Between Descriptors and Properties: Understanding the Ligand

May 10, 2017 - Although there is a controversy in the literature regarding both the mathematical validity and the performance ... Molecules 2017 22 (1...
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Between Descriptors and Properties: Understanding the Ligand Efficiency Trends for G Protein-Coupled Receptor and Kinase Structure−Activity Data Sets Jaroslaw Polanski* and Aleksandra Tkocz Institute of Chemistry, University of Silesia, 9 Szkolna Street, 40-006 Katowice, Poland S Supporting Information *

ABSTRACT: The chemical meaning of the ligand efficiency (LE) metrics is explained in this paper using a large G proteincoupled receptor (GPCR) and kinase structure−activity (IC50, Ki) data set. Although there is a controversy in the literature regarding both the mathematical validity and the performance of LE, it is in common use as an early estimator for drug optimization. Apparently, the numerous con arguments are not convincing enough. We show here for the first time that the main misunderstanding of the chemical meaning of LE is its interpretation as a molecular descriptor connected with a single molecule. Instead, LE should be interpreted as a statistical property. We show that the LE, which is designed as a regression of a binding property on the heavy atom count (HAC), is correlated to the reciprocal of the molecular weight because of Avogadro statistics. This indicates that the hyperbolic model of LE is basically a consequence of a nonbinding effect, an increase in the number of ligands that are available to a receptor for smaller molecules, and not a real increase in the binding potency for a single HAC as interpreted in the literature. Accordingly, we need to revisit and carefully reevaluate LE-based molecular comparisons.

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INTRODUCTION Free energy or another binding property averaged versus the heavy atom count (HAC), which is known as ligand efficiency (LE), has been used to explore the essentials of ligand−target binding.1 The popularity of LE has steadily increased, as illustrated by the number of Web of Knowledge records, which have increased from three in 1995 to 84 in 2016. LE-focused publications appear in multiple high-impact journals from Nature (Nature Reviews Drug Discovery) to JACS. LE is also used on a daily basis in the drug design process in the pharma industry, where it is hoped that it can be an early estimator for the potential of drug optimization not only in the context of biological activity but also in its HAC, molecular weight (MW), or log P by ameliorating the inf lation of these properties that has been observed in current medicinal chemistry practice.1 This hope is based on the fact that the LE monitors not only binding data but also the HAC or MW. Technically, the LE was designed to regress the in vitro potency on topological or physicochemical descriptors.2 However, the performance of the indicators is problematic. On the one hand, it has recently been shown that the LE can be used for efficient modeling of the activity of a compound.2,3 On the other hand, Sheridan4 debunked the idea that ligand efficiency indices are superior to pIC50 as quantitative structure−activity relationship alternatives. The performance of the LE is not the only reason for the substantial disagreement. Serious arguments against LE can be found in refs 5−7; however, this criticism appeared to be insufficiently convincing to be accepted by the drug design audience. The core point was expressed by Shultz,8 who questioned the normalization of size by the LE descriptor. The © 2017 American Chemical Society

answer to these criticisms required the application of car fuel efficiency to justify the mathematics of the LE trend,9 which evidently indicates small molecules as the ones with preferred binding potency. The hypotheses explaining this trend went as far as thermodynamics of binding.8 Actually, the LE strategy is an interesting option for drug design, in which the current procedures have resulted in the inflation of the MW (and also the HAC and log P) and the deterioration of the pharmacokinetic profiles of drug candidates. As in drug optimization we commonly use molar binding metrics (inhibitory concentration IC50 or inhibitory constant Ki), this might have contributed to molecular obesity.1,10 In this context, the LE descriptor, which clearly indicates a preference for small HACs (MWs), seems to be especially attractive. However, the controversies surrounding the LE signal stresses the need for increased vigilance, and it is important to properly understand the mathematics and chemistry that underlie the metrics. In particular, one puzzle of the LE trend is the fact that we cannot understand why the regression of the binding property on the HAC results in a substantial preference for the efficiency of small versus large molecules, i.e., it would be helpful to understand the chemical effects that generate this mathematical model. One of the latest hypotheses went as far as suggesting that the overall trend in ΔG-related LE is mainly a consequence of enthalpic efficiency and [that] conformational entropy may not be as signif icant as is commonly believed.1 Received: February 23, 2017 Published: May 10, 2017 1321

