Optimizing ligand conformations in flexible protein targets: amulti-objective strategy

Optimizing ligand conformations in flexible protein targets: a multi-objective strategy Esteban López-Camacho1 María Jesús García-Godoy1 José García-Nieto1 Antonio J. Nebro1 José F. Aldana-Montes1 Abstract Finding the orientation of a ligand (small molecule) with the lowest binding energy to the macromolecule (receptor) is a complex optimization problem, commonly called ligand–protein docking. This problem has been usually approached by minimizing a single objective that corresponds to the final free energy of binding. In this work, we propose a new multiobjective strategy focused on minimizing: (1) the root mean square deviation (RMSD) between the co-crystallized and predicted ligand atomic coordinates, and (2) the ligand–receptor intermolecular energy. This multi-objective strategy provides the molecular biologists with a range of solutions computing different RMSD scores and intermolecular energies. A set of representative multi-objective algorithms, namely NSGA-II, SMPSO, GDE3 and MOEA/D, have been evaluated in the scope of an extensive set of docking problems, which are featured by including HIV-proteases with flexible ARG8 side chains and their inhibitors. As use cases for biological validation, we have included a set of instances based on new retroviral inhibitors to HIV-proteases. The proposed multi-objective approach shows that the predictions of ligand’s pose can be promising in cases in which studies in silico are necessary to test new candidate drugs (or analogue drugs) to a given therapeutic target. Keywords Molecular docking · Multi-objective optimization · Metaheuristics 1 Introduction Ligand–protein docking is an optimization problem whose main objective is to predict a position and orientation of a smallmolecule(ligand)toalargerreceptormolecule(macromolecule)withminimumbindingenergy.Moleculardocking is based on two steps: (1) the prediction of the position and orientation of the ligand to the receptor and (2) the assessment of the ligand–receptor binding affinity that is obtained. This technique has been used to do studies in silico in drug discovery. In the last years, molecular docking has been focused on minimizing the final free energy binding using singleBJosé F. Aldana-Montes [email protected] 1Department of Computer Languages and Computing Sciences, Institute for Software Technologies and Software Engineering (ITIS), Biomedical Research Institute of Málaga (IBIMA), University of Málaga ETSI Informática, Campus de Teatinos, 29071 Málaga, Spain objective algorithms by using a force-field-based energy function (Ru et al. 2016; Peh and Hong 2016). In accordance with other past studies (López-Camacho et al. 2015), the differential evolution algorithm (DE) returned the best solutions in terms of energy and RMSD score compared to steady-state genetic algorithm (ssGA), generational genetic algorithm (gGA) and PSO (particle swarm optimization). A recent approach is DockThor (Leonhart et al. 2018), which involvesasingle-objectiveapproachbasedonminimizingthe final binding energy. DockThor was compared with a set of mono-objective approaches presented in the state of the art, including those of jMetalCpp (López-Camacho et al. 2014). The results showed that DockThor outperforms the selected jMetalCpp algorithm. In parallel, a series of studies based on multi-objective approaches to solve this problem have emerged (Gu et al. 2015; Janson et al. 2008; Sandoval-Perez et al. 2013). In López-Camacho et al. (2016), we presented a new multiobjective approach to minimize two objectives based on the RMSD score and the intermolecular energy (Einter). The first one corresponds to a similarity measure between the reference and the co-crystallized ligands. The second one is an while analysis of results and discussions are reported in Sect. 5. Section 6contains conclusions and future work. 2 Multi-objective ligand–protein docking Previously to describe the ligand–protein docking problem, we include a series of formal definitions concerning multiobjectiveoptimization.Withoutlossofgenerality,weassume that minimization is the goal for all the objectives. Definition 1 (Multi-objective Optimization Problem) Find a vector x∗=x∗ 1,x∗ 2,...,x∗ nwhich satisfies the minequality constraints gi(x)≥0,i=1,2,...,m,thepequality constraints hi(x)=0,i=1,2,...,p, and minimizes the vector function f(x)=[f1(x), f2(x),..., fk(x)]T, where x=[x1,x2,...,xn]Tis the vector of decision variables. The set of all values satisfying the constraints defines the feasible region Ωand any point x∈Ωis a feasible solution. We seek the Pareto optima. Definition 2 (Pareto Optimality) for every x∈Ωand I= {1,2,...,k}either ∀i∈I(fi(x)=fi(x∗))or there is at least one i∈Isuch that fi(x)>fi(x∗). This definition states that x∗is Pareto optimal if no other feasible vector xexists which would improve some criteria without causing a simultaneous worsening in at least one other criterion. Definition 3 (Pareto Dominance) A vector u=(u1,...,uk) is said to dominate v=(v1,...,v k)(denoted by uv)if and only if uis partially less than v.