Proteochemometric modeling in a Bayesian framework
- Isidro Cortes-Ciriano^{1},
- Gerard JP van Westen^{2},
- Eelke Bart Lenselink^{3},
- Daniel S Murrell^{4},
- Andreas Bender^{4}Email author and
- Thérèse Malliavin^{1}Email author
DOI: 10.1186/1758-2946-6-35
© Cortes-Ciriano et al.; licensee Chemistry Central Ltd. 2014
Received: 5 March 2014
Accepted: 18 June 2014
Published: 28 June 2014
Abstract
Proteochemometrics (PCM) is an approach for bioactivity predictive modeling which models the relationship between protein and chemical information. Gaussian Processes (GP), based on Bayesian inference, provide the most objective estimation of the uncertainty of the predictions, thus permitting the evaluation of the applicability domain (AD) of the model. Furthermore, the experimental error on bioactivity measurements can be used as input for this probabilistic model.
In this study, we apply GP implemented with a panel of kernels on three various (and multispecies) PCM datasets. The first dataset consisted of information from 8 human and rat adenosine receptors with 10,999 small molecule ligands and their binding affinity. The second consisted of the catalytic activity of four dengue virus NS3 proteases on 56 small peptides. Finally, we have gathered bioactivity information of small molecule ligands on 91 aminergic GPCRs from 9 different species, leading to a dataset of 24,593 datapoints with a matrix completeness of only 2.43%.
GP models trained on these datasets are statistically sound, at the same level of statistical significance as Support Vector Machines (SVM), with ${R}_{0}^{2}$ values on the external dataset ranging from 0.68 to 0.92, and RMSEP values close to the experimental error. Furthermore, the best GP models obtained with the normalized polynomial and radial kernels provide intervals of confidence for the predictions in agreement with the cumulative Gaussian distribution. GP models were also interpreted on the basis of individual targets and of ligand descriptors. In the dengue dataset, the model interpretation in terms of the amino-acid positions in the tetra-peptide ligands gave biologically meaningful results.
Keywords
Proteochemometrics Bayesian inference Gaussian process Chemogenomics GPCRs Adenosine receptors Applicability domainBackground
The advent of high-throughput (HT) technologies has contributed in the last decades to a vast data increase in proprietary and public bioactivity databases. In a parallel manner, a large amount of biological data has been collected on protein structure and sequence information for numerous species. Chemogenomic techniques [1–3] can capitalize on this large amount of information by modeling the relationships between the chemical and the biological space. This data integration permits the bioactivity prediction of compound-target combinations lying in regions of the drug-target interaction space which are sparsely sampled by experimental measurements. These techniques are based on the similarity principle [4, 5], which follows the premise that similar compounds (and targets) [6] are more likely to exhibit akin bioactivity profiles in comparison to structurally distant structures. Among others, chemogenomic approaches have enabled: (i) the prediction of protein targets for new compounds based on the bioactivity profiles of similar compounds, [7–9] (ii) the study of protein similarity on the basis of the similarity of their ligands, [10, 11] and (iii) receptor deorphanization [12].
In the field of chemogenomics, Proteochemometrics (PCM) [6] uses machine learning models to relate compounds to their biomolecular targets (usually proteins). PCM extends traditional Quantitative Structure-Activity Relationship (QSAR) [13] by allowing to both inter- and extrapolate on the target and/or chemical spaces. Therefore, compounds can be optimized not only with respect to their affinity on a target, [14] but also by taking into account their selectivity [15]. In that way, PCM also permits to detect compound substructures conferring inhibitory activity to a panel of related biomolecular targets [14].
Although the relevance of PCM has been confirmed by both in silico and experimental validation, [6, 16] current methods cannot: (i) inherently determine the applicability domain (AD) of a model, or (ii) provide individual confidence intervals for each prediction.
The applicability domain (AD) of a bioactivity model is defined as the range of chemical (and target in PCM) space to which the model can be reliably applied [17–19]. Therefore, the AD is a measure of the generalization properties of a given model: the volume of chemical (descriptor) space that can be reliably predicted [20]. Given that compounds are encoded with descriptors when training predictive models, it is important to distinguish between the chemical space (referring to chemical structures) and the chemical descriptor space. This distinction is important as in the calculation of some popular descriptors (e.g. Morgan fingerprints) [21], chemical substructures are hashed: different chemical substructures are mapped at the same descriptor position. Consequently, two different structures in the chemical space can be represented by the same descriptor values. A detailed discussion of the different methods proposed to assess models AD can be found in Ref., [20] to which the interested reader is referred. In PCM, the AD is an essential feature, as extrapolation has to be used to predict the bioactivity for new chemicals on new targets [6].
In parallel to the concern about the evaluation of individual bioactivity predictions, recent publications have aimed at establishing the level of uncertainty in public bioactivity databases [22–25]. In this vein, Brown et al.[26] highlighted the importance of including the uncertainty of bioactivity data into the evaluation of models quality. Hence, predictive models should be assessed through: the analysis of the experimental error of the data, the evaluation of the models AD as well as the definition of intervals of confidence for the predictions. However, acceptable levels of prediction errors are also determined by the context in which the model will be applied. Indeed, models exhibiting high prediction errors can be nevertheless useful in a high-throughput (HTS) campaign while not being suitable in lead optimization [26].
