Figure 1.
Mechanical models and templates.
(A) Geometry of a quadrilateral shell element for the finite element method. The thin three-dimensional surface is parametrized by a two-dimensional shell with implicit thickness and set of director vectors D (Text S1). (B) An element used in the triangular biquadratic spring model. and
represent positions and edge lengths in resting and deformed state, respectively. The strain tensor can be expressed in terms of edges of the element in resting and deformed states. (C) The quadrilateral patch used for comparing triangular biquadratic springs and finite element shell models. (D, E) Different templates representing selected plant-like geometries used in tissue pressure simulations.
Figure 2.
Comparing triangular biquadratic springs and finite element shell models.
(A, B) Uniaxial stretching test on a quadrilateral patch shows prefect agreement within numerical accuracy between both methods for principal stress and area ratio versus deflection of top right corner of the quad. Isotropic material (Young modulus = 400 , Poisson ratio = 0.2 and 0.4, thickness = 0.01
, size = 1
, force = 8
). (A) Principal stress. (B) Area ratio. (C) Principal stress value for isotropically loaded patch with
force for the same patch using TRBS method where Young modulus and Poisson ratio were varied. The difference between principal stress value in TRBS method and integrated principal stress over thickness in FEM shell model is less than 0.1% (Figure S1A). (D) First and second principal stress values for the same patch of anisotropic material with transverse and longitudinal Young modulus of 400 and 800
respectively and Poisson ratio of 0.2, under 0.8
and 0.2
anisotropic loading force. The anisotropy direction was varied between 0 deg (maximal force direction) and 180 deg. (E, F) Bending test results from pressurizing a patch of elements. (E) Principal stress direction and principal strain value for TRBS (left) and shell (right). The material is isotropic with Young modulus 400
and Poisson ratio 0.2. Number of elements is 400 and 250 for shells and TRBS, respectively. (F) Distribution of equivalent Mises strain value over elements. TRBS elements show slightly higher strain values because of the lack of bending energy. Average equivalent Mises strain over elements: 0.0527 and 0.0492 for TRBS and shell, respectively.
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%).
Figure 4.
Zonation properties of the stress feedback model in meristem-like geometries.
(A) The stress feedback together with fiber model for a paraboloid representing the geometry in the central zone and its close neighborhood results in two distinct zones where maximal stress and strain directions are either parallel (white) or perpendicular (black). The red bars(here and panel D) show fiber directions (B, C) Area expansion and material anisotropy (elasticity ratio) show different properties in these two regions. The elastic deformation is larger and radially oriented in the peripheral zone and the material is anisotropic whereas in the central zone deformation is less and the material becomes more isotropic. The blue lines (in the panels B and E) are showing the maximal strain directions. (D, E, F) The same results as A, B and C respectively for a meristem-like template. Maximal strain and stress directions are aligned at the apex and valley because of almost isotropic material and anisotropic stress respectively. For the meristem-like template due to the large variability of stress value in different regions the absolute stress anisotropy measure with is used. The parameters used for pressurized templates in Figure 1E were: thickness
= 1
, cell size about 10
,
for paraboloid = 0.05
and for meristem = 0.08
,
= 0.2,
for paraboloid = 40
and for meristem = 50
,
for paraboloid = 100
and for meristem = 150
, fiber model with
= 0.4,
= 2. The deformation is within 5% to 7% for paraboloid and within 1% to 9% for meristem.
Figure 5.
Stress and orthogonal strain feedback models impact on geometry.
(A, B, C) Comparing stress and orthogonal strain feedback models for a set of templates with different geometric anisotropies which is considered here as the ratio between principal axes. This ratio is 1 for the sphere and increases for more elongated templates. (A) Higher anisotropic growth can be seen for the stress feedback model (red) compared to orthogonal strain feedback model (white). (B) The deformed shape anisotropy versus resting shape anisotropy for different feedback models. Values are normalized corresponding simulations with isotropic material of the same overall elasticity. The results show that even for a low deformation the stress feedback model increases shape anisotropy whereas orthogonal strain feedback model decreases this value, indicating that strain based feedback results in more symmetric geometry. (C) Strain anisotropy averaged over elements for simulations with the two feedback mechanisms are plotted versus resting shape anisotropy. The values are normalized to the corresponding simulations with an isotropic material of the same overal elasticity. In case of stress feedback the results are consistently lower than orthogonal strain feedback. (D) Comparing deformations resulting from different feedback models for the meristem-like pressurized template with the same parameters as Figure 4. More anisotropic growth in the stress feedback model (red) compared to the orthogonal strain feedback model (white) promotes the outgrowth of the primordium. The material parameters used in simulation were: =
, thickness
= 0.01
, pressure
= 1.5
. The radius of the sphere is
= 1
, isotropic
= 8
,
= 12
,
= 4
.
Figure 6.
In the fiber model mechanical anisotropy is adjusted based on an anisotropy measure dependent on stress or strain in such way that the overall elasticity of the material is conserved. The plot shows result of using ,
and
between 0 and 1 in Equation 7. In our simulations model parameters were chosen such that material anisotropy was close to its maximum when stress anisotropy was about 0.5.