Mapper is a topological construction similar to a Reeb graph, and is used to summarize the shape of a dataset as a (generalized) graph. Formally, mapper can be constructed for any connected space and algorithms have been developed to compute mapper for point clouds and 2D images. In this paper, we extend mapper to 3D volumetric images. We use our algorithm to compute mapper for scans of barley generated using computed tomography. We demonstrate the flexibility of the construction by highlighting different aspects of the morphology through different choices of starting parameters. Applying mapper to this type of data provides an integrated means of visualization, segmentation and clustering, and can thus be used to study the topology of any 3D object.

Epicormic branches arise from dormant buds patterned during the growth of previous years. Dormant epicormic buds remain just below the surface of trees, pushed outward from the pith during secondary growth, but maintain vascular connections. Epicormic buds can be activated to elongate into a new shoot, either through natural processes or horticultural intervention, to potentially rejuvenate orchards and restructure tree architecture. Because epicormic structures are embedded within secondary growth, tomographic approaches are a useful method to study them and understand their development. We apply techniques from image processing to determine the locations of epicormic vascular traces embedded within secondary growth of sweet cherry (Prunus avium L.), revealing the juvenile phyllotactic pattern in the trunk of an adult tree. Techniques include the flood fill algorithm to find the pith of the tree, edge detection to approximate the radius, and a conversion to polar coordinates to threshold and segment phyllotactic features. Intensity values from magnetic resonance imaging (MRI) of the trunk are projected onto the surface of a perfect cylinder to find the locations of traces in the “boundary image”. Mathematical phyllotaxy provides a means to capture the patterns in the boundary image by modeling phyllotactic parameters. Our cherry tree specimen has the conspicuous parastichy pair (2,3), phyllotactic fraction 2/5, and divergence angle of approximately 143°. The methods described provide a framework not only for studying phyllotaxy, but also for processing of volumetric image data in plants. Our results have practical implications for orchard rejuvenation and directed approaches to influence tree architecture. The study of epicormic structures, which are hidden within secondary growth, using tomographic methods also opens the possibility of studying genetic and environmental influences such structures.

Iron is a metal essential for cellular metabolism. Excess or lack of iron can cause serious health conditions. To deal with these difficulties, the intracellular levels of iron are tightly constrained by a complex control network of proteins. Recently, Chifman et al. developed and validated a mathematical model in the form of five differential equations, of the core control system of intracellular iron homeostasis in normal breast epithelial cells. Their work was motivated by the fact that intracellular iron homeostasis can play a role in the pathogenesis of breast cancer. For any choice of parameters, their dynamical system has a unique equilibrium, and Chifman et al.’s simulations suggest it is globally stable. Here we introduce a biologically reasonable simplification of the Chifman model. For this valid approximation, we show that it has a unique steady state and that it is locally asymptotically stable. We also give evidence that this model seems to approach global stability by using a geometric analysis. This reduced version gives us an insight on how the original model behaves.

Phosphorylation systems are ubiquitous chemical mechanisms in biology. Multisite phosphorylation systems can be distributive or processive. Distributive systems have been shown to exhibit bistability, while processive systems exhibit global stability. However the processive result was proven for a specific mechanism of processive phosphorylation (namely, all catalytic reactions are reversible.) Accordingly, we generalize this result to allow for processive phosphorylation networks that are reversible or reversible or involve product inhibition. Specifically we create an all-encompassing processive system that encapsulates each of these schemes. By appealing to monotone systems theory we prove that the dynamical system arising from mass-action kinetics has a unique steady state and that it is a global attractor. We also establish the same result for a more general system using graph reductions and recent graph-theoretic stability criteria.