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

Abstract

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 twostate 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 kJ͞mol) 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 I n 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):[1]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 ...

Dynamics, Energetics, and Structure in Protein Folding

“…Figure 5 shows DSC thermograms for free energy surfaces with a high barrier (~ 18 kJ/mol at the midpoint, see inset), a marginal barrier (~3 kJ/mol at the midpoint, see inset), and fully downhill. The three DSC thermograms can be well fit to a chemical two-state model (continuous lines), as has been discussed before for experimental DSC data (62,65). However, comparison with the true theoretical DSC baseline reveals that the fitted "native" baseline becomes increasingly sloped and shifts to higher values (i.e.…”

“…The expected changes are highly sloped baselines that might even cross near the unfolding transition. This effect has been documented in CD baselines (72), but it is in the DSC experiment that becomes very informative (62). The reason is that the DSC baselines directly report the energy fluctuations of the system and the difference between the "native" and "unfolded" baselines renders the ΔC p of unfolding.…”

“…In the second method, thermodynamic barrier heights for folding are obtained from the analysis of DSC data (62). The fundamental idea here is to extract the probability density as a function of an order parameter by taking advantage of the known connection between the DSC thermogram of a protein and its folding partition function.…”

“…When handling a real protein DSC thermogram, however, one must fit the experimental data to an idealized mathematical description of the free energy surface. In this particular case the free energy surface was represented by a Landau polynomial using enthalpy as the order parameter (62). The procedure is quite simple and involves fewer fitting parameters than even a chemical two-state fit.…”

Direct Observation of Amyloid Nucleation under Nanomechanical Stretching

“…This hypothesis is widely used in modeling protein folding and is known as the two-state model. [78][79][80]. The probability to observe the nucleation after t th scan is then equal to the probability that the tip will successfully stretch the protein after t th trials.…”