Models of Mechanics and Growth in Developmental Biology: A Computational Morphodinamics approach

Recent evidence has revealed the role of mechanical cues in the development of shapes in organisms. This thesis is an effort to test some of the fundamental hypotheses about the relation between mechanics and patterning in plants. To do this, we develop mechanical models designed to include specific features of plant cell walls. These are heterogeneous stiffness and material anisotropy as well as rates and directions of growth, which we then relate to different domains of the plant tissue.In plant cell walls, anisotropic fiber deposition is the main controller of longitudinal growth. In our model, this is achieved spontaneously, by applying feedback from the maximal stress direction to the fiber orientation. We show that a stress feedback model is in fact an energy minimization process. This can be considered as an evolutionary motivation for the emergence of a stress feedback mechanism. Then we add continuous growth and cell division to the model and employ the strain signal directing large growth deformations. We show the advantages of strain-based growth model for emergence of plant-like organ shapes as well as for reproducing microtubular dynamics in hypocotyls and roots. We also investigate possibilities for describing microtubular patterns, at root hair outgrowth sites according to stress patterns. Altogether, the work described in this thesis, provides a new improved growth model for plant tissue, where mechanical properties are handled with appropriate care in the event of growth driven by either molecular or mechanical signals. The model unifies the patterning process for several different plant tissues, from shoot to single root hair cells, where it correctly predict microtubular dynamics and growth patterns. In a long-term perspective, this understanding can propagate to novel technologies for improvement of yield in agriculture and the forest industry

Auxin Acts through MONOPTEROS to Regulate Plant Cell Polarity and Pattern Phyllotaxis.

The periodic formation of plant organs such as leaves and flowers gives rise to intricate patterns that have fascinated biologists and mathematicians alike for hundreds of years [1]. The plant hormone auxin plays a central role in establishing these patterns by promoting organ formation at sites where it accumulates due to its polar, cell-to-cell transport [2-6]. Although experimental evidence as well as modeling suggest that feedback from auxin to its transport direction may help specify phyllotactic patterns [7-12], the nature of this feedback remains unclear [13]. Here we reveal that polarization of the auxin efflux carrier PIN-FORMED 1 (PIN1) is regulated by the auxin response transcription factor MONOPTEROS (MP) [14]. We find that in the shoot, cell polarity patterns follow MP expression, which in turn follows auxin distribution patterns. By perturbing MP activity both globally and locally, we show that localized MP activity is necessary for the generation of polarity convergence patterns and that localized MP expression is sufficient to instruct PIN1 polarity directions non-cell autonomously, toward MP-expressing cells. By expressing MP in the epidermis of mp mutants, we further show that although MP activity in a single-cell layer is sufficient to promote polarity convergence patterns, MP in sub-epidermal tissues helps anchor these polarity patterns to the underlying cells. Overall, our findings reveal a patterning module in plants that determines organ position by orienting transport of the hormone auxin toward cells with high levels of MP-mediated auxin signaling. We propose that this feedback process acts broadly to generate periodic plant architectures.The research leading to these results received funding from the Australian Research Council (M.G.H.) and European Research Council under the European Union’s Seventh Framework Programme ( FP/2007-2013 )/ERC grant agreement 261081 (M.G.H.). The work was also supported by the European Molecular Biology Laboratory (N.B., C.O., and M.G.H.), EMBL International PhD Programme (N.B.), Gatsby Charitable Foundation ( GAT3395/PR4 ) (H.J.), and Swedish Research Council ( VR2013-4632 ) (H.J.).This is the final version of the article. It first appeared from Elsevier (Cell Press) via https://doi.org/10.1016/j.cub.2016.09.04

Anisotropic growth is achieved through the additive mechanical effect of material anisotropy and elastic asymmetry.

Fast directional growth is a necessity for the young seedling; after germination, it needs to quickly penetrate the soil to begin its autotrophic life. In most dicot plants, this rapid escape is due to the anisotropic elongation of the hypocotyl, the columnar organ between the root and the shoot meristems. Anisotropic growth is common in plant organs and is canonically attributed to cell wall anisotropy produced by oriented cellulose fibers. Recently, a mechanism based on asymmetric pectin-based cell wall elasticity has been proposed. Here we present a harmonizing model for anisotropic growth control in the dark-grown Arabidopsis thaliana hypocotyl: basic anisotropic information is provided by cellulose orientation) and additive anisotropic information is provided by pectin-based elastic asymmetry in the epidermis. We quantitatively show that hypocotyl elongation is anisotropic starting at germination. We present experimental evidence for pectin biochemical differences and wall mechanics providing important growth regulation in the hypocotyl. Lastly, our in silico modelling experiments indicate an additive collaboration between pectin biochemistry and cellulose orientation in promoting anisotropic growth

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

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

Limitations of field-theory simulation for exploring phase separation: The role of repulsion in a lattice protein model

Field-theory simulation by the complex Langevin method offers an alternative to conventional sampling techniques for exploring the forces driving biomolecular liquid-liquid phase separation. Such simulations have recently been used to study several polyampholyte systems. Here, we formulate a field theory corresponding to the hydrophobic/polar HP lattice protein model, with finite same-site repulsion and nearest-neighbor attraction between HH bead pairs. By direct comparison with particle-based Monte Carlo simulations, we show that complex Langevin sampling of the field theory reproduces the thermodynamic properties of the HP model only if the same-site repulsion is not too strong. Unfortunately, the repulsion has to be taken weaker than what is needed to prevent condensed droplets from assuming an artificially compact shape. Analysis of a minimal and analytically solvable toy model hints that the sampling problems caused by repulsive interaction may stem from a loss of ergodicity

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

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