Geometric quantification of the plant endoplasmic reticulum

One of the most difficult obstacles to make biological sciences more quantitative is the lack of understanding the interplay of form and function. Each cell is full of complex-shaped objects, which moreover change their form over time. To tackle this problem, we suggest the use of geometric invariants that are able to produce precise reference points to compare a cell's different functional elements such as organelles under fixed and varying physiological conditions. In this paper, we look at the topology of an almost static sample of the plant cortical endoplasmic reticulum (ER) under close-to-normal physiological conditions using a multi-disciplinary approach combining confocal microscopy, image processing techniques, visualization, computational geometry and graph theory. Data collected from a series of optical sections taken at short, regular intervals along the optical axis are used to reconstruct the ER in three dimensions. A graph structure of the ER network is obtained after thinning the ER geometry to its essential features. The graph is the final and most abstract quantification of the ER and serves very well as a geometrical invariant, even and very importantly, in cases in which the ER sample is moving or slightly changing shape during image acquisition. Moreover, graph theoretic features, such as the number of nodes and their degrees and the number of edges and their lengths, are very robust against different kinds of small perturbations that should not change the ER function. We will also attach surface areas and volumes estimated for the plant ER network as weights to the graph, allowing an even more precise quantitative characterization of this organelle. In total, we have compared 28 different samples under similar experimental conditions. The methods used in this paper should also be applicable to the quantification of other organelles in which geometric abstraction is possible to analyse function. Finally, by the use of confocal microscopy, our techniques will be transferable to situations in which protein markers can move inside the organelle's lumen and/or on the membrane surface to test further aspects of protein distribution.