Structure-Reactivity Maps as a Tool for Visualizing
Xenobiotic Structure-Reactivity Relationships

Eric M. Gifford
The University of Toledo, Department of Chemistry, Toledo, OH 43606

Mark A. Johnson
Upjohn Laboratories, Kalamazoo, MI 49001

Douglas A. Smith
DASGroup Inc., Johnstown, PA 15905 and

Chun-che Tsai
Kent State University, Department of Chemistry, Kent, OH 44242


http://www.netsci.org/Science/Special/feature04.html

Introduction

Since biotransformation may profoundly affect the bioavailabilty and toxicity of a compound, the effects of probable biotransformations are now considered in the early stages of computer-aided drug discovery. Incorporation of biotransformation information into the drug-discovery process requires both experimental measurement and predictive methods.

Given the myriad of complex factors influencing drug distribution and metabolism, experimental measurement continues to be the only reliable method for learning the metabolic fate of a compound in any suitably restricted class of organisms. However, given the expense of metabolic studies, both in time and resources, it may be necessary to make predictive judgments as to likely sites of metabolic activity in the absence of experimental evidence on the relevant compound. Such judgments are often incorporated into toxicity predictions in both pharmacological and environmental toxicology, synthesis programs in the pharmaceutical industry, and in the choice of labeling sites and reference compounds in metabolic research. This review of our work describes our efforts toward formulating computational chemistry based models, largely reliant upon molecular similarity arguments, for deriving such judgments. We also propose statistical methods for reliably assessing the predictive capability of those judgments.

The metabolic predictive problem is difficult for many reasons. For example, suppose that two compounds can undergo the same metabolic transformation under the same linear kinetics rate constant. If these compounds have different elimination rate constants, (again under linear kinetics), the amount of metabolite formed in the two cases can differ dramatically. Unless one adjusts out the effect of elimination rate on the amount of metabolite formation, one is likely to associate structural differences in these two compounds with differences in the metabolic rate constants rather than with differences in their elimination rate constants. Thus, any method of predicting absolute metabolic rates or the occurrence or non-occurrence of a particular metabolic transformation is bound to be unreliable unless the elimination rate of the intact parent compound has been appropriately taken into account.

Under linear kinetics the rate at which the intact parent compound has been eliminated can be adjusted out by studying the relative occurrence of two metabolic reactions rather than the absolute occurrence of one metabolic reaction. It can be shown that, under linear kinetics, the ratio of the amounts formed of two direct metabolites of a parent compound is independent of the elimination rate of the parent compound. An example is shown in Figure 1.

Figure 1 Calculated distribution of metabolic products assuming various linear rate constants.

Suppose compound P1 is converted to metabolite M1 at a linear rate constant 0.5/hr (k1) and is eliminated from the system at a linear rate constant 1.0/hr ( ). The amount of M1 produced would be 33.3% of the total amount of compound P1 metabolized and/or eliminated. Now suppose compound P2 is converted to metabolite M2 at the same linear rate constant of 0.5/hr (k2) but is eliminated from the system at a linear rate constant 8.0/hr ( ). In this case, the amount of M2 produced would be only 5.9% of the total amount of compound P2 metabolized and/or eliminated. Obviously, the amounts produced of each of these two metabolites is dependent on the overall rate of elimination, . If the difference in elimination rates of the parent compounds is not taken into account, it would appear that P2 was not a good substrate for the enzyme responsible for the biotransformation, when in fact, it is just as good as P1 as judged by their common rate constants. P2 is simply eliminated faster than P1 and does not get the same chance to react.

An examination of the relative rates of two competing reactions adjusts out the elimination rates of the parent compounds. This enables the prediction of the relative occurrence of two reactions and may eventually lead to a ranking of the relative rates of many biotransformations.

To illustrate, consider two compounds, P3 and P4, both of which undergo the same metabolic transformation at the same linear rate constant (ki = 0.5/hr i=3,4) to metabolites M3 and M4 respectively. Compounds P3 and P4 also each have other possible reaction sites. So, in addition to forming metabolites M3 and M4, P3 and P4 may also undergo another common transformation to metabolites M'3 and M'4. In this example it is assumed that this metabolic transformation occurs with a linear rate constant k'i = 0.2/hr (I=3,4).

Compound P3 is eliminated at a linear kinetic rate constant, = 1.0/hr, is converted to M3 with k3 = 0.5, and is converted to M'3 with k'3 = 0.2. Given these rate constants it is calculated that metabolites M3 and M'3 comprise 29.4% and 11.8%, respectively, of the total amount of compound P3 metabolized and/or eliminated. The ratio of these two metabolites (M3/M'3) is 2.5.

Compound P4, on the other hand, is eliminated at a linear kinetic rate constant, = 8.0/hr, is converted to M4 with k4 = 0.5, and is converted to M'4 with k'4 = 0.2. Given these rate constants it is calculated that metabolites M4 and M'4 comprise only 5.7% and 2.3%, respectively, of the total amount of compound P4 metabolized and/or eliminated. Thus, for compound P4, the ratio of these two metabolites (M4/M'4) is 2.5. This is the same ratio as that found for compound P3 even though the rate of elimination of compound P4 was eight times faster than that of compound P3.

The rate of elimination of the parent compounds, which is often not available and definitely complicates any prediction of the absolute rates or absolute occurrences of a metabolic transformation, is effectively adjusted out by examining the relative rates of two competing reactions. If only the first of two reaction products is observed it can be assumed that the second product was not observed, not because the parent compound was eliminated before it had the opportunity to react, but because either the first reaction was faster or the chemical structural environment of the second reaction site of the parent compound is not compatible with the active site of the enzyme.

Based on these arguments we have restricted our studies to the development of methods for the prediction of the relative occurrences of metabolic transformation. It must be noted that easily accessible biotransformation rate data sets are extremely scarce. However, biotransformation occurrence data sets are readily available. These biotransformation occurrence data sets are the underlying source of information for most computer-based approaches to metabolic pathway prediction.

