Picard is surprised when Professor Richard Galen, his old archaeology professor whom he hasn't seen in 30 years, pays a visit to the U.S.S. Enterprise and asks Picard to accompany him on a mission. The renowned archaeologist has made a discovery so profound, the ultimate findings could reverberate throughout the galaxy, but will only reveal them to Picard if he agrees to join him on a lengthy, possibly year-long expedition. The offer is tempting to Picard, who once intended to make archaeology his life's work, but he chooses duty instead, prompting the professor to angrily leave the starship. Soon afterward, the Enterprise receives a distress call — Galen's ship is under attack by Yridians. After inadvertently destroying the attacking vessel with a phaser hit the crew is able to transport Galen aboard, but not before he takes a disruptor blast in the chest at point-blank range. Picard is at the professor's side when he dies in Sickbay, taking his secret with him to the grave.
Attempting to learn why Galen was attacked, the crew finds 19 strange blocks of numbers stored in his computer's memory. Picard decides the answer might be on Ruah IV, part of an unexplored star system Galen visited just before first contacting the Enterprise — and sets course despite the fact that he is due at a diplomatic conference elsewhere. When the search yields nothing, he opts to continue the investigation on Indri VIII, Galen's next intended destination.
Upon their arrival at Indri VIII, the crew finds every microbe of life on the planet is being destroyed before their eyes, leading Picard to believe that Professor Galen's number blocks may have something to do with organic matter. Picard and Dr. Crusher study the blocks, and learn that they are mathematical representations of DNA fragments, each from a different life form from 19 different worlds scattered across the quadrant. Picard decides to set a course for Loren III, the only planet capable of sustaining life in the area to which the professor was referring. There, they encounter two Cardassian war vessels and a Klingon attack cruiser, each on the trail of the same discovery.
After a tense initial confrontation, Picard is able to get the Cardassian Captain, Gul Ocett, and the Klingon captain, Nu'Daq, to share their respective organic matter samples and agree to solve the puzzle together. They soon learn that they are still missing one DNA fragment, but Picard initiates an elaborate computer search, which soon reveals the location where the missing DNA may be found. However, when the findings are announced, the Cardassian captain dematerializes and her ship fires on both the Enterprise and the Klingon vessel.
Picard sets course for the Vilmoran system, accompanied by the Klingon captain, whose ship was crippled in the attack. Soon after they arrive, the Cardassian captain appears on the planet, as do a group of Romulans, who have been following all along in a cloaked vessel. Gul Ocett threatens to destroy the few samples that still remain on the barren world rather than work with the Romulans, and as the Klingon leader joins the face-off, Picard and Beverly quietly retrieve a partially fossilized sample. Unnoticed, they feed the sample into their tricorder. The mysterious program is activated, and a humanoid hologram recorded billions of years ago appears before them. The hologram tells the surprised group that her race found itself alone in their travels of the galaxy. The genetic puzzle was created in hopes that those races would come together in cooperation and fellowship in order to activate the message. The humanoid tells the group that they all come from this common seed, and implores them to remember this bond. The message fades, leaving Nu'Daq and Gul Ocett unbelieving, and repulsed at the mere thought of having anything in common. The groups return to their vessels.