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Figure 1. Comparison of the trends for BEI vs MW with those for log Ki and log IC50 vs MW for the kinase and GPCR data: (a−c) BEI (=(log Ki)/ MW) vs MW; (d−f) log Ki vs MW; (g−i) BEI (=(log IC50)/MW) vs MW; (j−l) log IC50 vs MW. The hyperboles 2/MW are shown as red diamonds.

descriptors can be when interpreting the precise meaning of LE (BEI).

The G protein-coupled receptor (GPCR) (GPCRSARfari) and kinase (KinaseSARfari) data sets2,11 were used here to show that the basis for the LE puzzle can be cleared up by a careful analysis of its chemical meaning. Because the molar IC50 or Ki parameters are routinely used in drug design, they are believed to be the primal binding properties. In fact, we are measuring in the lab the binding property for a sample of a certain weight, which can be recalculated directly to the binding property per gram of the substance (Bgram). Interestingly, the binding eff iciency index (BEI), a molecular descriptor related to the LE and calculated as the molar binding property averaged versus MW, is identical with Bgram. Usually, the Bgram value is normalized to a molar scale using the MW as a multiplier. Accordingly, we show here that properly interpreted chemical effects in ligand−receptor interactions can solve the puzzle of the dominant LE (BEI) trend. Additionally, we show how informative the careful discrimination between properties and

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METHODS Molecular Descriptors versus Properties. Basically, chemical compounds, i.e., both molecules and substances, can be represented by molecular descriptors, i.e., indicators related to the molecules or molecular structures that can be calculated from the molecules or properties that are to be measured experimentally if there are real values or that require predictions during molecular design. However, it is not always easy to distinguish between these two data types.12 Let us analyze molecular weight (MW). It can be a property when it is measured for molecules, e.g., in mass spectrometry or even when we are weighing a mole of a substance (i.e., the Avogadro number of molecules) or its fraction, but alternatively, it can also be a descriptor when we are estimating the MW of a single molecule by simply summing the atomic mass contributions to 1322

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Figure 2. Comparison of the trends for the binned data of BEI vs MW with those for log Ki and log IC50 vs MW for the kinase and GPCR data: (a− c) mean BEI (=(log Ki)/MW) vs MW bin; (d−f) mean log Ki vs MW bin; (g−i) mean BEI (=(log IC50)/MW) vs MW bin; (j−l) mean log IC50 vs MW bin.

the total MW. The mass of 1 mole of a substance will be its MW [in g/mol], while the mass of a single molecule will be its MW [in Da]. The correlation between these two variables is 100% and creates the major normalization basis in chemistry when we are mapping substances to molecules and vice versa. In fact, we need an Avogadro number (NA), which is a chemical routine, for this transformation that is generally overlooked. Therefore, MW [in Da] × NA = MW [in g/mol]. The three parameters in the equation are the molecular descriptor, which is the mass of a single molecule (MW), and the properties, NA and the mass of a sample of 1 mole of the substance. Ligand and Binding Efficiency Metrics. Formal definitions of ligand and binding efficiency (LE and BEI, respectively) have previously been described in the literature, and various forms of these parameters were precisely described recently by Cortes-Ciriano.2 We will interpret these parameters in their widest sense as given below: LE = binding property/HAC

where the binding property is any property that is measured in order to define the interactions between a ligand and a receptor and the HAC is the heavy (non-hydrogen) atom count. BEI = binding property/MW

(2)

where the MW is the molecular weight in daltons. Generally, in the literature both LE and BEI are interpreted as molecular descriptors in the sense of physicochemical descriptors (for BEI, compare especially refs 2 and 13). According to the differentiation of the molecular descriptors and properties that was discussed in the previous paragraph, these are (binding) properties per molecular fragment defined by the MW [in Da] or the HAC. Alternatively, the BEI can be interpreted directly as a property, namely, the direct measure of binding of 1 gram of a substance. Accordingly, the LE is a property, namely, the direct measure of binding of 1 mole of HAC.

