Transcript Computational Chemistry - University of Hong Kong

Computational Chemistry
• Molecular Mechanics/Dynamics
F = Ma
• Quantum Chemistry
SchrÖdinger Equation
H = E
Molecular Mechanics Force Field
•
•
•
•
Bond Stretching Term
Bond Angle Term
Torsional Term
Non-Bonding Terms: Electrostatic Interaction &
van der Waals Interaction
Bond Stretching Potential
Eb = 1/2 kb (Dl)2
where, kb : stretch force constant
Dl : difference between equilibrium
& actual bond length
Two-body interaction
Bond Angle Deformation Potential
Ea = 1/2 ka (D)2
where, ka : angle force constant
D : difference between equilibrium
& actual bond angle

Three-body interaction
Periodic Torsional Barrier Potential
Et = (V/2) (1+ cosn )
where, V : rotational barrier
 : torsion angle
n : rotational degeneracy
Four-body interaction
Non-bonding interaction
van der Waals interaction
for pairs of non-bonded atoms
Coulomb potential
for all pairs of charged atoms
MM Force Field Types
•
•
•
•
•
MM2 Small molecules
AMBER Polymers
CHAMM Polymers
BIO
Polymers
OPLS
Solvent Effects
CHAMM FORCE FIELD FILE
########################################################
##
##
## TINKER Atom Class Numbers to CHARMM22 Atom Names ##
##
##
##
1 HA
11 CA
21 CY
31 NR3
##
##
2 HP
12 CC
22 CPT
32 NY
##
##
3 H
13 CT1
23 CT
33 NC2
##
##
4 HB
14 CT2
24 NH1
34 O
##
##
5 HC
15 CT3
25 NH2
35 OH1
##
##
6 HR1
16 CP1
26 NH3
36 OC
##
##
7 HR2
17 CP2
27 N
37 S
##
##
8 HR3
18 CP3
28 NP
38 SM
##
##
9 HS
19 CH1
29 NR1
##
##
10 C
20 CH2
30 NR2
##
##
##
########################################################
atom
atom
atom
atom
atom
atom
atom
atom
atom
atom
atom
atom
1
2
3
4
5
6
7
8
9
10
11
12
1
2
3
4
4
5
5
3
3
3
3
6
HA
HP
H
HB
HB
HC
HC
H
H
H
H
HR1
"Nonpolar Hydrogen"
"Aromatic Hydrogen"
"Peptide Amide HN"
"Peptide HCA"
"N-Terminal HCA"
"N-Terminal Hydrogen"
"N-Terminal PRO HN"
"Hydroxyl Hydrogen"
"TRP Indole HE1"
"HIS+ Ring NH"
"HISDE Ring NH"
"HIS+ HD2/HISDE HE1"
vdw
vdw
vdw
vdw
vdw
vdw
vdw
vdw
vdw
vdw
################################
##
##
## Van der Waals Parameters ##
##
##
################################
/Ao
1
2
3
4
5
6
7
8
9
10
1.3200
1.3582
0.2245
1.3200
0.2245
0.9000
0.7000
1.4680
0.4500
2.0000
/(kcal/mol)
-0.0220
-0.0300
-0.0460
-0.0220
-0.0460
-0.0460
-0.0460
-0.0078
-0.1000
-0.1100
bond
bond
bond
bond
bond
bond
bond
bond
bond
##################################
##
##
## Bond Stretching Parameters ##
##
##
##################################
/(kcal/mol/Ao2)
1
1
1
1
1
1
1
1
1
10
11
12
13
14
15
17
18
21
330.00
340.00
317.13
309.00
309.00
322.00
309.00
309.00
330.00
/Ao
1.1000
1.0830
1.1000
1.1110
1.1110
1.1110
1.1110
1.1110
1.0800
angle
angle
angle
angle
angle
angle
angle
angle
angle
angle
angle
angle
################################
##
##
## Angle Bending Parameters ##
##
##
################################
/(kcal/mol/rad2)
3
13
13
13
14
14
14
15
15
15
16
16
10
10
10
10
10
10
10
10
10
10
10
10
34
24
27
34
24
27
34
24
27
34
24
27
50.00
80.00
20.00
80.00
80.00
20.00
80.00
80.00
20.00
80.00
80.00
20.00
/deg
121.70
116.50
112.50
121.00
116.50
112.50
121.00
116.50
112.50
121.00
116.50
112.50
############################
##
##
## Torsional Parameters ##
##
##
############################
torsion
1
11
11
1
torsion
1
11
11
11
torsion
1
11
11
22
torsion
2
11
11
2
torsion
2
11
11
11
torsion
2
11
11
14
torsion
2
11
11
15
torsion
2
11
11
22
torsion
2
11
11
35
torsion
2
11
11
36
torsion
11
11
11
11
torsion
11
11
11
14
torsion
11
11
11
15
torsion
11
11
11
22
torsion
11
11
11
35
torsion
11
11
11
36
/(kcal/mol)
2.500
3.500
3.500
2.400
4.200
4.200
4.200
3.000
4.200
4.200
3.100
3.100
3.100
3.100
3.100
3.100
/deg
180.0
180.0
180.0
180.0
180.0
180.0
180.0
180.0
180.0
180.0
180.0
180.0
180.0
180.0
180.0
180.0
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
Algorithms for Molecular Dynamics
Runge-Kutta methods:
x(t+Dt) = x(t) + (dx/dt) Dt
Fourth-order Runge-Kutta
x(t+Dt) = x(t) + (1/6) (s1+2s2+2s3+s4) Dt +O(Dt5)
s1 = dx/dt
s2 = dx/dt [w/ t=t+Dt/2, x = x(t)+s1Dt/2]
s3 = dx/dt [w/ t=t+Dt/2, x = x(t)+s2Dt/2]
s4 = dx/dt [w/ t=t+Dt, x = x(t)+s3 Dt]
Very accurate but slow!
Algorithms for Molecular Dynamics
Verlet Algorithm:
x(t+Dt) = x(t) + (dx/dt) Dt + (1/2) d2x/dt2 Dt2 + ...
x(t -Dt) = x(t) - (dx/dt) Dt + (1/2) d2x/dt2 Dt2 - ...
x(t+Dt) = 2x(t) - x(t -Dt) + d2x/dt2 Dt2 + O(Dt4)
Efficient & Commonly Used!
General QM/MM scheme
Combined QM/MM method
QM
MM
1. QM is used to describe the site where
reactions occur, including those atoms
make important and direct interactions to
atoms undergoing valence change in the
reactions process.
2. MM is used to describe the rest of the
system. Presumably atoms in these regions
contribute to the reaction moieties through
a static and classical electrostatic fashion.
E  EQM  EQM / MM  EMM
Hao Hu, HKU
A simple approach: ONIOM method
• Our owN n-layered Integrated molecular
Orbital + molecular mechanics Method
+
=
E Q M (1)
E MM (1  2)
 E MM (1)  E MM ( 2 )  E MM (1 / 2 )
Hao Hu, HKU
Mechanical embedding model
EQM / MM (1  2)
 EQM (1)  E MM (2)  E MM (1 / 2 )
 EQM (1)  E MM (1  2)  E MM (1)
Electrostatic embedding model
H QM  MM  H QM  H QM / MM  H MM
 H QM  H QM / M M , ele  H QM / MM , vdW  H M M
 H QM  MM 
  H QM  H QM / MM ,ele  H QM / MM ,vdW  H MM 
QM
  H QM  H QM / MM ,ele   EQM / MM ,vdW  E MM
MM
When MM atoms are represented as point charges
H QM  H QM / M M , ele  H QM 