Bayesian inference provides a reliable theoretical framework to handle all previously mentioned aspects within a unique bioactivity model. Gaussian Processes (GP) are a non-parametric machine learning method based upon Bayesian inference: they thus permit an evaluation of the AD of a given model as well as providing the most objective estimation of the predictions uncertainty. Furthermore, the experimental bioactivity errors can be used as model input. A GP prediction of a given compound-target combination is a Gaussian distribution whose variance defines intervals of confidence: in principle, this variance measures the distance of the compound-target pair to the training set. GP models can be globally validated by traditional statistical metrics (e.g. R^{2} or Q^{2}) [27–29] while also providing individual assessment for predictions. GP were firstly introduced in the field of QSAR modeling by Burden et al.[30]. Later on, GP were also used for: (i) the modeling of ADMET properties, [31, 32] (ii) the prediction of electrolyte solubility, [33] (iii) the bioactivity prediction of small peptide datasets, [34–36] (iv) protein engineering, [37] and (v) the bioactivity prediction of bioactivity-focused (GPCRs) combinatorial chemolibraries [38]. The purpose of the present study is to propose Gaussian Process (GP) to simultaneously model chemical and multispecies protein information in the frame of PCM. GP models are validated by comparing their performance to that of SVM using a panel of kernels on two PCM datasets extracted from ChEMBL database, [39] involving adenosine receptors (10,999 data points, 8 sequences) and aminergic GPCRs (24,593 data points, 91 sequences), and on a third dataset extracted from the literature concerning the catalytic activity of four dengue virus NS3 proteases (199 data points, 4 sequences). GP perform as well as SVM, with statistically non-significant differences in performance. Nonetheless, GP provide additional information with respect to SVM, namely the uncertainties on individual bioactivity predictions. GP also permit the interpretation of the models with respect to the targets of adenosine receptors and GPCR datasets, and also with respect to the ligand descriptors.
Methods
Datasets
Aminergic GPCRs
Overview of the proteochemometric datasets modeled in this work
Adenosine receptors | Dengue virus NS3 Proteases | Aminergic GPCRs | |
---|---|---|---|
Datapoints | 10,999 | 199 | 24,593 |
Sequences | 8 | 4 | 91 |
Ligands | 4,419 | 56 | 11,121 |
Source Organisms | H. sapiens and Rattus norvegicus | Dengue virus | H. sapiens, Rattus norvegicus, Mus musculus, Bos taurus, Sus scrofa, Canis familiaris, Cavia porcellus, Chlorocebus aethiops, and Mesocricetus auratus |
Bioactivity | p K _{ i } | K _{ c a t } | p K _{ i } |
Matrix Completeness (%) | 31.11 | 88.84 | 2.43 |
Adenosine receptors
This dataset previously published by van Westen et al.[16] is composed of 10,999 bioactivity data points measured on the rat and human adenosine receptors, A_{1}, A_{2A}, A_{2B} and A_{3}. The dataset was extracted from ChEMBL 2. Only compounds tested on rat or human receptors by radio-ligand binding assays and for which p K_{ i } bioactivity values were annotated with a ‘=’ relationship were included in the final dataset. Bioactivity values range from 4.50 to 10.52 p K_{ i } units. Compounds were normalized and ionized at pH 7.4. Subsequently, they were assigned 2D coordinates and converted to fingerprints. See Table 1 for further details about the dataset.
Dengue virus NS3 proteases
This dataset was collected from the proteochemometric study published by Prusis et al., [41] which modeled the catalytic activity of the Dengue virus NS3 proteases from four viral serotypes using datapoints measured on 56 different tetra-peptide substrates (Table 1). These substrates were designed to evaluate the role amino acid residues located at P1’-P4’ in the sequence. The catalytic efficiency was measured as the turnover number (k_{ c a t }) for the cleavage of the substrate. In contrast to the two datasets presented above, the number of data points in this case was only 199.
Descriptors
Chemical compounds were described by Scitegic circular fingerprints (ECFP_6 type), [21, 42] calculated in PipelinePilot 8.5.0.200 [43]. For the calculation of keyed ECFP_6 fingerprints, each compound substructure, with a maximal diameter of three bonds, is treated as a compound feature. The substructures are then mapped into an unhashed array of counts, thus enabling the estimation of their contribution to bioactivity (see Results and Discussion). The efficiency of these fingerprints to identify chemical features relevant for bioactivity has been previously demonstrated [16, 44]. Pairwise compound similarity plots were calculated in R using the vegan package [45]. Protein amino acids of the GPCRs and adenosine receptors binding sites, as well as the Dengue virus NS3 proteases substrates, were described with five amino acid extended principal property scales (5 z-scales). The property calculation was conducted in R [46]via in-house scripts following the work of Sandberg et al.[47]. In the GPCRs dataset a descriptor accounting for the amino acids side chain charge at pH 7.4 was also added (with values of: +1 if the charge is positive, -1 if negative and 0 for neutral amino acids). The four Dengue virus NS3 protease variants were described with binary descriptors.
