Levinthal Paradox

Journal of Biomolecular Structure & Dynamics, ISSN 0739-1102 Volume 20, Issue Number 1, (2002) ©Adenine Press (2002) Express Communication Loop Fold Structure of Proteins: Resolution of Levinthal’s Paradox http://www.jbsdonline.com Abstract According to Levinthal a protein chain of ordinary size would require enormous time to sort its conformational states before the final fold is reached. Experimentally observed time of folding suggests an estimate of the chain length for which the time would be sufficient. This estimate by order of magnitude fits to experimentally observed universal closed loop elements of globular proteins – 25-30 residues. Key words: Levinthal’s paradox, protein folding, chain conformation, closed loops, loop fold structure Igor N. Berezovsky1,* Edward N. Trifonov2 1Department of Structural Biology The Weizmann Institute of Science P.O.B. 26 Rehovot 76100 Israel 2Genome Diversity Center Institute of Evolution University of Haifa Haifa 31905 In 1968 Levinthal in his report of which only brief summary is available (1) noted that reversibly denaturable proteins during transition from “random disordered state into a well-defined unique structure” have to go through conformational space with immense number of states, so that the time required for visiting all the states would also be very large. Indeed, (e.g. (2)) for a protein chain of length L = 150 (residues) with n = 3 alternative conformations for every residue, the time t required for sorting out all possible conformations of the chain is: t = nL ⋅ τ = 3150 ⋅ 10-12 s = 1048 yrs [1] Israel (τ = 10-12 s is time for elementary transition (2)). Observed values of t are in the range 10-1 to 103 s (2), that is, the full sorting as above is impossible. Thus, protein folding has to proceed along a certain path that would avoid most of the conformational space. The path should somehow be directed by an as yet unknown sequence-dependent folding rule(s). The size of the short chain for which the observed time span of 10-1 to 103 s would be sufficient for trying every possible state can be calculated from [1], with the same assumptions, as l0 = lg(t/τ) = 23 to 31 residues. In this case all conformlg n ations could be tried during the given time, and the lowest energy state(s) attended. Being logarithmic this estimate is rather insensitive to the choice of the values for n which according to various authors may change between 1.6 and 10 elementary conformations (3, 4). With these extreme values the above estimate spans the range l0 = 11 to 74 residues. The l0 value may, thus, serve as a rough estimate of the size of the units (chains or chain segments), which could attend all conformational states during observed time. The estimated size of the hypothetical unit is identical to the optimum of loop closure for polypeptide chains, 20 to 50 residues (5), and to the observed size of recently discovered closed loop elements, 25-30 amino acid residues (5-7), of Phone: 972-8-9343367 Fax: 972-8-9344136 E-mail: [email protected] 5 6 Berezovsky and Trifonov which globular proteins are universally built. One can hypothesize that the closed loops are also elementary folding units. In this case their linear arrangement within the protein folds (5, 8) would suggest a straightforward folding path: sequential formation of the closed loop units, along with their synthesis in the ribosome. If such successive formation of the stable folding units in the course of translation is assumed, it will require time proportional to the number of the units, that is, only several fold larger than required for a single unit. The above scenario is consistent with the typical rates of translation, 3 to 20 residues per second (9). Synthesis of the protein of length L = 150 takes, thus, 8 to 50 seconds, which is a fair match to the above range of folding rates. Thus, according to the estimates above the consecutive formation of the loop-like folding units of 25-30 residues is by the order of magnitude time-wise consistent with both folding and translational experiments. Acknowledgement We are grateful to Prof. A. Yu. Grosberg for valuable comments and enlightening discussions. References and Footnotes 1. 2. 3. 4. 5. 6. 7. 8. 9. Levinthal, C. J. Chim. Phys. Chim. Biol. 65, 44-45 (1968). Branden, C. and Tooze, J. Introduction to Protein Structure, Garland Publishing (1999). Bryngelson, J. D. and Wolynes, P. G. Proc. Natl. Acad. Sci. USA 84, 7524-7528 (1987). Pande, V. S., Grosberg, A. Yu. and Tanaka, T. Reviews of Modern Physics 72, 259-314 (2000). Berezovsky, I. N., Grosberg A. Y. and Trifonov, E. N. FEBS Letters 466, 283-286 (2000). Berezovsky, I. N. and Trifonov, E. N. J. Biomol. Struct. Dyn. 19, 397-403 (2001). Berezovsky, I. N. and Trifonov, E. N. J. Mol. Biol. 307, 1419-1426 (2001). Berezovsky, I. N. and Trifonov, E. N. Prot. Engineering 14, 403-407 (2001). Varenne, S., Buc, J., Lloubes, R. and Lazdunski, C. J Mol Biol. 180, 549-576 (1984). Date Received: July 6, 2002 Communicated by the Editor M. D. Frank-Kamenetskii to explain how a problem of sampling the impossibly large number of conformations by the folding protein chain can be avoided.. However. However. without any significant reconstruction. Issue Number 3. Key words: protein folding.protres. etc. As demonstrated by numerous chevron plots (5-8). folding nucleus. This refers to proteins Phone/Fax: (+7-095) 924 0493 Email: afinkel@vega. of only the native and the unfolded molecules during folding of moderate size (single-domain) proteins. mid-transition. and then the bigger structures obtained at the second stage serve as the building blocks for the next stage. in a visible quantity. two-state kinetics. Levinthal paradox. for the time comparable with that of its own formation.e. a nucleation mechanism can account for all these major features simultaneously and resolves the Levinthal paradox.e. many single-domain proteins (having up to 200–250 residues) successfully fold near and even in the point of thermodynamic equilibrium between their native and denatured states. Specifically.com Abstract A hierarchic scheme of protein folding does not solve the Levinthal paradox since it cannot provide a simultaneous explanation for major features observed for protein folding: (i) folding within non-astronomical time. Such a mechanism may work only when the native structure is much more stable than the unfolded or denatured state of protein chain. any hierarchic mechanism (including the one suggested by Berezovsky & Trifonov) implies that the formed folding unit has to preserve its once found form at least until the next unit will be formed. Russia Berezovsky & Trifonov (1) have recently revisited an attractive idea of a hierarchic protein folding (2. i. and (iii) co-existence. a high stability of the native structure is not obligatory for folding. Berezovsky & Trifonov in their clearly written paper (1) assume that local “closed loops” of 25-30 residues (the smallest folding units) find their lowest-energy structures by an exhaustive search of all their conformations and then stick together. A “hierarchic mechanism” means that some structures formed at the first stage serve. (ii) independence of the native structure on large variations in the folding rates of given protein under different conditions. Since the folding units then stick together. I would like to note that any hierarchic scenario cannot serve as a general solution of the Levinthal paradox. Pushchino Moscow Region. Long life of the once found form means that it is thermodynamically stable as compared to the initial denatured state of the same piece of the chain. when protein folding occurs near the point of thermodynamic equilibrium between the native and denatured states of the protein. Finkelstein Institute of Protein Research Russian Academy of Sciences 142290. It cannot work. (2002) ©Adenine Press (2002) An Opinion Piece: Conversation on Levinthal Paradox & Protein Folding #1 Cunning Simplicity of a Hierarchical Folding http://www.Journal of Biomolecular Structure & Dynamics. 