Presentation on theme: "13th lectureModern Methods in Drug Discovery WS07/081 Prediction of molecular properties (I) The general question of rational drug design How is the biological."— Presentation transcript:
13th lectureModern Methods in Drug Discovery WS07/081 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
13th lectureModern Methods in Drug Discovery WS07/082 Prediction of molecular properties (II) What are molecular properties? molecular weight MW (from the sum formula C 12 H 11 N 3 O 2 ) melting point boiling point vapour pressure solubility (in water) charge dipole moment polarizability ionization potential electrostatic potential Directly computable from the electronic wave function of a molecule observables
13th lectureModern Methods in Drug Discovery WS07/083 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 Atomic coordinates Molecular mechanics (MM) force fields Quantum mechanics (QM)
13th lectureModern Methods in Drug Discovery WS07/084 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: The (electrostatic) interaction between charged particles (electrons, cores) is expressed by Coulomb‘s law Atomic cores are > 1000 times heavier than the electrons und thus notice the electrons only as an averaged field
13th lectureModern Methods in Drug Discovery WS07/085 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: particlewave galvanic diffraction precipitation
13th lectureModern Methods in Drug Discovery WS07/086 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.
13th lectureModern Methods in Drug Discovery WS07/087 The wave function (I) 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). The wave function is a mathematical expression describing the spacial arrangement of the (fluctuating) electrons.
13th lectureModern Methods in Drug Discovery WS07/088 The Hamilton operator with the squared Nabla operator The Hamilton operator contains the kinetic (T) and the potential (V) energy operators of all considered particles i in the system with As a consequence of the Born-Oppenheimer approximation, also the Hamilton operator can be separated into a core and an electronic part.
13th lectureModern Methods in Drug Discovery WS07/089 The wave function (II) 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)) Any mathematical expression of the wave function must fulfill certain criteria to account for the physical nature of the electrons.
13th lectureModern Methods in Drug Discovery WS07/0810 Schrödinger equation (II) 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 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
13th lectureModern Methods in Drug Discovery WS07/0811 Molecular Orbital Theory (I) Common expression for a MO with the atomic orbitals Molecular orbitals can be constructed as a linear combination of atomic orbitals (LCAO approach) or other basis functions. e.g. for H 2
13th lectureModern Methods in Drug Discovery WS07/0812 Molecuar Orbital Theory (II) Applying the LCAO approach for the wave function we yield for H 2 =1 =1 overlap intergral S Due to the normalization of the wave function regarding the complete space:
13th lectureModern Methods in Drug Discovery WS07/0813 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 (c A, c B,...) that produces reasonable orbital energies variational principle Hartree-Fock equations Self Consistent Field (SCF) method
13th lectureModern Methods in Drug Discovery WS07/0814 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 p z -orbitals). The -electrons move in an electric field produced by the -electrons and the atomic cores.
13th lectureModern Methods in Drug Discovery WS07/0815 Hückel Theory (II) Example: ethene H 2 C=CH 2
13th lectureModern Methods in Drug Discovery WS07/0816 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 12341234
13th lectureModern Methods in Drug Discovery WS07/0817 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.
13th lectureModern Methods in Drug Discovery WS07/0818 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
13th lectureModern Methods in Drug Discovery WS07/0819 Hartree-Fock based methods Born-Oppenheimer approximation one-determinant approach H = E Hartree-Fock-equations ab initio methods with limited basis set RHF optimized basis sets all electron ECP Valence electrons multi-determinant approaches UHF spin ( , ) space CIMCSCFCASSCF semiempirical methods with minimal basis set ZDO-approximation valence electrons parameters semiempirical C.I. methods
13th lectureModern Methods in Drug Discovery WS07/0820 Semiempirical methods (I) The problem of ab initio calculation is their N 4 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.
13th lectureModern Methods in Drug Discovery WS07/0821 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. 1985und PM3 (Parameterized Method 3) J.P.P. Stewart 1989 This is partly also due to their availability of the wide spread MOPAC 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.
