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If the amino acid sequences in the proteins of two organisms are similar, why will their DNA also be similar?

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Amino acid sequences are determined by nucleotide sequences in DNA, through their transcribed expression in mRNA and its subsequent translation, codon after codon, determined by what is called the Genetic Code. In the examples that follow, similar will be understood as having a certain degree of colinearity. To understand this, compare:

PRINCECHARLES

PRINCESSDIANA

You will promptly recognise colinearity of the first 6 characters. Comparisons between sequences (be it of amino acids or nucleotides) can only be made if they are aligned together. Identity can be continuous or not. In this case, you might ask also whether the first A in each sequence is "conserved" or a coincidence. Any biologist would tend to argue for coincidence, but this is not always the case. It's a tough decision (see final part of this answer).

So, at least the first 6 "amino acids" are "similar" because they can be aligned. Let us now use the one-letter code for each amino acid (clarifying that PRINCE =...

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