Roberto Todeschini Viviana Consonni Manuela Pavan Andrea Mauri Davide Ballabio Alberto Manganaro chemometrics molecular descriptors QSAR multicriteria decision making environmetrics experimental design artificial neural networks statistical process control Milano Chemometrics and QSAR Research Group Department of Environmental Sciences University of Milano - Bicocca P.za della Scienza, Milano (Italy) Website: michem.unimib.it/chm/

Roberto Todeschini Milano Chemometrics and QSAR Research Group Molecular descriptors Constitutional descriptors and graph invariants Iran - February 2009

Content  Counting descriptors  Empirical descriptors  Fragment descriptors  Molecular graphs  Topological descriptors

Counting descriptors Each descriptor represents the number of elements of some defined chemical quantity. For example: - the number of atoms or bonds - the number of carbon or chlorine atoms - the number of OH or C=O functional groups - the number of benzene rings - the number of defined molecular fragments

Counting descriptors... also a sum of some atomic / bond property is considered as a count descriptor, as well as its average For example: - molecular weight and average molecular weight - sum of the atomic electronegativities - sum of the atomic polarizabilities - sum of the bond orders

A counting descriptor n is semi-positive variable, i.e. n  0 Its statistical distribution is usually a Poisson distribution. Counting descriptors Main characteristics simple the most used local information high degeneracy discriminant modelling power

Empirical descriptors Descriptors based on specific structural aspects present in sets of congeneric compounds and usually not applicable (or giving a single default value) to compounds of different classes.

It is a descriptor dedicated to the modelling of the benzene rings and is defined as the sum of the six lengths joining the adjacent substituent groups. HH HH CH 3 Cl Index of Taillander Empirical descriptors Taillander et al., 1983

Empirical descriptors It is a descriptor dedicated to the modelling of hydrophilicity and is based on a function of the counting of hydrophilic groups (OH-, SH-, NH-,...) and carbon atoms. nHynumber of hydrophilic groups nCnumber of carbon atoms ntotal number of non-hydrogen atoms -1  Hy  3.64 Hydrophilicity index (Hy) Todeschini et al., 1999

Empirical descriptors CompoundnHynCnHy hydrogen peroxide carbonic acid water butanetetraol propanetriol ethanediol methanol ethanol decanediol propanol butanol pentanol methane nHy = 0 and nC = 000N0.00 decanol ethane pentane decane alcane with nC =

Fragment approach  Parametric approach (Hammett – Hansch,1964)  Substituent approach (Free-Wilson, Fujita-Ban, 1976)  DARC-PELCO approach (Dubois, 1966)  Sterimol approach (Verloop, 1976)

Fragment approach The biological activity of a molecule is the sum of its fragment properties common reference skeleton molecule properties gradually modified by substituents Congenericity principle QSAR styrategies can be applied ONLY to classes of similar compounds

Biological response = f 1 (L) + f 2 (E) + f 3 (S) + f 4 (M) Corvin Hansch, 1964 Hansch approach Lipophilic properties Electronic properties Steric properties Other molecular properties

Hansch approach 1 Congenericity approach 2 Linear additive scheme 3 Limited representation of global molecular properties 4 No 3D and conformational information

Free-Wilson approach 12

Free-Wilson, 1964 F Br I F Br I Pos. 1Pos. 2 I ks absence/presence of k-th subst. in the s-th site

Fragment approach Fingerprints binary vector presence of a fragmentabsence of a fragment similarity searching

Molecular graph

Mathematical object defined as G = ( V, E ) set V set V vertices et E set E edges atomsbonds

Usually in the molecular graph hydrogen atoms are not considered H - depleted molecular graph Molecular graph

A walk in G is a sequence of vertices w = (v 1, v 2, v 3,..., v k ) such that {v j, v j+1 }  E. The length of a walk is the number of edges traversed by the walk. A path in G is a walk without any repeated vertices. The length of a path (v 1, v 2, v 3,..., v k+1 ) is k. v 1 v 2 v 3 v 2 v 5 walk of length 4 v 1 v 2 v 3 v 4 v 5 path of length Molecular graph

The topological distance d ij is the length of the shortest path between the vertices v i and v j d 15 = 2 The detour distance  ij is the length of the longest path between the vertices v i and v j.  15 = 4

Molecular graph A self returning walk is a walk closed in itself, i.e. a walk starting and ending on the same vertex. A cycle is a walk with no repeated vertices other than its first and last ones (v 1 = v k ). v 1 v 2 v 3 v 2 v 1 Self returning walk of length v 2 v 3 v 4 v 5 v 2

Molecular graph The molecular walk (path) count MWCk (MPCk) of order k is the total number of walks (paths) of k-th length in the molecular graph. MWC0 = nSK (no. of atoms) MWC1 = nBO (no. of bonds)  Molecular size  Branching  Graph complexity DRAGON MWC1, MWC2, …, MWC10

Molecular graph The self-returning walk count SRWk of order k is the total number of self-returning walks of length k in the graph. spectral moments of the adjacency matrix, i.e. linear combinations of counts of certain fragments contained in the molecular graph, i.e. embedding frequencies. SRW1 = nSK SRW2 = nBO DRAGON SRW1, SRW2, …, SRW10

Molecular graph Local vertex invariants (LOVIs) are quantities associated to each vertex of a molecular graph. Graph invariants are molecular descriptors representing graph properties that are preserved by isomorphism. ® characteristic polynomial ® derived from local vertex invariants