Modeling with Bayesian inference
Gaussian processes
where: (i) P(G P(D)|D) is the posterior probability distribution giving the bioactivity predictions, (ii) the likelihood P(y|G P(D),X) is the probability of the observations, y, given the training set, X and the model G P(D), and (iii) P(G P(D)) is the prior probability distribution of the functions G P(D) candidates to model the dataset D.
where δ(x_{ j },x_{ k }) is the Kronecker delta function and ${\sigma}_{d}^{2}$ is the noise of the datapoints (experimental error), which is assumed to be normally distributed with mean zero. The value of ${\sigma}_{d}^{2}$ accounts for the noise in the observed bioactivities, $\mathbf{\text{y}}=\mathit{\text{GP}}\left(\mathbf{\text{D}}\right)+\mathcal{N}\left(0,{\sigma}_{d}^{2}\right)$ which in turn reflects the trade-off between the quality and smoothness of the fitting.
C_{ X } is obtained by applying a positive definite kernel function (also known as statistic covariance) [49] to X,C_{ X }=C o v(X). Owing to the fact that the covariance function is based upon dot products, the kernel trick can be applied in a similar way as in SVM [50]. Kernel parameters are called hyperparameters since their values define the probability of each function of the prior probability distribution. The different kernels implemented in this study are listed in Additional file 1: Table S2.
Bioactivity prediction for new datapoints
and the uncertainty of the prediction corresponds to the standard deviation of this probability distribution.
where the best estimate for the bioactivity of x^{⋆} is the average value of y^{⋆}, ${\mu}_{{y}^{\star}}=\u3008P\left({y}^{\star}\right)\u3009$, and ${\sigma}_{{y}^{\star}}$, the standard deviation, its uncertainty.
As can be seen in Eq. 5, those compound-target combinations in X similar to x^{⋆}, contribute more to the prediction of y^{⋆}, as y is weighted by k^{ T }. This means that GP, as a kernel method, mainly infers the value of y^{⋆} from the most similar compound-target combinations in descriptor space present in the training set, X.
On the other hand, the predicted variance, ${\sigma}_{{y}^{\star}}^{2}$, is equal to the difference between the a priori knowledge about x^{⋆}: m=C o v(x^{⋆},x^{⋆}), and what can be inferred about x^{⋆} from similar compound-target combinations present in X: ${k}^{T}{C}_{X}^{-1}k$. Thus, in the case of x^{⋆} being similar to the compound-target combinations in X, the value of ${\sigma}_{{y}^{\star}}^{2}$ is small. By contrast, a high value of ${\sigma}_{{y}^{\star}}^{2}$ indicates that x^{⋆} is not similar (is distant) to the compound-target combinations in X. In that case, the GP cannot learn much about x^{⋆} from the training set, so the prediction should be consider as less reliable. Consequently, ${\sigma}_{{y}^{\star}}^{2}$ gives an idea of the applicability domain (AD) of the model and thus serves to evaluate the uncertainty of the prediction.
Computational details
Determining the kernel hyperparameters
where p is the descriptor index and P the total number of descriptors. Each length scale, l, is treated as a hyperparameter wich value needs to be optimized during model training. High length scale values will be assigned to irrelevant features for the model. Therefore, the inverse of the optimized l value obtained for a given descriptor gives an idea of its importance for the model. This inherent ability of Bayesian inference to infer the relevance of each descriptor is the so-called Automatic Relevance Determination (ARD) [48]. In the context of PCM, ARD can be exploited to provide a biologically meaningful interpretation of the models.
Several methods can be implemented to accomplish this multivariate optimization problem, such as a simplex method, Monte Carlo (MC) Sampling, [53] a genetic algorithm, nested sampling, [54] forward variable selection [31] or the conjugate gradient method [48].
In the present study, kernel hyperparameters were optimized by grid search and k-fold cross-validation (CV) in the case of the adenosine receptors and aminergic GPCRs datasets (section S1 of the Additional file 1), because of their large size and high number of descriptors. The experimental error, ${\sigma}_{d}^{2}$, (Equation 2) was considered as fixed with a value of 0.29 p K_{ i } units, this value being taken from the work of Kramer et al.[22] The same length scale value, l, was used for all descriptors to simplify the hyperparameter optimization.
In the case of the dengue virus dataset, due to its small size, and to the lack of information concerning the experimental uncertainty, the noise variance, ${\sigma}_{d}^{2}$, was optimized by conjugate gradient as implemented in the GPML toolbox [55]. As the number of descriptors is only 24, we optimized the length scales using the radial kernel. In the frame of Automatic Relevance Determination (ARD), the importance of each descriptor for the model was estimated using the inverse of the optimized l values, in the way described above.
GP Tolerance to noise
To better understand the influence of the experimental error in GP modeling, we trained 15 models for each dataset with increasing levels of noise with both the radial and the normalized polynomial (NP) kernel, thus leading to a total number of 90 models. Their predictive ability was monitored on the external set. The levels of added noise (noise variance) ranged from 0 to a maximum value of 10, which corresponds to a noise deviation of 3.2 p K_{ i } units for the adenosine receptors and GPCR datasets, and 3.2 log units for the dengue virus NS3 proteases dataset.
