Branch-and-Terminate: a combinatorial optimization algorithm for protein design D Benjamin Gordon 1 and Stephen L Mayo 2 *

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1 Reeah Atle 1089 Banh-and-Temnate: a ombnatoal optmzaton algothm fo poten degn D Benjamn Godon 1 and Stephen L Mayo 2 * Bakgound: Seveal detemnt and tohat ombnatoal optmzaton algothm have been appled to omputatonal poten degn and homology modelng. A tutual taget neae n ze, howeve, t ha beome neeay to fnd moe poweful method to adde the neaed ombnatoal omplexty. Reult: We peent a new detemnt ombnatoal eah algothm alled Banh-and-Temnate (B&T), whh deved fom the Banh-and-Bound eah method. The B&T appoah baed on the ontuton of an effent but vey ettve boundng expeon, whh ued fo the eah of a ombnatoal tee epeentng the poten ytem. The boundng expeon ued both to detemne the optmal oganzaton of the tee and to pefom a hghly effetve punng poedue named temnaton. Fo ome alulaton, the B&T method val the uent detemnt tandad, dead-end elmnaton (D), ometme fndng the oluton up to 21 tme fate. A moe gnfant featue of the B&T algothm that t an povde an effent way to omplete the optmzaton of poblem that have been patally edued by a D algothm. Conluon: The B&T algothm an effetve optmzaton algothm when ued alone. Moeove, t an neae the poblem ze lmt of amno ad dehan plaement alulaton, uh a poten degn, by ompletng D optmzaton that eah a pont at whh the D tea beome neffent. Togethe the two algothm make t poble to fnd oluton to poblem that ae ntatable by ethe algothm alone. Addee: 1 Dvon of Chemty and Chemal ngneeng, Calfona Inttute of Tehnology, Paadena, Calfona 91125, USA and 2 Howad Hughe Medal Inttute and Dvon of Bology, , Calfona Inttute of Tehnology, Paadena, Calfona 91125, USA. *Coepondng autho. -mal: teve@mayo.alteh.edu Key wod: Banh-and-Bound, dead-end elmnaton, global enegy mnmum, poten degn, otame Reeved: 8 Febuay 1999 Revon equeted: 23 Apl 1999 Revon eeved: 14 May 1999 Aepted: 18 May 1999 Publhed: 26 Augut 1999 Stutue Septembe 1999, 7: /99/$ ee font matte 1999 leve Sene Ltd. All ght eeved. Intoduton Sgnfant advane n poten degn [1,2] and poten dehan homology modelng [3] have aen fom the applaton of optmzaton algothm and pealzed potental to the dehan plaement poblem. In thee alulaton, one eahe fo the et of dehan onfomaton that podue the global mnmum enegy onfomaton (GMC) fo the gven poten bakbone. The enege of dehan nteaton ae evaluated ung empally baed enegy potental and, to edue the omplexty of the alulaton, the et of poble dehan oentaton detzed nto tattally epeentatve onfomaton alled otame [4,5]. The eah fo the optmal eleton of dehan otame fo a pefed poten fold neealy a ombnatoal optmzaton poblem; an exhautve eah though all ombnaton ntatable. A uh, the poblem ha been appoahed by eveal dffeent method, nludng Monte Calo [6,7] and mulated annealng [8], mean-feld [9,10], and dead-end elmnaton (D) [11 14]. In patula, D method have emeged a poweful tool fo moe dffult poten degn alulaton, n whh the optmal dehan ae eleted fom otame of many dffeent amno ad [1,2]. Thee ae optmzaton, howeve, fo whh D algothm ae not uffent, due to ethe the natue of the enegy dtbuton o the hee ze. Fo example, the optmzaton of long hydophl dehan on β heet typally ompoed of lage numbe of otame wth nteaton enege that ae vey mall n magntude. D able to edue the ombnatoal ze of the poblem gnfantly at the outet, but, oon afte, elmnaton beome neffent, elyng entely on omputatonally expenve D double alulaton [12,14]. Th behavo alo obeved n the late tage of vey lage alulaton, when afte eveal ound of unfaton [15] futhe elmnaton beome dffult and the numbe of upe-otame at upe-edue poton beome vey lage. To omplete uh alulaton, a tehnque ontng of exhautve ombnatoal buldup aded by D ha been debed [3]. Howeve, a the effetvene of the elmnaton tea poo n thee ae, t advantageou to ontut a method that not dependent on them. To adde thee dffult optmzaton poblem, we have developed an enhaned veon of a Banh-and-Bound (B&B) algothm [16] that we have dubbed Banh-and- Temnate (B&T). B&B algothm ompe a ubla