3) in an attempt to resolve the Levinthal paradox (4). this means that the native protein is. as building blocks at the next stage of folding.. ISSN 0739-1102 Volume 20. entering the tertiary protein fold in the already found form. though. co-existence of the native and the unfolded phases. in turn.ru 311 . Indeed. rate of folding Alexei V. In principle. i. a hierarchic mechanism can help to avoid sampling of the huge conformational space. more stable than the sum of these sable folded units. and thus much more stable than the denatured state of protein chain. On the contrary.jbsdonline. it is noteworthy that this question does not arise at all when the native fold is determined by its stability. The discovery of this fact in kinetics was actually a re-discovery of wellknown “all-or-none” thermodynamics of de. A hierarchic folding scheme discussed in this paper cannot satisfy this criterion. observed in some cases (15.and renaturation of single-domain proteins (9). observed for folding of single-domain proteins: (i) folding within non-astronomical time. according to the logic of Berezovsky & Trifonov (1). Also. The presented considerations give an important criterion of applicability of any protein-folding model to single-domain proteins: whether or not this model can explain protein folding near the point of thermodynamic mid-transition between the folded and unfolded states. The “all-or-none” transition means that only the native and denatured proteins are present (close to the mid-transition) in a visible quantity. as well as to those that have three-state folding kinetics (i. In its strict form.. crystallization) in macroscopic systems (9. It is worthwhile to note that the latter fact is easily explained if the choice of the native fold is determined by its stability (rather than by folding kinetics): it is not obligatory that every element of the lowest-energy fold has an enhanced stability. one has to explain why the folding with detectable folding intermediates (far from the mid-transition) is not drastically faster than the folding that does not have such intermediates. 14). 16). 13. On the other hand. Specifically. though most of them have to have an enhanced stability for pure statistical reason (17). “semi-native” forms. in a visible quantity. The only necessary prerequisite of such a transition is an energy gap between the lowest-energy native fold and misfolded structures (11. Therefore. (ii) independence of the native structure on large variations in the folding rates of given protein under different conditions. it has been shown that the nucleation mechanism (that pays main attention to the boundary between the folded and denatured phases within a protein molecule) can resolve the Levinthal paradox (11) and leads to realistic estimates of the protein folding rates (12). An “all-or-none” transition is a microscopic analog of first order phase transitions (i.e. it is not a surprise that this transition in proteins was shown to follow a nucleation mechanism (8) that is well known (in physics) to be specific for the first order phase transitions (10). 10). and (iii) co-existence.e. Moreover. that have detectable folding intermediates far from the equilibrium point). However..e. i. On the contrary. are unstable and therefore present in a very small quantity. the nucleation model of protein folding meets this criterion and resolves the Levinthal paradox. this does not mean that a hierarchic folding scenario is completely inconsistent with all proteins. the hierarchic mechanism means that the native protein structure is controlled by kinetics rather than by stability of the whole protein. has to lead to folding units of rather different sizes). Acknowledgements This work was supported by an International Research Scholar’s Award from the Howard Hughes Medical Institute and by a grant from the Russian Foundation for Basic Research. while others. The same refers (19) to the simplest versions of a funnel model of protein folding (20-22). one has to explain why the same native protein structure results from foldings held under different conditions and having 1000-fold difference in speed (which.312 Finkelstein having a simple two-state folding kinetics. A hierarchic folding does not provide a general solution of the Levintal paradox since it cannot simultaneously explain for major features. If so. the simplicity of a hierarchic folding is cunning. do not prevent protein from correct hierarchic folding. a hierarchic folding seems to take place in large multi-domain proteins whose denaturation is not an “all-or-none” transition but proceeds as a sum of “all-or-none” denaturations of their domains (18). of only the native and the denatured protein molecules during the folding near the point of thermodynamic mid-transition between these two states. and why the evidently non native-like intermediates.. . Nature Struct. 7. M. 5. V.... A. 24. V. J. and Dill K. R. Phys. 9.. Struct. Curr. Biol. Protein Chem.. Pergamon (1959). Protein Engineering 14. Ya. Goldstein. M. D. B. 21. A. V. Galzitskaya. 2. Sci. J. A. Nature 340. USA 92. Trends Biochem. 4. Kellis. A. 11. and Badretdinov. Biomol.. 8. L.. 33. D. N. Protein 23. O. A. 12. S. Akad. V. and Sugihara. L. Ivankov. Y. Bryngelson and P. Privalov. USA 89. Acad. Fersht. Folding & Design 2. London. and Fersht A.. 14. M. 20. Serrano. 142-150 (1995).. C. A. Statistical Physics. Wolynes. 77-83 (1999). 17. A.. E. Struct. Natl. O. Sci. Acad. Chim. Dokl. Z. 122-126 (1989). N. S.. D. A. N. Baldwin. and Wolynes.References and Footnotes 1. Badretdinov. Dyn. B. ibid. Proteins 30. and Bagchi. Bicout.. R...-I. Natl. 22. Proc. 6. Berezovsky. Biol. 2-33 (1998). G. Jr..... T.. Protein Science 9. Biopolymers 23. 2473-2488 (1984). G. 20-22 (1992).. 19. 3. R. Levinthal. USA 89 4918-4122. 9029-9033 (1992). Sci. 452-465 (2000). V. A.. Proc. A. Kiefhaber. 20. and Finkelstein A. 13. Protein Chem. Sci. Acad. J. Proc. L. 3-9 (1997). 7524 (1987). 5-6 (2002). 1-104 (1982). G. USA 84. J. T. Adv. and Gutin. Dyn... P. Acad.. Natl. 44-45 (1968). J. Biol. A. Natl. 1213-1215 (1973). A. 8.. 18. P. N. 167-241 (1979). Finkelstein. E. 15. 115-121 (1997). 16. 19. and Szabo. R. Privalov. Opin. Matouschek. 735-737 (2002). L. Fernandez. P. L. 2002 Communicated by the Editor Ramaswamy H Sarma .. I. A. and Finkelstein. Baldwin. Sci. A. Luthey-Schulten. and Lifshitz. Biomol. Zwanzig. D. 10. 26-33. FEBS Letters 489. L. 9029-9033 (1995). 92-94 (2001). and Trifonov. S. Nauk SSSR 210. R. 65. Struct. 35. 113-118 (2001). Proc. J. Landau. B. and Rose. 7. Bogatyreva. Finkelstein... Szabo. R.. Adv. Segawa. Chan H. Chim. 521-523 (2001). 313 Cunning Simplicity of a Hierarchical Folding Date Received: September 16. Ptitsyn. . chain-return nature of the units (14).com Entia non sunt multiplicanda praeter necessitatem (Occam) Abstract In response to the criticism by A. The same calculation in reverse as in (2) leads to what we would call the Levinthal limit – in other words the size of the polypeptide chain or part thereof for which the observed times of protein folding would be sufficient to sort out all possible conformations.* Edward N.Berezovsky@weizmann. Biomol. 2002) several issues are dealt with. Finkelstein (1). and every contribution towards their characterization (9-13) is important. 20.O. 2002) of our Communication (J.g. due to long-living native and non-native meta-stable states (5.ac. 20. More elaborate approaches should take into consideration both physical factors and biological circumstances that may influence the estimate. 