Analysis of tomato root initiation using a normal mixture distribution.

SUMMARY We attempt to identify the number of underlying physical phenomena behind tomato lateral root initiation by using a normal mixture distribution coupled with the Box-Cox power transformation. An initial analysis of the data suggested the possibility of two (possibly more) subpopulations, but upon taking reciprocals, the data appear to be very nearly Gaussian. A simulation study explores the possibility of erroneously detecting a second subpopulation by fitting data which are improperly scaled. A power calculation suggests that only unrealistically large sample sizes can detect the unbalanced mixtures one might expect with data of this type. In the course of the initiation of growth of tomato lateral roots, cells in the parent root begin to divide and form a lateral root primordium, a rounded cluster of cells that originates in an internal layer of the root called the pericycle and extends toward the surface of the root. From this apparently unorganized cluster of cells, a highly ordered structure emerges, with progenitor cells in the apex of the root dividing and differentiating to form the different cell types found in the mature root. These progenitor cells are, in turn, derived from a limited number of primary initial cells located in the root tip. Of those parent root cells that are stimulated to divide during lateral root initiation, only a subset will give rise to the primary initial cells which will form a lateral root. Taylor, Lee, and Rogers, (1994) have devised a method for marking a fraction of the cells in the parent root that contribute to the lateral root as a whole (the founder cell population) using a transposable element-reporter gene combination. When the transposable element is present in the reporter gene, GUS, the gene is not active. Transposition of the element, which is an infrequent, apparently random event, activates the gene, providing a permanent, heritable tag for the cell in which the event occurred and for its descendants (Finnegan et al., 1989). Root cells in which the gene is active can be detected with a stain that causes the marked cells to turn blue (Jefferson, Kavanagh, and Bevan, 1987). For a given segment of the lateral root, the relative proportion of cells that express GUS is indicative of the proportion of the initial cells expressing GUS at the time that segment of the root was laid down, since only the cells at the tip of the root divide. The inverse of the proportion of the lateral root which expresses GUS, which we will call Y, is a possible indicator of the number of initial cells in the lateral root. That this is not an exact indicator is due to factors such as non-uniform cell division and possible variability in the fidelity of staining. The sectors analyzed in this study originated in the parent root and extended into lateral roots that developed at the sector border, therefore this approach is also complicated by the possibility that more than one initial cell may be expressing GUS. Our primary interest is determining whether the observed proportions are the result of one or possibly two underlying physical phenomena. For example, if the roots had two or three initial cells in equal proportions, then one would expect the data to suggest a mixture of two subpopulations. The fitting of mixture models is an old problem of recurring interest, see Titterington, Smith, and Makov (1985) and McLachlan and Basford (1988). We have obtained observations on 40 roots (see Table 1 for data). We present a kernel density estimate and the normal Q-Q plot of Y (Figure 1); the