[riboswitch]

Hello guys! I'm trying to understand the Levinthal paradox. According to my book:

Assume that the 2n backbone torsional angles, Φ and ψ, of an n-residue protein each have three stable conformations. This yields 3²ⁿ ≈ 10ⁿ possible conformations for the protein, which is a gross underestimate, if only because the side chains are ignored.

Why does my book say "2n"? According to the PPT slides provided for us, there should be 2ⁿ torsional angles for an n-residue protein. So what's the difference in meaning between "2n" and "2ⁿ" and which is the correct one?

And why would a protein explore a particular conformation for every 10^-13 seconds? Where did that value come from?

[billnotgatez]

Have you read this
https://en.wikipedia.org/wiki/Levinthal%27s_paradox

[Yggdrasil]

The 2n backbone torsional angles comes from the fact that each amino acid has two bonds that are able to rotate to change the path of the polypeptide chain (the Φ and ψ angles).  The question says to assume that for each bond, there are three stable conformations.  Thus, the number of available conformations is 3²ⁿ.

The 2ⁿ estimate assumes that each amino acid can exist in one of two conformations.

In reality, Φ and ψ angles can exist in a number of different conformations (see http://en.wikipedia.org/wiki/Ramachandran_plot for more information).  Both statements are making simplifying assumptions in order to allow you to calculate the number of possible protein conformations.  I'd probably say a reasonable assumption would be that each amino acid can exist in one of three different conformations (alpha helix, beta sheet, random coil), so the number of available conformations would be 3ⁿ.  Nevertheless, whichever assumption you make, the number of potential conformations available for a typical protein is much larger than could be randomly sampled during the typical time a protein takes to fold.

The 10^-13 s figure is related to time required for bond rotations and again is a rough estimate.

[riboswitch]

I read the Wikipedia entry on Levinthal paradox.

If my understanding of the article is correct, then I can say that for a 50-residue polypeptide, we have 49 peptide bonds. Since every peptide bond has fixed Φ and Ψ angles, we can say that the molecule has 98 different Φ and Ψ torsional angles. If each of these bond angles can be in one of the three stable conformations, the molecule may fold into a maximum of 3⁹⁸ different conformations.

Therefore, I can say that a molecule (or protein) with n peptide bonds will have 3²ⁿ available conformations. I thought n represents the number of aminoacid residues in a protein, but I guess I'm wrong. If n represents the number of aminoacid residues of that particular molecule, then for a 50-residue polypeptide, we have at least 3¹⁰⁰ different conformations. (3¹⁰⁰ ≠ 3⁹⁸)

By "3²ⁿ" I'm saying that every torsional angle (may it be Φ or Ψ) can assume three different values. Am I right?

By "2ⁿ" I'm saying that every aminoacid residue can assume two different conformations. By conformation, are we talking about the stable secondary structure of proteins (alpha helix and beta sheets)? Because if it is, I can't make sense to it. It would be more reasonable for didactic purposes to assume that there are three stable conformations instead. I'll just ignore the PPT slides for now.

Thanks for the help.

[Yggdrasil]

I read the Wikipedia entry on Levinthal paradox.

If my understanding of the article is correct, then I can say that for a 50-residue polypeptide, we have 49 peptide bonds. Since every peptide bond has fixed Φ and Ψ angles, we can say that the molecule has 98 different Φ and Ψ torsional angles. If each of these bond angles can be in one of the three stable conformations, the molecule may fold into a maximum of 3⁹⁸ different conformations.

Therefore, I can say that a molecule (or protein) with n peptide bonds will have 3²ⁿ available conformations. I thought n represents the number of aminoacid residues in a protein, but I guess I'm wrong. If n represents the number of aminoacid residues of that particular molecule, then for a 50-residue polypeptide, we have at least 3¹⁰⁰ different conformations. (3¹⁰⁰ ≠ 3⁹⁸)

Yes, if you assume each Φ and Ψ can adopt one of three stable configurations, then the number of conformations available to a protein composed of n amino acids would be 3^2(n-1) not 3²ⁿ.  The exact numbers do not matter so much for the thought experiment, so biology textbooks will often be approximate with the numbers (n and n-1 are approximately the same for large n).

By "3²ⁿ" I'm saying that every torsional angle (may it be Φ or Ψ) can assume three different values. Am I right?

By "2ⁿ" I'm saying that every aminoacid residue can assume two different conformations. By conformation, are we talking about the stable secondary structure of proteins (alpha helix and beta sheets)? Because if it is, I can't make sense to it. It would be more reasonable for didactic purposes to assume that there are three stable conformations instead. I'll just ignore the PPT slides for now.

Yes, those are correct interpretations of those numbers.  Again, the formulas are approximations of reality.  Wikipedia uses 3^2(n-1), your textbook uses 3²ⁿ, your professor uses 2ⁿ, and some papers use 3^n-1 (for example, http://www.ncbi.nlm.nih.gov/pmc/articles/PMC48166/?page=1).  The exact formulas are not the important point of Levinthal's paradox; rather, the important point of the thought exercise is that polypeptide chains cannot fold by simply randomly sampling available conformations.