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1
Molecular Mechanics
Empirical Potencial
Force Field
Focused on biomolecules
Michal Otyepka
Department of Physical Chemistry
Faculty of Science, Palacky University Olomouc
4. 12. 2011
Why? There is QM.
Small molecule – tractable system at high-level QM
Medium size molecule – limited quality QM
(HF/6-31G(d); GGA/LDA - DFT)
Ordinary protein – intractable system by
QM, but … SCC-DFT-B, PM6-DH …
Compromise
Quality (reliability)
100.000 10.000 1.000 100 10
Model size Computer cost
incomplete model reliable method
representative model approximative method
MM LDA GGA hybrid DFT MP2 CCSD(T)
CCSD(T)/CBS the gold standard (thermochemical or higher accuracy, err. < 1 kcal/mol)
2
Born-Oppenheimer approximation
Separation of electronic and nuclear motions
Quantum electrons vs. Classical nuclei
RfE
Tractable by classical physics
molecular mechanics
Bond Distance Deformation
Behavior close minimum
Taylor expansion
2
2
02
0
2
0
0
2)(
...!2
1
!1
1
o
o
rrk
rE
rrr
rErr
r
rErErE
Pote
nciá
lní en
erg
ie
1.0 2.0vzdálenost 10 m
-10
0.74 = r
vazebná vzdálenost0
elektronická energie
distance
Bond distance
Pote
ntial energ
y
Total energy from SE
hookean approximation does not allow
bond dissociation!
Bond as Spring
E
rr0
2
2)( orrk
rE
kxF
xrr
rrkr
rEF
0
0force
deviation from equil. position
Hooke‘s law – strain is directly
proportional to stress
force (spring) constant
2)( 012
1 rra
D eEE
Morse pot.
k
r
rE
2
2
Ut tensio, sic vis, meaning, "As the extension, so the force".
3
Different Bonds Different Springs
various covalent bonds have various
r0, bond distance
k, force constant
molecule
H2
H35Cl
H79Br
H127I
k / N·m-1 r0 / pm
510 74.1
478 127.5
408 141.4
291 160.9
Bond Types
similar bonds X–Y share similar behavior in all
molecules, are insensitive to environment
(context)
- Parameters are transferable -Transferability: Application of empirical force field
parameters to molecules not explicitly included during the
parameter optimization.
To define similar bonds, to assign k a r0
Atom types are defined
Carbon Atom Types (in biomolecules – parm99/AMBER)
PARM99 for DNA,RNA,AA, organic molecules, TIP3P wat.
C sp2 C carbonyl group
CA sp2 C pure aromatic (benzene)
CB sp2 aromatic C, 5&6 membered ring junction
CC sp2 aromatic C, 5 memb. ring HIS
CD sp2 C atom in the middle of: C=CD-CD=C
CK sp2 C 5 memb.ring in purines
CM sp2 C pyrimidines in pos. 5 & 6
CN sp2 C aromatic 5&6 memb.ring junct.(TRP)
CQ sp2 C in 5 mem.ring of purines between 2 N
CR sp2 arom as CQ but in HIS
CT sp3 aliphatic C
CV sp2 arom. 5 memb.ring w/1 N and 1 H (HIS)
CW sp2 arom. 5 memb.ring w/1 N-H and 1 H (HIS)
C* sp2 arom. 5 memb.ring w/1 subst. (TRP)
CY nitrile C (Howard et al.JCC,16,243,1995)
CZ sp C (Howard et al.JCC,16,243,1995)
O
N
H
H
H
HH
N
CT
CT
HC
HC
HC
C
O
Ala
4
Bond Types
CT-CT 310.0 1.526
CT-HC 340.0 1.090
CT-H1 340.0 1.090
CT-H2 340.0 1.090
CT-H3 340.0 1.090
CT-HP 340.0 1.090
CT-N* 337.0 1.475
CT-N2 337.0 1.463
CT-OH 320.0 1.410
CT-OS 320.0 1.410
C*-HC 367.0 1.080
C*-CB 388.0 1.459
C*-CT 317.0 1.495
C*-CW 546.0 1.352
CB-CN 447.0 1.419
k r0
database of parameters –
AMBER force fields
How to Derive Parameters?
