Equilibrium unfolding
Encyclopedia
In biochemistry
, 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, adding chemical denaturants, or applying force as with an atomic force microscope
tip. 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.
in 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
with rate constants
and 
for the folding (
) and unfolding (
) reactions, respectively. The dimensionless equilibrium constant 
can be used to determine the conformational stability 
by the equation
where
is the gas constant
and
is the absolute temperature
in kelvin
s. Thus,
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
of a molecule with two-state folding is to measure its kinetic rate constants 
and 
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
is an emerging technique to directly measure conformational change
and
.
and 
, 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
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 Energy Model (LEM; by Nick Pace). 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 (
). 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:
where
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: 
,
, 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:
where
is the equilibrium constant for the exchange reaction and 
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 
for urea and 0.6 
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:
where the slope
is called the "m-value"(> 0 for the above definition) and 
(also called Cm) represents the denaturant concentration at which 50% of the molecules are folded (the denaturation midpoint
of the transition, where
).
In practice, the observed experimental data at different denaturant concentrations are fit to a two-state model with this functional form for
, together with linear baselines for the folded and unfolded states. The 
and 
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 
can be calculated for any denaturant concentration (including the stability at zero denaturant) from the fitted parameters 
and 
. 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.
and 
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), dual polarisation interferometry
(which reports the molecular size and fold density) 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 shift
s, 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
and 
, one generally assumes that the observable 
adopts one of two values, 
or 
, corresponding to the native or unfolded state, respectively. Hence, the observed value equals the linear sum
By fitting the observations of
under various solution conditions to this functional form, one can estimate 
and 
, as well as the parameters of 
. The fitting variables 
and 
are sometimes allowed to vary linearly with the solution conditions, e.g., temperature or denaturant concentration, when the asymptote
s of
are observed to vary linearly under strongly folding or strongly unfolding conditions.
, 
and 
provided one has knowledge on the 
of the system under investigation.
The thermodynamic observables of denaturation can be described by the following equations:
→
→
→
→
where
, 
and 
indicate the enthalpy
, entropy
and Gibbs free energy
of unfolding under a constant pH and pressure. The temperature,
is varied to probe the thermal stability
of the system and
is the temperature at which half of the molecules in the system are unfolded. The last equation is known as the Gibbs-Helmholtz equation
.
thermogram of the system assuming that the
is independent of the temperature. However, it is difficult to obtain accurate values for 
this way. More accurately, the 
can be derived from a the variations in 
vs. 
which can be achieved from measurements with slight variations in 
or protein concentration. The slope of the linear fit is equal to the 
. Note that any non-linearity of the datapoints indicates that 
is probably not independent of the temperature.
Alternatively, the
can be estimated very accurately from the calculation of the accessible solvent area (ASA) of a protein prior and after thermal denaturation as follows:
For proteins that have a known 3d structure, the
can be calculated through computer programs such as Deepview (also known as swiss PDB viewer). The 
can be calculated from tabulated values of each amino acid through the semi-empirical equation:
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
and 
through the following equation:
by comparing the calorimetric enthalpy of denaturation i.e. the area under the peak,
to the van 't Hoff enthalpy described as follows:
at
the 
can be described as:
When a two-state unfolding is observed the
. The 
is the height of the heat capacity peak.
, pH
, or by applying force with an atomic force microscope
tip.
Biochemistry
Biochemistry, sometimes called biological chemistry, is the study of chemical processes in living organisms, including, but not limited to, living matter. Biochemistry governs all living organisms and living processes...
, equilibrium unfolding is the process of unfolding a protein or RNA molecule
Protein folding
Protein folding is the process by which a protein structure assumes its functional shape or conformation. It is the physical process by which a polypeptide folds into its characteristic and functional three-dimensional structure from random coil....
by gradually changing its environment, such as by changing the temperature or pressure, adding chemical denaturants, or applying force as with an atomic force microscope
Atomic force microscope
Atomic force microscopy or scanning force microscopy is a very high-resolution type of scanning probe microscopy, with demonstrated resolution on the order of fractions of a nanometer, more than 1000 times better than the optical diffraction limit...
tip. 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.
