Design of chemical space networks incorporating compound distance relationships

Data sets

For network design, one large and three small compound sets (active against human targets with defined equilibrium constants) were taken from ChEMBL (version 21) (https://www.ebi.ac.uk/chembl/)⁷, as reported in Table 1. We note that there was no specific reason to focus on these sets; many others could have been selected instead.

Molecular representation and similarity metric

Compounds were represented using the MACCS fingerprint⁸ (consisting of 166 structural keys or patterns), which were generated using an in-house Python implementation. Pairwise similarity values were calculated using the Tanimoto coefficient (Tc)⁹. Fingerprint descriptors of different design might have been selected instead, but for our proof-of-principle investigation, the relatively simple MACCS fingerprint was readily sufficient.

Similarity vs. distance

Pairwise similarity values were transformed into distances using the formula

                      distance = 1 – CDF(similarity)

where CDF is the cumulative distribution function for an assumed normal distribution. For each compound set, the mean and standard deviation were calculated from its pairwise similarity values. The CDF was used to emphasize compound pairs with large Tc values and de-emphasize pairs with small values compared to a linear relationship.

Network layouts

Alternative CSN layouts were generated with in-house Java programs based upon the JUNG library (http://jung.sourceforge.net/doc/JUNG_journal.pdf). Please also see the ‘Data availability’ section.

Fruchterman-Reingold. The Fruchterman-Reingold (FR) algorithm¹⁰ has so far consistently been used for NSGs³ and CSNs⁵. FR is a force-directed algorithm that brings together subsets of densely connected objects and separates different subsets from each other through repulsion (until equilibrium positions are obtained). Only similarity values reaching a pre-defined threshold are considered in FR layout construction (all other similarity values are ignored). In FR-based network views, distances between compounds have no chemical meaning.

Kamada-Kawai. The Kamada-Kawai (KK) algorithm¹¹, adapted herein for CSN design, is also a force-directed layout method. However, KK uses all distances derived from similarity values as input, and optimizes (threshold-independent) edge lengths with respect to inter-compound distances. Thus, the KK approach incorporates distance relationships into network layouts. In principle, KK-based networks are completely connected. Thus, edges between distant compounds might be omitted for clarity. Although all similarity values and corresponding distance relationships are considered for network construction, for selective edge display, similarity threshold values can also be applied.

As similarity-based compound networks, KK network representations are covered by the general definition of CSNs^4,5 and are in the following also referred to as KK CSNs.

Kamada-Kawai network design

The characteristic feature of the KK approach is that it takes distances derived from all pairwise similarity values quantitatively into account during network construction. The resulting layout reflects relative compound distances, which principally increases the chemical information contained in KK CSNs compared to threshold-dependent FR CSNs. Independent of the KK network structure, which remains constant, edges in KK CSNs can be selectively displayed at varying similarity threshold values to optimize the clarity of the presentation.

Kamada-Kawai network of a model data set

For an initial proof-of-principle assessment, a model data set was generated by combining four subsets (A–D) of five hypothetical data points, each with well-defined intra-set similarity value ranges, as reported in Table 2. Subsets A–C contained highly similar data points with varying inter-subset similarity values (Table 2), whereas subset D consisted of dissimilar data points (singletons). The KK CSN of this model data set is shown in Figure 1. All three subsets of similar data points formed separate clusters in the network, whereas data points from subset D were widely distributed. Furthermore, clusters of subsets A and B, which displayed largest inter-subset similarity values (Table 2), were located close to each other and removed from the less similar subset C. Moreover, the KK CSN also correctly accounted for the smaller distance between A and C compared to B and C. Thus, the KK CSN incorporated for various distance relationships present in the model set; an encouraging finding.

Figure 1. Kamada-Kawai network of a model data set.

Shown is the KK CSN of the model data set according to Table 2. Data points are colored on the basis of compound subset membership: A, blue; B, red; C, green; D, gray.

Kamada-Kawai networks for different sets of bioactive compounds

Figure 2 shows KK CSNs for inhibitors of MAP kinase ERK2 and glutamate [NMDA] receptor subunit ε2 (data sets 11638 and 222, respectively) (Table 1). In each case, edges were selectively displayed at three different similarity threshold values, which enabled viewing edge distributions on a “sliding scale”. The KK CSN of the MAP kinase inhibitor set 11638 revealed a clear clustering of similar compounds with comparably high or low potency, corresponding to the presence of locally continuous SARs¹. By contrast, the KK CSN of the set of ligands of glutamate [NMDA] receptor subunit ε2 revealed a cluster of highly similar compounds with large potency variations, corresponding to a high degree of local SAR discontinuity¹. This cluster was distant from other compounds of this set, consistent with the presence of unique structural features.

