A New Multi-Objective Approach for Molecular Docking Based on RMSD and Binding Energy

A New Mul i-Objec i e App oach o Molecula Docking Based on RMSD and Binding Ene gy Es eban L´opez-Camacho, Ma ´ıa Jes´us Ga c´ıa Godoy, Jos´e Ga c´ıa-Nie o, An onio J. Neb o, and Jos´e F. Aldana-Mon es Khaos Resea ch G oup Depa amen o Compu e Sciences, Uni e si y o M´alaga, ETSI In o m´a ica, Campus de Tea inos, M´alaga - 29071, Spain {es eban,mjga ciag,jnie o,an onio,j am}@lcc.uma.es Abs ac . Ligand-p o ein docking is an op imiza ion p oblem based on p edic ing he posi ion o a ligand wi h he lowes binding ene gy in he ac i e si e o he ecep o . Molecula docking p oblems a e a- di ionally ackled wi h single-objec i e, as well as wi h mul i-objec i e app oaches, o minimize he binding ene gy. In his pape , we p opose a no el mul i-objec i e o mula ion ha conside s: he Roo Mean Squa e De ia ion (RMSD) di e ence in he coo dina es o ligands and he bind- ing (in e molecula ) ene gy, as wo objec i es o e alua e he quali y o he ligand-p o ein in e ac ions. To de e mine he kind o Pa e o on app oxima ions ha can be ob ained, we ha e selec ed a se o ep esen- a i e mul i-objec i e algo i hms such as NSGA-II, SMPSO, GDE3, and MOEA/D. Thei pe o mances ha e been assessed by applying wo main quali y indica o s in ended o measu e con e gence and di e si y o he on s. In addi ion, a compa ison wi h LGA, a e e ence single-objec i e e olu iona y algo i hm o molecula docking (Au oDock) is ca ied ou . In gene al, SMPSO shows he bes o e all esul s in e ms o ene gy and and RMSD ( alue lowe han 2 ˚ A o success ul docking esul s). This new mul i-objec i e app oach shows an imp o emen o e he ligand-p o ein docking p edic ions ha could be p omising in in silico docking s udies o selec new an icance compounds o he apeu ic a ge s ha a e mul id ug esis an . Keywo ds: Molecula Docking, Mul i-Objec i e Op imiza ion, Na u e Inspi ed Me aheu is ics, Algo i hm Compa ison 1 In oduc ion Ligand-p o ein docking is an op imiza ion p oblem which aims a p edic ing he posi ion o a small molecule (ligand) o a ecep o (mac omolecule) wi h he goal o inding he ligand posi ion o he ecep o wi h a minimum binding ene gy. Molecula docking p oblem has been ackled wi h single-objec i e algo i hms, o minimize he binding ene gy [11], as well as wi h mul i-objec i e app oaches, o minimize he in e molecula ene gy Ein e (ene gy in e ac ion be ween lig- and and he a ge ) and he in amolecula ene gy Ein a ( he in e nal ene gy compound) [4]. 2 E. L´opez-Camacho e al. In his ega d, a numbe o s udies based on he applica ion o mul i-objec i e algo i hms o he ligand-p o ein docking ha e been p oposed. A i s a emp was ca ied ou in 2006 by Oduguwa e al. [15], in which h ee e olu iona y mul i- objec i e algo i hms (NSGA-II, PAES, and SPEA) we e applied o e alua e h ee objec i es such as he Ein e , Ein a and shape complemen a i ies on h ee molecula complexes. G osdidie e al. [5] p oposed a new hyb id e olu iona y algo i hm called EADock ha op imizes wo di e en ene gy sco e unc ions ha e alua e he Ein e , Ein a and he sol a ion ee ene gy. In 2008, Janson e al. [7] designed a pa allel mul i-objec i e algo i hm using Au oDock 3.05 ene gy unc ion, called Clus MPSO, minimizing as objec i es he Ein e and Ein a when dealing wi h six