Computer-based approaches to metabolic prediction are being investigated by many workers [1,2,3,4,5,6,7]. Most of these computer-based approaches utilize databases containing information on the biotransformation pathways of xenobiotics. These databases, whether machine readable [2,3,4,8] or hard-copy documents [9], provide information necessary to develop the rules used for predicting xenobiotic metabolism. The predictive ability of computer-based methods for suggesting potential xenobiotic biotransformations will increase as further improvements in determining structure-reactivity relationships are made.

Metabolic pathways are represented as a series of pair-wise connected molecules. An arrow between the two molecules indicates that a biotransformation from the reactant to the product has been observed. Reactivity, defined in a simple sense as the observed occurrence or the non-occurrence of a given reaction, may be thought of as a molecular activity. By defining reactivity as a molecular activity, the data analysis techniques which are used in structure-activity relationship (SAR) studies become applicable. The following article reviews our efforts in developing computational structure-activity relationships in xenobiotic metabolism.

In order to determine structure-activity relationships it is usual to convert the chemical structural formula of a molecule to an appropriate SAR representation or descriptor. The selection of the representation is not always a simple matter and, even if a suitable representation is found, a method for effectively visualizing the relationships between these representations is needed. The representations developed in the following study, called "n-level reaction site representations", include information about the chemical structural environment of a reaction site on a molecule.

With a reaction site representation in hand, the next step is to visually inspect the data for structure-activity relationships. Although it is possible to simply list data in tables along with the associated chemical structures or representations, the tabular format does not necessarily indicate the relationships between the entries in the table. Of course the data table may be arranged in a defined order, for example alphabetical lists of names or increasing values of a property, but even with the order defined it may still be difficult, if not impossible, to show any relationship between the chemical structures and the experimental data. A means of presenting data in a manner which facilitates the discovery of structure activity-relationships is needed.

The application of molecular similarity concepts to determine structure-activity relationships [9] is well documented. Structure-activity relationships, which may be difficult to establish from data tables, should become apparent once suitable maps of the structure-activity space have been constructed. The notion that similar compounds have similar properties can also be applied to reactivity. An analysis of the similarities between potential reaction sites of molecules which undergo a given reaction may indicate fundamental structural requirements necessary for this reaction to occur.

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General Concepts of Structure-Activity Maps

An intuitive idea of a structure-activity map is motivated by intercity distance maps. An example of a distance matrix underlying an intercity distance map is given in Table 1 and the corresponding intercity distance map is shown in Figure 2. These maps (see Figure 1), which are graphical representations of a portion of the information contained in an intercity distance matrix (see Table 1), provide a concise representation of the distance relationships between cities. It is interesting to note that an edge is not drawn from each city to all of the other cities contained in the distance matrix even though this information is available. It is assumed that the minimum distance edges are the most relevant. Just as an intercity distance map helps us visualize the information in an intercity distance matrix, structure-reactivity maps help us visualize the information in a similarity-based structure-reactivity relationship.

The construction of an intercity distance map requires a distance matrix giving the pair-wise distances between the cities. From this intercity distance matrix, one can construct a graph whose vertices are cities and whose edges are labeled with the intercity distances. Here one is not entirely sure as to why some pair-wise distances are represented by edges in the intercity distance map and others are not represented. However, one certainly suspects that the edge representing the distance between Syracuse and Boston was omitted because Albany lies in between these two cities.

Albany Bangor Boston Buffal New Yo Philad Portlnd Scrant Syrac Watertwn
Albany 0 405 165 283 157 249 272 179 132 204
Bangor 405 0 240 688 451 543 133 581 537 609
Boston 165 240 0 448 211 303 107 341 297 369
Buffalo 283 688 448 0 418 418 555 288 151 223
New York 157 451 211 418 0 92 318 130 267 339
Philadelphia 249 543 303 418 92 0 410 130 267 339
Portland 272 133 107 555 318 410 0 448 404 476
Scranton 179 581 341 288 130 130 448 0 137 209
Syracuse 132 537 297 151 267 267 404 137 0 72
Watertown 204 609 369 223 339 339 476 209 72 0


Table 1 An example of an intercity distance matrix


Figure 2 An example of an intercity distance map.


A structure map bears many analogies with an intercity distance map. In both cases, the underlying graph is called a proximity graph. In a structure map, the vertices of the proximity graph are molecular structures rather than cities, and the edges denote pair-wise molecular similarities, distances or dissimilarities. The many molecular similarity indices that might be used to generate the underlying pair-wise distance or proximity matrix are reviewed by M.A. Johnson [11]. Moreover, the proximity graph comes in a variety of forms such as nearest neighbor graphs, minimum spanning trees, relative neighborhood graphs [12], Gabriel graphs [13], and sphere of influence graphs [14]. In our development of a structure map, we use a similarity measure based on maximum common substructures [15] and a proximity graph closely related to the relative neighbor proximity graph [15]. However, the concept of a structure map is general. It can be based on any method of computing the underlying proximity graph and the pair-wise matrix of molecular similarities from which that graph is derived.

The structure map seems to be an efficient way to visualize the increasing amount of information contained in chemical activity databases. The structure map of a set of compounds in conjunction with a given activity of these compounds constitute a "map" of the chemical structure-activity space. The collection of these structure-activity maps for an entire database constitute an "atlas" for the chemist interested in chemical structure-activity relationships. The task of predicting properties of new compounds is facilitated with a suitable set of structure-activity maps. The activity may be predicted either directly from the structure-activity map, using local averaging methods such as nearest neighbor predictors, or by using classical QSAR methods employing descriptors suggested by visual inspection of the structure-activity map.

In the following section, a structure-activity map will be developed in the area of xenobiotic metabolism [16,17]. The activity which will be examined is a chemical reactivity, the relative occurrence of N-demethylation and N-oxidation biotransformations. A structure-activity map also can be developed to examine the relative occurrence of N-demethylation and non-methyl N-dealkylation. Such maps should prove useful in the design of new drugs. The predictive success of a structure-reactivity map depends on the underlying structure-activity similarity principle which, in this case, states that compounds with structurally similar reaction sites generally have similar reactivities. It should be apparent from a structure-activity map which relatively small structural changes are responsible for dramatic changes in reactivity.