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also. Therefore, in Figure 1d−f,j−l we have shown the dependence of the molar Ki and IC50 on the MW. The relationship here is much more chaotic. In order to visualize more clearly the differences between the molar and BEI metrics, in Figure 2 we have additionally organized the IC50 and Ki data in a binning mode. We can see that the BEI metric evidently displays data here in the form of the hyperbolic-like plot (Figure 1a−c,g−i for BEI versus Figure 1d−f,j−l). At the same time, the binned statistics for the molar metrics (Figure 2) indicate large deviations in binding potency without an evident trend in the regions of MW below ca. 200 Da and above 600 Da, whereas in the region of medium MW values the trends are less chaotic and, similar to the BEI, the IC50 and Ki tend to increase with decreasing MW. Interestingly, we can explain chemical effects that control binding using a molar metric. For example, a strictly constant value of the binding property (IC50 or Ki) for a series of ligands of different sizes would be explained by a simple pharmacophoric model, i.e., the same molecular moiety is always engaged in the ligand−receptor interactions for each ligand, and therefore, independent of the increasing ligand size the binding remains essentially constant. At the same time, we cannot understand the chemical effect that would explain the trends in the BEI metrics so clearly. Why is the fragmental efficiency (physicochemical descriptor) so high for small MWs? In Figure 3 we present the molecular effect, the Avogadro statistics of a 1 gram substance that does not depend upon

Molar metrics of ligand binding (inhibitory concentration IC50, inhibitory constant Ki, or their log or pK values) if used here are in the direct form without transformation to the negative values. Data Sets. Two data sets, GPCRSARfari and KinaseSARfari, were assembled by gathering bioactivity information from the ChEMBL database, version 3.00,11 as previously described.2 This includes 40 826 IC50 values for 22 882 distinct compounds and 142 GPCRs and 75 614 Ki values for 33 541 compounds and 136 GPCRs as well as 28 835 compounds for 374 kinases, which makes a total of 74 204 data points. Both of these databases contain easily accessible information that can be downloaded in many file formats. The records were carefully inspected, curated, and prepared for further processing (e.g., notations with an activity unit equal to “nM” were collected). All of the data sets report the compound potency or binding affinity as IC50 and Ki values. Different IC50 or Ki values for the same compound were treated as separate compounds or alternatively binned versus MW values. Calculations were performed using the KNIME Analytics Platform, version 3, on an Intel Core 2 Duo CPU 1.80 GHz computer system with 4.00 GB of RAM and a 64-bit Windows 10 operating system. Graphs were plotted using MATLAB version R2015b.

