The incorporation of pyrrolysine, also known as lysine methylamine methyltransferase cofactor adduct, into a peptide; uses a special tRNA that recognizes the UAG codon as a modified lysine, rather than as a termination codon. Pyrrolysine may be synthesized as a free amino acid or synthesized from a lysine charged tRNA before its incorporation; it is not a posttranslational modification of peptidyl-lysine; this modification is found in several Methanosarcina methylamine methyltransferases.