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Methodology

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The metabolism of a tertiary amine may result in N-dealkyl and/or N-oxide products [18]. According to Ziegler [19], the N-oxides of tertiary aliphatic amines are often less toxic and pharmacologically less active than the parent amine. Several structure-reactivity relationships for N-dealkylation as well as for N-oxidation have been proposed. Among the parameters used for relating the rates of N-dealkylation to chemical structure are lipophilicity, the pKa of the N-alkyl nitrogen, and steric descriptors [20,21,22,23,24,25,26,27,28]. Structure-reactivity relationships between some of these same parameters and the N-oxidation of tertiary amines have also been investigated. These studies have either focused on the rates of dealkylation with respect to a structural descriptor(s) [29,30] or the differences in enzymology of N-oxidation based on the pKa of the nitrogen which is being oxidized [31,32]. Gorrod [32,33] divides all amines into three classes based on the pKa of the parent amine. Basic amines, pKa between 8 and 11, are primarily oxidized by a flavin-containing monooxygenase [20]. Non-basic amines, pKa below 1, are primarily oxidized by the cytochrome P450 system. Intermediate amines are oxidized by both enzyme systems. It is also suggested that the most basic nitrogen atom in a compound is the one to be oxidized [33].

In this study we examine structure-reactivity relationships of N-methyl tertiary amines to determine the dominant correlative variables influencing N-demethylation and N-oxidation. Our approach to modeling the relative occurrence of these two reactions is based on a top down statistical modeling method which relies on representative sampling of many reaction sites followed by statistical modeling of this data. A top down modeling approach should provide information about the dominant structural features influencing the reactivity of a class of compounds. This is in contrast to a bottom up mechanistic modeling method, in which the mechanistic detailed pharmacokinetics of the biotransformation of one compound is examined [32]. Our study is confined to structurally-based predictions which are readily computed from a compound's chemical graph.

For this study, we have examined the relative occurrence of N-demethylation and N-oxidation of N-methyl tertiary amines (see Figure 3). We have chosen these two competitive reactions because they involve the simplifying feature of having identical reaction sites, and because they occur frequently in drug design problems.

Figure 3. The N-demethylation and N-oxidation reactions.


The compounds in the study were obtained from volumes 1 and 2 of D.R. Hawkins' Biotransformations [9]. This is a survey of recently documented biotransformations in animals. The N-demethylation/N-oxidation data set is a subset of the biotransformation pathways contained in these volumes. The criteria for a parent compound's inclusion in the N-demethylation/N-oxidation data set are as follows:

  1. The parent compound must be N-demethylated or N-oxidized at the nitrogen of an N-methyl tertiary amine.
  2. The study referenced must have been conducted in one of, the following four test systems: human in vivo, human microsomes, rat in vivo, rat microsomes.

The first criterion was set to investigate the prediction of relative occurrences of these biotransformations. This criterion enables one to study the relative occurrence of a pair of reactions, but not the absolute occurrence of a single reaction. Primary and secondary amines are excluded from the analysis since N-oxide products are not observed for these substrates.

The second criterion was set to investigate the correlation between animal and human models from the microsomal level to the in vivo level. Both of the above criteria must be met for a compound to be included in our data set. The biotransformation pathways selected for inclusion to our data set are then further analyzed to determine the methods employed to isolate and/or characterize metabolic products. Each observation in our data set corresponds to an N-methyl reaction site in a documented metabolic transformation pathway study [9], not necessarily a unique compound.

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Construction of the Structure-Reactivity Map [34]

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There are four basic steps in constructing a structure-reactivity map. First, a suitable structure representation is defined. Second, a similarity distance between each of these representations is calculated. Third, the reactivity space is mapped onto a planar surface so as to effectively visualize inter-structure distance relationships. Finally, the structure map is labeled to indicate properties associated with each structure representation.

1. Reaction Site Representations

A structure-reactivity map may be constructed once a representation for a chemical structure or, as in this study, a chemical transformation site has been chosen. The choice of chemical structure representations is dependent on several factors including, but not limited to, the ability of the representation to be easily manipulated by a computer, the ease with which a chemical structure can be converted to a representation and then back to a chemical structure, and finally a chemical rationale for choosing the important structural features to include in the representation. Each N-methyl reaction site in the data set is converted to a chemical reaction site representation. Thus a molecule may have multiple reaction site representations associated with it; one for each N-methyl site which is N-demethylated and/or N-oxidized. Both the N-demethylation and N-oxidation reactions may be represented as having the same reaction center, namely the N-methyl nitrogen atom. This representation describes the overall chemical structural change not necessarily those atoms involved in an actual reaction mechanism. Our N-dealkylation/N-oxidation reaction site representations include the reaction site (the methyl substituted nitrogen atom), surrounding atoms, and a steric bulk descriptor for the remainder of the molecule. This type of representation follows from our previous work in this area [35,36] and is similar to the "focus" concept in the DARC system [37].

Specifically, the reaction site of a molecule is described by an n-level representation. This type of representation is defined by labeling the reaction site of a molecule. The atoms are numbered according to their bond distance away from the reaction center (see Figure 4). Atoms and bonds with a distance greater than n are then replaced with graph-theoretic steric bulk descriptors. Although the 1-level and 2-level reaction site representations were investigated in preliminary studies, the reaction site representation chosen for this analysis is a 3-level reaction site representation. It has been observed that a 1-level reaction site representation is useful for differentiating primary, secondary, and tertiary amines. But since our data set contains only tertiary amines, separation between reaction sites is achievable only by the differences in bulk descriptors. The 2-level reaction site representation contains, in addition to all of the information contained in the 1-level reaction site representation, information about the hybridization of the atoms alpha to the reaction center along with the atom type of the beta atoms. Although the 2-level reaction site representation yields interesting results in the analysis of the N-demethylation/N-oxidation data set, more satisfying results were obtained using the 3-level reaction site representation. The 3-level representation contains the reaction center (the methyl substituted nitrogen atom) and all non-hydrogen atoms connected by a bond path distance of less than or equal to 3 bonds. This 3-level representation contains electronic information about the reaction site encoded in the atom labels and bond types.