13th lectureModern Methods in Drug Discovery WS07/0822 Non commerical programs MOPAC 7.1 (and MOPAC2007) J.J.P. Stewart http://openmopac.net/ GHEMICAL http://www.bioinformatics.org/ghemical/ghemical/index.html
13th lectureModern Methods in Drug Discovery WS07/0823 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) - NO 2 containing compounds
13th lectureModern Methods in Drug Discovery WS07/0824 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 + NO 2 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
13th lectureModern Methods in Drug Discovery WS07/0825 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)
13th lectureModern Methods in Drug Discovery WS07/0826 Heats of formation Computation of heats of formation at 25° C atomization energies Heats of formation of the elements Experimentally known Only the electronic energy has to be computed
13th lectureModern Methods in Drug Discovery WS07/0827 Comparison of the methods Calculated heats of formation at 25° C for different compounds Average mean error (in kcal/mol) Number of compoundsmethod (C, H, N, O, and)MNDOAM1 PM3 MNDO/d Al (29)22.110.516.44.9 Si (84)12.0 8.5 6.06.3 P (43)38.714.517.17.6 S (99)48.410.3 7.55.6 Cl (85)39.4126.96.36.199 Br (51)16.215.2 8.13.4 I (42)25.421.713.44.0 Zn (18)21.016.914.74.9 Hg (37)13.7 9.0 7.72.2 Mg (48) 9.315.412.09.3
13th lectureModern Methods in Drug Discovery WS07/0828 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)
13th lectureModern Methods in Drug Discovery WS07/0829 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 o permanent dipol moment of the molecule (if present) polarizability (first) hyperpolarizability 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 ind is obtained as:
13th lectureModern Methods in Drug Discovery WS07/0830 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 0000energy 1000electric dipol moment 0 100magnetic dipol moment 0010hyperfine coupling constant (EPR) 0001energy gradient (geom.optimization) 2000electric polarizability 3000(first) hyperpolarizability 000 2harmonic vibration (IR) 1001IR absorption intensities 1100circular dichroisms (CD) 0020nuclear spin coupling const. (NMR) 0110nuclear magnetic shielding (NMR)
13th lectureModern Methods in Drug Discovery WS07/0831 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.
13th lectureModern Methods in Drug Discovery WS07/0832 Molecular electrostatic potential (II) To determine the MEP at a point r the integration is practically replaced by a summation of sufficiently small 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.
13th lectureModern Methods in Drug Discovery WS07/0833 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) 361 CHELPG approach Singh & Kollman J.Comput.Chem. 5 (1984) 129 RESP approach atomic charges for the AMBER force field
13th lectureModern Methods in Drug Discovery WS07/0834 Quantum mechanical descriptors (selection) atomic charges (partial atomic charges) No observables ! Mulliken population analysis electrostatic potential (ESP) derived charges WienerJ (Pfad Nummer) dipole moment polarizability HOMO / LUMO energies of the frontier orbitals given in eV 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
13th lectureModern Methods in Drug Discovery WS07/0835 Molecular properties from semiempirical methods (II) Which method for which purpose ? structural properties only (molecular geometries): PM3esp. for NO 2 compounds, otherwise AM1 electronic properties: MNDO for halogen containing compounds (F, Cl, Br, I) AM1for hypervalent elements (P,S), H-bonds Do not mix descriptors computed from different semiempirical methods ! e.g. PM3 for NO 2 containing molecules and AM1 for the remaining compounds in the set.
13th lectureModern Methods in Drug Discovery WS07/0836 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)
13th lectureModern Methods in Drug Discovery WS07/0837 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.
13th lectureModern Methods in Drug Discovery WS07/0838 BBB-model with 12 descriptors Lit: M. Hutter J.Comput.-Aided.Mol.Des. 17 (2003) 415. Descriptors mainly from QM calculations: electrostatic surface, principal components of the geometry, H-bond properties
13th lectureModern Methods in Drug Discovery WS07/0841 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.
13th lectureModern Methods in Drug Discovery WS07/0842 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)
13th lectureModern Methods in Drug Discovery WS07/0843 Molecular properties from molecular dynamics simulations Binding affinities (actually free energies of binding) G 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.