Why do we choose a grammar from a lot of grammars that can describe structure of proteins? Usually, there are a lot of (maybe infinite) grammars that can describe one language. Why do we choose a specific one? Especially the language of proteins of gene just has finite sentences or words.
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I did not understand your question completely but this is what I guess you want to know:
Answer to first question cant be definitive. The textbook explanation for why a 3-letter code exists is because a 2-letter code cannot describe all the 22 amino acids, with all its permutations. But then comes another question: why only 22 amino acids. I can't even speculate the answer to this question; the simple amino acids are fine but for an example why lysine and not a similar amino acid with a shorter side chain - no idea.
So, given 22 amino acids, a 3-letter code and 4 nucleotides, the multiple codons for an amino acid are similar in sequence (usually the first two bases are conserved in accordance with the wobble hypothesis. The last base usually doesnt base pair strongly with the tRNA and is of minimal importance).
The answer to second question is somewhat straightforward. The sequence can give rise to infinitely many structures but in a given environment (consisting of the solvent, other solutes etc), the sequence adopts a particular structure in accordance with the free energy minimization principle. There is a possibility that given an environment a protein can have two different optimal structures and can switch between these two structures.