molecula complexes. In he same yea , Boisson e al. [1] implemen ed a pa allel e olu iona y bi-objec i e model based on op imizing wo objec i es: he sum o Ein e and Ein a and a su ace e m o he docking o six ins ances. Sando al-Pe ez e al. [16] used he implemen a ion o NSGA-II p o ided by he jMe al amewo k o op imize bound and non-bound ene gy e ms o ligand/ ecep o as objec i es applied o ou docking ins ances. Gu e al. [6] de eloped a new mul i-objec i e app oach based on op imizing he solu ions gene a ed by an agg ega ed sco ing unc ion ha includes e ms om o ce- ield, empi ical and knowledge-based sco ing unc ions. In all hese p e ious publica ions, a se ies o di e en mul i-objec i es o mu- la ions ha e been p oposed ha ocus on ene gy sco ing unc ion. Howe e , hey do no conside guiding he sea ch wi h a new objec i e when he co-c ys allized ligand is known, which could complemen he adi ional ene gy unc ion. Wi h his mo i a ion, we p opose in his wo k a no el mul i-objec i e app oach consis ing minimizing: (1) he binding ene gy ( he unbound and bound ene gy e ms o he ligand/ ecep o complex), and (2) he Roo -Mean-Squa e-De ia ion (RMSD) sco e, when he co-c ys allized ligand pose is known. These wo main objec i es ha e been used o e alua e he quali y o he ligand-p o ein in e ac ions. Wi h his aim, we compa e and analyze he pe o mance o ou mul i-objec i e me aheu is ics when sol ing 11 lexible ligand- ecep o docking complexes aken om he Au oDock 4.2 benchma k [12]. This da ase includes lexible ligands wi h di e en sizes and lexible side-chains o HIV-p o ease ecep o s o mo e ealis ic esul s. The algo i hms used in his s udy a e: Nondomina ed So ing Gene ic Algo i hm II (NSGA-II) [2], Speed Modula ion Mul i-Objec i e Pa icle Swa m Op imiza ion (SMPSO) [13], Thi d E olu ion S ep o Gene alized Di e en ial E olu ion (GDE3) [8], and Mul i-Objec i e E olu iona y Algo i hm Based on Decomposi ion (MOEA/D) [18]. These algo i hms cons i u e a a ied se o e olu iona y and di e ence- ec o mul i-objec i e echniques ep esen a i e o he s a e o he a , pe o ming di e en lea ning p ocedu es and inducing di e en beha io s in e ms o con e gence and di e si y. This pape is o ganized as ollows: Sec ion 2 desc ibes he molecula dock- ing p oblem om a mul i-objec i e o mula ion. S udied algo i hms a e b ie ly desc ibed in Sec ion 3. Sec ion 4 epo s he expe imen a ion me hodology and Sec ion 5 analyzes he esul s ob ained. Finally, Sec ion 6 con ains concluding ema ks and u u e lines o esea ch. A new MO App oach o Molecula Docking 3 2 The p oblem: Mul i-Objec i e Docking A mul i-objec i e op imiza ion p oblem is cha ac e ized by wo spaces: he decision and he objec i e spaces. The o me in ol es all he possible easible solu ions, and he la e includes hei co esponding objec i e alues. Decision space. The main objec i e in he molecula docking p oblem is o ind an op imized con o ma ion be ween he ligand ( L ) and he ecep o ( R ) ha esul s in he lowes binding ene gy. The ligand- ecep o in e ac ion is e alua ed by an ene gy unc ion calcula ed h ough h ee componen s ep esen ing deg ees o eedom: (1) he ansla ion o he ligand molecule, in ol ing he h ee axis alues ( x, y, z ) in ca esian coo dina e space; (2) he ligand o ien a ion, modeled as a ou a iables qua e nion including he angle