Molecules as Graphs

The chemical markup language (CML) is based on the extensible markup language (XML) in order to code chemical structures [21-24], Besides chemical structures and reactions, CML is capable of drawing spectra, etc. 

In graph theory, a graph is an ensemble of points, some of them connected with lines. We call the points vertices and the lines edges. A graph is suitable to represent a molecule. Graph theory plays an important role in searching of structures and substructures. 

The representation of molecules according to graph theoretical rules plays a role in the development of structure property relationships for organic molecules. 

Gasteiger, J. (ed.) Handbook of Chemoinformatics. From Data to Knowledge, vol. 1-4. WUey-VCH, Weinheim (2003) 

Basic overview of chemoinformatics. J. Chem. Inf. Model. 46(6), 

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The paper is divided into several sections. Section 2 discusses representation of chemical molecules as graphs. 

Fig. 6-11 Small molecules as chemical node (atom), the type of edge (bond), and graphs [81]. Representation of the structure the connectivity of nodes. Hydrogen atoms of small molecules as graphs encoded by an are not considered as nodes in the graph, adjacency matrix that specifies the type of...

Clearly, the next step is the handling of a molecule as a real object with a spatial extension in 3D space. Quite often this is also a mandatory step, because in most cases the 3D structure of a molecule is closely related to a large variety of physical, chemical, and biological properties. In addition, the fundamental importance of an unambiguous definition of stereochemistry becomes obvious, if the 3D structure of a molecule needs to be derived from its chemical graph. The moleofles of stereoisomeric compounds differ in their spatial features and often exhibit quite different properties. Therefore, stereochemical information should always be taken into ac-count if chiral atom centers are present in a chemical structure. 

Sketch the molecules on graph paper to help in determining the atomic coordinates. This is the first use of molecular geometry, a property that will become increasingly important as we go on. At this stage, the geometries are approximate the difference, for example, between cis and trans isomer s is ignored. 

Figure 10.7 The population of excited energy states relative to that of the ground state for the CO molecule as predicted by the Boltzmann distribution equation. Graph (a) gives the ratio for the vibrational levels while graph (b) gives the ratio for the rotational levels. Harmonic oscillator and rigid rotator approximations have been used in the calculations. The dots represent ratios at integral values of v and 7. which are the only allowed values.

How the structural information in molecules is represented is crucial to the types of chemical questions that can be asked and answered. This is certainly true in MSA where different representations and their corresponding similarity measures can lead to dramatically different results (2). Four types of mathematical objects are typically used to represent molecules—sets, graphs, vectors, and functions. Sets are the most general objects and basically underlie the other three and are useful in their own right as will be seen below. Because of... 

As seen in the last step, protein models are often developed based on graphs. Graphs may be used to nondirectional molecular interactions and associations between two or more molecules. However, graphs may rapidly grow in complexity and overwhelm even the most powerful supercomputers today. 

Lederberg, J., "Dendral-6k, a System for Computer Construction, Enumeration, and Notation of Organic Molecules as Tree Structures and Cyclic Graphs, Part I," NASA Scientific and Technical Aerospace Report, STAR No. N65-I3158 and CR 57029, 19611. 

Nowadays, more than 4000 types of descriptors are known.17 There exist different ways to classify them. With respect to the type of molecular representation used for their calculations—chemical formula, molecular graph, or spatial positions of atoms—one speaks about ID, 2D, and 3D descriptors, respectively. Descriptors can be global (describing the molecule as a whole) and local (only selected parts are considered). One could distinguish information-based descriptors, which tend to code the information stored in molecular structures, and knowledge-based (or semiempir-ical) descriptors issued from the consideration of the mechanism of action. Most of those descriptors can be obtained with the DRAGON, CODESSA PRO, and ISIDA programs. 

Fig. 3.4 Definition of scaffold (Sc) and reduced scaffold (ReSc). In this work we defined the scaffold of a molecules as the side-chain depleted molecular graph without annotation of atom types. A reduced scaf-...

Chemical hardness and softness are much newer ideas than electronegativity, and they were quantified only fairly recently. Parr and Pearson (1983) proposed to identify the curvature (i.e. the second derivative) of the E versus N graph (e.g. Fig. 7.10) with hardness, rj [151]. This accords with the qualitative idea of hardness as resistance to deformation, which itself accommodates the concept of a hard molecule as resisting polarization - not being readily deformed in an electric field if we choose to define hardness as the curvature of the E versus N graph, then... 

The Flory principle is one of two main assumptions underlying the ideal kinetic model of any processes of synthesis and chemical modification of polymers. The second assumption is the neglect of the reaction between any active centers belonging to the same molecule. Clearly, in the absence of such intramolecular reactions, molecular graphs of all components of a reaction system will not contain cycles. The last affirmation applies just to sol molecules. As for the gel, in the framework of the ideal model, the cyclization reaction is admissible. 

Fig. 5. Examples of molecules whose graphs contain non-planar Ks (37-43) or K3 3 (44) graphs as a subgraph. In each case, a possible subgraph is shown by the means of starred atoms...

One of the first topologically chiral molecule whose graph contains K5 as a subgraph is probably the completely asymmetric cluster 49 Rh8(CO)19C (Fig. 7) synthesized in 1974 by P. Chini et al. [78]. (Note that the C —Rh bonds are not... 

In the most general sense, a graph represents a network of nodes (called vertices, singular vertex) connected by links (called edges). As such, it can only describe the connectivity of atoms, not their other properties, such as the number of protons, electronic configuration, and so on, or the properties of the molecule as a whole—its chirality and so on. These features have to be added on top of the graphic representation of the molecule. 

Figure 11.8 An example of the stochastic trajectory from Monte Carlo simulation according to the Gillespie algorithm for reaction system given in Equation (11.19) and corresponding master equation graph given in Figure 11.4. Here we set Ns = 100 and Nes = 0 at time zero and total enzyme number Ne = 10. (A) The fluctuating numbers of S and ES molecules as functions of time. (B) The stochastic trajectory in the phase space of (m, n).

The chirality of the compounds discussed so far, especially of those in Section 2 is caused entirely by the chirality of their substituents. Due to their nonplanar shape, calixarenes offer various additional possibilities to produce chiral derivatives, the chirality of which is not based on a chiral group or subunit but on the absence of a symmetry plane, an inversion center, or an alternating axis in the molecule as a whole. This means, that opening of the macrocyclic structure would lead to an achiral linear molecule. Such molecules may therefore be called inherently chiral which should not be confused with the term intrinsically chiral,110 A graph whose chirality is independent of its embedding in the three-dimensional space is intrinsically chiral, while the inherent chirality defined above is due to the three-dimensional structure. 

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