Growth in Root Apices. I. The Kinematic Description of Growth

Notational conventions for spatial (Eulerian) and referential (Lagrangian) descriptions of growth in a root apex are discussed Root growth is assumed to be one-dimensional with generation of new length occurring as the result of cell division and cell expansion in the apical growth zone Growth in such a system may be described either in terms of activities at fixed distances from the apex or in terms of activities associated with elements of the root which move away from the tip during growth The first approach leads to a spatial description where growth variables, such as the spatial velocity field or cell density, are expressed as functions of position on the axis and of time The second approach gives a referential description of growth: here, growth variables are associated with marked particles of root material and are expressed as functions of the position that a particle occupies at a reference time, and of current time A number of concepts that arise from these definitions, including the use of material derivatives and interconversion between spatial and referential descriptions, are discussed The concepts are illustrated with data from Erickson and Sax's classical papers on the growth of seedling roots of Zea mays L