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As a particular example, how does the backbone of a peptide look at $\mathrm{pH}~2$? Where does the backbone get protonated in such an acidic solution?

I know that at low $\mathrm{pH}$ the free amino acids have $\ce{-NH3+}$ and $\ce{-COOH}$ groups, but when the residues are linked together into peptide bonds, where do the extra protons go?

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In the backbone of a polypeptide, you would find amide bonds formed from the amino (NH2) and carboxyl (COOH) groups of two amino acids. Water is lost in this condensation process:

Condensation of two amino acids/peptides

The peptide backbone is considered to simply be the repeating Cα–C–N units. In the peptide shown below (a hexapeptide), the backbone is bolded. The side chains of each amino acid residue (R1, R2, ...) are not part of this backbone.

Hexapeptide backbone

Amides are only very weakly acidic and very weakly basic, and do not undergo appreciable protonation or deprotonation under aqueous conditions.

Therefore, you only consider the terminal amino and carboxyl groups. So, at pH 2, both the N- and C-termini will be protonated:

Hexapeptide at pH 2

Of course, if any of your residues contain a group that could become protonated, you need to consider its pKa as well.

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I suggest to visualize the question on your own using the Avogadro molecule editor.

I'm not adding screenshots from my own computer since I run Avogadro with a localized German menu.

  1. Generate some tripeptides
    From the menu chose Build --> Insert --> Insert peptide
    Chose three amino acids. Knowing the three-letters codes is an advantage ;-)
    You might want to try this with amino acids that contain additional $\ce{-NH2}$ groups, such as Lys and those having additional $\ce{-COOH}$, such as Glu.

  2. Adjust structure for pH
    From the menu chose Build --> Add Hydrogens for pH...
    Select a pH of 2 and have a look for changes.

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