These 3-level representations are augmented with a bulk descriptor to describe the size of that portion of the molecule which is not directly included in the 3-level representation. A series of "bullet" atoms (solid circles) was added to each of the terminal atoms of the representation. The number of bullet atoms indicates the number of atoms in the chemical structure which are not included in the 3-level representation. A range of atoms was defined as follows (see Figure 4): If between 1 and 3 atoms are deleted, a single bullet atom is added. If between 4 and 9 atoms are deleted, two bullet atoms are added. If between 10 and 27 atoms are deleted, three bullet atoms are added. And if over 27 atoms are deleted, four bullet atoms are added. The bullet atoms are a crude approximation of the position and steric bulk of that portion of the molecule not explicitly contained in the n-level representation. The reaction sites on non-cyclic molecules can be effectively described in this straight forward manner. An example is provided in Figure 4.

Figure 4. The reaction site is identified and labeled "­". Each atom is then labeled based on its bond distance from this reaction center. Each n-level is then constructed by deleting atoms with labels greater than n. The deleted atoms are then represented by bullet atoms. The number of bullet atoms added is based on the number of deleted atoms and the above scale. If 1 to 3 atoms are deleted from the structure to obtain the n-level representation then 1 bullet atom is added to the n-level atom of the representation. If 4 to 9 atoms are deleted from the structure to obtain the n-level representation then 2 bullet atoms are added. If 10 to 27 atoms are deleted then 3 bullet atoms are added and if more than 27 atoms are deleted then four bullet atoms are added. The bullet atoms are a crude approximation of the position and steric bulk of that portion of the molecule not explicitly contained in the n-level representation.

Reaction sites on cyclic molecules require a method for representing the positioning about the ring system of the low level atoms and the steric bulk of the high level atoms. The approach used is illustrated in Figure 5. The reaction site is determined, the N-methyl nitrogen atom, and the n-level representation is constructed with the following modifications. The bullet atoms are not added until the ring closure has been indicated. If the terminal atoms of the n-level representation are part of a ring system then they are connected via bonds to a bullet atom, this maintains the cyclic nature of the graph. The ring closure is indicated by dashed bonds from the terminal atoms of the n-level representation to the ring closure bullet atom. The second step accounts for those atoms not included in the newly formed representation. The size and location of the remaining portion of the molecule is represented by bullet atoms. The remaining atoms are assigned to either a terminal atom or the ring closure bullet based on the shortest bond path distance. The range of atoms not in the n-level representation is shown with the same bullet atom scale previously described. If a terminal atom, at the n-level ring closure step, has two atoms as its nearest neighbors, then that terminal atom is bonded to a single bullet atom. Nearest neighbor atoms are those atoms for which the bond path distance between the terminal atom and the neighbor atom(s) is a minimum. Each atom not explicitly contained in the n-level representation is assigned to the nearest terminal or ring closure bullet atom based on bond path distance. It is possible to describe where on the ring system the remaining portion of the molecule is located by assigning bullet atoms to terminal or ring closure bullet atoms.

Figure 5. Step 1: The reaction center is identified and labeled "*". Each atom is then labeled based on its bond distance from the reaction center. Each n-level is then constructed by deleting atoms and bonds between atoms with labels greater than n. If the n-level atoms were part of a ring they are connected by bonds to a ring closure bullet atom. Step 2: The deleted atoms are then represented by bullet atoms based on the previously defined scale. The arrangement of the bullet atoms depends on the location of the deleted atoms. Each deleted atom is assigned to is nearest neighbor n-level atom. The sum of the deleted atoms assigned to each n-level determines the number of bullet atoms assigned. Atoms which are equidistant from each n-level atom are assigned to the ring closure bullet atom.

An example of the construction of the n-level representations of N-methyl piperidine, assuming the nitrogen atom is the reaction site, is shown in Figure 5. Atoms which are equidistant from both terminal atoms are assigned to the ring closure bullet atom. Figure 6 illustrates the possible n-level degeneracies obtained from different structures.

Figure 6. The n-level representations are constructed for these simple 5 and 6 member ring systems.


2. Computing Distances Between Reaction Site Representations

The distances between each pair of reaction site representations of the data set were determined as follows. Each N-methyl reaction site was converted to it's 3-level representation. For example, for the nitrogen reaction site, the nitrogen is labeled N*, the alpha-carbons are labeled C1, the beta-carbons are labeled C2, and the gamma-carbons are labeled C3 and so on. The representations of the reaction site retain these labels throughout the analysis. The number of non-superimposible bonds between each pair of representations is computed. This number is called the edge deletion distance as it is the smallest number of bonds that can be deleted so as to obtain a common substructure of the two representations. During this superpositioning, atom and bond labels are preserved. A C1 will only be matched with another C1. A double bond will only be matched with another double bond.

Graphs (molecular structures or 3-level representations) with an edge deletion distance of zero are identical, while an increase in edge deletion distance indicates a decrease in similarity between the graphs. For example, reaction sites 4, 5, 23, 29, 33, 34, 35, and 36 are all represented by the same 3-level representation (a) as shown in Figure 7. Their edge deletion distance from each other is zero. The representation for reaction site 42 (b) is shown in Figure 7. The distance between these two representations is determined by counting the number of non-superimposible bonds between them (the circled bonds in Figure 8).

Figure 7. The N-methyl reaction sites for reaction sites 4, 5, 23, 29, 33, 34, 35, and 36 are all represented by the non-cyclic 3-level representation (a) shown in the figure. The reaction site number indicates a documented N-demethylation/N-oxidation biotransformation site, not necessarily a unique chemical structure. The N-methyl reaction site for each of these observations is represented by the same non-cyclic 3-level descriptor.