slope ( θ ); and (3) he lexibili ies, ep esen ed by he ee o a ion o o sion (dihed al angles) o he ligand and sidechains o he ecep o . Each p oblem solu ion o Au oDock and jMe al ( he ools we ha e used) is encoded by a eal- alue ec o o 7 + n a iables, in which he i s h ee alues co espond o he ligand ansla ion, he nex ou alues co espond o he ligand and/o mac omolecule o ien a ion, and he emaining n alues a e he ligand o sion dihed al angles. Fu he mo e, in o de o allow a apid e alua ion o he ene gy con o ma ions, a g id-based me hodology is implemen ed. The ene gy in e ac ion is calcula ed and assigned o each g id poin and is e alua ed o ob ain he ene gy o a gi en ligand pose [12]. Objec i e space. Ou bi-objec i e o mula ion consis s o : he Ein e and he RMSD sco e. The Ein e is he ene gy unc ion as used in Au odock, ha is calcula ed as ollows: Ein e =QR−L bound +QR−L unbound (1) QR−L bound and QR−L bound a e he s a es o bound and unbound o he ligand- ecep o complex, espec i ely. Q=W dw X i,j (Aij 12 ij − Bij 6 ij ) + Whbond X i,j E( ) Cij 12 ij − Dij 10 ij !+ +Welec X i,j qiqj ε( ij) ij +Wsol X i,j (SiVj+SjVi)e(− 2 ij /2σ2) (2) Each pai o ene ge ic e alua ion e ms includes e alua ions ( Q ) o dispe - sion/ epulsion ( dw ), hyd ogen bonds ( hbond ), elec os a ics ( elec ) and des- ol a ion ( sol ). Weigh s W dw , Whbond , Wcon , Welec , and Wsol o Equa ion 2 a e cons an s o Van de Waals, hyd ogen bonds, o sional o ces, elec os a ic in e ac ions and desol a ion, espec i ely. ij ep esen s he in e a omic dis ance, Aij and Bij in he i s e m a e Lenna d-Jones pa ame e s aken om he Ambe o ce ield. Simila ly, Cij and Dij in he second e m a e Lenna d-Jones pa ame e s o maximum well dep h o po en ial ene gies be ween wo a oms, and E ( ) ep esen s he angle-dependen di ec ionali y. The hi d e m uses a 4 E. L´opez-Camacho e al. Coulomb app oach o elec os a ics. Finally, he ou h e m is calcula ed om he olume ( V ) o he a oms ha a e su ounding a gi en a om weigh ed by S , and an exponen ial e m which in ol es a om dis ances. An ex ended explana ion o all hese a iables can be ound in [12]. The RMSD is a measu e o simila i y be ween he eal ligand posi ion in he ecep o and he compu ed posi ion o he docking ligand, ha akes in o accoun symme y, pa ial symme y (e.g. symme y wi hin a o a able b anch) and nea -symme y in a simple heu is ic way. Ideally, he lowe RMSD sco e he be e solu ion is. A ligand- ecep o docking solu ion wi h a RMSD sco e below 2 ˚ A is conside ed as a solu ion wi h high docking accu acy. I is wo h no ing ha o he docking solu ions can be e u ned wi h highe RMSD sco es and low alues o Ein e , indica ing ha o he possible in e ac ion ligand si es should be conside ed. The RMSD sco e o wo iden ical s uc u es a and b is de ined as ollows: RMSDab =max(RMSD0 ab, RMSD0 ba), wi h RMSD0 ab =s1 NX i min j ij 2 (3) The sum is o e all N hea y a oms in s uc u e a , he minimum is o e all a oms in s uc u e awi h he same elemen ype as a om iin s uc u e b. 3 Algo i hms We ha e included in ou s udy ou algo i hms which a e ep esen a i e o he s a e-o - he-a in he mul i-objec i e op imiza ion ield. A b ie desc ip ion o each one o hem is gi en nex : NSGA-II: NSGA-II [2] is a gene a ional gene ic algo i hm, which uses he ypical gene ic ope a o s (selec ion, c osso