(A) Overview of C. elegans embryonic development. Three epidermal cell types are found around the circumference: dorsal, ventral and seam cells. (B) Schemes showing a C. elegans crossn section of the embryo. Small yellow arrows in the left image indicate the contraction force that occurred in the seam cell. Four muscle bands under the epidermis and actin bundles surround the outer epidermis.

Architecture of the program. The program reflects the framework of the research. On the one hand, the proposed model explains the early and late elongation of the C. elegans, on the other hand, the early myosin activity is estimated, the deformations (bending, twisting, rotation) occurring in the late period are recovered, and the estimation of the energy dissipated during muscle activity is achieved.

Adopted real size parameters of the C. elegans (Vuong-Brender et al. (2017a); Ben Amar et al. (2018)).

(A) Cylindrical structure in the reference configuration (left) with a vertical centerline and its deformation in the current configuration (right). The deformed configuration is fully parameterized by the centerline r (Z) (resulting from the distortion of the central axis) and the deformation of each cross section. (B) Schematic representation of the body shape of the C. elegans embryo with the cortical epidermis and the four muscles. The fibers are embedded in the cortex. The blue part representing the epidermis shows the outer distribution of the actin organized in horizontal hoop bundles when the muscles are not activated. The yellow part includes the vertical red muscles, represented by axial fibers.

(A) Bending of a giraffe’s neck. (B) Torsion of an elephant’s trunk. (B) Twisting of a plant vine. (D) to (F) Deformation configuration under different activations obtained by our simulations for bending and torsion of large rods, twisting and torsion of thin rods.

(A) Schematic of early elongation and the cross section of C. elegans. In the cross-section, the black circular part is the actin region (R2′ < R < R3′ with shear modulus μa), the blue part is the epidermis layer (R2 < R < R2′ with shear modulus μe). The central or inner part (0 < R < R2 in white has shear modulus μi, except for the four muscles, the shear modulus of the four muscles μm is much larger than the inner part. (B) Predictions of normalized embryo radius evolution during early elongation by the pre-strain model compared to experimental data from (Vuong-Brender et al. (2017a)). For the model, see Eq.(20) in Appendix 2. (C) Blue dots: extraction of the parameter g0(t) from Eq.(30) and Eq.(32) in Appendix 2. Blue dash line, see Eq.(3).

(A) Schematic diagram of C. elegans muscle fibers and its cross section, and it does not show the actin fibers. There are four muscle bands exist in the yellow layer. However, the yellow region is not an actual tissue layer and it is only used to define the position of the muscles. (B) Deformation diagram, left side muscles M1 and M2 are activated. (C) Deformation diagram, right side muscles M3 and M4 are activated. (D) Schematic diagram of C. elegans actin fibers and cross section. (E) Once the muscle is activated, the actin fiber orientation changes from the ‘loop’ to the ‘slope’, resulting in torque. (F) Schematic diagram of torsional and bending deformation.

Deformed configurations for different activation for muscles, (A) gm = −0.02, αa = π/3, ga = −0.01. (B) gm = −0.05, αa = π/4, ga = −0.01. (C) gm = −0.08, αa = π/6, ga = −0.01. (D) The graphs were captured from the Hymanlab, and the website: https://www.youtube.com/watch?v=M2ApXHhYbaw. The movie was acquired at a temperature of 200C using DIC optics. (E) gm = −0.1, αa = π/4, ga = −0.7. (F) Curvature is plotted as a function of muscle activation. (G) Torsion is plotted as a function of the actin activation and angle of actin fibers.

(A) The elongation for each contraction varies with time. Black line: all energy converted to the elongation, blue line: partial energy converted to the elongation. The activation: gm = −0.15, ga = −0.01. (B) The model predicted results agree well with the experimental data of wild-type and different mutant C. elegans embryos (Lardennois et al. (2019)). Activation of the wild-type model (blue dashed line): gm = −0.15, ga = −0.01. The activation of unc-112(RNAi) (brown dashed line): gm = 0, ga = 0. In the case of pre-stretch failure (green dashed line), λ decreases from 1.8.

Parameters adopted in this work

Simplified ross-sectional model with four scattered muscle sections simplified to thin layers (R1 = 0.7, R1′ = 0.768, R2 = 0.7, R2′ = 0.96).

Schematic diagram of energy conversion.