Follow-up: Prospective compound design using the ‘SAR Matrix’ method and matrix-derived conditional probabilities of activity

SAR matrices

To generate SARMs compounds are subjected to a systematic two-step fragmentation procedure yielding matched molecular pairs (MMPs)⁸. An MMP is defined as a pair of compounds that only differ at a single site. In the first step, compounds are fragmented into core structures and substituents. In the second step, resulting core structures are subjected to fragmentation. This two-step fragmentation protocol identifies series of compounds with related core structures (forming “core MMPs”). Series of compounds with cores forming MMP relationships are organized in individual SARMs, as illustrated in Figure 1. Each matrix cell defines a unique combination of a core and substituent (reminiscent of yet distinct from R-group tables). Following MMP terminology, the core is called key fragment and the substituent value fragment⁸. Each filled cell represents an actual compound color-coded by activity or potency and each empty cell a VC representing a previously unexplored core-substituent (key-value) combination. Accordingly, VCs are thought to generate a “chemical space envelope” around structurally related compound series. Depending on the structural relationships that are present within a given compound set, varying numbers of SARMs are obtained that systematically organize available analog series and provide many VCs for further consideration. The more similar data set compounds are to each other, the more SARMs are typically obtained.

Figure 1. SAR matrix.

A schematic representation of a SARM is shown. Compound fragmentation (indicated by thick lines in matrix cells) yields three analog series with structurally related cores (keys). Each series consists of analogs that share a core and differ by a single substituent (value, blue). Structural differences between the cores of the three series are highlighted in red. Each SARM combines all analog series with structurally related cores available in a compound set. Rows and columns represent compounds sharing the same core and substituent, respectively. In each cell, the combination of a core and a substituent defines a unique molecular structure. Compounds present in the data set are represented by filled cells that are color-coded according to activity. In addition, empty cells represent virtual compounds (i.e., previously unexplored key-value combinations resulting from MMP fragmentation).

Matrix-based local QSAR models

A compound neighborhood (NBH) approach was developed for potency prediction of VCs based on known potencies of structural analogs⁴, as illustrated in Figure 2. A qualifying NBH consists of two known active compounds that contain the key and value fragment of a given VC, respectively (D and G in Figure 2a), and a third active compound (E) that consists of the key of D and value of G. The potency of a VC can then be predicted from its neighbors by applying the additivity assumption underlying Free-Wilson analysis⁹ using the equation shown in Figure 2a. For a given VC, all qualifying NBHs are identified across all SARMs, as illustrated in Figure 2b, and for each NBH, an individual potency prediction is carried out using a local “mini-QSAR” model. The average potency over all NBHs is then calculated to yield the final prediction.

Figure 2. Neighborhood-based activity prediction.

(a) A NBH of virtual compound X is marked in blue in a model SARM and compounds forming this NBH are displayed. Compounds D and G share the same substituents and core with X, respectively, and the third neighbor E consists of the core of D and substituent of G. At the lower left, the equation to predict the potency of X from the values of D, E, and G is shown. (b) The process of NBH mining is illustrated. For X, the set of all qualifying NBHs (marked in blue) in a given SARM are identified and potency values are predicted for individual NBHs (indicated by color-coded squares). “act” stands for activity (in this case, numerical potency values are used).

The NBH approach is based upon numerical values and thus well suited for potency prediction during compound optimization considering multiple analog series. Principal limitations of QSAR modeling also apply to the NBH methodology, given its Free-Wilson foundation. Hence, meaningful potency predictions can only be expected in the presence of SAR continuity (when small structural changes are accompanied by gradual changes in potency). By contrast, SARMs capturing discontinuous SARs or activity cliffs¹⁰ fall outside the QSAR applicability domain. Because potency predictions are carried out over multiple NBHs in different SARMs, standard deviations of predictions provide a simple yet effective indicator of prediction reliability. High and low standard deviations indicate the presence of SAR discontinuity and continuity, respectively, for compound subsets involved in the predictions. When standard deviations are low, accurate SARM-based potency predictions can be expected⁴.

Predictions based on conditional probabilities of activity

A conceptually different approach was developed for hit expansion from screening data based upon conditional probabilities of activity derived from SARMs, as outlined in Figure 3. In contrast to NBH-based prediction of numerical potency values, the conditional probability method can utilize approximate potency measurements (e.g., primary screening data) leading to a binary classification of inactive vs. active data set compounds and ensuing prediction of a probability of activity for VCs.

Figure 3. Predictions based on conditional probabilities of activity.

Steps and equations required to derive probabilities of activity from SARMs for prediction of virtual compound X are shown using a model SARM with nine compounds (A–I) that contain three cores (c1–c3) and four values (v1–v4). Matrix cells are color-coded according to compound activity (red, inactive; green, active). In the first step, initial class probabilities are calculated for all cores and values using equation 2 and 1, respectively. Value- and core-weighted matrices are then derived via equations 4 and 3. The class contribution of core c3 is obtained from the value-weighted matrix using equation 5. Analogously, the class contribution of value v1 is obtained from the core-weighted matrix using equation 6. The value and core class contributions are then normalized using equations 7 and 8. Finally, the activity probability p_x of 0.13 is obtained for X by combining the normalized core c3 and value v1 class probabilities using equation 9.

