Evolutionary algorithms (EA) are based on the principles of biological evolution and have often been successful in solving complex search and optimization problems. Majority of bioinformatics applications of EA have been in the discovery of motifs such as transcription factor binding sites 4950515253. Yet, only a few researchers have used EA for the prediction of peptides binding to protein sequences 36.

An EA consists of (1) representing input variables as individuals or chromosomes (binary or real valued) in a population, (2) formulating the fitness (objective function) to evaluate individuals, (3) generating a new population by genetic operations (such as reproduction, crossover, and mutation) on the current population, and (4) determining if the population has reached the optimal fitness. The algorithm begins with an initial population and evolves over time. At a particular instance of evolution, every individual is evaluated by its fitness. New populations (offspring) are produced from highly fit individuals (parents) selected, which undergo genetic operations. Each offspring is paired and compared to its parents. Highly fit individuals are retained in the population while less fit individuals are discarded. Search mechanisms such as elitism, constraint-handling, and multi-objective optimization are available for finding a better spread of solutions, depending on the needs of the optimization problem 54555657.

Multi-objective evolutionary algorithms (MOEA) are used to solve problems which require simultaneous optimization of a number of competing objective functions 58596061. MOEA maintains a set of solutions ranked by their dominance at a given instant of the evolution. A solution is said to dominate another if it is better or equal with respect to all objectives and strictly better in at least one objective 58. Often, there are more than one non-dominated solutions, representing the best ones, collectively known as the Pareto front. MOEA algorithms result in a Pareto optimal set of solutions.

Non-dominated Sorting Genetic Algorithm II (NSGA-II) was recently introduced to incorporate several new genetic mechanisms for better convergence, such as non-dominated sorting, elitism, diversity preservation, and constraint handling 58. In NSGA-II, a population is subjected to several rounds of non-dominated sorting. That is, all the non-dominated individuals are identified and assigned the same fitness value until a new set of non-dominated solutions is found. The solutions found in subsequent rounds are assigned fitness values lower than those in the previous rounds. This process continues until the whole population is partitioned into non-dominated fronts with diverse fitness values. The elitism prevents the loss of fit individuals encountered in earlier generations by allowing earlier solutions to survive in the subsequent generations. The diversity of Pareto-optimal solutions is maintained by imposing a measure referred to as crowding distance. A solution that satisfies the constraints defined by the objective functions is called a feasible solution.

In this paper, we attempt to find an optimal motif describing peptide binding to MHC class II molecules, using experimentally determined binding data. There are several factors that impede the derivation of such a consensus motif. The first is the strong resemblance among the peptides isolated in a single experiment and the second is the diversity among different datasets. A motif derived from a dataset lacking diversity indicates a bias towards the dataset used in deriving the motif. Such motifs are difficult to generalize on other experimental or previously unseen datasets. The MOEA based motif detection algorithm is designed to find a consensus motif on I-A^g7 datasets, which alleviates the influences arising from biased datasets and thereby predicts binding peptides more accurately in new datasets.

We use our approach to find a consensus motif on seven experimental datasets of peptides binding to I-A^g7 molecules, obtained from literature 40414243626364. The motif is validated using an independent testing set generated from the Stratmann dataset 46. The overall quality of prediction was measured using area under curve (AUC) of the receiver operating characteristics (ROC) curve 656667. AUC values of all feasible solutions in the final population of EA were evaluated and the solution with the highest AUC was chosen as the consensus motif (see Additional file 1).

Table 1 shows the information of the datasets extracted from literature, which were used in the training. A blank '-'indicates the unavailability of a particular information. As an example, the details of the experimental motif of Reizis et al are given in Table 2. Table 3 shows the performance when an experimental motif is used to predict peptide binders in other datasets. As seen, a motif of a particular experiment does not characterize peptide binding of I-A^g7 molecules in other datasets. Table 4 shows the cross-validation performance of two motifs (by self-discovery and guided-discovery) derived using MOEA; in a particular cross-validation run, one experimental dataset was excluded and the motif was derived using the information of the remaining datasets. The motif was tested for predicting binders and non-binders of the left-out dataset. The self-discovery approach uses only the binding information whereas the guided-discovery uses both binding information as well as information associated with experimental motifs. As seen in Table 4, by achieving AUC values greater than 0.7 for all cross-validation runs, MOEA derived motifs demonstrate better generalization capabilities compared to experimentally determined motifs. The binding motifs derived from self-discovery and guided-discovery are illustrated as sequence logo plots 68 in the Additional file 2.

