Structure-Activity 3D-QSAR Relationships3D.pdf1 Computer-Chemie-Centrum Universität...

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Computer-Chemie-Centrum Universität Erlangen-Nürnberg

3D3D--QSARQSAR

Tim ClarkComputer-Chemie-Centrum

Universität Erlangen-NürnbergNägelsbachstraße 25

91052 Erlangen


StructureStructure--ActivityActivityRelationshipsRelationships

ChemicalChemicalStructureStructure

BiologicalactivityQSAR

Physicalproperty

QSPR


Molecules

Gases

Perfect Crystals

Liquids

Polymers

Crystal Defects

Amorphous Solids

Easy

Diificult to impossible

Small

Medium

Large

Organic

Inorganic

Hybrid

Equilibrium

Fast (τ < ns)

Intermediate

Size

Structure

Energy

Enthalpy

Dipole Moments

Polarizability

Binding Energy

IR Spectra

Transition States

Activation Energy

NMR Spectra

Elastic Modulus

uv Spectra

Free Energy


QSPR QSPR MethodsMethods forfor PolymersPolymers• The Van Krevelen Method

o D. W. Van Krevelen, Properties of Polymers, 3rd ed., (Amsterdam, Elsevier, 1990).

• The Askadskii Methodo Andrey A. Askadskii, Physical Properties of

Polymers: Prediction and Control (Amsterdam, Gordon and Breach Publishers,1996).

• Connectivity Indiceso Jozef Bicerano, Prediction of Polymer

Properties (New York, Marcel Dekker, Inc., 1993).

2


Van Van KrevelenKrevelen• The Van Krevelen method is a

group-additive method• Each group in the monomer is

assigned an additive increment• The target property is obtained

by simply summing the increments due to each fragment in the monomer


Van Van KrevelenKrevelen

HC CH2

n

Polystyrene:

HC

CH2+

Group 1

Group 2


Van Van KrevelenKrevelen (1)(1)Polystyrene:

HC

CH2+

M=90.12V=82.15

M=14.03V=15.85

( )

1( )

1

N groups

ii

N groups

ii

MMV V

ρ =

=

=∑

∑

Density, ρ

-314.03 90.12 1.06 g cm15.85 82.15

ρ += =

+

Exp. = 1.05 g cm-3


Van Van KrevelenKrevelen (2)(2)Polystyrene:

HC

CH2+

M=90.12Yg=3500

M=14.03Yg=2700

( )

,1

( )

1

N groups

g ig i

g N groups

ii

YYT

V M

=

=

=∑

∑

Glass-transitiontemperature, Tg

2700 3500 362 K14.03 90.12gT +

= =+

Exp. = 373±2 K

3


Van Van KrevelenKrevelen (3)(3)• Advantages

o Fast, easyo Usually accurate

• Disadvantageso Missing parameters for new groupso Not applicable for random polymers or

copolymers


AskadskiiAskadskii• The Askadskii method treats each

monomer as a series of harmonic oscillators

• The thermal movement related to each harmonic oscillator is in turn related to the glass-transition temperature

• After some manipulation, this concept leads to a simple additive model


AskadskiiAskadskii

( / )

1( / ) ( / )

1 1

N atoms groups

ii

g N atoms groups N atoms groups

i i ii i

VT

a V b

=

= =

∆=

∆ +

∑

∑ ∑

Glass-transitiontemperature, Tg

= van der Waals volume of atom or group , semiempirical coefficients

i

i i

V ia b∆

=


Typical errorsTypical errors

±13.23%-Surface tension

±2.99%-Thermal decomposition temperature

±6.09%±5.22%Dielectric constant

±5.82%±3.71%Tg

±4.32%±7.21%Heat capacity (solid)

±5.12%±5.62%Heat capacity (liquid)

±1.02%±0.66%Refractive Index

±3.42%±1.58%Density

AskadskiiVan KrevelenProperty

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BiceranoBicerano• The Bicerano method is based on

“electrotopological indices”, which were introduced by Kier and Hall:

o Molecular Structure Description, L. B. Kier and L. H. Hall, Academic Press, San Diego, 1999.

• Topological indices are derived from molecular bonding graphs


DescriptorsDescriptors: 2D: 2D• Topological Descriptors

o e.g. Kier und Hall:oχn :

n different types of descriptor that describe mainly the branching in the molecule

oκn:“shape” descriptors

oE-States :“electronic“ descriptors that describe the acceptor properties of the atoms.


