How does formylation, of the sort shown here:
(from this Wikipedia page) affect the hydrophobicity of the target protein? Can we say anything general about adding a formyl group to a nitrogen?
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Formylmethionine is used as the start amino acid for a protein chain in certain organisms (the prokaryotes.)
As the alpha nitrogen is formylated, it cannot be in the middle of the chain. This is never seen in real proteins, partly because there would be two carbonyl groups attached to the nitrogen, which would have a very strong electron withdrawing effect and be quite unstable.
Other organisms (the eukaryotes, including the animals and humans) use methionine as the start amino acid of the chain.
As there can only ever be one N-formylated methionine in a real protein (and it is often removed after protein synthesis in a step called post-translational modification) the effect on hydrophobicity will not be very much.
If on the other hand, you were to take an existing protein and formylate the -(CH2)4-NH2 side chains on all the lysine amino acid residues, I would expect a slight increase in hydrophobicity in neutral solution. In acid solution I would expect the difference to be more marked, as the formylated NH2 groups would lose their ability to become ionised by accepting a proton from the acid.