Machine learning analyses and implementation
Machine learning models were built in R using the caret package [56]. Non-default kernels for GP were introduced in the caret framework by in-house R scripts and by the definition of custom models (custom option in the caret package) implementing kernel functions from either the kernlab[57] package or in-house kernel functions. Source code is available from the authors upon request. Likewise, The Gaussian Process for Machine Learning (GPML) Toolbox version 3.2 [55] was used to build GP models in Matlab version 7.15 [58] to assess the importance of ligand descriptors (Automatic Relevance Determination). The python package infpy[59] helped to generate Additional file 1: Figure S1. The data pre-processing and the in silico modeling pipeline are described in Additional file 1, along with model training and validation.
Assessment of maximum model performance
The Tropsha validation criteria, [27–29] (Equations S7-S10 in Additional file 1) were used for accepting or dismissing the model (section Internal validation of Additional file 1). Hence, the distributions of minimum RMSEP_{ext} and maximum, ${Q}_{\mathit{\text{ext}}}^{2}$, ${R}_{0\phantom{\rule{2.22144pt}{0ex}}\mathit{\text{ext}}}^{2}$, and ${R}_{\mathit{\text{ext}}}^{2}$ (Equations S3-S6 in Additional file 1) were calculated for each dataset in the following way. Firstly, a random sample, A, of the same size of the external set was drawn from the experimental bioactivity values. Secondly, the sample B was calculated by adding to A a random noise with mean zero and standard deviation equal to the experimental error. Then, the statistical metrics were calculated for A with respect to B. The calculation of statistical metrics on 1,000 generations of random samples A and noisy samples B provided a distribution of statistical metrics for each dataset. These maximum and minimum values of the distribution were then used to validate the metrics values obtained when evaluating the bioactivities predicted for the external sets. If the obtained metrics were beyond the maximum values (for ${Q}_{\mathit{\text{ext}}}^{2}$, ${R}_{0\phantom{\rule{2.22144pt}{0ex}}\mathit{\text{ext}}}^{2}$, and ${R}_{\mathit{\text{ext}}}^{2}$) or the minimum values (for RMSEP_{ext}) of the distribution, the model is likely to be over-optimistic. The experimental errors required to define the random samples B were determined in the following way. For adenosine and GPCR datasets, the experimental error of p K_{ i } data was considered to be approximately 0.29 p K_{ i } units, which corresponds to the average standard deviation value for public K_{ i } datasets estimated by Kramer et al.[22] The experimental error of the dengue dataset was inferred from the data by considering its uncertainty as a hyperparameter of the GP model since we could not find information about the experimental uncertainty in the study of Prusis et al.[41].
Interpretation of ligand substructures
To calculate the influence of a given feature (chemical substructure) to p K_{ i }, we iteratively set the count of the feature equal to zero in all compound descriptors presenting it, in order to virtually remove the substructure. Bioactivity values were then predicted using the modified compound descriptors, and the difference between the predicted values in the presence or absence of a given feature were calculated. The average value of these differences, weighted by the number of counts of the feature in each compound, corresponds to the contribution of that feature to bioactivity. The contribution was estimated for all compound features considered in the model. The sign of the difference ({+/-}) indicates if the feature is respectively beneficial or deleterious for compound bioactivity. This approach is closely related to the method proposed by van Westen et al., [14] although two modifications have been made: (i) the weighting of the average difference between predicted and observed bioactivities, and (ii) the calculation of descriptor importance on a per target basis.
