Thermodynamic analysis of protein-ligand binding using differential scanning
calorimetry
Here we present the detailed derivation of the equations used to analyze the differential
scanning calorimetry curves for different protein/ligand molar ratios. We have used a
simple model with two coupled equilibria, i.e., the protein-ligand binding/dissociation
equilibrium and the two-state folding/unfolding of the protein:
NL 
 N  L 
 U  L
In this scheme N is the native free protein, U is the unfolded protein, L is the free ligand
and NL is the protein-ligand complex. The relevant binding and unfolding equilibrium
constants are defined as:
Kb 
NL
N · L
Ku 
U 
N 
and using the native state, N, as the reference state of the protein subsystem, the
partition function of the protein and its temperature derivative are given by:
Q
dQ
dT
N   U   NL
 1  K u  K b L 
N 
 Ku
H u
RT 2
 K b L 
H b
RT 2
 Kb
d L 
dT
The molar fractions of protein in each state can be obtained as:
xN 
1
xU 
Q
Ku
Q
x NL 
K b L 
Q
and the total concentration of ligand, L0, is:
 K C 
L0  L  NL  L·1  b 0 

Q 

where C0 is the total protein concentration in the solution. Substituting Q and solving
for the free ligand concentration, [L]:
L  
 B  B 2  4 AC
2A
where A, B and C are, respectively:
C   L0 1  K u 
B  K b C0  L0   K u  1
A  Kb
The temperature derivatives of each of these quantities are:
dA

dT
dB
dT

dK b
dT
C0  L0  
dC
dT
dK b
dT
dK u
dT
  L0
 Kb
 Kb
dK u
dT
H b
RT 2
H b
RT 2
C0  L0   K u
  L0 K u
H u
RT 2
H u
RT 2
and the temperature derivative of [L] is then given by:


 12  dB
 dB
dA
dC 
2

 B


B

4
AC
 2C
 2A

d L  dT


dT
dT
dT

dT
2A
1
dA 
2
 B  B 2  4 AC 2 


 dT
4 A2


We define the average enthalpy of the whole system as:
H  H N N   H NL NL  HU U   H L L
where HN, HNL, HU and HL and the molar enthalpies of each species in the solution. If
we use as the reference state for the whole system a hypothetical state where all the
protein is in its native and free state and all the ligand is also free, the enthalpy of this
reference state would be:
H
ref
 H N C0  H L L0
and the excess enthalpy relative to this reference state is:
H  H N N   C0   H U U   H NL NL  H L L  L0 
 H U  H N U   H NL  H N  H L NL  H u U   H b NL
Dividing by the total protein concentration, C0:
H  H u xU  H b ·x NL
which is expressed per mole of protein.
The excess heat capacity, Cp, is the temperature derivative of the excess enthalpy:
C p 
d H
dxU
 C p,u ·xU  H u
dT
dT
 C p,b ·xnL  H b
dx NL
dT
where the temperature derivatives of the mole fractions are given by:
dxU
dT
dx NL

dT

K u H u
Q RT
2

dQ K u
·
dT Q 2
H b
1
d L  K b L dQ
 
 K b
L   K b
Q
RT 2
dT 
Q 2 dT
It is necessary to define the molar heat capacity functions for each state of the system.
We have assumed linear functions for the native state of the protein and the proteinligand complex and a quadratic function for the unfolded protein. This last function can
be calculated from the protein sequence using the parametrization of Makhatadze and
Privalov (Makhatadze, G. I. & Privalov, P. L. (1990) J. Mol. Biol. 213, 375-384). For
the free ligand, we determined experimentally its Cp function, which is accurately
described by a 4th order polynomial.
C p ( N )  n0  n1·T
C p ( NL)  nl 0  nl1·T
C p (U )  u0  u1·T  u 2·T 2
C p ( L)  l 0  l1·T  l 2·T 2  l 3·T 3  l 4·T 4
Accordingly, the temperature functions for the heat capacity changes of unfolding and
binding are:
C p ,u  (u0  n0)  (u1  n1)·T  u 2·T 2
C p,b  ( nl 0  n0  l 0)  ( nl1  n1  l1)·T  l 2·T 2  l 3·T 3  l 4·T 4
and the temperature dependences of the enthalpy, entropy and Gibbs energy changes as
well as of the equilibrium constant for the unfolding process are given by:
H u  H u (Tu )  (u0  n0)·(T  Tu ) 
S u 
H u (Tu )
Tu
 (u0  n0) ln
T
Tu
(u1  n1)
2
·(T 2  Tu2 ) 
 (u1  n1)·(T  Tu ) 
u2
2
u2
3
(T 3  Tu3 )
(T 2  Tu2 )
Gu  H u  T ·Su
Ku  e

