Protein Structure and Folding
Folding and unfolding by temperature and pressure
Ultimately one is interested in protein stability and folding under physiological conditions. However, such
investigations are troubled in many ways and will thus not reveal salient properties of the system. Here we will examine
a simple system: a reversible single step transition from the native state (N) to the denatured state (D) governed by an
equilibrium constant K.
K
N 
D
Initially we will outline thermodynamics for temperature changes and subsequently expand the treatment to include
pressure changes. Its worth to remind that protein stability is perturbed also by denaturants and pH.
1.
Effect of temperature
The Gibbs energy of the transition is
G(T )  G D  G N  H (T )  TS (T )   RT ln K
where we have partitioned to the enthalpy to express the role of interactions responsible for the stability and to the
entropy to express the role of the available states to the stability. At the midpoint of the transition
G(Tm )  H (Tm )  Tm S (Tm )  0
In the vicinity of the transition point denoted by Tm the enthalpy can be written as
T
H (T )  H (Tm )   C P (T )dT
Tm
 H (T ) 
C P  

 T  P
where we have employed the definition for the heat capacity change under constant pressure. The integration over the
temperature range is simple when the heat capacity change can be regarded as a constant to give
H (T )  H (Tm )  T  Tm C P
Likewise we may derive the entropy of the transition
 S (T ) 
S (T )  S (Tm )   
 dT
T  P
Tm 
T
The first term and the integrant can be re-expressed using the relations to the enthalpy at the transition
S (Tm ) 
H (Tm )
,
Tm
1
 S (T ) 
 S (T )   H (T ) 

 
 
  CP
 T  P  H  P  T  P T
 dS 
C P
dT
T
to obtain
S (T ) 
H (Tm ) T
  C P (T )d ln T
Tm
Tm
1
where d(lnT) =(1/T)dT has been used. Once again assuming the constant heat capacity change over the temperature
range of interest we have
S (T ) 
H (Tm )
T
 C P ln
Tm
Tm
By collecting the enthalpy and entropy contributions the Gibbs energy of the transition (the principle result) is finally
obtained
G (T )  H (T )  TS (T )
 H (Tm )  T  Tm C P  T

T
 1 
 Tm
H (Tm )
T
 TC P ln
Tm
Tm

T
H (Tm )  T  Tm C P  TC P ln
Tm

It is of interest to analyze the result vs. temperature. First we attempt to obtain the shape of the curve simply by
plotting it. To that end we must set values of the enthalpy, transition temperature and the heat capacity keeping in mind
that proteins are marginally stable, the transition temperature is usually below 100 C and that the heat capacity is
positive.
We obtain a convex curve that implies that there is a maximum stability at T = Ts when (the slope of tanget is zero)
H (Tm )
G(T )
T

 C P ln
0
T
Tm
Tm
That can be solved to give
 H (Tm ) 

Ts  Tm exp  
 Tm C P 
A plot of the derivative
shows that near Ts the function can be approximated by ln(1+x) = x - x2/2 + …
T
T
 T  T T  Tm  T 1  Tm  T 
 ln
 ln m  ln  m
 
 
 
Tm
T
T
T
2 T 
 T
2
2
to give the transition temperature in the first order

H (Tm )
T T
H (Tm )
T
 C P m

 C P  C P m  0
Tm
T
Tm
T
 Ts 
Tm
H (Tm )
1
Tm C P
The convex shape of the curve implies that there is another transition temperature Tm’ < Ts that should be the other
solution to G = 0. Using the second order approximation (above)

T 
T
G (T )  1  H (Tm )  T  Tm C P  TC P ln
Tm
 Tm 
 Tm  T 1  Tm  T  2 

T 
 1  H (Tm )  T  Tm C P  TC P 
 
 
2  T  
 Tm 
 T
 H (Tm ) 1
T T 
 Tm  T 
 C P m
0
2
T 
 Tm
H (Tm ) 1
T T
Tm

 C P m
 Tm' 
2H (Tm )
Tm
2
T
1
Tm C P
From this result the dependence of the cold denaturation becomes apparent: an increase in Tm leads to increase in Tm’,
decrease in the enthalpy leads to increased Tm’ and a larger CP also results in larger Tm’.
All in all G(T) is a convex curve that is due to the subtle balance between H and S as well as due to the CP >
0. This means that the heat capacity is larger in the denatured states above Tm and below Tm’ than in between i.e. in the
native state.
2.
Effect of pressure
The observation of cold denaturation is in practice hampered because Tm’ is usually below the freezing point of water.
Thus it is a common practice to reduce the stability of a protein by pH or denaturants. In our plot of G above this
means that the level G = 0 will be lifted upwards. Consequently Tm will become lower and Tm’ higher. Alternatively
anti-freezing agents or super-cooled water can be used. The effect of pressure will both affect the stability of a protein
and serve to lower the freezing point of water.
We will now analyze the phase diagram in the T-P-plane. The contributions of temperature and pressure to the free
energy of the transition will obtained by integration of
T P
T P
To Po
To Po
G  Go    dG  Go     SdT  VdP 
from an arbitrary reference point Go at To, Po to the second order
G  Go  T  To C P  TC P ln
T
 S o T  To 
Tm
  P  Po T  To   Vo P  Po  
1
2
 P  Po 
2
3
where the first three terms give the familiar entropy contributions due to temperature as computed earlier above and the
second line gives the effects of pressure.  = (V/P)T denotes the changes in compressibility during denaturation
and  = (V/T)P = -(S/P)T denotes the changes in expansivity.
To make sense out of the equation we once again resort to the second order approximation of the natural logarithm
to obtain a quadratic equation
G  Go 
1 C P
T  To 2  S o T  To 
2 To
  P  Po T  To   Vo P  Po  
1
2
 P  Po 
2
The second order curve in the T-P plane is in general a conic section, i.e. hyperbola or ellipse. It turns out that for
proteins the curve is elliptic. We may analyze the curve mathematically
U ( x, y )  Ax 2  By 2  2Cxy  2 A' x  2 B' y  D
x  T  To ; y  P  Po 
A
1 C P
1
; B :  ; 2C   ; 2 A'  S o ; 2 B'  Vo ; D  Go ;
2 To
2
In our case the necessary condition for the ellipse is
det M  AB  C 2  0  
C P 
  2  0
To
The positive CP (which is usually the case) implies a negative  the isothermal compressibility. The sign and
magnitude of  has been bit of a puzzle referred to as the protein volume paradox. Upon unfolding hydrophobic
groups are transferred to water that is expected to result in a large decrease in the specific volume. However  is
usually small (~1%) and in some cases even positive.
Finally we investigate whether the pressure will destabilize or stabilize. The center of the second order curve is
obtained from the pair of equations
 Axo  Cyo  A'  0

Cxo  By o  B'  0
that can be solved in our case of interest for yo i.e. pressure to give
yo 
AB' A' C
C 2  AB
Note that above the sign of the denominator (<0) was obtained. Thus when yo > 0 pressure will destabilize and when yo
< 0 pressure will stabilize and depend on the sign of the nominator

C P
Vo  S o   0
To

C P
Vo  S o   0
To
4
Note
a) Make sure you are able to show that G(T) is convex and that there is the most stable temperature and
temperature for cold denaturation
b) Make sure you understand that the G(T,P) is approximately an ellipse and that the pressure may also
stabilize a protein.
5