I understand extremophiles as organisms adapted to extreme conditions like the ocean deeps or Nuclear reactors for example.
What is the best theory for their evolution ?
Natural selection.
Such organisms probably adapt to such conditions gradually. Individuals with mutations and genetic rearrangements that allow them to tolerate a little more extreme conditions than the rest of the population gain access to areas where they have no competition, and their populations increase. Eventually through new mutations and genetic changes they acquire the ability to tolerate even more extreme conditions, and so on.
I’m not sure why you think that natural selection doesn’t cover this. As living forms expand their territories to escape competition or find new sources of food they will naturally experience sites that are inimical. Most will die. Over time a very few will have mutations that allow them to survive longer and in more extreme conditions. Eventually life will be found everywhere.
ETA: My answer is a near copy of Colibri’s, a true expert. I’m patting myself on the back.
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I’m not sure why you think that natural selection doesn’t cover this
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I did not think that. Most versions of natural selection examples, I have read, were about an established colony of organisms where the environment changed on them and how they adapted to the new environment. I have not read about species actively seeking out harsher environments - and that shows my limits :).
So in the case of ocean deeps, organisms from shallow depths started seeking the deeps (although why I do not understand) and some of them made it and followed the natural selection route. So if this is true - we will have a trend of organisms from the deepest to the shallowest part - with a gradient of species that are more adapted as the depth goes more. Presumably, the gradient will show that there are less number of species as the depth increases since at lower depths species have had more time to evolve. Is this indeed the case ? Also - the species at higher depths will be more primitive - right ?
I should leave this to Colibri, but I have to say that there is no such thing as “more primitive” species. A species either exists or it doesn’t. There is no hierarchy. Humans are not the peak that bacteria aspire to.
They are not “actively seeking out” harsher environments. Species will tend to expand throughout the areas they can survive in. Naturally, some individuals will be at the limits of the species’ tolerance. If the populations at the edge of the species’ range acquire genetic changes that allow them to survive beyond the previous limit, they will expand there because there is no competition.
Also, conditions in any one area are likely to fluctuate. Let’s say a hot spring becomes temporarily hotter. It will kill most of the bacteria, but the ones that survive are able to tolerate hotter conditions. These will be able to colonize new hot springs that are hotter than the original one was.
I’m not sure I follow this. However, it is theorized by some that life may have actually originated at the hot volcanic vents on the ocean floor, because the extremophiles that live there (bacteria-like organisms called Archaea) belong to some of the most ancient lineages of life that still exist. (Life may also have survived there during the “Snowball Earth” phase of Earth history, when the entire planet was covered in ice.)
You’re thinking too much about “seeking” as if extremophiles had some choice about where they are.
Think about it like this: a lake no longer receives fresh water, so it starts drying up, becoming saltier and saltier (possibly with concentrations of other potentially toxic substances). After X generations, the surviving populations are going to be the most tolerant examples possible. As long as conditions aren’t so harsh that they kill 100% of the organisms, those who survive continue to adapt.
Or this: Once a year, event X sweeps tons of organisms into the deep oceans. As with Lysol, 99.99% die… but the 1 in 10,000 survivors still represent millions of individuals who might find a niche in which they can cope until they adapt to thrive there.
It’s also worth pointing out that some experts think the extremophiles may have been first, especially around deep-sea volcanic vents. So the argument may be that life started off adapted for high pressure and boiling temperatures and the “extreme” environment is this dry, cold, low-pressure hell we’re coping with now. 
“To get to the other side” :rolleyes:
I’d guess that the OP already knows it was natural selection. Just saying that is obvious, and doesn’t tell him anything useful.
Here’s what you should do. Dawkins explains how the eye evolved. He doesn’t just say “natural selection.” He doesn’t just say “adapt to conditions gradually.” What he does ius show a series of steps, describing how each one made the eye better.
I think the OP would probably like a similar description, with each step in the process explained.
“Extreme” is relative. To an organism adapted to very cold temperatures, what we consider a mild temperature is “extreme”.
That’s what we did. Neither of us just said “natural selection” and left it at that.
The OP was sufficiently vague that it was not clear whether he understood how natural selection works.
If you think you can do a better job of explaining it, go for it.
Years ago I did an experiment with guppies to see how much chlorine they could stand. I would slowly add enough chlorine to kill off about 90% of the group and then let them breed back up. At that particular time I wasn’t writing things down but after about 6 generations they were considerably more resistant but not dramaticaly so. The chlorine was introduced to kill and then allowed to evaporate out so the fish would remain vigorous durring their growing and breeding cycle.
Also, I recall reading that the specially heat adapted proteins they use appear to be what the proteins of more “normal” organisms are based on and not the other way around.
Actually it’s not all that relative I understand; the conditions that the extreme-cold extremophiles in Antarctic rocks live in are so harsh that even they can barely survive. Their metabolism is so slow that they are in constant danger of simply breaking down.
In the history of life and the billions (?) of species that have evolved I’m sure that every scenario you can think of has occurred many times over. However, I’d guess that seeking out harsher environments is a rare one. Most organisms aren’t very mobile. They either live their lives near where they are “born” or they are carried off by wind or currents or whatever and have to try to survive and reproduce where they find themselves. Most will end up in environments different, maybe “harsher”, than where they started. A very few of those will have adaptations that allow them to thrive; most will just die.
This is exactly what I am seeking
I think it’s been pretty well explained already. Any population of organisms are going to have a range of genetic variation. Some of the variants will be better than others at surviving in any given environment. Individuals that are able to open a new niche - that is, live in an environment or lifestyle that is so far unoccupied - will have a huge selective advantage due to reduced competition. Rinse, repeat for a few hundred million years.