Gu
RT
where Tu is the unfolding temperature of the free protein, i.e., Ku(Tu) = 1.
Similarly, the temperature dependences of the enthalpy change and the equilibrium
constant of binding are given by:
(T 2  Tb2 ) l 2 3
H b  H b (Tb )  ( nl 0  n0  l 0)·(T  Tb )  ( nl1  n1  l1)·
 (T  Tb3 )
2
3
l3 4
l
4
 (T  Tb4 )  (T 5  Tb5 )
4
5
ln K b  ln K b (Tb )

H b (Tb )

( nl 0  n0  l 0)
R

R
( nl 0  n0  l 0)
R

l4
20 R
ln
T

·Tb 
( nl1  n1  l1)
( nl1  n1  l1)
Tb
2R
2R
·Tb2 
(T  Tb ) 
l2
6R
l2
3R
·Tb3 
l3
4R
(T 2  Tb2 ) 
·Tb4 
l3
12 R
1 1 
·Tb5   
5R  T Tb 
l4
(T 3  Tb3 )
(T 4  Tb4 )
where Tb is reference temperature where Kb(Tb) and Hb are known.
Finally, the molar partial heat capacity of the whole system, Cp, expressed per mole of
protein, is:
C p  C p ( N )  C p ( L)
L0
C0
 C p
from which the apparent heat capacity curve measured in a DSC experiment relative to
the baseline obtained for the buffer, C app
p , can be derived as:
C app
p  Cp 
C p (N ) 
vM
v H 2O
vM
v H 2O
C p ( H 2O ) 
vL
v H 2O
C p ( H 2O )
L0

