Prediction of molecular properties (I). The general question of rational drug design. How is the biological space (activity) of a compound connected to the chemical space (structure) ?. Is it possible to make predictions based on the molecular structure ?. QSAR and QSPR.
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The general question of rational drug design
How is the biological space (activity) of a compound connected to the chemical space (structure) ?
Is it possible to make predictions based on the molecular structure ?
QSAR and QSPR
Modern Methods in Drug Discovery WS07/08
What are molecular properties?
molecular weight MW (from the sum formula C12H11N3O2)
melting point
boiling point
vapour pressure
solubility (in water)
charge
dipole moment
polarizability
ionization potential
electrostatic potential
observables
Directly computable from the electronic wave function of a molecule
Modern Methods in Drug Discovery WS07/08
All molecular properties that can be measured by physico-chemical methods (so called observables) can also be computed directly by quantum chemical methods.
Required: A mathematical description of the electron distribution e.g. by the electronic wave function of the molecule
Electron distribution
Quantum mechanics (QM)
Molecular mechanics (MM)force fields
Atomic coordinates
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To make the mathematical formalism practically useable, a number of approximations are necessary. One of the most fundamental consists in separating the movement of the atomic cores from that of the electrons, the so called
Born-Oppenheimer Approximation:
Atomic cores are > 1000 times heavier than the electrons und thus notice the electrons only as an averaged field
The (electrostatic) interaction between charged particles (electrons, cores) is expressed by Coulomb‘s law
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As electrons are particles, their movement can be described by classical mechanics according to Newtons 2nd law:
As electrons are also very small particles (quanta), they exhibit properties of particles as well as those of waves:
particle wave
galvanic diffraction precipitation
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Electrons can be described in the form of a wave function by the time-dependent Schröder equation
If the Hamilton operator H is time-independent, the time-dependence of the wave function can be separated as a phase factor, which leads to the time-independent Schrödinger equation. Here, only the dependence from the coordinates remains.
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The wave function is a mathematical expression describing the spacial arrangement of the (fluctuating) electrons.
The squared wave function holds the propability P to find the particle (electron) at a given place in space.
P is a so-called observable, whereas the wave function itself is no observable, physical quantity.
Thus, integration over the complete space must yield 1 (= total propability to find the electron somewhere in space).
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The Hamilton operator contains the kinetic (T) and the potential (V) energy operators of all considered particles i in the system
with the squared Nabla operator
with
As a consequence of the Born-Oppenheimer approximation, also the Hamilton operator can be separated into a core and an electronic part.
Modern Methods in Drug Discovery WS07/08
Any mathematical expression of the wave function must fulfill certain criteria to account for the physical nature of the electrons.
As a simplification the wave function of all electrons in a molecule is assumed to be the product of one-electron functions which themselves describe a single electron.
Modern Methods in Drug Discovery WS07/08
According to the Schrödinger equation there must be several different energetic levels for the electrons within an atom or molecule. These (orbital) energies can be obtained by integration and rearrangement to
The resulting energies are, however, dependend on the quality of the applied wave function and thus always higher or, in the best case, equal to the actual energy.
In the simplest case we chose 1s orbitals as basis set to describe the wave function
Modern Methods in Drug Discovery WS07/08
Molecular orbitals can be constructed as a linear combination of atomic orbitals (LCAO approach) or other basis functions.
e.g. for H2
Common expression for a MO
with the atomic orbitals
Modern Methods in Drug Discovery WS07/08
Applying the LCAO approach for the wave function we yield for H2
Modern Methods in Drug Discovery WS07/08
Common notation of the Sekular equations using matrices:
The solutions of these Sekular equations for E yield the energies of the bonding and anti-bonding MOs
The main numerical effort consists in the iterative search for suitable coefficients (cA, cB, ...) that produces reasonable orbital energies
variational principle
Hartree-Fock equations
Self Consistent Field (SCF) method
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The -orbitals are obtained as linear combinations of atomic orbitals (LCAO of pz-orbitals). The -electrons move in an electric field produced by the -electrons and the atomic cores.
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Within the Hückel theory the Fock matrix contains as many columns, respectively rows, as atoms are present in the molecule. All diagonal elements correspond to an atom i and are set to the value . Off-diagonal elements are only non-zero if there is a bond between the atoms i and j. This resonance parameter is set to (<0). Values for can be obtained experimentally from UV/VIS-spectra ( -4.62 eV).
Example butadiene:
1 2 3 4
1 2 3 4
Modern Methods in Drug Discovery WS07/08
For a cyclic -system as in benzene, the orbital energies and orbital coefficients results to
This also yields the Hückel rule:a system of [4n+2] -electrons is aromatic.
