There's actually a fan theory that none of the current Lannister flock are actually Lannisters. In the books, it's strongly implied that the Targaryens would regularly consort with the wives of their nobles, and Jaime, Cersei and Tyrion are described as having platinum hair, only a few shades darker than the Targaryen white. Add in the mental instabilities, propensity for incest and some remarks Tywin makes to Tyrion about not being able to prove that Tywin isn't the father, and it starts to look like not a one of Tywin Lannister's kids might be his own.
This would also explain why Cersei and Jaime are relatively normal aside from the paranoia (which is also attributable to their environment) and incest, while Joffrey got hit with the insanity stick so bad...an infusion of fresh blood diluted the genetic causes of their mental issues, but going right back to family lovin' compounded them again. It's likely that their other children will start manifesting similar symptoms as they grow older.