Re: How far has been Leventhal's paradox in protien folding resolved?

Question:	How far has been Leventhal’s paradox in protein folding 
resolved?
From:	Harikrishnan 
Grade:	grad (science)
City:	No city entered., State/Prov.: No state entered. Country: No 
	country entered.
Area:	Biochemistry Message ID Number: 1100999555.Bc

Despite many different representation [golf-course analogy] and 
concepts put forward to explain the 
situation, no straight answers or at least the way to proceed, seem to 
have put forward? What is the 
current situation and what are the strategies currently employed to 
resolve the problem?

Dear Harikrishnan,

This is an advanced question, so I’ll give you a relatively sophisticated answer to what is still a very active area of research. The issue at hand is really centered on understanding the driving forces behind protein folding, and the rate limiting steps that exist between an ensemble of unfolded states and a narrow ensemble of conformations around the native state. A complete description of the problem, much less a review of the current consensus models of protein folding is far beyond the scope of this forum, so forgive me in advance for leaving out a fair number of details. The short answer is that researchers agree that Levinthal’s paradox is essentially “solved” due to overwhelming evidence of fast protein folding, though there is still disagreement on the details describing the general mechanism of protein folding. (By analogy, evolution is a “theory” with overwhelming evidence in support of it, but due to the complexity of the problem there will always be arguments over the details).

First, let us discuss Levinthal’s paradox. Borrowing from Stryer’s Biochemistry textbook [1]:

   “The cooperative folding of proteins is a thermodynamic
   property; its occurrence reveals nothing about the
   kinetics and mechanism of protein folding. How does a
   protein make the transition from a diverse ensemble of
   unfolded structures into a unique conformation in the
   native form? One possibility a priori would be that all
   possible conformations are tried out to find the
   energetically most favorable one. How long would such a
   random search take?

   “Consider a (hypothetical) small protein with 100 residues.
   Cyrus Levinthal calculated that, if each residue can assume
   three different conformations, the total number of
   structures would be 3100, which is equal to 5 × 10^47. If it
   takes 10-13 s to convert one structure into another, the
   total search time would be 5 × 1047 × 10-13 s, which is
   equal to 5 × 1034 s, or 1.6 × 1027 years. Clearly, it would
   take much too long for even a small protein to fold properly
   by randomly trying out all possible conformations.  The
   enormous difference between calculated and actual folding
   times is called Levinthal’s paradox.”

To put it another way, the problem is approached from a statistical perspective, and is an estimate of the maximum amount of time it would take a protein of 100 amino acids in length to sample every possible conformation of each amino acid in each one of three possible rotameric states.

Borrowing again, from a recent review article by Yawen Bai [2]:

   “In the early 1960s, Anfinsen and co-workers observed that
   small protein molecules could fold spontaneously to the
   native state. This observation has led to a thermodynamic
   hypothesis: the native state has the lowest free energy [3].
   While the thermodynamic hypothesis has become a paradigm of
   protein folding, a simple and general model for describing
   how an unfolded protein molecule finds its native state is
   still not available. In 1968, Levinthal [4] invoked an
   influential argument that came to be known as the Levinthal
   paradox: an unfolded protein molecule does not have the time
   needed to search all possible conformations, yet it reaches
   the single native state rapidly.”

To put it another way, Afinsen’s observations of fast protein folding are at odds with Levinthal’s paradox that suggests astronomically slow protein folding should occur. Indeed, Afinsen is not the only researcher who has observed fast protein folding, which is clearly a fact of life. Intuitively, the solution to Levinthal’s paradox should tell us that his original hypothesis was incomplete, and that an answer to the problem should include some fundamental detail that makes fast protein folding not only possible but rather makes fast folding an inevitability.

The problem with Levinthal’s paradox fundamentally is that the process of protein folding reduces the number of possible conformations available to a protein. Ignoring flexibility in the native state for simplicity, there is a single fixed conformation for the 100 amino acid protein in achieving the final folded state. If we back up one step and allow a single amino acid to be “unfolded”, assuming again three rotameric states available to each amino acid in the unfolded state, our protein now has three conformations; unfolding two residues, the protein has 3^2 = 6 conformations available; unfolding five residues gives us 5^2 = 25 available conformations, and so on. As we move farther away from the native state, more conformations become available; conversely as the protein folds, the complexity of its conformational ensemble is reduced. Why is this such an important point? Levinthal’s paradox is an oversimplification of the problem, and as such drastically overestimates the scope of the problem. Using another analogy, this time billiard balls, Levinthal’s picture is like describing the random process of billiard balls exploring the top of a pool table. The real problem of protein folding is more like a rough funnel, where all billiard balls will eventually fall to the bottom.