Results
Model validation
PCM GP models agree with the validation criteria
Internal and external validation metrics for the PCM models
Adenosine Receptors Dataset | ||||
---|---|---|---|---|
${R}_{\mathit{\text{int}}}^{2}$ | RMSEP_{int} | ${R}_{0\phantom{\rule{1em}{0ex}}\mathit{\text{ext}}}^{2}$ | RMSEP_{ext} | |
GP Bessel | 0.64 | 0.70 | 0.70 | 0.63 |
GP Laplacian | 0.67 | 0.68 | 0.67 | 0.66 |
GP Norm. Polynomial (NP) | 0.69 | 0.65 | 0.75 | 0.58 |
GP Polynomial | 0.70 | 0.64 | 0.70 | 0.63 |
GP PUK | 0.57 | 0.79 | 0.56 | 0.77 |
GP Radial | 0.65 | 0.69 | 0.65 | 0.68 |
PLS | 0.29 | 0.97 | 0.30 | 1.00 |
SVM Norm. Polynomial (NP) | 0.70 | 0.64 | 0.73 | 0.60 |
SVM Polynomial | 0.71 | 0.63 | 0.71 | 0.62 |
SVM Radial | 0.68 | 0.65 | 0.70 | 0.64 |
Family QSAR | 0.31 | 0.70 | 0.31 | 0.96 |
Aminergic GPCRs Dataset | ||||
${R}_{\mathit{\text{int}}}^{2}$ | RMSEP _{ int } | ${R}_{0\phantom{\rule{1em}{0ex}}\mathit{\text{ext}}}^{2}$ | RMSEP _{ ext } | |
GP Bessel | 0.56 | 0.83 | 0.56 | 0.80 |
GP Laplacian | 0.62 | 0.78 | 0.63 | 0.75 |
GP Norm. Polynomial (NP) | 0.69 | 0.68 | 0.72 | 0.66 |
GP Polynomial | 0.68 | 0.71 | 0.70 | 0.68 |
GP PUK | 0.46 | 0.93 | 0.46 | 0.90 |
GP Radial | 0.69 | 0.69 | 0.71 | 0.66 |
PLS | 0.69 | 0.69 | 0.27 | 1.05 |
SVM Norm. Polynomial (NP) | 0.69 | 0.68 | 0.72 | 0.66 |
SVM Polynomial | 0.69 | 0.69 | 0.71 | 0.66 |
SVM Radial | 0.69 | 0.69 | 0.72 | 0.66 |
Family QSAR | 0.38 | 0.98 | 0.38 | 0.97 |
Dengue virus NS3 proteases Dataset | ||||
${R}_{\mathit{\text{int}}}^{2}$ | RMSEP _{ int } | ${R}_{0\phantom{\rule{1em}{0ex}}\mathit{\text{ext}}}^{2}$ | RMSEP _{ ext } | |
GP Bessel | 0.91 | 0.43 | 0.92 | 0.44 |
GP Laplacian | 0.88 | 0.54 | 0.91 | 0.50 |
GP Linear | 0.91 | 0.45 | 0.91 | 0.48 |
GP Norm. Polynomial (NP) | 0.88 | 0.50 | 0.91 | 0.48 |
GP Polynomial | 0.91 | 0.42 | 0.92 | 0.44 |
GP PUK | 0.77 | 1.10 | 0.81 | 1.13 |
GP Radial | 0.91 | 0.45 | 0.91 | 0.45 |
PLS | 0.90 | 0.45 | 0.91 | 0.49 |
SVM Norm. Polynomial (NP) | 0.86 | 0.54 | 0.91 | 0.46 |
SVM Polynomial | 0.89 | 0.46 | 0.90 | 0.51 |
SVM Radial | 0.90 | 0.48 | 0.90 | 0.48 |
Family QSAR | 0.29 | 1.19 | 0.48 | 1.13 |
GP statistics are within the limits of the theoretical maximum model performance
The distributions of maximum ${R}_{\mathit{\text{ext}}}^{2}$, ${R}_{0\phantom{\rule{1em}{0ex}}\mathit{\text{ext}}}^{2}$, and ${Q}_{\mathit{\text{ext}}}^{2}$ and minimum RMSEP_{ext} theoretical values, obtained as described in subsection Assessment of maximum model performance in Methods, are given in Additional file 1: Figure S2 for the three datasets. The mean value of the distribution of maximum ${R}_{0\phantom{\rule{1em}{0ex}}\mathit{\text{ext}}}^{2}$ values are equal to 0.80, 0.68 and 0.96 for the adenosine, GPCRs, and dengue virus NS3 proteases datasets, which highlights that the maximum correlation values that can be obtained when modeling public data are far from the optimal maximum correlation value of one. This is not surprising given the noise levels in public bioactivity data [22, 23]. The best RMSEP_{ext} and ${R}_{0\phantom{\rule{1em}{0ex}}\mathit{\text{ext}}}^{2}$ values (Table 2) obtained with GP are respectively: 0.58 and 0.75 (adenosine receptors), 0.66 and 0.72 (GPCRs), and 0.44 and 0.92 (dengue virus NS3 proteases), which remain in the limits of these extreme theoretical values (Additional file 1: Figure S2), thus supporting the suitability of our modeling pipeline to handle data uncertainty. The mean values of the theoretical RMSEP distribution were close to the experimental uncertainty on bioactivity, for the adenosine receptors and the dengue virus NS3 proteases datasets, with respective mean RMSEP_{ext} values of 0.54 p K_{ i } units and 0.27 log units (Additional file 1: Figure S2). However, the mean RMSEP_{ext} value increases up to 0.68 p K_{ i } units for the GPCRs dataset owing to its larger size and sparsity.
PCM outperforms QSAR on the studied datasets
A comparison between models trained on only compound descriptors (‘Family QSAR’) [62] and PCM permits to assess whether the use of GP improved the bioactivity modeling, by simultaneously modeling the target and the chemical spaces within a PCM study [6]. Indeed, radial kerneled Family QSAR models with ligand descriptors (Table 2) failed to model the data, being the RMSEP_{ext} and ${R}_{0\phantom{\rule{1em}{0ex}}\mathit{\text{ext}}}^{2}$ values respectively: 0.96 and 0.31 (adenosine receptors), 0.97 and 0.38 (GPCRs), and 1.13 and 0.48 (dengue virus NS3 proteases).
