Pattern formation of transcellular ionic currents in plant cells

Stationary and non-stationary spatio-temporal pattern formations are a common feature of biological cells and tissues. The propagation of an action potential along an excitable cell is a well-known example of non-stationary spatio-temporal pattern. Stationary patterns of ion concentration and electrical potential occur in plant cells and tissues. They are correlated to steady ionic currents through the cell membrane. For instance, in chara, a freshwater plant, periodic spatial variations of pH appear along the cylindrical internodal and whorl excitable cells (length from few millimetres to 15 cm) breaking the translation symmetry. The characteristic wavelength and time vary respectively from 1 mm to 2 cm and from 5 min to 60 min. This spatial pH pattern occurs upon activation of the plasma membrane H+-ATPase. In the absence of any external cue, the first pH band appears at a random position suggesting that the initial symmetric state is unstable. The cable model is widely used to understand spatio-temporal pattern formation in excitable cells. The success of this model rests on a typical feature of excitable cells: a negative differential conductance in part of the current-voltage characteristic of their membrane. It cannot explain stationary pattern formation of transcellular ionic current. We have used an electrodiffusive model coupled with cellular electric activity. We report here an unexpected property of the resulting model: a phase space domain that gives rise to stationary spatio-temporal patterns in the presence of a positive total differential membrane conductance. The model can explain stationary spatio-temporal pattern formation in both excitable and non-excitable cells. Our results indicate that these patterns occur if the H+-ATPase is coupled either to voltage-dependent ion channels or to a symport