∀i∈{1,...,k},ui≤ vi∧∃i∈{1,...,k}:ui<v i. 2.1 Ligand–protein docking The main objective in this problem is to find a ligand’s conformation (L) to a receptor (R) that results with minimum energy. An energy scoring function is then calculated to evaluate the ligand–receptor interaction according to three components, which represent degrees of freedom: – The first one corresponds to the translation of the ligand, which is represented by the three position values (x,y,z) in the axis of the Cartesian coordinate space; – The second one includes the ligand’s orientation, which is modeled as a four variables quaternion including the angle slope (θ); – The third one represents the dihedral angles referring to the free rotation of torsion of the ligand and side chains of the receptor’s residues. aggregation of bound and unbound states of the molecular complex. This approach was applied to a set of 11 instances that involve flexible receptors and inhibitors. In that study, a set of representative multi-objective algorithms were compared and SMPSO (Nebro et al. 2009) was the optimizer that obtained the best overall results for the tested molecular docking instances. In the present study, we follow our previous research line by performing a thorough experimentation, with the motivation of assessing whether our initial claims could be generalized or not in the context of a large set of problems. We have performed an extended research from our previous study (López-Camacho et al. 2016), involving 73 instances used as benchmark by a previous study such as Morris et al. (2009). The macromolecules are therapeutic targets in which flexibility was applied. The ligands have different sizes (small, medium and large), and the obtained results can be analyzed taking into account the ligand size. Our motivation is then to provide molecular biologists with a set of trade-off solutions with different RMSD scores and Einter. It is worth noting that the optimized ligand structure, even with outstanding RMSD scores (measured in Å), does not necessary correspond to the already known best solution. From a pharmacological point of view, the macromolecule (the therapeutic target) might have other different ligand binding sites that do not correspond to the co-crystallized ligand binding site obtained from in silico studies, for example, the case of allosteric modulation, which refers to the alteration of the macromolecule functionality by the binding of a modulator (a ligand) at a distant binding site (Townsend et al. 2015). Other examples refer to those cases where the ligand docking avoids the enzyme–substrate interaction and its binding site is different from the co-crystallized ligand. In this context, the approach proposed in this paper provides a wide range of solutions that the practitioner can choose, i.e., a solution that corresponds to a ligand conformation with a higher RMSD score and lower Einter.The ligand related to this solution can have a position close to the entrance of the macromolecule’s active site or to some of molecular sites with known allosteric modulation. For this approach, the search methods used are several representative multi-objective algorithms which are: SMPSO, GDE3, MOEA/D and NSGA-II. Finally, in order to show the usability of this multi-objective strategy, we have chosen a set of flexible instances that correspond to flexible macromolecule– ligand complexes. In these examples, several solutions from the front of non-dominated solutions have been chosen and analyzed in terms of ligand binding site and molecular interactions. This paper is organized as follows: Sect. 2 describes the ligand–protein docking problem from a multi-objective formulation. Section 3 describes the optimization techniques evaluated. Experimental methodology is explained in Sect. 4, Each problem solution is encoded by a real-value vector of 7 +nvariables, in which the first three values are used for ligand translation, the following four variables represent the ligand and/or macromolecule orientation, and the last nvariables represent the dihedral angles of ligand and receptor positions. In addition, a grid-based methodology is implemented (Morris et al. 2009) to perform the energy computation of molecular conformations. More specifically, the Einter is one of the force-field-based energy function terms used in AutoDock 4.2. This energy term is calculated as follows: Einter =QR−L bound +QR−L unbound (1) QR−L bound and QR−L bound are the states of bound and unbound of the ligand–receptor complex, respectively. Q=Wvdw  i,jAij r12 ij −Bij r6 ij  +Whbond  i,j E(t)Cij r12 ij −Dij r10 ij  +Welec  i,j qiqj ε(rij)rij +Wsol  i,j (SiVj+SjVi)e(−r2 ij/2σ2) (2) These energetic evaluation terms (Q) take into account dispersion/repulsion (vdw), hydrogen bonds (hbond), electrostatics (elec) and desolvation (sol). In Eq. 2, constants for van der Waals, hydrogen bonds, torsional forces, electrostatic interactions and desolvation are included by means of weights Wvdw,Whbond,Wconf,Welec, and Wsol, respectively. Value rij is the interatomic distance. Lennard–Jones parameters taken from the Amber force field are represented by Aij and Bij values, whereas Cij and Dij indicate the maximum well depth of potential energies between two atoms. The angle-dependent directionality is represented by E(t).The third term computes a Coulomb approach for electrostatics, and the fourth term involves the volume (V) of surrounding atoms of a given one weighted by S, with an exponential factor computing atom distances. A further explanation of the Einter computation can be obtained from Morris et al. (2009). TheRMSDisa quantitative measure ofsimilaritybetween