Fig 1.
Biophysics of root growth in soil.
(A) Roots grow when turgor pressure exceeds the mechanical resistance of cell walls and the friction and compression from the soil. (B) In cellular models of plant tissues (top), cells are divided into cell wall segments and the balance of forces is computed on each vertex [9]. A simplifying approach is to represent the cell as a point in space (particle) and to define the interactions between other cells as a function of the size of the cell, its distance to neighbors and its shape (bottom). Mathematically, a kernel function is defined to interpolate field variables such as mechanical stress (turgor pressure or tensile stress in cell walls) or velocity, and numerical approximation of particle dynamics reduces to the weighted sum of variables attached to each particle, with the weight given by a kernel function.
Fig 2.
(A) The interaction between cells is described through the kernel function (blue) associated with the particle (see Method A in S1 Text), which is a function that decreases with the distance from the particle center with the smoothing length. . It can be used to recover various field quantities, here the wall potential which indicates the likeliness of the position of the cell wall. (B) Ellipsoids define the size and shape of the cell. An ellipsoid is defined by its principal axes (here long in green, short in red) and used to generate the wall potential (background color, yellow being the most probable location of a wall and orange being the ridge). (C) The cell division occurs along the longest axis and split the mother cell into equally sized daughter cells.
Fig 3.
Impact of anisotropy of mechanical properties on the morphology of a growing tissue.
(A) Three cases were used to test the ability of the model to describe tissue morphogenesis. The isotropic case (A, left) was modeled using a radial Young modulus ER equal to the axial Young modulus EA. The anisotropic case (A, right) was modeled using ER equal to the maximal radial Young modulus EH. The balanced case (A, center) was modeled using a ER increasing smoothly from EA at the tip to EH at the base, using parameters shown in Table 1. Arrows describe the resulting strain rates in the growing tissue with the axial strain εA shown in red and radial strain εR shown in green. Simulations were initiated with particles distributed on a 2D cartesian lattice. (B) Comparison of SPH predictions with analytical solutions for the isotropic, anisotropic, and balanced cases for axial strain (B, left) and radial strain (B, right). Dashed curves represent analytical solutions and solid curve numerical results. The isotropic, balanced, and anisotropic cases are represented in blue, green, and orange, respectively. Analytical solutions are computed from Eq 8.
Fig 4.
Simulation of unimpeded growth of Arabidopsis thaliana roots using axial strain rate data, cell length data and cell division rate data derived from [20].
(A) Simulations of the growth of a root for 50 hours. Colors indicate the number of cell divisions from the beginning of the simulation. Blue cells have not divided during the simulation. Red cells have divided three times. Simulations were initiated with particles distributed on a 2D cartesian lattice. (B) Comparison between model and experimental data. Comparison between experimental and SPH predictions for cell division rate (Left). Comparison between experimental and SPH predictions for axial strain rate (Center). Comparison between experimental and SPH predictions for cell lengths (Right). Symbols are mean ± SE over 40 consecutive time steps. Experimental observations are represented by orange dashed lines and numerical simulations by blue solid lines.
Table 1.
Description of model parameters.
When cases involve more than one set of parameters, differing parameters are listed into brackets.
Fig 5.
Simulation showing how the softening and anisotropy coefficients vary in the roots of Pisum sativum in response to increases in soil strength using strain rate data from [15] and root radius data from [16].
(A) Results of SPH simulation for a pressure differential of 1 (top) and 0.4 (bottom). Simulations were initiated with particles distributed on a 2D cartesian lattice. Only the mechanical parameters of the model were adjusted to data. The cell division rate was set to a constant threshold of 120 μm which produced reasonable average cell sizes and numerical stability. (B) To match experimental data, the softening coefficient λ was found to reach a maximum closer to the tip and tail off quicker for harder soils (top left). Such variations in the softening coefficient along the root allowed the SPH model to match the axial strain rate observed experimentally (top right). The radial modulus ER was found to take smaller values at the tip but increased quicker to reach its maximum for harder soils (bottom left). These variations of radial modulus along the root allowed the SPH model to match the variations of root radius observed experimentally (bottom right). Experimental observations are represented by dashed lines and numerical solutions by solid lines. Increase in the pressure differential is represented from light blue to dark blue.
Fig 6.
Future applications for SPH models in root developmental biology.
Image processing pipelines currently available to analyze 3D live microscopy data can extract the geometrical properties of cells (top left). These were used to compute the shape matrix of particles Qi and were used as input for SPH simulation (top right). The model then could compute the growth of plant roots, including cell size (bottom left, colored ellipsoids) and strain rate (bottom left, arrows show radial velocity). Kernel functions could then be used to compute cell walls (bottom right). Here, the root is represented either as a whole (top) or following a cross section (bottom).