A threshold model for polydactyly

Learning spatio-temporal patterns with Neural Cellular Automata

Neural Cellular Automata (NCA) are a powerful combination of machine learning and mechanistic modelling. We train NCA to learn complex dynamics from time series of images and Partial Differential Equation (PDE) trajectories. Our method is designed to identify underlying local rules that govern large scale dynamic emergent behaviours. Previous work on NCA focuses on learning rules that give stationary emergent structures. We extend NCA to capture both transient and stable structures within the same system, as well as learning rules that capture the dynamics of Turing pattern formation in nonlinear PDEs. We demonstrate that NCA can generalise very well beyond their PDE training data, we show how to constrain NCA to respect given symmetries, and we explore the effects of associated hyperparameters on model performance and stability. Being able to learn arbitrary dynamics gives NCA great potential as a data driven modelling framework, especially for modelling biological pattern formation.

Spatial waves and temporal oscillations in vertebrate limb development

The mesenchymal tissue of the developing vertebrate limb bud is an excitable medium that sustains both spatial and temporal periodic phenomena. The first of these is the outcome of general Turing-type reaction-diffusion dynamics that generate spatial standing waves of cell condensations. These condensations are transformed into the nodules and rods of the cartilaginous, and eventually (in most species) the bony, endoskeleton. In the second, temporal periodicity results from intracellular regulatory dynamics that generate oscillations in the expression of one or more gene whose products modulate the spatial patterning system. Here we review experimental evidence from the chicken embryo, interpreted by a set of mathematical and computational models, that the spatial wave-forming system is based on two glycan-binding proteins, galectin-1A and galectin-8 in interaction with each other and the cells that produce them, and that the temporal oscillation occurs in the expression of the transcriptional coregulator Hes1. The multicellular synchronization of the Hes1 oscillation across the limb bud serves to coordinate the biochemical states of the mesenchymal cells globally, thereby refining and sharpening the spatial pattern. Significantly, the wave-forming reaction-diffusion-based mechanism itself, unlike most Turing-type systems, does not contain an oscillatory core, and may have evolved to this condition as it came to incorporate the cell-matrix adhesion module that enabled its pattern-forming capability.

Making and shaping endochondral and intramembranous bones

Skeletal elements have a diverse range of shapes and sizes specialized to their various roles including protecting internal organs, locomotion, feeding, hearing, and vocalization. The precise positioning, size, and shape of skeletal elements is therefore critical for their function. During embryonic development, bone forms by endochondral or intramembranous ossification and can arise from the paraxial and lateral plate mesoderm or neural crest. This review describes inductive mechanisms to position and pattern bones within the developing embryo, compares and contrasts the intrinsic vs extrinsic mechanisms of endochondral and intramembranous skeletal development, and details known cellular processes that precisely determine skeletal shape and size. Key cellular mechanisms are employed at distinct stages of ossification, many of which occur in response to mechanical cues (eg, joint formation) or preempting future load‐bearing requirements. Rapid shape changes occur during cellular condensation and template establishment. Specialized cellular behaviors, such as chondrocyte hypertrophy in endochondral bone and secondary cartilage on intramembranous bones, also dramatically change template shape. Once ossification is complete, bone shape undergoes functional adaptation through (re)modeling. We also highlight how alterations in these cellular processes contribute to evolutionary change and how differences in the embryonic origin of bones can influence postnatal bone repair.

Compares and contrasts Endochondral and intramembranous bone development

Reviews embryonic origins of different bones

Describes the cellular and molecular mechanisms of positioning skeletal elements.

Describes mechanisms of skeletal growth with a focus on the generation of skeletal shape

How unique is unique? Quantifying geometric differences in stripe patterns of Cape mountain zebra, Equus zebra zebra (Perissodactyla: Equidae)

Quantified coat pattern dissimilarity provides a visible surface for individual animal traceability to populations. We determined the feasibility in quantifying uniqueness of stripe patterns of Cape mountain zebra (CMZ; Equus zebra zebra) using geometric morphometrics. We photogrammetrically created dense surface models of CMZ (N = 56). Stripe edges were landmarked, superimposed and compared for shape variation across replicates and the population. Significant allometry in stripe patterns prompted allometric correction to remove increased curvature of stripes at the rump, belly and back with larger adult individuals, to facilitate equilibrated comparison between individuals. Re-landmarked replicates showed lower dissimilarity (Di) than non-replicates (Dp), representing minimal landmarking error. Individuals were 78.07 ± 1.79% unique (U=1−DiDp×100%) relative to the study population. Size, the number of torso stripes and degree of branching in four rear torso stripes described the most shape variation (36.79%) but a significant portion could only be distinguished with geometric morphometrics (41.82%). This is the first known use of geometric morphometrics to quantify coat pattern uniqueness, using a model species to provide baseline individual morphological variation. Measures of coat pattern similarity have a place in phenotypic monitoring and identification.