Topological indicesTopological indices• Graph-theoretical invariants

• W Wiener Indexo Oldest topological indexo Corresponds to surface area of moleculeo Dij is the bond distance between atoms i

and j

1 1

12

N N

iji j

W D= =

= ∑∑Computer-Chemie-Centrum Universität Erlangen-Nürnberg

Topological indicesTopological indices• χ molecular connectivity index (Kier, Hall)

o Possibility of molecules for bimolecular interaction

o σi number of sigma electrons, hi number of connected hydrogens

… and many more• Used frequently in published models but

often of limited use in practical application due to difficult interpretation of descriptorso The inverse QSAR problem: going from model

to compound

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KierKier and Hall and Hall TopologicalTopologicalIndicesIndices

• Molecular connectivity chi and kappa indices (1995)

o L. H. Hall and L. B. Kier, The Molecular Connectivity Chi and Kappa Shape Indexes in Structure-Property Modeling, in Reviews in Computational Chemistry, K. B. Lipkowitz and D. B. Boyd (eds), VCH, New York, 1999.

o Connectivity indices intended primarily to describe the molecular shape.


χχ (connectivity) Indices: (connectivity) Indices: DefinitionsDefinitions

= number of skeletal (non-hydrogen) neighbor atoms to atom i iδ

( )0

1

1N atoms

i i

χδ=

= ∑

Zeroth order chi index:


χχ (connectivity) Indices: (connectivity) Indices: DefinitionsDefinitions

= number of skeletal (non-hydrogen) neighbor atoms to atom and are the two atoms involved in bond i i

i j ijδ

( )1

1

1N bonds

ij i j

χδ δ=

= ∑First order chi index:


χχ Indices: Heat of Indices: Heat of Atomization for Atomization for AlkanesAlkanes

1 4

4 5 5

286.38 12.46 1.515

1.142 2.474 2.026 114.38atom C P

PC C PC

H N χ χ

χ χ χ

∆ = − +

+ − − +

Higher order indices depend on paths (P) or clusters (C) in the molecular graph.

Standard deviation to experiment = 0.46 kcal mol-1

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κκ (shape) Indices: Paths(shape) Indices: Paths

141076

131176

6666

3756

4656

3556

4556

3456

3P2P1PNCMolecule


κκ (shape) Indices: (shape) Indices: DefinitionsDefinitions

max min

1

, = maximum and minimum possible indicesof order for a given

is the first order path number for molecule

m m

C

i

P Pm N

P i

( )( )( )

21 11 max min

2 21 1

12 C C

i i

N NP P

P Pκ

−= =

First order kappa index:



max min

1

, = maximum and minimum possible indicesof order for a given

is the first order path number for molecule

m m

C

i

P Pm N

P i

( )( )( )

( )

22 22 max min

2 22 2

1 22 C C

i i

N NP P

P Pκ

− −= =

Second order kappa index:



( )3 3

3 max min23

4

i

P P

Pκ =

Third order kappa index:

( )( )( )

23

23

1 3 for is oddC C

C

i

N NN

Pκ

− −=

( )( )( )

23

23

2 3 for is evenC C

C

i

N NN

Pκ

− −=

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ElectrotopologicalElectrotopological Indices; the EIndices; the E--StateState

= number of hydrogens bonded to atom

,

where is the number of valence electrons for atom

iv vi i i

vi

h i

Z h

Z i

δ = −

O

O

= 1, 1vδ δ =

= 1, 6vδ δ =

= 3, 4vδ δ =

= 2, 6vδ δ =

= 2, 2vδ δ =

= 1, 1vδ δ =


δδ and and δδvv

17sp3

16sp2

26sp15sp2

25sp3

35sp3

24sp

34sp2

44sp3

δδvHybridizationAtomC

C

C

N

N

N

O

O

F


Intrinsic (I) StatesIntrinsic (I) States

( )212 v

NIδ

δ

+

=

i jij

ij

I II

r−

∆ =

= principal quantum numberN

= number of bonds between

atoms and (the topological distance)

ijr

i j


The The ElectrotopologicalElectrotopological (E(E--)State)State

( )

1

N atoms

i i ijj

S I I=

= + ∆∑= the E-State for atom S i

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EE--StatesStates

O

O

1.78

0.48

1.38 4.41

-0.20

9.82


BiceranoBicerano• The Bicerano method uses chi and

kappa indices, I-states and E-statesto calculate the properties of polymers.

• E.g molar volume:

0 0 1

1

3.64277 9.798697 8.85282921.693912 0.978655

v

vMV

VN

χ χ χ

χ

= + −

+ +

( )

( ) ( ) ( ) ( ) ( )

where24 18 5 7 16

2 3 5 5 11 7( 1)MV Si S silfone Cl Br

backbone ester ether carbonate C C cyc fused

N N N N N N

N N N N N N− −

=

= − − − −

+ + + + − − −


Typical errorsTypical errors

±13.23%

±2.99%

±6.09%

±5.82%

±4.32%

±5.12%

±1.02%

±3.42%

Askadskii

--Surface tension

--Thermal decomposition temperature

±1.77%±5.22%Dielectric constant

±5.57%±3.71%Tg

±5.57%±7.21%Heat capacity (solid)

±5.55%±5.62%Heat capacity (liquid)

