Solving Levinthal's Paradox Trivia Quiz

Answer: Free energy

Proteins fold to reduce their Gibbs free energy (G), which can be done by either decreasing the protein's enthalpy (H), or by increasing the system's entropy (S). This is represented by the equation G = H - (S x T), where T is the temperature of the system. Gibbs free energy can be defined as thermodynamic potential, and the lower this is, the more stable the protein is.

While temperature is usually constant (i.e. physiological temperature), enthalpy can be decreased by forming strong interactions between the different amino acids of the protein. Entropy is defined as the number of possible conformations the protein can adopt.

It is entropy that lies at the heart of the Levinthal Paradox. Increasing the entropy of the system is one of the strongest explanations of how the Levinthal Paradox can be resolved.

Answer: Phi and Psi

The phi bond is between the alpha (central) carbon atom and the amino nitrogen atom. The psi bond is between the alpha carbon and the carbonyl carbon. Each amino acid residue in the peptide chain therefore has one phi and one psi bond, each of which can exist in three stable bond angles.

Therefore, for a 100-residue peptide, there are 198 phi and psi bonds and so 3^198 possible conformations. These numbers are already mind-boggling, and they are exacerbated by the fact that the rotatable bonds found in the variable (R) groups of amino acids have not yet been taken into consideration. Bond rotations are rapid (one billion bond rotations can occur in just one second!), but even this is insufficient to explain the short time it takes for most proteins to fold.

Answer: The theory of hierarchical folding

The idea that proteins fold by only acquiring native interactions (i.e. interactions that exist in their correctly folded state) seemed promising. With each native contact made, the number of possible conformations dramatically decreases and so does the time taken to search for the correctly folded state.

These folding dynamics can be visualised by "folding funnels". These visual representations show a three-dimensional energy surface representing the energies of the differently folded states of a single protein.

There is a high energy plateau which represents the numerous high energy (unstable) folded conformations, with a gradual slope leading to a central trough representative of the lowest energy folded state (usually taken to be the native conformation).

This theory is often accompanied by the "blind golfer analogy" - if a blind golfer were to hit golf balls, it would take an unimaginably long time to make a hole in one. However, if the green were arranged so that the hole lay in a central trough, the ball hit by the golfer would simply roll down the hill into the hole regardless of where the ball landed.

Here, the golf ball's progress to the hole can be described as being guided. This theory contributed to solving the Levinthal Paradox, but was still insufficient to explain the speed of protein folding. Moreover, the discovery of proteins which acquired non-native interactions as part of their normal folding process (e.g. beta-lactalbumin) showed that this explanation was not universal.

Answer: The peptide bond

This finding was made by Linus Pauling, who is arguably unmatched in his contribution to protein structure. In the peptide bond, the lone electron pair of the nitrogen atom becomes partially delocalised, giving a partial double bond which cannot freely rotate (unlike the single bonds of phi and psi). On top of this, the observation that many of the theoretical conformations could not exist due to steric strain reduced the number of actual conformations by an even greater extent.

This was another step in solving the Levinthal Paradox.

Answer: That "unfolded" proteins have some structure

The finding that apparently unfolded proteins possess pre-existing structure was a huge step forward in understanding how proteins fold so rapidly. By getting somewhat of a head start in the search for its native state, the number of possible conformations is reduced yet further. The experimental approaches alluded to in the question are astounding and I will mention just a few of them here. Firstly, a technique called ultraviolet circular dichroism can be used in its "far" form to probe secondary structure (e.g. the appearance of alpha helices and beta sheets), while its "near" form probes tertiary structure. Secondly, hydrogen exchange experiments can be performed. Exchange of hydrogen for (heavier) deuterium occurs at exposed residues, but not at residues that are buried in the protein interior.

The time taken for residues to become "protected" from this exchange therefore can tell us a lot about the order of folding in proteins.

Answer: Hydrogen bonds

This theory suggests that all proteins experience a rapid and automatic folding step, determined by their peptide backbone. Since the backbones of all proteins are the same, this theory is applicable to all folding situations. It has been theorised (though the topic remains controversial) that the hydrogen bonds which form between amides are stronger and more stable than those which form between amides and water.

This promotes the rapid formation of alpha helices (which is indeed commonly seen in protein folding) and acts to reduce the enthalpy of the system and therefore reduce the Gibbs free energy.

A following rate-limiting folding step then occurs which gives the protein its unique structure, as determined by the variable sequence of amino acids in the peptide chain.

Answer: Arginine

Isoleucine, leucine and phenylalanine are all hydrophobic amino acids. Conversely, arginine is positively charged and so is less likely to be found in protein interiors. The hydrophobic effect is ubiquitous in biochemistry and is essential to life. It was first recognised by Kauzmann and has remained a leading theory in protein folding.

Its importance relative to the hydrogen bond theory (described in the previous question) has been a matter of contention for decades. On review of evidence, it seems likely that the two occur concomitantly. Far ultraviolet circular dichroism shows the presence of secondary structure from early stages of protein folding, thus supporting the hydrogen bond theory of folding.

However, the appearance of a hydrophobic core also occurs early in protein folding.

This has been shown using a molecule known as ANS, which fluoresces in hydrophobic environments, and does so in the early stages of protein folding. The early stages of protein folding remain enigmatic due to the inability of current experimental techniques to measure changes over such short timescales.

Answer: Protein folding increases the entropy of the surrounding solvent

This theory ties in nicely with Kauzmann's hydrophobic collapse model. As hydrophobic residues accrete in the protein interior, water is excluded from the same site and so even though the protein becomes more ordered, the surrounding solvent becomes less ordered.

This disorder increases the entropy of the system which, as we can see from the equation G = H - (S x T), reduces the Gibbs free energy and so promotes folding. As mentioned, there is contention between the hydrogen bond and hydrophobic collapse models of protein folding. Even in the former, the importance of water is recognised.

It is thought that water acts as an essential hydrogen bond donor and acceptor, "lubricating" the movement of the proteins and assisting the formation of hydrogen bonds between amide groups.

Answer: The Topomer Search Model

It seems apparent that there is not one theory which explains the Levinthal Paradox. Instead, it is likely that several factors contribute to explaining how proteins fold into their native states so quickly despite the huge number of folding routes. For example, it has been predicted that for a 27-residue protein, around 10^16 "moves" are required to reach the correct folded state (if this process were random).

The hydrophobic collapse would reduce this value to around 10^10. A rate-limiting search for an intermediate conformation then occurs (as described by the Topomer Search Model). If there were 10^3 such intermediates, the number of "moves" required to reach the native state falls to 10^7. Clearly this example is highly hypothetical, but it gives a clear representation of how these different theories can be pieced together using our knowledge of protein folding to help resolve the Levinthal paradox.

Answer: Because enzymes catalyse the formation of disulphide bonds

An enzyme known as protein disulphide isomerase increases the rate of formation and breakage of disulphide bonds in vivo, meaning that its rate-limiting effects are less pronounced than when studied in vitro. Similarly, cis-trans isomerisation of proline residues is a major rate-limiting step when studying protein folding outside of the body. Again, this is overcome in vivo by the action of an enzyme known as peptidyl prolyl cis/trans isomerase. Characterising protein folding as it happens in vivo remains an important task.

These investigations could help to resolve the Levinthal Paradox yet further. For example, it has been predicted that "unfolded" proteins in vivo have an even greater degree of preorganization than in vitro (cf. Q5).