To compare the performance of our method with earlier methods, a training dataset was created by combining all the experimental datasets given in Table 1. Motifs derived on the training dataset were tested on an independent test dataset – a balanced set generated from Stratmann dataset. The Stratmann dataset was balanced by adding randomly generated non-binders. Twenty five such balanced test datasets were assembled by generating random samples starting from different seeds and adding them to the Stratmann dataset. The results reported are based on the average AUC values over all balanced test sets. Figure 1 shows comparison of performances of motifs derived by MOEA and by earlier motif prediction approaches such as MEME and RANKPEP. An increase of 4–10% in predictive performance is observed with MOEA over the other approaches.

Comparison of performances of MOEA derived motifs for BM-Set1 (see Table 5) with enhanced Gibbs sampler 32, TEPITOPE 35, SVRMHC 8 and ARB 10, is given in Table 6. As seen, MOEA shows comparable or superior performance with Gibbs sampler on all datasets except for the Southwood dataset. Out of the ten non-redundant (NR) datasets, the MOEA outperformed Gibbs sampler, TEPITOPE, SVRMHC and ARB by seven, nine, eight and ten datasets, respectively.

The performance of MOEA on BM-Set2 (see Table 7) was compared with Gibbs sampler 32, TEPITOPE 35, SVRMHC 8, ARB 10 and NetMHCII 11. Each allele dataset was subjected to five-fold cross-validation and the results are given in Table 8. The present method shows comparable or superior performance on majority of allele datasets compared to Gibbs sampler, SVRMHC, TEPITOPE, and NetMHCII. A fair comparison of ARB method cannot be drawn because the method has been trained on quantitative data obtained from IEDB 10.

Several I-A^g7 datasets were extracted from literature 40414243626364 and from Brusic, V.(unpublished data). The numbers of binders and non-binders in each dataset are given in Table 1. The datasets consist of short peptides ranging from 9–30aa in length. Their binding affinities had been experimentally determined by independent studies and classified as binders or non-binders based on IC₅₀ values according to the following scheme 41: good binder (IC₅₀ = 100 nM); weak binder (IC₅₀ = 2000 nM); non-binder (IC₅₀ = 50000 nM). The datasets in 40414243626364 were combined into a single training dataset and curated by removing duplicates and redundancy as follows: if a binder is a subsequence of another binder sequence, the longer binder sequence is discarded; if a non-binder is a subsequence of another non-binder, the shorter subsequence is discarded. Let the curated whole dataset be referred to as training dataset here onwards and it be denoted by D = {(x_i, v_i): i = 1, 2,.... N} where N is the number of total peptide sequences and x_i is the i-th peptide sequence with the label v_i ε {b, nb} indicating whether the sequence x_i is a binder (b) or a non-binder (nb). The number of peptides in the training set N = 438 in which the number of binders N_b = 304 and the number of non-binders N_nb = 134.

The set of experimentally validated I-A^g7motifs 38394041424344 derived largely from uncorrelated datasets 40414243 was extracted and is illustrated in Table 1 with the distribution of binders and non-binders in each dataset. Table 2 illustrates an experimentally validated motif of I-A^g7 reported by Reizis et al 40. Experimental motifs are described by the anchor positions and binding affinities of amino acids of the motif. The residues which contribute significantly to the peptide binding are called primary anchor residues and positions they reside are called anchor positions. An amino acid occupying a specific position within a motif is characterized as well tolerated, weakly tolerated, or non-tolerated based on its involvement in the binding process.

An independent dataset was generated from binders of Stratmann dataset 46, consisting of a diverse set of I-A^g7 binding peptides with their binding affinities, to find the test accuracies in predicting binders and non-binders. The Stratmann dataset was balanced with randomly generated 9-mer non-binders so that for testing dataset, N_b = N_nb = 112.