Strong mapping power of the normalized polynomial kernel
Radial and polynomial kernels have been traditionally used in QSAR and PCM modeling, [16, 63] but the versatility of other kernels for bioactivity modeling has been recently demonstrated [63–65]. To investigate this point in the frame of GP models, we compared the performance of various kernels (Bessel, Laplacian, NP, and PUK) with the radial and polynomial kernels.
As described above, in contrast to Huang et al., [63] we found the normalized polynomial (NP) kernel to have enough mapping power to model the three datasets (Table 2). Nonetheless, in the case of the dengue virus NS3 proteases dataset, although NP kernel produces a statistically correct modeling with RMSEP_{ext} and ${R}_{0\phantom{\rule{1em}{0ex}}\mathit{\text{ext}}}^{2}$ values of 0.48 and 0.91, it is slightly outperformed by the Bessel kernel, which displays respective RMSEP_{ext} and ${R}_{0\phantom{\rule{1em}{0ex}}\mathit{\text{ext}}}^{2}$ values of 0.44 and 0.92 (Table 2). The PUK kernel [65] exhibited strong mapping power in previous studies of HIV-1 proteases and histone deacetylases (HDAC) inhibitors, [63, 64] but in the present study we could not obtain satisfactory models for none of the three datasets. The Laplacian and Bessel kernels allow a proper mapping of the three datasets with ${R}_{0\phantom{\rule{1em}{0ex}}\mathit{\text{ext}}}^{2}$ values within the range 0.60–0.90 (see Table 2 for further details).
For the adenosine receptors dataset, different statistics values are observed between the internal and external validation, as the RMSEP_{ext} values are larger for the radial kernel (0.68) than for the polynomial and Bessel kernels (0.63 in both cases). Nonetheless, a different picture is observed for RMSEP_{int}, as the values for the radial, polynomial and Bessel kernels are 0.69, 0.64 and 0.70 p K_{ i } units. Although RMSEP_{ext} and RMSEP_{int} values are similar, the small increase of RMSEP_{ext} with the Bessel kernel might suggest a slight degree of overfitting [66].
GP and SVM perform on par
To probe the linearity of the datasets, we trained linear PLS models. For two datasets, PLS appears unable to infer the complex (non-linear) relationships within the data, leading to RMSEP_{ext} and ${R}_{0\phantom{\rule{1em}{0ex}}\mathit{\text{ext}}}^{2}$ of 1.00 and 0.30 for the adenosine receptors, and 1.05 and 0.27 for the GPCRs datasets, respectively (Table 2). At contrary, the dengue NS3 proteases dataset presents a clearly linear relationship, with RMSEP_{ext} and ${R}_{0\phantom{\rule{1em}{0ex}}\mathit{\text{ext}}}^{2}$ values of the PLS model of 0.49 and 0.91. However, on the same dataset, the model obtained with a linear kerneled GP model slightly outperformed PLS, with respective RMSEP_{ext} and ${R}_{0\phantom{\rule{1em}{0ex}}\mathit{\text{ext}}}^{2}$ values of 0.48 and 0.91.
Noise influence on GP depends on the kernel
Predicted confidence intervals follow the cumulative density function of the Gaussian distribution
GP predictions mostly follow the cumulative Gaussian distribution
GP determine the applicability domain of the model
Interestingly, while the average differences between predicted and observed bioactivities are close to zero for the subsets of GP errors of 0.25, 0.5 and 0.75, this average value is biased towards few tenths of a p K_{ i } unit (Figure 4) for the subset displaying the largest GP error. This observation indicates that errors on bioactivities are underestimated by the GP model for compound-target pairs distant from the training set. GP models with the NP and radial kernels provide prediction errors in agreement with the cumulative Gaussian distribution, which is the maximum theoretical precision attainable. Furthermore, the applicability domain of GP models can be determined from the errors predicted by GP.
Analysis of GP performance pertarget
To further understand the predictive capability of GP models on each analyzed target, we trained ten GP models with the NP kernel. Different seed values were used for the generation of the training and the external sets. Once the GP predictions have been obtained, we divided the external set into subsets grouped by target, and calculated average ${R}_{0\phantom{\rule{1em}{0ex}}\mathit{\text{ext}}}^{2}$ and RMSEP_{ext} values on these subsets. This analysis per target was conducted only on the datasets of adenosine receptors and GPCRs, because of their large sizes and numbers of involved targets.
Adenosine receptors
On the other hand, the results cannot be related to the chemical diversity of the compounds, analyzed with pairwise Tanimoto similarity (Additional file 1: Figure S3). Indeed, the two targets displaying the largest variability in the range of 0.50–0.75 Tanimoto similarity are rat A_{3} (AA3R RAT) and human A_{2b} (AA2BR HUMAN), for which quite different performances are observed (RMSEP_{ext} in the 0.70–0.75 range and in the 0.59–0.61 range respectively: Figure 5). Similarly, human A_{1} (AA1R HUMAN) and A_{2a} (AA2AR HUMAN) receptors, display the smallest variability for compounds, and show quite different levels of performance (${R}_{0\phantom{\rule{1em}{0ex}}\mathit{\text{ext}}}^{2}$ in the 0.56–0.60 range and in the 0.70–0.74 range respectively).
