1 Computer-Chemie-Centrum Universität Erlangen-Nürnberg 3D 3D- QSAR QSAR Tim Clark Computer-Chemie-Centrum Universität Erlangen-Nürnberg Nägelsbachstraße 25 91052 Erlangen Computer-Chemie-Centrum Universität Erlangen-Nürnberg Structure Structure-Activity Activity Relationships Relationships Chemical Chemical Structure Structure Biological activity QSAR Physical property QSPR Computer-Chemie-Centrum Universität Erlangen-Nürnberg 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 Computer-Chemie-Centrum Universität Erlangen-Nürnberg QSPR QSPR Methods Methods for for Polymers Polymers • The Van Krevelen Method o D. W. Van Krevelen, Properties of Polymers, 3rd ed., (Amsterdam, Elsevier, 1990). • The Askadskii Method o Andrey A. Askadskii, Physical Properties of Polymers: Prediction and Control (Amsterdam, Gordon and Breach Publishers,1996). • Connectivity Indices o Jozef Bicerano, Prediction of Polymer Properties (New York, Marcel Dekker, Inc., 1993).
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Computer-Chemie-Centrum Universität Erlangen-Nürnberg
3D3D--QSARQSAR
Tim ClarkComputer-Chemie-Centrum
Universität Erlangen-NürnbergNägelsbachstraße 25
91052 Erlangen
Computer-Chemie-Centrum Universität Erlangen-Nürnberg
• 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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Computer-Chemie-Centrum Universität Erlangen-Nürnberg
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.
Computer-Chemie-Centrum Universität Erlangen-Nürnberg
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
Computer-Chemie-Centrum Universität Erlangen-Nürnberg
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)
Computer-Chemie-Centrum Universität Erlangen-Nürnberg
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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Computer-Chemie-Centrum Universität Erlangen-Nürnberg
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.
Computer-Chemie-Centrum Universität Erlangen-Nürnberg
MolecularMolecular Orbital Orbital MethodsMethods• Linear Combination of Atomic Orbitals (LCAO)
Computer-Chemie-Centrum Universität Erlangen-Nürnberg
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.
Computer-Chemie-Centrum Universität Erlangen-Nürnberg
HHüückelckel--TheoryTheory: : EthyleneEthylene
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Computer-Chemie-Centrum Universität Erlangen-Nürnberg
HHüückelckel--MatrixMatrix
C2 C3
C4
C1
H
H
H
H
H
H
1
αβ00C4
βαβ0C3
0βαβC2
00βαC1
C4C3C2C1
Computer-Chemie-Centrum Universität Erlangen-Nürnberg
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
Computer-Chemie-Centrum Universität Erlangen-Nürnberg
ButadieneButadiene--MOsMOs
Computer-Chemie-Centrum Universität Erlangen-Nürnberg
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.
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Computer-Chemie-Centrum Universität Erlangen-Nürnberg
MolecularMolecular Orbital Orbital MethodsMethods• Self Consistent Field (SCF)
e-
e-e-
Computer-Chemie-Centrum Universität Erlangen-Nürnberg
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
Computer-Chemie-Centrum Universität Erlangen-Nürnberg
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
Computer-Chemie-Centrum Universität Erlangen-Nürnberg
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+
=+ +
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Computer-Chemie-Centrum Universität Erlangen-Nürnberg
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
Computer-Chemie-Centrum Universität Erlangen-Nürnberg
INDOINDO--IntegralsIntegrals• Of all the possible integrals (µν⏐λσ), only (µµ⏐λλ) are used
• 5 Types :
•Gss
•Gsp
•Gpp
•Gp2
Computer-Chemie-Centrum Universität Erlangen-Nürnberg
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
Computer-Chemie-Centrum Universität Erlangen-Nürnberg
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
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Computer-Chemie-Centrum Universität Erlangen-Nürnberg
Semiempirical Semiempirical MOMO--MethodsMethods
Computer-Chemie-Centrum Universität Erlangen-Nürnberg
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
Computer-Chemie-Centrum Universität Erlangen-Nürnberg
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
Computer-Chemie-Centrum Universität Erlangen-Nürnberg
o Is included by scaling the one- and two-center integrals.
• Semiempirical CI-Calculationso Therefore only treat static correlationo … and are therefore easily interpreted
Computer-Chemie-Centrum Universität Erlangen-Nürnberg
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))
Computer-Chemie-Centrum Universität Erlangen-Nürnberg
abab initioinitio MO Theory: MO Theory: ApproximationsApproximations
• Linear Combination of Atomic Orbitals (LCAO)
+
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Computer-Chemie-Centrum Universität Erlangen-Nürnberg
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)
Computer-Chemie-Centrum Universität Erlangen-Nürnberg
HartreeHartree--FockFock--LimitLimit
Ener
gy →
SCF-energies
Hartree-Fock limit
Correlation energy
Experiment
Computer-Chemie-Centrum Universität Erlangen-Nürnberg
ab ab initioinitio ComputationalComputationalLevelsLevels
Correlation →
Basi
s Se
t →
Computer-Chemie-Centrum Universität Erlangen-Nürnberg
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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Computer-Chemie-Centrum Universität Erlangen-Nürnberg
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
Computer-Chemie-Centrum Universität Erlangen-Nürnberg
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
Computer-Chemie-Centrum Universität Erlangen-Nürnberg
•Elapsed time (single 2 GHz Xeon under Windows) 60 minutes
Computer-Chemie-Centrum Universität Erlangen-Nürnberg
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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Computer-Chemie-Centrum Universität Erlangen-Nürnberg
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
Computer-Chemie-Centrum Universität Erlangen-Nürnberg
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
Computer-Chemie-Centrum Universität Erlangen-Nürnberg
ZieglerZiegler--NattaNatta
ZrRActivity depends onthis angle (linear QSAR)
Local model,only works forzirconium!
Computer-Chemie-Centrum Universität Erlangen-Nürnberg
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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Computer-Chemie-Centrum Universität Erlangen-Nürnberg
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=
⋅
∫∫ ∫
Computer-Chemie-Centrum Universität Erlangen-Nürnberg