A k-mer motif of amino acids is characterized by a PSSM Q = {q_ia}_{k × 20} where q_ia denotes the binding strength of the site i when it is occupied by amino acid a. The binding score of a putative motif is computed by adding the binding scores assigned to each amino acid at the respective positions. The binding score indicates the likelihood of the motif binding to the molecule. The binding score s_i of sequence x_i = (x_i,1, x_i,2,...x_{i, n}) of length n is determined by the maximum value of binding scores computed for all k-mer subsequences in x_i:

s i = max ⁡ j { s i j : j = 1 , 2 , ⋯ n − k + 1 } MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaem4Cam3aaSbaaSqaaiabdMgaPbqabaGccqGH9aqpdaWfqaqaaiGbc2gaTjabcggaHjabcIha4bWcbaGaemOAaOgabeaakiabcUha7jabdohaZnaaBaaaleaacqWGPbqAcqWGQbGAaeqaaOGaeiOoaOJaemOAaOMaeyypa0JaeGymaeJaeiilaWIaeGOmaiJaeiilaWIaeS47IWKaemOBa4MaeyOeI0Iaem4AaSMaey4kaSIaeGymaeJaeiyFa0haaa@4BBC@

where s_ij denotes the binding score of the subsequence beginning at location j of the sequence i, which is given by

s i j = ∑ l = 1 , 2 , ⋯ k q ( j + l ) , x i ( j + l ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaem4Cam3aaSbaaSqaaiabdMgaPjabdQgaQbqabaGccqGH9aqpdaaeqbqaaiabdghaXnaaBaaaleaacqGGOaakcqWGQbGAcqGHRaWkcqWGSbaBcqGGPaqkaeqaaOGaeiilaWIaemiEaG3aaSbaaSqaaiabdMgaPjabcIcaOiabdQgaQjabgUcaRiabdYgaSjabcMcaPaqabaaabaGaemiBaWMaeyypa0JaeGymaeJaeiilaWIaeGOmaiJaeiilaWIaeS47IWKaem4AaSgabeqdcqGHris5aaaa@4D17@

and assuming that only one motif instance exists in every sequence, the location j* of the motif is given by

j ∗ = arg ⁡ max ⁡ j { s i j : j = 1 , 2 , ⋯ n − k + 1 } MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaemOAaO2aaWbaaSqabeaacqGHxiIkaaGccqGH9aqpcyGGHbqycqGGYbGCcqGGNbWzdaWfqaqaaiGbc2gaTjabcggaHjabcIha4bWcbaGaemOAaOgabeaakiabcUha7jabdohaZnaaBaaaleaacqWGPbqAcqWGQbGAaeqaaOGaeiOoaOJaemOAaOMaeyypa0JaeGymaeJaeiilaWIaeGOmaiJaeiilaWIaeS47IWKaemOBa4MaeyOeI0Iaem4AaSMaey4kaSIaeGymaeJaeiyFa0haaa@4F4D@

That is, the most likely motif instance of sequence x_i, say m_i, is given by the sequence m_i = (x_ij*·x_{ij* + 1},... x_{ij* + k-1}).

We derive a consensus motif from the training dataset which consists of peptides from several experiments and of varying lengths. The positions of binding cores within the peptides are unknown. The elements of the PSSM are represented as 20k-tuples (q_ia, : i = 1,... k; a ε Ω) where Ω represents the amino acid alphabet. Each element in the k-tuple is converted to a real number representation using a binary word of size θ so that q_ia ∈ [0, 2^θ-1]. The k-mer motif is therefore represented by an individual of 20kθ long string in the EA. Let the population at t-th iteration of the evolution is denoted by q(t) = {q₁(t), q₂(t),..... q_M (t)} where q_j(t) represents an individual in a population of size M.

