Development of a mathematical method for classifying and comparing tree architecture using parameters from a topological model of a trifurcating botanical tree

This paper describes a model for the topological mapping of trifurcating botanical trees. The model was based on a system of modular units that represented the interconnectivity of shoot meristems (terminal segments) and internodes (internal segments) within whole plant canopies, organized with increasing centrifugal ordering. The model was capable of describing the dynamics of plant growth as expressed by changes in topological parameters over time. Preliminary calculations for experimental trees indicated that the model represents growth in a biologically sound manner. Methods are described for the calculation of the architecture parameters size, size-complexity, structural complexity, and tree asymmetry index (TAI). Parameter calculations were based on the mathematical principles developed for the classification of bifurcating dendrite trees, and were designed to both extract structural information, and to enable statistical comparison between trees of different size. Parameters were mathematically adjusted for trifurcation, and appeared to be able to represent quantitatively the architectural properties of tree structures. In addition to the calculation of the TAI for trifurcating trees, new methods were developed to enable comparisons to be made of the architectural complexity of trifurcating trees of differing size. These were based on the principle of the pair-wise comparison of the mean centrifugal order number (MCON) with respect to segments against highest order number. We argue and illustrate that this principle can be more informative than that of pair-wise comparison of the MCON against tree degree (topological size). Further improvements to this method were made by examining branching points (vertices) rather than segments (links) to calculate the MCON.

A continuum mechanics approach to determining the cellular velocity field with a wool follicle

A model, based on the principles of continuum mechanics, is presented for the analysis of cell-velocity fields within wool follicles. The model requires specification of three follicle characteristics in the form of spatially varying fields: viscosity, cell density and cell production rate. The viscosity is introduced as an attempt to model both complex intercellular interactions and individual cell deformation as the cells move. It is demonstrated that the distribution of cell production is more important than axial variation in viscosity in determining the overall flow pattern.

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.