Figure 8. The edge deletion distance (equal to 2) between the 3-level representations (a) and (b) from Figure 7 is determined from the number of non-superimposible bonds between the two representations. The circled bonds in (b) of this figure indicate these non-superimposible bonds.

3. Mapping the Space of Reaction Sites

Similarity analysis of reaction site representations and their associated reactions requires the computation of a proximity measure for each pair of representations. One such measure is the edge deletion distance. The edge-deletion distance can be shown to be a metric on any set of nonisomorphic, connected graphs [38]. It is the smallest number of bonds that can be deleted so as to obtain a common subgraph of two graphs. An example of an edge deletion distance calculation is given in Figure 8.

3-level representations with an edge deletion distance of zero are identical, while an increase in edge deletion distance indicates a decrease in similarity between the graphs. For example, reaction sites 5, 23, 33, and 34 are all represented by the same 3-level representation as shown in Figure 7. Their edge deletion distance from each other is zero. On the other hand, we see from Figure 8 that the distance between site representation 42 and representations 5, 33, 34, and 23 is two.

The relationships between 3-level representations can be represented by a metagraph in which the vertices are the 3-level representations and the labeled edges indicate the edge deletion distances between representations. The complete graph quickly becomes very complicated. The number of edges of a complete graph is m(m-1)/2, where m is the number of vertices. We are primarily interested in edges which represent the more interesting structural changes as defined by the nearest neighbor proximity graph. The proximity graph is obtained using the triangle inequality. Each triangle, a complete graph with three vertices, is selected from the complete proximity graph. Each edge is labeled with the edge deletion distance between the two adjacent vertices. An edge is marked if the other two edge labels are of lower value. After all triangles have been examined, all edges receiving a mark are deleted. (See Figure 9) The remaining edges largely represent those smaller structural differences into which the larger structural changes can be decomposed. It is this decomposition of complex structural changes into simpler structural changes that makes the edges of the proximity graph interesting.

Figure 9. This figure illustrates the construction of a structure map for a portion of the data set based on the method described in the text. The complete graph is obtained for the data set (a). Each vertex is a 3-level representation and each edge is labeled with the edge deletion distance between vertices. The complete graph is then analyzed using the triangle inequality (b). Each triangle of the complete graph is examined and the longest edge of the triangle is deleted. The deleted edges are indicated by an "X". Once an edge is delete that edge is removed from the complete graph. The remaining edges are then redrawn in the form of the structure map (c). The resulting structure map links the 3-level representations (vertices) by the minimum edge deletion distance edges of the data set.

Minimum spanning trees and relative neighborhood graphs are examples of proximity graphs [39]. The edges of the proximity graph largely represent those smaller structural differences into which the larger structural changes can be decomposed. It is this decomposition of complex structural changes into simpler structural changes that makes the edges of the proximity graph interesting.

By itself, the proximity graph is largely an analytical tool. It is only when this graph is aesthetically drawn out on a sheet of paper that it becomes a powerful visual tool. This embedding of the vertices of a graph on a plane is called a mapping.

A usefully arranged proximity graph with its vertices replaced by the corresponding structural representations, regardless of how this embedding was obtained, will be called a structure map or, more specifically for this case, a 3-level reaction-site map.

Multidimensional scaling [40] provides useful mappings that emphasize global distances and relationships [41]. Mappings that minimize the number of edge crossings clarify and emphasize local relationships. The principle that similar structures tend to have similar properties is founded on local relationships. As a consequence, we seek an aesthetically pleasing map under the constraint of minimizing the number of edge crossings.

The resulting proximity graph may be drawn in any arrangement as long as the topology (connections between vertices) remains the same. Since the proximity graph is a topological representation it is important that the edges of the graph be easily recognized. In order to clearly see the edges of the graph edge crossings should be kept to a minimum. An aesthetically pleasing, e.g. a minimum of edge crossings, arrangement of the proximity graph in this study (Figure 10) was obtained using an interactive computer graphics program. One can use multidimensional scaling methods for arranging a proximity graph on a plane [41], or one can order the vertices according to property of their corresponding graphs. However, the purposes of those methods often conflicts with the aesthetic goal of minimizing the number of edge crossings.

The edge deletion distance between each vertex is shown as the edge label. The dashed lines indicate connections in which the edge deletion distance between representations is greater than 10. A small edge deletion distance indicates similar representations. A large edge deletion distance arises from a lack of intervening representations between two representations.

A usefully arranged proximity graph with its vertices replaced by the corresponding structural representations is called a structure map or, more specifically for this case, a 3-level site map. A structure map for the N-demethylation/N-oxidation data set is shown in Figure 10. The closed curves represent an aspect of relative reactivity that will be discussed later. Each vertex of the proximity graph in this example is a 3-level representation.

Since reaction sites on different molecules may have the same 3-level representation, each vertex may contain information about many compounds. Figure 11 is obtained by replacing the 3-level site representations in Figure 10 with the set of reaction sites having those respective representations. It will be shown that the structure map, constructed using the 3-level reaction site representations, effectively clusters the N-oxidation reaction sites. These N-oxidation reaction sites are found within the closed curves in Figure 10.

Figure 10. The structure map is constructed based on the 3-level representations of the N-demethylation/N-oxidation data set. Solid edges indicate relationships between representations linked by an edge deletion distance less than or equal to 10. Dashed edges indicate the link between representations differing by an edge deletion distance greater than 10. The reaction center of each 3-level representation is labeled with an "­". The edge deletion distance is shown as the edge label.

Figure 11. The structure map arranged as in Figure 10, but now displaying the actual structures included in the data set. The reaction sites are labeled with a "­".


