Final answer:
The minimum codon length for encoding 200 amino acids would be four bases, as this provides 256 possible unique codons, which is more than the 200 needed.
Step-by-step explanation:
If there were 200 amino acids instead of 20, the minimum number of bases in a codon would need to increase to accommodate the larger number of amino acids. Using the logic of combinatorial mathematics, we know a two-base codon system provides 4², or 16 possible codons, which is not enough even for the current 20 amino acids. A three-base codon system offers 4³, or 64 possible codons, which is sufficient for coding the present 20 amino acids but not for 200. Hence, to predict the minimum codon length if there were 200 amino acids, we would need at least a four-base codon system, which would yield 4⁴ or 256 possible codons. This would be the minimum length required since it is the smallest number of bases that can provide a sufficient number of unique codons for 200 amino acids.