Like glucose, amino acids are also insulinogenic as well. So, presumably, just like glucose, the body would also like to keep amino acids levels in the blood stream below (or within) some certain concentrations as well.

This question Why is too much glucose harmful? details some of the means by which high blood sugar levels can affect various cells and body function. But, would excess blood amino acid(s) levels have similarly adverse effects?

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    $\begingroup$ Your question is unclear and contains dubious assumptions. First the title refers to proteins, whereas the question refers to amino acids. Please correct that. Proteins are not insulinogenic. Second your assumption that because high blood glucose is harmful the primary purpose of insulin is to prevent this harm is at odds with the obvious accepted idea that it does it so the organism can obtain energy from or store this. Hence the supposition that insulin stimulates amino acid uptake to prevent harm, rather than to utilize them is completely unjustified. $\endgroup$
    – David
    May 10 '18 at 14:53