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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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prediction of molecular properties i
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

Modern Methods in Drug Discovery WS08/09

prediction of molecular properties ii
Prediction of molecular properties (II)

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 WS08/09

prediction of molecular properties iii
Prediction of molecular properties (III)

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

Modern Methods in Drug Discovery WS08/09

quantum mechanics i
Quantum mechanics (I)

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

Modern Methods in Drug Discovery WS08/09

quantum mechanics ii
Quantum mechanics (II)

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

Modern Methods in Drug Discovery WS08/09

schr dinger equation i
Schrödinger equation (I)

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.

Modern Methods in Drug Discovery WS08/09

the wave function i
The wave function (I)

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).

Modern Methods in Drug Discovery WS08/09

the hamilton operator
The Hamilton operator

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 WS08/09

the wave function ii
The wave function (II)

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.

  • These function must obey some rules:
  • electrons are indistinguisable
  • they repell each other
  • the Pauli principle (two electrons with different spin can share a common state (orbital))

Modern Methods in Drug Discovery WS08/09

schr dinger equation ii
Schrödinger equation (II)

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 WS08/09

molecular orbital theory i
Molecular Orbital Theory (I)

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 WS08/09

molecuar orbital theory ii
Molecuar Orbital Theory (II)

Applying the LCAO approach for the wave function we yield for H2

      • 
  • =1 =1 overlap intergral S
  • Due to the normalization of the wave function regarding the complete space:

Modern Methods in Drug Discovery WS08/09

molecular orbital theory iii
Molecular Orbital Theory (III)

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

Modern Methods in Drug Discovery WS08/09

h ckel theory i
Hückel Theory (I)
    • (1931) Limited to planar, conjugated -systems,-orbitals are neglected.
  • The original aim was to interpret the non-additive properties of aromatic compounds (e.g. benzene compared to “cyclohexatrien”) regarding their heats of combustion.

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.

Modern Methods in Drug Discovery WS08/09

h ckel theory ii
Hückel Theory (II)
  • Example: ethene H2C=CH2

Modern Methods in Drug Discovery WS08/09

h ckel theory iii
Hückel Theory (III)

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 WS08/09

h ckel theory iv
Hückel Theory (IV)

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.

Modern Methods in Drug Discovery WS08/09

h ckel theory v
Hückel Theory (V)
  • Application of the Hückel method to predict and interpret UV/VIS spectra
  • Different parameters  for different atoms (C,N,O) allow the application of the Hückel theory to further compounds
  • Orbital energies can be determined experimentally by photo electron spectroscopy (PES) and thus also  (the respective ionization potential) and 

Modern Methods in Drug Discovery WS08/09

hartree fock based methods
Hartree-Fock based methods

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 WS08/09

semiempirical methods i
Semiempirical methods (I)

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.

Modern Methods in Drug Discovery WS08/09

semiempirical methods ii
Semiempirical methods (II)

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.

Modern Methods in Drug Discovery WS08/09

non commerical programs
Non commerical programs

MOPAC 7.1 (and MOPAC2007) J.J.P. Stewart

http://openmopac.net/

GHEMICAL

http://www.bioinformatics.org/ghemical/ghemical/index.html

Modern Methods in Drug Discovery WS08/09

am1 austin model 1
AM1 (Austin Model 1)

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

  • Deficiencies of AM1 (and all other methods based on NDDO):
  • - hypervalent elements in general, because no d-orbitals
  • compounds with lone electron pairs (esp. anomeric effect)
  • NO2 containing compounds

Modern Methods in Drug Discovery WS08/09

pm3 parameterized method 3
PM3 (Parameterized Method 3)

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

  • Disadvantages compared to AM1:
  • All other nitrogen containing compounds worse
  • higher atomic charges lead to a more polar character of the molecules
  • Not all parameterized elements (e.g. Mg and Al) yield reliable results for all substance classes