±0.63%±0.66%Refractive Index

±2.10%±1.58%Density

BiceranoVan KrevelenProperty


Internet Internet SourcesSources• Properties available

o http://www.dtwassociates.com/?phb_list_of_properties

• Interactive demo for Askadskiicalculationso http://mzchem.com/index.wm?opt=8&subopt=0&page=main_1_7.h

tm

• Handbook with calculation detailso http://www.chemcad.fr/produits/documentation/dtwassociates-

ppphand/ppphb_dug.pdf

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BCUT BCUT DescriptorsDescriptorsAdjacency matrix:

•Diagonal elements:•Atomic number•Atomic charge•Atomic polarizabililty•H-bond properties

•Off-diagonal elements•(Lewis bond orders)/10 for bonded atoms•0.001 for all other elements


BCUT BCUT DescriptorsDescriptors

8.001.1.001.001.001.001.001.001.001

8.2.001.001.001.001.001.001.001

6.001.001.001.1.001.001.001

6.001.001.001.001.1.001

7.15.001.001.001.15

6.15.001.001.001

6.15.001.001

6.15.001

6.15

6

N

O

O H


BCUT BCUT DescriptorsDescriptors• Eigenvalues of the adjacency matrix• The highest and the lowest eigenvalues are useful ADME descriptors

• BCUTs may be either 2D, as described above, or 3D

• BCUTs are also known as Burden Eigenvalue descriptors


3D3D--QSAR and QSPRQSAR and QSPR

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WhatWhat isis ChemicalChemicalStructureStructure??

• 2D-Structureo Atoms, Bonds (“Connection Tables“)

• 3D-Structureo Atomso Coordinates

• Molecular surfaces


Molecular StructureMolecular Structure

CH3

HH2N

HO OSMILES: N[C@@H](C)C(=O)O (L-Alanine)

CH3

HH2N

HO O





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3D3D--QSARQSAR

• 3D must be better than 2D (?)o We know the “real” structure of the

moleculeo Therefore, we also know exactly its

binding propertieso ….. but do we ??


Multiple MinimaMultiple Minima


TheThe TargetTarget MoleculeMolecule

N

O

ZrN

O

CH2Ph

PhH2C


TheThe TargetTarget MoleculeMolecule

-142 -144 -146 -148 -150 -152 -154 -156 -158 -160

Heat of Formation (kcal mol-1)

0

1

2

-161.9

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An Easy MoleculeAn Easy MoleculeCH3

Cl

Br O

CH3

H3C

H3C


C8

C9

C10

C10a

C7a

C7

C11

C11a

C6a

C6

C12

C12a

C5a

C5

C1

C2

C3

C4

OH

H3C

O OH

NMe2

OHO

OH

O

NH2

OH

C8

C9

C10

C10a

C7a

C7

C11

C11a

C6a

C6

C12

C12a

C5a

C5

C1

C2

C3

C4

OH

H3C

O OH

NMe2

OHO

O

O

NH2

OH

Tetracycline Tetracycline –– a nota not--soso--easy easy MoleculeMolecule

D AC B


TwoTwo ConformationsConformations(just (just forfor thethe rings)rings)

“Extended”• Favored by Solvation• More stable in solution

“Twisted”• More stable in vacuo• Consistently 2.5 –3.0

kcal mol-1 less stablethan extended in water


SixSix TautomersTautomers

OH O OOHOH

O

NH2

OH

NMe2HH

CH3HO

OH O OOHOH

O

NH2

O

NHMe2HH

CH3HO

OH OH OOOH

O

NH2

O

NHMe2HH

CH3HO

OH O OOOH

O

NH2

OH

NHMe2HH

CH3HO

OH O OOOH

OH

NH2

O

NHMe2HH

CH3HO

OH O OHOOH

O

NH2

O

NHMe2HH

CH3HO

N

1.7

Energy (kcal mol-1)

0.0

2.6

~ 6

6.4

Ze

Zd

Zc

Zb

Za

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BombykolBombykol• Sexual Pheromone of the silkworm bombyx

morio 11 rotatable bondso Roughly 8,000 conformations

OH


Biological ConformationBiological Conformation


3D 3D SimilaritySimilarity TechniquesTechniques• Search for similarity with the target

pharmacophore• Pure shape similarity (www.eyesopen.com)• Electrostatic similarity and similarity of the

electron density (Sanz, Carbo, Richards)Carbo Index:

2 2

A BAB

A B

P P dR

P d P d

τ

τ τ=

⋅

∫∫ ∫


How Much Difference Does How Much Difference Does Conformation Make?Conformation Make?

350 400 450 500 550Predicted boiling point

-17

-16

-15

-14

-13

-12

-11

-10

-9

AM

1 H

eat o

f For

mat

ion

(kca

l mol

-1)

H2NNH

NH2

Boltzmann-averagedpredicted boiling point = 444±36°

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Where do we get 3DWhere do we get 3D--Structures?Structures?

