Polyglutamylase which modifies tubulin, generating polyglutamate side chains of variable lengths on the gamma-carboxyl group of specific glutamate residues within the C-terminal tail of tubulin (PubMed:16901895, PubMed:25959773). Mediates both ATP-dependent initiation and elongation steps of the polyglutamylation reaction (PubMed:16901895, PubMed:25959773). Preferentially modifies the beta-tubulin tail over an alpha-tail (PubMed:16901895, PubMed:25959773). Competes with monoglycylase TTLL3 for modification site on beta-tubulin substrate, thereby creating an anticorrelation between glycylation and glutamylation reactions (By similarity). Required for neurite growth; responsible for the strong increase in tubulin polyglutamylation during postnatal neuronal maturation (By similarity). 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 Interacts with both alpha- and beta-tubulin (via C-terminal tubulin tails). In cells with primary cilia, found in both cilia and basal bodies. In neuronal cells, found in dendrites and perikaryon. Highly expressed in the nervous system including spinal cord, thalamus, hippocampus, hypothalamus and cerebellum. The enzyme uses its core to engage the disordered anionic tails of alpha- and beta-tubulin and the flexible c-MTBD (cationic microtubule binding domain) region to bind the microtubule and position itself for beta-tail modification. The c-MTBD region is positively charged and becomes ordered when bound to microtubules: it interacts with a negatively charged patch on alpha-tubulin. The presence of positive charges in the c-MTBD region is essential for proper binding. Belongs to the tubulin--tyrosine ligase family.