4. Coloring a Reaction-Site Map

Just as the region surrounding a city in an inter-city distance map might be colored so as to indicate the altitude at that city, we can color a structure map so as to indicate the value of some property associated with the structures. In general, there are many properties that might be associated with the structures giving rise to many different colorings of a structure map. We can combine these different colorings using one of the many methods available for representing multivariate data [42].

We modify the usual presentation of "snowflake" representations of multivariate data [43] in two regards. We replace the snowflake by a regular polygon in which each sector represents one property or unit of the multivariate data. The shading of the sector represents the value of the corresponding property. For example, in Figure 12, the shading of the upper-right sector of a compound's hexagon indicates whether a value was available for that compound in an oral rat study, and if so, whether N-demethylation, N-oxidation, or both were observed. Figure 10, together with Figure 12 defines an equivalence structure-activity map, or more specifically an equivalence 3-level site-reactivity map.

Figure 12 is colored to indicate the observed reaction for each observation. It has been observed that there is a high proportion of N-oxidation reaction sites in some equivalence classes and not in others. This observation is exemplified by the cluster of reaction sites 4, 5, 23, 29, 33, 34, 35, and 36 (see Figure 10) all of which are N-oxidized. Moreover, visual inspection strongly suggests that similar reaction site representations have similar reaction products.

The 3-level site representations associated with the relative occurrence of N-oxidation are circled in Figure 10. Edges which cross these curves, for example the edge between site 22 and site 12 in the top center portion of the structure map, suggests N-oxidation is less likely to occur relative to N-demethylation in sterically hindered sites. This suggestion is supported by a large majority of other edges that cross the closed curves.

Figure 12 also provides information about which animal test system was used in the documented study. For example, it was noted that both N-oxidation and N-demethylation are observed in a rat and a human in vivo study for reaction site 9 (roughly the center of Figure 12). Comparisons between test systems and the observed products are possible using such colorings. In this particular instance, the structure-reactivity map can only show us that there are too few data points to support any definitive conclusions regarding the relationships between animal models and human metabolism.

The mode of administration of a xenobiotic may also influence the metabolic fate of that compound. This attribute of the documented metabolic pathway study is also easily included in the reactivity map. It has been observed that only N-oxide products (or no N-demethylation products) are formed in those studies in which the compound was administered intravenously. (see reaction sites 4, 13, and 23).

The structure map is also very useful for summarizing and displaying the multiplicity of analytical methods used in each study. The structure map in Figure 13 has been colored to indicate the analytical methods employed in determining the N-demethylation/N-oxidation products in our data set.

Figure 12. The test system structure-reactivity map for the N-demethylation/N-oxidation data set. Each sector of the hexagon vertex of the structure-reactivity map is shaded to indicate which test system was employed for each reaction site in the data set.


Figure 13. The experimental method structure-reactivity map for the N-demethylation/N-oxidation data set. Each section of the hexagon vertex of the proximity graph is shaded according to the experimental techniques used to determine the metabolic products for each observation in the data.

Five techniques of metabolic product determination were commonly employed, mass spectrometry (MS), nuclear magnetic resonance (NMR), ultraviolet spectroscopy (UV) or infra-red spectroscopy (IR), thin layer chromatography (TLC) or high performance liquid chromatography (HPLC), and gas chromatography (GC). It was also of interest as to whether the N-oxide reference compound was synthesized. Since these six parameters were relevant to the analysis, a hexagon was chosen as the label for each observation.

Each sector of the polygon corresponds to one of the parameters. If a technique was employed in the documented observation, then the corresponding sector of the polygon is shaded. The shading may be a simple yes or no, shaded or not shaded, as in the NMR, UV/IR, TLC/HPLC, GC, or reference N-oxide sectors. The sector may also contain more detailed information in the form of a gradient of shading indicating variations of the same technique as illustrated in the MS sector of the vertex labels of Figure 13. The shading in this block is used to distinguish between electron impact (EI), chemical ionization (CI), and fast ion bombardment (FAB) mass spectroscopy. In Figure 13, a completely filled hexagon indicates a study which included all of the indicated techniques (e.g. observation 38) while a hexagon such as observation 18 indicates a study in which only one technique (in this case, liquid chromatography) was employed for metabolite characterization.

The choice of analytical methods used to characterize metabolites depends on the metabolite(s) which interest an investigator. If several techniques were employed in a study one would expect that the chances of observing even minor metabolites should increase. If a study indicates that a reference compound of a specific metabolite has been included in the study one has greater confidence that the investigators were actually looking for that compound. Figure 13 indicates that in all but two cases (sites 8 and 39) an N-oxide product was observed when the N-oxide reference compound was available in the study.

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Visual Analysis and QSAR Model

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The various colorings of the structure-activity map are useful for determining relationships between many aspects of xenobiotic metabolism data. A quantitative structure-activity model of the relative occurrence of N-demethylation and N-oxidation from an analysis of Figure 10 and Figure 11 as been developed. But first a brief analysis of the experimental conditions coloring (Figure 12) and the analytical methods coloring (Figure 13) of the N-demethylation/N-oxidation structure-activity maps is presented.

1. Experimental Conditions Coloring

Comparisons between test systems and the observed products are possible using an experimental conditions coloring (Figure 12). Given enough data it may be possible to determine similarities between animal models and human metabolism based on structure maps of this type. Unfortunately, in this particular instance, the structure-reactivity map can only show us that there are too few data points to support any definitive conclusions regarding the relationships between animal models and human metabolism.

The experimental conditions coloring also indicates the mode of administration of a xenobiotic. This aspect of the metabolic study also influences the metabolic fate of that compound. For example, when only N-oxide products (or no N-demethylation products) are formed, the compound was always administered by IV, thus circumventing first pass metabolism by the liver (see reaction sites 4, 13, and 23).