• X-Ray crystal structures• 2D-3D conversion

o CORINAo CONCORD

• Geometry optimizationo Force-Fieldo QM

o Semiempiricalo Density Functionalo Ab initio


ClassicalClassical MechanicsMechanics (Force (Force Fields)Fields)

• “Atoms and springs” mechanical model of molecules


Potential Functions Potential Functions bond stretch bond stretch undund angle bendangle bend

• Bond stretch

• Angle bend

0 2( )stretch stretchV k r r= −

0 2( )bend bendV k θ θ= −


Potential Potential FunctionsFunctionsTorsionsTorsions

N = Periodicity of the barrier (e.g. Ethane = 3)

One torsionalcontribution per ABCD combination

[ ]1 cos( )tors torsV k Nφ= +

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Potential Potential FunctionsFunctionsvan der van der WaalsWaals//repulsionrepulsion

D = van derWaals-well depth,

R = van derWaals-Radius

12 6

. 2A B A BvdW A B

AB AB

R R R RV D Dr r

⎡ ⎤⎛ ⎞ ⎛ ⎞+ +⎢ ⎥= −⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦


Potential Potential FunctionsFunctionsvan der van der WaalsWaals//repulsionrepulsion

Distance

Pote

ntia

l Ene

rgy


Potential Functions Potential Functions Coulomb InteractionsCoulomb Interactions

• Charge-charge

• Dipole-Dipole

i jCoulomb

ij

q qV

rε=

Bond dipole


ForceForce--FieldField methodsmethods• Molecular Mechanics:

o Structures and energies can be more exact than experiment.

o Very well suited for conformational problems, relative stability of isomers etc.

o Cannot extrapolate; are only aplicable for classes of compounds that are experimentally well characterized.

o Usually not suitable for reactions.

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ForceForce--FieldField methodsmethods• Conjugated π-systems:

o Each π-System requires its own force fieldo The force field for a π-bond depends on the

bond ordero A simple MO-technique is used to calculate

bond orders


ForceForce--FieldField methodsmethods

• Molecular dynamics:o Long simulations are necessary in order

to obtain good statistical samplingo Systems (e.g. enzyme + water) are often

very large (> 10.000 atoms)


Conformational SearchingConformational Searching• A molecule with N three-fold rotatable bonds

has 3N possible conformations that must be searched

• If we need 1 µsec for each conformation, we need one hour for a molecule with 20 rotatable bonds, 8×1013 years for one with 50

• A molecule with 50 rotatable bonds corresponds to (Gly)25


Conformational SearchingConformational Searching• Simulated annealing to find the most stable (“global”) minimum

• “Dead-end” search algorithms to eliminate high-energy conformations early

• Stochastic search algorithms such as GAs

• Still only possible with force fields

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ForceForce--FieldField methodsmethods• Molecular dynamics:

o Movements of the atoms (or molecules) are calculated from the forces and velocities

o Integration over long simulation times gives thermodynamic quantities

o “Global” minima can be found by Simulated Annealing

o reliable thermodynamic quantities can be obtained from Free Energy Perturbation (FEP) calculations


Molecular DynamicsMolecular Dynamics• Solve Newton’s equations of motion by numerical

integration for the classical mechanical molecular model• Need to include solvent molecules for biological systems• Often use periodic boundary conditions to avoid edge

effects• “long” simulations are of the order of 10 nanoseconds• “interesting” protein movements are of the order of

microseconds to milliseconds• Bottleneck are the long-range Coulomb interactions (use

Particle-Mesh Ewald, PME)


ForceForce--FieldField methodsmethods• Monte-Carlo (MC):

o Random movements are tried and selected according to a thermodynamic test (Boltzmann-distribution).

o Simulations usually reach equilibrium fatserthan MD.

o No kinetic information is available.o Can be used for Simulated Annealing or Free

Energy Perturbation –calculations.


Monte Carlo Calculations: an exampleMonte Carlo Calculations: an example

• The “dartboard method” for calculating π

x

y

• green area = π r2/4• integrate over the greenarea for r=1 → π /4

• ∴

•

( )1 1

0 0

4 ,x y

f x y dxdyπ= =

= ∫ ∫

( ) ( )2 2, 1 if 1f x y x y= + ≤

0 otherwise=

•

••

•

• •

•

•

•

•

•

••

•

•

•

18



• Calculate π

• where

( )1 1

10 0

44 ( , ) ,N

i iix y

f x y dxdy f x yN

π== =

= ≈ ∑∫ ∫

( ) 2 2, 1 if 1f x y x y= + ≤0 otherwise =



3.141592654


ErrorsErrors

1 500 1000 1500 2000

Number of Cycles/106

-15

-13

-11

-9

-7

Log 10

(erro

r)


Calculational Techniques: Calculational Techniques: Property PredictionProperty Prediction

1. Quantitative Structure-Activity and Structure-Property-Relationships (QSAR and QSPR)

2. Free-Energy Perturbation Calculations (MD or MC)

3. Kinetic or mesoscopic modeling

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MolecularMolecular Orbital Orbital MethodsMethods• Linear Combination of Atomic Orbitals (LCAO)o Molecular orbitals (MOs) are calculated

as linear combinations of atomic orbitals(AOs) .

o AOs are usually known as the basis set .o This approximation was introduced by

Erich Hückel.