Modern Methods in Drug Discovery WS08/09

molecular properties from semiempirical qm calculations i
Molecular properties from semiempirical QM calculations (I)
  • In contrast to ab initio calculations the semiempirical methods MNDO, AM1, and PM3 were calibrated to reproduce experimental data:
  • heats of formation [Bildungswärmen]
  • molecular geometries (bond lengths, bond angles)
  • dipole moments
  • ionization potentials

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 WS08/09

heats of formation
Heats of formation

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

Modern Methods in Drug Discovery WS08/09

comparison of the methods
Comparison of the methods

Calculated heats of formation at 25° C for different compoundsAverage mean error (in kcal/mol)

  • Number of compounds method
  • (C, H, N, O, and) MNDO AM1 PM3 MNDO/d
  • Al (29) 22.1 10.5 16.4 4.9
  • Si (84) 12.0 8.5 6.0 6.3
  • P (43) 38.7 14.5 17.1 7.6
  • S (99) 48.4 10.3 7.5 5.6
  • Cl (85) 39.4 29.1 10.4 3.9
  • Br (51) 16.2 15.2 8.1 3.4
  • I (42) 25.4 21.7 13.4 4.0
  • Zn (18) 21.0 16.9 14.7 4.9
  • Hg (37) 13.7 9.0 7.7 2.2
  • Mg (48) 9.3 15.4 12.0 9.3

Modern Methods in Drug Discovery WS08/09

new semiempirical methods since 1995
New semiempirical methods since 1995

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)

Modern Methods in Drug Discovery WS08/09

electronic molecular properties i
Electronic molecular properties (I)

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

Modern Methods in Drug Discovery WS08/09

electronic molecular properties ii
Electronic molecular properties (II)

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)

Modern Methods in Drug Discovery WS08/09

molecular electrostatic potential i
Molecular electrostatic potential (I)

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.

Modern Methods in Drug Discovery WS08/09

molecular electrostatic potential ii
Molecular electrostatic potential (II)

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 WS08/09

molecular electrostatic potential iii
Molecular electrostatic potential (III)

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

Modern Methods in Drug Discovery WS08/09

quantum mechanical descriptors selection
Quantum mechanical descriptors (selection)

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

Modern Methods in Drug Discovery WS08/09

molecular properties from semiempirical methods ii
Molecular properties from semiempirical methods (II)

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 WS08/09

prediction of molecular properties examples
Prediction of Molecular Properties, Examples

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.

  • in vitro mutagenicity of MX compounds
  • Blood-brain distribution (logBB)
  • CNS permeability of substances
  • QT-interval prolongation (hERG channel blockers)

Modern Methods in Drug Discovery WS08/09

quantum qsar
Quantum QSAR

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.

Modern Methods in Drug Discovery WS08/09

bbb model with 12 descriptors
BBB-model with 12 descriptors

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.

Modern Methods in Drug Discovery WS08/09

cns permeability

96%

mde34

95%

ar5

CNS Permeability

CNS–

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 WS08/09

Lit.: C.Andres & M.Hutter QSAR Comb.Sci.25 (2006) 305.

96%

100%

decision tree for qt prolonging drugs

90%

89%

mghbd

Decision tree for QT-prolonging drugs

100%

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 WS08/09

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%

common structural features of qt prolonging drugs
Common structural features of QT-prolonging drugs

Derived common substructure expressed as SMARTS string

Lit.: M.Gepp & M.Hutter Bioorg.Med.Chem.14 (2006) 5325.

Modern Methods in Drug Discovery WS08/09

molecular properties from force fields
Molecular properties from force fields

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.

  • Due to the low computational effort, force fields are well suited to allow conformational searches.
  • 4D-QSAR (different docked conformations,

e.g. in cytochrome P450)

Modern Methods in Drug Discovery WS08/09

molecular properties from molecular dynamics simulations
Molecular properties from molecular dynamics simulations

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.

Modern Methods in Drug Discovery WS08/09