Well it depends. Most I think would say it would be relatively easy to prove that it is related to Earth life. But not super easy. It wouldn’t be enough for instance to show that it uses the same amino acids, though that would be a good sign. Even DNA would not really be enough evidence.
IDEA THAT EARTH LIFE IS OPTIMAL - A KIND OF CONVERGENT EVOLUTION TO DNA
The problem is that some biologists think that Earth based life is optimal, and that any other life would be the same in many respects, even use the same amino acids and bases. There’s a lot of redundant coding and no reason why life can’t use more amino acids, so why doesn’t it? Maybe the ones it uses already are the best possible?
If that was true, it might be hard to prove it is related until you know a lot about it. Now if every detail of the encoding of amino acids for instance, and the cell machinery exactly copied Earth life, then that would surely be strong evidence for a common origin. But at an early stage if all you know is that it uses the same amino acids and the same bases and has the same chirality - I think exobiologists would probably start to think that it is very likely related but it wouldn’t yet be 100% proof because of this convergence argument.
I don’t find that argument very plausible myself, I think that Earth life is probably optimal - but you can have a local optimum. It’s impossible that evolution explored all possible biochemistries, and so it might well just have optimized over a finite set of the biochemistries it did explore.
So anyway if you accept that, that there are many possible forms of life and that the particular choice of amino acids for instance and bases, is to some extent arbitrary, then it would be easy to get at least a high probability assessment that the life was related. Still just probabilities though. And you can get intermediate positions here where you think that there can be alternative biologies, but only a few of them. E.g. if you think there are only two or three alternative biologies, and you have arguments to suggest they all have the same chirality (from initial chiral imbalances in the forming gas cloud - there is some evidence for an initial chiral bias in the inorganic chemistry before life) - it still wouldn’t be very strong evidence quite yet.
Also, that’s not the first thing you’d do, be able to do a complete enumeration of all the nucleotides and all the amino acids it uses - you are talking here about quite a late stage in the biological exploration of Mars. Probably anyway. Especially so if it is past life as past organics might be severely degraded by cosmic radiation. Present day life also may occur in only very low concentrations and may be hard to isolate and study, especially if it only grows in vivo and won’t flourish in vitrio in the experimental conditions on a rover as is the case for the vast majority of microbes on Earth.
So - we might well get to the stage where we have strong evidence of life on Mars, past or present day, perhaps years before we reach the stage where we can decide whether it is related to Earth life or not, even if in principle it seems quite easy to verify. Just because the life is hard to find, degraded, and because of limitations on how many expeditions we can send to Mars to search for it, given that there’s a land mass there the size of the land area of Earth, with just a few slow moving rovers over the entire surface. If we can speed this up for instance with telerobotics and numerous rovers on Mars we might find out quite quickly.
NOT A BINARY RESULT THAT IT EITHER ALL IS RELATED OR ALL IS UNRELATED - MAY BE SOME RELATED, SOME UNRELATED
Another thing that complicates this is that it is not a binary related / unrelated. There’s a good chance that some Earth life had got to Mars at some point in the past - though not at all certain. So suppose for instance that Chroococcidiopsis (one of our best candidates for a lifeform in present day Earth able to survive on Mars) somehow got to Mars on a meteorite after a giant impact on Earth large enough to send debris to Mars, survived ejection from Earth, the century minimum of transit in the vacuum and cold of space and cosmic radiation - and that it found a habitat there and flourished. This might seem even quite probable in early Mars when it still had surface oceans.
Well - would that make all native Mars life extinct? Perhaps not. Local Mars life might be better adapted to conditions there ven than Chroococcidiopsis. It might be that Chroococcidiopsis does the photosynthesis and native Mars life then eats it or co-exists with it. Or even that native Mars life is better at photosynthesis than any Earth life but that Chroococcidiopsis survives in some places there and our rovers happen to find it first.
Indeed it might be that all we prove is that a single species on Mars derives from Earth, leaving open the possibility that other species on Mars are unrelated, maybe based on a totally different biology. They could be unrelated even in the same habitat, or perhaps some habitats that are preferred by Mars originated life and some preferred by Earth originated life.
OTHER COMPLICATIONS - VERY EARLY LIFE
Also - another thing that makes it less binary Earth / Mars - what if it is early life? On the face of it at least, since Mars had its oceans for a much shorter time than Earth, this seems a strong possibility.
An RNA based lifeform using ribozymes (RNA catalysis) instead of ribosomes for instance, - that’s one strong hypothesis for what we might find on Mars. In that case, it might be related to a precursor to Earth life. But since we don’t know what the precursors were and have none surviving, it might take a lot of detective work to prove that it is indeed related to a DNA precursor rather than independently evolved.
Also there are ideas that life precursors on Earth might have used more stable polymers like PNA or TNA instead of RNA. So even if we find PNA based life on Mars, say, it might be related to Earth life, it might be that some of its ancestors are also our ancestors making us distant cousins.
Another idea that complicates this considerably is the idea that life in our solar system might have been seeded by life that evolved on a planet around a precursor star - perhaps one of the far more common red dwarf stars. It’s a bit of a puzzle, given how much more common red dwarf stars are and given that red dwarf planetary systems are thought to be quite habitable now - why did we evolve around one of the much rarer yellow dwarf stars? Well what if we actually did evolve originally around a red dwarf star? Habitability of red dwarf systems
If so it might be a cousin ,but a very distant cousin. Perhaps its ancestors aren’t from Earth, and our ancestors aren’t from Mars, but both of us have ancestors around another star that went through the nebula as our solar system was forming.
So, in short, I think this will be a long process. I think it is possible that we find some lifeform on Mars early on that looks identical to an Earth lifeform, e.g. we find Chroococcidiopsis there and then we are able to do tests to confirm that it is essentially the same, so much so that it must be related to Earth life. Even that would not prove that all Mars life is related to Earth life though, even in the present never mind in the past. And I think it’s also quite likely that we are simply unable to answer this question until we are a good way into our biological exploration of Mars.
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