Workshop on TRANSPORT AND SELF-ORGANIZATION IN COMPLEX SYSTEMS TSOCS’2004 Porto Alegre, Brasil, agosto 2004 A model for the closure of stomata in a leaf Carmen P. C. Prado (IFUSP) Marcus Cima Ferraz (IFUSP, student) Gustavo Maia Souza (UNOESTE, SP)
What is a stomata? Stomata are pores responsible for water and gas exchange in a leaf CO 2 H 2 O
How do estomata control gas exchange (water loss)? Cells turgid – stomata open Cells flacid – stomata close
Stomatal patchiness distribution of stomatal apertures over a leaf surface is not always homogeneous “patchy stomatal conductance” Recent studies have shown that the distribution of stomatal apertures over a leaf surface is not always homogeneous, and under some circumstances show complex spatial patterns known as “patchy stomatal conductance”, indicating a nontrivial collective behavior. Fluorescence images of a leaf for which gas exchange measurements are being taken. Brighter areas = high stomatal conductance. There are oscillations. This movie was created by time lapse photography of chlorophyll fluorescence. Total elapsed time was approximately 5 hours. Patchiness was initiated by a decrease in ambient humidity (Vicia faba).
Epidermis of zebrina, light microscope view (X 150). The stomatal pore, guard cells and adjacent cells are visible Scanning electron micrograph of the epidermis of a cactus Stomatal Unit Guard cells surrounding epidermal tissue (SU) areola leaf veins Flow among epidermis Flow between guard and epidermis J. W. Haefner et al, Plant, Cell and Environment 20, (1997)
Hypothesis of the model (a) water moves among water compartments the leaf in response to water potential gradients; (b) Water flows from the xylem to an evaporating site, that experiences a reduction of water potential proportional to the transpiration rate; (c) Stomatal aperture is a linear combination of epidermal and guard cell turgor pressures; (d) The evaporating sites are in close hydraulic contact with the epidermis; (e) The osmotic pressure of the guard cell is a function of the water potential (turgor) of evaporating site, so stomata will respond to w (water deficit)
Model equations e epidermis water potential g guard cell water potential e epidermis osmotic pressure g guard cell osmotic pressure For each SU we define: g = e e = constant
Changes e depend on (a) in / out water flux and (b) loss by transpiration Changes e depend on (a) in / out water flux and (b) loss by transpiration g I conductance, ~ stomatal aperture A i W mole fraction difference between leaf interior and ambient air For each SU i Water potential e P e,g Turgor pressure C e,g Mechanical influence coefficients if positive otherwise if positive otherwise Same equation for P e ; g,e osmotic pressure
To calculate the osmotic pressure, we use hypothesis (e): g ss - g. To calculate the osmotic pressure, we use hypothesis (e): turgor pressure of epidermis controls (by metabolic means) the osmotic pressure of the guard cell. A high turgor pressure in the epidermic cells causes the solute to be pumped into the guard cell and vice- versa. So, P e defines the stationary sate of g : The osmotic pressure of the guard cell, then, approximates its stationary state with a velocity that is proportional to g ss - g. where Osmotic pressure g
Summarizing … E i A i, A i P g - i P e (if > 0), P e,g e - e,g (if > 0), For each stomata unit i A “leaf” is defined by the way SU are put toghether and get in touch with veins (boundary conditions)patchiness
Our leaf Areola model In this model Areolas are isolated Final patchiness is result of initial disorder and of i distribution random vein model Periodic BC e of veins = 0 (constant)
Results and Conclusions Mean stomatal conductance ww % veins = 20; c= 0.08; = 0.1 gsgs w Data for a leaf of Xantium strumarium, Plant, Cell and Env., 20 (1997)
Leaves Regime 1: oscillations
Regime 2 : steady state