There isn't really a domestication gene. (There is something called domestication syndrome, which refers to the physical changes that are genetically linked to breeding for tameness - the "symptoms" include physical neoteny especially of the face, floppy ears, and patched or piebald coats).
Researchers studying the horse genome, though, have found 125 domestication genes. Or to be more precise, 125 genes involved in the physical and behavioral traits we favor. Stuff like, you know, not bucking people off, paying more attention to humans, having good withers to support a saddle and, most likely, size (wild horses are smaller than most domestic breeds).
I find this particular topic fascinating. One day I'll get a convention panel together on it.