A mathematical model on water redistribution mechanism of the seismonastic movement of Mimosa pudica

Abstract

A theoretical model based on the water redistribution mechanism is proposed to predict the volumetric strain of motor cells in Mimosa pudica during the seismonastic movement. The model describes the water and ion movements following the opening of ion channels triggered by stimulation. The cellular strain is related to the angular velocity of the plant movement, and both their predictions are in good agreement with experimental data, thus validating the water redistribution mechanism. The results reveal that an increase in ion diffusivity across the cell membrane of <15-fold is sufficient to produce the observed seismonastic movement.

Figure 1

The deformation of the primary pulvinus and the bending of the petiole. (a) Original position. (b) Final position.

Figure 2

The angular velocity versus time obtained from the experiment (M1–M3) with the prediction from the mathematical model. M1–M3 are the measurements from three different days and the prediction includes the upper and lower bound for the measurements. The values of constants and variables are L_w = 2.5 × 10⁻¹¹ ms⁻¹ Pa⁻¹; $c_{K_i}=250$ mM; $c_{C{l}_i}=100$ mM; θ₀ = 45°; d₀ = 4 mm; ϕ_i = 11 and 13 m s⁻¹; E = 3.5; and 7.0 MPa for upper and lower bound, respectively. The remaining constants are given in the last column of Tables 1 and 2.

Figure 3

The microscopic pictures of a straight primary pulvinus. (a) Whole longitudinal section (bar, 200 μm). (b) Magnified upper part. (c) Magnified lower part of the pulvinus (bar, 50 μm).

Figure 4

The microscopic pictures of a curved primary pulvinus. (a) Whole longitudinal section (bar, 200 μm). (b) Magnified upper part. (c) Magnified lower part of the pulvinus (bar, 50 μm).

Figure 5

The approximated deformation of the longitudinal section of the primary pulvinus. The section deformed from rectangular into a fan-shaped plane with the neutral axis remaining in the middle.

Figure 6

A single spherical extensor cell in the model. The cell contains the cell wall, membrane, and intracellular space. R and t are the radius of the cell and the cell wall thickness. P_i/o and Π_i/o are the water pressure and osmotic pressure in the intracellular or extracellular space, and both of the spaces contain K⁺, Cl⁻, water, and fixed charges.

Figure 7

Predicted angular velocity versus time with varying constants. (a) Elastic modulus E and (b) Poisson’s ratio v of cell wall; (c) hydraulic permeability L_w; (d) initial permeability of ions (K⁺ and Cl⁻) L_K, L_Cl; and (e) ratio between the triggered permeability to the initial permeability of ions ϕ_K, ϕ_Cl.

Figure 8

Predicted angular velocity versus time with different initial conditions. (a) Varying initial intracellular concentration of K⁺; (b) Cl⁻; (c) both K⁺ and Cl⁻; and (d) varying initial radius of the cell R₀.