Levinthal’s paradox

Christian Anfmsen s experiments demonstrated that proteins can fold reversibly. A corollary result of Anfmsen s work is that the native structures of at least some globular proteins are thermodynamically stable states. But the matter of how a given protein achieves such a stable state is a complex one. Cyrus Levinthal pointed out in 1968 that so many conformations are possible for a typical protein that the protein does not have sufficient time to reach its most stable conformational state by sampling all the possible conformations. This argument, termed Levinthal s paradox, goes as follows consider a protein of 100 amino acids. Assume that there are only two conformational possibilities per amino acid, or = 1.27 X 10 ° possibilities. Allow 10 sec for... 

Levinthal s paradox led protein chemists to hypothesize that proteins must fold by specific folding pathways, and many research efforts have been devoted to the search for these pathways. 

Label-free optical detection 199 Lactate dehydrogenase 465 -469, 472 A-repressor folding kinetics 551 Lattice simulations 597 Leaving group 81, 86, 88-93 Levinthal s paradox 575,576, 598, 599,600... 

Levinthal s paradox [1,2], first introduced in the 1960 s (early in the childhood of simulations in Chemistry), serves as a good illustration of the limitations we still face in the application of molecular d3mamics (MD). Levinthal reasoned that if we were to assume that every residue in a polypeptide has a least two stable conformations, then a small 100 residue polypeptide would have possible states. If we were to study such a protein using traditional, state of the art, MD techniques, the native state would only be deduced after a little more than a billion years. 

Levinthal s paradox As there are an astronomical number of conformations possible, an unbiased search would take too long for a protein to fold, and yet most proteins fold in seconds ... 

Folding Paradox - Levinthal s paradox states that there are approximately 10 possible conformations for a protein, such as ribonuclease (124 residues). If one new conformation could be attempted every 10" seconds, it would still take over 10 years to randomly test all of the possibilities, yet ribonuclease... 

This problem is known as Levinthal s paradox (as it was formulated by Cyrus Levinthal (1922-1990) at Columbia University in New York) protein molecule certainly cannot search through all of its conformations, yet it does find the particular one with lowest energy. It is a paradox, isn t it How does the chain manage to find the equilibrium ... 

The long-standing problem of the polymer conformational transitions is an open issue of the cross-disciplinary research work and interest [1—4]. It is usually referred to as the Levinthal s paradox with an emphasis on the substantial discrepancy between the phenomenological data and the theoretical background of the issue [5]. The original Levinthal s analysis [5] has led to extensive search for the preferred pathways (trajectories) in the conformation space of a macromolecule. Recently, the problem is sharpened by the new approach that calls for the fiinnel-like form of the conformation space [6], probably offering the possibility to overcome the Levinthal s paradox. 

In this chapter, we offer a new approach to the problem. Actually, we show that the fuUy quantum-mechanical approach within the decoherence theory [7] offers both, existence and stability of the molecules conformations, and the rather fast decoherence-like transition between the different conformations. Within our approach, the Levinthal s paradox completely disappears. 

Actually, as Levinthal emphasizes, the conformational transitions can be realized through a sequence of local rotations (cf. Figure 9.3) eventually giving rise to another (energetically preferable) conformation. The core of the Levinthal s paradox can be presented as follows ... 

The Levinthal s paradox raised the search for the preferred (ordered set of) pathways in K-space. The core of the research work in this regard refers to this task essentially pointed out by Levinthal. However, some recent approaches shed some new fight in this concern. 

The Levinthal s paradox emerges from the (semi-)classical picture of the molecules conformations defined in the f-space of the one-dimensional model of Figure 9.4. Within this strategy, the particle bears a definite position h in every instant of time. Thus every conformational change can be represented by a trajectory (path) in K-space, following the shape of V(k). 

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