Intercellular signal essential for a variety of patterning events during development. Desert hedgehog protein N-product: Cell membrane; Lipid-anchor; Extracellular side. The N-terminal peptide remains associated with the cell surface. Desert hedgehog protein C-product: Secreted, extracellular space. The C-terminal peptide diffuses from the cell. The desert hedgehog protein N-product binds calcium and zinc ions; this stabilizes the protein fold and is essential for protein-protein interactions mediated by this domain. The C-terminal domain displays an autoproteolysis activity and a cholesterol transferase activity. Both activities result in the cleavage of the full-length protein and covalent attachment of a cholesterol moiety to the C-terminal of the newly generated N-terminal fragment (N-product). This covalent modification appears to play an essential role in restricting the spatial distribution of the protein activity to the cell surface. The N-product is the active species in both local and long-range signaling, whereas the C-product has no signaling activity (By similarity). Belongs to the hedgehog family.