Final answer:
The gene unique to T regulatory cells that prevents self-antigen T-cell activation is Foxp3. This gene is crucial for Treg cell function in immune response regulation.
Step-by-step explanation:
The gene that is unique to T regulatory (Treg) cells and is associated with the inhibitory mechanisms that prevent self-antigen-specific T-cell activation is Foxp3. Treg cells, also known as suppressor T cells, are specialized lymphocytes that help to regulate the immune response to prevent autoimmune diseases. These cells express the CD4 molecule, but the presence of Foxp3 is a distinguishing feature that is critical for their function in suppressing other T cell immune responses and maintaining immune tolerance.