A thought provoking question! 

I think, RNases which are very difficult to denature are mostly RNA themselves, i.e. Ribozymes. Protein could still be degraded by chaotropic agents, heat denaturation etc. But no matter how much you heat, RNA tends to renature back to their native catalytically active structure. Use of RNases to destroy those ribozymes would be a suicide mission. So, RNA-RNases are difficult to get rid off. 

I think the evolutionary basis of the shift from the RNA world to a DNA world centres around this. If the degrading enzyme is extremely stable, ubiquitous in nature and can't be eliminated, then you probably can't put full faith on that biomolecule's stability. Giving such biomolecules the responsibility to carry all the genetic information for survival, sustenance and growth would be futile. Hence, DNA emerged as the genetic material replacing RNA. 

RNase, such as pancreatic RNase (RNase A), is a small protein with 4 disulfide bonds. The small size means it can rapidly fold into its native structure, even when denatured (Nobel Prize to Anfinsen), and the disulfide bonds help maintain the native structure. This is an enzyme that functions in a harsh environment in the digestive tract. It is therefore not surprising that it is very stable. Adding to the mystique of invulnerability of RNase is the lack of a simple additive for inhibiting it, unlike DNase, which can be inhibited by adding EDTA to chelate Mg2+.

I don't think the stability of RNase A can be connected to the RNA world hypothesis. Its stability is an evolutionary consequence of its having to perform its function in a challenging environment. By the way, it is a bit of an exaggeration to say "We know that RNA came 1st in evolutionary process."

Finally, RNase A is not a ribozyme. It is a protein.

Thanks Adam B Shapiro! I just want to clarify few things. I know RNase A is a protein enzyme and that it is highly stable. We all use it in our lab. Although I, myself have not worked with RNA, but seen people working with it. Why then do they take the extra precaution? Why wear mask, gloves and all. As we know, RNase A is likely to be present in digestive tract. Then it should not be omnipresent or is it? Even after DEPC treatment (which as far as I know covalently modifies His, Lys and makes protein catalytically inactive) people do get RNA degradation? 

That is precisely it – the omnipresence of RNase on skin and finger prints and on flakes of dust from head and dermis, in sweat, on water droplets from the mouth and nose, etc., hence the need to take barrier precautions.  Essentially to preserve RNA integrity one needs to extract RNA from cells and tissues in the presence of chaotropic denaturing agents – classically phenol and guanidinium thiocyanate – and then to physically separate the RNA from the denatured proteins including RNases that would re-nature on dilution of the denaturing agents.  This was classically achieved by ultracentrifugation through a dense layer of Caesium Chloride, leaving proteins floating above and a 'contact lens' of purified RNA at bottom of the tube. but now much more easily achieved using commercial column-based kits.  Of course, having obtained purified RNA one then needs to avoid reintroducing RNase-bearing contamination, hence the barrier method approach and the use of sterile glassware baked at greater than 160oC to carbonise any existing protein and RNase free-plasticware achieved by a combination of high temperature at which the plastic is moulded and gamma irradiation and not subsequent contaminating the plasticware with sticky fingers!

 As Dr Shapiro says, RNases are proteins, which are not especially stable in the sense that they can be denatured by heat, alkali, chemical modification, etc., but are extremely resilient in terms of their ability to self-assemble (and reassemble following denaturation), actually a property of all proteins in collaboration with the cellular machinery to eliminate incorrectly folded molecules, but enhanced for RNases by the small number of disulphide bonds and thus the smaller member of crosslinks to go wrong.

 Dr Shapiro cautions that RNA may not have come first in the evolutionary process and we certainly need to know much more about molecular evolution to be able to describe an accurate ontogeny.  However, it is clear that if one heats a 'soup' of simple carbon and nitrogen-based chemicals, ribonucleotides will emerge, some of which will polymerise randomly to form short RNA molecules which possess catalytic activity – the original ribozymes – and can promote the assembly or degradation or RNA or amino acids also present in the mix.  You should not overlook that 18S and 28S RNA remain at the core of ribosome protein synthetic machinery today.  At some point, the evolutionary pressure to preserve the 'good experiments' of nature emerging from this soup would have exploited the information coding and transfer capacity of certain RNA molecule combinations 'coding' RNA and 'transfer' RNAs.  However, at some point the information coding role passed to DNA, a double stranded molecule allowing recombination of information and thus the emergence of sexual reproduction.  Meanwhile the chemical stability of RNA leading to the coding signal being 'on' all the time would have been a barrier to the emergence of 'regulation', the answer being to rapidly degrade RNA through enzymatic means to 'turn it off'.  I could go on, but as you begin to appreciate there are books to be written about this and indeed many that have already been written that you should seek out yourself to read.  However, you should not lose sight of the basic facts of RNA, a chemically stable molecule than can be extremely rapidly degraded by enzymes that follow all the rules of protein chemistry but are highly resilient because of the way that they have evolved.

 Hope that helps and I look forward to Dr Shapiro adding his take on the story!

Thank you so much Dr. Shapiro and Dr. Morley for valuable and logical answers. Dr. Morley has mentioned, "if one heats a 'soup' of simple carbon and nitrogen-based chemicals, ribonucleotides will emerge". if we consider the structural differences between a simplest amino acid glycine and a ribonucleotide, definitely glycine is simpler. I wonder if an amino acid being simpler, is produced first from soup, how did RNA become predominant in the primordial soup?