two superimposed atomic coordinates of the co-crystallized ligand’s conformation and the predicted position of the ligand. In a nutshell, the lower the RMSD score, the better the solution is (if the co-crystallized ligand is considered as reference). Accordingtotheliterature(TrottandOlson2010;Kufareva and Ruben 2010), the RMSD cutoff of 2 Å is used as a criterion of the correct bound structure prediction. Ligands’ conformationswith an RMSDscore above 2 Åindicate a prediction that is not very accurate. However, it is worth noting that other docking solutions with high RSMD but low Einter values would indicate alternative ligand–macromolecule interactions. These solutions should be indeed considered, since they can be interesting from a pharmacological point of view, as mentioned in Sect. 1. RMSD =    1 N n  i d2 i(3) where the averaging is performed over the npairs of equivalent atoms and diis the distance between the two atoms in the i-th pair. 2.2 Proposed multi-objective formulation for ligand–protein docking As mentioned, our goal is to formulate molecular docking as a bi-objective optimization problem by considering the two terms separately, although following a Pareto dominance scheme as follows: – Objective1:The Einter asoneoftheenergyfunctionterms used in AutoDock 4.2, and expressed in Eq. 1. – Objective 2: The RMSD (expressed in Eq. 3) as quantitative measure of similarity between two superimposed atomic coordinates of the co-crystallized ligand’s conformation and the predicted position of the ligand. In this way, we can now take advantage of specific learning models of Pareto optimality-based techniques to deal with the molecular docking, which will result in sets of nondominated solutions with different choices of energy and RMSD.Inaddition,weavoidtheuseofadditionalweightfactorsthatcouldbiasthesearchproceduretooneofthedifferent terms, as usually is seen in single-objective approaches. 3 Optimization strategy Our optimization strategy includes two main processes acting in parallel: an optimization algorithm and a molecular docking procedure. The optimization part is carried out through a multi-objective algorithm provided by jMetalCpp (López-Camacho et al. 2014), an open-source framework of single/multi-objective optimization metaheuristics. The molecular binding procedure is performed by AutoDock 4.2 (Morris et al. 2009), a widely used tool for virtual drug discovery involving rigid and flexible macromolecule–ligand docking simulations. Algorithm 1 Pseudocode of Common Algorithmic Template for Multi-objective Molecular Docking 1: alg ∈{NSGA-II, SMPSO, GDE3, MOEA/D} 2: pop ←initializePopulation() 3: evaluateFitness(pop, Autodock()) // Autodock thread 4: pareto_front ←initializeParetoFront(pop) 5: while not StopCondition() do 6: pop ←performNewAlgorithmIteration(pop, alg) 7: evaluateFitness(pop, Autodock()) 8: pareto_front ←updateParetoFront(pareto_front, pop) 9: end while 10: return pareto_front For this study, as we are interested on evaluating our multi-objective approach for molecular docking, we have focusedonfourrepresentativemulti-objectivemetaheuristics in the state of the art. In concrete, we have selected the most widely used algorithm in the field (NSGA-II), a swarm intelligence optimizer (SMPSO), a solver based on differential evolution (GDE3), and an algorithm based on decomposition (MOEA/D). A brief description of these techniques is given as follows: – NSGA-II (Deb et al. 2002). The Non-dominated Sorting Genetic Algorithm II performs a generational strategy to obtain a new offspring population from the original one, by means of the application of genetic operators: selection, crossover and mutation. It uses a non-dominated sorting procedure based on Pareto ranking together with crowding distance density estimator to promote convergence and diversity, respectively. In spite of being one of the classical EMO algorithms in the literature, it still shows a prominent performance in multiple benchmarking and real-world problems. – SMPSO (Nebro et al. 2009)(Speed Modulation Particle Swarm Optimization) is a multi-objective particle swarm optimization algorithm. Its main feature is a bounding strategy of particles velocities to guide new particle’s positions within the problem search space of variables. SMPSO uses the polynomial mutation as perturbation factor to avoid fast convergence. It uses an external archive to store those non-dominated solutions obtained during the search. – GDE3 is the Generalized Differential Evolution 3 (Kukkonenand Lampinen 2005), whichfollows the algorithmic design of NSGA-II, although using differential mutationandselectionoperators(ofDE),insteadofthose topically used by NSGA-II. GDE3 uses an adapted version of crowding distance to generate better distributed sets of solutions. – MOEA/D (Zhang and Li 2007)(Multi-objective Evolutionary Algorithm Based on Decomposition) has become the classical representative archive-less optimization Fig. 1 Overall scheme of the optimization strategy. Solutions generated by multi-objective algorithms in jMetalCpp are then evaluated by AutoDock and finally sent back to jMetalCpp. Once the stop condition has been met, the best solution is returned to the AutoDock code which in turn generates an output file with the results. These results can be visualized with any tool that is suitable for visualizing AutoDock format (DLG) like AutoDockTool (ADT) Therefore, jMetalCpp and AutoDock are integrated in such a way that the algorithms available in jMetalCpp can communicate with AutoDock to cooperate on the molecular docking optimization. This integration process consists of running AutoDock and jMetalCpp in two different threads inside the same task. This way, as the internal memory is shared, the two threads can communicate with each other and they synchronize using mutexes. This approach is efficient and flexible, allowing any of the algorithms included in jMetalCpp to be easily used for solving docking problems. Figure 1 illustrates the optimization process, which algorithmic steps are also organized through the pseudocode of Algorithm 1. Therefore, after population and Pareto front are initialized (lines 2 to 4), while a given stop condition is not met (e. g., maximum number of iterations), a given metaheuristic algorithm in jMetal is executed to iteratively generate new solutions (line 6). Whenever the binding quality of a new solution has to be numerically quantified, it is sent to AutoDock to be evaluated in terms of binding energy and RMSD. After this evaluation, AutoDock returns the corresponding objective values to the algorithm, which are assigned to the evaluated solution (line 7). After this, the Pareto front is updated (line 8). This process is repeated until reaching a given stop condition. As a result, those non-dominated solutions found so far by the multi-objective algorithms are returned (Line 10). Finally, AutoDock is instanced to generate the output data concerning the optimized docking. Thus, the final results follow the standard format tailored to AutoDock users. technique, where a multi-objective problem is decomposed into a set of single-objective subproblems that are optimized simultaneously. In particular, we have focused in this study on MOEA/D-DE (Li and Zhang 2009), which is a variant that applies differential evolution as variation operators. In addition, this algorithm applies a polynomial mutation to avoid premature convergence, hence to improve its exploration capability. 4 Experimentation This section is devoted to explain the set of ligand–protein instances used for benchmarking the evaluated algorithms. The experimental methodology is also explained, as well as the set of parameters used to configure the algorithms. 4.1 Ligand–protein instances In order to perform a through experimentation, we have selectedanextensivebenchmarkof73complexescomprising both ligand and receptor flexibility. These complexes have been selected to cover an heterogeneous range of molecular samples, comprising from small-size ligand to large inhibitors. Our aim is to obtaining unbiased conclusions concerning which algorithm performs the best on a wide and heterogeneous set of complexes, hence to be suggested as optimizer to the decision-maker. Other past studies carried out with these molecular instances (Morris et al. 2009) showed that those complexes involving small ligands usually represent more difficult problems, since the flexibility added to the receptor side chains (ARG8) increases the space of ligand interaction. The ligand–protein instances have been extracted from the PDB database,1so they have been lightly adapted for being used in our docking simulations for solution evaluation. The benchmark of ligand–protein instances selected, with the PDB accession code, the X-ray crystal structures names, andthestructureresolution(Å),isshowninTable1.Forall of them, the torsional degrees of freedom (flexibility) are 10 for ligands and 6 for macromolecules (receptors). These torsion movements are selected to allow the fewest number of atoms to move around the ligand core. This implies a number of 23 problem variables (n) in solution vectors, consisting of: 3 for translation, 4 for rotation quaternion, and 16 for torsional degrees. 4.2 Methodology A series of 31 independent executions have been launched for each combination of algorithm and molecular instance, 1In URL http://www.rcsb.org/pdb/home/home.do. Table 1 X-ray crystal structure coordinates taken from the PDB database and used in docking experiments. They consist of 73 complexes with accession codes from the PDB database. The range of resolution (Å) of each subgroup is also shown in the last column Ligand type PDB code Resolution (Å) Small size 1A9M, 1AAQ, 1B6L, 1B6M, 1BDL, 1BDQ, 1BDR, 1GNM, 1GNN, 1GNO, 1HBV, 1HEG, 1HIH, 1HPV, 1HSG, 1HTE, 1KZK, 1SBG, 1TCX, 1ZIH, 1ZIR, 3AID 1.09–2.8 Medium size 1B6J, 1B6K, 1B6P, 1D4K, 1D4L, 1HEF, 1HPS, 1HXW, 1IZH, 1IZI, 1JLD, 1K6C, 1K6P, 1K6T, 1K6V, 1MTR, 1MUI, 2BPX, 4HVP, 4PHV, 5HVP 1.75–2.8 Large size 1A94, 1HIV, 1HOS, 1HTG, 1HVI, 1HVJ, 1HVK, 1HVL, 1HVS, 1HWR, 1ODY, 1VIJ, 1VIK, 3TLH, 7HVP, 9HVP 1.8–2.8 Cyclic urea 1BV7, 1BV9, 1BWA, 1BWB, 1DMP, 1G35, 1HPO, 1MES, 1MEU, 1PRO, 1QBR, 1QBT, 1QBU, 7UPJ 1.8–2.5 summing up a total number of: 31 ×73 ×4=9052 runs in the experimentation phase. To carry out the experiments, we have used a Condor2middleware platform for distribute computing, which comprises close to 400 cores acting task scheduler, each task devoted to one independent run. Theresultingdistributionshavebeencomputedintermsof themedianandinterquartilerange(IQR).Wehavefocusedon two main quality indicators to assess performance: hypervolume (IHV) and unary additive Epsilon (I+)(Deb2001). The first indicator quantifies the convergence and diversity of the resulting Pareto fronts, whereas the second one (I+) measures the convergence degree. For each molecular instance, a reference Pareto front has been computed with all the nondominated solutions obtained in all the executions of all the algorithms. This is a useful practice when calculating the two indicators, since we are dealing with a real-world optimization problem for which the Pareto fronts are not known. 