311-314. and the necessity of further characterization of the units for the elucidation of the protein folding in vivo are discussed. Key words: Levinthal paradox. and their fate during protein folding may follow many different scenaria (6.Journal of Biomolecular Structure & Dynamics. its size and structure. and it is not about hierarchical folding. Calculations for the ordinary protein sizes on the basis of Levinthal’s original idea (15) lead to astronomical times.jbsdonline. The passage in our note (2) “If such successive formation of the stable folding units in the course of translation is assumed.g.7)). Igor N. Rehovot 76100. Biomol.B.il 315 . The very notion of independently folding units is an important concept. Israel The main points of our Communication (2) – the structural units of folding and closed loops as likely candidates for that role – are not challenged by A. The structural units of folding may or may not remain intact once formed. folding units. thus. Israel 2Genome Diversity Center Institute of Evolution University of Haifa Haifa 31905. or larger (e. 311-314. Dyn. Berezovsky1. The respective simple formalism provides a rough estimate of the size of such a chain. But again. the discussion on these scenaria is well beyond the scope of our original Communication (2). become either smaller (e. along with the synthesis of the polypeptide chain. due to Phone: 972-8-9343367 Fax: 972-8-9344136 Email: Igor. This initiated decades of theoretical and experimental research (16-20) to figure out what is a time-wise agreeable path of folding of the full-size proteins. …” is not a statement. Issue Number 3. Struct. apparently. 26. Trifonov2 1Department of Structural Biology The Weizmann Institute of Science P. under the assumption that there are no returns to the same conformations during the sorting. Finkelstein (J. and the structural component to it . It may. The criticism (J. biological relevance. 5-6. 7). Dyn. Dyn. (2002) ©Adenine Press (2002) An Opinion Piece: Conversation on Levinthal Paradox & Protein Folding #2 Back to Units of Protein Folding http://www. ISSN 0739-1102 Volume 20. Biomol. This biological dimension is. Importance of the notion of elementary folding unit. Our study suggests a rather narrow size range for the hypothetical folding units. 2002) on the hierarchical protein folding is also briefly addressed. But his piece rather invites to a separate chapter in protein folding studies – a hierachical folding (3-8). Struct. but rather a simple illustration of how the elementary structural units may partake in the initial stages of folding process during translation. protein folding. Struct. a crucial component in protein folding problem. 20. G. Natl. E. R. V. N. 20.. USA 90. Sci. A.. 573-582 (2002). Panchenko. V.316 Berezovsky and Trifonov special sequence organization (4. Gething. Nature 355. Nat. The balance is still to be found. J. 3-9 (1997). Biol. A. Proc. Z. S. Curr.. D. 20.. 19. R. 2. D. I. and Rose. J. Evol. I. A. Ivankov. N. Sci. A. Sci. Biol. Natl. Biomol. and Trifonov. A. I. References and Footnotes 1. 1213-1215 (1973). Grosberg. as the one suggested by the polymer statistics of the polypeptide chains (14) and by the observed rates of translation (2). Fersht. Shakhnovich. 53. 113-118 (2001). Trends Biochem. O. N. Mol. Acad. T. J. 283-286 (2000). Yu. E. 12. E. Shakhnovich. Phys. J. G. Luthey-Schulten. 20. 10. 1525-1529 (2000). Baldwin.. Luthey-Schulten. 24.. FEBS Letters 466. P. Pande. Berezovsky. 6. R. Fersht. O.. Yu. J. and Bedows. Baldwin. Opin... Curr. 735-737 (2002). Natl. 24. P.. R... A. Berezovskii. 7. V. Biol. I. 26-33 (1999). N. N. J. Comparative and Functional Genomics. B. 394401 (2001). Acad. E. 5. Esipova. 557-565 (1997). J. 311-314 (2002).. G. E. 21-24). 6. Z. 7524-7528 (1993). 65. and Sambrook. I. N.. 33-45 (1992). V. N. J. R. 13. Sci. Chem. USA 91. V. M. Struct. L. 814-817 (1999). Ptitsyn. S. Berezovsky.. Natl. Biomol. Finkelstein. 2002 Communicated by the Editor Ramaswamy H Sarma . W. R. R. 29-40 (1997). J. and Trifonov. 12972-12975 (1994). Possible clues are apparent evolutionary fingerprints in protein sequences (23. Biomol. Dokl. Cell 108. Galzitskaya. Struct. Ruddon. M. Berezovsky. L. N. 814-817 (1999). We believe. 5-6 (2002)... 4. 95105 (1997). Gutin. in press (2002).. and Gutin. Opin. Biol. Y. 21. V. Proc. Struct. Struct. Biophysics 42. and Tumanyan. N. and Rose. I.. 24. E. 24). Biol. Acad. R. 25. 77-83 (1999). 26)). 15. 22. and Wolynes. V. Kirzhner. A. 17. Struct. 19. Panchenko. Sci. Dyn. E. and Tanaka. Abkevich. Baldwin. Baldwin. and Tanaka. N. A.. A. Dyn. 18. R. 3125-3128 (1997). 272. 25-30 amino acid residues. A. 2008-2013 (1996).. Grosberg A. T.. 6. G.. Chim. R. 10026-10036 (1994). that both experimental and theoretical studies on the units of folding is the right way to elucidation of how exactly the protein folds in vivo. 9.. Reviews of Modern Physics 72. 16.. Kirzhner. A. V. 7. L.. 3. 272. Struct. Dyn.. Pande. Levinthal. V.. D. Nauk SSSR 210. Sci. Proc. N. Fersht... Chim.. M. A. C. 44-45 (1968). 259-314 (2000). Biochemistry 33.. Grosberg. Proc. Cole. R.. G. Trends Biochem. USA 93. and Trifonov. 7. and Finkelstein A. Struct.. G. Akad. Nat.. Trifonov. Mol. and Berezovsky. and Wolynes. I. I. 11. 8. M. 14. FEBS Lett 489. Shakhnovich.. USA 97. 26. Biol. I. E. Fernandez.. Biol. Acad. 23. and possible structural guidance within the ribosome or in chaperones (25. and Daggett. that indicate the same size range. Date Received: October 25. L. and their intent was to test systematically all different sorts of wine. Trifonov (IBET for brevity). But there is also another more interesting reason: the time necessary for the random walk to visit all M sites does not scale as Mτ. Issue Number 3. that cooperative (all-ornone) character of folding and unfolding transition is indicative of the sequences selected. (2002) ©Adenine Press (2002) An Opinion Piece: Conversation on Levinthal Paradox & Protein Folding #3 A Few Disconnected Notes Related to Levinthal Paradox http://www. such that reliable folding does not require exhaustive conformation sampling. Many biophysicists deem themselves experts. This simple property of random walks and diffusion processes seems to be underappreciated in the current discussions revolving around the celebrated Levinthal paradox (1). This reinforces the understanding of Levinthal paradox which emerged in the last decade. and assuming there were some M pavilions. simply because random walk visits some sites great many times before the first visit to some other sites. MN 55455. advertised as a merry traditional peasant holiday. In fact. Before long. because M is astronomically large. Fraunfelder (3) suggests to Email: grosberg@physics. where M is the number of distinct conformations. namely. The opinion is formulated that the discussions of Levinthal paradox should now fly to the new spheres. and τ is the time necessary to sample one conformation. and the real time was much longer than that. for instance. USA 2Institute of Biochemical Physics Russian Academy of Sciences Moscow 117977. even if they were to complete the exhaustive testing. it can be significantly larger than that. particularly in the ones initiated by the recent very clearly written article (2) by I. and the major event was the all-Georgian wine testing festival. the month was October. a large biophysics meeting was held in the then Soviet Republic of Georgia. In the most standard formulation. Such expectation was proved to be terribly wrong. A. The few participants (perhaps bad biophysicists) who were able to continue scientific observations realized soon that wine testing continued for an unexpectedly long time. The site was a rural place in the center of a famous wine producing region.Journal of Biomolecular Structure & Dynamics. H.jbsdonline. however. M is so large that. Assuming visit to one pavilion takes time τ. well before the breakup of Soviet Union. and each offering (for free!) a glass of young wine. Grosberg1.umn. Levinthal paradox arises from the idea that the time required for a protein molecule to sample all of its conformations is at least