Molecular graph and more Molecular graph Topological matrix Algebraic operator Local Vertex Invariants Graph invariants Molecular descriptors

molecular graph graph invariants Wiener index, Hosoya Z index Zagreb indices, Mohar indices Randic connectivity index Balaban distance connectivity index Schultz molecular topological index Kier shape descriptors eigenvalues of the adjacency matrix eigenvalues of the distance matrix Kirchhoff number detour index topological charge indices Wiener index, Hosoya Z index Zagreb indices, Mohar indices Randic connectivity index Balaban distance connectivity index Schultz molecular topological index Kier shape descriptors eigenvalues of the adjacency matrix eigenvalues of the distance matrix Kirchhoff number detour index topological charge indices total information content on..... mean information content on..... total information content on..... mean information content on..... Kier-Hall valence connectivity indices Burden eigenvalues BCUT descriptors Kier alpha-modified shape descriptors 2D autocorrelation descriptors Kier-Hall valence connectivity indices Burden eigenvalues BCUT descriptors Kier alpha-modified shape descriptors 2D autocorrelation descriptors D-Wiener index 3D-Balaban index D/D index D-Wiener index 3D-Balaban index D/D index topological information indices topostructural descriptors topochemical descriptors molecular geometry x, y, z coordinates topographic descriptors

Molecule graph invariants Numerical chemical information extracted from molecular graphs. The mathematical representation of a molecular graph is made by the topological matrices: adjacency matrix atom connectivity matrix atom connectivity matrix distance matrix distance matrix edge distance matrix edge distance matrix incidence matrix incidence matrix... more than 60 matrix representations of the molecular structure

Local vertex invariants (LOVIs) are quantities associated to each vertex of a molecular graph. Examples: atom vertex degree atom vertex degree valence vertex degree valence vertex degree sum of the vertex distance degree sum of the vertex distance degree maximum vertex distance degree maximum vertex distance degree Local vertex invariants

Topological matrices Adjacency matrix Derived from a molecular graph, it represents the whole set of connections between adjacent pairs of atoms. a ij = 1 if atom i and j are bonded 0 otherwise

Bond number B It is the simplest graph invariant obtained from the adjacency matrix. It is the number of bonds in the molecular graph calculated as: where a ij is the entry of the adjacency matrix. Topological matrices

atom vertex degree It is the row sum of the vertex adjacency matrix Local vertex invariants

number of valence electrons of the i-th atom number of hydrogens bonded to the i-th atom valence vertex degree for atoms of the 2nd principal quantum number (C, N, O, F)

Local vertex invariants the vertex degree of the i-th atom is the count of edges incident with the i-th atom, i.e. the count of  bonds or  electrons. valence vertex degree

Local vertex invariants total number of electrons of the i-th atom (Atomic Number) for atoms with principal quantum number > 2

Topological descriptors Zagreb indices (Gutman, 1975)  i vertex degree of the i-th atom

Topological descriptors Kier-Hall connectivity indices (1986) Randic branching index (1975) They are based on molecular graph decomposition into fragments (subgraphs) of different size and complexity and use atom vertex degrees as subgraph weigth. is called edge connectivity

Topological descriptors mean Randic branching index

Topological descriptors atom connectivity indices of m-th order m Pnumber of m-th order paths qsubgraph type (Path, Cluster, Path/Cluster, Chain) n = mfor Chain (Ring) subgraph type n = m + 1 otherwise The immediate bonding environment of each atom is encoded by the subgraph weigth. The number of terms in the sum depends on the molecular structure. The connectivity indices show a good capability of isomer discrimination and reflect some features of molecular branching.

They encode atom identities as well as the connectivities in the molecular graph. valence connectivity indices of m-th order Topological descriptors

Kier-Hall electronegativity correlation with the Mulliken-Jaffe electronegativity: principal quantum number Kier-Hall relative electronegativity electronegativity of carbon sp 3 taken as zero

Distance matrix vertex distance matrix degree s i It is the row sum of the vertex distance matrix The distance d ij between two vertices is the smallest number of edges between them sisi ii s i is high for terminal vertices and low for central vertices

The eccentricity  i of the i-th atom is the upper bound of the distance d ij between the atom i and the other atoms j Local vertex invariants

Topological descriptors Petitjean shape index (1992) A simple shape descriptor I PJ = 0for structure strictly cyclic I PJ = 1for structure strictly acyclic and with an even diameter

Topological descriptors Wiener index (1947) high values for big molecules and for linear molecules low values for small molecules and for branched or cyclic molecules The Average Wiener index is independent from the molecular size. d ij topological distances

Topological descriptors Balaban distance connectivity index (1982) B number of bonds C number of cycles s i sum of the i-th row distances one of the most discriminant indices average sum of the i-th row distances number of atoms

Edge descriptors abc de f abcdef b a c d e f EsiEsi EiEi a b c d e f atom bond

Some geometrical descriptors are derived from the corresponding topological descriptors substituting the topological distances d st by the geometrical distances r st. topographic descriptors They are called topographic descriptors. Topographic descriptors For example, the 3D-Wiener index:

The geometry matrix G (or geometric distance matrix) is a square symmetric matrix whose entry r st is the geometric distance calculated as the Euclidean distance between the atoms s and t: Molecular geometry

Department of Environmental Sciences University of Milano - Bicocca P.za della Scienza, Milano (Italy) Website: michem.disat.unimib.it/chm/ THANK YOU Roberto Todeschini Viviana Consonni Manuela Pavan Andrea Mauri Davide Ballabio Alberto Manganaro chemometrics molecular descriptors QSAR multicriteria decision making environmetrics experimental design artificial neural networks statistical process control Milano Chemometrics and QSAR Research Group

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Goal

Molecular graph

Molecule graph invariants

Molecular graph

Hansch molecular descriptors partition coefficients - logP, logKow chromatog. param. - Rf, RT, Solubility …. Hammett constants molar refraction dipole moment HOMO, LUMO Ionization potential …. molecular weight VDW volume molar volume surface area …. lipophilic properties steric properties electronic properties Hansch approach

Molecular graph