SummaryIn the present paper, we summarize and further develop recent research in the estimation of the variance of stereological estimators based on systematic sampling. In particular, it is emphasized that the relevant estimation procedure depends on the sampling density. The validity of the variance estimation is examined in a collection of data sets, obtained by systematic sampling. Practical recommendations are also provided in a separate section.

In eukaryotes, the interphase nucleus is organized in morphologically and/or functionally distinct nuclear “compartments”. Numerous studies highlight functional relationships between the spatial organization of the nucleus and gene regulation. This raises the question of whether nuclear organization principles exist and, if so, whether they are identical in the animal and plant kingdoms. We addressed this issue through the investigation of the three-dimensional distribution of the centromeres and chromocenters. We investigated five very diverse populations of interphase nuclei at different differentiation stages in their physiological environment, belonging to rabbit embryos at the 8-cell and blastocyst stages, differentiated rabbit mammary epithelial cells during lactation, and differentiated cells of Arabidopsis thaliana plantlets. We developed new tools based on the processing of confocal images and a new statistical approach based on G- and F- distance functions used in spatial statistics. Our original computational scheme takes into account both size and shape variability by comparing, for each nucleus, the observed distribution against a reference distribution estimated by Monte-Carlo sampling over the same nucleus. This implicit normalization allowed similar data processing and extraction of rules in the five differentiated nuclei populations of the three studied biological systems, despite differences in chromosome number, genome organization and heterochromatin content. We showed that centromeres/chromocenters form significantly more regularly spaced patterns than expected under a completely random situation, suggesting that repulsive constraints or spatial inhomogeneities underlay the spatial organization of heterochromatic compartments. The proposed technique should be useful for identifying further spatial features in a wide range of cell types.

Minkowski functionals encompass standard geometric parameters such as volume, area, length and the Euler-Poincaré characteristic. Software tools for computing approximations of Minkowski functionals on binary 2D or 3D images are now available based on mathematical methods due to Serra, Lang and Ohser. Minkowski functionals can not be used to describe spatial heterogeneity of structures. This description can be performed by using Minkowski measures, which are local versions of Minkowski functionals. In this paper, we discuss how to extend the computation of Minkowski functionals to the computation of Minkowski measures. Approximations of Minkowski measures are computed using filtering and look-up table transformations. The final result is represented as a grey-level image. Approximation errors are investigated based on numerical examples. Convergence and non convergence of the measure approximations are discussed. The measure of surface area is used to describe spatial heterogeneity of a synthetic structure, and of an image of tomato pericarp.

The shoot apical meristem (SAM) is a small group of dividing cells that generate all of the aerial parts of the plant. With the goal of providing a framework for the analysis of Arabidopsis meristems at the cellular level, we performed a detailed morphometric study of actively growing inflorescence apices of the Landsberg erecta and Wassilewskija ecotypes. For this purpose, cell size, spatial distribution of mitotic cells, and the mitotic index were determined in a series of optical sections made with a confocal laser scanning microscope. The results allowed us to identify zones within the inflorescence SAM with different cell proliferation rates. In particular, we were able to define a central area that was four to six cells wide and had a low mitotic index. We used this technique to compare the meristem of the wild type with the enlarged meristems of two mutants, clavata3-1 (clv3-1) and mgoun2 (mgo2). One of the proposed functions of the CLV genes is to limit cell division rates in the center of the meristem. Our data allowed us to reject this hypothesis, because the mitotic index was reduced in the inflorescence meristem of the clv3-1 mutant. We also observed a large zone of slowly dividing cells in meristems of clv3-1 seedlings. This zone was not detectable in the wild type. These results suggest that the central area is increased in size in the mutant meristem, which is in line with the hypothesis that the CLV3 gene is necessary for the transition of cells from the central to the peripheral zone. Genetic and microscopic analyses suggest that mgo2 is impaired in the production of primordia, and we previously proposed that the increased size of the mgo2 meristem could be due to an accumulation of cells at the periphery. Our morphometric analysis showed that mgo2 meristems, in contrast to those of clv3-1, have an enlarged periphery with high cell proliferation rates. This confirms that clv3-1 and mgo2 lead to meristem overgrowth by affecting different aspects of meristem function.

The shoot apical meristem (SAM) is a small group of dividing cells that generate all of the aerial parts of the plant. With the goal of providing a framework for the analysis of Arabidopsis meristems at the cellular level, we performed a detailed morphometric study of actively growing inflorescence apices of the Landsberg erecta and Wassilewskija ecotypes. For this purpose, cell size, spatial distribution of mitotic cells, and the mitotic index were determined in a series of optical sections made with a confocal laser scanning microscope. The results allowed us to identify zones within the inflorescence SAM with different cell proliferation rates. In particular, we were able to define a central area that was four to six cells wide and had a low mitotic index. We used this technique to compare the meristem of the wild type with the enlarged meristems of two mutants, clavata3-1 (clv3-1) and mgoun2 (mgo2). One of the proposed functions of the CLV genes is to limit cell division rates in the center of the meristem. Our data allowed us to reject this hypothesis, because the mitotic index was reduced in the inflorescence meristem of the clv3-1 mutant. We also observed a large zone of slowly dividing cells in meristems of clv3-1 seedlings. This zone was not detectable in the wild type. These results suggest that the central area is increased in size in the mutant meristem, which is in line with the hypothesis that the CLV3 gene is necessary for the transition of cells from the central to the peripheral zone. Genetic and microscopic analyses suggest that mgo2 is impaired in the production of primordia, and we previously proposed that the increased size of the mgo2 meristem could be due to an accumulation of cells at the periphery. Our morphometric analysis showed that mgo2 meristems, in contrast to those of clv3-1, have an enlarged periphery with high cell proliferation rates. This confirms that clv3-1 and mgo2 lead to meristem overgrowth by affecting different aspects of meristem function.