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Editorial Delving into the Fundamental Aspects of Drug-Receptor Interaction “Good tests kill flawed theories; we remain alive to guess again.” Karl Popper
1. INTRODUCTION Molecules interact with specialized enzymes/receptors in appropriate physiological compartments and thereby produce their effect(s). Biological properties of molecules may be looked upon as the result of ligand-biotarget interactions and can be expressed by the relationship: BR = f (S, B)
(1)
In Eq. 1, BR represents biological response produced by the ligand (drug or toxicant) in the target biological system, and B represents the relevant biochemical part of the target system which is perturbed by the ligand to produce the measurable effect. The structure (S) becomes the sole determinant of the variation of the measured BR from one chemical to another when the biological system, B, remains practically the same during the course of the experiment and there is alternation only in the structure of ligands. Eq. 1 under such a condition approximates to: BR = f (S)
(2)
When characterization of BR is done using chemical structure alone following Eq. 2, we really attempt to understand which characteristics of the chemical structure are recognized by the biomolecular target. We ask: Which structural factors are more important and which have a marginal impact on BR? This is often accomplished by computed molecular structural descriptors which quantify various aspects of molecular structure such as shape, size, symmetry, chirality, stereo-electronic nature, hydrogen bonding capability, etc. using numerous mathematical techniques.
2. GRAPH THEORY AS A TOOL FOR CHEMICAL STRUCTURE QUANTIFICATION “Chemistry has the same quickening and suggestive influence upon the algebraist as a visit to the Royal Academy, or the old masters may be supposed to have on a Browning or a Tennyson. Indeed it seems to me that an exact homology exists between painting and poetry on the one hand and modem chemistry and modem algebra on the other. In poetry and algebra we have the pure idea elaborated and expressed through the vehicle of language, in painting and chemistry the idea enveloped in matter, depending in part on manual processes and the resources of art for its due manifestation.” — James Joseph Sylvester During the past half century or so graph theoretical approaches for the characterization of molecular structure are gaining momentum [1-7]. A graph, G, is defined as an ordered pair consisting of two sets V and R,G = [V(G), R], where V(G) represents a finite nonempty set of points, and R is a binary relation defined on the set V(G). In a molecular graph, V represents the set of atoms and R or E represents the set of edges or bonds present in the molecule. It should be noted, however, that the set E is not limited to covalent bonds. In fact, elements of E may symbolize any type of bonds, viz., covalent, ionic, or hydrogen bonds, etc. It was emphasized by Basak et al. [8] that weighted pseudographs constitute a very versatile model for the representation of a wide range of chemical species. In depicting a molecule by a connected graph G = [V(G), E(G)], V(G) may contain either all atoms present in the empirical formula or only non-hydrogen atoms. Hydrogen-filled graphs are preferable to hydrogen-suppressed graphs when hydrogen atoms are involved in critical steric or electronic interactions intramolecularly or intermolecularly or when hydrogen atoms have different physicochemical properties due to differences in their bonding neighborhoods. The majority of stable chemical species can be represented by simple graphs or multigraphs. The structural formula, labeled hydrogen-filled and the labeled hydrogen-suppressed graphs for acetamide are shown in Fig. (1).
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Fig. (1). Structural formula (G0), labeled hydrogen-filled graph (G1), and labeled hydrogen-suppressed graph (G2) of acetamide. Many topological indices can be conveniently derived from various matrices including the adjacency matrix A(G) and the distance matrix D(G) of a chemical graph G [9]. These matrices are usually constructed from labeled graphs of hydrogensuppressed molecular skeletons. For such a graph G with vertex set {v1, v2, …, vn}, A(G) is defined to be the n x n matrix (aij), where aij may have only two different values as follows: aij = 1, if vertices vi and vj are adjacent in G, aij = 0, otherwise. The distance matrix D(G) of a nondirected graph G with n vertices is a symmetric n x n matrix (dij), where dij is equal to the distance between vertices vi and vj in G. Each diagonal element dij of D(G) is equal to zero. Since topological distance in a graph is not related to the chemically related weight associated with each edge (bond), D(G) does not represent valence bond structures of molecules containing more than one covalent bond between adjacent atoms. The adjacency matrix A(G2) and the distance matrix D(G2) for the labeled graph G2 in Figure 1 may be written as follows:
