Estimating position-time relationships in steady-state, one-dimensional growth zones.

Two methods are described for estimating position-time relationships (pathlines) in steady, one-dimensional growth zones. Pathlines can be used to provide a time base for spatial data in developmental studies. The methods apply within extension-only zones (zones of growth without cell division) and require data for cell-number densities, or cumulative cell numbers, or mean cell lengths, and for the overall elongation rate of an organ. The first method ("continuous-pathline" method) can be used to estimate spatial velocity fields within extension-only zones and pathlines can then be obtained by integration of the velocity data. This method is based on the continuity equation for cell-number densities. Pathlines can also be estimated using a simple graphical version of this method. The second method ("pathline-coordinate" method) is based on the approximation that a cell of mean length remains of mean length as it moves through the extension-only zone, and can be used to estimate the coordinates of wall pathlines at discrete intervals. The methods are illustrated using published data from studies of apical growth in Zea mays L. roots and of intercalary growth in Triticum aestivum L. leaves.