MolecularMolecular Orbital Orbital MethodsMethods• Linear Combination of Atomic Orbitals (LCAO)


HHüückelckel--TheoryTheory

• “π-only”-Theory (each atom is represented by a single p-Orbital, hydrogens are ignored).

• Overlap (β) between bonded atoms is constant, otherwise zero.

• Hückel-theory is a one-electron theory.


HHüückelckel--TheoryTheory: : EthyleneEthylene

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HHüückelckel--MatrixMatrix

C2 C3

C4

C1

H

H

H

H

H

H

1

αβ00C4

βαβ0C3

0βαβC2

00βαC1

C4C3C2C1


HHüückelckel--MatrixMatrix

αβ00C4

βαβ0C3

0βαβC2

00βαC1

C4C3C2C1

Diagonal-isation

-.37170.6015-.60150.3717ϕ4

0.6015-.3717-.37170.6015ϕ3

-.6015-.37170.37170.6015ϕ2

0.37170.60150.60150.3717ϕ1

α+1.618β

α+0.618β

α-0.618β

α-1.618β

ψ4ψ3ψ2ψ1


ButadieneButadiene--MOsMOs


MolecularMolecular Orbital Orbital MethodsMethods• Self Consistent Field (SCF)

o Each electron “feels” the mean field of all the others (also known as the mean-field approximation).

o The SCF-problem ca.o Elektron-Elektron-Abstoßung wird durch

die SCF-Methode überschätzt.

21


MolecularMolecular Orbital Orbital MethodsMethods• Self Consistent Field (SCF)

e-

e-e-


MOMO--MethodsMethods• Pople-Pariser-Parr (PPP)

o SCFo π-onlyo For planar moleculeso Used mainly for absorption spoectra

(still used extensively in industry!)o Very strongly parameterized


MOMO--MethodsMethods• Complete Neglect of Differential Overlap

(CNDO)o J. A. Pople, R. Segal, J. Chem. Phys. 1965, 43,

S136-S149. o 3-dimensional theory (σ- and π-systems)o LCAO-SCFo Only the repulsion integrals (µµ|λλ) are

considered and are all equal for a given elemento p-Orbitals are treated as if they were s- for

the two-electron integrals


CNDOCNDO--IntegralsIntegrals• Of all the possible integrals (µν⏐λσ), only (µµ⏐λλ) are used

AB(µµ λλ)=γ

AA A AIP EAγ = −

( )2AA BB

ABAB AA BBrγ γγ

γ γ+

=+ +

22


MOMO--MethodsMethods• Intermediate Neglect of Differential

Overlap (INDO)o J. A. Pople, D. L. Beveridge und P. A. Dobosh, J.

Chem. Phys. 1967, 47, 2026 – 2033.o 3-dimensional theorie (σ- und π-systems)o LCAO-SCFo Only the repulsion integrals (µµ|λλ) are

considered and are all equal for a given elemento One-center integrals are parameterized

according to their type


INDOINDO--IntegralsIntegrals• Of all the possible integrals (µν⏐λσ), only (µµ⏐λλ) are used

• 5 Types :

•Gss

•Gsp

•Gpp

•Gp2


MOMO--MethodenMethoden• Neglect of Diatomic Differential Overlap (NDDO)o J. A. Popleo 3-dimensional theory (σ- and π-systems)o LCAO-SCFo Of all the repulsion integrals, only

(µν|λσ) (µ and ν are on the same atom and λ and σ are also on one atom) are used


NDDONDDO--IntegralsIntegrals• Of all the possible integrals (µν⏐λσ), only those in which µ and ν are on the same atom and λ and σare also centered on one atom are considered.

• The same 5 types (for an sp-basis set) as for INDO

• Integrals are calculated as a multipole-multipoleinteraction (up to quadrupole)

• Also available for d-orbitals

23


Semiempirical Semiempirical MOMO--MethodsMethods


NDDONDDO--Methods(Methods(s,ps,p))MNDO MNDO/H§

AM1§ PM5§,¶PM3§,¥≡ ≅

§ Gaußian functions added to the core-core repulsion¥ Classical torsional potential used for amide bonds (C-N) to correct the rotation barrier¶ Classical two-center dispersion potential


MNDOMNDO• M.J.S. Dewar, W. Thiel, J. Am. Chem. Soc., 99, 4899, (1977).o NDDO-based methodo Element-specific parameterizationo Multipole approximation for the two-

electron integrals o s-, p-Basis seto “Frozen core”-approximation


MNDO: MNDO: ImprpovementsImprpovementsoverover MINDO/3MINDO/3

o Geometries – especially bond angles -are reproduced better than in MINDO/3.

o Heats of formation are generally more accurate.

o MINDO/3’s strong tendency to make non-classical bridged structures is corrected.