Mτ. ISSN 0739-1102 Volume 20. one could have naively expected that after time close to Mτ the testing would be over.2 1Department of Physics University of Minnesota Minneapolis. with each individual in the cloud undergoing random walks. When biophysicists arrived. Russia A long time ago. are unlikely to realize the completion of the task and to stop at that.edu 317 . the paradox goes. each representing a particular village.com Abstract We estimate that the longest protein chain capable of exhaustive sampling of all its conformations within a millisecond is shorter than 15 residues. One possible reason is trivial: wine testers. the cloud of biophysicists seemed perfectly obeying the diffusion equation. Then. unguided folding into one particular (native) state requires at least time of order Mτ which is far too long. they found a large plaza with dozens of pavilions. Berezovsky and E. one millisecond. This increases the result for N0 to between 11 and 16. and random walk is forced to come back. will increase the exhaustion time (because exhaustion requires visiting all the tops of all energy barriers) – unlike folding time. M is so large because it is exponential in the degree of polymerization of the protein chain. [1] This number. .he said it was at least Mτ. (Note that energy landscape. which seems to rule out most of the speculations by IBET. one pavilion after another. such as N = 150.g. reads t ≅ τM2 / 1nM. which. At d > 2. of course. the mechanism of sampling changes. the exhaustion time is larger than Mτ. never returning to the once visited conformation. Thus. they decided to estimate the length of protein chain N0 such that at N < N0 protein can exhaustively sample all of its conformations within some specified time T. More sophisticated estimate for one dimensional case. (4)).318 Grosberg call it a “biological number. Berezovsky and E. Trifonov (2) turned the Levinthal’s argument up-side-down. let us discuss the estimate of N0 more closely. [2] twice smaller than (1). because random walk tends to leave behind large unvisited regions. They argued that while exhaustive sampling of all conformations is out of reach for large N. it is definitely possible for sufficiently small N. the result depends on d. accurate estimate of exhaustive sampling time by a random walk is not completely trivial. Based on τM2. they wrote τesN0 = T and obtained N0 = (1/s) 1n (T/τ) . IBET would have obtained N0 = (1/2s) 1n (T/τ) . suppose it is an unbiased random walk. N: M = esN. when present. which means we cover all M sites when √t/τ ≅ M. but sober model is unrealistic for the wine tester. For the random walk in the space of higher dimension d. where s is a constant close to unity (see. In fact. can be decreased). Needless saying. This is the length between 9 and 15. When d crosses over 2. Assuming exhaustive sampling time Mτ. How can we estimate the exhaustive sampling time for the unbiased random walk model? Consider first that wine testing pavilions arranged along a line.. Levinthal did not say that the time of exhaustive conformation sampling (or wine testing) was Mτ . say. in other words. Equally unrealistic is the model of protein chain dynamics which orderly samples all conformational states. such as a millisecond to a second? Comparison with wine testing makes it immediately clear that the answer depends on how the sampling is organized. a sober person can do that. Clearly. Mτ is the time required to visit all pavilions in an orderly fashion. in terms of wine testing.” where biological numbers dwarf astronomical ones. and perhaps more realistic. Of course. which means there is no conformation dependent (free) energy landscape involved. never returning to the already visited place. turns out to be somewhere between 23 and 31. model of protein dynamics (and also of wine testing) would be purely random walk in the space of conformations. This led IBET to a series of attractive speculations. which will not be derived here. he said it is ≥ Mτ. exhaustive sampling is only possible because the overall volume is restricted. I. To begin with. and the time of exhaustive sampling is t ≅ τΜ2. the very question is perfectly legitimate: what is the maximal length of the chain which can exhaustively sample all of its conformations within the specified time interval T. First of all. Indeed. Leaving speculations aside for a moment. the difference between τM and τM2 is very significant. In fact. e. This was sufficient for him to conclude that exhaustive sampling is impossible for realistic N. Then random walk of longevity t brings us as far as about √ t/τ. For all other sampling strategies. The opposite. according to (2). It is also well understood that high cooperativity is the property of proteins which is due to their peculiar selected sequences. It seems safe to say that this length is smaller than 15. it appears that IBET significantly overestimated the length of a protein capable of exhaustive conformation sampling. the small value of N0 makes the stability of the presumed folding units questionable. even the globule as a whole is not particularly stable. under the conditions of thermodynamic equilibrium between folded and unfolded states. First of all. There are quite a few other factors (5. This estimate. the all-or-none cooperative mechanism of folding. as already said. because d = M corresponds to the situation where each conformation (site in conformation space. not the real three-dimensional space. it is important to emphasize that the fact of non-cooperative folding in the majority of sequences is well understood beyond lattice models. or wine pavilion) can be equally probably reached from every other conformation in just one step τ.. while the parts – supposedly the units of hierarchical scenario – are not stable at all. IBET suggested that exhaustively sampling blocks combine together to form hierarchically folding large proteins. in this case. high cooperativity is a well established experimental fact (12). We leave it for the reader to decide whether the concept of folding should be applied to such a short chain. which means that it relies on the transition between denatured and native forms being highly cooperative. That means.4 and 4. Please do not forget that d here is the dimension of the abstract space of protein conformations. Indeed. as the scaling of folding time. in the review article (4). The result close to (1) would be correct for the dimension d as high as M. Under such conditions. all reducing the result for N0. which is very significantly smaller than Levinthal time proportional to τ exp (sN). Measurements in different regions of conformation space yield results for d between 1. exhaustion time would have scaled as τM ln M. Since everything related to the lattice models is perceived with a large dose of (healthy?) skepticism in protein community. This possibility was critically reviewed recently by Finkelstein (7). it is interesting to mention that experimental observations do not provide any evidence on the folding (under equilibrium conditions) time dependence on the chain length. it was foreseen by Bryngelson and Wolynes a long time ago (14). The relevant dimension was measured for the vicinity of the native conformation of a lattice model protein (5). Among random sequences. 319 Disconnected Notes Related to Levinthal Paradox .What is d in reality is anybody’s guess. of all-or-none type. To conclude this part. is valid under the conditions of thermodynamic equilibrium. It should be noted that the understanding of Levinthal paradox has progressed very significantly since it was first formulated (1). it is found that the folding time. At the next stage. e.5. Actually.g. This latter fact has been extensively tested using lattice models (as described. yielding N0 between 18 and 25. vast majority would not have exhibited any signs of cooperativity. in the point of thermodynamic equilibrium between native and denatured states. as it was first established by Shakhnovich and Gutin (13). Clearly. 