Transfers putrescine to 5-phosphomethyl-2'-deoxyuridine (5-PmdU) to produce 5-Nalpha-putrescinylthymidine (Nalpha-PutT) as a step in the pathway leading to thymidine hypermodifications in the viral genome (PubMed:34522950, PubMed:7260058, PubMed:7109031). As a final result of the pathway of hypermodification, Nalpha-PutT substitutes for about 50% of thymidines in the viral DNA (PubMed:7260058). These modifications probably prevent degradation of viral genome by the host restriction-modification antiviral defense system (PubMed:34522950). 5-phosphomethyl-dUMP in DNA + putrescine = 5-N(alpha)-putrescinyl-dTMP in DNA + phosphate Mutant am37 contains a normal amount of thymine, no alpha-putrescinylthymine, and hydroxymethyluracil (PubMed:7260058, PubMed:7109031). Mutants am36 and am42 contain more thymine and less a-putrescinylthymine than wild-type (PubMed:7109031). Belongs to the thymidine aminotransferase family.