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Modern Methods in Drug Discovery WS07/08
HY = EY
Born-Oppenheimer approximation one-determinant approach
Hartree-Fock-equations
ZDO-approximation valence electrons parameters
optimized basis sets
all electron
RHF
semiempirical methods with minimal basis set
ab initio methods with
limited basis set
multi-determinant approaches
Valence electrons
UHF
ECP
spin (a,b)
space
semiempirical C.I. methods
CI
MCSCF
CASSCF
Modern Methods in Drug Discovery WS07/08
The problem of ab initio calculation is their N4 dependence from the number of two-electron integrals. These arise from the number of basis functions and the interactions between electrons on different atoms.
In semiempirical methods the numerical effort is strongly reduced by assumptions and approaches:
1. Only valence electrons are considered, the other electrons and the core charge are described by an effective potential for each atom (frozen core).
2. Only a minimal basis set is used (one s and three p-orbitals per atom), but using precise STOs that are orthogonal to each other.
3. More or less stringent use of the Zero Differential Overlap (ZDO) approach.
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Since 1965 a series of semiempirical methods have been presented from which still some are in use today for the simulation of electromagnetic spectra: CNDO/S, INDO/S, ZINDO
Following methods have shown to be particularly successful in predicting molecular properties:
MNDO (Modified Neglect of Diatomic Overlap) Thiel et al. 1975,
AM1 (Austin Model 1) Dewar et al. 1985 und
PM3 (Parameterized Method 3) J.P.P. Stewart 1989
This is partly also due to their availability of the wide spreadMOPAC program package and its later commerical sucessors. All three method are based on the same NDDO approach and differ in the parameterization of the respective elements.
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MOPAC 7.1 (and MOPAC2007) J.J.P. Stewart
http://openmopac.net/
GHEMICAL
http://www.bioinformatics.org/ghemical/ghemical/index.html
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Dewar, Stewart et al. J.Am.Chem.Soc.107 (1985) 3902
Advantages compared to MNDO:
+ better molecular geometries esp. for hypervalent elements (P, S)
+ H-bonds (but with a tendency towards forking)
+ activation energies for chemical reactions
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J.J.P. Stewart J.Comput.Chem.10 (1989) 209
Parametrization was performed more rigerously using errror minimization than in previous methods.
Advantages compared to AM1:
+ better molecular geometries for C, H, P and S
+ NO2 containing compounds better
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The results of semiempirical methods regarding these properties are therefore often better than that of ab initio calculations at low level (with comparable computational effort)
Modern Methods in Drug Discovery WS07/08
Computation of heats of formation at 25° C
atomizationenergies
Heats of formation of the elements
Experimentally known
Only the electronic energy has to be computed
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Calculated heats of formation at 25° C for different compoundsAverage mean error (in kcal/mol)
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MNDO/d
Thiel & Voityuk J.Phys.Chem.100 (1996) 616
Expands the MNDO methods by d-obitals and is “compatible” with the other MNDO parameterized elements
PM3(tm), PM5
d-orbitals for transition elements (transition metals)
SAM1 Semi ab initio Method 1
Certain integrals are thouroghly computed, therefore also applicable to transition metals (esp. Cu and Fe)
AM1*
Winget, Horn et al. J.Mol.Model.9 (2003) 408.d-orbitals for elements from the 3rd row on (P,S, Cl)
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Besides the structure of molecules all other electronic properties can be calculated. Many of those result as response of the molecule to an external disturbance:Removal of one electron ionization potential
In general a disturbance by an electric field can be expressed in the form of a Taylor expansion. In the case of an external electrical field F the induced dipole moment mind is obtained as:
mo permanent dipol moment of the molecule (if present)
a polarizability
b (first) hyperpolarizability
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Selection of properties that can be computed from the n-th derivative of the energy according to external fields
electr. magn. nuc.spin coord. property
0 0 0 0 energy1 0 0 0 electric dipol moment0 1 0 0 magnetic dipol moment0 0 1 0 hyperfine coupling constant (EPR)0 0 0 1 energy gradient (geom.optimization)2 0 0 0 electric polarizability3 0 0 0 (first) hyperpolarizability0 0 0 2 harmonic vibration (IR)1 0 0 1 IR absorption intensities1 1 0 0 circular dichroisms (CD)0 0 2 0 nuclear spin coupling const. (NMR)0 1 1 0 nuclear magnetic shielding (NMR)
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Due to the atomic cores Z and the electrons i of a molecule a spacial charge distribution arises. At any point r the arising potential V(r) can be determined to:
While the core part contains the charges of the atomic cores only, the wave function has to be used for the electronic part.
Remember: In force fields atomic charges (placed on the atoms) are used to reproduce the electric multipoles and the charge distribution.
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To determine the MEP at a point r the integration is practically replaced by a summation of sufficientlysmall volume elements.
For visualization the MEP is projected e.g. onto the van der Waals surface.
Other possibilities are the representation of surfaces with the same potential (isocontour)
From: A. Leach, Molecular Modelling,2nd ed.
