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Since Tyrosine and Tryptophan are amino acids, their polarity is determined on their side chains or R groups. If their R groups are polar, the amino acid is polar. Both Tyrosine and Tryptophan are listed as non-polar molecules. However, Tryptophan has nitrogen in its side chain and Tyrosine has oxygen, both of which have a high electronegativity compared to the carbon that makes up the bulk of both molecules, which is something commonly seen in polar amino acids like Serine, which has oxygen in its side chain. This should mean that Tyrosine and Tryptophan are polar, so why are they non-polar, even with an electronegative atom in their side chains?

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Unlike that in organic chemistry, polar and non-polar compounds determination is done differently in biochemistry, probably because most biochemical reactions are done in aqueous medium. Therefore, the classification of them as polar and non-polar is somewhat different from source to source. In general, amino acids, which can make hydrogen bonding in aqueous solutions are called polar, and ones repel from water are non-polar. According to Ref.1:

All polar amino acids have either an $\ce{OH}$ or $\ce{NH2}$ group (when in aqueous environment), and can therefore make hydrogen bonds with other suitable groups.

Following image shows 20 amino acids categorized to polar (10) and non-polar groups (10):

20 essential amino acids

All 10 non-polar amino acids are hydrophobic and 10 polar amino acids are hydrophilic. Polar amino acid residues have a tendency to be on the outside of a protein, due to the hydrophilic properties of the side chain (Ref.1). Keep in mind that tryptophan has indole function, but its lone pair of nitrogen is involved in the aromatic system. Thus, it make only weal H-bonding, which are not good enough to categorize as "polar."

References:

  1. Bruce Alberts, Alexander Johnson, Julian Lewis, Martin Raff, Keith Roberts, Peter Walter, In Molecular Biology of the Cell; 5th edition, Garland Science: New York, NY, 2008, p. 125-129 (ISBN-13: 978-0815341055).
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