I actually think the tattoos are subliminal messages just like "would you kindly"
everytime Jack does something that you can't control, or HAVE to do, he either hears the phrase or looks at his tattoo.
For instance, when you first get your electovolt plasmid, jack's first instinct is to jam a gigantic needle into his arm just for the hell of it.. except that he holds it up, looks at both of his arms, and THEN jams the gigantic needle into his arm.
Also, you can see his tattoos shortly before the plane goes down, so jack was probably responsible for that too
maybe. There's no real mention of them or what they do in the game so it's all speculation.