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Stress and Strain Provide Positional and Directional Cues in Development

Figure 3

Comparison between stress and orthogonal strain based feedback models.

The results of the three distinct relations between mechanical stress/strain and anisotropy of the material in different loading force situations are analyzed. The first row (A, B, and C) pertains to the predefined and static direction of material anisotropy. The second row (D, C and F) describes the results of stress feedback model and the third row (G, H and K) the orthogonal strain feedback model. The first column (A, D and G) presents the results of the simulation of anisotropic biaxial loading of a square patch from Figure 1C. For varied anisotropy of the loading force (vertical axis in the graphs) and the ratio of Young moduli along each of the load directions (horizontal axis in the graphs), the cosine of the angle between maximal stress and strain directions is plotted with a gray-scale map. Force anisotropy and elasticity ratio in A, D and G are calculated by and , respectively. Force anisotropy 0 corresponds to isotropic loading and elasticity ratio 1 to an isotropic material. The gray dashed line in panel A and circles in panel D are discussed in the main text. The second column (B, E and H) shows the equilibrium state of fiber directions (red bars) in the cylindrical part of the tissue pressure model simulation for the template shown in the Figure 1D. The third column (C, F and K) pictures the distributions of the stress, strain and fiber directions in the cells with respect to the circumferential (horizontal) direction resulting from the tissue pressure model simulation. (A) For the fixed anisotropy direction (no feedback mechanism present) we observe distinct regions in the parameter space where maximal stress and strain directions are either mutually parallel (white) or perpendicular (black). (B) In the stem template simulations the anisotropy (fiber) direction is prealigned and set to circumferential. (C) This results in a maximal stress direction parallel to the fiber direction (circumferential) and maximal strain direction orthogonal to the fiber direction (longitudinal). (D) In the stress feedback model the identity of the regions of mutually parallel (black) or orthogonal (white) relation between the maxima stress and strain directions is maintained from the no-feedback case A. The yellow circle in D shows the approximate value for force and material anisotropy on the side of a cylinder where anisotropic curvature results in force anisotropy about 0.5. (E) In this model fibers are dynamically aligned in the direction of the maximal stress and the circumferential orientation of them arises spontaneously in the stem template simulation. (F) Similarly to the static case (first row) the maximal strain direction is perpendicular to the stress and fiber directions ie. longitudinal. (G) For the orthogonal strain feedback model the maximal stress and strain directions are always parallel in contrast to A and D. (H) In this case fibers are dynamically updated to match the direction orthogonal to maximal strain. This results in unstable initial circumferential alignment of fibers which realign in the longitudinal direction. (K) Both maximal stress and strain directions are perpendicular to the fiber directions ie. circumferential. The parameters used in the simulation with the pressurized template in Figure 1D were: thickness = 1 , cell size 10 to 20 , = 0.1 , = 0.2, = 50 , = 120 , fiber model with = 0.4 and = 2, deformation is between 5% to 10% (B)6%, (E) 6%, (H) 10%).

doi: https://doi.org/10.1371/journal.pcbi.1003410.g003