C0
L

vL

C p ( H 2O )  C p ( L ) 
C p ( H 2 O )  0  C p


v H 2O

 C0
We have considered the partial specific volumes of the ligand and the protein equal to
0.73 ml g1.
Table 1: Ambiguous interaction and intermolecular NOEderived distance restraints
AIRs of protons of SH3 to all atoms of ligand within 6 Å
Leu12.HA, Leu12.HB
Tyr13.HA, Tyr13.HB
Tyr15.HD
Gln16.NH
Lys18.NH
Ala21.NH
Glu22.NH
Asn38.HB, Asn38.HD2
Asp40.NH, Asp40.HB
Trp41.NH, Trp41.HB, Trp41.HE3, Trp41.HE1
Trp42.NH
Lys43.HA, Lys43.HB
Phe52.NH, Phe52.HA, Phe52.HB, Phe52.HE
Pro54.HA, Pro54.HB, Pro54.HD
Ala55.NH
Ala56.NH, Ala56.HB
Tyr57.NH, Tyr57.HB, Tyr57.HD
Intermolecular NOEs: R21A-SH3 - P41 ligand
Tyr15.HE# - Pro7.HD#
Asn38.HD21 - Ala1.HB#
Asn38.HD22 - Ala1.HB#
Asp40.HB1 - Pro6.HD1
Asp40.HB1 - Pro6.HD2
Trp41.HE3 - Ala1.HA
Trp41.HD1 - Ala1.HB#
Trp41.HE1 – Ser3.HA
Trp41.HE1 - Tyr4.HD#
Trp41.HH2 - Tyr4.HE#
Trp41.HZ2 - Tyr4.HD#
Trp41.HZ2 - Tyr4.HE#
Trp41.HD1 - Ser5.HA
Trp41.HE1 - Ser5.HA
Trp41.HZ2 - Ser5.HA
Trp41.HE1 - Pro6.HA
Trp41.HH2 - Pro6.HA
Trp41.HZ2 - Pro6.HA
Trp41.HD1 - Pro6.HD1
Trp41.HD1 - Pro6.HD2
Trp41.HE1 - Pro6.HD1
Trp41.HE1 - Pro6.HD2
Trp41.HZ2 - Pro6.HD1
Trp41.HZ2 - Pro6.HD2
Trp41.HH2 - Pro7.HD#
Trp41.HZ2 - Pro7.HD#
Phe52.HD# - Ace0.HA#
Phe52.HE# - Ace0.HA#
Phe52.HZ – Ace0.HA#
Phe52.HD# - Ala1.HB#
Phe52.HE# - Ala1.HA
Phe52.HE# - Ala1.HB#
Phe52.HE# - Ala1.HN
Phe52.HZ - Ala1.HN
Phe52.HZ - Ala1.HA
Phe52.HZ - Ala1.HB#
Tyr57.HD# - Pro9.HD#
Tyr57.HE# - Pro9.HD#
Table 2. Apparent amide hydrogen-deuterium exchange rate constants and apparent Gibbs
energies for the R21A Spc-SH3 domain at pH* 3.0 and 27.1 ºC, in its free form and in the
presence of a 96% saturating concentration of the p41 peptide. Uncertainties in the values
correspond to 95% confidence intervals for the khx values.
Free R21A Spc-SH3
R21A Spc-SH3 + p41
Residue
khx · 10-3
(min-1)
Ghx
(kJ·mol1)
khx · 10-3
(min-1)
Ghx
(kJ·mol1)
Leu 8
Val 9
Leu 10
Ala 11
Leu 12
Tyr 13
Asp 14
Tyr 15
Gln 16
Glu 17
Ser 19
Glu 22
Val 23
Thr 24
Met 25
Lys 26
Gly 28
Asp29
Ile 30
Leu 31
Thr 32
Leu 33
Leu 34
Asn 35
Thr 37
Asn 38
Asp40
Trp 41
Trp 42
Lys 43
Val 44
Glu 45
Val 46
Arg 49
Gln 50
Gly 51
Phe 52
Val 53
Ala 55
Ala 56
Tyr 57
Val 58
Lys 59
Lys 60
Leu 61
8.5  0.6
1.13  0.05
1.18  0.07
2.2  0.3
1.34  0.11
1.42  0.05
3.2  0.5
20.3  1.0
23.9  0.5
35  14
28  2
5.05  0.16
14.4  0.8
8.7  0.3
9.3  0.6
9.9  1.8
83
5.9  0.3
1.26  0.06
2.9  0.3
2.06  0.19
1.5  0.3
93
49  12
26  5
3.1  1.9
5.83  0.19
0.69  0.05
1.85  0.07
1.08  0.04
4.08  0.19
2.05  0.10
10.8  0.4
11.36  0.3
8.6  1.2
6.61  0.4
1.42  0.21
3.1  0.6
10.8  1.8
1.83  0.12
0.96  0.03
2.81  0.11
26  8
8.3  0.4
5.29  0.19
6.81  0.12
7.58  0.14
9.1  0.3
8.22  0.21
8.56  0.09
10.1  0.4
4.98  0.12
63
5.7  1.1
4.62  0.24
5.87  0.08
4.02  0.14
7.46  0.10
6.67  0.17
8.3  0.5
10.5  1.0
5.95  0.11
6.78  0.12
7.6  0.3
7.86  0.23
6.8  0.4
7.9  0.7
3.4  0.6
7.6  0.5
12.5  1.5
7.85  0.08
9.90  0.19
9.38  0.10
8.40  0.10
8.54  0.12
8.07  0.12
8.93  0.09
7.33  0.06
9.1  0.4
6.97  0.15
7.4  0.4
8.1  0.5
6.4  0.4
9.05  0.17
8.29  0.07
8.23  0.10
4.2  0.8
4.12  0.12
5.2  0.3
0.044  0.004
0.035  0.008
0.01  0.03
0.031  0.011
0.042  0.005
5.1  0.6
0.097  0.023
4.41  0.16
29  5
6.8  0.4
1.00  0.03
4.42  0.17
0.33  0.03
2.54  0.08
1.80  0.18
1.69  0.08
0.033  0.005
0.137  0.011
0.072  0.008
0.059  0.009
0.58  0.07
9.3  0.6
5.5  0.4
0.269  0.021
0.026  0.009
0.047  0.007
0.030  0.007
0.102  0.007
0.079  0.004
2.05  0.05
3.63  0.09
0.22  0.04
0.31  0.05
0.034  0.007
0.13  0.03
3.8  0.3
0.047  0.009
0.020  0.007
0.074  0.007
11  3
5.3  0.2
6.50  0.15
14.73  0.21
16.2  0.6
16.8  0.9
17.4  0.8
17.2  0.3
10.9  0.3
18.6  0.6
8.74  0.09
6.2  0.5
8.15  0.13
9.85  0.09
6.95  0.10
15.51  0.20
9.87  0.08
12.52  0.24
9.01  0.12
15.7  0.4
15.27  0.22
16.1  0.3
14.8  0.4
14.7  0.3
7.55  0.17
11.4  0.17
15.4  0.19
17.9  0.8
18.4  0.4
17.2  0.4
17.58  0.18
16.04  0.14
13.01  0.06
10.16  0.06
18.1  0.4
14.4  0.4
16.5  0.5
15.9  0.6
8.94  0.18
18.0  0.5
17.8  0.8
17.1  0.2
6.3  0.6
5.24  0.10