Polyglutamylase which modifies both tubulin and non-tubulin proteins, generating alpha-linked polyglutamate side chains on the gamma-carboxyl group of specific glutamate residues of target proteins (By similarity). Preferentially mediates ATP-dependent long polyglutamate chain elongation over the initiation step of the polyglutamylation reaction (By similarity). Preferentially modifies the alpha-tubulin tail over a beta-tail (By similarity). Mediates microtubule polyglutamylation in cilia axoneme, which is important for ciliary structural formation and motility (PubMed:17761526, PubMed:22246503). Polyglutamylates olfactory cilia, necessary for the regulation of ciliary sructure and beating (PubMed:17761526). ATP + L-glutamate + L-glutamyl-[protein] = ADP + gamma-L-glutamyl-L-glutamyl-[protein] + H(+) + phosphate (L-glutamyl)(n)-gamma-L-glutamyl-L-glutamyl-[protein] + ATP + L-glutamate = (L-glutamyl)(n+1)-gamma-L-glutamyl-L-glutamyl-[protein] + ADP + H(+) + phosphate CEP41 is required for its transport between the basal body and the cilium axoneme. The flexible c-MTBD (cationic microtubule binding domain) region mediates binding to microtubules. It is positively charged and becomes ordered when bound to microtubules: it interacts with a negatively charged patch on tubulin. The presence of positive charges in the c-MTBD region is essential for proper binding. Gln-174 is the main determinant for regioselectivity, which segregates between initiases and elongases in all tubulin--tyrosine ligase family. A glutamine residue at this position is found in elongases TTLL6, TTLL9, TTLL11, TTLL13, TTLL10 and favors glutamate-chain elongation, whereas an arginine residue is found in initiases TTLL2, TTLL4, TTLL5, TTLL3, TTLL8 and favors initiation. Fish display a lack of tubulin polyglutamylation and cilia formation in olfactory placodes. Belongs to the tubulin--tyrosine ligase family.