The other plays out across the generations in the titular spaceship. The crew's first encounter with it came at approximately the year 10,000,000 by the Doctor's calculations, just as the Earth was being consumed by the Sun's final novic gasp (something that he would later glimpse firsthand in his ninth incarnation). Fleeing the planet, the remaining humans have packed all of her lifeforms, plant and animal, into a gigantic space ship en route to the planet Refusis II. This planet was chosen for it's wonderfully Earth-like conditions, though it is suspiciously populated by the ethereal, disembodied Refusians.
Joining the humans on their trip is an alien race known as Monoids, who are somewhat amphibious-looking and have a single eye where the human mouth would be. However, this docile, quasi-servant race is the least of anybody's concern. Dodo, it seems, brought the common cold into this 57th Segment of time, where it has been extinct for so long that it quickly accelerates into a fullblown plague. Paranoid that this is a germ warfare plot by the Refusians, the humans try the trio of TARDIS time travelers in a McCarthyite kangaroo court.
Only the intervention of the aged human commander, who is ill himself, buys the Doctor the time required to reinvent the cure for the cold. Successful, the supicions of the conservative faction are relieved and the visitors are given the grand tour of the ship, which Dodo dubs "The Ark," including the towering statue of Homo sapiens due to be completed by the end of their 700 year voyage to Refusis II.
Bidding their hosts adieu, the TARDIS dematerializes... only to rematerialize on the same spot a moment later. A moment in relative time for the Doctor, Dodo and Steven, however, equates to 700 years for The Ark! The TARDIS has brought them back to see the consequences of their intervention. The first ominous sign is the completed statue with the head of a Monoid rather than the head of a man.
Now, as the ship bears down on its destination, the Doctor and his companions learn that they did not cure the common cold as such. Rather, it mutated into a less deadly but more insidious form, sapping the will of the human population and allowing the Monoids to revolt. The tables have turned for the former slaves, though how that bears out with the incoporeal Refusians remains to be seen.
Overall, The Ark is a very good story that takes advantage of the unique characteristics of time travel. Many of these were later picked up by the most recent Doctors in the "New Earth Trilogy": The End of the World, New Earth, and Gridlock. It's doubtful that the New Earth of New Earth, and it's New New York (learning as we did during The Chase, courtesy of Vicki, that the original New York was destroyed in The Dalek Invasion of Earth), is Refusis II, but it's a provocative thought. Nevertheless, we are still dealing with the destruction of the Earth and colonization of a new one through the Doctor's repeat visits over time.