Taegan Smith was killed 15 years before the FAYZ, when a meteorite hit the nuclear power plant and dragged his body deep underground. The impact of the meteorite also lead to radiation from the power plant flowing down the hole it had made. An alien virus that was on the meteorite mixed with Taegan's DNA. The radiation caused the virus and human DNA to mutate and grow into a single creature. Its growth was sped up rapidly by the radiation. The creature, the Gaiaphage, quickly transformed to feed off of radiation, since it had easy access to it. This means that the Gaiaphage, in a way, is Sam and Caine’s father.
This was the reason for Caine and Sam having four bars, due to the genetic connection, the Gaiaphage had a stronger hold on them. They should have been the most powerful, but Little Pete was easier to tamper with, due to his autism.
Teagan Smith was not originally believed to be both Sam and Caine's father, because Connie was also having an affair (which is hinted at that Taegan was the father of one of the twins, and the unknown man-the other). However, it is revealed in Light that Connie took blood samples of Caine and Sam after Caine started attending Coates Academy, and discovered that Taegan is in fact the father of both boys.