6). real protein chains are nowhere near this extreme. and the whole of the hierarchical scheme even more difficult to imagine. scales as τ exp (s´N2/3) (9-11). His major point is that reasonably fast folding is frequently observed (see (8) and references therein) under the conditions where native state is not significantly lower in free energy (or not lower at all!) than fully denatured state – that is. Speaking about the relation between sequence selection. see also references therein). and all fundamentally arising from the fact that we are dealing with a polymer chain in which all units are linearly connected. In addition to this convincing argument by Finkelstein. this estimate is completely unrealistic. His analysis seems quite convincing. However. . Lett... related to the mutation stability (see also in the review article (4)). 13. 1828-1831 (2000).. Finkelstein. A. A. Tanaka. Du. C. B. 87. although sufficient to rule out any paradoxes. 65. 33. 594-600 (2001). 16). Understanding this was a remarkable achievement of the last decade. Dyn. Biomol. October. I. V. Barthèlèmy. Proc. Nat. 16.. J. 4670-4673 (1999). D. 9. Chem. Proc. J.. I. and does not need exhaustive conformation sampling to do so. Pande for critical reading of the first draft of this manuscript. E. Sci. Folding and Design 1. Proc..320 Grosberg the above mentioned theoretical prediction. Berezovsky I. How does the sequence selection work (or worked) in real evolution? What are the specific scenario of folding dynamics for selected sequences – how specific is the nucleation. T. Grosberg A. Privalov. USA 90. S. namely. I thank also V. motor and other functions of proteins. Phys. 2002. M. E. By contrast. in other words. 6.. Sci. 20. Wolynes.. A. . Opin. Gutin. it seems that the Levinthal’s question – how can protein sample “biologically large” number of conformations – has been answered: protein does not sample them. Rev.. Shakhnovich. M. 55. L. Shakhnovich. Struct. Finkelstein. E. A. 5-6 (2002).. Acad.. 7195 (1993).. the selected sequences – the same ones which exhibit highly cooperative folding-unfolding transition! – are reliable in the sense that their native state with high probability survives and remains stable even after several mutations. it is getting increasingly clear that there are many sequences which meet the sufficient criteria of reliable folding. Reference and Footnotes 1. 187. V. Adv. Yu. Biomol. To summarize. Europhys. Grosberg.. Protein Physics. Academic Press. Levinthal.. Yu. in the light of all the findings of the last decade... Nunes Amaral. Rev. Protein. 44-45 (1968). 3. Acad. 311-314 (2002). 115 (1997). how precisely do these selected sequences slide down their folding funnels (17-20)? What are the physical principles behind the selection of certain spatial structures. Shakhnovich. Ac. 1989. A. 11.. Phys. Shakhnovich. 20. where every “good” sequence is capable of folding. 167-241 (1997). In the majority of sequences. Chim. 259-314 (2000). I am indebted to I. Du. or folds and fold families (21)? What are the general physical principles behind the enzymatic. I. A. 2. Grosberg.. V. Pande. Chem. A. Ptitsyn.. T. 4. Tanaka. Fersht.. While the real mechanisms of evolutionary sequence selection remain unknown. T. However many questions remain open. Colloquium talk at the University of Minnesota Physics Department.. T. 7. Folding & Design 2. it seems clear (to the present author. what is the reaction coordinate associated with folding. Letters 84. Opin... Struct. 67 (1998). L. 15.. Dyn. Trifonov E. Curr. may be still an overestimate. Fraunfelder. Berezovsky and E. Biol. Struct. J. Biophys. Sci. Trifonov. A. R. P. Folding & Design 3. Biol. A. S.. H. and my personal discussions with them were useful and pleasant. Natl. R. how many and which conformations belong to the transition state. Badretdinov. 7. Scala.. USA 91. 14. Nat. of proteins? There are very many works on these subjects... 2002. Their paper (2) initiated the present note. Biol. Pande. 12. and Tanaka. 8. V. 34. 7524-7528 (1987). to make the list of them is a daunting task far beyond the framework of the present note. 7. Tanaka. A. P. Finkelstein. Yu. Gutin.. Bryngelson. G. 50-52 (1996). or mechanisms preventing aggregation. and while the computational models of sequence selection keep improving since the first suggestions (15. Letters 83. 10. The role of sequence selection is also well understood from a different view point.. 12972 (1994). A. Struct. Curr. 3-9 (1997). Phys. O. there are sufficiently many “good” sequences for the evolution to select from. Reviews of Modern Physics 72. 5. τ exp (s´N2/3). every mutation breaks the stability of the native state with the probability very close to 100%. E. Most importantly. J. at least) that the discussions about Levinthal paradox must now move forward to the new spheres. and do they have any relation to the principles involved in folding? What are the mechanisms of aggregation. 29-40 (1997). Grosberg. 17. A. 666 (1996). Tanaka. Curr. Wingreen.. D. K.. Struct. P. N. Socci. 4.. 10-19 (1997). Pande. Opin.. Rokhsar.. C. H. 19. Biol. Li. Folding and Design 1. V. 20. S. A. Chan. Tang. R. 2002 Communicated by the Editor Ramaswamy H Sarma . N.. G. Z.. Struct. Onuchic. Helling. Science 273. Yu. T.18. Wolynes. Nat. 441-450 (1996). J.. 321 Disconnected Notes Related to Levinthal Paradox Date Received: October 27.. 8. Biol. 68-79 (1998). H. A.. Dill. Luthey-Schulten. S. D.. Grosberg.. N. S... 21. . Jernigan* Baker Center for Bioinformatics and Biological Statistics Iowa State University 123 Office and Lab Bldg. but a highly interconnected network of such loop contacts. with probably the most fundamental paper on this “paradox” written ten years ago by Zwanzig et al. Hierarchical folding usually refers to the old views on protein folding. so that the ball rolls almost always (except for kinetic traps) downhill (4.jbsdonline. Rather it looks more like a bumpy funnel. Trifonov (1) follows a long line of work. Contrary to Berezovsky and Trifonov (J. 2002) the loops important in protein folding usually are much larger in size than 23-31 residues. being instead comparable to the size of the protein for single domain proteins. Key words: protein folding. Berezovsky and Edward N. 5). It suggests that the Levinthal paradox really does exist and has yet not been resolved. Instead he explains the folding pathways through formation of a folding nucleus and a delicate thermodynamic balance between the native and denaturated states of the protein (10). 5-6. which is untrue. closed loops. (2002) ©Adenine Press (2002) An Opinion Piece: Conversation on Levinthal Paradox & Protein Folding #4 Loop Folds in Proteins and Evolutionary Conservation of Folding Nuclei http://www. Issue Number 3. Biomol. The folding landscape does not resemble a flat golf course with a single hole corresponding to the native state. In our opinion the terminology “hierarchical folding” used by Finkelstein is improper. He also classified the idea of Berezovsky and Trifonov as a “hierarchical folding” and shows that such hierarchical folding would lead to a native state that is too stable to unfold. The funnel-like landscape of folding is. the result of hydrophobic collapse. There is significant literature on this problem. Ames. folding nucleus Andrzej Kloczkowski Robert L. Finkelstein (9) does not support this simplest funnel-like mechanism of folding. The common opinion in the protein folding community is that the Levinthal paradox (3) of finding a needle in a haystack doesn’t exist because proteins do not fold by randomly searching all possible (extremely large) numbers of conformations. Proteins fold to the native state from non-native (denaturated) states that are already substantially structured and make all the arithmetic supporting the Levinthal paradox irrelevant. In the letter “Cunning simplicity of a hierarchical folding” Alexei V. IA 50011-3020 The title “Loop Fold Structure of Proteins: Resolution of Levinthal paradox” of the communication by Igor N. looking for order or regularity in proteins. Struct. that the primary structure (sequence) leads to the formation of the protein secondary strucPhone: (515) 294-3833 Fax: (515)294-3841 Email: jernigan@iastate. (2).Journal of Biomolecular Structure & Dynamics. Levinthal paradox. ISSN 0739-1102 Volume 20.com Abstract We show that loops of close contacts involving hydrophobic residues are important in protein folding. and the assumption of a random coil as a starting point for the folding process is the basis of the Levinthal paradox. which provides extra stability to a protein fold and which leads to their conservation in evolution. A protein does not fold from random coil conformations. which greatly reduces the total conformational space (6-8). 