e and mu a ion) o ob ain new indi iduals om he o iginal popula ion. To p omo e con e gence, a non-domina ed so ing p ocedu e based on Pa e o anking is used, while he c owding dis ance densi y es ima o is applied o os e he di e si y o he se o ound solu ions. GDE3: The Gene alized Di e en ial E olu ion (GDE) algo i hm [8] is based on NSGA-II, bu he gene ic mu a ion and selec ion ope a o s a e eplaced by hei di e en ial e olu ion coun e pa s. Fu he mo e, GDE3 modi ies he c owding dis ance o NSGA-II as well o gene a e a be e dis ibu ed se o solu ions. SMPSO: SMPSO [13] is a mul i-objec i e pa icle swa m op imiza ion algo i hm. I s main ea u e is he limi a ion o he pa icle speed o allow new e ec i e pa icle posi ions o be p oduced when he speed becomes oo high. SMPSO uses he polynomial mu a ion as he u bulence ac o and an ex e nal a chi e ha s o es he non-domina ed solu ions ound du ing he sea ch. A new MO App oach o Molecula Docking 5 MOEA/D: MOEA/D [18] has become he ypical ep esen a i e decomposi ion- based mul i-objec i e algo i hm, whe e a mul i-objec i e p oblem is decomposed in o a se o single-objec i e subp oblems ha hen op imized simul aneously. In his s udy we ha e used he a ian MOEA/D-DE [9], which applies di e en- ial e olu ion as a ia ion ope a o s. This algo i hm also applies a polynomial mu a ion ope a o o imp o e i s sea ch capabili y. In sho , we ha e selec ed he mos widely used algo i hm in he ield (NSGA- II), a sol e based in di e en ial e olu ion (GDE3), a PSO (SMPSO) and an algo i hm based on decomposi ion (MOEA/D). 4 Expe imen a ion In his sec ion, we include he selec ed benchma k p oblems, he expe imen a ion me hodology we ha e ollowed, and he pa ame e se ings o he algo i hms. 4.1 Benchma k P oblems In his s udy, we ha e selec ed a benchma k composed o 11 complexes ha ing ecep o and ligand lexibili y. The selec ion o hese complexes has been mo i a ed as hey a e ac ually di icul docking p oblems con aining a wide ange o ligand sizes ( om small o la ge inhibi o s). The ecep o s o hese complexes ha e a unnel-shaped ac i e si e ha w aps a ound a pep idomime ic inhibi o [12]. The ecep o is a dime whose subuni s a e b idged by an a ginine-aspa a e sal b idge a he end o he unnel. The docking s udies pe o med wi h hese ins ances in [12] o es he ene gy unc ion o Au oDock 4.2 demons a ed ha he mos di icul p oblems a e hose which in ol e smalle ligands. This is due o he lexibili y added o he ecep o side-chains (ARG-8) ha inc eases he space o ligand in e ac ion. These ins ances ha e been aken om he PDB da abase 1 and hey ha e been p ope ly p epa ed o he docking simula ions. Table 1 summa izes he se o p oblems selec ed showing he PDB accession code, he X- ay c ys al s uc u es names and he s uc u e esolu ion ( ˚ A). Fo all ins ances, he o sional deg ees o eedom ( lexibili y) o ligands and mac o- molecules a e 10 and 6, espec i ely, selec ing hose o sions ha allow he ewes numbe o a oms o mo e a ound he ligand co e. The e o e, he o al numbe o solu ion a iables ( n ) is 23 (3 o ansla ion, 4 o o a ion qua e nion, and 16 o o sional deg ees). 