Movable type method applied to protein-ligand binding
Disclosed herein is a method of estimating the pose of a ligand in a receptor comprising identifying all possible atom pairs of protein-ligand complexes in a given configuration space for a system that comprises proteins; creating a first database and a second database; where the first database comprises associated pairwise distant dependent energies and where the second database comprises all probabilities that include how the atom pairs can combine; combining the first database with the second database via statistical mechanics to accurately estimate binding free energies as well as a pose of a ligand in a receptor; and selecting a protein-ligand complex for further study.
1. A method of estimating a pose of a ligand in a protein receptor comprising:
identifying all possible atom pairs of protein receptor-ligand complexes in a given configuration space for a system that comprises proteins;
creating a first database and a second database; where the first database comprises associated pairwise distant dependent energies and where the second database comprises all probabilities, interaction energy parameters, and/or distances that include how the atom pairs can combine;
combining the first database with the second database via statistical mechanics to accurately estimate binding free energies as well as the pose of the ligand in the protein receptor; and
selecting a protein receptor-ligand complex for further study.
2. The method of claim 1 , further comprising determining a ranking of the protein receptor-ligand complexes in relation to experimentation, manufacturing, use in drugs, use in therapy, use in artificial body parts, or a combination thereof.
3. The method of claim 1 , where the probabilities that include how the atom pairs can combine comprises using bonds, angles, dihedrals, non-bonded interactions, or a combination thereof for the protein-receptor and the ligand.
4. The method of claim 1 , where the method samples all of configuration space of a selected region in a single effort without using schemes involving Monte Carlo, genetic algorithms or molecular dynamics simulations.
5. The method of claim 1 , where a Helmholtz free energy can be obtained as the sum of energies of all ligand poses divided by the total number of all poses along with an estimate of integration volume of Boltzmann factors:
G
≈
A
=
-
RT
ln
[
Z
]
=
-
RT
ln
[
〈
e
-
β
E
〉
]
=
-
RT
ln
[
∑
i
e
-
β
E
i
N
]
(
9
)
where
Z represents a canonical ensemble partition function;
β is the reciprocal of the thermodynamic temperature k B T;
G is the Gibbs free energy;
A is the Helmholtz free energy;
T is the temperature in degrees Kelvin;
k B is the Boltzmann constant;
R is the ideal gas constant and has the value (8.314 J·K −1 ·mol −1 );
E i is the energy of protein-ligand interaction; and
N is the total number of poses.
6. The method of claim 5 , where a Helmholtz free energy can be obtained from equation (11)
G
≈
A
=
-
RT
ln
[
Z
]
=
-
RT
ln
[
〈
e
-
β
E
〉
]
=
-
RT
ln
[
∑
i
Q
i
e
-
β
E
i
]
,
(
11
)
where Q i is a weighted probability of different Boltzmann factor to differentiate the binding pocket preference from ligand poses as shown in equation (10)
Q
i
=
e
-
β
E
i
∑
i
e
-
β
E
i
(
10
)
and where Z is given in equation (12)
Z
=
∑
i
All
Poses
Q
i
e
-
β
E
i
=
∑
p
All
Combinations
∏
q
All
atom
pairs
Q
pq
e
-
β
E
pq
=
∑
α
Bond
Distance
Range
∏
a
No
.
of
Bonds
Q
a
α
e
-
β
E
a
α
×
∑
β
Angle
Distance
Range
∏
b
No
.
of
Angles
Q
b
β
e
-
β
E
b
β
×
∑
γ
Torsio
n
Distance
Range
∏
c
No
.
of
Torsions
Q
c
γ
e
-
β
E
c
γ
×
∑
δ
vdw
-
elec
Distance
Range
∏
d
No
.
of
vdw
-
elec
interactions
Q
d
δ
e
-
β
E
d
δ
where when the protein-ligand system is broken down to the atom-pair level, “q” indicates all atom pairs in the molecular system, and “p” indicates each possible combination of all atom pairs each of which is at a pre-chosen distance, a, b, c and d refer to each atom pair as a bond, angle, torsion or long-range (van der Waals or electrostatic) interaction in a canonical system, respectively, and α, β, γ and δ refers to each sampled separation distance between the corresponding atom pair.