The conceptual basis of the approach is provided by the following ideas: based on the observed frequency of occurrence of given core and value fragments in active versus inactive compounds (in the following referred to as the active versus inactive class), probabilities of activity and inactivity can be derived for cores and values. Importantly, the contributions of cores and values are thought to be influenced by each other because compounds are represented in SARMs as combinations of individual core and value fragments. Considering the conditional nature of core and value contributions to activity, initial probabilities are weighted to derive class probabilities for any core and value. For a given VC, probabilities of its core and value are then combined to yield a final probability of activity.

Key steps of the methodology are summarized in Figure 3 (and for each step, the respective equation is provided). To illustrate the approach in an intuitive manner, we will go through an exemplary probability calculation for a given VC, guided by Figure 3.

Core and value class probabilities

The SARM in Figure 3 contains nine compounds (A–I) that comprise three cores (c1–c3) and four values (v1–v4). The probability of activity will be predicted for virtual compound X that shares core c3 with compounds G, H, and I and value v1 with compounds A and D.

Given the distribution of individual values v and cores c in active and inactive compounds, probabilities of activity and inactivity are calculated using equation 1 and equation 2. Here, P(y|v) and P(y|c) are the conditional probabilities that describe how likely it is to observe a given specific class y ∈{active, inactive} for a value v and a core c, respectively. If c^(x), v^(x), y^(x) is the core, value, and class of a given compound x, we can express the conditional probabilities as the fraction of compounds with a core c or value v and class y relative to all compounds containing this core or value. In case of value v1, both class probabilities are equal (i.e., 1/2) because v1 is contained in one active and one inactive compound. By contrast, the probability of inactivity is two times higher for core c3 than its probability of activity (2/3 vs. 1/3).

Core- and value-weighted matrices

These initial estimates are further refined by taking information from all SARM compounds into account. For this, the inverse of value and core class probabilities is used to derive the value-weighted matrix and core-weighted matrix, respectively. In case of the value-weighted matrix, the inverse class probabilities of the values are mapped to the compounds that represent the corresponding value and class. Analogously, the core-weighted matrix is derived by mapping the inverse class probabilities of the cores to the compounds that represent the corresponding core and class. The value-weighted matrix results from the assignment of a weight to each compound using equation 3 and the core-weighted matrix is obtained using equation 4.

Refinement of core and value class contributions

In this step, core probabilities using value-weighted matrices and value probabilities using core-weighted matrices are derived. The underlying idea is to statistically assess if a core or value contributes more to activity or inactivity. This rationalizes the calculation of weights from the previous step: the less frequently observed class for a core or value is assigned a higher weight, which leads to a larger class contribution of the corresponding value or core of a compound, respectively. For example, the class probability of core c3 is updated by considering information from values v2, v3, and v4 in compound G, H, and I, respectively. All compounds containing value v2 are active (2/2); hence, the core class probabilities of compounds B and G are assigned a weight of 1.0 (through value-weighting). For value v3, the compounds show equal class frequency of (in)activity (1/2); thus, both active and inactive compounds are assigned the same weight of 2.0. Finally, two of three compounds containing value v4 are inactive. Accordingly, inactive compound I receives a lower weight of 1.5 indicating that its inactivity is more likely due to v4. It follows that with increasing frequency of inactivity for a given value, core weights of inactive compounds decrease (and vice versa), indicating that the value is likely to be responsible for inactivity. Analogous considerations apply to assess probabilities of activity.

From the value-weighted matrix, core class contributions are calculated with equation 5. For core c3, contributions of 0.34 and 1.12 to activity and inactivity are obtained, respectively, using a smoothing factor of α=0.1 (this factor is applied to prevent zero probabilities when no compound is available to represent a possible core or value class):

Through normalization using equation 7 core class probabilities between 0 and 1 are obtained; for c3 values of 0.23 (activity) and 0.76 (inactivity).

Analogously, value class probabilities are refined using the core-weighted matrix (generated using equation 4). For example, class probabilities of value v1 are adjusted by considering information from cores c1 and c2 in compounds A and D that contain v1. Compound A is active and belongs to the majority class of c1 and is thus assigned a lower weight than D, which is inactive and belongs to the minority class of c2. The higher weight assigned to compound D means that its inactivity is statistically more likely to result from value v1 than core c2. Weighted value class contributions calculated using equation 6 give activity and inactivity contributions of 0.72 and 1.40, respectively, for value v1 (applying a smoothing factor of α=0.1):

Normalization using equation 8 then yields updated v1 class probabilities of 0.34 (activity) and 0.66 (inactivity).

Combined activity probability

Finally, the normalized core and value probabilities are combined via equation 9 yielding an activity probability p_x (ranging from 0 to 1) for any core-value combination representing a VC. Increasing p_x values indicate an increasing probability of activity. For classification, a threshold value of activity must be set (e.g., 0.5). In our example, the normalized core and value class probabilities for c3 and v1 result in an activity probability p_x of 0.13 for virtual compound X representing this core-value combination. Thus, given the low probability of activity, this VC is predicted to be inactive. In benchmark calculations on sets of known active and inactive compounds, conditional probability calculations yielded reasonably accurate predictions of activity, at least comparable to current state-of the-art machine learning approaches⁵.

Because the conditional probability approach is statistically grounded, prediction accuracy is expected to increase with sample sizes and matrix density⁶. Therefore, it makes sense to exclude SARMs from the calculations that contain only a small number of data set compounds or have limited row overlap (accounting for shared substitution patterns among structurally related series)⁶. Accordingly, SARMs with more than 50% row overlap are typically considered informative and prioritized for probability calculations.

Different from the NBH approach, the conditional probability method is generally applicable and not confined to compound subsets representing continuous SARs. Thus, QSAR applicability domain restrictions do not apply in this case.