Alpha-Helix Structure and Hydrogen Bonds

An α‐helix is a fundamental secondary structure found in proteins. It is stabilized by hydrogen bonds that form between the backbone atoms of the polypeptide chain.

These hydrogen bonds occur specifically between the carbonyl oxygen (\(\text{CO}\)) of one amino acid residue and the amide hydrogen (\(\text{NH}\)) of an amino acid residue located four positions further along the chain towards the C-terminus. This is often referred to as an i to i+4 hydrogen bond.

Calculating Hydrogen Bonds in a 15-Residue Alpha-Helix

In an α‐helix consisting of \(\text{N}\) amino acid residues, the hydrogen bonds involve the \(\text{CO}\) group of residue \(\text{i}\) and the \(\text{NH}\) group of residue \(\text{i+4}\). For a bond to form completely within the defined helix, both residues \(\text{i}\) and \(\text{i+4}\) must be part of the helix.

For a helix with \(\text{N}\) residues, the \(\text{CO}\) groups that can participate in this type of bond within the helix are from residue \(\text{i}=1\) up to residue \(\text{N}-4\).

In this question, the helix has 15 amino acid residues, so \(\text{N} = 15\). The residues whose \(\text{CO}\) groups form hydrogen bonds are from \(\text{i}=1\) up to \(\text{i}=15-4=11\).

The pairs of residues involved in hydrogen bonds are:

• CO of residue 1 with NH of residue 5
• CO of residue 2 with NH of residue 6
• CO of residue 3 with NH of residue 7
• CO of residue 4 with NH of residue 8
• CO of residue 5 with NH of residue 9
• CO of residue 6 with NH of residue 10
• CO of residue 7 with NH of residue 11
• CO of residue 8 with NH of residue 12
• CO of residue 9 with NH of residue 13
• CO of residue 10 with NH of residue 14
• CO of residue 11 with NH of residue 15

To find the total number of hydrogen bonds, we count the number of pairs. The residue \(\text{i}\) ranges from 1 to 11. The number of values from 1 to 11 is \(11 - 1 + 1 = 11\).

Thus, there are 11 hydrogen bonds involving the backbone \(\text{CO}\) and \(\text{NH}\) groups in a 15-residue α‐helix.

Summary

For an α‐helix of \(\text{N}\) residues, the number of backbone hydrogen bonds between the \(\text{CO}\) of residue \(\text{i}\) and the \(\text{NH}\) of residue \(\text{i+4}\) is \(\text{N}-4\).

Given \(\text{N}=15\), the number of hydrogen bonds is \(15 - 4 = 11\).

Therefore, 11 hydrogen bonds involving the backbone \(\text{CO}\) and \(\text{NH}\) can be observed in an α‐helix consisting of 15 amino acid residues.