The lack of connection between the performance and the chemical diversity could arise from the binding site residue selection, which might not be equally suited for all adenosine receptors. This is supported by two other facts, namely: (i) the differences in extracellular loop length that are known for the adenosine receptor paralogues; and (ii) secondly the knowledge that these loops are important for ligand binding [67–69].
GPCRs
In the GPCR dataset, the best RMSEP_{ext} (Additional file 1: Figure S4) and ${R}_{0\phantom{\rule{1em}{0ex}}\mathit{\text{ext}}}^{2}$ (Additional file 1: Figure S5) values are obtained on target subsets with a number of annotated compounds larger than 200 (in grey in Additional file 1: Figures S4 and S5). Between the subsets, no major differences in performance are observed for an amount of annotated compounds between several hundreds and over 1500. It is however noticeable that the predictive ability of the models increased as the target space included in the training dataset broadened. Indeed, a bioactivity selection previously done including information from 26 human aminergic GPCRs (4,951 datapoints), marked with an asterisk in Additional file 1: Table S2, did not produce any sound statistical metrics, as ${R}_{0\phantom{\rule{1em}{0ex}}\mathit{\text{ext}}}^{2}$ values lower than 0.40 were obtained whatever the kernel or machine learning algorithm used. But, the addition to the first selection of the bioactivities measured on mammal orthologues improved the prediction, although some of the additional bioactivity sets were singletons (Additional file 1: Table S2).
A large diversity of performance with RMSEP_{ext} values in the range of 0.00–2.50 p K_{ i } units is observed for the targets annotated with one compound (Additional file 1: Figure S4). A relationship can be nevertheless established between these performances and the number of annotated compounds on orthologues proteins. For example, the 5-HT2C mouse receptor (5HT2C MOUSE) annotated with three compounds exhibits a mean RMSEP_{ext} value between 0.00 and 0.20 p K_{ i } units (Additional file 1: Figure S4), because 345 and 558 compounds are respectively annotated on the orthologue rat and human 5-HT2C receptors. The good performance obtained for this mouse receptor is probably due to the similarity of the 345 and 558 compounds to the ones annotated to the 5-HT2C mouse receptor. The importance of various targets for GP prediction was assessed for the adenosine receptors and GPCRs datasets. To obtain statistically validated models, a balance has to be found between two trends: (i) the inclusion of bioactivity information from orthologues improves the predictive ability of the models for both datasets, but (ii) an increase of the chemical diversity might hamper the acquisition of sound models as shown for the adenosine receptors dataset.
Model interpretation of ligand descriptors
Compounds bioactivity depends on multiple weak contributions of chemical substructures
The influence of the substructures on compound bioactivities, for both the adenosine receptors and the GPCRs, was analyzed as described in section Interpretation of ligand substructures. In the present study, the contribution of more than 90% of substructures to the p K_{ i } values is close to zero (black regions in Additional file 1: Figure S6). We observed similarly (data not shown) that chemical substructures contributing in a very variable way to the p K_{ i } values (average contribution equal to zero and standard deviations in the range of 0.50 - 1.00 p K_{ i } units), are present in sets of compounds displaying large variability in experimental bioactivity on a given target.
Hence, more than 90% of the substructures from the datasets analyzed here, display alternatively the following properties: (i) they are not implicated in compound bioactivity as their presence or absence does not influence compounds bioactivity, (ii) their contribution to the p K_{ i } values, is conditioned to the presence or absence of other substructures [70].
The highest contributions to the p K_{ i } values, on both the GPCRs and the adenosine receptors datasets, is close to 1 p K_{ i } units (Additional file 1: Figure S6), in the range similar to those obtained by van Westen et al.[15]. Therefore, even those few substructures with a large contribution, highlighted in Additional file 1: Figure S6, do not explain a large proportion of the bioactivity.
ARD provides a biologically meaningful interpretation of PCM models
GP models were interpreted on the basis of ligand descriptors. For datasets where ligands are compound descriptors (GPCRs and adenosine receptors datasets), the interpretation was not conclusive. By contrast, the interpretation of GP models according to the amino acids of the tetra-peptide ligands in the dengue datasets gave biologically meaningful results, in agreement with the scientific literature [41]. In that way, ARD can be applied to biologically interpret systems: e.g. identify residues responsible for compound binding. Additionally, ARD with the radial kernel can model non-linear relationships, which is not possible with PLS without the introduction of (not easily interpretable) cross-terms [6, 41].
Discussion
In the present study, we have demonstrated that Gaussian Processes (GP) allow to predict compound bioactivities on biomolecular targets. The statistically soundness of GP models is observed for a broad panel of kernels, among which the NP and radial kernels display the best results. GP and SVM display statistically similar performance for the modeling of multispecies proteochemometric datasets of different sizes. Moreover, Family QSAR and PCM models were trained on the same number of datapoints and PCM produced much better results than Family QSAR, due to the introduction of target descriptors.
