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Notes -
One thing unique to DNA/RNA is that they can be used in two distinct ways: directly copied or interpreted as instructions for building stuff. This is a pretty fundamental property because it allows constructing https://en.wikipedia.org/wiki/Quine_(computing) , with the cell corresponding to the program and the DNA corresponding to the text constant in the program that the cell uses to construct a copy of the cell and also copies and inserts directly. Also it avoids the issue of, how do you replicated a hammer without using a stronger hammer to disassemble it -- instructions for building a hammer don't possess the strength of a hammer and can be examined and replicated easily.
I'm not sure that something that doesn't have this duality can be a somewhat general purpose replicator, not by default at least, and I'd expect any good paper proposing some replicator mechanism to be aware of this.
As far as I can tell, no one has seriously tackled that question in full - I'm not aware of any paper for now that confidently advances a novel system explaining how an alternative replicator mechanism can be interpreted as instructions for building stuff. The way DNA/RNA is translated into building an organism is a fairly convoluted multi-step process and building such a system for any hypothetical replicator is probably very difficult.
Most of the papers I come across are at the very basic level of "how can a sequence of information robustly self-reproduce and transmit its characteristics in a way that Darwinian selection can operate on it", that additional layer of complexity surrounding translation is unfortunately not touched on (either because it's not part of their intention to create a general purpose replicator, or because they can't propose one).
This isn't physical, but consider game of life constructors and replicators. Machines in GOL are turing complete, and can also interact in various ways with their environment, so they can pretty much do anything - and one of 'anything' here is being a quine / selfreplicating. The 0E0P metacell seems cool.
I don't think the original question is fundamentally interesting, tbh - any system capable of universal computation tied to some sort of action will be capable of self-replicating in all sorts of bizzare ways, comparable to turing tarpits.
Hadn't heard about the 0E0P metacell before, reading about it now and it's certainly cool.
I suppose the question was less "would there be other usable self-replication methods" - because the answer's almost certainly yes - and more "has anyone else posited one and would that specific system be capable of significant emergent complexity". The question was asked for completely trivial worldbuilding purposes where specific details are crucial - I have a tendency to get bogged down in detail analysis to an unreasonable degree.
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