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

Architecture of the program. The program reflects the framework of 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) happening in the late period is recovered, and the estimation of 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 central axis 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 into horizontal hoop bundles when the muscles are not activated. The yellow part includes the vertical red muscles, represented by axial fibers. (C) Bending of a Giraffe neck. (D) Torsion of an Elephant trunk. (E) Plant vine twisting. (F) to (H) Deformation configuration under different activations obtained by our simulations for bending and torsion of large rods, twisting and torsion of thin rods.

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

(A) Schematic diagram of C. elegans muscle fibers and its cross section, and it does not show the actin fibers. Four muscle bands exist in the yellow layer. But, the yellow region is not an actual tissue layer and it is simply to define the position of muscles. (B) Deformation diagram, when left side muscles M1(θ1 ≤ θ0 ≤ θ2) and M2(θ3 ≤ θ0 ≤ θ4) (C) Deformation diagram, when right side muscles M3(θ5 ≤ θ0 ≤ θ6) and M4 (θ7 ≤ θ0 ≤ θ8). (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’, which results 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)). The activation of 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 pre-stretch failure case (green dashed line), λ will decrease from 1.8.

Parameters adopted in this work

Cross-sectional simplified model with four scattered muscle sections simplified to thin layers (R1′ = 0.768).

Schematic diagram of energy conversion.