4.2 Me hodology Fo his wo k, we ha e ca ied ou a ho ough expe imen a ion consis ing in pe o ming 30 independen uns o each combina ion o algo i hm and molecula ins ance. F om hese execu ions, we ha e calcula ed he median and in e qua - ile ange (IQR) as measu es o cen al endency and s a is ical dispe sion, 1In URL: h p://www. csb.o g/pdb/home/home.do 6 E. L´opez-Camacho e al. Table 1. The accession codes, he X- ay c ys al s uc u e and esolu ion aken om PDB da abase a e p esen ed. PDB Code P o ein-ligand complexes Resolu ion (˚ A) 1AJV HIV-1 p o ease/AHA006 2.00 1AJX HIV-1 p o ease/AHA001 2.00 1BV9 HIV-1 p o ease/α-D-glucose 2.20 1D4K HIV-1 p o ease/Mac ocyclic pep idomime ic inhibi o 8 1.85 1G2K HIV-1 p o ease/AHA047 1.95 1HIV HIV-1 p o ease/U75875 2.00 1HPX HIV-1 p o ease/KNI-272 2.00 1HTF HIV-1 p o ease/GR126045 2.20 1HTG HIV-1 p o ease/GR137615 2.00 1HVH HIV-1 p o ease/Q8261 1.80 2UPJ HIV-1 p o ease/U100313 3.00 espec i ely. We ha e conside ed wo quali y indica o s o assess he algo i hm pe o mance: Hype olume ( IHV ) and Una y Addi i e Epsilon Indica o ( I+ ) [3]. The i s indica o akes in o accoun bo h con e gence and di e si y, whe eas he second one ( I+ ) gi es a measu e o he con e gence deg ee o he ob ained Pa e o on app oxima ions. In his sense, i is wo h no ing ha we a e dealing wi h a eal-wo ld op imiza ion p oblem, and he e o e he ue Pa e o on s o calcula e hese wo me ics a e no known. To cope wi h his issue, we ha e gene a ed a e e ence Pa e o on o each ins ance by combining all he non-domina ed solu ions compu ed in all he execu ions o all he algo i hms. As men ioned, we ha e used he implemen a ion o he ou algo i hms s udied p o ided in he jMe alCpp amewo k [10] in combina ion wi h Au oDock 4.2 o e alua e he new gene a ed solu ions. To cope wi h he high compu a ional equi emen s needed by ca y ou ou expe imen s, we ha e used he Condo 2 sys em, a middlewa e pla o m managing close o 400 co es ha ac s as a dis ibu ed ask schedule (each ask dealing wi h one independen un). 4.3 Pa ame e Se up The selec ed algo i hms ha e been con igu ed wi h a popula ion size o 150 indi iduals (pa icles in he case o SMPSO). The s opping condi ion has been se o compu e a numbe o 1,500,000 unc ion e alua ions. These alues we e chosen because hey a e he de aul se ings used by Au oDock and hey ha e been used in o he s udies [14]. Each algo i hm has been con igu ed using he pa ame e se up ecommended in he esea ch s udy whe e i was p oposed, and hese pa ame e s a e used as de aul in he jMe al amewo k. In pa icula , SBX c osso e and polynomial mu a ion a e he a ia ion ope a o s used in NSGA-II. The dis ibu ion indexes o bo h ope a o s a e ηc = 20 o c osso e , and ηm = 20 o mu a ion. The 2In URL: h p:// esea ch.cs.wisc.edu/h condo / A new MO App oach o Molecula Docking 7 Table 2. Median and in e qua ile ange o IHV o each algo i hm and ins ance. Bes and second bes median esul s ha e da k and ligh g ay backg ounds, espec i ely. NSGAII SMPSO GDE3 MOEAD 1AJV 0.00e+ 000.0e+00 3.51e−014.0e−02 0.00e+ 000.0e+00 0.00e+ 002.9e−01 1AJX 0.00e+ 000.0e+00 5.52e−012.0e−02 0.00e+ 000.0e+00 7.47e−036.8e−01 1D4K 0.00e+ 000.0e+00 4.93e−011.3e−01 0.00e+ 000.0e+00 0.00e+ 000.0e+00 1G2K 0.00e+ 000.0e+00 3.32e−013.3e−02 0.00e+ 000.0e+00 0.00e+ 004.1e−01 1HIV 0.00e+ 000.0e+00 5.96e−011.3e−01 0.00e+ 000.0e+00 0.00e+ 000.0e+00 1HPX 0.00e+ 000.0e+00 2.04e−011.8e−01 1.27e−016.5e−01 0.00e+ 001.1e−01 1HTF 0.00e+ 000.0e+00 5.26e−021.3e−01 0.00e+ 004.6e−03 2.78e−023.3e−01 1HTG 0.00e+ 000.0e+00 3.51e−025.6e−02 0.00e+ 000.0e+00 0.00e+ 001.9e−01 1HVH 0.00e+ 000.0e+00 7.67e−013.7e−02 0.00e+ 000.0e+00 5.31e−017.7e−01 1VB9 0.00e+ 000.0e+00 7.34e−016.5e−02 0.00e+ 000.0e+00 0.00e+ 001.4e−01 2UPJ 0.00e+ 000.0e+00 5.86e−019.8e−02 0.00e+ 000.0e+00 1.90e−015.8e−01 c osso e p obabili y is pc = 0 . 