The BFW model introduced by Bohman, Frieze, and Wormald [Random Struct. Algorithms, 25, 432 (2004)] and recently investigated in the framework of discontinuous percolation by Chen and D'Souza [Phys. Rev. Lett., 106, 115701 (2011)], is studied on the square and simple-cubic lattices. In two and three dimensions, we find numerical evidence for a strongly discontinuous transition. In two dimensions, the clusters at the threshold are compact with a fractal surface of fractal dimension d f = 1.49 ± 0.02. On the simple-cubic lattice, distinct jumps in the size of the largest cluster are observed. We proceed to analyze the tree-like version of the model, where only merging bonds are sampled, for dimension two to seven. The transition is again discontinuous in any considered dimension. Finally, the dependence of the cluster-size distribution at the threshold on the spatial dimension is also investigated.

Intra-cortical bone remodelling is a cell-driven process that replaces existing bone tissue with new bone tissue in the bone cortex, leaving behind histological features called secondary osteons. While the scaling of bone dimensions on a macroscopic scale is well known, less is known about how the spatial dimensions of secondary osteons vary in relation to the adult body size of the species. We measured the cross-sectional area of individual intact secondary osteons and their central Haversian canals in transverse sections from 40 stylopodal bones of 39 mammalian species (body mass 0.3–21 000 kg). Scaling analysis of our data shows that mean osteonal resorption area (negative allometry, exponent 0.23,R2 0.54,p<0.005) and Haversian canal area (negative allometry, exponent 0.31,R2 0.45,p<0.005) are significantly related to body mass, independent of phylogeny. This study is the most comprehensive of its kind to date, and allows us to describe overall trends in the scaling behaviour of secondary osteon dimensions, supporting the inference that the osteonal resorption area may be limited by the need to avoid fracture in smaller mammalian species, but the need to maintain osteocyte viability in larger mammalian species.

Bone adaptation is modulated by the timing, direction, rate and magnitude of mechanical loads. To investigate whether frequent slow, or infrequent fast, gaits could dominate bone adaptation to load, we compared scaling of the limb bones from two mammalian herbivore clades that use radically different high-speed gaits, bipedal hopping (suborder Macropodiformes; kangaroos and kin) and quadrupedal galloping (order Artiodactyla; goats, deer and kin). Forelimb and hindlimb bones were collected from 20 artiodactyl and 15 macropod species (body mass M 1.05–1536 kg) and scanned in computed tomography or X-ray microtomography. Second moment of area (Imax) and bone length (l) were measured. Scaling relations (y = axb) were calculated for l versus M for each bone and for Imax versus M and Imax versus l for every 5% of length. Imax versus M scaling relationships were broadly similar between clades despite the macropod forelimb being nearly unloaded, and the hindlimb highly loaded, during bipedal hopping. Imax versus l and l versus M scaling were related to locomotor and behavioural specializations. Low-intensity loads may be sufficient to maintain bone mass across a wide range of species. Occasional high-intensity gaits might not break through the load sensitivity saturation engendered by frequent low-intensity gaits.

Research software is often developed with expedience as a core development objective because experimental results, but not the software, are specified and resourced as a project output. While such code can help find answers to specific research questions, it may lack longevity and flexibility to make it reusable. We reimplemented BoneJ, our software for skeletal biology image analysis, to address design limitations that put it at risk of becoming unusable. We improved the quality of BoneJ code by following contemporary best programming practices. These include separation of concerns, dependency management, thorough testing, continuous integration and deployment, source code management, code reviews, issue and task ticketing, and user and developer documentation. The resulting BoneJ2 represents a generational shift in development technology and integrates with the ImageJ2 plugin ecosystem.