2. Analytical Methods Coloring

The analytical methods coloring is useful for evaluating each documented study. The choice of analytical methods used to characterize metabolites depends on the metabolite(s) which interest an investigator. If several techniques were employed in a study one would expect that the chances of observing even minor metabolites should increase. It is also of interest as to whether the N-oxide reference compound was synthesized. First it suggests the metabolic focus of the investigator, and second, there is an apparent correlation between the relative occurrence of N-oxidation and the use of a reference N-oxide in the experimental metabolism studies of our data. It can be seen from Figure 13 that in all but two cases (sites 8 and 39) an N-oxide product is observed when the N-oxide reference compound was available in the study.

3. N-Oxidation Structure-Reactivity Relationships

The most fundamental observation to be made from Figure 10 is that the N-demethylation product is observed for all but three sites (4, 13 and 23, IV route of administration) in this data set. This suggests that if one observes N-oxidation at a N-methyl site one would predict N-demethylation to also occur.

Examining Figure 12 it becomes apparent that there is a high proportion of N-oxidation reaction sites in some equivalence classes and not in others. This observation is exemplified by the cluster of reaction sites 4, 5, 23, 29, 33, 34, 35, and 36 (see Figure 12) all of which are N-oxidized. Moreover, visual inspection of Figure 10 strongly suggests that similar reaction site representations have similar reaction products. This suggests that the dominant structural features affecting the relative occurrence of N-demethylation and N-oxidation are accounted for with the 3-level reaction site representation.

To simplify the analysis, if an N-oxide product was observed for any reaction site within an equivalence class, then that reaction site representation is associated with N-oxidation. The 3-level site representations associated with the relative occurrence of N-oxidation are circled in Figure 10. Edges which cross these curves, for example the edge between site 22 and site 12 in the top center portion of the structure map, indicate important structural features which influence the relative occurrence of the N-oxide product.

In general it seems that the N-methyl reaction sites which undergo N-oxidation are less sterically hindered than those sites which do not undergo N-oxidation. This suggestion is supported by the structural changes reflected in edges that cross the closed curves of Figure 10. For example, the edge connecting site 22 at the top of Figure 10 to site 12 defines a change from a relatively unhindered N-methyl reaction site to a more sterically hindered N-methyl reaction site. This suggestion is supported by a large majority of other edges that cross the closed curves. This steric effect can also be seen by comparing those reaction site representations within the N-oxidation areas with those reaction site representations outside these areas. These general observations will now be incorporated into a QSAR model based on a single descriptor.

4. QSAR Model

The local steric environment of the 3-level reaction site representations will be described by a very simple steric index. The number of anti, rotational, and syn interactions with each N-methyl site was counted. Several examples are given in Figure 14.

Anti interactions occur when the N-methyl carbon atom and the 2-level atoms are either anticlinical or antiperiplanar as defined by Johnson et al. [44]. Syn interactions arise from synclinical or synperiplanar [45] arrangements of the N-methyl carbon and the 2-level atoms. Rotational interactions arise from 2-level atoms which may assume both syn and anti conformations. Each anti interaction is assigned a value of 0, representing no steric hindrance, each rotational interaction is assigned a value of 1, representing moderate steric hindrance and each syn interaction is assigned a value of 2, indicating significant steric hindrance. The steric index of the 3-level reaction site representation is the sum of each of the values of each of these interactions.

Figure 14. A syn-stericity index which indicates the steric hindrance of the N-methyl reaction site. Each syn interaction is worth 2, each rotation interaction is worth 1, and each anti interaction is worth 0. The sum of all these interactions for a 3-level reaction site representation is its steric index.

The steric index for each reaction site is given in Table 2. The occurrence and non-occurrence of the N-oxide is plotted against the value of the site index in the upper plot of Figure 15.

Figure 15. Plots of the occurrence of the N-oxide versus the steric index for the N-demethylation/N-oxidation data set (top plot) and an independent data set of N-methyl tertiary amines (lower plot).


Site Steric Index (SI) N-Oxide Observed Predicted Probability
3 0 0 0.84972
4 1 1 0.60161
5 1 1 0.60161
6 1 1 0.60161
7 1 0 0.60161
8 1 0 0.60161
9 2 1 0.28739
10 3 0 0.09723
11 3 0 0.09723
12 2 0 0.28739
13 0 1 0.84972
14 2 1 0.28739
15 1 0 0.60161
16 1 0 0.60161
17 4 0 0.02796
18 2 0 0.28739
19 4 0 0.02796
20 4 0 0.02796
21 2 0 0.28739
22 0 1 0.84972
23 1 1 0.60161
24 4 0 0.02796
25 1 0 0.60161
26 3 0 0.09723
27 2 0 0.28739
28 1 1 0.60161
29 1 1 0.60161
30 0 1 0.84972
32 2 0 0.28739
33 1 1 0.60161
34 1 1 0.60161
35 1 1 0.60161
36 1 1 0.60161
37 1 1 0.60161
38 1 1 0.60161
39 1 0 0.60161
40 1 1 0.60161
41 0 0 0.84972
42 0 1 0.84972
43 4 0 0.02796
44 4 0 0.02796


Table 2. The steric index, observed N-oxidation (yes = 1, no = 0), and the predicted probability of N-oxidation from equation 2 for the N-demethylation/N-oxidation data set (Biotransformations, Vols. 1 & 2) [9]


Logistic regression [45] was used to investigate the relationship between the probability of observing N-oxidation and the steric index. The linear logistic model has the form:

logit(p) = log(p / (1-p) ) = alpha + beta'x ..... Equation 1.

where p is the probability of observing the N-oxide given the occurrence of N-demethylation, alpha is the intercept parameter, and beta is the vector of slope parameters. The following equation was obtained:

logit(p) = 1.73244 (+/-0.7011) - 1.3203 (+/-0.4808)SI ..... Equation 2.


        n = 41       r = 0.7390       P**2 = 7.5418       sig. = 0.0060

here SI is the steric index, n is the number of reaction sites, r is the rank correlation coefficient, P**2 is the statistic for testing the need for SI in the model and chance of observing a higher r value if SI was not related to relative activity. The line in the upper plot of Figure 15 is the probability of observing the N-oxide versus the steric index. The probability of observing the N-oxide is obtained from Equation 2.