20.edu 323 . according to the most popular theories. Dyn. Additionally what is important are not single loop contacts. evolutionary conservation. loop fold structure. Leu(94) and Tyr(97). Kloczkowski and Jernigan (18) developed a theory of evolutionary conserved residues based on the hydrophobic-polar HP lattice model of protein and showed that a highly connected network of hydrophobic contacts provides extra stability to a protein fold and may be crucial in reaching the lowest energy native state. by maximizing the number of hydrophobic contacts. We have developed a model for locating the evolutionarily conserved residues in proteins. but there is a possibility that these loop contacts inside a protein core might be a part of the folding nucleus (14). all of these conserved residues are obtained only by including phylogenetically diverse cases. The theory was also applied with success for identifying the conserved residues reported by Mirny and Shakhnovich (19) who used COC (conservatism of conservatism) method. the conserved residues are Gly(6). What is however important are not the single loop contacts of regular size 23-31 . measured by number of contacts. such as a nucleation mechanism. They identified 19 conserved hydrophobic non-functional residues. Such terminology leads him to conclude that the closed loops with nearly standard size segments of 23-31 residues reported by Trifonov and coworkers cannot be combined with more modern mechanisms of folding kinetics. Trp(14). pointing up the uncertainties involved in sequence comparisons. Phe(10). We studied the non-functional evolutionarily conserved residues in several subfamilies of proteins. Recently Kai-Li Ting and Jernigan (17) studied the conservation of the non-functional residues in the lysozyme/α-lactalbumin family and identified the possible folding nucleus. We first find a core of a protein with the known structure based on the packing of residues. According to Ptitsyn this network of conserved contacts could be the folding nucleus for cytochrome c. Then we find the so called “supercore” inside the core. Ile(111). All these hydrophobic residues form a network of conserved contacts which is possibly the folding nucleus. He came to the conclusion that there is a common folding nucleus composed of four residues. Another possibility is that there are present three conserved subclusters.324 Kloczkowski and Jernigan ture. 10-97 and 94-97. Using the horse cytochrome c (1hrc) (which has the total length on 105 residues) as the reference for the numbering. but not necessarily only those of almost regular size 23-31 as reported by Berezovsky et al. 6-97. Using sperm whale myoglobin (1mbd) as the reference the structurally conserved residues are Val(10). The predicted conserved residues for 1hrc and 1mbd are exactly the same as those reported by Ptitsyn and Ptistyn and Ting. All these residues are hydrophobic and form a network of five conserved contacts 6-94. Importantly. might be quite important in protein folding. The formation of such closed loops in the protein native state does not mean that residues forming such loop contacts always remain intact during the folding process. respectively. This work was initiated in our Lab by the late Oleg Ptitsyn who studied different subfamilies of c-type cytochromes (15). where each subcluster is composed of a network of conserved contacts. 10-94. especially if several of such loop contacts involving hydrophobic residues are located near one another inside the core. which in turn leads to the formation of the protein tertiary structure. As a matter of fact closed loops. in the series of their earlier papers (1113). That is probably too many due to a lack of substantial evolutionary diversity in the studied protein family. The second subfamily of proteins studied by Ptitsyn and Kai-Li Ting was globins (16). The non-functional conserved residues are located inside a protein core and their detection might give us valuable information about the mechanism of protein folding. which is of critical importance for its folding. Leu(115). Met(131) and possibly Leu(135). Note that the sizes of these loops are mostly substantially larger than the 23-31 proposed by Berezovsky et al. These results show that looping contacts are indeed important for protein folding. A. 655-666 (1998). L. Mol. 142-150 (1995). Dill. Acad. 54.. E.. 44-45 (1968). J.. 15. Proc. A. Biol. R. Berezovsky.. A. 20-22 (1992). References and Footnotes 1. K. 9...-L. 8. Biomol. Struct. S. 7524-7528 (1987). 18. and Jernigan. Sci. A. 12. R. Mol. Dyn. B. to be published. and Dill. Biol. Natl. Kloczkowski. Proc. and Dill. J. Proc. and Trifonov. while in the case of 1mbd the size of the loops is 100-120. J. Mol. Biomol. Evol. E. 19. Dyn. I. J. Proc. H. 278. USA 92. J. 65. K. Ting. but possibly involving also short helical contacts. 325 Loop Folds in Proteins and Conservation of Folding Nuclei Date Received: November 6.-L. I. G. N.. 5. A. L. Mirny. Berezovsky. 311-314 (2002). and Dill. and Trifonov. B. I. Natl. 2002 Communicated by the Editor Ramaswamy H Sarma . USA 89. K. Biophys. Biol. V.. A. 397-403 (2001). B. 66. Struct. V. E. Chim. Evolutionary Conserved Residues and Protein Folding. 2. 10. Biol. 14. 291. Natl. 8. 291. and Trifonov... Natl. N. J. Phys. 283-286 (2000). J. 11. Ptitsyn. 20. In the case of 1hrc such a loop is of size 90 residues (which is nearly the total length 105 of the protein). K. Finkelstein. Mol. O. 307.. J.. 17. Y.. USA 87. J. K. Mol. USA 84. Yue. 3. J. Lau. K.. K. Acad. Phys. K. also approaching the total length 153 of the protein. 5-6 (2002). and Gutin. I.. 13.. 1213-1215 (1973). Protein Sci. 1419-1426 (2001). 7. Sci.. A. Biomol. Proteins 23. Nauk SSSR 210. Chem. N. F. L.. C. 95.. and Shakhnovich. P. Dokl. J. Chim. N.residues. 20. Biol. Berezovsky. FESB Letters 466. I. N. 177-196 (1999). Acad. J. Bryngelson.. Ptitsyn.. M. Ptitsyn. N. and Jernigan. and Trifonov. 638-642 (1990). N. Akad. E. and Wolynes. Y. L. O. 19. B. 671-682 (1999). O. 20. 16. I. Badretdinov. R. 4. Acad. D. Bahar. 146-150 (1995). Chan. and Bagchi. Levinthal. Sci. reported by Berezovsky et al. 3775-3787 (1991). Berezovsky. Sci. R. Dyn. 1166-1180 (1999). and Ting. Szabo. and Jernigan.. A. A. 6. A. A. N. Grosberg A. Finkelstein. 454-466 (1994). Zwanzig. E.. but rather an interconnected network (or cluster) of such contacts of loops of much larger size. These are more consistent with the frequently remarked upon feature of proteins – that the ends of the chain are close together (20). Struct.. 