Stress and Strain Provide Positional and Directional Cues in Development

<div><p>The morphogenesis of organs necessarily involves mechanical interactions and changes in mechanical properties of a tissue. A long standing question is how such changes are directed on a cellular scale while being coordinated at a tissular scale. Growing evidence suggests that mechanical cues are participating in the control of growth and morphogenesis during development. We introduce a mechanical model that represents the deposition of cellulose fibers in primary plant walls. In the model both the degree of material anisotropy and the anisotropy direction are regulated by stress anisotropy. We show that the finite element shell model and the simpler triangular biquadratic springs approach provide equally adequate descriptions of cell mechanics in tissue pressure simulations of the epidermis. In a growing organ, where circumferentially organized fibers act as a main controller of longitudinal growth, we show that the fiber direction can be correlated with both the maximal stress direction and the direction orthogonal to the maximal strain direction. However, when dynamic updates of the fiber direction are introduced, the mechanical stress provides a robust directional cue for the circumferential organization of the fibers, whereas the orthogonal to maximal strain model leads to an unstable situation where the fibers reorient longitudinally. Our investigation of the more complex shape and growth patterns in the shoot apical meristem where new organs are initiated shows that a stress based feedback on fiber directions is capable of reproducing the main features of in vivo cellulose fiber directions, deformations and material properties in different regions of the shoot. In particular, we show that this purely mechanical model can create radially distinct regions such that cells expand slowly and isotropically in the central zone while cells at the periphery expand more quickly and in the radial direction, which is a well established growth pattern in the meristem.</p></div

Mechanical models and templates.

<p>(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 <i>D</i> (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003410#pcbi.1003410.s008" target="_blank">Text S1</a>). (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.</p

Comparing triangular biquadratic springs and finite element shell models.

<p>(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% (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003410#pcbi.1003410.s001" target="_blank">Figure S1A</a>). (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.</p

Zonation properties of the stress feedback model in meristem-like geometries.

<p>(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 <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003410#pcbi-1003410-g001" target="_blank">Figure 1E</a> 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.</p

Comparison between stress and orthogonal strain based feedback models.

<p>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 <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003410#pcbi-1003410-g001" target="_blank">Figure 1C</a>. 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 <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003410#pcbi-1003410-g001" target="_blank">Figure 1D</a>. 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 <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003410#pcbi-1003410-g001" target="_blank">Figure 1D</a> 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%).</p

Stress and orthogonal strain feedback models impact on geometry.

<p>(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 <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003410#pcbi-1003410-g004" target="_blank">Figure 4</a>. 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 .</p

The fiber model.

<p>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 <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003410#pcbi.1003410.e084" target="_blank">Equation 7</a>. In our simulations model parameters were chosen such that material anisotropy was close to its maximum when stress anisotropy was about 0.5.</p

Orientation of the network.

<p>Distribution of the orientation of unit vectors (local orientation of microtubules) located close to the top and bottom faces. Top and bottom faces refer to the largest faces of the square shape represented in C, and to arbitrary opposite large faces of the cube or the long shapes. The reference angle 0 corresponds to the longest axis of the long cell and is parallel to one of the edges of the top face in the two other cases. (A,B) Distribution of orientations for sharp cube, sharp square, and sharp long shapes, as labeled in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006011#pcbi.1006011.g001" target="_blank">Fig 1</a>. (A) Weak anchoring and (B) strong anchoring. (C) Arrow showing the reference for angle measurement for the long shape (top) and for the square and cube shapes (bottom). The shapes are not at the same scale. Model parameters have the default values (see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006011#sec017" target="_blank">Methods</a>).</p

Shapes of simulated cells.

<p>(A) Top left: Two time steps of a simulation when the microtubule grows freely inside the cell and simultaneously shrinks from the minus end; at the plus side (arrowhead), the new vector is calculated by adding a small random deviation to the previous vector. Bottom left: Example of a zippering process—When the new microtubule encounters an existing one at a shallow angle (if the angle <i>a</i> is smaller than threshold <i>α</i>), the new direction of the vectors follows that of the leading microtubule; a steep angle would lead to plus end shrinkage. Top right: Strong anchoring to the plasma membrane—The microtubule grows tangentially to the local plane. Bottom right: Weak anchoring to the membrane—When the microtubule encounters the membrane at a shallow angle (<i>a</i> < <i>α</i>), it continues tangentially until the random deviation leads it to leave the membrane. (B) Shapes of the envelope of the cell: cube, “square”, and “elongated”. First row: smooth shapes. Second row: sharp shapes. Scales are not respected. (C,D) Example of a simulation in a sharp square with weak anchoring and default parameters (see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006011#sec017" target="_blank">Methods</a>); the length of the simulation is 50000 timesteps, corresponding to approximately 100 minutes, also see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006011#pcbi.1006011.s004" target="_blank">S2 Video</a>. (C) Snapshot showing all microtubules (left), microtubules located in a central slice that is ∼2.4<i>μ</i>m-thick (center), and the projection of this slice in 2D (right). (D) Snapshots taken after 10000 timesteps intervals (∼20 min); the pictures show a confocal microscopy-like <i>z</i>-projection (see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006011#sec017" target="_blank">Methods</a>).</p

Anisotropy of the network.

<p>(A) Effect of global shape: Anisotropy for the three shapes (sharp/smooth and weak/strong anchoring were pooled together). (B) Effect of curvature: Anisotropy for sharp shapes vs. smooth shapes with strong anchoring. (C) Effect of curvature: Anisotropy for sharp shapes vs. smooth shapes with strong anchoring. (D) Effect of anchoring strength: Anisotropy for all shapes (pooled). In all panels, the data are pooled except for the parameter that is varied in the subfigure; model parameters have the default values (see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006011#sec017" target="_blank">Methods</a>).</p