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