4.3 Parameter setup For the common parameters of all the evaluated algorithms, wehavechosenapopulationsizeof150individuals(particles in the case of SMPSO) and 1,500,000 function evaluations as stop condition. These values were chosen, as they are the default settings used by AutoDock in the reference study (Norgan et al. 2011). 2In URL: http://research.cs.wisc.edu/htcondor/. In the case of specific parameters of each algorithm, a different parameter setup has been set by following recommendations of our previous studies (López-Camacho et al. 2015,2016), as follows: – NSGA-II: The distribution indexes for SBX crossover andmutationoperatorsareηc=20andηm=20,respectively. For these two operators, pc=0.9 is the crossover provability and pm=1/nthe mutation probability, with nthe number of decision variables. It applies binary tournament selection. – SMPSO: In this algorithm, ϕ1and ϕ2are the acceleration coefficients, which are set to 1.5. The inertia weight is ω=0.9, and polynomial mutation is applied to a set of particles (1/6) in the swarm. – GDE3 (variant rand/1/bin), the differential mutation and crossover parameters μand Crtake a value of 0.5. – MOEA/D μis set to 0.5 and Cris set to 1.0. The polynomial mutation is configured with the same settings applied in NSGA-II. SMPSO obtains the best results in 60 of the 73 instances and the second best in 6 instances. MOEA/D obtains the best results in 6 instances and the second best in 61 instances. Finally, GDE3 obtains the best results in 7 instances although the second best in 6 instances. Moreover, with the aim at providing the results obtained with statistical confidence, we have used a formal procedure by means of nonparametric statistical tests, since in some cases the distribution of results did not follow the conditions of normality and heteroscedasticity required (Sheskin 2007). In this way, the analyses and comparisons are focused on the distributions of each of the two metrics studied. To this end, we have used Friedman’s ranking and Holm’s post hoc multicomparison tests to check which algorithms are statistically worsethanthebestrankedtechnique,whichisusedascontrol one. In accordance with this procedure, Table 4shows the results of these tests, where SMPSO has the best Friedman’s ranking value of IHV (with a value of 1.20), followed by MOEA/D, GDE3, and NSGA-II. This means that SMPSO is set as the control technique for IHV in the post hoc Holm test, and hence, it is compared with the remaining algorithms. In this regard, the adjusted P-values (HolmAp in Table 4) resulting from these comparisons are for the last three algorithms (MOEA/D, GDE3 and NSGA-II), lower than the confidence level, set toα=0.05 in thisstudy.In the caseof I+, SMPSO also shows the best ranking and is statistically better than MOEA/D,GDE3and NSGA-II, for thebenchmarkofmolecular instances used here. From a graphical point of view, Figs. 2,3,4and 5plot the approximated reference Pareto fronts from all executions (in continuousline), withregardtotheParetofrontsofeachcompared algorithm, for all the molecular instances with small size, medium size large size and cyclic urea, respectively. In these graphs, each solution from the four studied algorithms is represented using different shapes and colors as shown in the plot legend. An interesting observation can be made with regards to the convergence region of fronts for each algorithm, where, for example, MOEA/D seems to converge to the region promoting solutions with low energies, while SMPSO is more focused on regions biased to the RMSD objective. This gives us insights into the different learning procedures induced by SMPSO and MOEA/D, which search in different regions of the problem landscape, therefore obtaining complementary solutions covering different parts in the Pareto fronts approximated. 5.2 Analysis on ligand binding sites and molecular interactions In order to show the applicability of this multi-objective strategy, we have chosen a set of complexes that include HIV-protease therapeutic targets. The 1BDR and 1QBU cor5 Results In this section, the obtained results are presented and analyzed. First, we compare the performance of the algorithms. A second analysis with selected solutions is also carried to provide insights into biological validation. 