Two major different views on how Levinthal’s paradox may be solved have been proposed. A classical view suggests that existence of partially unfolded intermediates may be essential to solve the paradox since they allow folding to occur in a stepwise manner and effectively reduce the scale of conformational search [5]. Richard Dawkins used an analogy in “The Blind Watchmaker” to address how long it would take a monkey poking randomly at a typewriter to reproduce Hamlet's remark to Polonius, “Methinks it is like a weasel”. An astronomically large number of keystrokes, of the order of 10 to the 50th, would be required (estimated from 28^36 = 10^52 assuming 26 letters and ten numbers possible for each or the 28 positions in the phrase). However, suppose that we preserved each correct character and allowed the monkey to retype only the wrong ones. In this case, only a few thousand keystrokes, on average, would be needed. The crucial difference between these cases is that the first employs a completely random search, whereas, in the second, partly correct intermediates are retained.

A recently proposed view suggests that the paradox can be solved if unfolded molecules search the native state on a funnel-like energy landscape [6, 7, 8 and 9]. In this new view, the existence of intermediates is not essential for unfolded molecules to reach the native state; rather, these intermediates populate as a result of the ruggedness of the energy landscape. In earlier experimental studies, intermediates were commonly found, supporting the classical view [10]. However, the proteins studied in these cases are relatively large (>120 amino acids). More recent studies on smaller single domain proteins with the size of ~100 amino acids or less show that intermediates are generally not detectable in kinetic folding experiments [11 and 12]; thus, the classical view has dropped from favor from those who are actively researching the paradox [13].

The subtle difference between the classical view of protein folding and the “new view” is revealed in comparing folding kinetics with studies that probe the equilibrium states explored in protein folding. Folding of small proteins very often is kinetically a two-state process, that is no intermediates accumulate between fully unfolded and the final folded state. These same proteins have been shown experimentally to populate distinct equilibrium intermediate states that describe the protein folding pathway [14]. Intermediates exist, but there is only one rate limiting step to folding and so they do not show up in kinetic experiments.

I strongly encourage you to do further reading as this description only scratches the surface of the problem. In addition to the references I’ve cited, you should explore the NIH “Entrez” database: http://www.ncbi.nlm.nih.gov/Entrez/

(Try “Levinthal Paradox” as keywords and see what you come up with).

Thanks for the interesting question.

Regards,
Dr. James Kranz

References:

1. L. Stryer, J L. Tymoczko, J. M. Berg. Biochemistry, 5th ed., New York: W. H. Freeman Company. QP514.2 .S66 2002

2. Y. Bai. Hidden intermediates and Levinthal paradox in the folding of small proteins. Biochem Biophys Res Commun. 305 (2003), pp. 785-788.

3. C.B. Anfinsen, Principles that govern the folding of protein chains. Science 181 (1973), pp. 223–230.

4. C. Levinthal, Are there pathways for protein folding?. J. Chim. Phys. 65 (1968), pp. 44–45.

5. R.L. Baldwin, The problem was to find the problem (1.3 mb PDF). Protein Sci. 6 (1997), pp. 2031–2034.

6. R. Zwanzig, A. Szabo and B. Bagchi, Levinthal’s paradox . Proc. Natl. Acad. Sci. USA 89 (1992), pp. 20–22.

7. P.G. Wolynes, J.N. Onuchic and D. Thirumalai, Navigating the folding routes. Science 267 (1995), pp. 1619–1620.

8. A. Sali, E. Shakhnovich and M. Karplus, How does a protein fold? . Nature 369 (1994), pp. 248–251.

9. K.A. Dill and H.S. Chan, From Levinthal to pathways to funnels. Nat. Struct. Biol. 4 (1997), pp. 10–19.

10. P.S. Kim, R.L. Baldwin, Intermediates in the folding reactions of small proteins Ann. Rev. Biochem. 59 (1990) 631–660.

11. S.E. Jackson, How do small single- domain proteins fold?. Fold. Des. 3 (1998), pp. R81–R91.

12. B.A. Krantz, L. Mayne, J. Rumbley, S.W. Englander and T.R. Sosnick, Fast and slow intermediate accumulation and the initial barrier mechanism in protein folding. J. Mol. Biol. 324 (2002), pp. 1–13.

13. V. Daggett and A.R. Fersht, Is there a unifying mechanism for protein folding. Trends Biochem. Sci. 28 (2003), pp. 18–25.

14. J. Rumbley, L. Hoang, L. Mayne, S. W. Englander. An amino acid code for protein folding. Proc. Natl. Acad. Sci. 98 (2001), pp. 105-12.