the actin filament is submitted to external loading perpendicular to its axis, the detachment can be helped 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 delay the detachment. This energy has to 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 l_a of radius r_b = 0.05µm estimated from (Lardennois et al. (2019)) and shear modulus µ_a = 5 KP a is given by . This result must be compared to the individual energy of detachment times the number of myosin motors on a cable. This number is uneasy to fix but estimation is given by the length of the cable l_a divided by the distance between 2 anchoring sites of myosin which is about 5nm while the attachment energy per motor is about 6k_bT. These indications are for skeletal muscle myosins (Howard and Clark (2002)) and have to be taken with caution. Nevertheless, the order of magnitude of the debonding energy for a collection of myosin heads from actin cable can be estimated to 4.8l_a10⁻¹²J. Then the ratio between both quantities is of order 4(1 −g₀(t)) which explains that the time-scale of debonding for a cable in compressive stress in the orthogonal direction of its axis is then:

where p₃ is a positive constant of order one which cannot be predicted exactly. This time scale justifies the exponential correction in Eq.(3) of the manuscript.

Appendix 4

Modelling details of without pre-strain case

In the paper, we discuss the case with pre-strain. Here, more details about without prestrain (G₀ = I) case are displayed. To lowest order, the solution of the Euler–Lagrange equations is obviously given by a⁽⁰⁾ = (R, Θ, 0), and p⁽⁰⁾ = 1 so there is no deformation and the zero and first order of energy vanish. At order 𝒪(ε²) of the elastic energy, the solution for the active strain components are:

where f_1,2 are functions related to the active stress g and fiber angles α and β:

The coefficients for determining stiffness:

The coefficients for determining deformation:

Modelling details of with pre-strain case

The first order solutions of the theory with the pre-strain case, h_1,2,3 (R) are related to activation g and fiber angles α and β:

The coefficients for determining stiffness:

The coefficients for determining deformation:

Appendix 5

Energy transformation calculations

The energy transformation process is depicted in Fig.1.

Appendix 5–figure 1.

To obtain the elongation δζ_i after each muscle contraction, we need to calculate the energy, and the total energy takes the following form:

where the integration region S is related to each part of the cylinder with a different shear modulus µ, so the model must be divided into 3 different parts for integration.

The final part of the energy conversion per unit volume is then:

where V₁ is not the whole first order energy, we only consider the energy induced by activation of acto-myosin g_a and muscles g_m. After obtaining solutions a⁽⁰⁾,a⁽¹⁾ and deformations from Eq.(11)-(12), the accumulated energy during the contractile period W_c that we define is:

When the muscles are relaxed and only acto-myosin is activated, the total increase of volumetric energy W_r is then:

By calculating energy conversion, we obtain g_a1 = −0.66 at the beginning of the late elongation phase, Fig.7 of the manuscript shows the elongation for each contraction and total elongation varies with time.