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biochemistry Biochemistry or biological chemistry is the study of chemical processes within and relating to living organisms. A sub-discipline of both chemistry and biology, biochemistry may be divided into three fields: structural biology, enzymology ...
, equilibrium unfolding is the process of unfolding a protein or RNA molecule by gradually changing its environment, such as by changing the temperature or
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country a ...
, pH, adding chemical denaturants, or applying force as with an
atomic force microscope Atomic force microscopy (AFM) or scanning force microscopy (SFM) is a very-high-resolution type of scanning probe microscopy (SPM), with demonstrated resolution on the order of fractions of a nanometer, more than 1000 times better than the opt ...
tip. If the equilibrium was maintained at all steps, the process theoretically should be reversible during equilibrium folding. Equilibrium unfolding can be used to determine the thermodynamic stability of the protein or RNA structure, i.e. free energy difference between the folded and unfolded states.


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 in 1945, but is believed to hold only for small, single
structural domain In molecular biology, a protein domain is a region of a protein's polypeptide chain that is self-stabilizing and that folds independently from the rest. Each domain forms a compact folded three-dimensional structure. Many proteins consist of se ...
s 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 constant In chemical kinetics a reaction rate constant or reaction rate coefficient, ''k'', quantifies the rate and direction of a chemical reaction. For a reaction between reactants A and B to form product C the reaction rate is often found to have the f ...
s k_ and k_ for the folding (U -> N) and unfolding (N -> U) reactions, respectively. The dimensionless equilibrium constant K_ \ \stackrel\ \frac = \frac can be used to determine the conformational stability \Delta G^o by the equation : \Delta G^ o = -RT \ln K_ where R is the
gas constant The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment p ...
and T is the
absolute temperature Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics. Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynamic ...
in
kelvin The kelvin, symbol K, is the primary unit of temperature in the International System of Units (SI), used alongside its prefixed forms and the degree Celsius. It is named after the Belfast-born and University of Glasgow-based engineer and ph ...
. Thus, \Delta G^o 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 \Delta G^o of a molecule with two-state folding is to measure its kinetic rate constants k_ and k_ 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.
Dual polarisation interferometry Dual-polarization interferometry (DPI) is an analytical technique that probes molecular layers adsorbed to the surface of a waveguide using the evanescent wave of a laser beam. It is used to measure the conformational change in proteins, or oth ...
is an emerging technique to directly measure
conformational change In biochemistry, a conformational change is a change in the shape of a macromolecule, often induced by environmental factors. A macromolecule is usually flexible and dynamic. Its shape can change in response to changes in its environment or oth ...
and \Delta G^o.


Chemical denaturation

In the less extensive technique of equilibrium unfolding, the fractions of folded and unfolded molecules (denoted as p_ and p_, 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 Urea, also known as carbamide, is an organic compound with chemical formula . This amide has two amino groups (–) joined by a carbonyl functional group (–C(=O)–). It is thus the simplest amide of carbamic acid. Urea serves an important ...
. (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_ = \frac : p_ = 1 - p_ = \frac = \frac 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 John Schellman) and the Linear Extrapolation Model (LEM; by Nick PaceMyers JK, Pace CN, Scholtz JM, Denaturant m values and heat capacity changes: relation to changes in accessible surface areas of protein unfolding, Protein Sci. 4(10), 2138–2148 (1995)). 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 (\Delta 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_ - RT \Delta n \ln \left(1 + k \right) where \Delta G_ is the stability of the protein in water and is the denaturant concentration. Thus the analysis of denaturation data with this model requires 7 parameters: \Delta G_,\Delta 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_ - RT \Delta n \ln \left(1 + (K-1) X_ \right) where K is the equilibrium constant for the exchange reaction and X_ 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^ for urea and 0.6 M^ 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 it is purely empirical, though it is widely used in interpreting solvent-denaturation data. It has the general form: : \Delta G = m \left( - \right) where the slope m is called the "''m''-value"(> 0 for the above definition) and \left D \right (also called ''Cm'') represents the denaturant concentration at which 50% of the molecules are folded (the '' denaturation midpoint'' of the transition, where p_ = p_ = 1/2). In practice, the observed experimental data at different denaturant concentrations are fit to a two-state model with this functional form for \Delta G, together with linear baselines for the folded and unfolded states. The m and \left D \right 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 \Delta G can be calculated for any denaturant concentration (including the stability at zero denaturant) from the fitted parameters m and \left D \right. 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.


Structural probes

Unfortunately, the probabilities p_ and p_ 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 Tryptophan (symbol Trp or W) is an α-amino acid that is used in the biosynthesis of proteins. Tryptophan contains an α-amino group, an α-carboxylic acid group, and a side chain indole, making it a polar molecule with a non-polar aromati ...
and
tyrosine -Tyrosine or tyrosine (symbol Tyr or Y) or 4-hydroxyphenylalanine is one of the 20 standard amino acids that are used by cells to synthesize proteins. It is a non-essential amino acid with a polar side group. The word "tyrosine" is from the Gr ...
), far-
ultraviolet Ultraviolet (UV) is a form of electromagnetic radiation with wavelength from 10 nm (with a corresponding frequency around 30  PHz) to 400 nm (750  THz), shorter than that of visible light, but longer than X-rays. UV radiati ...
circular dichroism (180-250 nm, which reports on the secondary structure of the protein backbone),
dual polarisation interferometry Dual-polarization interferometry (DPI) is an analytical technique that probes molecular layers adsorbed to the surface of a waveguide using the evanescent wave of a laser beam. It is used to measure the conformational change in proteins, or oth ...
(which reports the molecular size and fold density) and near-ultraviolet
fluorescence Fluorescence is the emission of light by a substance that has absorbed light or other electromagnetic radiation. It is a form of luminescence. In most cases, the emitted light has a longer wavelength, and therefore a lower photon energy, ...
(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_ and p_, one generally assumes that the observable A adopts one of two values, A_ or A_, corresponding to the native or unfolded state, respectively. Hence, the observed value equals the linear sum : A = A_ p_ + A_ p_ By fitting the observations of A under various solution conditions to this functional form, one can estimate A_ and A_, as well as the parameters of \Delta G. The fitting variables A_ and A_ are sometimes allowed to vary linearly with the solution conditions, e.g., temperature or denaturant concentration, when the
asymptote In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...
s of A are observed to vary linearly under strongly folding or strongly unfolding conditions.


