See also: Denaturation (biochemistry).
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, pH, adding chemical denaturants, or applying force as with an atomic force microscope tip.[1] [2] 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.
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,[3] 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 Uwith rate constants
kf
ku
\DeltaGo
\DeltaGo=-RTlnKeq
where
R
T
\DeltaGo
The most direct way to measure the conformational stability
\DeltaGo
kf
ku
\DeltaGo
In the less extensive technique of equilibrium unfolding, the fractions of folded and unfolded molecules (denoted as
pN
pU
pN=
1 | |
1+e-\Delta |
pU=1-pN=
e-\Delta | |
1+e-\Delta |
=
1 | |
1+e\Delta |
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[4]) and the Linear Extrapolation Model (LEM; by Nick Pace[5]). 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 (
\Deltan
\DeltaG=\DeltaGw-RT\Deltanln\left(1+k[D]\right)
where
\DeltaGw
\DeltaGw
\Deltan
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:
\DeltaG=\DeltaGw-RT\Deltanln\left(1+(K-1)XD\right)
where
K
Xd
M-1
M-1
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.[5] 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:
\DeltaG=m\left([D]1/2-[D]\right)
where the slope
m
\left[D\right]1/2
pN=pU=1/2
In practice, the observed experimental data at different denaturant concentrations are fit to a two-state model with this functional form for
\DeltaG
m
\left[D\right]1/2
\DeltaG
m
\left[D\right]1/2
Unfortunately, the probabilities
pN
pU
To convert these observations into the probabilities
pN
pU
A
AN
AU
A=ANpN+AUpU
By fitting the observations of
A
AN
AU
\DeltaG
AN
AU
A
Assuming a two state denaturation as stated above, one can derive the fundamental thermodynamic parameters namely,
\DeltaH
\DeltaS
\DeltaG
\DeltaCp
The thermodynamic observables of denaturation can be described by the following equations:
\begin{align}\DeltaH(T)&=\DeltaH(Td)+
T | |
\int | |
Td |
\DeltaCpdT \\ &=\DeltaH(Td)+\DeltaCp[T-Td] \\ \DeltaS(T)&=
\DeltaH(Td) | |
Td |
+
T | |
\int | |
Td |
\DeltaCpdlnT \\ &=
\DeltaH(Td) | |
Td |
+\DeltaCpln
T | |
Td |
\\ \DeltaG(T)&=\DeltaH-T\DeltaS \\ &=\DeltaH(Td)
Td-T | |
Td |
+
T | |
\int | |
Td |
\DeltaCpdT-
T | |
T\int | |
Td |
\DeltaCpdlnT \\ &=\Delta
H(T | ||||
|
\right)-\DeltaCp\left[Td-T+Tln\left(
T | |
Td |
\right)\right] \end{align}
where
\DeltaH
\DeltaS
\DeltaG
T
Td
In principle one can calculate all the above thermodynamic observables from a single differential scanning calorimetry 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 the variations in vs. which can be achieved from measurements with slight variations in pH or protein concentration. The slope of the linear fit is equal to the . Note that any non-linearity of the datapoints indicates that
\DeltaCp
Alternatively, the can also be estimated from the calculation of the accessible surface 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:[6]
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,
Apeak
\DeltaHvH(T)=-R
dlnK | |
dT-1 |
at
T=Td
\DeltaHvH(Td)
\DeltaHvH(Td)=
| ||||||||||||||||
Apeak |
When a two-state unfolding is observed the
Apeak=\DeltaHvH(Td)
\Delta
max | |
C | |
p |
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.[7] Such equations cannot be derived for pentamers of higher oligomers because of mathematical limitations (Abel–Ruffini theorem).