A(G2) =
D(G2) =
(1)
(2)
(3)
(4)
1
0
1
0
0
2
1
0
1
1
3
0
1
0
0
4
0
1
0
0
(1)
(2)
(3)
(4)
1
0
1
2
2
2
1
0
1
1
3
2
1
0
2
4
2
1
2
0
Harry Wiener was the first to put forward the idea of a structural index (topological index) for the estimation of properties of molecules from their structure [10]. This index is popularly known as the Wiener index, W. It can be calculated from the distance matrix D(G) of a hydrogen-suppressed graph G as the sum of entries in the upper triangular submatrix [11]:
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Current Computer-Aided Drug Design, 2017, Vol. 13, No. 2
W=
1
2
d = h g ij
ij
89
(3)
h
h
where gh is the number of unordered pairs of vertices whose distance is h. 3. SIMPLE CONNECTIVITY INDICES TO VALENCE CONNECTIVITY INDICES TO ELECTROTOPOLOGICAL INDICES – PROGRESSIVE INCORPORATION OF CHEMICALLY RELEVANT INFORMATION INTO MOLECULAR GRAPHS Matrices of molecular graphs, including the adjacency and distance matrices mentioned above, may be used for the computation of graph-theoretic molecular descriptors. From the adjacency matrix of a graph with n vertices, it is possible to calculate i, the degree of the ith vertex, as the sum of all entries in the ith row: n
i = aij
(4)
j=1
The zero order connectivity index, 0 , is defined as:
= ( i )
0
12
(5)
i
First order connectivity index, 1 , is defined as [12]: 1
=
all edges
( ) i
12
(6)
j
Based on these two indices, Kier et al. [2] developed a generalized connectivity index h considering paths of type v0, v1, … vh of length h in the molecular graph: h
= ( v0 , v1 ,…, vh )
12
(7)
where the summation is taken over all paths of length h. Kier and Hall [2] extended this method to include cluster (hc), path-cluster (hpc), and cyclic (hch) types of simple connectivity indices which encode information about branching patterns, paths extending from branch points, and the presence of ring fragments, respectively. Electrotopological state indices are another group of descriptors which were developed taking into consideration the electronic states of atoms. This progressive incorporation of chemically relevant information into molecular graphs led to the quantification of chemically meaningful information and this, in turn, resulted in great success in the formulation of many useful QSARs for chemistry and biology [1]. 4. EVOLUTION OF DESCRIPTOR SPACES: ONE AT A TIME TO A FEW AT A TIME TO MANY DESCRIPTORS FOR FEWER COMPOUNDS In the early phases of descriptor formulation, a single or a handful of descriptors were used for correlation analyses [1-6]. But currently available software [13-16] can calculate many more descriptors as compared to the number of data points to be modeled. In statistics this is called a “rank deficient” situation. Under such circumstances one has to use robust statistical methods to characterize the “intrinsic dimensionality” of descriptor spaces [17, 18]. Beginning at the early 1980s, Basak and coworkers have studied methods which can extract useful information from diverse sets of calculated chemodescriptors. Basak et al. [4] carried various hierarchical QSAR (HiQSAR studies) using topostructural, topochemical, 3-D, and quantum chemical indices for various physicochemical, pharmacological, and toxicological property prediction. They divided the graph theoretic chemodescriptors into two major categories: a.
Numerical invariants defined on simple molecular graphs representing only the adjacency and distance relationship of atoms and bonds; such invariants are called topostructural (TS) indices.
b.
Topological indices derived from weighted molecular graphs, called topochemical (TC) indices.
Results of HiQSARs show that TC indices alone or a combination of TS and TC indices gave best predictive models. Whereas the simple connectivity indices fall into the TS category, the valence connectivity indices and electrotopological indices belong to the more important TC group. It is notable that addition of 3-D and quantum chemical indices after the use of TS and TC descriptors did very little improvement in model quality. How can we explain the above-mentioned trend in HiQSAR?