9 and he mu a ion p obabili y is pm = 1 /n , being n he numbe o decision a iables o he ackled p oblem. NSGA-II applies bina y ou namen selec ion. In he case o GDE3 ( a ian and/1/bin), he wo DE con ol pa ame e s µ and C ake a alue o 0.5, whe eas in MOEA/D µ is se o 0.5 and C is se o 1.0. Bo h MOEA/D and SMPSO use he polynomial mu a ion wi h he same se ings applied in NSGA-II. In SMPSO, he accele a ion coe icien s ϕ1 and ϕ2 a e se o 1.5, he ine ia weigh is W = 0 . 9, and he polynomial mu a ion is applied o one six h o he pa icles in he swa m. 5 Resul s This sec ion is de o ed o p esen ing and analyzing he esul s ob ained in ou s udy. We s a by assessing he pe o mance o he algo i hms and hen hey a e compa ed wi h he alues o a single-objec i e app oach. 5.1 Pe o mance Compa isons We s a ou analysis by discussing he esul s yielded by applying he IHV indica o . Le us emind ha his indica o is he sum o he con ibu ed olume o each poin o a on in espec o a e e ence poin , and he highe he con e gence and di e si y deg ees o a on , he highe i s IHV alue. Table 2 shows he median and in e qua ile ange o he compu ed solu ions o IHV quali y indica o s o he se o 11 docking ins ances and he ou algo i hms being compa ed. Acco ding o hese esul s, SMPSO achie es he bes IHV alues in all he ele en conside ed p oblems and MOEA/D is he second bes pe o ming echnique. We ha e o no e ha many cells ha e a IHV alue equal o ze o; his happens when all he poin s o he p oduced on s a e beyond he limi s o he e e ence poin . This happens in mos o he p oblems in all he algo i hms excep ing SMPSO, which indica es we a e acing a e y ha d op imiza ion p oblem. A simila beha io can be obse ed in Table 3 wi h ega ds o I+ , which is a con e gence measu e. Acco ding o hese esul s, SMPSO ob ains he bes I+ 8 E. L´opez-Camacho e al. Table 3. Median and in e qua ile ange o I+ o each algo i hm and ins ance. Bes and second bes median esul s ha e da k and ligh g ay backg ounds, espec i ely. NSGAII SMPSO GDE3 MOEAD 1AJV 5.23e+ 001.2e+00 5.60e−019.8e−02 5.00e+ 001.0e+00 3.87e+ 004.4e+00 1AJX 3.43e+ 002.4e+00 2.61e−017.2e−02 1.49e+ 003.3e−01 1.01e+ 002.0e+00 1D4K 8.06e+ 002.7e+00 4.56e−011.4e−01 8.56e+ 005.7e−01 4.65e+ 002.8e+00 1G2K 4.28e+ 001.4e+00 5.71e−011.2e−01 3.93e+ 001.3e+00 2.69e+ 003.6e+00 1HIV 5.12e+ 001.2e+00 2.63e−012.1e−01 4.69e+ 001.4e+00 4.07e+ 001.6e+00 1HPX 1.42e+ 013.6e+00 6.32e−012.8e−01 6.71e−011.1e+01 1.03e+ 011.3e+01 1HTF 1.76e+ 005.5e−01 9.30e−013.0e−01 1.13e+ 008.0e−01 7.94e−019.2e−01 1HTG 7.48e+ 007.1e−01 9.63e−016.6e−02 6.82e+ 008.7e−01 5.03e+ 006.4e+00 1HVH 5.94e+ 001.5e+00 1.34e−012.7e−02 4.93e+ 001.7e+00 4.16e−012.1e+00 1VB9 8.59e+ 002.4e+00 1.33e−015.6e−02 7.85e+ 001.3e+00 7.04e+ 004.9e+00 2UPJ 3.42e+ 002.4e+00 3.03e−016.6e−02 3.56e+ 001.1e+00 7.64e−012.7e+00 Table 4. A e age F iedman’s ankings wi h Holm’s Adjus ed p - alues ( α = 0 . 