From experiments
bond geometries
X-ray and neutron scattering, NMR, rotational
spectroscopy
force constants
vibrational spectroscopies
From calculations
fit of energy curves
2/1
2
1 ~
effm
k
c
21
21
mm
mmmeff
Structures (free)
Protein Data Bank (PDB) – http://www.pdb.org
IR data (free)
NIST - http://webbook.nist.gov/chemistry/
Molecular Mechanics
potential energy as a function of nuclei positions
vdwctnoncovalen
tabcovalent
tnoncovalencovalent
EEE
EEEE
EEfE
R
Additive model
Simple form of the potential energy function
5
Internal Terms
tabcovalent EEEE
Bond Stretching
Angle Bending
Torsion/Dihedral Rotation
Angle Bending
2
02
k
E
9.122
degkcal/mol. 80
0
2
k
Torsions/Dihedrals
E
Degrees of Rotation
30060 120 180 2400 360
HH
HH
HH
H
HH
H
H H
2.9 kcal/mol
0cos12
nk
E t
n = 3
0 = 180.0°
kt = 2.9/2*9 (IDIVF=1)
kt = 2.9/2 (IDIVF=9)
i
iii
tin
kE 0,
,cos1
2
6
Dihedrals
C-C-C-C
H-C-C-H
Improper Torsion
geometry of aminogroup
AMBER
2-3-1-4
phase 180°
n = 2
CHARMM
2
02
k
E
Conformational Behavior
conformational behavior of biomacromolecules is given mostly by dihedral parameters (and non-bonded terms)
parameterization Deduced/derived from experimental data (NMR, X-ray -
distributions) – „knowledge based“ – CHARMM
QM profiles – QM in gas phase // QM in solvent (er = 80)
CMAP – 2D dihedral energy correction map, CHARMM (some dihedrals are correlated)
7
CMAP
The popular CHARMM 22 FF (C22) is shown on the left, giving a poor Ramachandran
plot. In 2004, Alexander MacKerell added a grid-based φ/ψ term (CMAP) to the
CHARMM 22 (C22) force field (Mackerell 2004 J. Comput. Chem. 25:1584),
Ramachandran plot looks acceptable to a crystallographer.
Jane Richardon’s MOLPROBITY describing the contour lines for the allowed
regions of the Ramachandran plot (Lovell et al 2003 Proteins 50:437):
Nonbonded Terms
vdwctnoncovalen EEE
Coulomb (electrostatic) interaction
Van der Waals forces
Nonbonded Terms
model of pair potencial
0,,...,
,,...,
...,,,,...,
1
1
1
ji
jieffN
ji
jiN
kji
kjiji
jiN
uU
uU
uuU
RRRR
RRRR
RRRRRRR
effective pair potencial
=??? + +
8
Non-Covalent Interactions
H-bonds (bridges); blues-shifting H-bond
Salt bridges
π-π interaction, ion-π interaction
Van der Waals forces
Halogen bond
Dihydrogen bonds
Hydrophobic effect
MESS!
The Four Basic NC Interactions
Coulombic (Electrostatic)
p. multipole – p. multipole (+, -)
Polarization
p. multipole - ind. multipole (-)
London (dispersion) forces
inst. multipole – inst. multipole (-)
Repulsion
(Pauli) electron exchange (+)
(+) repulsive
(-) attractive
Coulombic Interaction
multipole-multipole interaction in monopole expansion
atomic centred parcial charges (how to derive them?)
Coulomb law
r
qqE
ji
04
1
e
9
Electrostatics
Partial Charges
Nonobservable!
