Equilibrium unfolding

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In biochemistry, equilibrium unfolding is the process of unfolding a protein or RNA molecule by gradually changing its solution conditions, i.e., its environment. Since equilibrium is maintained at all steps, the process is reversible (equilibrium folding). Equilibrium unfolding is used to determine the conformational stability of the molecule.

1 Theoretical background
2 Chemical denaturation
3 Structural probes
4 Thermal denaturation
5 Other forms of denaturation
6 References

[edit] Theoretical background

In its simplest form, equilibrium unfolding assumes that the molecule may belong to only two thermodynamic states, the folded state (typically denoted N for "native" state) and the unfolded state (typically denoted U). This "all-or-none" model of protein folding was first proposed by Tim Anson (1945), but is believed to hold only for small, single structural domains of proteins (Jackson, 1998); larger domains and multi-domain proteins often exhibit intermediate states. As usual in statistical mechanics, these states correspond to ensembles of molecular conformations, not just one conformation.

The molecule may transition between the native and unfolded states according to a simple kinetic model


$N \!\$		$U \!\$

with rate constants $k f$ and $k u$ for the folding ( $U \rightarrow N$ ) and unfolding ( $N \rightarrow U$ ) reactions, respectively. The dimensionless equilibrium constant $K_{eq} \ \stackrel{\mathrm{def}}{=}\ \frac{k_{u}}{k_{f}} = \frac{\left[ U \right]_{eq}}{\left[ N \right]_{eq}}$ can be used to determine the conformational stability $Δ G$ by the equation

Δ G = - R T ln K e q

where $R$ is the gas constant and $T$ is the absolute temperature in kelvins. Thus, $Δ G$ is positive if the unfolded state is less stable (i.e., disfavored) relative to the native state.

The most direct way to measure the conformational stability $Δ G$ of a molecule with two-state folding is to measure its kinetic rate constants $k f$ and $k u$ under the solution conditions of interest. However, since protein folding is typically completed in milliseconds, such measurements can be difficult to perform, usually requiring expensive stopped-flow or (more recently) continuous-flow mixers to provoke folding with a high time resolution.

[edit] Chemical denaturation

In the less expensive technique of equilibrium unfolding, the fractions of folded and unfolded molecules (denoted as $p N$ and $p U$ , respectively) are measured as the solution conditions are gradually changed from those favoring the native state to those favoring the unfolded state, e.g., by adding a denaturant such as guanidinium hydrochloride or urea. (In equilibrium folding, the reverse process is carried out.) Given that the fractions must sum to one and their ratio must be given by the Boltzmann factor, we have

$p_{N} = \frac{1}{1 + e^{-\Delta G/RT}}$

$p_{U} = \frac{e^{-\Delta G/RT}}{1 + e^{-\Delta G/RT}}$

Protein stabilities are typically found to vary linearly with the denaturant concentration. A number of models have been proposed to explain this observation prominent among them being the denaturant binding model, solvent-exchange model (both by Schellman, JA) and the Linear Energy Model (LEM; Pace, NC). All of the models assume that only two thermodynamic states are populated/de-populated upon denaturation. They could be extended to interpret more complicated reaction schemes.

The denaturant binding model assumes that there are specific but independent sites on the protein molecule (folded or unfolded) to which the denaturant binds with an effective (average) binding constant k. The equilibrium shifts towards the unfolded state at high denaturant concentrations as it has more binding sites for the denaturant relative to the folded state ( $Δ n$ ). In other words, the increased number of potential sites exposed in the unfolded state is seen as the reason for denaturation transitions. An elementary treatment results in the following functional form:

$\Delta G = \Delta G_{w} - RT \Delta n \ln \left(1 + k [D] \right)$

where $Δ G w$ is the stability of the protein in water and [D] is the denaturant concentration. Thus the analysis of denaturation data with this model requires 7 parameters: $Δ G w$ , $Δ n$ , k, and the slopes and intercepts of the folded and unfolded state baselines.

The solvent exchange model (also called the ‘weak binding model’ or ‘selective solvation’) of Schellman invokes the idea of an equilibrium between the water molecules bound to independent sites on protein and the denaturant molecules in solution. It has the form:

$\Delta G = \Delta G_{w} - RT \Delta n \ln \left(1 + (K-1) X_{D} \right)$

where $K$ is the equilibrium constant for the exchange reaction and $X d$ is the mole-fraction of the denaturant in solution. This model tries to answer the question of whether the denaturant molecules actually bind to the protein or they seem to be bound just because denaturants occupy about 20-30 % of the total solution volume at high concentrations used in experiments, i.e. non-specific effects – and hence the term ‘weak binding’. As in the denaturant-binding model, fitting to this model also requires 7 parameters. One common theme obtained from both these models is that the binding constants (in the molar scale) for urea and guanidinium hydrochloride are small: ~ 0.2 $M - 1$ for urea and 0.6 $M - 1$ for GuHCl.

