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If I understand correctly, solubility of organic compounds depends mainly on the presence of polar groups plus putting in the factor of the number of carbon atoms in the molecule.

So putting that in mind I do not understand why Glycine, an amino acid with two polar groups, the amine and the carboxyl groups, is considered non-polar and insoluble in water .

Glycine

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Most amino acid molecules are polar in the sense that they have polar functional groups. Even the "least polar" amino acids like leucine and phenylalanine are likely soluble in water.

However, we also classify amino acids by the polarity of their side chains. When incorporated into proteins, clusters of amino acids with nonpolar side chains can create pockets of hydrophobicity in an otherwise aqueous environment.

Glycine is the only amino acid without a side chain. As such, it is not appropriate to classify its side chain as polar or nonpolar.

Polar acidic side chain:             aspartic acid, glutamic acid
Polar basic side chain:              arginine, histidine, lysine, 
Other polar side chain:              asparagine, glutamine, serine, threonine
Nonpolar weakly acidic side chain:   cysteine, tyrosine
Other nonpolar side chain:           alanine, isoleucine, leucine, methionine,
                                     phenylalanine, proline, tryptophan, valine
No side chain:                       glycine
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According to Wikipedia and my experience, glycine is soluble in water and is a polar molecule. The reference you saw, uses polar and non-polar to mean if there is another group other than the carboxylic acid and the amine group.

Solubility in water 24.99 g/100 mL (25 °C)

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