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The Enterprise is on the way to a starbase for shore leave when they get a garbled distress call, apparently from the Intrepid. The Intrepid is the only Starfleet starship crewed exclusively by Vulcans. Shortly afterward, Spock looks up in shock and pain; he has felt the Intrepid die. Kirk doesn't publicly doubt him, but he's obviously skeptical and sends Spock to Sickbay. Spock explains to McCoy that the sheer magnitude of the event was able to reach him telepathically, even though he wasn't in physical contact.
Then the Enterprise gets a message from Starfleet, ordering them to investigate the loss of communications with not only the Intrepid, but all the planets in the Gamma 7A planetary system. Oddly, even though Kirk seems to be in real-time communication with Starfleet, he does not mention Spock's claim that the Intrepid has been destroyed.
By the time Spock returns to the bridge, the Enterprise has found a strange void in space, at the same location where the planetary system and the Intrepid should be. When a probe enters the void, it causes a strange noise throughout the ship, and many of the crew feel sick or dizzy. Those that aren't obviously sick are tired and irritable, demonstrated by Kirk as he loses patience with Spock's repeated declarations of lack of information. McCoy reports that not only are people feeling ill, but their life energy is gradually draining away - everyone is dying.
The Enterprise moves toward the void and abruptly crosses into it, causing another round of illness. When the void envelopes them, they cannot see out of it to the stars around the region. Spock finally speculates that the void is a negative energy field. Certain things now seems to be operating in reverse, such as the ship's engines. They are being pulled farther into the void, and in order to work against the pull, they have to fire the engines in reverse.
The source of the negative energy field comes to them: a giant, one-celled organism akin to an amoeba. This amoeba is 11,000 miles long. (This episode, like others, mixes English and metric units, giving the length here in miles, but later giving distances in kilometers.) Spock has no idea how or why the amoeba is generating the negative energy field, but is positive that this is the case. And the longer the Enterprise stays in the field, the weaker everyone gets, until they will die. The problem is that the ship no longer has enough power to escape whatever is pulling them in.
Despite the peril to the ship and crew, McCoy and Spock are nearly beside themselves with excitement in studying the giant amoeba. They immediately jump to the conclusion that someone needs to go into the amoeba in a shuttlecraft in order to study it and figure out how to kill it so the Enterprise can escape. Kirk initially vetoes this idea, since it's likely that whoever goes in the shuttlecraft will die. However, as Spock and McCoy vie for who is best qualified, we can see Kirk re-evaluating the situation and realizing that someone has to go. He retires to his quarters to decide.
McCoy has the specific medical expertise for studying the amoeba. But Spock has general scientific expertise, plus superior physical stamina. Kirk regretfully designates Spock for the mission, which McCoy's help in planning the equipment and tests that will be needed. McCoy doesn't take the "rejection" well, and snipes about Spock's cool attitude, which he interprets as superiority. We realize that it's Spock's way of dealing with the potential deadliness of the mission.
Spock pilots the shuttlecraft into the amoeba and relays information back to the Enterprise, despite increasing interference. He reports the location of the "chromosome bodies" and that they appear to be prepared to reproduce. The Enterprise loses contact with Spock, and given his limited life support, they assume he is dead. We do in fact witness Spock leaving a last log entry commending the captain and crew of the Enterprise.
Since the amoeba seems to be feeding on energy, Kirk realizes any conventional attack on it would be unsuccessful. He decides to inject a problem with antimatter into the amoeba near the chromosomes. In order to make sure the probe gets to the right location, the Enterprise will travel into the amoeba and place the probe.
They drop off the probe inside the amoeba and try to get out as fast as possible. They encounter Spock's shuttlecraft on the way and bring it along with a tractor beam. The bomb explodes, and the ship and shuttlecraft are thrown outward. Fortunately, everyone survives and they discover that Spock is alive. The amoeba is destroyed.
The chemistry between the characters is really the high point of this episode. Kirk becomes extremely irritable, but Spock and McCoy are understanding and tolerant of it, and Kirk himself realizes it. Spock and McCoy have some classic shots at each other, although McCoy's side is distinctly underlined with the desperation he feels; clearly he would rather be doing something, anything to help rather than sit helpless on the ship. Later, when McCoy and Spock are sniping over the signal when Spock is on the shuttlecraft, Kirk actually gives them a weary grin. And Spock's comment about "Captain McCoy" is classic.
The episode suffers a bit on logic, though. First is the "negative energy field" idea. Negative energy is not an impossible idea: energy is a scalar property and the scale one measures it by is arbitrary. Scientists routinely call some energies negative, such as the binding energy of an atom. I think the idea of negative energy that is intended is that it's energy that does the opposite of normal energy.
So what does normal energy do? Well, a very simplified definition is that energy is what gives matter the ability to move. So I suppose that negative energy would drain matter of the ability to move instead of provided matter with the ability to move. This roughly fits with the idea that people are gradually dying from a lack of energy. However, it doesn't fit with the idea of some actions being reversed, such as the engines. The ship should gradually be drained of power (which does happen) - the directionality should be unaffected. And really, this part of the episode is superfluous - once they realize they have to use the engines in reverse, that's that.
The other big issue I have with the suspension of disbelief is the idea that a 11,000 mile long one-celled organism could exist. The "one-celled organism" idea isn't just some analogy - the characters talk about the cells chromosomes, nucleus, and protoplasm, which implies it's literally a one-celled organism. I do not believe it would be possible for such a scaled-up one-celled organism to exist.
If you scale up any three-dimensional object, both the volume and the surface area of the object increase. Since the volume is proportional to the length of the object cubed, and the surface area is proportional to the length of the object squared, the volume of the object increases at a faster rate than the surface area. This disparity in the rate of increase will cause major problems.
I am not an expert in biology, but let's assume that the amount of nutrients the organism needs is proportional to its volume, since that's the most direct measure of the amount of matter in the organism (assuming its density is the same as any normal cell). Let's also assume that the amount of wastes the organism produces is also proportional to its volume. As you make this cell bigger and bigger, the amount of nutrients needed and wastes produced increases at a faster rate than the surface area. That means that at some point, the cell doesn't have enough surface area to take in the nutrients it needs and expel its wastes. The wastes will build up inside, and the nutrients will not get in fast enough. The cell could not function.
Even if my assumptions about nutrients and wastes are not correct, I know that the heat produced by the cell (from its metabolism) will be proportional to its volume. The larger the cell gets, the more heat will be trapped inside because there is not enough surface area to release it from; the cell will overheat, and again it could not function. This idea is like comparing a hot potato to a hot pea: since the potato is large, it will off slowly, while the pea will cool off very quickly.
All of this means that there's a reason we don't see one-celled organisms larger than a certain (small) size: they can't function. This makes the creature in the episode unbelievable.
One smaller issue I have has to do with Spock's sensing of the Intrepid's death. As he describes it, and as it seems to happen, the death was sudden - it was the deaths of all 400 of the crew at once that created such strong feelings that Spock could sense it. And he described them as astonished, which fits with this. However, from what the Enterprise experienced, the effects of the amoeba were an extended draining of energy. Presumably, had the Enterprise not escaped, crewmembers would gradually begin dying. It would take a period of hours or maybe days for everyone to die. This just doesn't fit with the implied fate of the Intrepid.