Mulliken – many drawbacks
RESP – „Restrained ElectroStatic Potential fit“ – based on fit of electrostatic potencial of molecule
HF/6-31G* - overestimated dip. moments, compensation of missing induction (parm99)
B3LYP/cc-pVTZ/PCM(er=4) … (ff03)
Polarization can be treated explicitely
RESP
10
RESP
RESP
j jresp
i j ij
j
iiiesp
irespesp
bbqa
r
qVVV
qg
2/1222
22
222
ˆ,ˆ
Induction/Polarization
In many classical FF neglected
FF with polarisation
fluctuating charges, induced dipole, Drude
oscilators
slow, small improvement
WIREs Comput Mol Sci 2011, 1, 844‐854 : Polarization effects in molecular interactions by F. Javier Luque, et al. DOI:
10.1002/wcms.32
11
London Dispersion Forces
Van der Waals Term
covers dispersion and
repulsion int.
-0.2
-0.1
0
0.1
0.2
2 3 4 5 6 7 8 9 10
Distance (10-10 m)
repulsion
dispersion
vdW profile
Minimum well
Lennard-Jones Potencial
2/ ,min2 ,0
24
6
612612
vdw
vdwvdw
uu
rrrrru
e
e
e
van der Waals radius
Why is LJ potencial 12-6
computationally effective?
Enumeration of square is quick
r –12 = (r –6)2.
Repulsion increases
exponentically!
-0.2
-0.1
0
0.1
0.2
2 3 4 5 6 7 8 9 10
f(r –12)
f(r –6)
12
Mixing Rules
How to derive and e for atoms A and B?
Lorentz/Berthelot mixing rule
BBAAAB
BBAAAB
eee
2
612612
22
22
ij
ji
ij
ji
ji
ijij
ijrrrr
ru
ee
e
Minimum – vdw complex
Large Compensation of Errors – parm99
Zgarbova M, Otyepka M, Sponer J, Hobza P, Jurecka* P: Large-scale compensation of errors in pairwise-additive empirical
force fields: comparison of AMBER intermolecular terms with rigorous DFT-SAPT calculations. Phys. Chem. Chem. Phys.,
12, 10476-10493, 2010
Molecular Mechanics
2
02
rrk
E rb
2
02
k
Ea
0cos12
nk
E tt
ij
ji
r
cr
qqE
ee 04
1
126
2
ij
ij
ij
ij
ij
ijvdwrr
E
e
e
13
Molecular Topology
defines, which bonds, angle, dihedrals etc. are applicable
ALA INT 1
CORR OMIT DU BEG
0.00000
1 DUMM DU M 0 -1 -2 0.000 0.000 0.000 0.00000
2 DUMM DU M 1 0 -1 1.449 0.000 0.000 0.00000
3 DUMM DU M 2 1 0 1.522 111.100 0.000 0.00000
4 N N M 3 2 1 1.335 116.600 180.000 -0.41570
5 H H E 4 3 2 1.010 119.800 0.000 0.27190
6 CA CT M 4 3 2 1.449 121.900 180.000 0.03370
7 HA H1 E 6 4 3 1.090 109.500 300.000 0.08230
8 CB CT 3 6 4 3 1.525 111.100 60.000 -0.18250
9 HB1 HC E 8 6 4 1.090 109.500 60.000 0.06030
10 HB2 HC E 8 6 4 1.090 109.500 180.000 0.06030
11 HB3 HC E 8 6 4 1.090 109.500 300.000 0.06030
12 C C M 6 4 3 1.522 111.100 180.000 0.59730
13 O O E 12 6 4 1.229 120.500 0.000 -0.56790
O
N
H
H
H
H
H
5
4
7
68
11
12
Ala
9
1013
distance bond dihedral parc. charge name at. type conec.
What is enumerated?
binding – covalent bonds, angles, dihedrals only
coulomb – 1-2, 1-3 neglected; 1-4 scaled (2.0), all (but PME)
vdW – 1-2 and 1-3 neglected; 1-4 scaled (1.2), all (but …)
to reduce number of non-covalent interactions – cutoff 1 4
3 2
Cutoff
Number of nonbonding terms ~ 2 × N(N–1)/2
for 10.000 atoms (smaller system) ~108 pairs
vdW term quickly vanishes with distance (r–6)
vdW inter. at 2.4x minimum distance is ~100× smaller than in minimum
cutoff for vdw interaction; pairs over rmax neglected
problem: a discontinuity is introduced (infinite derivations at rmax)
switching function
14
Switching function
Graph of van der Waals potential with and without the application of the
switching function. With the switching function active, the potential is smoothly
reduced to 0 at the cutoff distance. Without the switching function, there is a
discontinuity where the potential is truncated.
