Exploring protein-folding ensembles: A variable-barrier model for the analysis of equilibrium unfolding experiments

Exploring protein-folding ensembles: A variable-barrier model for the analysis of equilibrium unfolding experiments
approved October 28, 2004 (received for review August 9, 2004)
Published online before print December 9, 2004
Victor Mu“oz *, , and Jose M. Sanchez-Ruiz , ,
PNAS | December 21, 2004
*Department of Chemistry and Biochemistry and Center for Biomolecular Structure and Organization, University of Maryland, College Park, MD 20742; Departamento de Quœmica Fœsica, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain; and Instituto de Biocomputaci€n y Fœsica de Sistemas Complejos, Universidad de Zaragoza, 50009 Zaragoza, Spain
Edited by Michael Levitt, Stanford University School of Medicine, Stanford, CA
Recent theoretical and experimental results point to the existence of small barriers to protein folding. These barriers can even be absent altogether, resulting in a continuous folding transition (i.e., downhill folding). With small barriers, the detailed properties of folding ensembles may become accessible to equilibrium experiments. However, further progress is hampered because folding experiments are interpreted with chemical models (e.g., the two-state model), which assume the existence of well defined macrostates separated by arbitrarily high barriers. Here we introduce a phenomenological model based on the classical Landau theory for critical transitions. In this physical model the height of the thermodynamic free energy barrier and the general properties of the folding ensemble are directly obtained from the experimental data. From the analysis of differential scanning calorimetry data alone, our model identifies the presence of a significant (>35 kJ/mol) barrier for the two-state protein thioredoxin and the absence of a barrier for BBL, a previously characterized downhill folding protein. These results illustrate the potential of our approach for extracting the general features of protein ensembles from equilibrium folding experiments.
experimental analysis | free energy barrier | downhill folding | two-state folding | phenomenological model
In contrast to the situation in many fields of modern physics, experimental results in protein folding are seldom directly interpretable by analytical theory or computer simulations. The interpretation typically involves a simple phenomenological model with which experimental results are analyzed, and the outcome of such ad hoc analysis is used to extract conclusions regarding experiments. A paradigm of this principle is the two-state model, in which protein-folding reactions are analyzed in terms of a chemical equilibrium between two independent species, native (N) and unfolded (U):
In Eq. 1, species with intermediate degree of structure are ignored, and the transition from one state to the other is of the first order. The use of a two-state model and its obvious generalization (i.e., a series of chemical equilibria between n structurally defined macrostates) is deeply rooted in the tradition of describing chemical transformations of small molecules as reaction schemes.
Despite the obvious limitations of comparing protein folding with simple chemical reactions, the two-state protein-folding model enjoys tremendous popularity. One of the reasons is that this model seems to accommodate the folding behavior of a large set of single-domain proteins (1). Two-state folding could also provide proteins with a significant biological advantage by conferring upon them kinetic stability in vivo (see ref. 2 for a recent discussion). However, to make sure that the two-state character of proteins is not a self-fulfilling prophecy (3), it is important to analyze experimental data with a procedure that does not make assumptions about the existence of a free energy barrier. A more general procedure would also provide an opportunity to extract information about protein-folding ensembles that is discarded in the traditional two-state analysis.
The need for better procedures to analyze folding experiments has become more urgent in light of recent developments. Theory predicts that folding free energy barriers arise from the nonsynchronous compensation between energy and entropy (4, 5), and are small in the chemical sense (6). Accordingly, protein-folding transitions are expected to be of the first order (i.e., type I scenario in the energy landscape language) or continuous (i.e., type 0 scenario, or downhill) depending on experimental conditions (4). Kinetic experiments in very fast-folding proteins (7) and thermodynamic analysis of the folding kinetics of several two-state-like proteins (8) suggest that for many natural proteins folding barriers are, indeed, rather small. Computer-designed proteins fold faster than their natural templates, although no selection for folding efficiency was included in the design strategy (9). Therefore, the higher folding barriers of the natural proteins might be the result of natural selection, rather than an intrinsic feature of protein folding. The theoretical analysis of protein polymer models also indicates that it might be harder for proteins to achieve cooperativity (i.e., a large free energy barrier) than a stable folded structure (10). Furthermore, folding barriers can be reduced by mutations resulting in a populated "activated" complex (11) or can even disappear when mutations are combined with extrinsic stabilizing agents (12). In fact, in some proteins the folding free energy barrier can be altogether absent in thermodynamic terms, resulting in global downhill folding and continuous unfolding transitions (13, 14). Proteins with such features could even have an important biological role as molecular rheostats (14).
The problem of describing processes that, depending on conditions, behave either as first-order or continuous transitions arises in a well known branch of thermodynamics: the theory of critical transitions. In the classical Landau theory of critical transitions (see chapter 10 in ref. 15), this phenomenon is described with a free energy functional expressed as a series expansion in powers of an "order parameter" (the property that exhibits large fluctuations near critical conditions) and truncating the expansion at the quartic level. The truncated expansion produces a free energy functional with one or two free energy minima, depending on the sign of the coefficient of the quadratic term
Here, using the Landau free energy as a starting point, we introduce a simple phenomenological model for the analysis of equilibrium protein-folding experiments. The great advantage of this model is that the height of the free energy barrier and the general properties of the folding ensemble are not preassumed but are obtained directly from the experimental data. We have used this variable-barrier model to analyze differential scanning calorimetry (DSC) experiments of protein unfolding. Because DSC data are directly related to the relevant protein partition function (16), their analysis with the model highlights the potential of this approach for investigating equilibrium folding ensembles. The model produces a large barrier when used to analyze the DSC thermogram of the two-state protein thioredoxin (17) and a barrierless free energy profile when used to analyze the DSC thermogram of the downhill folding protein BBL (14).
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