Chemical space representations dimensionality

The table shows a number of representative descriptor types (there are many more) that can be used to define chemical spaces. Each descriptor adds a dimension (with discrete or continuous value ranges) to the chemical space representation (e.g., selection of 18 descriptors defines an 18-dimensional space). Axes of chemical space are orthogonal only if the applied molecular descriptors are uncorrelated (which is, in practice, hardly ever the case). 

Briefly, DynaMAD is designed to map database compounds to activity-specific consensus positions in chemical space representations of stepwise increasing dimensionality [38] and ACCS-FP is utilized in conventional fingerprint search calculations using multiple reference compounds [61]. 

In chemoinformatics research, partitioning algorithms are applied in diversity analysis of large compound libraries, subset selection, or the search for molecules with specific activity (1-4). Widely used partitioning methods include cell-based partitioning in low-dimensional chemical spaces (1,3) and decision tree methods, in particular, recursive partitioning (RP) (5-7). Partitioning in low-dimensional chemical spaces is based on various dimension reduction methods (4,8) and often permits simplified three-dimensional representation of... 

An alternative to dimension reduction is the use of composite and uncorrelated descriptors that are suitable for the design of information-rich yet low-dimensional chemical spaces. An elegant example is presented by the popular BCUT (Burden-CAS-University of Texas) descriptors (Pearlman and Smith 1998). BCUTs are a set of uncorrelated descriptors that combine information about molecular connectivity, inter-molecular distances, and other molecular properties. BCUT spaces used for many applications are typically only six-dimensional and can frequently be further reduced to 3D representations for visualization purposes by identifying those BCUT axes around which most compounds map. 

The dimensionality of chemical structure space exceeds that of known biological functional space. The dimensionality of biological functional space has increased in recent years due to the discovery of a multitude of genes, largely from the Human Genome Project. This chapter, however, will focus on chemical diversity rather than functional diversity. Quantification of chemical diversity involves two areas first, the predefmition of a chemical space, accomplished by selection of a diversity metric and a compound representation (i.e., molecular descriptors) and second, a rational subset selection, or classification, method dependent on efficient dimensionality reduction. Here, we describe these methods, prerequisites for a definition... 

The number of features combined in a vector-type representation is indicative of the dimensionality of the problem space. Low-dimensional representations, on the one hand, allow easy visualization but are most often not very discriminative. Highdimensional representations, on the other hand, such as those encoded in Daylight fingerprints [23], MACCS keys [24], or UNITY fingerprints [25], provide more detailed accounts on structural or chemical variations. However, this is achieved at the cost of visualization. Part of these high-dimensional representations describe specific local features of molecules, and because not all molecules in the data contain these features, gaps or zeros are introduced in the data representation. For certain data mining methods, this could be problematic. In many cases, dimensionality reduction procedures are applied to reduce the complexity of the representation. The reduction of the dimensionality is accomplished by means of 1) variable selection procedures, 2)... 

The main requirement in the determination of bond orders is to derive rules on how to measure the number of electrons shared between two atoms. For this purpose, a definition of an atom in a molecule is required, which, however, cannot be formulated in a unique and unambiguous way [169]. Quantum chemical calculations are typically performed in the Hilhert-space analysis, where atoms are defined by their basis orbitals. Such an analysis, however, strongly depends on both the atomic basis set chosen and the type of wave function used. The position-space representation, on the other hand, where atoms are defined as basins in three-dimensional physical space does not suffer from these insufficiencies. In this chapter, we present one option for a three-dimensional atomic decomposition scheme and the reader is referred to Refs. [170-173] for further examples. 

FIGURE 3.3 Two-dimensional representation of the chemical space distribution of the 30 drug molecules tested for phospholipidosis activity. The red filled circles indicate drugs that induce phospholipidosis the blue filled circles represent drugs that do not induce phosphohpidosis. The numbers indicate which specific drug (see Table 3.6) is associated with each data point. For color details, please see color plate section. 

Since most molecular representations are actually of high dimension (cf [74]), their corresponding chemical spaces are intrinsically of high dimension as well. The spatial properties of high-dimensional spaces can, in some cases, give rise to surprising problems since they tend to behave in a manner that is uncharacteristic of low-dimensional spaces [158, 159]. It is possible, however, to construct lower dimension representations of chemical spaces by computing the similarity of or... 

The chemical constitution of a molecule or an ensemble of molecules (EM) of n atoms is representable by a symmetric n X n BE-matrix and corresponds accordingly to a point P in TR ( +D/a an n(n +1)/2 dimensional Euclidean space, the Dugundji space of the FIEM(A). The "city block distance of two points P i and P 2 is twice the number of electrons that are involved in the interconversion EMi EM2 of those EM that belong to the points Pi and P2. This chemical metric on the EM of an FIEM provides not only a formalism for constitutional chemistry, but also allows us to use the properties of Euclidean spaces in expressing the logical structure of the FIEM, and thus of constitutional chemistry 3e>32c>. 

The NOESY spectrum relies on the Nuclear Overhauser Effect and shows which pairs of nuclei in a molecule are close together in space. The NOESY spectrum is very similar in appearance to a COSY spectrum. It is a symmetrical spectmm that has the Iff NMR spectmm of the substance as both of the chemical shift axes (Fi and F2). A schematic representation of NOESY spectmm is given below. Again, it is usual to plot a normal (one-dimensional) NMR spectmm along each of the axes to give reference spectra for the peaks that appear in the two-dimensional spectmm. 

The statistical approach to chemical kinetics was developed by Li et al. (2001, 2002), and high-dimensional model representations (HDMR) were proposed as efficient tools to provide a fully global statistical analysis of a model. The work of Feng et al. (2004) was focused on how the network properties are affected by random rate constant changes. The rate constants were transformed to a logarithmic scale to ensure an even distribution over the large space. 

Fig. 2.8. Three-dimensional representation of stationary-state locus, with the growth and development of the surrounding limit cycle, in a-b-p space for the pool chemical model.

Maehr, H. Graphic representation of configuration in two-dimensional space. Current conventions, clarifications and proposed extensions. Journal of Chemical Information and Computer Sciences 2002, 42, 884-902. 

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