24


MNDOMNDO--WeaknessesWeaknesses• Rotation barriers are too low

• π-Systems are often calculated to be non-planar


MNDOMNDO--WeaknessesWeaknesses• Rotation barriers are too low• π-Systems are often calculated to be non-planar


MNDOMNDO--WeaknessesWeaknesses• Repulsion between lone pairs is too weak


MNDOMNDO--WeaknessesWeaknesses• Hydrogens bonds do not exist in MNDO

25


MNDOMNDO--WeaknessesWeaknesses• Rings are generally too flat with inversion barriers that are too low. Cyclobutane is predicted to be planar.


MNDOMNDO-- ““Periodic TablePeriodic Table““

H, He, Li, Be, B, C, N, O, FNa, Mg, Al, Si, P, S, Cl K, Ca,

Zn, Ge, Br Cd, Sn, IHg, Pb


AM1AM1• (Austin Model 1) M.J.S. Dewaret.al. J. Am. Chem. Soc.,107 3902 (1985).o Quantum mechanically almost identical

to MNDOo Core-core repulsion modified by

additional Gaussian functions as introduced in MNDO/H by Burstein and Isaev


AM1: AM1: ImprovementsImprovements OverOverMNDOMNDO

o Rotation barriers are higher than in MNDO, but still too low.

o π-Systems are reproduced better than in MNDO, but are often still not completely planar.

o Hydrogens bonds give roughly the right energies – however, the geometry is wrong.

26


AM1AM1--WeaknessesWeaknesses• Geometries of hydrogen bonds


AM1AM1--WeaknessesWeaknesses• Very poor geometries for P- and S-compounds

• Very poor energies for hypervalent compounds including sulfones


AM1AM1--WeaknessesWeaknesses• The energies of nitro-compounds are reproduced poorly.

• Alkyl amines are too flat.


AM1AM1--””Periodic TablePeriodic Table““

HB, C, N, O, F

Na, Mg, Al, Si, P, S, Cl Zn, Ge, Br

Sn, IHg

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PM3PM3• (Parameterized Method 3) J.J.P. Stewart, J. Comp. Chem., 10, 209 (1989); 12, 320 (1991).o Quantum mechanically identical to AM1o Automatic parameterisation with more

degrees of freedom than for AM1o Parameterized with special attention

paid to hypervalent compounds, hydrogen bonds and nitro-compounds


PM3: PM3: ImprovementsImprovements OverOverAM1AM1

o Geometries for P- und S-compounds are better

o Geometries for hydrogen bonds are improved over AM1

o Results optimized for nitro-compounds


PM3PM3--WeaknessesWeaknesses• Amide-CN-rotation barrieren are extremely small (force-field correction).

• Amide-nitrogens are calculated to be pyramidal.


PM3PM3--WeaknessesWeaknesses

• Rotation barriers are far too low.

• π-Systems are often calculated to be non-planar.

• Rings are too flat.

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PM3PM3--””Periodic TablePeriodic Table““

H, He, Li, Be, B, C, N, O, FNa, Mg, Al, Si, P, S, Cl

Ca,Zn,Ga,Ge,As,Se,BrCd,In,Sn,Sb, Te, IHg, Tl,Pb, Bi


NDDONDDO--MethodsMethods ((s,p,ds,p,d))MNDO MNDO/d

AM1

PM3

AM1(d)

Voityuk und Rösch, nur Mo

AM1(d)

V, Fe, Cu, Mo, Pd, Ag, Pt

FujitsuPM3-tmWave-

functionFirst-row transition metals(only parameterized for geometries)

Al, Si, P, S, Cl, Br, I

AM1*

Erlangen, H-F, Al-Cl, Ti, Zr, Mo


Ab Ab initioinitio--MOMO--MethodsMethods


HartreeHartree--FockFock--LimitLimitenergy

experiment

SCF-energies

Hartree-Fock limit

Correlation energy

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CorrelationCorrelation• Dynamic correlation

o Results directly from the overestimation of electron-electron repulsion in SCF-Theory.

• Non-dynamic (static) correlationo Only significant in systems with near-

degenerate partially occupied orbitals(e.g. biradicals).


Dynamic CorrelationDynamic Correlation• Semiempirical MO-Methods

o Is included by scaling the one- and two-center integrals.

