How would it be possible to have a change in a single base of DNA, but have the protein not change and be functional?

The change in single base results in the substitution by a chemically similar amino acid

Explanation:

Whenever a change in one of the bases on the DNA sequence leads to the synthesis of a chemically similar amino acid (e.g glutamate substituted by aspartate, a fellow amino acid with acidic side chain) the protein formed is not altered, and still retains its function. Hence, this type of single base change is known as Silent mutation

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