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My textbook in biochemistry ("Biochemistry" - Berg,Tymoczom Gatto, Stryer) tells me that:

"Almost all peptide bonds in proteins are trans" (p. 40)

At the same time, it states the following regarding the structure of the $\alpha$-helix:

"A tightly coiled backbone forms the inner part of the rod and the side chains point outwards" (p.42)

How do I reconcile these two statements?

If the peptide sequence in an $\alpha$-helix is predominantly trans, that would imply that every other amino acid residue would point inwards, towards the center of the helix.

Does this imply that peptide sequences involved in $\alpha$-helixes are predominantly cis, or that the amino acid residues pointing towards the center of the helix must be very short, like for instance glycine?

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    $\begingroup$ It is neither. See, the whole thing is 3D. Take a paper sheet and draw an all-trans peptide chain along a straight line. Now roll up the paper into a cylinder and bend all residues outwards. $\endgroup$ Sep 22 '21 at 6:54
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    $\begingroup$ At this moment there are nine editions of your textbook. When citing a source please make sure your citation is complete and unambiguous. On Chemistry.SE the preferred citation style is ACS. $\endgroup$
    – andselisk
    Sep 22 '21 at 7:30
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[Ivan Neretin in the comments] It is neither. See, the whole thing is 3D. Take a paper sheet and draw an all-trans peptide chain along a straight line. Now roll up the paper into a cylinder and bend all residues outwards.

That is so true. Here is an animation to illustrate:

enter image description here

The peptide planes are still planar, and wrap around an imaginary cylinder. Consecutive planes are at an angle with respect to one another because there is a tetrahedral carbon, the alpha carbon, in every amino acid. The side chain (shown as gray methyl group here) stick out away from the axis of the cylinder.

[OPs cited statements] "Almost all peptide bonds in proteins are trans"

All peptide bonds in an alpha helix are trans.

[OPs cited statements] "A tightly coiled backbone forms the inner part of the rod and the side chains point outwards" (p.42)

You could call it a rod if you show a spacefilling model, or a toilet paper roll (or paper towel roll for longer helices) if you show an all-bonds model like above. Notice it says the side chains point outwards, not the amino acids.

[OP] How can all amino acids point outwards from an alpha helix, if peptides are mostly trans?

They don't. The side chains point outwards, and that is compatible with trans-peptide bonds. In fact, different secondary structures (turn, strand, helix) are possible by changing the torsion angles on the tetrahedral alpha carbon while keeping the peptide bonds all-trans.

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