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RESULTS AND DISCUSSION Interestingly, the LE was originally developed to compare maximal ligand−target affinities.11 As small, non-hydrogen ligand cations or anions are among the ligands that have been investigated, the heavy (i.e., non-hydrogen atom) count (HAC) was the initial normalization parameter. It is worth noting that the atom count (AC), hydrogen count, or molecular weight (MW) generate analogous metrics that have fully comparable trends. BEI, which is a variation of LE, can only rarely be found in the literature. However, the complete identity of BEI with the binding per gram (Bgram) and the obvious relation to the LE make this parameter an interesting LE model. Figure 1 shows the distributions of BEI for the GPCRSARfari and KinaseSARfari data sets. The typical trend of a large BEI increase for small MWs that has been widely reported in literature can also be identified here. The effect that the dependence of the binding efficiency on the molecular size is different from the one expected theoretically was previously observed already by Kuntz et al.11 However, this could have not been explained therein. The definition of BEI determines that mathematically BEI is the interaction of the binding property and 1/MW, i.e., BEI = binding property × 1/MW. Therefore, the hyperbolic-like mathematics of BEI versus MW should not be a surprise. For example, if the binding property would be essentially a constant value, then the BEI would form a hyperbolic model (constant/ MW versus MW). Interestingly, the trends in Figure 1a−c,g−i are clearer for the smaller and larger MW ranges, while a large deviation from the definite model can be observed for the medium MWs, where the observables are dif f used over a relatively large area. Mathematically, this can be explained by the fact that for the superposition (interaction) of the binding property and 1/MW, the low and high values of 1/MW make this term especially dominant in the regions of low and high MWs. As the binding property on the molar scale is a second term determing a value of BEI, it is important to analyze its trend

Figure 3. (a) Two molecules with a virtual 1 Da fragment (molecular descriptor) indicated within two different molecules and (b) a schematic view of the two 1 gram samples formed by the two molecules. The number of ligands available for the ligand−receptor binding in a 1 g sample (property) amounts to NA × 1/MW, which is significantly larger for smaller MWs (in proportion to 1/MW).

binding and thus arranges the BEI versus MW in the form of a hyperbolic plot. Originally, the efficiency metrics were designed as a regression of a property on molecular descriptors, which has aroused controversy on the normalization of size. However, basically BEI is a property (see Methods) measured for statistical effect determined in the population of ligands interacting with the population of receptors; therefore, it should be interpreted accordingly. A fully consistent BEI interpretation refers to the combination of two properties: the first is a molar binding property, and the second is the 1/MW factor, a property deciding the number of molecules in a 1 gram sample of the ligand substance [mol/g]. Accordingly, the BEI as a property has a precise chemical meaning, i.e., it is a measure of binding for a 1 gram sample. Furthermore, the dependence of the BEI on the MW is as follows. A mole of 1 Da fragments builds 1 gram of a substance formed by NA × 1/ 1324

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Figure 4. Comparison of the trend for LE vs HAC with those for log Ki and log IC50 vs HAC for the kinase and GPCR data: (a−c) LE (=(log Ki)/ HAC) vs HAC; (d−f) log Ki vs HAC; (g−i) LE (=(log IC50)/HAC) vs HAC; (j−l) log IC50 vs HAC. The hyperboles 2/HAC are shown as red diamonds. (m) MW vs HAC for all databases.

MW molecules; therefore, the number of ligands that are available to bind the target (i.e., to form ligand−receptor complexes) depends on 1/MW. Accordingly, an increase in the fragmental binding property results first of all not from a real increase in the binding efficiency but from the Avogadro 1/MW statistics determining the number of molecules that are available for ligand−receptor interactions. The surprise here is only due to the fact that we are not accustomed to the BEI (1 gram) property metric.

It would be interesting now to carefully analyze why the interpretation of BEI as a physicochemical descriptor fails to explain properly the BEI trend. The reason is that the property−property interpretation that we present here includes substantial information on the ligand−receptor binding. Although, we usually do not remember this, binding and BEI relate to the property of the ligand−receptor complex, and therefore, the number of ligands presented to the receptors should comply for the whole series of molecules analyzed. The physicochemical descriptor interpretation of BEI fails to predict 1325

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Figure 5. (a−c) BEI (=pKi/MW) vs MW. The two hyperboles in (a) are constructed from the edge of negative observables (negative hyperbole), which is then reflected vs the y = 0 axis to form a positive-edge data representation (first hyperbole). (d−f) pKi vs MW. (g−i) BEI (=(pKi + 8)/MW) vs MW. (j−l) Mean (pKi + 8) vs MW bin. (m−o) Mean BEI (=(pKi + 8)/MW) vs MW bin.