i M M
qi
r  ri
The forces on MM atoms
  H QM  H QM / M M , ele 
Hao Hu, HKU
 ri
  q i / r  ri 

 ri
Dirty details: QM/MM boundary
• Dangling bond
Linked hydrogen
Local orbital
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Pseudo atom
Monte Carlo Metropolis sampling
v
v’
v’’
1
DEvv '  0

wvv '  
exp( DEvv ' ) DEvv '  0
Hao Hu, HKU
Free energy, enthalpy, & Entropy
Partition function of canonical ensemble
Z   exp( E / kT )dpdx
E  Ekinetic  E potential
In classical mechanics, potential energy is independent to kinetic
energy

Z   exp( Ekin / kT )dp  exp( E pot / kT )dx

 C ( N )  exp( E pot / kT )dx
C(N) is a constant for the same N
Z   exp( E / kT )dx
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Free energy, enthalpy, & Entropy
Free energy (Helmholtz)
A  kT ln Z
Internal energy
<>: Ensemble average
U  E 
  E exp( E / kT )dx
 exp( E / kT )dx
1
  E exp( E / kT )dx
Z
Entropy
S  U  A T
Hao Hu, HKU
Difficulty for absolute Free energy simulation
Absolute free energy requires converged
integration on 3N dimensions
Z   exp( E / kT )dx 3 N
kBT
No experimental absolute free energies available
Relative free energies are what really matters
Hao Hu, HKU
Free energy perturbation
Free energy difference between two states
State 1; energy E1
State 2; energy E2
A2  A1  kT ln Z2  ln Z1 
 kT ln  exp( E2 / kT )dx  ln  exp( E1 / kT )dx 


Hao Hu, HKU
Free energy perturbation
Free energy difference between two states
A2  A1  kT ln  exp( E2 / kT )dx  ln  exp(  E1 / kT )dx 


DA12
exp( E

 kT ln
 exp( E
exp( E

 kT ln
2
/ kT )dx
1
/ kT )dx
2
/ kT ) exp( E1 / kT ) exp( E1 / kT )dx
 exp( E
1
exp(  E

 kT ln
2
/ kT )dx
 E1  / kT ) exp( E1 / kT )dx
 exp( E
1
/ kT )dx
 kT ln exp(  E2  E1  / kT ) 1  kT ln exp( DE1 2 / kT )
Hao Hu, HKU
Zwanzig, R. W., J. Chem. Phys. 1954, 22:1420-1426
1
Free energy perturbation
More details for the solvation free energy case
State 1; energy E1
State 2; energy E2
E  EX  ES  ES  X
E1  EX  ES
E2  EX  ES  EX S
A2  A  kT ln exp(  E2  E1  / kT ) 1  kT ln exp(  E X  S  / kT )
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1
Complex system
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