425-436 (2002). . ac. even paradoxical. Belgium During a meeting held in 1969 in Monticello. Indeed. (2002) ©Adenine Press (2002) An Opinion Piece: Conversation on Levinthal Paradox & Protein Folding #5 What is Paradoxical about Levinthal Paradox? http://www. whereas the number of conformations sampled by a natural protein before reaching its final state is of the order of 108 (1). a few residues can only present a very marginal stability. According to Anfinsen (2). Marianne Rooman* Yves Dehouck Jean Marc Kwasigroch Christophe Biot Dimitri Gilis Ingénierie Biomoléculaire Université Libre de Bruxelles CP 165/64. which has essentially never been refuted. he proposed that “protein folding is speeded and guided by the rapid formation of local interactions which then determine the further folding of the peptide. Levinthal solved his own paradox. In particular. Issue Number 3. that the only way of achieving correct folding is by the sequential growth of the polypeptide chain on the ribosome. under suitable conditions. Levinthal estimated the number of different conformations accessible to a 150-residue protein to be roughly of the order of 10300.” Of course. Levinthal raised an interesting problem about protein folding. In other words. In the 70’s. according to Honig et al. this conclusion was reached on the basis of the experimental observation that small denatured proteins were able to refold in vitro (3).be 327 .Journal of Biomolecular Structure & Dynamics. He did not seem to find this paradoxical and immediately proposed a straightforward solution. there are two words in Levinthal’s sentence that are questionable: local and stable. Let us start with the last one. In the same report (usually incorrectly referenced). most current scenarios involve a huge number of parallel pathways possibly sharing a number of key steps. (5) in 1976. Levinthal himself solved what could at that time appear as nonintuitive. this suggests local amino acid sequences which form stable interactions and serve as nucleation points in the folding process. “proteins fold by following a multiply branched pathway”. native protein *Phone: 32-2-650 2067/5572 Fax: 32-2-650 3606 Email: mrooman@ulb. as he realized that proteins have no time to explore exhaustively their conformational space on the way to their native structure. The number of pathways was (and is still) a subject of debate. In good agreement with this assumption. 50 avenue Roosevelt B-1050 Bruxelles. It was realized at about the same time that the folding of a protein into its native structure essentially occurs independent of the initial conditions. Indeed. ISSN 0739-1102 Volume 20. Anfinsen clarified this point in 1973 by introducing the concept of ‘flickering equilibria’ (2): “it seems reasonable to suggest that portions of a protein chain that serve as nucleation sites for folding will be those that can ‘flicker’ in and out of the conformation that they occupy in the final protein. This ruled out the hypothesis. to our knowledge originally due to Chantrenne (4). and that they will form a relatively rigid structure stabilized by a set of cooperative interactions. Anfinsen did not restrict nucleation to local interactions along the protein chain. Clearly.jbsdonline. Another point of discussion concerned the question of whether protein folding is under thermodynamic or kinetic control. since he stated that “the nucleation centers might be expected to involve substructures as helices.com Abstract We would be tempted to state that there has never been a Levinthal paradox.” Moreover. the prevailing view was that folding follows pathways along which nucleation events take place. pleated sheets or β-bends”. 328 Rooman et al. Therefore. What have we learned since then? Many new folding models. frequently opposing the partisans of different models. on the contrary. depending or not on the environmental conditions. and that some trace of this evolution is left in the current proteins. It is difficult to believe that all current proteins exhibit this property. Both tendencies are probably often conjugated. We feel however that the originality of these ‘new’ concepts is often somewhat overrated by comparing to a hypothetical ‘classical view’ of protein folding which would involve well defined pathways. Hence. passionate. numerous experimental and theoretical studies have revealed protein sequences exhibiting a strong signal towards the native structure. which are much less numerous than high energy states. encoded locally along the sequence or. But Levinthal (1) argued that “the final conformation has not necessarily to be the one of lowest free energy. predominance of local interactions and an absolute necessity for stable intermediates. where typically some secondary structure elements or loops form first. However. in specific tertiary contacts. hierarchic folding units must not be viewed as rigid but rather as flickering entities. which actually means that they flicker in and out of a specific conformation. theories and concepts have been proposed. Moreover. it has been recently suggested that proteins are made up of closed loops that fold separately (9). . this simplistic view is very unlikely to correctly reflect the thoughts prevailing in those pioneering days. First. On the other hand. Another much debated issue is whether nucleation centers consist of local interactions along the chain or of tertiary contacts. and indisputably provide valuable precisions and clarifications on the mechanisms of protein folding. rather. it is obvious that all residues along the chain are not equally prone to constitute nucleation centers. often based on interesting ideas or experimental observations. vision of the folding mechanism. where folding is funneled towards low energy states. these approaches should be considered as complementary. it has been suggested that the energy gap between the native conformation and the other conformations that are structurally unrelated to it must be sufficiently large (10). as nicely pointed out by Anfinsen (2). with this softened view. It obviously must be a metastable state which is in a sufficiently deep energy well to survive possible perturbations in a biological system. In the context of hierarchic folding. Indeed. This unquestionably yields a very nice. This is probably. structures correspond to global free energy minima. there has been a continuing dispute between the supporters of nucleation and those of hierarchic folding. which have been assembled during evolution. with for example small flickering secondary structure elements forming a tertiary contact and thereby inducing nucleation. much of the controversy vanishes. intuitive. the lack of contradiction between old and new views has not prevented ongoing. a necessary condition. in general. These two views easily reconcile when considering that small structure elements can only flicker in and out. the idea that original proteins were small closed-loop peptides with flickering stability. Again. at least in a slightly modified form taking into account the existence of proteins adopting several folded structures. Another concept proposed a few years ago is that protein energy landscapes have the shape of a funnel (11). some specific protein residues have been observed to form native tertiary contacts earlier than others and to stabilize folding nuclei (8). is quite attractive.” These chosen extracts show that the folding problem was well understood in the early 70’s. As stressed by Honig (6). The answer seems obvious: it is proteindependent. This is supported by the experimental observation that some peptides are more structured in solution than others (7). debates around Levinthal’s paradox. To achieve rapid folding towards the native state. we do not feel that hierarchic folding must be opposed to nucleation. Curiously however. London. H. A. Dyson. Sci. A. Onuchic. N. Biol. and Trifonov. Mol. Proc.. R. recent advances should somewhat be relativized. C. Science 181. V. aggregate. B. are Research Assistant and Research Director. 2. J. Biol. Struct. 3. this basically corresponds to requiring the independence towards initial conditions and the absence of insurmountable energy minima on the ways to the native state. J. Monticello. 