GP were applied on the following datasets: two large datasets involving GPCRs and adenosine receptors and one small dataset (199 datapoints) comprising four dengue NS3 proteases. The dengue dataset exhibits a high degree of linearity, as demonstrated by the high performance of both PLS and GP with a linear kernel on this dataset. Unsurprisingly, a better performance of GP is observed with different kernels for the dengue dataset than for the two other ones, due to the high matrix completeness in the dengue dataset and to its linearity. The satisfactory results obtained for the dengue dataset encourages the application of GP to model relatively small datasets issued from a single laboratory. The use of such in-house datasets would reduce the bias introduced by annotation errors and by the use of non-normalized experimental conditions.
The inclusion of chemical and target information from several organisms (orthologues) increases model performance and the applicability of models to predict bioactivity for new compound target-combinations. These observations are in favor for the routine inclusion of multispecies bioactivity information in PCM settings. These results disagree with Gao et al.[72], who stated that the addition of orthologues to human aminergic GPCRs would reduce the AD. Our understanding of the results obtained here is that the incorporation of bioactivity data from a wide range of species led to a significant increase of models performance given that binding patterns tend to be conserved among orthologues [73]. We have seen on the GPCR dataset, that the inclusion of singletons compounds bioactivities on human orthologues helps to increase models performance. This may be of tremendous relevance in the often encountered cases where limited bioactivity information is known on a given human target, but a much larger number of bioactivities have been measured on orthologues of this target [16, 73, 74]. Our results suggest that the chemical diversity considered and the number of datapoints have to be balanced to obtain sound models while exhibiting proper predictive abilities.
An additional outcome of GP with respect to SVM is the estimation of the uncertainty of predictions. Indeed, the Bayesian formulation of GP permits to obtain intervals of confidence for individual predictions defined from the GP predicted variance. These intervals were shown to be in agreement with the cumulative Gaussian distribution when using the radial and NP kernels, but not always for the Bessel or Laplacian kernels, highlighting that the kernel choice has to be made in the light of both models performance and reliability of the predicted variances. We have also shown here that GP using as covariance function the polynomial or the NP kernel can handle noisy datasets, as the GP performance is only slightly affected when noise is introduced in the data. Nonetheless, each kernel should be chosen in the light of underlying structure of the dataset, as the kernel controls the prior distribution over functions, and thus the models generalization properties [48, 75]. It is noteworthy to mention that we have implemented a broad, though not exhaustive, panel of kernels, which is susceptible to be further completed with other base kernels or kernel combinations (composite kernels) [48, 75, 76].
GP can consider individual experimental errors as input for the probabilistic model which may constitute a preeminent advantage when gathering information from diverse sources, each of which including distinct levels of experimental uncertainty [33]. In the present study, an approximation of the experimental uncertainty of heterogeneous p K_{ i } values, recently reported by Kramer et al.[22] to exhibit a standard deviation of 0.54 p K_{ i } units, has been introduced in the model. Nonetheless, GP allow the inclusion of the uncertainty of each individual datapoint into the model, which might lead to a more accurate modeling pipeline in cases where the experimental uncertainty of each datapoint is available.
Traditionally, the application of GP to model large datasets has been limited since the inversion of the covariance matrix scales with the cube of its dimension, i.e. GP is an algorithm of complexity O(N^{3}) [31, 48]. In the present study, we have not reported training times since models have been trained with GP implementations coded in different programming languages (subsection Machine learning analyses and implementation). In the experience of the authors, the application of ARD is limited by the size of the datasets, being not applicable in practice to datasets with more than several thousands of datapoints, or with more than several hundreds of descriptors. Nevertheless, new GP implementations have proved to seemingly decrease calculation times, [77–79] which might increase the applicability of GP to large PCM datasets in the future.
Overall, we have shown here that GP simultaneously provides bioactivity predictions and assessment of their reliability. The application of GP to PCM datasets, gives the insight that GP could also be very useful in the drug discovery for personalized medicine, when the target space includes several mutants of a given target [15, 80]. In the same way, GP could even be used in the context of decision making in clinics [81].
Conclusion
Gaussian Processes (GP) have been proposed and tested for the prediction of bioactivity measurements, and found to perform at the same level of statistical significance as Support Vector Machines (SVM). In addition, GP is the only method, up to now, to give predictions as probability distributions, thus permitting the estimation of errors on the bioactivity predictions as well as an estimation of the applicability domain. Moreover, GP are tolerant to noisy bioactivities. GP models trained on PCM datasets can also be used to analyze the effect of ligand features (compound substructures or peptide residues).
Declarations
Acknowledgments
Cesar Roberto de Souza from the Federal University of São Carlos (UFSCar) is acknowledged for fruitful discussions about kernels. ICC thanks the Paris-Pasteur International PhD Programme for funding. GvW thanks EMBL (EIPOD) and Marie Curie (COFUND) for funding. ICC and TM thanks CNRS, Institut Pasteur and ANR bipbip for funding. EBL thanks the Dutch Research Council (NWO) for financial support (NWO-TOP #714.011.001). AB thanks Unilever and the European Research Commission (Starting Grant ERC-2013-StG 336159 MIXTURE) for funding.
Authors’ Affiliations
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