Cutoff
T. Schlik
Cutoff for Electrostatics?
electrostatic interaction of two monopoles vanishes slowly (r–1) at 10r is equal to 10%Eels at r
When periodic boundaries (PBC) are used - Ewald summation can be used PME – particle mesh Ewald
Interaction of two particles is summed at „short“ distances directly and at „long“ distanced in Fourier space, Fast Fourier Transform is used
k
kkk
~
~
~~ 2
,
,
lr
lrlr
ji
jisrsr
ji
lrsrjitot
E
rrE
EErrE
Fourier pair of potecial
Charge density
15
More Approximative FF
United Atoms Non-polar hydrogens are not treated explicitely
E.g. –CH3 is treated as one „pseudoatom“
Membrane simulations
Coarse grained models (CG) Beads
For very large systems
Rough approximation
Applications: membranes, protein complexes
CGFF – MARTINI
http://troll.med.unc.edu/ifoldrna/ (RNA CG)
Coarse Grained
MARTINI The forcefield has been parametrized in a systematic way, based on the
reproduction of partitioning free energies between polar and apolar phases of a large number of chemical compounds.
The model is based on a four-to-one mapping, i.e. on average four heavy atoms are represented by a single interaction center. In order to keep the model simple, only four main types of interaction sites are defined: polar (P), non-polar (N), apolar (C), and charged (Q). Each particle type has a number of subtypes, which allow for an accurate representation of the chemical nature of the underlying atomistic structure.
Currently topologies are available for many lipids and surfactant molecules, including cholesterol, and for all amino acids. Scripts are furthermore available to build topologies for arbitrary peptides and proteins.
CG - MARTINI
16
CG - MARTINI
MARTINI
Vesicle Formation and Fusion
Ripple Phase
Hexagonal Phase Formation
http://md.chem.rug.nl/~marrink/science.html
Common Force Fields
organic molecules
(Allinger) MM2, MM3,
MMFF (Merck Molecular FF), CVFF ...
biomacromolecules
AMBER
parm99, ff03, ff10 …
CHARMM
OPLS
17
AMBER
18
Ramachandran Plot
these regions are R for -helices (right-handed), s for -sheets, and ' for tight turns.
AMBER FF - proteins
parm99 – error for Gly, sampling of
disallowed region in RP
corrected parm99SB
parm03 (ff03) – seems to overstabilize -
helices
FF99, 99SB a 03
chignolin
19
D(R)NA backbone
Average Torsion Angles for Nucleic Acid Helices (in °)
Structure Type Alpha Beta Gamma Delta Epsilon Zeta Chi
A-DNA (fibres) -50 172 41 79 -146 -78 -154
GGCCGGCC -75 185 56 91 -166 -75 -149
B-DNA (fibres) -41 136 38 139 -133 -157 -102
CGCGAATTCGCG -63 171 54 123 -169 -108 -117
Z-DNA (C residues) -137 -139 56 138 -95 80 -159
Z-DNA (G residues) 47 179 -169 99 -104 -69 68
DNA-RNA decamer -69 175 55 82 -151 -75 -162
A-RNA -68 178 54 82 -153 -71 -158
Source: Blackburn and Gait, Nucleic acids in chemistry and biology, Oxford University Press New York 1996.
AMBER FF - NA
parm99 – / flips (pathological irreversible
-trans), slow degradation of B-DNA
parmbcs0 – repar. /
parm99+parmbsc0 – resonably good
description of RNA, but „ladder-like“
structures in A-RNA duplexes
AMBER FF - RNA parm99/parmbsc0 –„ladder-like“ structures
20
Torsion profiles for the χ angle (on the left, ff99-optimized geometries) and the χ dihedral terms (on the right) of ff99 (black), Ode et al.