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 AnsonMortimer Louis Anson
Mortimer Louis Anson was an early protein scientist.He is famous for having proposed that protein folding was areversible, two-state reaction, and for being the foundingeditor of the journal Advances in Protein Chemistry....
in 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
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
, 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
and for the folding () and unfolding () reactions, respectively. The dimensionless equilibrium constant can be used to determine the conformational stability by the equationwhere
is the gas constantGas constant
The gas constant is a physical constant which is featured in many fundamental equations in the physical sciences, such as the ideal gas law and the Nernst equation. It is equivalent to the Boltzmann constant, but expressed in units of energy The gas constant (also known as the molar, universal,...
and
is the absolute temperatureThermodynamic temperature
Thermodynamic temperature is the absolute measure of temperature and is one of the principal parameters of thermodynamics. Thermodynamic temperature is an "absolute" scale because it is the measure of the fundamental property underlying temperature: its null or zero point, absolute zero, is the...
in kelvin
Kelvin
The kelvin is a unit of measurement for temperature. It is one of the seven base units in the International System of Units and is assigned the unit symbol K. The Kelvin scale is an absolute, thermodynamic temperature scale using as its null point absolute zero, the temperature at which all...
s. Thus,
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
of a molecule with two-state folding is to measure its kinetic rate constants and 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 interferometryDual Polarisation Interferometry
Dual polarization interferometry is an analytical technique that can probe molecular scale layers adsorbed to the surface of a waveguide by using the evanescent wave of a laser beam confined to the waveguide...
is an emerging technique to directly measure conformational change
Conformational change
A macromolecule is usually flexible and dynamic. It can change its shape in response to changes in its environment or other factors; each possible shape is called a conformation, and a transition between them is called a conformational change...
and
.Chemical denaturation
In the less expensive technique of equilibrium unfolding, the fractions of folded and unfolded molecules (denoted as and , 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 denaturantDenaturation (biochemistry)
Denaturation is a process in which proteins or nucleic acids lose their tertiary structure and secondary structure by application of some external stress or compound, such as a strong acid or base, a concentrated inorganic salt, an organic solvent , or heat...
such as guanidinium hydrochloride
Guanidine
Guanidine is a crystalline compound of strong alkalinity formed by the oxidation of guanine. It is used in the manufacture of plastics and explosives. It is found in urine as a normal product of protein metabolism. The molecule was first synthesized in 1861 by the oxidative degradation of an...
or urea
Urea
Urea or carbamide is an organic compound with the chemical formula CO2. The molecule has two —NH2 groups joined by a carbonyl functional group....
. (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
Boltzmann factor
In physics, the Boltzmann factor is a weighting factor that determines the relative probability of a particle to be in a state i in a multi-state system in thermodynamic equilibrium at temperature T...
, we have
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 Energy Model (LEM; by Nick Pace). 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 (
). 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:where
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: ,, 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:
where
is the equilibrium constant for the exchange reaction and 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 for urea and 0.6 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:
where the slope
is called the "m-value"(> 0 for the above definition) and (also called Cm) represents the denaturant concentration at which 50% of the molecules are folded (the denaturation midpointDenaturation midpoint
Assuming two-state protein folding, denaturation midpoint is defined as that temperature or denaturant concentration at which both the folded and unfolded states are equally populated at equilibrium....
of the transition, where
).In practice, the observed experimental data at different denaturant concentrations are fit to a two-state model with this functional form for
, together with linear baselines for the folded and unfolded states. The and 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 can be calculated for any denaturant concentration (including the stability at zero denaturant) from the fitted parameters and . 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 and 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 tryptophanTryptophan
Tryptophan is one of the 20 standard amino acids, as well as an essential amino acid in the human diet. It is encoded in the standard genetic code as the codon UGG...
and tyrosine
Tyrosine
Tyrosine or 4-hydroxyphenylalanine, is one of the 22 amino acids that are used by cells to synthesize proteins. Its codons are UAC and UAU. It is a non-essential amino acid with a polar side group...
), far-ultraviolet
Ultraviolet
Ultraviolet light is electromagnetic radiation with a wavelength shorter than that of visible light, but longer than X-rays, in the range 10 nm to 400 nm, and energies from 3 eV to 124 eV...
circular dichroism (180-250 nm, which reports on the secondary structure of the protein backbone), dual polarisation interferometry
Dual Polarisation Interferometry
Dual polarization interferometry is an analytical technique that can probe molecular scale layers adsorbed to the surface of a waveguide by using the evanescent wave of a laser beam confined to the waveguide...
(which reports the molecular size and fold density) and near-ultraviolet fluorescence
Fluorescence
Fluorescence is the emission of light by a substance that has absorbed light or other electromagnetic radiation of a different wavelength. It is a form of luminescence. In most cases, emitted light has a longer wavelength, and therefore lower energy, than the absorbed radiation...
