A unifying modeling of plant shoot gravitropism with an explicit account of the effects of growth

Abstract

Gravitropism, the slow reorientation of plant growth in response to gravity, is a major determinant of the form and posture of land plants. Recently a universal model of shoot gravitropism, the AC model, was presented, in which the dynamics of the tropic movement is only determined by the conflicting controls of (1) graviception that tends to curve the plants toward the vertical, and (2) proprioception that tends to keep the stem straight. This model was found to be valid for many species and over two orders of magnitude of organ size. However, the motor of the movement, the elongation, was purposely neglected in the AC model. If growth effects are to be taken into account, it is necessary to consider the material derivative, i.e., the rate of change of curvature bound to expanding and convected organ elements. Here we show that it is possible to rewrite the material equation of curvature in a compact simplified form that directly expresses the curvature variation as a function of the median elongation and of the distribution of the differential growth. By using this extended model, called the ACĖ model, growth is found to have two main destabilizing effects on the tropic movement: (1) passive orientation drift, which occurs when a curved element elongates without differential growth, and (2) fixed curvature, when an element leaves the elongation zone and is no longer able to actively change its curvature. By comparing the AC and ACĖ models to experiments, these two effects are found to be negligible. Our results show that the simplified AC mode can be used to analyze gravitropism and posture control in actively elongating plant organs without significant information loss.

Keywords: control; gravitropism; growth; morphogenesis; proprioception.

Figure 1

Geometric description of a rod-like plant organ. (A) The arc length s is defined along the median line of the organ with s = 0 referring to the base and s = L the apex. The total length of the organ L can be broken down into the length of the growth zone L_gz, which is moving with the apex, and L_f, the length of the zone which is not growing. This fixed portion elongates with time as elements leave the active growth zone. A(s) is the local orientation with respect to the vertical. (B) To simplify this, only two parts of the organ are considered to be elongating, the red part, at a position defined at its base s = s₁, and the green part, at a position defined at its base s = s₂. During elongation, the curvatures of these two parts are modified. Due to elongation, their distance relative to each other is modified. The position s = s₂ no longer refers to the green element. To account for the variation in the material element, the derivative must follow the material element.

Figure 2

A small element of an organ is considered as a cylinder of radius R with the upper side of length δs₁ and the lower side of length δs₂. The sides are growing with a relative elongation rates of $\dot{ϵ}$₁ and $\dot{ϵ}$₂, respectively. After time t the curvature of the organ has changed.

Figure 3

The elongation rate of an organ. (A) When the organ is shorter than the length of the growth zone, L < L_gz, it is expected that the whole organ is elongating. For simplicity, it is considered that the elongation rate is constant along the organ Ė₀. (B) If the length of the organ is longer than the elongation zone, L > L_gz, the elongation rate is equal to 0 outside the growth zone.

Figure 4

An element of a curved organ is elongating with the same growth rate $\dot{ϵ}$ on each side. The increment of length does not affect the curvature of the organ but the final orientation of the element is changed.

Figure 5

Simulations of gravitropic growth. The color (from blue to red) codes for the elapsed simulation time. (A) Solution of the ACĖ model during exponential growth when B = 10, $\tilde{\gamma }$ = 0.1 and L₀/R = 100. The simulated organ does not reach a steady state. The size of the curved zone is expanded by growth and the organ cannot regulate its posture (Movie 1). (B) Solution of the ACĖ model during exponential growth when B = 10, $\tilde{\gamma }$ = 10 and L₀/R = 100. The simulated organ reaches a steady state even if the organ is elongating (Movie 2). (C) Solution of the ACĖ model during subapical growth when B = 10, $\tilde{\gamma }$ = 0.1, and L_gz/R = 100. The black curve is the part of the organ that is outside the growth zone where the curvature cannot be modified. As the simulated organ reaches the vertical, oscillations are fixed on the final shape due to the elements convected outside the growth zone (Movie 3). (D) Solution of the ACĖ model during subapical growth when B = 10, $\tilde{\gamma }$ = 10, and L_gz/R = 100. The black curve is the part of the organ that is outside the growth where the curvature cannot be modified. The simulated organ reaches a steady state before the elements of the organ are convected outside of the growth zone. No oscillations are fixed on the final shape (Movie 4).

Figure 6

B = L_eff/L_c as a function of the effective length of the organ L_eff/R. Each point denotes an individual plant. The red line represents $B=\sqrt{L_{eff}/R}$. All the points are under this line $B<\sqrt{L_{eff}/R}$, showing that the destabilizing effects of growth on the whole gravitropic movement are negligible. The green line accounts for B = L_eff/6.2R. Most of the points are under this line, showing that it is not possible to distinguish the steady state of the ACĖ model from the AC model.