Accurate prediction of major histocompatibility complex class II epitopes by sparse representation via ℓ ₁-minimization

Data

Encoding scheme

The most common way of amino acid encoding is the binary encoding scheme represented by a 20-bit binary vector, where 19 bits are set to zero and one bit is set to 1. Property encoding, on the other hand, is based on a vector containing one or more amino acid properties. Property encoding has two main advantages over binary encoding. First, physicochemical properties play an important role in biomolecular recognition; therefore, this type of encoding is more informative. Secondly, property encoding mitigates the problem of flexible lengths. To test the reliability of property encoding, we used classical binary encoding and compared it against two property encoding methods, 11-factor encoding and divided physicochemical property scores (DPPS). The 11-factor encoding is calculated from physicochemical properties of amino acids as described by [12]. The properties were obtained from general physicochemical properties of amino acids and a number of properties identified in 3-D quantitative structure-activity relationship (QSAR) analysis [13]. The DPPS scheme was proposed by [14]. The DPPS descriptor was obtained by applying principal component analysis (PCA) to thousands of amino acid structural and property parameters. The resulting transformation yielded score vectors involving significant nonbinding properties of each of the 20 amino acids.

Thus, every vector correlates directly with the physicochemical properties of amino acids, allowing the prediction of class II-peptide interaction. Additionally, we mitigated the problem of flexible lengths since every epitope is represented as a vector of size 10 or 11.

Classification via sparse representation

We applied the selective nature of sparse representation to perform classification. As presented in [8], ℓ ₁-minimization techniques provide a satisfactory method to solve sparse representation problems. We propose a classifier based on the solution of an ℓ ₁-minimization problem for classification. A supervised learning system performing classification is commonly called a classifier.

Formally, given an input dataset, W = {w ₁,…,w _n }, a set of labels/classes T = {t ₁,…,t _n }, and a training dataset D = {(x _i ,t _i ):i = 1,…,n}, such that t _i is the label/class associated to the sample x _i , a classifier is a mapping from W to T, assigning the correct label t ∈ T to a given input w ∈ W, that is, $\mathcal{F}(\mathbf{w},\mathbf{D})=t.$

Let us consider a training data set $\left\{\right({\mathbf{x}}_i,t_i):i=1,\dots ,n\},{\mathbf{x}}_i\in {\mathbb{R}}^d,t_i\in \{1,2,\dots ,N\},$ where n is the number of samples and N the number of classes. The vector ${\mathbf{x}}_i\in {\mathbb{R}}^d,$ represents the i th sample (for instance containing “gene expression” values, special features, etc), and t _i denotes its corresponding label (in our case, binding or non-binding). Assume that d < n, that is, the length of each sample is less than the number of elements in the training dataset.

that is, the collection of sets of indices ${\left\{{\Omega }_i\right\}}_{i=1}^N$ forms a partition of the set {1,2,…,n}, where n is the amount of samples available in the training dataset.

and conclude that the testing sample y has label t=s. In this manner, we identify the class of the test sample y based on how effectively the coefficients associated with the training samples of each class recreate y.

Support vector machines (SVM)

We compared the results of our proposed method for classification problems with the well known SVM strategy that has been commonly used in different pattern recognition and machine learning applications. SVMs are a set of related supervised learning methods that analyze data and recognize patterns commonly used for classification and regression analysis. The original SVM algorithm was proposed by Vladimir Vapnik and the current standard implementation was proposed by Corinna Cortes and Vladimir Vapnik, [15]. Standard SVM takes a set of input data and predicts for each given input which of two possible classes the input is a member of, which makes the SVM a non-probabilistic binary linear classifier. Intuitively, an SVM model is a representation of the samples as points in space, mapped so that the samples of the separate categories are divided by a clear gap that is as wide as possible.

Slow training is a possible drawback of SVM approaches because SVMs are trained by solving quadratic programming problems where the number of variables is equal to the number of samples in the training data set. When a large number of training data is available, the training process might turn slow. More information about the different strategies used in SVM for classification problems are described in [16]. Here we use the implementation of SVM available in MATLAB as part of the Statistics Toolbox, and report the results for the best possible setup (using radial basis functions) found after an appropriate parameter tuning stage (model selection).

Evaluation of method performance

To evaluate the prediction performance and robustness of our algorithm, we performed a 10-fold (n-fold) cross-validation. An illustration of the 10-fold cross validation partition process is shown in Figure 1. In the n-fold cross-validation, all the binding and non-binding epitopes were mixed and then divided equally into n parts, keeping the same distribution of binders and non-binders in each part. Then n-1 parts were merged into a training data set while the remnant was taken as a testing data set. This process was repeated 10 times and the average performance of n-fold cross-validation computed. We then measured sensitivity (Sn), specificity (Sp), accuracy (Acc) and Matthew’s Correlation Coefficient (MCC) for every fold and then took the average (Avg) as shown in Table 2. In addition, we performed a ROC curves analysis using different I C ₅₀ thresholds.

Prediction accuracy

Techniques for predicting MHC binding include ANNs (NetMHCpan [6] and NN-Align [17]), position Specific Scoring Matrices (PSSMs) (RANKPEP [18]), and amino acid pairwise contact potentials as input vector for SVM (EpicCapo [10]). These methods have typical prediction accuracies of almost 70-90%[19]. Overall, our binding prediction accuracies are comparable to the reported 70-90% accuracies. Table 2 shows a comparison of the three encoding schemes with two alleles. While our method consistently favors DPPS encoding with the alleles tested, SVM shows a slightly better accuracy with 11-factor encoding. The experiments performed indicate that the physicochemical properties of amino acids are more informative in predicting MHC-II binding peptides. These results are also consistent with the MCC obtained for binary encoding, which yielded negative and zero scores on various occasions, implying that the predictions were not better than random predictions.

ROC curves analysis

We also applied ROC analysis to examine the performance of the ℓ ₁-minimization and SVM classifiers. An ROC graph is a plot with the false positive rate on the x axis (1 - specificity) and the true positive rate on the y axis (sensitivity). The point (0,1) is the perfect classifier: it classifies all positive cases and negative cases correctly. The point (0,0) represents a classifier that predicts all cases to be negative, while the point (1,0) is the classifier that is incorrect for all classifications. The ROC curves were calculated using the thresholds shown in Table 1 to distinguish binders from non-binders.

In Figures 2, 3 and 4 we present the corresponding ROCs for the sparse representation method and ROCs were calculated using different IC ₅₀ cutoff values.The area under the ROC curve (AUC) provides a measure of overall prediction accuracy. An AUC value of 0.5 indicates random choice; while values close to 1 indicate excellent predictive capabilities of the method used. AUC values were computed using trapezoidal rule for numerical integration. With DPPS encoding, the ℓ ₁-minimization method for predicting epitopes on H2-IA^d and HLA-DPA1*0103/DPAB1*0201 molecules rendered AUC values of 0.729 and 0.764, respectively, higher than any of the AUC of SVM for the same encoding scheme. However, with 11-factor encoding, the AUC obtained by SVM was 0.806 for molecule HLA-DPA1*0103/DPAB1*0201, a higher value than any AUC obtained by ℓ ₁-minimization. Tables 4, 5, 6, 7 and 8 show the values of sensitivity (Sn), specificity (Sp) and accuracy (Acc) for each of the IC ₅₀ cutoff points, when using both the 11-factor and DPPS encoding schemes.