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 im portant structural component of cell walls. Based on a geometrical theory p roposed earlier [A. M. C. Emons. Plant, Cell and Environment 17, 3-14 ( 199 4)]. the present mode I describes the space-time evolution of the density o f the so-called rosettes, the CMF synthesizing complexes. The motion of the se 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 deposit ed, to the local density of rosettes. The rosettes are created inside the c ell in the endoplasmatic reticulum and reach the cell-membrane via vesicles derived from Golgi-bodies. After bring inserted into the plasma membrane t hey are assumed to be operative for a fixed, finite lifetime. The plasma me mbrane domains within which rosettes are activated are themselves also supp osed to be mobile. We propose a feedback mechanism that precludes the densi ty of rosettes to rise beyond a maximum dictated by the geometry of the cel l. The above ingredients lead to a quasi-linear first order PDE Tor the ros ette-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 a nalytic solutions of the model that give rise to helicoidal, crossed polyla mellate, helical, axial and random textures, since all cell walls are compo sed of (or combinations of) these textures.