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Processes that regulate size and patterning along an axis must be highly integrated to generate robust shapes; relative changes in these processes underlie both congenital disease and evolutionary change. Fin length mutants in zebrafish have provided considerable insight into the pathways regulating fin size, yet signals underlying patterning have remained less clear. The bony rays of the fins possess distinct patterning along the proximodistal axis, reflected in the location of ray bifurcations and the lengths of ray segments, which show progressive shortening along the axis. Here, we show that thyroid hormone (TH) regulates aspects of proximodistal patterning of the caudal fin rays, regardless of fin size. TH promotes distal gene expression patterns, coordinating ray bifurcations and segment shortening with skeletal outgrowth along the proximodistal axis. This distalizing role for TH is conserved between development and regeneration, in all fins (paired and medial), and between Danio species as well as distantly related medaka. During regenerative outgrowth, TH acutely induces Shh-mediated skeletal bifurcation. Zebrafish have multiple nuclear TH receptors, and we found that unliganded Thrab—but not Thraa or Thrb—inhibits the formation of distal features. Broadly, these results demonstrate that proximodistal morphology is regulated independently from size-instructive signals. Modulating proximodistal patterning relative to size—either through changes to TH metabolism or other hormone-independent pathways—can shift skeletal patterning in ways that recapitulate aspects of fin ray diversity found in nature. 

Digital images enable quantitative analysis of material properties at micro and macro length scales, but choosing an appropriate resolution when acquiring the image is challenging. A high resolution means longer image acquisition and larger data requirements for a given sample, but if the resolution is too low, significant information may be lost. This paper studies the impact of changes in resolution on persistent homology, a tool from topological data analysis that provides a signature of structure in an image across all length scales. Given prior information about a function, the geometry of an object, or its density distribution at a given resolution, we provide methods to select the coarsest resolution yielding results within an acceptable tolerance. We present numerical case studies for an illustrative synthetic example and samples from porous materials where the theoretical bounds are unknown.