7167-7175 (1988). R. or to its pathological consequences. pp 22-24. Honig B. Pretending that nothing has evolved since the early 70’s is certainly an exaggeration. R. L. A. C.However.. edited by Vogel.. Berezovsky. M.. Y. Otzen D. Houghten R. N. In Informational Macromolecules. Bryson. FEBS Lett. Dyson. 1419-1426 (2001). the impression that some earlier contributions in the disciplne have been forgotten or misinterpreted and that in light of these. General remarks on protein structure and biosynthesis. Trifonov.. 11. exhibit domain swapping or several folded structures. J.. Chen. V. respectively.. S. C.. pp 153-166. Sci. C. 2002 Communicated by the Editor Ramaswamy H Sarma . C.. 10.. Abkevich. (University of Illinois Press.. Science 167. M. 201. 254. A. N. G. however. Mol. 1974-1978 (1976). Bogatyreva. P. B. 235. Proteins 21. Wilson. Edited by Debrunner P.. The development of potent experimental and theoretical techniques have led to support and clarify many of the proposed views. is supported by a BioVal research program of the Walloon Region. H.. A. S.. P. J. Anfinsen. 283-286 (2000).. A.. 260-288 (1995). Finally. Nature 318. evolution singled out a tiny fraction of possible amino acid sequences.. J. 9. J. Rance. and Lerner. Shakhnovich... 283-293 (1999). 223-230 (1973).. and Karplus. M. 8... 521-523 (2001). Natl. 329 What is Paradoxical about Levinthal Paradox? Date Received: October 29. C. 7. Illinois. O.. Gutin.. E. J. J. 6. R.. Biomol. A. Proceedings of a meeting held at Allerton House... I.. C.. Itzaki. Bryngelson. Socci. J. E. Biochemistry 27.. N. D. These considerations bring us to think about the folding mechanism that evolution tends to favor.. 4. P. 1309-1314 (1961). H. Haber. J. Ray.. and Lerner. N. Proc. J. and Fersht. Natl. and Lampen. having adequate folding and functional properties. B. Natl. Biol. B. and Finkelstein.. N. 14. We have. J. Mol.. and that there is probably also not a unique answer to the question whether native structures correspond to relative or absolute free energy minima. B. J.. A. at the Belgian National Fund for Scientific Research (FNRS). 5. V. This issue is probably related to the possible biological role played by the multiplicity of conformations. Protein Eng. a funnel-like shape has been shown to constitute an insufficient condition for ensuring consistent folding (12). I. H. Levinthal. USA 73. 20. A. I. Mol.. and Levinthal. p 122. 201-217 (1988). R. with the finding of proteins that polymerize. New-York & Paris. Sali. 307. Chantrenne H. Anfinsen.. 1961). G. Berezovsky. Illinois. it was realized that folding is even more complex to predict and simulate than originally hoped for. and Wolynes P. NewYork & London. Acknowledgments D.. E. Wright. by a grant from the Fonds de la Recherche pour l’Industrie et l’Agriculture (FRIA). Grosberg. Acad. References and Footnotes 1. USA 92. R. The Biosynthesis of Proteins (Pergamon Press. and Lerner. Mol. 12. N. Dyson. Tsibris J. I. But most certainly. A. Besides. 5-6 (2002). E.. M. 167-195 (1995).. A. Biol. Wright... F.. D. K. and M. J. E. and Anfinsen. USA 47. and Y. 886-887 (1970). A. How to Fold Graciously. 466. Dyn. B. and Shakhnovich. 293. Schechter. 1963). Proc. (Academic Press. and White F. Acad. Oxford. Wright. H.. E. & Munck E. Cross. Biol.. 1282-1286 (1995). D. Sela. R. Urbana. A. and to translating the ‘old view’ in the framework of statistical mechanics. Anfinsen. 1614-1636 (1994). E. Honig.. I. 1969). E.. In Mossbauer spectroscopy in biological systems. Houghten. N. 480-483 (1985). Acad. E.. . while the attractive pairwise contributions dramatically increase the probability of other conformations.com Abstract In this contribution we shall try to argue that no folding scenario – be it hierachical. nonhierarchical.jbsdonline. he will be forced to think carefully about something he might have never thought about otherwise. Here is. In this way. (2002) ©Adenine Press (2002) An Opinion Piece: Conversation on Levinthal Paradox & Protein Folding #6 Protein Folding: Where is the Paradox? http://www. after this thinking induced by the paradox. Ariel Fernández1. nor are their probabilities constant in time.e. etc. And. he might be surprised at some of Zeno’s conclusions. like “an arrow never reaches the target”. the exhaustive exploration of conformation space in a finite time of biological relevance is practically impossible since it is also exponential in N. Levinthal argued that since the number of possible conformations of a protein chain may be estimated to be exponential in the number (N) of aminoacids. the repulsive LennardJones terms in the intramolecular potential energy – to name a single contribution – dramatically reduce the probability of certain conformations (i. ISSN 0739-1102 Volume 20. Thus. the layman might not know that the sum of infinite rational numbers may yield a finite result. we decided to coin one of our own: A paradox is a logically consistent construction which sprouts from a false premise but one which is not too obviously so. the number of a-priori possible conformations of the chain might indeed be exponential in N. – needs to be invoked to solve Levinthal’s paradox: It fails on its own grounds. Since the layman would find the latter statement more striking than the original premise (we assume he has not been exposed to Calculus). Thus. He assumed the number of possible conformations for each individual aminoacid to be small and fixed. causing surprise and forcing us to revise the starting premise.e.edu 331 .Journal of Biomolecular Structure & Dynamics. and that the conformations available to each individual aminoacid may be assigned constant and equal probabilities (although the equality condition may be relaxed) at all times throughout the exploration of conformation space. i. we believe. those producing an over-all structure at odds with excluded volume). and Economics Program Columbia University Business School 3022 Broadway #8G New York. and produces a logical conclusion which is very ostensibly false. NY 10027 *Phone: 773 834 4782 Fax: 773 702 0439 Email: ariel@uchicago. IL 60637 2Finance Since we could not find a satisfactory definition of paradox. paradoxes help us to think more rigorously and be more critical of our own thoughts. Not quite in the ancient tradition of Zeno’s paradoxes. say from 2 to 100. nucleation. NY 10027 3100 Morningside Drive New York. This premise is false: the conformations of an individual aminoacid are not equally probable in time. he might be a bit less of a layman. The probability of a conformation of an individual aminoacid within the chain depends on the geometric or structural constraints and basins of attraction the chain generates as each aminoacid picks its coordinates. where the Socratic or moral content of the paradox resides.* Alejandro Belinky2 María de las Mercedes Boland3 1Institute for Biophysical Dynamics The University of Chicago Chicago. that the sum of infinite rational numbers might be finite. Issue Number 3. Thus. and for that reason. but this does not imply that the time to exhaustively visit all accessible conformations of the chain is also exponential in N. Date Received: October 5.332 Fernández et al. 2002 Communicated by the Editor Ramaswamy H Sarma . Moreover. Thus. The former might not be much of a defect in a paradox. the latter certainly is. these are baseless hypotheses. In a nutshell: The very existence of an intramolecular potential renders the starting premise of the Levinthal paradox false. Why is the thermodynamic limit even relevant to protein folding? Where does the idea that the native structure is the free energy minimum come from? To the best of our knowledge. even the idea that the protein chain exhaustively explores all conformations available along a successful folding pathway is false. we fail to see why the Levinthal paradox attracts attention: Its premises are too obviously false and the striking conclusion it purports to reach is uncalled for.