Because edge densities can be selectively displayed, KK CSNs generated for larger compound data sets can be readily analyzed. In addition, for SAR analysis, one may initially select a high threshold value to primarily focus on intra-cluster relationships and then gradually decrease the threshold to concentrate more on inter-cluster connections, as illustrated in Figure 2.

Figure 2. Chemical space networks with compound distance relationships.

KK CSNs are displayed for two data sets (11638 and 222 according to Table 1) at three similarity threshold values of 0.8, 0.7, and 0.6, respectively. Nodes are colored on the basis of potency values applying a color gradient from green (highest potency) over yellow (intermediate) to red (lowest potency).

Comparison of Kawada-Kawai and Fruchterman-Reingold networks

Figure 3 compares the KK and FR CSNs for a set of inhibitors of the apoptosis regulator Bcl-W (data set 100476), revealing the presence of distinct layouts. In the KK CSN a larger cluster of similar –and mostly weakly potent– compounds emerged that was distant from other data set compounds. The corresponding FR CSN provided a completely different view of the compound set with several clusters that were essentially evenly distributed across the layout (consistent with its threshold-dependent force-directed design). For each of these clusters, a corresponding cluster was also identified in the KK CSN. In three cases, the corresponding compounds were so similar –and the resulting distances so small– that these clusters needed to be magnified for a detailed inspection, as shown in Figure 4a. Hence, the KK and FR CSNs also provided complementary network views of the data set.

Figure 3. Comparison of Kamada-Kawai and Fruchterman-Reingold layouts.

For data set 100476, KK and FR CSNs are compared at a similarity threshold value of 0.8 for selective edge display (KK) and network generation (FR). Nodes are colored according to Figure 2. At the bottom, similarity values and corresponding network distances of all compound pairs are compared in scatter plots and correlation coefficients are reported.

Figure 4. Comparison of compound communities.

In (a), corresponding compound communities are highlighted in the KK and FR CSNs from Figure 3 and enlarged. Compounds in each community are numbered. In (b), compounds forming the top cluster in (a) are shown.

The scatter plots in Figure 3 reveal that there was no correlation between similarity values and network distances in the FR CSN, consistent with its design principles. By contrast, with a correlation coefficient of -0.79, significant inverse correlation (i.e. large similarity values corresponding to small distances) was observed for the KK CSN, which was largely determined by compound pairs with similarity values greater than 0.5. For small similarity values, correlation was only weak. This observation was consistent with the use of the CDF in the distance function, which emphasized distance relationships between similar compounds, as discussed above. For data sets 222 and 11638 (Figure 2), KK CSNs yielded correlation coefficients of -0.84 and -0.88, respectively.

Comparison of compound communities and series

In Figure 4a, corresponding compound communities in KK and FR CSNs are compared in detail. FR CSN clusters contain edges of comparable length and have similar topology, which is a characteristic feature of this layout. By contrast, KK CSN clusters display different topologies and contain edges of different length that further differentiate intra-cluster similarity relationships and position similar compounds closely together. For example, compounds 3, 4, and 5 from the cluster at the top in Figure 4a only differ by the (ortho, meta, or para) position of a benzene ring and are more similar to each other than to compounds 1, 2, 6, and 7 that have different substituents (Figure 4b).

Figure 5 shows a KK CSN representation for three analog series (A, B, and C) that were extracted from compounds active against the serotonin 1a receptor (data set 51). Series A and B had chemically related core structures, whereas the core of series C was distinct from A and B. In the KK CSN, the three series formed communities that were separated from each other. Consistent with the structural relationship between their cores, series A and B were positioned closer to each other than to series C. A single compound from series A was found to form a bridge between the communities of A and B. This compound contained a cyclobutyl substituent at R1 and thus closely resembled the core of series B. Taken together, these observations indicated that the KK CSN captured similarity relationships between these analog series in a meaningful way.

Figure 5. Exemplary compound series.

Shown are three analog series from the KK CSN of data set 51. For clarity, a similarity threshold value of 0.88 was applied for edge display. Each analog series is encircled and its common core structure is displayed. Compounds in each series were distinguished by substituents at a single site (R1).