7. The method of claim 1 , where an atom pairwise radial distribution function for the protein-ligand complexes is represented in equation (14):
g
ij
(
r
)
=
n
ij
(
r
)
n
ij
*
(
r
)
=
n
ij
(
r
)
N
ij
V
4
π
r
a
Δ
r
(
14
)
where n ij (r) is the number of protein-ligand pairwise interactions between an atom pair type i and j in a bin (r, r+Δr), with the volume 4πr a Δr collected from a training set, n ij *(r) in the denominator mimics a number of protein-ligand atom type pairs i and j in a same distance bin in an ideal gas state, N ij is the total number of atom pairs of type i and j, an average volume V of the protein-ligand binding sites is given as
4
a
+
1
π
R
a
+
1
,
and where a is an average volume factor and is less than 8.
8. The method of claim 7 , where the radial distribution frequency is then normalized by dividing the sum of radial distributions of all the atom pairs in the system as shown in equation (15):
q
ij
(
r
)
=
g
ij
(
r
)
∑
i
∏
j
g
ij
(
r
)
=
R
a
+
1
n
ij
(
r
)
(
a
+
1
)
N
ij
r
a
Δ
r
∑
i
∏
j
R
a
+
1
n
ij
(
r
)
(
a
+
1
)
N
ij
r
a
Δ
r
(
15
)
and where a cutoff distance R is assigned to each atom type pair defining a distance at which an atom pairwise interaction energy can be regarded as zero.
9. The method of claim 1 , where a binding energy in solution is as follows:
Δ
G
b
s
=
Δ
G
b
g
-
Δ
Δ
G
solv
Δ
G
b
s
=
Δ
G
b
g
+
Δ
G
solv
PL
-
Δ
G
solv
P
-
Δ
G
solv
L
=
-
RT
ln
[
a
V
〈
e
-
β
E
PL
(
r
)
〉
8
C
〈
e
-
β
E
P
(
r
)
〉
〈
e
-
β
E
L
(
r
)
〉
]
-
RT
ln
[
〈
e
-
β
E
PLS
(
r
)
〉
〈
e
-
β
E
PL
(
r
)
〉
]
+
RT
ln
[
〈
e
-
β
E
PS
(
r
)
〉
〈
e
-
β
E
P
(
r
)
〉
]
+
RT
ln
[
〈
e
-
β
E
LS
(
r
)
〉
〈
e
-
β
E
L
(
r
)
〉
]
=
-
RT
ln
[
aV
〈
e
-
β
E
PLS
(
r
)
〉
8
C
〈
e
-
β
E
PS
(
r
)
〉
〈
e
-
β
E
LS
(
r
)
〉
]
=
-
RT
(
ln
[
aV
8
C
]
+
ln
[
∑
ijk
I
+
J
+
K
(
Q
~
i
P
Q
~
j
L
Q
~
k
PL
exp
[
-
β
(
E
i
P
+
E
j
L
+
E
k
PL
+
E
s
PLpsol
+
NABA
PL
)
]
)
(
∑
i
I
Q
i
P
exp
[
-
β
E
i
P
+
E
s
P
psol
+
NABA
P
]
)
(
∑
js
J
+
S
Q
j
L
exp
[
-
β
(
E
j
L
+
E
s
Lpsol
+
NABA
L
)
]
)
]
)
where
Δ
G
b
g
=
-
RT
ln
[
aV
〈
e
-
β
E
PL
(
r
)
〉
8
C
〈
e
-
β
E
P
(
r
)
〉
〈
e
-
β
E
L
(
r
)
〉
]
=
-
RT
(
ln
[
aV
8
C
]
+
ln
[
∑
ijk
I
+
J
+
K
(
Q
~
i
P
Q
~
j
L
Q
~
k
PL
exp
[
-
β
(
E
i
P
+
E
j
L
+
E
k
PL
)
]
)
(
∑
i
I
Q
i
P
exp
[
-
β
E
i
P
]
)
(
∑
j
J
Q
j
L
exp
[
-
β
E
j
L
]
)
]
)
;
where
Δ
G
solv
L
≈
Δ
A
solv
L
=
-
RT
ln
[
Z
LS
Z
L
]
=
-
RT
ln
[
∫
e
-
β
E
LS
(
r
)
dr
∫
e
-
β
E
L
(
r
)
dr
]
=
-
RT
ln
[
〈
e
-
β
E
LS
(
r
)
〉
〈
e
-
β
E
L
(
r
)
〉
]
where
Δ
G
solv
P
≈
Δ
A
solv
P
=
-
RT
ln
[
Z
PS
Z
P
]
=
-
RT
ln
[
∫
e
-
β
E
PS
(
r
)
dr
∫
e
-
β
E
P
(
r
)
dr
]
=
-
RT
ln
[
〈
e
-
β
E
PS
(
r
)
〉
〈
e
-
β
E
P
(
r
)
〉
]
;
and
Δ
G
solv
PL
≈
Δ
A
solv
PL
=
-
RT
ln
[
Z
PLS
Z
PL
]
=
-
RT
ln
[
∫
e
-
β
E
PLS
(
r
)
dr
∫
e
-
β
E
PL
(
r
)
dr
]
=
-
RT
ln
[
〈
e
-
β
E
PLS
(
r
)
〉
〈
e
-
β
E
PL
(
r
)
〉
]
.