The model, Equation 2, was used to predict the relative occurrence of N-oxidation in an independent N-methyl tertiary amine data set obtained from D.R. Hawkins' Biotransformations Vol. 3 [9]. The N-methyl reaction sites included in this testing data set were selected using the same criteria as previously described. The structures and corresponding 3-level reaction site representations are shown in Figure 16. If an N-oxide product is observed it is marked "yes" in this figure. The probability of observing the N-oxide product, obtained from Equation 2 is also indicated. Correct predictions are marked "C" and incorrect predictions are marked "X". The assignment of correct and incorrect is based on a 0.5 probability cut-off. For probability greater than 0.5 N-oxidation is predicted; for probability less than 0.5 N-oxidation is not predicted. The steric index, observed product, and predicted probability for each reaction site is listed in Table 3. The occurrence and non-occurrence of the N-oxide is plotted against the steric index in the lower plot of Figure 15. The line in this plot indicates the predicted probability of observing the N-oxide versus the steric index.

Site Steric Index (SI) N-Oxide Observed Predicted Probability
106 1 1 0.60161
131 1 0 0.60161
202 2 0 0.28739
204 2 1 0.28739
221_1 4 0 0.02796
224 1 1 0.60161
287 0 1 0.84972
292 4 0 0.02796
303 0 0 0.84972
304 0 1 0.84972
331 4 0 0.02796
343_1 4 0 0.02796
343_2 2 0 0.28739
344_1 4 0 0.02796
344_2 4 0 0.02796
344_3 2 0 0.28739
360 4 0 0.02796
369 4 0 0.02796
379 2 0 0.28739


Table 3. The steric index, observed N-oxidation (yes = 1, no = 0), and the predicted probability of N-oxidation from Equation 2) for an independent (Biotransformations, Vol 3.) [9] N-demethylation/N-oxidation data set.


Figure 16. The data set of N-methyl tertiary amines from Biotransformations Vol. 3 [9]. Reported N-oxides are marked "yes", the predicted probability (from the model) and the results (correct "check" or incorrect "X") are also indicated.


From Figure 16 it is seen that only three reaction sites are incorrectly predicted. Sites 131 and 303 are predicted to undergo N-oxidation, but the N-oxide product is not reported in Biotransformations Vol. 3 [9]. Site 204 is not predicted to undergo N-oxidation, but the N-oxide product is reported in Biotransformations Vol. 3 [9]. Three errors out of 19 predictions yields a very satisfying predictive performance of 84% for this independent data set.

Because of our concentration on the relative occurrence of the N-oxide, it almost goes unnoticed that the N-oxide is never predicted to occur in the absence of the N-demethylated metabolite. This prediction holds true for all 19 sites listed in Volume 3.

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Summary

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A new family of reaction site descriptors, the n-level reaction site representations, have been presented. The reaction site representations contain both electronic (atom labels and bond types) and steric (the bullet atoms) information. In this study, a 3-level reaction site representation effectively described the local environment of the N-demethylation/N-oxidation reaction site of tertiary N-methyl amines.

The structure maps constructed using a 3-level reaction site representation have shown very promising results in indicating structural features which may play a role in differentiating between the relative occurrence of the N-demethylation and N-oxidation of tertiary amines. In addition, the structure maps facilitate the analysis of documented metabolic pathway information by indicating in one drawing the values of several experimental parameters (i.e. test system, route of administration, characterization techniques etc.). The structure map with its associated "colorings" has the ability to display complex data sets in a concise manner and is much easier to interpret than data presented in a tabular format.

The structure map's effectiveness in indicating clear structure-activity relationships is useful for suggesting what information to include in a database. In addition to the observed biotransformations, it may be useful to include information about the test system (animal model or human metabolism), the level of the system studied (i.e. microsomes, hepatocytes, in vivo) and the experimental conditions employed for the analysis of metabolites in a metabolic pathway database. Whether or not reference compounds were used in the documented study is of interest for two reasons. First it suggests the metabolic focus of the investigator, and second, there is an apparent correlation between the relative occurrence of N-oxidation and the use of a reference N-oxide in the experimental metabolism studies of our data.

A striking feature of the structure map is the observation that given a data set composed of compounds which undergo either N-demethylation or N-oxidation, N-demethylation is observed in the vast majority of cases. The only exception is if the compound is administered intravenously. This observation may be expressed in the form of the following predictive rule. If N-oxidation at the N-methyl site is observed, N-demethylation is predicted to occur so long as the compound is administered orally.

The prediction of the relative occurrence of the N-oxidation reaction is not as simple. N-oxidation is observed some of the time, but not every time N-demethylation takes place. The structure maps constructed using a 3-level reaction site representation suggest that the local environment of the N-methyl nitrogen must be sterically unhindered for N-oxidation to occur. The model more specifically predicts that the steric index for a reaction site must be less than two for N-oxidation to occur.

It will take additional studies to reconcile these results with earlier work on the roles of pKa and lipophilicity as determinants of N-demethylation and N-oxidation rates for a number of reasons. First, we are modeling the relative and not the absolute occurrence of these metabolic reactions. Second, we have restricted attention to N-methyl tertiary amines. Third, we are confining our attention to purely structural determinants directly computed from the stereochemical structures of the compounds. The potential utility of these structural determinants in metabolic prediction are born out by the accuracy of the predictions on volume 3. We expect these determinants to have additional utility in ascertaining the physico-chemical determinants of N-oxidation and N-demethylation as the relationships between absolute and relative occurrence data are better sorted out.

Although the structure maps we have presented have been used to provide a visualizable method for the analysis of the relative occurrence of N-demethylation and N-oxide formation in xenobiotic metabolism, the methodology presented clearly applies to structure-activity relationships in general.

An atlas of structure-reactivity maps constructed from a metabolic pathway database would provide a useful tool for researchers interested in predicting drug metabolism. By examining these structure maps it may be possible to determine which structural modifications are necessary to influence the metabolism of a new compound.

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