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Introducing the Levinthal's Protein Folding Paradox and Its Solution

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Introducing the Levinthal's Protein Folding Paradox and Its Solution Article pubs.acs.org/jchemeduc Introducing the Levinthal’s Protein Folding Paradox and Its Solution Leandro Martínez* Department of Physical Chemistry, Institute of Chemistry, University of Campinas, Campinas, Saõ Paulo 13083-970, Brazil *S Supporting Information ABSTRACT: The protein folding (Levinthal’s) paradox states that it would not be possible in a physically meaningful time to a protein to reach the native (functional) conformation by a random search of the enormously large number of possible structures. This paradox has been solved: it was shown that small biases toward the native conformation result in realistic folding times of realistic-length sequences. This solution of the paradox is, however, not amenable to most chemistry or biology students due to the demanding mathematics. Here, a simplification of the study of the paradox and its solution is provided so that it is accessible to chemists and biologists at an undergraduate or graduate level. Despite its simplicity, the model captures some fundamental aspects of the protein folding mechanism and allows students to grasp the actual significance of the popular folding funnel representation of the protein energy landscape. The analysis of the folding model provides a rich basis for a discussion of the relationships between kinetics and thermodynamics on a fundamental level and on student perception of the time-scales of molecular phenomena. KEYWORDS: Upper-Division Undergraduate, Graduate Education, Biochemistry, Biophysical Chemistry, Molecular Biology, Proteins/Peptides, Theoretical Chemistry nderstanding and controlling protein folding is arguably From a general physicochemical point of view, the protein U the most important challenge in structural biology. The folding problem is stated in Levinthal’s paradox:9 How can ability to predict, and thus model, protein structure and proteins assume their functional structures given the enormous function from sequence may, in the future, transform amount of unfolded structures possible? Levinthal’s paradox completely the molecular biology, synthetic chemistry, and was introduced from the supposition that the folding medicinal chemistry fields. Achieving this goal requires both the comprehension of the general physicochemical aspects of mechanisms of proteins occur by random search. Proteins, as folding and of the specificities leading each protein, with its linear chains of amino acids, should randomly search particular sequence, to adopt a specific functional structure. configurations until the native, functional arrangement is Presently, there have been great advances on the comprehen- reached. Such a search can be shown9,10 to be inconsistent sion of the physics of folding mechanisms in general, although with experimental folding rates in such a way that proteins with the actual predictive ability of current models for particular a few hundred residues would require astronomical scales of proteins is small. Thus, the protein folding problem can be time to reach, in average, the folded state. regarded at the same time solved as its physics is understood by Of course, real proteins fold in time-scales ranging from simple models, or completely open, as methods for prediction 11 1 microseconds to minutes, and therefore, only a minimal of structure from sequence are still very limited. fi Protein structures are complex enough to stimulate educa- fraction of the con gurational space is sampled. The general tional efforts in many areas. For example, a continuous area of requirements for folding mechanisms that solve this problem 12,13 educational research is the proposition of mechanical models of have been proposed; Bagchi and co-workers have shown representation of protein structures2,3 and the classification of that small energetic biases toward the native state reduce the folds in periodic-table-like charts.4 Experiments and macro- conformational search to the point that realistic folding times scopic models for specific unfolding events have been proposed can be obtained. The analysis presented here is based on this 5,6 for undergraduate students. Another amazing tool with both simple argument but avoids the use of mathematical concepts fi 7 educational and scienti c relevance is FoldIt, which provides not familiar to most chemistry or biology students. This an interactive game for people with no particular scientific analysis naturally leads to the “folding funnel” picture, which is background to play with protein folding by manipulating widespread in both scientific and didactic books, many times structures, with remarkable achievements in solving and 10 designing protein folds.7,8 Nevertheless, the general theory of with incomplete explanations or misinterpreted implications. protein folding has not been published in simplified forms accessible for a more general audience. Published: August 4, 2014 © 2014 American Chemical Society and Division of Chemical Education, Inc. 1918 dx.doi.org/10.1021/ed300302h | J. Chem. Educ. 2014, 91, 1918−1923 Journal of Chemical Education Article ■ AUDIENCE accordance with this experimental observation, the first This analysis was presented during a graduate course on assumption is “Thermodynamics of Biological Systems”. Most of the students Once a structure reaches the folded state, it remains folded. who were exposed to the present arguments, particularly Therefore, regardless of the protein folding path, reaching the experimental molecular biology students, did not have strong folded state would give rise to an increase of the folded mathematical backgrounds, a reason why the present derivation population, thus explaining function. is intended to be very simple. At the same time, this analysis At this point, the relationships between kinetics and addresses some fundamental concepts behind the proposal of a thermodynamics are discussed; the assumption above implies, physical theory of chemical reactions that graduates in physics thermodynamically, that the equilibrium population of the also appreciate. In particular, it stimulates discussions about (1) folded state is unitary. Therefore, the equilibrium population the differences between thermodynamically and kinetically cannot provide information on the folding mechanism and the controlled reactions and their intricate correlation in the folding kinetics must be studied. As with the determination of protein folding mechanism, (2) the validation of the simple mechanisms in physical chemistry, one may guess a particular models by intuitive guesses on the nature of molecular mechanism and deduce its kinetics, to be validated by comparison with experimental data. movements and energetics, and (3) validation or refusal of a fi theoretical model using experimental data. Similar to using this assumption is the calculation of the rst Therefore, the theory presented here can be included in passage time of the structure through the folded state. Here, ff however, the stability assumption is used because it clearly di erent undergraduate or graduate courses, for example, (1) as fi a illustrative example of chemical kinetics in a physical identi es the folding paradox as a kinetic rather than chemistry undergraduate course, (2) as an additional topic in thermodynamic problem, thus implying that the discussion any molecular biology, structural biology or biophysical course leading to its solution will be based on an assumption of its dealing with protein folding or protein structures, and (3) an mechanism. A fruitful discussion on the nature of protein native extra topic in a general chemistry course that focuses on structure stability is introduced by this assumption: most students with an experimental background know, from biochemistry or biophysics, after the introduction of basic fi chemical kinetics and thermodynamics. experience, that the activities of their puri ed proteins are Some suggested questions are available as Supporting rarely preserved for more than a few days. Thus, one may argue that many proteins may be kineticallybut not actually Information as a follow-up assignment, an assessment for learning or to stimulate classroom discussions. thermodynamically stable, thus justifying either the use of the first passage time or the assumption above to analyze ■ KINETIC DEFINITION OF THE PARADOX folding rates without additional complications related to unfolding reactions. The analysis of the protein folding problem is simplified by representing the sequence of a protein in a unidimensional way. ELAPSED TIME TO THE FOLDED STATE The “protein” is defined by a sequence of letters ■ At this point, the correct question is posed: given the first cicicciciciccicic... assumption on the stability of the folded state, how much time where c corresponds to a letter that corresponds to an “amino is needed for a protein to fold? The following model for the acid residue in the correct conformation”, that is, in the protein fold mechanism is proposed: conformation in which it should be in the native fold, and i is an (1) A protein starts with any unfolded structure. An unfolded “amino acid residue in the incorrect conformation”, that is, one structure, in the representation used here, is any structure different from that of the native fold. The first argument toward the comprehension of the for which at least one residue is in the incorrect, i, “ ” conformation. Levinthal paradox is that the correct fold is characterized by τ ff a unique conformation, in this case the sequence (2) At every time, , each residue (each letter) su ers a ccccccccccccccc.... perturbation.
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