Fig 1.
Schematic of root parameters and growth kinematics.
The mechanism of differential growth for a single time step (Δt) is summarized for our modification of the 2-dimensional model of Tsutsumi [31]. (a) The root before and after a bend of θ. The red dot identifies the tip of the root, and the blue line highlight the diameter D. (b) A close view of the region with the bending. The differential growth is achieved by a different growth rate (Gleft>Gright in this example) obtained at opposite sides of the root. (c) A 3-D view of the cross-section of the root, the local coordinate system is shown before Ox0y0z0 and after Oxyz the differential growth. This 3-D section is representative for the shape of segments defined on the root at each iteration. Outside of the root is drawn the reference curve location (black line in the direction of the y axis), the point having highest growth rate (green dot) and the corresponding ϕ angle, which corresponds to the angle between the y axis and the point of highest growth rate. (d) The kinematics for the same growth step.
Fig 2.
Schematic depicting key parameters under gravitropic signal.
(a) A bending root responding to gravity, showing the root tip orientation , the gravity vector
, the angle αg gives the deviation of the root from the gravity, and the signal velocity
. (b) A cross-section of the root showing the angle ϕg in the circumference along which the maximal signal is propagated. The reference line is drawn in black in (a) and with a black dot in (b).
Fig 3.
(a) An example of root apex intersecting with an obstacle depicted by a horizontal gray line. The root is covered with tactile sensory points, colored in green when no intersection is detected and colored in red when an intersection is detected (e.g., here at the very tip). A black line depicts the reference curve. (b) The corrected root (a) after a minimalistic bending to remove the intersection with the obstacle. (c) Tactile sensory points, shown from the bottom (tip) view on the root, are equidistantly spaced w.r.t. the center of the root in this view. The points are located at different angles ϕ w.r.t. the reference curve (ϕ = 0, black dot) depicted by the black line in (a, b). The mean of the intersected (red) points depicted by a black star represents an estimated location of the occurred touch. (d) The root shape is defined by a green line which illustrates how the root (y-axis) radius changes as a function of the distance from the tip (x-axis). x and y represent the multiplier for D as unitary value. Knowing the position of the mean of the activated sensory points (black star) is sufficient to estimate the αt angle w.r.t. the obstacle (represented in the figure with the light gray triangle) as the angle between the centerline of the root and the tangent (dashed black line) at the starred point.
Fig 4.
Effects of signal speed, sensitivity, and zonation on root growth.
Two different root zonation peaks are compared (in colored scale, where red represents the location of maximum response): (a) the peak of the growth rate (%) is located at a distance of 1.5 times the average diameter of the formed root (d = 1.5D) from the tip; (b) the peak of the growth rate (%) is located at a distance of d = 3.0D from the tip. All the roots present the result after 16 h of gravitropic stimulations with a different propagation speed (vg) and sensitivity (wg). Magenta lines depict the trajectories of the tip. Colors show the rate (%) of elongation per h.
Fig 5.
The interplay of gravity and touch responses.
The sensitivity to gravitropic (wg) and thigmotropic (wt) signals was set at (a) 0.5 and 0.5, (b) 0.1 and 0.9, and (c) 0.9 and 0.1, respectively. Colors show the rate (%) of elongation per h. Magenta lines depict the trajectories of the tip. (d) Root angle w.r.t obstacle over time for roots with different wg and wt = 1−wg for τs = Δt (immediate update of touch signal) and τs = 10 (immediate increase but a slow decrease of touch signal).
Fig 6.
Roots navigate towards regions with higher resource concentration with circumnutation movements driven by an internal oscillatory apparatus.
The resource concentration ranges from 0 (blue color) and 1 (red color). In the rows are shown the roots navigation (a, b, c) in the presence of only the resource signal active (wg = wt = 0, ws≠0) and (d, e, f) as a result of the interplay between gravity and resource stimulus responses (wt = 0, wg≠0, ws≠0). (a) A sample of navigation with root settings ws = 1, vs = 5D/hour, Ts = 0.75π, d = 3.0D. (b) The path for a root with settings ws = 0.25, vs = 10D/hour, Ts = 0.94π, d = 2.0D. (c) A sample of a non-regular pattern of circumnutation movements obtained with root settings ws = 0.75, vs = 2.5 D/hour, Ts = 1.07 π, d = 4.5D. For simulations in (d, e, f), stimulus and gravity signals originate at the tip and propagate along the root with speed vs = vg = 5D/hour. The internal oscillator outputs ss = 1 at d = 0 and ϕ which changes over time with frequency Ts = π. Weights of resources and gravity for each root were: (d) ws = 0.9, wg = 0.1; (e) ws = 0.6, wg = 0.4; (f) ws = 0.1, wg = 0.9.
Fig 7.
An example of root specialization.
We investigated whether the complex ramification of the plant root systems could be approximated by the same relatively simple model for each root type, but with different root-specific (e.g., primary, secondary, crown roots) and task-specific (e.g., gravity-, resource-sensitive) parameter values. (a) Different roots (crown, primary, seminal and lateral roots) initialized with different sensitivity to gravitropic wg, thigmotropic wt, and resource ws signals to very roughly approximate their possible behavior (e.g., the crown is initialized with negative gravitropism and upward growth by setting wg = −1, seminal roots could be stimulated to circumnutate towards better environmental conditions barely affected by gravity when setting ws = 0.7). All roots could avoid obstacles when setting wt = 0.2. (b) Simulation of root specialization with parameters given in (a). Different kinds of roots were differently affected by the presence of the environmental resource, and the roots with greater ws to the resource concentration C tended to grow towards the middle of the box where the maximum of C scaled between 0 (blue color) and 1 (red color).