Thermal denaturation

Assuming a two state denaturation as stated above, one can derive the fundamental thermodynamic parameters namely, \Delta H, \Delta S and \Delta G provided one has knowledge on the \Delta C_p of the system under investigation. The thermodynamic observables of denaturation can be described by the following equations: :\begin \Delta H(T)&=\Delta H(T_d)+ \int_^T \Delta C_p dT \\ &=\Delta H(T_d)+ \Delta C_p -T_d\\ \Delta S(T)&=\frac+ \int_^T \Delta C_p d \ln T \\ &=\frac+ \Delta C_p \ln \frac \\ \Delta G(T)&=\Delta H -T \Delta S \\ &=\Delta H(T_d) \frac+ \int_^T \Delta C_p dT - T\int_^T \Delta C_p d \ln T \\ &=\Delta H(T_d)\left(1-\frac\right) - \Delta C_p\left _d -T +T \ln \left(\frac\right)\right \end where \ \Delta H, \ \Delta S and \ \Delta G indicate the
enthalpy Enthalpy , a property of a thermodynamic system, is the sum of the system's internal energy and the product of its pressure and volume. It is a state function used in many measurements in chemical, biological, and physical systems at a constant ...
,
entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodyna ...
and
Gibbs free energy In thermodynamics, the Gibbs free energy (or Gibbs energy; symbol G) is a thermodynamic potential that can be used to calculate the maximum amount of work (physics), work that may be performed by a closed system, thermodynamically closed system a ...
of unfolding under a constant pH and pressure. The temperature, \ T is varied to probe the thermal stability of the system and \ T_d is the temperature at which half of the molecules in the system are unfolded. The last equation is known as the Gibbs–Helmholtz equation.


Determining the heat capacity of proteins

In principle one can calculate all the above thermodynamic observables from a single differential scanning calorimetry thermogram of the system assuming that the \ce is independent of the temperature. However, it is difficult to obtain accurate values for \ce this way. More accurately, the \ce can be derived from the variations in \ce vs. \ce which can be achieved from measurements with slight variations in pH or
protein concentration Proteins are large biomolecules and macromolecules that comprise one or more long chains of amino acid residues. Proteins perform a vast array of functions within organisms, including catalysing metabolic reactions, DNA replication, respon ...
. The slope of the linear fit is equal to the \ce. Note that any non-linearity of the datapoints indicates that \Delta C_p is probably not independent of the temperature. Alternatively, the \ce can also be estimated from the calculation of the
accessible surface area The accessible surface area (ASA) or solvent-accessible surface area (SASA) is the surface area of a biomolecule that is accessible to a solvent. Measurement of ASA is usually described in units of square angstroms (a standard unit of measurement ...
(ASA) of a protein prior and after thermal denaturation as follows: :\ce = \ce - \ce For proteins that have a known 3d structure, the \ce can be calculated through computer programs such as Deepview (also known as
swiss PDB viewer Swiss may refer to: * the adjectival form of Switzerland *Swiss people Places *Swiss, Missouri *Swiss, North Carolina * Swiss, West Virginia *Swiss, Wisconsin Other uses * Swiss-system tournament, in various games and sports * Swiss Internationa ...
). The \ce can be calculated from tabulated values of each amino acid through the semi-empirical equation: :\ce = \left( a_\ce \times \ce\right) + \left( a_\ce \times \ce \right) + \left( a_\ce \times \ce\right) where the subscripts polar, non-polar and aromatic indicate the parts of the 20 naturally occurring amino acids. Finally for proteins, there is a linear correlation between \ce and \ce through the following equation: :\ce = 0.61 \times \ce


Assessing two-state unfolding

Furthermore, one can assess whether the folding proceeds according to a two-state unfolding as described above. This can be done with differential scanning calorimetry by comparing the calorimetric enthalpy of denaturation i.e. the area under the peak, A_\text to the van 't Hoff enthalpy described as follows: :\Delta H_(T)= -R\frac at T=T_d the \Delta H_(T_d) can be described as: :\Delta H_(T_d)= \frac When a two-state unfolding is observed the A_\text=\Delta H_(T_d). The \Delta C_p^ is the height of the heat capacity peak.


Generalization to

protein complexes A protein complex or multiprotein complex is a group of two or more associated polypeptide chains. Protein complexes are distinct from multienzyme complexes, in which multiple catalytic domains are found in a single polypeptide chain. Protein ...
and multi-domain proteins

Using the above principles, equations that relate a global protein signal, corresponding to the folding states in equilibrium, and the variable value of a denaturing agent, either temperature or a chemical molecule, have been derived for homomeric and heteromeric proteins, from monomers to trimers and potentially tetramers. These equations provide a robust theoretical basis for measuring the stability of complex proteins, and for comparing the stabilities of wild type and mutant proteins. Such equations cannot be derived for pentamers of higher oligomers because of mathematical limitations (Abel–Ruffini theorem).


References


Further reading

* 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 for 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. * Jackson SE. (1998) "How do small single-domain proteins fold?", ''Folding & 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. {{Chemical equilibria Protein structure Equilibrium chemistry