The gene whose mutation can cause extra toes in mice and humans is Sonic hedgehog, commonly abbreviated SHH. It encodes a signaling protein that helps organize embryonic body patterning, including the formation and positioning of fingers and toes.
During limb development, SHH is produced in a specialized region called the zone of polarizing activity. Its concentration and timing help cells interpret whether they should become thumb-side or little-finger-side digits. If SHH activity appears in the wrong place or at the wrong level, an extra digit can form—a condition called polydactyly.
A common mix-up is that many human cases involve mutations in a distant regulatory enhancer rather than a change to the SHH protein-coding sequence itself. The best-known limb enhancer, called ZRS, lies roughly 1 megabase away and controls SHH expression in developing limbs. Other genes, including GLI3 and HOXD13, can also be involved in polydactyly, but SHH is the canonical answer among the choices.