Role of hysteresis in stomatal aperture dynamics

Stomata are pores responsible for gas exchange in leaves. Several experiments indicate that stomata synchronize into clusters or patches. The patches’ coordination may produce oscillations in stomatal conductance. Previous studies claim to reproduce some experimental results. However, none was able to explain the variety of behavior observed in the stomatal dynamics. Recently, Ferraz and Prado suggested a realistic geometry of vein distribution. Although it reproduces the patches, no oscillation was observed and the patches remain static. Without exploring significant details, the authors stated that hysteresis in stomatal aperture could explain several experimental features. In this paper, the hysteresis hypothesis is further explored through the concept of hysteretic operators. We have shown that the hysteresis assumption is sufficient to obtain dynamical patches and oscillations in stomatal conductance. The robustness of this hypothesis is tested by using different hysteresis operators. The model analysis reveals a dependence between the period of oscillation in stomatal conductance and the water deficit between the leaf and the environment. This underlying feature of the model might inspire further experiments to test this hypothesis.

Figure 1

The surface of the leaf is represented by a square lattice where some sites are defined as stomatal units (SU). Each SU includes the stomata itself together with adjacent cells. Other sites are defined as veins (boundary conditions).

Figure 2

Input/output diagrams of hysteresis operators (a) relay type I, (b) relay type II, and (c) relay type III relating the aperture (units of area) with pressure (units of pressure). Each operator has threshold values $\alpha$ and $\beta$. The thick lines represent the set of possible input-output pairs.

Figure 3

In (a) and (b) we used relay type I, in (c) and (d) we used relay type II, and in (e) and (f) we used relay type III. Figures on the left show the time series of the spatial average aperture (units of area). On the right are the spatial average trajectories of $\langle \Psi \rangle \left(t\right)$ and $\langle \Pi \rangle \left(t\right)$ (units of pressure). In the dashed lines hysterons display hysteresis with thresholds $\alpha =0.3$ MPa and $\beta =0.7$ MPa. In the solid lines hysterons do not have hysteresis ($\alpha =\beta =0.3$ MPa). In all cases the SUs have ${\Psi }_0=-1.6$ MPa and ${\Pi }_0=-2.8$ MPa.

Figure 4

(a) Time evolution of $\langle \Psi \rangle$ and $\langle \Pi \rangle$ (units of MPa) from different initial conditions on a $\ell =1024$ lattice with $104816$ veins and $\Delta w=15$. (b) Time evolution of stomatal conductance $\langle A\rangle$ (units of area). The curves 2 (solid) and 3 (dashed) are associated with oscillation of the $\langle A\rangle$ time series, while curve 1 (dash-dotted) is associated with zero stomatal conductance.

Figure 5

Dynamical pattern formation among SUs in the steady state. The crossed red squares are the water reservoir (color online). We use relay type II, so that the gray scale represents the degree of aperture of SUs, white is for closed, and dark for open. The snapshots of the $32\times 32$ lattice reveal the spatial extent of patches in the following numerical time steps: (a) 35 000, (b) 35 120, (c) 35 280, and (d) 35 400.

Figure 6

Relation between the time average of $\langle A\rangle$ (units of area) and the water vapor mole difference $\Delta w$. The fitted curve suggests a power-law behavior $1/\Delta w$.

Figure 7

Influence of $\Delta w$ on the time dependence of $\langle A\rangle$ (units of area). Here we use the nonideal relay operator in a $\ell =1024$ lattice with $104816$ veins(10$\%$ of the lattice sites); $\tau =0.001$ s; ${\Psi }_0=-1.2$ MPa; ${\Pi }_0=-2.4$ MPa.

Figure 8

The greater the water vapor deficit $\Delta w$, the lower the period (time steps) of oscillation of $\langle A\rangle$ (units of area). In this case we used the relay type I.