Intuitively, the difference in the number of binding sites between the folded and unfolded states is directly proportional to the differences in the accessible surface area. This forms the basis for the LEM which assumes a simple linear dependence of stability on the denaturant concentration. The resulting slope of the plot of stability versus the denaturant concentration is called the m-value. In pure mathematical terms, m-value is the derivative of the change in stabilization free energy upon the addition of denaturant. However, a strong correlation between the accessible surface area (ASA) exposed upon unfolding, i.e. difference in the ASA between the unfolded and folded state of the studied protein (dASA), and the m-value has been documented by Pace and co-workers. In view of this observation, the m-values are typically interpreted as being proportional to the dASA. There is no physical basis for the LEM and is purely empirical, though it is widely used in interpreting solvent-denaturation data. It has the general form:

$\Delta G = m \left( [D]_{1/2} - [D] \right)$

where the slope $m$ is called the "m-value"(> 0 for the above definition) and $\left[ D \right]_{1/2}$ (also called C_m) represents the denaturant concentration at which 50% of the molecules are folded (the denaturation midpoint of the transition, where $p N = p U = 1 / 2$ ).

In practice, the observed experimental data at different denaturant concentrations are fit to a two-state model with this functional form for $Δ G$ , together with linear baselines for the folded and unfolded states. The $m$ and $\left[ D \right]_{1/2}$ are two fitting parameters, along with four others for the linear baselines (slope and intercept for each line); in some cases, the slopes are assumed to be zero, giving four fitting parameters in total. The conformational stability $Δ G$ can be calculated for any denaturant concentration (including the stability at zero denaturant) from the fitted parameters $m$ and $\left[ D \right]_{1/2}$ . When combined with kinetic data on folding, the m-value can be used to roughly estimate the amount of buried hydrophobic surface in the folding transition state.

[edit] Structural probes

Unfortunately, the probabilities $p N$ and $p U$ cannot be measured directly. Instead, we assay the relative population of folded molecules using various structural probes, e.g., absorbance at 287 nm (which reports on the solvent exposure of tryptophan and tyrosine), far-ultraviolet circular dichroism (180-250 nm, which reports on the secondary structure of the protein backbone) and near-ultraviolet fluorescence (which reports on changes in the environment of tryptophan and tyrosine). However, nearly any probe of folded structure will work; since the measurement is taken at equilibrium, there is no need for high time resolution. Thus, measurements can be made of NMR chemical shifts, intrinsic viscosity, solvent exposure (chemical reactivity) of side chains such as cysteine, backbone exposure to proteases, and various hydrodynamic measurements.

To convert these observations into the probabilities $p N$ and $p U$ , one generally assumes that the observable $A$ adopts one of two values, $A N$ or $A U$ , corresponding to the native or unfolded state, respectively. Hence, the observed value equals the linear sum

A = A N p N + A U p U

By fitting the observations of $A$ under various solution conditions to this functional form, one can estimate $A N$ and $A U$ , as well as the parameters of $Δ G$ . The fitting variables $A N$ and $A U$ are sometimes allowed to vary linearly with the solution conditions, e.g., temperature or denaturant concentration, when the asymptotes of $A$ are observed to vary linearly under strongly folding or strongly unfolding conditions.

[edit] Thermal denaturation

A similar formalism may be applied to other types of denaturation by changing the assumed functional form of $Δ G$ . For example, in thermal denaturation, the following functional form is often adopted

$\Delta G = \left(1 - \frac{T}{T_{mid}} \right) \Delta H_{mid} - \Delta C_{p} \left[ T_{mid} - T + T \ln \frac{T}{T_{mid}} \right]$

where $T m i d$ is the thermal unfolding midpoint temperature (in kelvins), $Δ H m i d$ is the enthalpy of folding at the midpoint, and $Δ C p$ is the difference in specific heats at constant pressure between the folded and unfolded states.

[edit] Other forms of denaturation

Analogous functional forms are posssible for denaturation by pressure, pH, etc.

[edit] References

Pace CN. (1975) "The Stability of Globular Proteins", CRC Critical Reviews in Biochemistry, 1-43.

Santoro MM and Bolen DW. (1988) "Unfolding Free Energy Changes Determined by the Linear Extrapolation Method. 1. Unfolding of Phenylmethanesulfonyl α-Chymotrypsin Using Different Denaturants", Biochemistry, 27, 8063-8068.

Privalov PL. (1992) "Physical Basis ffor the Stability of the Folded Conformations of Proteins", in Protein Folding, TE Creighton, ed., W. H. Freeman, pp. 83-126.

Yao M and Bolen DW. (1995) "How Valid Are Denaturant-Induced Unfolding Free Energy Measurements? Level of Conformance to Common Assumptions over an Extended Range of Ribonuclease A Stability", Biochemistry, 34, 3771-3781.

Anson ML. (1945) "Protein Denaturation and the Properties of Protein Groups", Advances in Protein Chemistry, 2, 361-386.

Jackson SE. (1998) "How do small single-domain proteins fold?", Folding and Design, 3, R81-R91.

Schwehm JM and Stites WE. (1998) "Application of Automated Methods for Determination of Protein Conformational Stability", Methods in Enzymology, 295, 150-170.

Protein structure determination methods
High resolution:	X-ray crystallography \| NMR \| Electron crystallography
Medium resolution:	Cryo-electron microscopy \| Fiber diffraction \| Mass spectrometry
Spectroscopic:	NMR \| Circular dichroism \| Absorbance \| Fluorescence \| Fluorescence anisotropy
Translational Diffusion:	Analytical ultracentrifugation \| Size exclusion chromatography \| Light scattering \| NMR
Rotational Diffusion:	Fluorescence anisotropy \| Flow birefringence \| Dielectric relaxation \| NMR
Chemical:	Hydrogen-deuterium exchange \| Site-directed mutagenesis \| Chemical modification
Thermodynamic:	Equilibrium unfolding
Computational:	Protein structure prediction \| Molecular docking
←Tertiary structure		Quaternary structure→