5.1 Performance comparisons In order to evaluate the performance of each algorithm, we have taken in consideration the selected quality indicators. The first indicator we applied is the IHV, which is calculated by obtaining the sum of the contributed volume of each point of an approximated front with regard to a given reference point. Therefore, the higher the converge and diversity degrees, the better the IHV value is. The medians and interquartile ranges of the obtained solutions in terms of IHV, for the set of 73 docking instances and for the algorithms compared, are shown in Table 2. In accordance with these results, SMPSO obtains the best hypervolume for 63 out of the 73 instances, and the second best results in 5 instances. MOEA/D obtains the best results in 2 instances and the second best results in 31 instances. GDE3 obtained the best results in 8 instances and the second best results in 2 instances. It is observable that many cells have a IHV equal to zero. This happens when all the obtained solutions are out of the limits of the reference point, which indicates we are facing a very hard optimization problem. Table 3 shows the median and interquartile ranges computed in terms of I+. Similarly to the previous observations, Table 2Median and interquartile range of I(HV)for each algorithm and instance SMPSO GDE3 MOEA/D NSGA-II 1A9M 5.48e−01 (3.5e−02) 0.00e+00 (0.0e+ 00) 0.00e+00(3.3e−01) 0.00e+00(0.0e+ 00) 1AAQ 6.86e−01 (2.5e−02) 0.00e+00 (7.3e−01) 5.31e−01 (7.6e−01) 0.00e+00 (0.0e+ 00) 1B6L 6.86e−01 (8.3e−02) 0.00e+00 (0.0e+ 00) 0.00e+00(6.4e−01) 0.00e+00(0.0e+ 00) 1B6M 4.77e−01 (5.5e−02) 0.00e+00 (0.0e+ 00) 0.00e+00(2.6e−01) 0.00e+00(0.0e+ 00) 1BDL 5.87e−01 (4.8e−02) 7.81e−01 (9.3e−02) 5.52e−01 (3.8e−01) 5.21e−01 (6.2e−01) 1BDQ 6.87e−01 (2.9e−02) 0.00e+0 0(0.0e+ 00) 2.44e−01 (6.1e−01) 0.00e+00(0.0e+ 00) 1BDR 5.15e−01 (6.8e−02) 0.00e+00 (0.0e+ 00) 4.80e−01 (5.7e−01) 0.00e+00 (0.0e+ 00) 1GNM 8.13e−01 (3.2e−02) 0.00e+00 (0.0e+ 00) 0.00e+00(7.6e−01) 0.00e+00(0.0e+ 00) 1GNN 6.99e−01 (4.2e−02) 0.00e+00 (0.0e+ 00) 0.00e+00(3.5e−01) 0.00e+00(0.0e+ 00) 1GNO 7.78e−01 (3.4e−02) 0.00e+00 (0.0e+ 00) 0.00e+00(0.0e+ 00) 0.00e+00 (0.0e+ 00) 1HBV 6.07e−01 (4.5e−02) 0.00e+00 (0.0e+ 00) 4.27e−01 (6.5e−01) 0.00e+00 (0.0e+00) 1HEG 5.18e−01 (5.1e−02) 6.63e−01 (4.2e−02) 5.76e−01 (2.3e−01) 9.12e−02 (4.2e−01) 1HIH 3.54e−01 (5.6e−02) 1.20e−01 (4.2e−01) 1.55e−01 (5.6e−01) 0.00e+00(6.7e−02) 1HPV 6.57e−01 (6.0e−02) 0.00e+00(0.0e+00) 4.28e−01 (6.6e−01) 0.00e+00(0.0e+ 00) 1HSG 4.71e−01 (7.1e−02) 0.00e+00(0.0e+ 00) 0.00e+00(0.0e+ 00) 0.00e+00(0.0e+ 00) 1HTE 5.55e−01 (4.6e−02) 9.59e−02 (5.7e−01) 4.02e−01 (5.4e−01) 0.00e+00(0.0e+ 00) 1KZK 0.00e+00(0.0e+00) 1.13e−01 (7.0e−02) 4.04e−01 (2.3e−01) 0.00e+00 (6.2e−02) 1SBG 5.48e−01 (8.6e−02) 0.00e+00(0.0e+ 00) 1.56e−01 (4.9e−01) 0.00e+00(0.0e+ 00) 1TCX 5.62e−01 (8.7e−02) 0.00e+00(0.0e+ 00) 5.19e−01 (5.7e−01) 0.00e+00 (0.0e+00) 1ZIH 8.35e−01 (4.0e−02) 0.00e+00 (0.0e+ 00) 0.00e+00(8.9e−01) 0.00e+00 (0.0e+ 00) 1ZIR 5.21e−01 (1.4e−01) 0.00e+00 (0.0e+ 00) 0.00e+00(7.7e−01) 0.00e+00 (0.0e+ 00) 3AID 6.67e−01 (2.8e−02) 0.00e+00 (0.0e+ 00) 5.17e−01 (7.2e−01) 0.00e+00 (0.0e+ 00) 1B6J 8.12e−01 (2.3e−02) 0.00e+00 (0.0e+ 00) 0.00e+00(6.0e−01) 0.00e+00(0.0e+ 00) 1B6K 9.47e−01 (9.9e−03) 0.00e+00 (0.0e+ 00) 0.00e+00(7.9e−01) 0.00e+00 (0.0e+ 00) 1B6P 4.53e−01 (1.4e−01) 0.00e+00 (0.0e+00) 0.00e+00(3.2e−03) 0.00e+00 (0.0e+ 00) 1D4K 6.25e−01 (5.5e−02) 0.00e+00 (0.0e+ 00) 0.00e+00(0.0e+ 00) 0.00e+00(0.0e+ 00) 1D4L 7.15e−01 (5.5e−02) 0.00e+00 (0.0e+ 00) 0.00e+00(0.0e+ 00) 0.00e+00(0.0e+ 00) 1HEF 8.90e−01 (1.0e−02) 9.16e−01 (2.7e−02) 9.00e−01 (3.8e−02) 8.76e−01 (6.4e−01) 1HPS 7.54e−01 (4.7e−02) 0.00e+00(0.0e+ 00) 0.00e+00(4.7e−01) 0.00e+00(0.0e+ 00) 1HXW 5.57e−01 (6.0e−02) 6.08e−01 (4.8e−01) 4.88e−01 (5.2e−01) 0.00e+00 (5.0e−01) 1IZH 7.60e−01 (3.1e−02) 0.00e+00 (0.0e+ 00) 3.59e−01 (7.0e−01) 0.00e+00 (0.0e+ 00) 1IZI 7.26e−01 (4.4e−02) 0.00e+00(0.0e+ 00) 5.21e−01 (8.4e−01) 0.00e+00 (0.0e+ 00) 1JLD 5.99e−01 (3.4e−02) 0.00e+00 (6.8e−01) 5.52e−01 (6.3e−01) 0.00e+00 (0.0e+ 00) 1K6C 5.72e−01 (3.9e−02) 0.00e+00 (0.0e+ 00) 0.00e+00(4.1e−01) 0.00e+00(0.0e+ 00) 1K6P 3.73e−01 (1.2e−01) 0.00e+00 (0.0e+ 00) 0.00e+00(0.0e+ 00) 0.00e+00(0.0e+ 00) 1K6T 4.45e−01 (2.2e−02) 0.00e+00 (0.0e+00) 0.00e+00(0.0e+ 00) 0.00e+00(0.0e+ 00) 1K6V 5.90e−01 (6.4e−02) 0.00e+00 (0.0e+ 00) 0.00e+00(0.0e+ 00) 0.00e+00(0.0e+ 00) 1MTR 5.53e−01 (7.9e−02) 0.00e+00(0.0e+ 00) 0.00e+00(6.6e−01) 0.00e+00 (0.0e+ 00) 1MUI 7.57e−01 (4.6e−02) 0.00e+00 (0.0e+ 00) 0.00e+00(7.1e−01) 0.00e+00 (0.0e+ 00) 2BPX 8.53e−01 (8.2e−02) 0.00e+00(0.0e+ 00) 0.00e+00(0.0e+ 00) 0.00e+00(0.0e+ 00) 4HVP 6.27e−01 (1.8e−01) 0.00e+00 (0.0e+ 00) 4.13e−01 (7.5e−01) 0.00e+00 (0.0e+ 00) 4PHV 6.64e−01 (9.3e−02) 0.00e+00(0.0e+ 00) 0.00e+00(7.3e−01) 0.00e+00(0.0e+ 00) 5HVP 6.53e−01 (4.1e−02) 0.00e+00 (0.0e+ 00) 0.00e+00(0.0e+ 00) 0.00e+00(0.0e+ 00) 1A94 6.42e−01 (4.8e−02) 0.00e+00 (0.0e+ 00) 0.00e+00(5.4e−01) 0.00e+00(0.0e+00) 1HIV 6.13e−01 (9.2e−02) 0.00e+00 (0.0e+ 00) 0.00e+00(3.8e−01) 0.00e+00 (0.0e+ 00) 1HOS 4.40e−01 (5.5e−01) 0.00e+00 (4.4e−03) 9.52e−01 (9.2e−02) 0.00e+00 (1.5e−01) 1HTG 5.16e−01 (4.4e−02) 0.00e+00 (0.0e+ 00) 0.00e+00(0.0e+00) 0.00e+00 (0.0e+ 00) 1HVI 6.30e−01 (5.0e−02) 0.00e+00 (0.0e+ 00) 2.93e−01 (7.5e−01) 0.00e+00 (0.0e+ 00) 1HVJ 5.17e−01 (3.1e−02) 0.00e+00 (0.0e+ 00) 2.59e−01 (5.2e−01) 0.00e+00 (0.0e+ 00) 1HVK 5.46e−01 (1.5e−02) 0.00e+00 (0.0e+00) 2.57e−01 (2.4e−01) 0.00e+00 (0.0e+ 00) 1HVL 4.99e−01 (1.9e−02) 0.00e+00 (0.0e+ 00) 2.13e−01 (4.8e−01) 0.00e+00 (0.0e+ 00) 1HVS 3.92e−01 (5.3e−02) 0.00e+00 (0.0e+ 00) 1.43e−01 (2.6e−01) 0.00e+00 (0.0e+ 00) 1HWR 8.51e−01 (1.3e−02) 8.69e−01 (9.0e−01) 7.23e−01 (3.9e−01) 0.00e+00(7.5e−01) 1ODY 5.84e−01 (1.4e−01) 0.00e+00 (0.0e+ 00) 8.45e−02 (5.2e−01) 0.00e+00 (0.0e+ 00) 1VIJ 6.58e−01 (3.9e−02) 0.00e+00 (0.0e+ 00) 4.78e−01 (6.5e−01) 0.00e+0 0(0.0e+ 00) 1VIK 5.60e−01 (3.3e−02) 0.00e+00 (0.0e+ 00) 0.00e+00(5.1e−01) 0.00e+00 (0.0e+ 00) 3TLH 4.88e−01 (3.8e−02) 6.57e−01 (4.1e−02) 5.37e−01 (1.7e−01) 1.89e−01 (5.3e−01) 7HVP 6.45e−01 (1.8e−02) 6.79e−01 (2.0e−01) 6.73e−01 (1.9e−01) 4.80e−03 (6.2e−01) 9HVP 9.16e−01 (1.4e−02) 0.00e+00 (0.0e+ 00) 5.66e−01 (7.3e−01) 