05) o compa ed algo i hms (SMPSO, GDE3, MOEA/D, and NSGA-II) o he es se o 11 docking ins ances. Symbol * indica es he con ol algo i hm and column a igh con ains he o e all anking o posi ions wi h ega ds o IHV and I+. Hype olume (HV) Epsilon (I+) Algo i hm F iRank HolmAp Algo i hm F iRank HolmAp *SMPSO 1.02 -*SMPSO 1.09 - MOEA/D 2.68 2.24e-03 MOEA/D 2.00 9.87e-02 GDE3 3.09 1.45e-04 GDE3 3.09 2.79e-04 NSGA-II 3.22 5.21e-05 NSGA-II 3.81 7.25e-07 alues in en ou o he ele en p oblems, while MOEA/D ge s he bes alue in one p oblem (1HTF) and he second bes in all o hem bu 1HPX. In o de o p o ide hese esul s wi h s a is ical meaning (in his s udy α = 0 . 05), non-pa ame ic s a is ical es s ha e been applied because in se e al cases he dis ibu ions o esul s did no ollow he condi ions o no mali y and homoscedas ici y [17]. The e o e, he analyses and compa isons ocus on he en i e dis ibu ion o each o he wo me ics s udied. Speci ically, we ha e applied F iedman’s anking and Holm’s pos -hoc mul icompa e es s [17] o know which algo i hms a e s a is ically wo se han he con ol one (wi h he bes anking). In his ega d, as shown in Table 4, SMPSO is he bes anked echnique acco ding o IHV (wi h a alue o 1.02), ollowed by MOEA/D, GDE3, and NSGA-II. The e o e, SMPSO is es ablished as he con ol algo i hm o IHV in he pos -hoc Holm es , which is compa ed wi h he emaining algo i hms. The adjus ed p- alues ( HolmAp in Table 4) esul ing om hese compa isons a e, o he las h ee algo i hms (MOEA/D, GDE3, and NSGA-II), lowe han he con idence le el, meaning ha SMPSO is s a is ically be e han hese algo i hms. In he case o I+ , SMPSO is be e anked han he emaining compa ed algo i hms, al hough wi hou s a is ical di e ences in he case o MOEA/D. SMPSO is s a is ically be e han GDE3 and NSGA-II. In summa y, SMPSO shows he o e all bes balance o he wo quali y indica o s, ollowed by MOEA/D. These esul s a e g aphically suppo ed by wo examples included in Fig. 1, whe e he e e ence on s ob ained o wo A new MO App oach o Molecula Docking 9 −30 −29 −28 −27 0.0 0.2 0.4 0.6 1D4K Ene gy (kcal/mol) RMSD (Å) MOEA/D RF −30 −29 −28 −27 0.0 0.2 0.4 0.6 1D4K Ene gy (kcal/mol) RMSD (Å) SMPSO RF −27.0 −26.5 −26.0 −25.5 −25.0 −24.5 0.0 0.2 0.4 0.6 0.8 1.0 1HIV Ene gy (kcal/mol) RMSD (Å) MOEA/D RF −27.0 −26.5 −26.0 −25.5 −25.0 −24.5 0.0 0.2 0.4 0.6 0.8 1.0 1HIV Ene gy (kcal/mol) RMSD (Å) SMPSO RF Fig. 1. Re e ence on con ibu ions o docking ins ances 1D4K and 1HIV. SMPSO and MOEA/D con ibu e wi h p ac ically all he solu ions o he e e ence on s. ep esen a i e ins ances 1D4K and 1HIV a e plo ed. In hese g aphs, he con- ibu ions, in e ms o solu ions, o each algo i hm o he global e e ence on a e plo ed wi h di e en poin s and colo s. As i is easily obse able, SMPSO and MOEA/D con ibu e wi h almos all solu ions aking pa o he e e ence on . In e es ingly, SMPSO con e ges o he egion biased owa ds he RMSD objec i e, whe eas MOEA/D gene a e non-domina ed solu ions in a di e en egion o he ones o SMPSO, he eby gi ing cue o he ene gy op imiza ion. We can s a e ha he speci ic lea ning p ocedu es induced by SMPSO and MOEA/D lead hese algo i hms o sea ch in di e en egions o he p oblem landscape, hence gene a ing solu ions in complemen a y pa s o he e e ence on . 5.2 Compa ison Single Ve sus Mul i-objec i e A e he pe o mance compa ison o mul i-objec i e algo i hms, we a e now in e es ed in knowing how compe i i e hei solu ions a e agains hose yielded