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One of the authors [19] of this article (LBK) is the originator of the concept of pharmacophore. A pharmacophore consists of a set of structural features in a molecule that is recognized by a drug receptor and is responsible for that molecule's biological activity. One plausible explanation of the results of HiQSAR described above is that for the recognition of a receptor, e.g., the specific recognition of dibenzofuran by the Ah receptor, the dibenzofurans probably need some specific geometrical and stereoelectronic arrangements or a specific pharmacophore. But once this minimal requirement of the recognition process is present in the molecule, changes of bioactivities from one molecule to another in the same structural class are governed by more general structural features which are quantified reasonably well by the TS and TC indices derived from the conventional bonding topology of molecules and features like sigma bond, bond, lone pair of electrons, hydrogen bond donor acidity, hydrogen bond acceptor basicity, etc. More studies with different groups of molecules with diverse bioactivities are needed to validate or falsity this hypothesis in line with the falsifiability principle of Sir Karl Popper [20]. 5. INTIMATE MODE OF DRUG ACTION: BACK TO BIOLOGY “We are all agreed that your theory is crazy. The question which divides us is whether it is crazy enough to have a chance of being correct. My own feeling is that it is not crazy enough.” Niels Bohr Even if the structural factor (S) or the computed indices explain the differential effects of various drug molecules on the biological system, we have to ask the question: What exactly happens when a drug meets the receptor? In the drug-receptor communication one may look upon the drug as the sender and the receptor as the receiver in line with the engineering principles of communication networks. Kier and Hall [21] proposed that “proton hopping” could be the ultimate trigger for the initiation of biological response in a drug-receptor interaction process. This brings us back to biology as depicted in Eq. 1 above which involves interactions between chemical structure (S) and biological receptors (B) for the initiation of effects of drugs. REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
Kier, L.B.; Hall, L. Molecular Structure Description: The Electrotopological State. Academic Press: San Diego, CA, 1999. Kier, L.B.; Hall, L. Molecular Connectivity in Chemistry and Drug Research. Academic Press: New York, NY, 1976. Trinajsti, N. Chemical Graph Theory. CRC Press: Boca Raton, FL, 1992. Basak, S.C. Mathematical descriptors for the prediction of property, bioactivity, and toxicity of chemicals from their structure: A chemical-cumbiochemical approach. Curr. Comput. Aided Drug. Des., 2013, 9, 449-462. Basak, S.C.; Restrepo, G.; Villaveces, J.L. Eds. Advances in Mathematical Chemistry and Applications, 1st Ed.; vol. 1-2 (Revised Edition); Elsevier, 2015. Devillers, J.; Balaban, A.T. Eds. Topological Indices and Related Descriptors in QSAR and QSPR; Gordon and Breach: Amsterdam, Netherlands, 1999. Basak, S.C. Philosophy of mathematical chemistry: A personal perspective. Hyle- Int. J. Phil. Chem., 2013, 19, 3-17. Basak, S.C.; Magnuson, V.R.; Niemi, G.J.; Regal, R.R. Determining structural similarity of chemicals using graph-theoretic indices. Discrete Appl. Math., 1988, 19, 17-44. Janei, D.; Milievi, A.; Nikoli, S.; Trinajsti, N. Graph-Theoretical Matrices in Chemistry; CRC Press: Boca Raton, FL, 2015. Wiener, H. Structural determination of paraffin boiling points. J. Amer. Chem. Soc., 1947, 69, 17-20. Hosoya, H. Topological index. A newly proposed quantity characterizing the topological nature of structural isomers of saturated hydrocarbons. Bull. Chem. Soc. Jpn., 1971, 44, 2332-2339. Randic, M. Characterization of molecular branching. J. Amer. Chem. Soc., 1975, 97, 6609-6615. Basak, S.C.; Harriss, D.K.; Magnuson, V.R. POLLY v2.3; Copyright of the University of Minnesota, 1988. MolconnZ v4.05; Hall Ass. Consult.: Quincy, MA, 2003. Basak, S.C.; Grunwald, G.D.; Balaban, A.T. TRIPLET; Copyright of the Regents of the University of Minnesota, 1993. Todeschini, R.; Consonni, V. Molecular Descriptors for Chemoinformatics; Wiley-VCH: New York, 2009. Basak, S.C.; Magnuson, V.R.; Niemi, G.J.; Regal R.R.; Veith, G.D. Topological indices: their nature, mutual relatedness, and applications. Math. Modelling, 1987, 8, 300-305. Majumdar, S.; Basak, S.C. Exploring intrinsic dimensionality of chemical spaces for robust QSAR model development: A comparison of several statistical approaches. Curr. Comput. Aided Drug Des., 2016, 12, 294-301. Van Drie, J.H. Monty Kier and the origin of the pharmacophore concept. IEJMD, 2007, 6, 271-279. Popper, K. The Logic of Scientific Discovery. Taylor & Francis e-Library, 2005. Kier, L.B.: Hall, L.H. The creation of proton hopping from a drug-receptor encounter. Chem. Biodivers., 2013, 16, 2221-2225.
Dr. Subhash C. Basak (Editor-in-Chief) Natural Resources Research Institute Department of Chemistry & Biochemistry University of Minnesota Duluth Duluth, MN 55811 USA E-mail:
[email protected]
Dr. Lemont B. Kier (Co-Editor) Virginia Common Wealth University Richmond, VA USA E-mail:
[email protected]