Modern Methods in Drug Discovery WS07/08
Knowledge of these surface charges enables computation of atomic charges (e.g. for use in force fields) ESP derived atomic charges
These atomic charges must in turn reproduce the electric multipoles (dipole, quadrupole,...).Therefore the fitting procedures work iteratively.
literature:
Cox & Williams J.Comput.Chem.2 (1981) 304
Bieneman & Wiberg J.Comput.Chem.11 (1990) 361CHELPG approach
Singh & Kollman J.Comput.Chem.5 (1984) 129RESP approach atomic charges for the AMBER force field
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atomic charges (partial atomic charges) No observables !
Mulliken population analysis electrostatic potential (ESP) derived charges
dipole moment
polarizability
HOMO / LUMO
energies of the frontier orbitals given in eV
WienerJ (Pfad Nummer)
covalent hydrogen bond acidity/basicity difference of the HOMO/LUMO energies compared to those of water
Lit: M. Karelson et al. Chem.Rev.96 (1996) 1027
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Which method for which purpose ?
structural properties only (molecular geometries):
PM3 esp. for NO2 compounds, otherwise AM1
electronic properties:
MNDO for halogen containing compounds (F, Cl, Br, I)
AM1 for hypervalent elements (P,S), H-bonds
Do not mix descriptors computed from different semiempirical methods !e.g. PM3 for NO2 containing molecules and AM1 for the remaining compounds in the set.
Modern Methods in Drug Discovery WS07/08
Descriptors from semiempirical methods(ionization potential, dipole moment ...)along commonly used variables in QSAR equations and classification schemes.Often much more qualitative experimental data than quantitative date are available.
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Generation of molecular properties as descriptors for QSAR-equations from quantum mechanical data.
Example: mutagenicity of MX compounds
ln(TA100) = -13.57 E(LUMO) –12.98 ; r = 0.82
Lit.: K. Tuppurainen et al. Mutat. Res.247 (1991) 97.
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Descriptors mainly from QM calculations: electrostatic surface, principal components of the geometry,H-bond properties
Lit: M. Hutter J.Comput.-Aided.Mol.Des. 17(2003) 415.
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mde34
95%
ar5
CNS PermeabilityCNS–
CNS+
91%
vxbal
qsum+
96%
100%
99%
hlsurf
82%
qsum+
100%
99%
qsum+
100%
qsumo
72%
100%
qsum–
88%
100%
100%
pcgc
100%
qsum–
mpolar
79%
100%
100%
89%
77%
cooh
dipdens
83%
100%
qsum–
89%
100%
80%
hbdon
100%
size & shape
99%
qsum+
100%
86%
dipm
electrostatic
100%
kap3a
89%
100%
H-bonds
mde13
92%
94%
Modern Methods in Drug Discovery WS07/08
Lit.: C.Andres & M.Hutter QSAR Comb.Sci.25 (2006) 305.
96%
100%
89%
mghbd
Decision tree for QT-prolonging drugs100%
size & shape
88%
chbba
electrostatic
100%
hlsurf
75%
MR
89%
73%
hy
100%
H-bonds
96%
MR
hacsurf
mde23
99%
100%
96%
100%
mpolar
86%
71%
100%
sgeca
95%
logP
100%
MR
89%
SMARTS
100%
qsumn
dipdens
87%
100%
92%
100%
82%
t1e
MR
83%
100%
QT+
QT–
88%
t2e
100%
99%
logP
100%
93%
MR
Level of accuracy in %
100%
Modern Methods in Drug Discovery WS07/08
qsumn
96%
100%
90%
89%
mghbd
100%
88%
chbba
100%
hlsurf
75%
MR
89%
QT+
QT–
73%
hy
100%
96%
MR
hacsurf
mde23
99%
100%
96%
100%
mpolar
86%
71%
100%
sgeca
95%
logP
100%
MR
89%
SMARTS
100%
qsumn
dipdens
87%
100%
92%
100%
82%
t1e
MR
83%
100%
88%
t2e
100%
99%
logP
100%
93%
MR
100%
qsumn
96%
100%
Derived common substructure expressed as SMARTS string
Lit.: M.Gepp & M.Hutter Bioorg.Med.Chem.14 (2006) 5325.
Modern Methods in Drug Discovery WS07/08
As as principal consequence force fields show an even more emphasized dependence from the underlying parameterization.
Thus only predictions regarding structure ( docking), dynamics ( molecular dynamics) and, rather limited, about spectra (vibrational Infra Red) can be made.
e.g. in cytochrome P450)
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Binding affinities (actually free energies of binding)
DG for ligand binding to enzymes from free energy perturbation calculations
Advantage: quite precise predictions
Disadvantage: computationally very demanding, thus only feasible for a small number of ligands
Lit.: A.R. Leach Molecular Modelling, Longman.
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