(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
Nuclear magnetic resonance
Nuclear magnetic resonance is a physical phenomenon in which magnetic nuclei in a magnetic field absorb and re-emit electromagnetic radiation...
chemical shift
Chemical shift
In nuclear magnetic resonance spectroscopy, the chemical shift is the resonant frequency of a nucleus relative to a standard. Often the position and number of chemical shifts are diagnostic of the structure of a molecule...
s, intrinsic viscosity
Intrinsic viscosity
Intrinsic viscosity \left[ \eta \right] is a measure of a solute's contribution to the viscosity \eta of a solution. Intrinsic viscosity is frequently referred to as "Inherent Viscosity" in macromolecular literature...
, 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
and , one generally assumes that the observable adopts one of two values, or , corresponding to the native or unfolded state, respectively. Hence, the observed value equals the linear sumBy fitting the observations of
under various solution conditions to this functional form, one can estimate and , as well as the parameters of . The fitting variables and are sometimes allowed to vary linearly with the solution conditions, e.g., temperature or denaturant concentration, when the asymptoteAsymptote
In analytic geometry, an asymptote of a curve is a line such that the distance between the curve and the line approaches zero as they tend to infinity. Some sources include the requirement that the curve may not cross the line infinitely often, but this is unusual for modern authors...
s of
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,, and provided one has knowledge on the of the system under investigation.The thermodynamic observables of denaturation can be described by the following equations:
→
→
→
→
where
, and indicate the enthalpyEnthalpy
Enthalpy is a measure of the total energy of a thermodynamic system. It includes the internal energy, which is the energy required to create a system, and the amount of energy required to make room for it by displacing its environment and establishing its volume and pressure.Enthalpy is a...
, entropy
Entropy
Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...
and Gibbs free energy
Gibbs free energy
In thermodynamics, the Gibbs free energy is a thermodynamic potential that measures the "useful" or process-initiating work obtainable from a thermodynamic system at a constant temperature and pressure...
of unfolding under a constant pH and pressure. The temperature,
is varied to probe the thermal stabilityThermal stability
Thermal stability is the stability of a molecule at high temperatures; i.e. a molecule with more stability has more resistance to decomposition at high temperatures....
of the system and
is the temperature at which half of the molecules in the system are unfolded. The last equation is known as the Gibbs-Helmholtz equationGibbs-Helmholtz equation
The Gibbs–Helmholtz equation is a thermodynamic equation useful for calculating changes in the Gibbs energy of a system as a function of temperature...
.
Determining the heat capacity of proteins
In principle one can calculate all the above thermodynamic observables above from a single differential scanning calorimetryDifferential scanning calorimetry
Differential scanning calorimetry or DSC is a thermoanalytical technique in which the difference in the amount of heat required to increase the temperature of a sample and reference is measured as a function of temperature. Both the sample and reference are maintained at nearly the same temperature...
thermogram of the system assuming that the
is independent of the temperature. However, it is difficult to obtain accurate values for this way. More accurately, the can be derived from a the variations in vs. which can be achieved from measurements with slight variations in or protein concentration. The slope of the linear fit is equal to the . Note that any non-linearity of the datapoints indicates that is probably not independent of the temperature.Alternatively, the
can be estimated very accurately from the calculation of the accessible solvent area (ASA) of a protein prior and after thermal denaturation as follows:For proteins that have a known 3d structure, the
can be calculated through computer programs such as Deepview (also known as swiss PDB viewer). The can be calculated from tabulated values of each amino acid through the semi-empirical equation: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
and through the following equation:Assessing two-state unfolding
Furthermore, one can assess whether the folding proceeds according to a two-state unfolding as described above. The can be done with differential scanning calorimetryDifferential scanning calorimetry
Differential scanning calorimetry or DSC is a thermoanalytical technique in which the difference in the amount of heat required to increase the temperature of a sample and reference is measured as a function of temperature. Both the sample and reference are maintained at nearly the same temperature...
by comparing the calorimetric enthalpy of denaturation i.e. the area under the peak,
to the van 't Hoff enthalpy described as follows:at
the can be described as:When a two-state unfolding is observed the
. The is the height of the heat capacity peak.Other forms of denaturation
Analogous functional forms are possible for denaturation by pressurePressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...
, pH
PH
In chemistry, pH is a measure of the acidity or basicity of an aqueous solution. Pure water is said to be neutral, with a pH close to 7.0 at . Solutions with a pH less than 7 are said to be acidic and solutions with a pH greater than 7 are basic or alkaline...
, or by applying force with an atomic force microscope
Atomic force microscope
Atomic force microscopy or scanning force microscopy is a very high-resolution type of scanning probe microscopy, with demonstrated resolution on the order of fractions of a nanometer, more than 1000 times better than the optical diffraction limit...
tip.