the right trend because it cannot control a population of ligands and consequently ligand−receptor complexes on the same level for each MW. From the mathematical point of view, a careful analysis of the untransformed and binned molar IC50 or Ki versus BEI trends (Figures 1 and 2) shows that, paradoxically, the chaotic trend of the molar binding properties does not disorder the BEI plots, which appear to be significantly better correlated than molar metric plots (Figure 1a−c,g−i versus Figure 1d−f,j−l). The reason is that in a BEI value the 1/MW multiplier dominates

over the molar binding potency factor, especially as the MW approaches zero (eq 2), which smoothes a hyperbole landscape (Figure 1a−c,g−i). On the other hand, this makes a hyperbole hard for statistical analysis in this region. In practice, in drug design we generally use the LE and only very rarely the BEI. Therefore, Figure 4 presents the LE versus HAC plots. The trend here is similar to the one in the BEI plots. Mathematically, the LE is given by the formula LE = BEI × MW/HAC. Therefore, the relation of the MW and HAC explains the difference between the LE and BEI. Figure 4m 1326

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Figure 6. (a−c) BEIHAC (=[(log Ki)/MW] × HAC) vs MW. (d−f) BEIHAC (=[(log IC50)/MW] × HAC) vs MW. (g−i) BEIHAC (=[(log Ki)/MW] × HAC) vs HAC. (j−l) BEIHAC (=[(log IC50)/MW] × HAC) vs HAC.

data values and indicates a clear trend in the binding potency, namely, the increase in binding parameters with decreasing MW. Accordingly, as an increase in IC50 or Ki means a decrease in the binding potency of the ligand, the mean binding potency here is decreasing with decreasing MW. Paradoxically, although we observe precisely the same hyperbolic-like trend of the BEI or LE (Figures 1, 2, and 4) as previously described in the literature, the exact meaning of IC50 or Ki binding potency identifies quite the opposite regularity to that commonly described in the literature, i.e., increasing binding potency with decreasing MW. In order to analyze this effect more clearly, in Figure 5 we have transformed the Ki values into the respective pKi values, the advantage of which is that their increase means also an increase in binding potency of the ligands. Interestingly, pKi values here take both positive and negative values (Figure 5d−f). Actually, in the BEI plots (Figure 5a−c), along the negative and positive edges of the plots we can observe that positive and negative values map their individual positive and negative hyperboles, which are reflected versus the y = 0 axis (Figure 5a). The negative hyperbole can be identified more easily

shows that the plot of MW versus HAC for the analyzed data is more-or-less linear. Accordingly, the hyperbolic trend here is in full correspondence to the form of the BEI plot. It is important to note that the hyperbolic trend for both the BEI and LE can basically be explained by the Avogadro statistics and not necessarily by an increase in the contribution to the binding by the HAC or MW fragment. An interesting effect can be observed from a comparison of the binned statistics, i.e., mean BEI versus the mean molar IC50 and Ki potencies (Figure 2). Generally, the molar IC50 and Ki plots are more chaotic than the BEI plots, the effect that we explained in the latter paragraph. In the ranges below ca. 200 and above ca. 600, large fluctuations of the binding potency are observed, i.e., both ligand iso-MW series of very high but also very low molar potency can be found. We can observe higher regularity within the molar parameters in the range of 200 to 600 Da. This area is highly populated by observables, as can be precisely found in the MW histograms of the data distribution (see the Supporting Information), therefore indicating that a large survey provides reliable statistics that smoothes the mean 1327