The fitness function is designed to arrive at an optimal consensus of the motif, by using the training dataset. A solution is evaluated based on its ability to maximize the accuracies in identifying true binders (TP) and true non-binders (TN) as well as to widen the gap between the total score for binders and non-binders. This is achieved by two fitness functions: f₁ to minimize the sum of false positives (FP) and false negatives (FN), and f₂ to minimize the ratio between the average cumulative scores of non-binders and binders:

f₁ = FN + κ₁ FP

f 2 = N b N nb ∑ i = 1 N s ( m i ) δ ( v i = nb ) ∑ i = 1 N s ( m i ) δ ( v i = b ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@62AE@

Eqs. (4) and (5) are minimized and subjected to following two constraints:

FP N nb ≤ 1 α 1 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaqcfa4aaSaaaeaacqqGgbGrcqqGqbauaeaacqWGobGtdaWgaaqaaiabb6gaUjabbkgaIbqabaaaaOGaeyizImAcfa4aaSaaaeaacqaIXaqmaeaaiiGacqWFXoqydaWgaaqaaiabigdaXaqabaaaaaaa@38F1@

FN N b ≤ 1 α 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaqcfa4aaSaaaeaacqqGgbGrcqqGobGtaeaacqWGobGtdaWgaaqaaiabbkgaIbqabaaaaOGaeyizImAcfa4aaSaaaeaacqaIXaqmaeaaiiGacqWFXoqydaWgaaqaaiabikdaYaqabaaaaaaa@378C@

where s(m_i) denotes the score computed for the most likely motif instance m_i of sequence x_i of the training dataset, and Kronecker δ is one when the argument is satisfied and otherwise is zero. N_b and N_nb are the total counts of binders and non-binders in the dataset. The constant κ₁ (>N_b/N_nb for N_b > N_nb, or vice versa) was empirically determined to minimize the number of false positives. The two parameters α₁ (<<N_nb) and α₂ (<<N_b) are set to minimize FP and FN rates, respectively. If none of the individuals satisfies the above constraints, MOEA reports no feasible solution. Given the training set, a few trial runs with different initializations are necessary to determine the best values of α₁ and α₂.

The description of an experimental k-mer motif conveys three kinds of information at each site: (1) the amino acid occupied, (2) the tolerance level of the amino acid, and (3) the strength of binding. Let us denote a k-mer motif validated in experiment "e" by m(e) and the tolerance level of the residue at site j by ρ_j where ρ_j ∈ {well, weak, unknown, non – tolerated}. The binding strength of site j is expressed by σ_j ∈ {primary – anchor, secondary – anchor, other}. Then, the binding score for a k-mer experimental motif is given by

s ( m ( e ) ) = ∑ j = 1 k ρ j ⋅ σ j MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaem4CamNaeiikaGIaemyBa0MaeiikaGIaeeyzauMaeiykaKIaeiykaKIaeyypa0ZaaabCaeaaiiGacqWFbpGCdaWgaaWcbaGaemOAaOgabeaakiabgwSixlab=n8aZnaaBaaaleaacqWGQbGAaeqaaaqaaiabdQgaQjabg2da9iabigdaXaqaaiabdUgaRbqdcqGHris5aaaa@4482@

In this algorithm, we assume that experimentally determined motifs are available along with the experimental datasets. An MOEA is proposed to determine a motif closer to experimental motifs. An objective function f₃ is proposed to best represent the characteristics of the motif that is close to the knowledge embedded in the experimental motifs:

f 3 = ∑ e | Q ^ − Q ( m ( e ) ) | MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaemOzay2aaSbaaSqaaiabiodaZaqabaGccqGH9aqpdaaeqbqaamaaemaabaGafmyuaeLbaKaacqGHsislcqWGrbqucqGGOaakcqWGTbqBcqGGOaakcqqGLbqzcqGGPaqkcqGGPaqkaiaawEa7caGLiWoaaSqaaiabbwgaLbqab0GaeyyeIuoaaaa@3FA7@

where Q^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGafmyuaeLbaKaaaaa@2D0E@ denotes the estimated PSSM of the motif. We use the same objective function in Eq. (4) to accurately predict binders of the training dataset. The MOEA minimizes the objective functions given in Eqs. (4) and (9), subjected to the two constraints given in Eqs. (6) and (7). The summation in Eq. (9) is taken over all the experimental motifs and |Q^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGafmyuaeLbaKaaaaa@2D0E@ - Q(m(e))| is the sum of squares of differences between individual elements of weight matrices Q^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGafmyuaeLbaKaaaaa@2D0E@ and Q(m(e)). The knowledge of the experimental motif is incorporated to the consensus motif adaptively with the distance function used in f₃. Further, the fitness f₁ optimizes the specificity and sensitivity of the prediction of binders.