• Semiempirical CI-Calculationso Therefore only treat static correlationo … and are therefore easily interpreted


abab initioinitio MO TheoryMO Theory• Approximate solution to the time-independent

electronic Schrödinger equationo Linear Combination of Atomic Orbitals (LCAO)o Usually single Hartree-Fock reference configuration

based on a single Slater determinanto Correlation included either perturbationally (MPn) or

using Coupled-Cluster theory (e.g. CCSD(T))


abab initioinitio MO Theory: MO Theory: ApproximationsApproximations

• Linear Combination of Atomic Orbitals (LCAO)

+

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abab initioinitio MO Theory: MO Theory: ApproximationsApproximations

• Slater Determinants and Self-Consistent-Field Theoryo Multi-electron wavefunction is approximated as a

series of one-electron wavefunctions (orbitals)o Each electron interacts with the mean field of

all other electrons (Hartree-Fock Theory)


HartreeHartree--FockFock--LimitLimit

Ener

gy →

SCF-energies

Hartree-Fock limit

Correlation energy

Experiment


ab ab initioinitio ComputationalComputationalLevelsLevels

Correlation →

Basi

s Se

t →


abab initioinitio MO TheoryMO Theory• The method can be improved systematically so that

convergence of the results can be recognized• Therefore, extrapolation schemes give very high

accuracy• Scaling of methods with correlation is typically worse

than <O> N4

• Linear scaling (often local) methods are now available for many techniques

• Limit for problems that need extensive geometry optimizations or second derivatives lies by about 200 atoms

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Density Functional Theory Density Functional Theory (DFT)(DFT)

• The properties of a molecule can be derived from its ground-state electron density (1st Hohenburg-Kohn theorem)o Correlation is treated implicitly as a correction to the energy of

a uniform electron gaso Usually necessary to integrate the density numericallyo The energy is given by a functional of the electron densityo This functional is unknowno DFT is usually performed analogously to Hartree-Fock theory

using Kohn-Sham orbitals• Moderately parallel because of the numerical

integrations (4-8 processors)• Roughly 102 faster than comparable ab initio for large

systems


Semiempirical MOSemiempirical MO--TheoryTheory• Usually based on the NDDO approximation

o Current methods introduced in the 70’so Up to 104 faster than DFTo Scales with N3 but most implementations are closer to N2

o Applications with 1,000 atoms are not unusual, 500 standardo Linear scaling can be attained either by divide-and-conquer or by

localized MO-techniqueso Correlation is treated implicitly by scaling the two-electron

integralso Heavily parameterized to fit experimental data

o Heats of Formationo Ionization potentialso Dipole momentso Molecular structures


Semiempirical Semiempirical GeometryGeometryOptimizationOptimization

• 177 atoms

• no symmetry

• initial geometry from a GUI-builder

•Elapsed time (single 2 GHz Xeon under Windows) 60 minutes


Weaknesses of Semiempirical Weaknesses of Semiempirical MOMO--TheoryTheory

• Parameterized – extrapolation can lead to wild and unpredictable errors

• Weak interactions (dispersion) not reproduced at allo but not in DFT either

• Hydrogen bonds either not reproduced (MNDO), wrong geometry (AM1) or wrong energy (PM3)

• Bond rotation barriers are too low• Nitrogen pyramidalization etc. is a problem

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How do we use 3DHow do we use 3D--Information?Information?

• QSAR usually requires that we describe each molecule with a fixed number of descriptors

• …. but molecules have different numbers of atoms

• Three possible strategies:o Specific descriptorso Global descriptorso Grids


Specific DescriptorsSpecific Descriptors• Require a knowledge of what is important. E.g.o “Bite” angles for diphosphine ligandso HOMO energies (or coefficients) for

reactions with electrophileso Spin densities for radical reactionso Atomic charges for important atomso Double-bond order


ZieglerZiegler--NattaNatta

ZrRActivity depends onthis angle (linear QSAR)

Local model,only works forzirconium!


Global DescriptorsGlobal Descriptors• Describe a fundamental property of the

molecule that hopefully is related to the target property or activityo Molecular weight, volume, surface area,

polarizability, dipole moment, refractive index ……

o Descriptors constructed (invented) to describe molecular properties

o Similarity

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3D 3D SimilaritySimilarity TechniquesTechniques• Search for similarity with the target

pharmacophore• Pure shape similarity (www.eyesopen.com)• Electrostatic similarity and similarity of the

electron density (Sanz, Carbo, Richards)Carbo Index:

2 2

A BAB

A B

P P dR

P d P d

τ

τ τ=

⋅

∫∫ ∫


DescriptorsDescriptors: 3D: 3D• Atomic coordinates

o Autocorrelationo MORSE-Codes

• Molecular surfaceso Polar surface areao Statistical descriptors of the

electrostatioc potential at the surface (Politzer, Murray)

o Surface Autocorrelations (Gasteiger)


Surface descriptorsSurface descriptors• Fast Polar Surface Area Calculation (Ertl)• Calculate a local property (usually the MEP)

at the surface of the molecule (triangulated)

• “Murray-Politzer” descriptorso Use the statistical properties of the distribution

of the values of the local property as descriptors• Autocorrelation (Gasteiger)

o Use the distance between triangulation points to create a “spectrum”