10. A system comprising:
a device;
where the device is effective to estimate a pose of a ligand in a receptor comprising:
identifying all possible atom pairs of protein-ligand complexes in a given configuration space for a system that comprises proteins;
creating a first database and a second database; where the first database comprises associated pairwise distant dependent energies and where the second database comprises all probabilities, interaction energy parameters, and/or distances that include how the atom pairs can combine;
combining the first database with the second database via statistical mechanics to accurately estimate binding free energies as well as the pose of the ligand in the receptor; and
selecting a protein-ligand complex for further study.
11. The method of claim 1 , further comprising generating a mean potential between specific atom pairs (ω (2) (r 12 )) from the frequency of occurrence of the atom pairs contained in a large database of molecules:
ω
ij
(
2
)
(
r
12
)
=
-
1
β
ln
(
g
(
2
)
(
r
12
)
)
=
-
1
β
ln
(
ρ
ij
(
r
12
)
ρ
ij
*
(
r
12
)
)
(
1
n
)
where:
g (2) is a correlation function;
β=1/k B T;
k B is the Boltzmann constant;
T is the temperature;
ρ ij (r) is the number density for the atom pairs of types i and j observed in the known protein or ligand structures; and
ρ ij *(r) is the number density of the corresponding pair in the background or reference state.
12. The method of claim 1 , further comprising generating a matrix of Boltzmann-weighted energies of each of the atom pairs at different distances using formula 3n:
k
L
=
[
e
-
β
E
k
L
(
r
1
)
e
-
β
E
k
L
(
r
i
+
1
)
…
e
-
β
E
k
L
(
r
n
-
i
+
1
)
e
-
β
E
k
L
(
r
2
)
e
-
β
E
k
L
(
r
i
+
2
)
…
e
-
β
E
k
L
(
r
n
-
i
+
2
)
⋮
⋮
⋱
⋮
e
-
β
E
k
L
(
r
i
)
e
-
β
E
k
L
(
r
j
)
…
e
-
β
E
k
L
(
r
n
)
]
(
3
n
)
where: Z k L (Z-matrix) is a Boltzmann-weighted energy (Boltzmann factor) matrix for the kth atom pair in the observed molecule L containing energies ranging from distance r 1 to r n .
13. The method of claim 1 , further comprising generating a solvation free energy as follows:
Δ
G
solv
L
≈
-
RT
ln
[
Z
LS
Z
L
]
=
-
RT
ln
[
∫
e
-
β
E
LS
(
r
)
dr
∫
e
-
β
E
L
(
r
)
dr
]
=
-
RT
ln
[
DOF
LS
〈
e
-
β
E
LS
(
r
)
〉
DOF
L
〈
e
-
β
E
L
(
r
)
〉
]
(
7
n
)
where:
DOF LS and DOF L are degrees of freedom of the protein receptor-ligand complex in solution and in the gas phase; and
E LS is the energy of the molecule in solution obtained as follows:
E LS ( r )= E L ( r )+ E L-S interaction ( r ) (8n).
14. A method of estimating a pose of a ligand in a protein receptor comprising:
identifying all possible atom pairs of protein receptor-ligand complexes in a given configuration space for a system that comprises proteins;
creating a first database and a second database; where the first database comprises associated pairwise distant dependent energies in solution and where the second database comprises all probabilities, interaction energy parameters, and/or distances that include how the atom pairs can combine;
combining the first database with the second database via statistical mechanics to accurately estimate binding free energies in solution as well as the pose of the ligand in the protein receptor; and
selecting a protein receptor-ligand complex for further study;
where a binding energy in solution is obtained from formula 14n:
Δ
G
solv
L
≈
-
RT
ln
[
Z
LS
Z
L
]
=
-
RT
ln
[
〈
e
-
β
E
LS
(
r
)
〉
〈
e
-
β
E
L
(
r
)
〉
]
=
-
RT
ln
[
sum
(
total
LS
)
sum
(
total
L
)
]
=
-
RT
ln
[
sum
(
total
LS
total
LS
)
sum
(
total
L
total
L
)
]
.
15. The method of claim 1 , wherein the interaction energy parameters comprise atom pair bond, angle, torsion, and/or long-range non-covalent forces.