0.00e+00 (0.0e+ 00) 1BV7 5.00e−01 (3.4e−02) 0.00e+00 (0.0e+ 00) 0.00e+00(6.4e−01) 0.00e+00(0.0e+ 00) 1BV9 4.62e−01 (6.5e−02) 0.00e+00 (0.0e+ 00) 1.91e−01 (5.5e−01) 0.00e+00 (0.0e+ 00) 1BWA 5.42e−01 (7.4e−02) 0.00e+00(0.0e+ 00) 0.00e+00(6.9e−01) 0.00e+00(0.0e+ 00) 1BWB 7.50e−01 (4.4e−02) 0.00e+00 (0.0e+ 00) 3.69e−01 (7.3e−01) 0.00e+0 0(0.0e+ 00) 1DMP 8.30e−01 (3.2e−02) 0.00e+00 (0.0e+ 00) 0.00e+00(5.5e−01) 0.00e+00 (0.0e+ 00) 1G35 5.16e−01 (4.6e−02) 0.00e+00 (0.0e+ 00) 0.00e+00(0.0e+ 00) 0.00e+00(0.0e+ 00) 1HPO 6.31e−01 (4.5e−02) 0.00e+00(0.0e+ 00) 0.00e+00(4.9e−01) 0.00e+00(0.0e+ 00) 1MES 3.66e−01 (9.7e−02) 0.00e+00 (0.0e+ 00) 1.04e−01 (4.7e−01) 0.00e+00 (0.0e+ 00) 1MEU 6.19e−01 (7.7e−02) 5.13e−01 (3.3e−01) 4.73e−01 (4.5e−01) 0.00e+00 (0.0e+ 00) 1PRO 4.39e−01 (4.4e−02) 0.00e+0 0(0.0e+ 00) 1.21e−02 (4.6e−01) 0.00e+00(0.0e+ 00) 1QBR 7.20e−01 (7.8e−02) 0.00e+00 (0.0e+ 00) 5.48e−01 (7.5e−01) 0.00e+00 (0.0e+ 00) 1QBT 3.63e−01 (8.3e−02) 0.00e+00 (0.0e+ 00) 0.00e+00(4.2e−01) 0.00e+00(0.0e+ 00) 1QBU 9.18e−01 (1.1e−02) 9.18e−01 (9.4e−01) 8.52e−01 (9.1e−01) 0.00e+00 (0.0e+ 00) 7UPJ 7.23e−01 (8.1e−02) 0.00e+00(0.0e+ 00) 0.00e+00(4.2e−01) 0.00e+00(0.0e+ 00) Best and second best median results have dark and light gray backgrounds, respectively Table 3Median and interquartile range of I(+)for each algorithm and instance SMPSO GDE3 MOEA/D NSGA-II 1A9M 3.90e−01 (4.6e−02) 2.79e+00 (2.7e−01) 1.22e+00(9.7e−01) 2.42e+00(1.5e+ 00) 1AAQ 2.11e−01 (5.2e−02) 1.61e+00 (1.5e+ 00) 2.82e−01 (2.0e+ 00) 6.03e+00 (5.9e+ 00) 1B6L 2.26e−01 (1.8e−01) 1.89e+00 (1.7e+ 00) 1.77e+00(2.1e+ 00) 3.35e+0 0(2.2e+ 00) 1B6M 4.19e−01 (6.9e−02) 3.23e+00 (8.7e−02) 1.47e+00(1.8e+ 00) 3.46e+00(1.0e+ 00) 1BDL 2.94e−01 (8.3e−02) 7.48e−02 (8.8e−02) 2.74e−01 (3.4e−01) 3.47e−01 (2.1e+ 00) 1BDQ 2.59e−01 (5.0e−02) 7.83e+00 (5.8e+00) 6.26e−01 (3.8e+ 00) 8.70e+00 (6.3e+ 00) 1BDR 4.75e−01 (7.2e−02) 1.61e+00 (1.1e+ 00) 5.08e−01 (4.7e−01) 2.02e+00 (8.8e−01) 1GNM 1.06e−01 (3.3e−02) 6.21e+00 (1.2e+ 00) 4.16e+00(5.7e+ 00) 6.70e+00 (2.1e+ 00) 1GNN 2.17e−01 (7.2e−02) 5.43e+00 (1.9e−01) 4.52e+00(4.3e+ 00) 5.51e+00(8.8e−01) 1GNO 1.09e−01 (4.4e−02) 6.12e+00 (3.5e−02) 3.85e+00(2.5e+ 00) 6.71e+00(1.1e+ 00) 1HBV 3.36e−01 (7.8e−02) 1.92e+00 (4.2e−02) 4.02e−01 (1.8e+ 00) 3.53e+00 (2.0e+ 00) 1HEG 3.99e−01 (6.1e−02) 1.63e−01 (1.1e−01) 2.67e−01 (2.6e−01) 7.42e−01 (5.7e−01) 1HIH 5.85e−01 (7.4e−02) 7.23e−01 (1.5e+00) 6.65e−01 (1.5e+ 00) 2.32e+00(2.3e+ 00) 1HPV 2.87e−01 (7.6e−02) 1.72e+00(9.4e−01) 3.51e−01 (1.6e+ 00) 2.67e+00 (1.8e+ 00) 1HSG 4.07e−01 (1.3e−01) 4.88e+00(5.1e−01) 3.31e+00(2.2e+ 00) 4.45e+00(1.7e+ 00) 1HTE 3.40e−01 (6.8e−02) 7.85e−01 (5.7e−01) 3.85e−01 (9.0e−01) 1.60e+00(7.3e−01) 1KZK 1.14e+00(1.4e−01) 8.87e−01 (7.0e−02) 5.74e−01 (2.4e−01) 1.04e+00(2.6e−01) 1SBG 3.14e−01 (1.6e−01) 8.31e+00(6.7e−01) 6.13e−01 (3.1e+ 00) 7.62e+00(4.6e+ 00) 1TCX 3.11e−01 (1.8e−01) 5.01e+00(3.3e+ 00) 2.28e−01 (7.5e−01) 5.45e+00 (3.8e+00) 1ZIH 1.07e−01 (7.2e−02) 5.36e+00 (8.1e−01) 2.24e+00(4.2e+ 00) 8.54e+00(4.2e+ 00) 1ZIR 4.41e−01 (1.6e−01) 5.33e+00 (1.1e+ 00) 1.02e+00(2.8e+ 00) 6.58e+00 (2.2e+ 00) 3AID 3.00e−01 (3.1e−02) 3.77e+00 (1.5e+ 00) 3.50e−01 (1.2e+ 00) 4.12e+0 0(2.6e+ 00) 1B6J 1.13e−01 (3.6e−02) 5.86e+00 (1.9e+ 00) 2.77e+00(4.9e+ 00) 6.26e+00 (1.7e+ 00) 1B6K 2.63e−02 (9.4e−03) 5.55e+00 (4.4e+ 00) 1.14e+00(3.1e+ 00) 7.21e+00 (1.9e+ 00) 1B6P 2.85e−01 (1.8e−01) 5.83e+00 (5.0e−01) 3.19e+00(4.9e+00) 6.19e+00 (1.2e+ 00) 1D4K 2.81e−01 (9.2e−02) 8.22e+00 (1.1e−01) 5.24e+00(5.5e+ 00) 8.49e+00 (2.7e+ 00) 1D4L 1.98e−01 (1.1e−01) 3.69e+00 (2.2e+ 00) 3.62e+00(1.2e+ 00) 5.58e+00(3.0e+ 00) 1HEF 7.93e−02 (5.8e−03) 5.85e−02 (1.2e−02) 5.68e−02 (1.3e−02) 6.66e−02 (6.5e−01) 1HPS 1.41e−01 (4.9e−02) 7.27e+00(1.2e−01) 2.57e+00(2.6e+ 00) 7.32e+00(7.7e−01) 1HXW 3.71e−01 (9.6e−02) 2.39e−01 (4.7e−01) 2.96e−01 (4.5e−01) 1.78e+00(2.2e+ 00) 1IZH 1.05e−01 (3.2e−02) 6.32e+00(1.1e−02) 4.92e−01 (3.5e+ 00) 6.22e+00 (3.1e+ 00) 1IZI 2.64e−01 (5.0e−02) 2.12e+00(6.5e−02) 2.82e−01 (2.0e+ 00) 3.75e+00 (1.9e+ 00) 1JLD 3.02e−01 (7.5e−02) 2.03e+00 (5.4e+ 00) 2.43e−01 (9.9e−01) 5.58e+0 0(5.1e+ 00) 1K6C 2.33e−01 (8.7e−02) 4.28e+00 (8.8e−01) 2.06e+00(2.8e+ 00) 3.67e+00 (1.6e+ 00) 1K6P 4.92e−01 (1.5e−01) 1.85e+00(1.7e+ 00) 2.73e+00(2.1e+ 00) 3.57e+00(1.7e+ 00) 1K6T 4.24e−01 (5.6e−02) 1.94e+00(1.7e+ 00) 2.02e+0 0(1.7e+ 00) 2.54e+00 (1.3e+ 00) 1K6V 2.30e−01 (7.9e−02) 4.19e+00 (8.9e−01) 2.86e+00(1.5e+ 00) 4.37e+00(8.9e−01) 1MTR 3.08e−01 (1.5e−01) 4.73e+00(3.7e−01) 1.73e+00(2.6e+ 00) 4.65e+00(1.4e+ 00) 1MUI 1.90e−01 (8.8e−02) 7.41e+00 (2.5e+00) 3.02e+00(3.5e+ 00) 7.62e+00(3.3e+ 00) 2BPX 1.13e−01 (1.2e−01) 3.98e+00(3.4e+ 00) 3.29e+00(2.3e+ 00) 4.98e+00 (2.3e+ 00) 4HVP 3.02e−01 (2.0e−01) 4.15e+00 (8.1e−02) 3.02e−01 (2.1e+ 00) 4.31e+00 (6.4e−01) 4PHV 3.13e−01 (1.4e−01) 1.00e+01(2.2e+ 00) 4.06e+00(6.3e+ 00) 1.03e+01(3.3e+ 00) 5HVP 1.92e−01 (7.7e−02) 5.58e+00 (6.5e−01) 2.06e+00(2.2e−01) 6.67e+00(9.4e−01) 1A94 2.68e−01 (7.1e−02) 7.53e+00 (6.8e−03) 3.88e+00(7.0e+ 00) 9.19e+00 (2.4e+ 00) 1HIV 2.72e−01 (1.4e−01) 2.21e+00 (5.1e−01) 1.33e+00(3.1e+ 00) 3.72e+00(1.3e+ 00) 1HOS 5.59e−01 (5.7e−01) 1.01e+00 (1.9e−01) 4.70e−02 (9.3e−02) 1.16e+00 (6.0e−01) 1HTG 4.42e−01 (5.6e−02) 1.05e+01 (2.0e−02) 8.43e+00(7.0e+00) 9.98e+00 (2.6e+ 00) 1HVI 2.18e−01 (8.4e−02) 7.41e+00 (1.5e−01) 5.08e−01 (1.0e+ 00) 8.25e+00 (1.2e+ 00) 1HVJ 2.74e−01 (9.6e−02) 5.71e+00 (3.4e−01) 5.58e−01 (1.4e+ 00) 5.90e+00 (9.2e−01) 1HVK 1.90e−01 (3.0e−02) 7.59e+00 (3.7e+00) 5.21e−01 (3.8e−01) 8.02e+00 (7.6e−01) 1HVL 2.69e−01 (3.6e−02) 8.55e+00 (7.5e−02) 6.06e−01 (1.2e+ 00) 9.14e+00 (9.9e−01) 1HVS 3.84e−01 (1.1e−01) 1.05e+01 (1.0e−01) 7.10e−01 (7.3e−01) 1.17e+01 (1.5e+ 00) 1HWR 1.03e−01 (3.3e−02) 6.23e−02 (1.5e+00) 1.59e−01 (3.5e−01) 1.68e+00(5.5e+ 00) 1ODY 3.08e−01 (1.5e−01) 1.04e+01 (1.4e+ 00) 8.27e−01 (8.1e+ 00) 1.06e+01 (2.2e+ 00) 1VIJ 3.17e−01 (3.7e−02) 4.61e+00 (2.7e−02) 3.08e−01 (4.3e+ 00) 5.02e+0 0(8.5e−01) 1VIK 3.40e−01 (5.1e−02) 6.74e+00 (1.4e−01) 4.79e+00(6.2e+ 00) 6.90e+00(1.0e+ 00) 3TLH 4.52e−01 (4.0e−02) 2.18e−01 (3.2e−02) 3.39e−01 (1.7e−01) 6.49e−01 (7.0e−01) 7HVP 3.13e−01 (2.5e−02) 1.89e−01 (1.7e−01) 1.97e−01 (1.8e−01) 9.33e−01 (4.2e+ 00) 9HVP 6.74e−02 (2.9e−02) 8.42e+00 (1.1e+ 00) 3.55e−01 (4.1e+ 00) 9.53e+00 (2.3e+ 00) 1BV7 4.80e−01 (3.6e−02) 4.56e+00 (8.1e−01) 4.17e+00(4.7e+ 00) 5.89e+00 (1.6e+ 00) 1BV9 5.10e−01 (7.0e−02) 5.21e+00 (2.4e+ 00) 6.89e−01 (6.8e+ 00) 7.75e+00 (2.4e+ 00) 1BWA 4.32e−01 (8.9e−02) 3.49e+00(1.3e+ 00) 3.22e+00(4.4e+ 00) 5.24e+00(1.1e+ 00) 1BWB 1.76e−01 (7.0e−02) 2.92e+00 (2.5e+ 00) 4.49e−01 (5.1e+ 00) 6.32e+00 (2.5e+00) 1DMP 1.03e−01 (2.8e−02) 4.41e+00 (4.9e+ 00) 2.57e+00(4.0e+ 00) 7.94e+00(4.4e+ 00) 1G35 2.13e−01 (1.1e−01) 5.05e+00 (8.2e−01) 4.63e+00(3.7e+ 00) 5.25e+00 (1.2e+ 00) 1HPO 1.67e−01 (1.1e−01) 1.10e+00(1.3e+ 00) 1.82e+00(2.3e+ 00) 2.77e+0 0(1.4e+ 00) 1MES 5.71e−01 (1.3e−01) 5.05e+00 (4.0e+ 00) 7.24e−01 (5.1e+ 00) 8.30e+00 (4.1e+ 00) 1MEU 3.38e−01 (8.5e−02) 3.83e−01 (3.1e−01) 3.81e−01 (4.2e−01) 6.33e+00(3.0e+ 00) 1PRO 4.86e−01 (7.0e−02) 7.86e+00 (3.6e+ 00) 9.60e−01 (6.4e+ 00) 8.57e+00 (3.5e+ 00) 1QBR 2.69e−01 (9.7e−02) 3.57e+00 (8.4e−01) 3.50e−01 (3.3e+ 00) 4.87e+00 (1.3e+ 00) 1QBT 5.87e−01 (1.1e−01) 5.14e+00(6.2e−01) 5.84e+00(6.9e+ 00) 6.85e+00(1.4e+ 00) 1QBU 7.07e−02 (1.5e−02) 5.15e−02 (1.9e+00) 1.11e−01 (1.9e+ 00) 4.08e+00(2.2e+ 00) 7UPJ 1.66e−01 (9.5e−02) 2.43e+00(6.6e−01) 1.49e+00(1.9e+ 00) 2.37e+00(5.6e−01) Best and second best median results have dark and light gray backgrounds, respectively Table 4Friedman’s rankings with Holm’s Adjusted p-values (with confidence level α=0.05) of the techniques compared for the test set of 73 docking instances Hypervolume (IHV) Epsilon (I+) Algorithm FriRank HolmAp Algorithm FriRank HolmAp *SMPSO 1.20 – *SMPSO 1.30 – MOEA/D 2.52 7.54E−10 MOEA/D 1.99 1.34E−03 GDE3 2.93 1.01E−15 GDE3 2.83 1.76E−12 NSGA-II 3.33 6.28E−23 NSGA-II 3.86 1.81E−32 Symbol * indicates the control algorithm, and column at right contains the overall ranking of positions with regard to indicators IHV and I+ −24 −22 −20 0.0 0.4 0.8 1.2 1A9M Energy (kcal/mol) RMSD (Å) NSGAII GDE3 SMPSO MOEAD −22 −20 −18 −16 −14 0.0 0.4 0.8 1AAQ Energy (kcal/mol) RMSD (Å) NSGAII GDE3 SMPSO MOEAD −19.0 −18.0 −17.0 −16.0 0.0 0.4 0.8 1B6L Energy (kcal/mol) RMSD (Å) NSGAII GDE3 SMPSO MOEAD −22.5 −21.5 −20.5 −19.5 0.0 0.4 0.8 1.2 1B6M Energy (kcal/mol) RMSD (Å) NSGAII GDE3 SMPSO MOEAD −20 −19 −18 −17 −16 0.0 0.4 0.8 1BDL Energy (kcal/mol) RMSD (Å) NSGAII GDE3 SMPSO MOEAD −20.5 −19.5 −18.5 0.0 0.2 0.4 1BDQ Energy (kcal/mol) RMSD (Å) NSGAII GDE3 SMPSO MOEAD −21.5 −20.5 −19.5 −18.5 01234 1BDR Energy (kcal/mol) RMSD (Å) NSGAII GDE3 SMPSO MOEAD −36 −32 −28 −24 0.0 0.4 0.8 1GNM Energy (kcal/mol) RMSD (Å) NSGAII GDE3 SMPSO MOEAD −36 −34 −32 −30 −28 0.0 0.4 0.8 1GNN Energy (kcal/mol) RMSD (Å) NSGAII GDE3 SMPSO MOEAD −38−34−30−26 0.0 0.4 0.8 1GNO Energy (kcal/mol) RMSD (Å) NSGAII GDE3 SMPSO MOEAD −22 −20 −18 0.0 0.4 0.8 1HBV Energy (kcal/mol) RMSD (Å) NSGAII GDE3 SMPSO MOEAD −14 −12 −10 −8 −6 0.0 1.0 2.0 1HEG Energy (kcal/mol) RMSD (Å) NSGAII GDE3 SMPSO MOEAD −24.0 −23.0 −22.0 −21.0 0.0 0.5 1.0 1.5 1HIH Energy (kcal/mol) RMSD (Å) NSGAII GDE3 SMPSO MOEAD −21.0 −20.0 0.0 0.5 1.0 1.5 1HPV Energy (kcal/mol) RMSD (Å) NSGAII GDE3 SMPSO MOEAD −24.0 −23.0 −22.0 0.0 0.4 0.8 1HSG Energy (kcal/mol) RMSD (Å) NSGAII GDE3 SMPSO MOEAD −17 −16 −15 −14 −13 0.0 0.4 0.8 1HTE Energy (kcal/mol) RMSD (Å) NSGAII GDE3 SMPSO MOEAD 0 2000 6000 10000 345678 1KZK Energy (kcal/mol) RMSD (Å) NSGAII GDE3 SMPSO MOEAD −22.0 −21.5 −21.0 −20.5 0.0 0.2 0.4 0.6 1SBG Energy (kcal/mol) RMSD (Å) NSGAII GDE3 SMPSO MOEAD −21.5 −20.5 −19.5 0.0 0.2 0.4 0.6 1TCX Energy (kcal/mol) RMSD (Å) NSGAII GDE3 SMPSO MOEAD −19.5 −19.0 −18.5 0.0 0.2 0.4 1ZIH Energy (kcal/mol) RMSD (Å) NSGAII GDE3 SMPSO MOEAD −21.6 −21.2 −20.8 0.0 0.4 0.8 1.2 1ZIR Energy (kcal/mol) RMSD (Å) NSGAII GDE3 SMPSO MOEAD −18 −17 −16 −15 0.0 0.4 0.8 3AID Energy (kcal/mol) RMSD (Å) NSGAII GDE3 SMPSO MOEAD Fig. 2 Reference fronts from all executions (in continuous line) with regard to the Pareto fronts of the studied algorithms with best hypervolume (in dotted lines) for small-size molecular instances: 1A9M, 1AAQ, 1B6L, 1B6M, 1BDL, 1BDQ, 1BDR, 1GNM, 1GNN, 1GNO, 1HBV, 1HEG, 1HIH, 1HPV, 1HSG, 1HTE, 1KZK, 1SBG, 1TCX, 1ZIH, 1ZIR, 3AID respond to instances from the benchmark used in this study. The 3EKX and 3WSJ consist in interesting instances that have HIV-proteases as targets, and their inhibitors are being tested in resistant HIV-proteases. The 1BDR corresponds to the complex HIV-1 protease withinhibitorSB203386,atripeptideanalogueinhibitor. The chimeric HIV-protease crystal is formed by replacing amino acids in residues in the HIV type 1 with the corresponding residues from the HIV type 2 (31-37) and in the active site. It contains the mutations T31s, L33V, E34A, E35G, M36I and S37E. Some studies (Swairjo et al. 1998)have reported that these mutations can affect the dimensions of