Murray/Murray/PolitzerPolitzer--DescriptorsDescriptors

m ethane trim ethylam ine bis-T rifluorom ethylPhosphinic acid

σ 2tot = 5 .4 σ 2

tot = 446.6 σ 2tot = 651.0

ν = 0 .144 ν = 0 .009 ν = 0 .246

Total variance = σ2tot ; balance parameter = ν


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AutocorrelationAutocorrelation• Usually used for time-series:

• Can be used with distances r:

( )1

n

j j ij

i a aρ +=

=∑

( )1 1

( )n n

i j iji j

r a a f r rρ= =

= −∑∑


Surface AutocorrelationsSurface Autocorrelationsdifferentorientations

differentside-chainconformations

differentpoint densities

differentdistanceintervals

differentatomic radii

differentsurfaces


PCA of Surface AutocorrelationPCA of Surface Autocorrelation

High activity

* Intermediate activity

+ Low activity


MolecularMolecular SurfacesSurfaces

Van der Waals Conolly (SES)

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De NovoDe Novo Ligand DesignLigand Design


PharmacophoresPharmacophores

• Pharmacophores are two- or three-dimensional arrays of binding features that are associated with the desired biological activity

• The following example shows the use of a pharmacophore search for 17β-hydroxysteroid dehydrogenase

• The natural substrate is estradiol


EstradiolEstradiol DockedDocked in in 1717ββ--hydroxysteroid dehydrogenasehydroxysteroid dehydrogenase


EstradiolEstradiol PharmacophorePharmacophore

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FlavoneFlavone HitHit


Multiple HitsMultiple Hits


PharmacophorePharmacophore in in thethe Binding Binding PocketPocket


ComparativeComparative MolecularMolecularFieldField Analysis (Analysis (CoMFACoMFA))

N

N

N

O

N

N

N

H-bond acceptors

Hydrophobicgroup

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CoMFACoMFA GridGrid

www.kubinyi.de


CoMFACoMFA AnalysisAnalysis

www.kubinyi.de


CoMFACoMFA ResultsResults

www.kubinyi.de


CoMFACoMFA

• Steric• Green: + • Yellow: –

• Electrostatic (positive)• Blue: +• Red: –

H. Lanig, W. Utz, P. Gmeiner, J. Med. Chem. 2001, 44, 1151-1157.

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HighHigh--ThroughputThroughput DockingDocking

• Dock rigid or flexible ligands into the receptor (usually rigid)

• Precalculate grid of properties for the receptor to speed up searching

• Evaluate the results using a scoring function

• What is a scoring function?


ScoringScoring FunctionsFunctionsH2O H2O H2Oreceptor ligand receptor:ligand+

( ) ( ) ( ) ( )0

/ /

HB Ion

lipo lipo rot rot aro aro aro aro

G G G f r f G f r fG A G N G N

α α∆ = ∆ + ∆ ∆ ∆ + ∆ ∆ ∆

+∆ + ∆ + ∆ ∆∑ ∑

LUDI Scoring Function:


Salt bridgesH-Bonds

ScoringScoring FunctionsFunctions

-10 0 .69 kcal m olG∆ = −

( ) ( ) ( ) ( )0

/ /

HB Ion



α α∆ = ∆ + ∆ ∆ ∆ + ∆ ∆ ∆

+∆ + ∆ + ∆ ∆∑ ∑

-10.76 kcal molHBG∆ = −

-11 .45 kcal m olIonG∆ = −


Rotatable bonds

π-Stackinghydrophobic

( ) ( ) ( ) ( )0

/ /

HB Ion



α α∆ = ∆ + ∆ ∆ ∆ + ∆ ∆ ∆

+∆ + ∆ + ∆ ∆∑ ∑

ScoringScoring FunctionsFunctions

-10.03 kcal mollipoG∆ = −

-1/ 0.00 kcal molaro aroG∆ =

-10 .22 kcal m olrotG∆ = −

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FreeFree--EnergyEnergy PerturbationPerturbation (FEP)(FEP)

A(gas) B(gas)

A(bound) B(bound)

experimentallyknown

target

calculate

calculate?Computer-Chemie-Centrum Universität Erlangen-Nürnberg

FreeFree--EnergyEnergy PerturbationPerturbation


FreeFree--Energy perturbationEnergy perturbation

• Mutate as slowly as possible• Mutate in both directions (the results must be the same)

• Can only do very small changes in structure

• If it’s good, it’s very, very good


SummarySummary• 2D-Descriptors may be able to describe

geometry changes• 3D-Descriptors can be very sensitive to

conformation for QSAR but are often less so for QSPR

• 3D-Methods often require alignment• … and the problem of multiple

conformations and/or tautomers remains unsolved

Structure-Activity 3D-QSAR Relationships3D.pdf1 Computer-Chemie-Centrum Universität...

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