(blue), Yildirim et al. (green), and parameters derived herein (χOL-DFT orange, χOL red) for ribonucleosides. The dihedral term was offset
to χ = 250°, and idealized geometries were used to calculate the χ dihedral terms on the right to facilitate comparison with published data (see
also text).
Published in: Marie Zgarbova; Michal Otyepka; Jiri Sponer; Arnost Mladek; Pavel Banas; Thomas E. Cheatham III; Petr Jurecka; J. Chem. Theory
Comput. 2011, 7, 2886-2902.
DOI: 10.1021/ct200162x
Copyright © 2011 American Chemical Society
Parameterization
Bonds, Angles – r0, k – from QM
calculations / by analogy
Dihedrals – significantly affects
conformational behavior of molecules;
before any productive application parameters
should be thoroughly tested
Parameterization of Dihedrals
Calculate QM profile
Enough reliable QM method (MP2/cc-pVTZ)
Which environment? Gas phase or solvent?
1,2-dichloroethane (in gas phase – trans; in water –
gauge)
Calculate MM profile
without parameters of the parameterized torsion
Developed parameters will inherently include used
partial charges and LJ parameters – straightforward
transferabity between FF is impossible
21
Parameterization of Dihedrals
Traditional way
Derive „pure dihedral profile“
Edih=QM – MMQMgeom
Fit the profile by Fourier series
Alternative way
QM in water (CPCM),
MM geometries are relaxed (PB), full FF
SP without parameterized torsion (PB)
Zgarbova et al. J. Chem. Theory Comput., 7 (9), 2886-2902, 2011
MM Applications
Minimization of molecules
Conformational search
Classical MD
NMR, X-ray refinement
Scoring functions (drug design)
Normal mode analysis (entropy estimation)
Exploring PES
PES of larger molecules are complex
ifE R
163dim NE
22
PES in 3D
Important Points on PES
stacionary points
Hess matrix
Eigenvalues of Hess matrix
PES
23
Reaction coordinate – PES dissection
PES vs thermodynamics and kinetics
‡G
G
RTGB eh
Tkk
/‡
KRTG ln
RTEaAek/
STHG
TSHG
Tp
,
Chemisty is a hiking on PES
24
Catalysis Kinetically driven reaction
reaction coordinate reaction coordinate
intermediate
with
catalyst
without
catalyst
Photochemistry
reactant
transition state
product
reaction coordinate
conical intersection
Applicability of MM
„common“ FF does not allow bond
dissociation and formation
Electronic states cannot be considered
explicitly
Exploring of conformational space
But – combination with QM – QM/MM
methods
25
Exploring PES
Straightforward minimization
Simplex, Newton method … - closest local
minimum is often localized
Grid based methods (e.g. Torsional driving)
– not effective time consuming
Monte Carlo
Simulated annealing
Genetic algorithms
Failure of MM
Not able to describe halogen bond
D-X ... Y
X – halogen
Cl < Br < I
8.6 kJ/mol 10.1 kJ/mol
MM Failure
The PM6-D2X optimized structure (red lines) of the inhibitor bound to
the CK2a active site agrees well with the 2OXY X-ray structure (blue
lines), B: whereas the AMBER optimized structure (red lines) shows
substantial differences from the X-ray geometry (blue lines). The failure
of the AMBER empirical potential is caused by its inability to describe
halogen bonds (shown by black dotted lines).
Dobes et al. J. Phys Chem. B, 115(26), 8581-8589, 2011
26
… calculation of U and G
• machinery of statistical
thermodynamics
Q
eA
A i
ii
e
QkT
US ln
tranrotvibel QQQQQ
QM or MM calculation
from Hess matrix
moment of inertia
from mass
… calculation of U and G
• detailněji
Tkh
Tkh
v
B
TJJ
r
B
t
B
B
r
e
eq
k
hcBeJq
Tmk
hVq
/
2/
r
/)1(
3
1
,12
2 ,
N
qnRTGG mln0
evrtm qqqqq
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