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Journal of Chemical Information and Modeling because negative values are typical for the pKi values in the lowest-MW area. Now a careful reanalysis of the BEI (LE) trends calculated for the original Ki and IC50 values (Figure 1 and 4) identifies a similar splitting effect of the negative and positive hyperboles. Paradoxically, the increase or decrease in the BEI (LE) with increasing MW depends here on the negative or positive sign of the molar binding property. This paradox clearly indicates that the BEI (LE) trend is a superposition of two properties, namely, the nonbinding statistical Avogadro 1/MW effect and the real molar binding potency. In order to further emphasize the superposition nature of the BEI (LE) in Figure 5g−i we show the BEI trend for the pKi data scaled to positive values. Paradoxically, this results in a single hyperbolic-like trend that now complies with the expectations common in the literature, i.e., smaller molecules have privileged BEI (LE) for single MW or HAC fragments. Summing up, this indicates the artifactual nature of the BEI (LE), the values of which depend first of all on the statistical Avogadro term and not on a real binding potency. Therefore, we should beware of the use of the efficiency descriptors (LE or BEI) unless their statistical nature related to ligand−target binding is precisely understood. Eventually, an interesting aspect of artifactual effects in current drug design were indicated by anonymous reviewer of this publication. Thus, false hits can reach as much as 80−100% of initial hits and can result from pan-assay interference compounds (PAINS) or colloidal aggregators.14 This also means that the same effects are expected to be present in the large data sets probed in this research. The question arises whether we can design a parameter that could replace the LE descriptor. This should be a parameter that would preserve a constant ratio of the ligands interacting with the receptor population. Consequently, this parameter would have thermodynamic meaning because the concentration of ligands would be normalized for molecules of different sizes. Basically, molarity is the only metric that obeys the requirement of the normalized concentration of the ligands. Accordingly, in Figure 6 we show a possible metric based on the HAC molarity, in which the MW is rescaled into HAC. We can now estimate the influence of HAC on binding, but once more on the scale of full molecules (molar metric) and not molecular fragments.

LE or BEI should obey the same statistical rules as the molar properties IC50 or Ki. We have shown here that a fully coherent property-based interpretation of the BEI and LE explains their observed trends. Accordingly, we show that such a property relates to the 1/MW × NA factor, i.e., the number of ligands that interact with a receptor, a property that can be measured experimentally. Therefore, the hyperbolic model of the BEI (LE), the binding property for a 1 gram (1 mole of HAC) sample is basically first of all a consequence of the different number of ligands that are available to a receptor (i.e., capable of forming ligand−receptor complexes). If this is also a real increase in the binding property for a molecular fragment, as it is usually interpreted in the literature, we need to compare this increase on the hyperbolic basis. This indicates that we need to reevaluate LE-based comparisons carefully.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jcim.7b00116. Figures S1−S11 (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Jaroslaw Polanski: 0000-0001-7361-2671 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The research presented in this publication was supported by NCBR Grants ORGANOMET PBS2/A5/40/2014 and TANGO1/266384/NCBR/2015. We are grateful to Dr. M. Vieth and G. Keseru for the helpful discussions.

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ABBREVIATIONS MW, molecular weight; HAC, heavy atom count; AC, atom count; LE, ligand efficiency (the indices, e.g., MW or HAC, refer to the scaling factor, e.g., MW or HAC); B, binding property; NA, Avogadro number; BEI, binding efficiency index

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CONCLUSIONS In conclusion, a series of replicated molecules can be normalized by a fixed number of molecules (usually the Avogadro number NA), or alternatively, we can use a fixed weight metric (e.g., 1 gram of a substance). This paper has discussed the mathematical and chemical meaning of the BEI and LE metrics, which were designed to describe the efficiency of molecular fragment binding but at the same time are direct measures of the binding property for 1 gram or 1 mole of HAC samples, respectively. Accordingly, the BEI and LE designed as parameters normalizing the binding property on the basis of regression on a molecular descriptor (MW, HAC) are understood in the literature more in the sense of a descriptor connected strictly with a single molecular fragment, whereas the binding property is a statistical effect measured in the experiment for the population of molecules. Therefore, LE and BEI should be analyzed in this context as a property. Although formally they are properties of the molecular fragment of 1 Da, such a fragment constructs a whole molecule and in consequence the fragmental property

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