A dynamical model for plant cell wall architecture formation

We discuss a dynamical mathematical model to explain cell wall architecture in plant cells. The highly regular textures observed in cell walls reflect the spatial organisation of the cellulose microfibrils (CMFs), the most important structural component of cell walls. Based on a geometrical theory proposed earlier [A. M. C. Emons, Plant, Cell and Environment 17, 3-14 (1994)], the present model describes the space-time evolution of the density of the so-called rosettes, the CMF synthesizing complexes. The motion of these rosettes in the plasma membrane is assumed to be governed by an optimal packing constraint on the CMFs plus adherent matrix material, that couples the direction of motion, and hence the orientation of the CMF being deposited, to the local density of rosettes. The rosettes are created inside the cell in the endoplasmatic reticulum and reach the cell-membrane via vesicles derived from Golgi-bodies. After being inserted into the plasma membrane they are assumed to be operative for a fixed, finite lifetime. The plasma membrane domains within which rosettes are activated are themselves also supposed to be mobile. We propose a feedback mechanism that precludes the density of rosettes to rise beyond a maximum dictated by the geometry of the cell. The above ingredients lead to a quasi-linear first order PDE for the rosette-density. Using the method of characteristics this equation can be cast into a set of first order ODEs, one of which is retarded. We discuss the analytic solutions of the model that give rise to helicoidal, crossed polylamellate, helical, axial and random textures, since all cell walls are composed of (or combinations of) these textures.