The elements in the PSSM of experimental motifs are set to values within the same range [0, 2^θ-1] as before. The following procedure is adopted to determine the elements of Q(m(e)): a well tolerated amino acid at an anchor position of the motif receives the highest possible score of 2^θ-1; the lowest score of zero is assigned to a non-tolerated residue; weakly tolerated residues and residues at secondary anchor positions receive of (2^θ-1)/2; and all the other unknown positions receives a score of (2^θ-1)/3.

The binding scores of I-A^g7 experimental motifs were computed using Eq. (8) by assigning the following values for binding strengths: primary = 4, secondary = 2, and others = 1, and for anchor positions: well = 4, weak = 2, non-tolerated = -4, and unknown = 0. The experimentally determined motifs were used with peptide data in the guided-discovery of motifs.

We used AUC to compare performance of the proposed methods with earlier approaches 2834 and experimental motifs 38394041424344. Whether a peptide is a binder or a non-binder is determined by a threshold of the binding score. By varying this threshold, the ROC curve was plotted, from which AUC value was obtained. A comparison of performances of the methods is given in Figure 1.

In order to compare to the MEME method, only binders in the I-A^g7 training set were submitted to MEME motif discovery tool at the prediction server 71. The motif of 9-mer length was obtained with the following options: zero or one motif per sequence, minimum and maximum width = 9. The performance accuracy of RANKPEP approach on the testing dataset was carried out by uploading the dataset to the online prediction server at 72 with a 4% binding threshold 34.

The proposed self-discovery approach was tested on BM-Set1, i.e., HLA-DRB1*0401, which consists of one training set and 10 testing datasets and had been earlier used to benchmark a number of motif finding algorithms 25263273. The performance of MOEA was compared with earlier methods 8103235.

The training set consisting of binders and non-binders was assembled as follows: an ensemble of 532 unique binding peptides were extracted from SYFPEITHI 44 and MHCPEP 63 databases and a set of 177 unique non-binders were extracted from the MHCBN database 20. The datasets were pre-processed by removing peptides that did not allow a hydrophobic residue at P1 position of all putative 9-mer binding cores and unnatural peptides containing more than 75% alanine 32. The preprocessed binder set has 456 unique peptides with a length distribution ranging from 9 to 30 amino acid residues.

Of the 10 testing datasets, 8 datasets were taken from the MHC-bench as described in 74. The other 2 datasets were extracted from experiments described by Southwood 75 and Geluk 76. An affinity of (IC₅₀ = 1000 nM) was taken as the threshold for peptide binding as described in 75. Homology reduction had been carried out on all datasets in order to reduce the chances of over-fitting due to the redundancy of datasets. The peptides in the non-redundant (NR) datasets had sequence similarities less than 90%. The number of binders and non-binders in the original and NR datasets are given in Table 5.

We tested our method on BM-Set2 comprising of 3 mouse alleles and 13 HLA alleles made available at 77. These quantitative peptide datasets had been extracted from the IEDB at 78. The number of binders and non-binders in each dataset is given in Table 7. The DRB3-0101 allele dataset was excluded from the benchmark dataset because of the significant imbalance between binders and non-binders (3 binders and 99 non-binders). With this dataset, we compared our method with 810113235.

The range of positional scores was set with θ = 7. For each run of MOEA, the population size M = 500, crossover probability p_c = 0.9, and mutation probability p_m = 0.005 were used. The process was terminated after 300 generations as no significant improvement in the convergence was observed during the experimental trial sessions. The parameters of the fitness functions were empirically determined for optimum performance within the following ranges: κ₁ = 1~2.5, α₁ = 5.0–6.0, and α₂ = 1.0–2.0. The parameters κ₁ = 2.5, α₁ = 6.0, and α₂ = 2.0 were found to work well empirically for both datasets.