2 1090 Stutue 1999, Vol 7 No 9 of baktak algothm that utlze nfomaton about ot (o enege) of omplete and patal oluton. Baktak algothm ae ommonly ued n atom-level mulaton to ontut elf-avodng han, and they have been ued n poten degn to engnee metal-bndng te nto poten [17]. B&B algothm ae ommonly appled to theoetal ombnatoal and hedulng poblem, and, moe eently, to ombnatoal poblem of tutual bology angng fom equene algnment [18] and tutual ompaon [19], to maomoleula pakng [20], lgand degn [21] and, eently, poten tetay tutue pedton [22]. Regadng the tudy of poten dehan, Samudala and Moult [23] have debed a gaph-theoet appoah to the loely elated poblem of ompaatve modelng, n whh they epeent the eah a a lque-fndng poblem that they olve ung a B&B algothm. In addton, Leah and Lemon [24] have ued a B&B algothm (alled A* ) to exploe the onfomatonal enegy ufae of poten dehan. It taghtfowad to fomulate the dehan optmzaton poblem fo det optmzaton by a B&B algothm. All that neeay to debe the poblem a a eah of a ombnatoal tee whee one eahe fo the ngle path though the banhe that oepond to the GMC et of otame. The B&B algothm effetve beaue t multaneouly pune the tee whle eahng; eah banh teted wth a quanttatve boundng expeon befoe beng eahed. In mplementng a B&B algothm fo dehan eleton, we have nopoated ome novel algothm tehnque that neae the optmzaton peed damatally. Ft, we debe a boundng funton that maxmze the effeny of punng fo poblem n whh the total enegy an be deompoed nto nteaton between pa of otame. We alo debe a poe we all temnaton, n whh we ue the boundng funton to detemntally emove otame at all amno ad poton, theeby edung the oveall ze of the tee befoe eahng. Temnaton addtonally effetve when pefomed at evey level of euon of the eah, ometme neang the oveall peed of the optmzaton by an ode of magntude. Lat, we demontate how the eneget nfomaton podued by the temnaton poe an be ued to detemne the optmal eah ode fo the emande of the tee. Beaue temnaton effetvely eplae the uual boundng poe, the eultng beadth-ft algothm alled Banh-and-Temnate. We alo debe a vaaton of the B&T method that an apdly fnd appoxmate oluton loe to the GMC. The depton of the B&T algothm that follow taloed fo otame eleton, but the algothm an n fat be genealzed to any ombnatoal optmzaton poblem n whh all the nteaton enege ae pawe and peomputable. The boundng expeon we debe mlaly geneal. Although the B&T algothm an be ued by telf, geate beneft an often be obtaned by ung t n onet wth a D algothm. Togethe, the algothm an olve optmzaton poblem muh moe qukly than ethe an aomplh alone. Th may make t poble to qukly fnd the GMC fo poten degn poblem that wee pevouly noluble by ethe algothm. Reult Banh-and-Bound When a ombnatoal tee ued to debe the dehan optmzaton poblem, the oot of the tee plaed at the top and banhe extend downwad. ah level of depth of the tee oepond to an amno ad poton, and eah node epeent a patula otame hoe at that poton. Thu a path that extend all the way fom the tee oot though all level of banhe to a leaf debe a omplete otame equene. The poblem, then, to eah fo the path oepondng to the equene wth the lowet enegy. A patal path fom the oot debe a otame equene that nompletely pefed. Altenatvely, the path an be ntepeted phyally a pefyng a unque ompote otame, o upe-otame, that oupe a ubet of the amno ad poton. xtendng the path deepe nto the tee oepond to appendng addtonal otame to the upe-otame, whh an be epeated untl all poton ae pefed. Aodng to th ntepetaton, a full eah of the tee would ental the ontuton of all poble upe-otame to ompleton. It often poble, howeve, to detemne that a patula patally pefed upe-otame not pat of the GMC. In uh a ae, t unneeay to exploe any ombnaton that would eult fom buldng up the upe-otame futhe. Appled euvely, uh obevaton pune ubtee fom node thoughout the tee, theeby enablng an exhautve eah wthout omplete enumeaton of all poble upe-otame. The punng detemnaton aomplhed by ompang a lowe enegy bound fo the patally pefed otame equene to a known efeene enegy. Gven a efeene enegy of any plauble equene, t mut be tue that the enegy of the GMC le than o equal to the enegy of any plauble equene: GMC efeene One may theefoe dedue that the global mnmum doe not ontan a patula upe-otame upon obevng that (1)

3 Reeah Atle Banh-and-Temnate Godon and Mayo 1091 the enegy upe,bet of the equene eultng fom optmal ompleton of the anddate upe-otame geate than the efeene enegy: upe,bet > efeene Fndng the optmal ompletng equene, howeve, an be a dffult a the ognal poblem, o we ntead ontut an expeon fo a lowe enegy bound, upe,bound. The expeon ontuted to ompute an nexpenve lowe enegy bound baed on the patally pefed equene, a well a on the otame that ae avalable at the unpefed poton. By defnton, the bound mut atfy the nequalty upe,bet upe,bound Wth th quantty n hand, we may pune any ubtee fo whh we obeve that the lowe bound geate than the efeene enegy: > upe,bound efeene Th the boundng teon. The B&B algothm ont of an exhautve taveal of the ombnatoal tee, applyng th teon to eah node a t enounteed. Wheneve the eah podue a omplete path wth an enegy lowe than the uent efeene enegy, the efeene enegy updated. In th way, the effetvene of the boundng teon neaed ove the oue of the optmzaton. Moeove, upon ompleton of the eah, the efeene enegy the global mnmum enegy. The oepondng equene alo toed dung eah update, whh podue the oepondng GMC. Boundng expeon The ueful mplementaton of a B&B type of algothm depend lagely on the ontuton of the boundng expeon. A boundng expeon that vey tngent wll podue lowe bound that ae hgh n enegy, and theefoe wll eult n moe ubtee that an be puned by the boundng teon. The ze of the eultng tee wll be malle than one puned by a le tngent expeon, and the eah wll be fate. It theefoe mpotant to degn the boundng expeon to mot fully utlze the equene nfomaton avalable. On the othe hand, tngeny obtaned at the ot of tme. A maxmally tngent bound mght pune all ubtee exept fo the one ontanng the global mnmum, but t would take an mpatal amount of tme to ompute. It theefoe alo neeay to tempe tngeny wth peed ondeaton n ode to obtan (2) (3) (4) a boundng expeon that popely balaned fo effent eahng. We debe the ontuton of uh a boundng expeon n the Mateal and method eton. Gven a patally ontuted upe-otame and the avalable otame at the emanng poton, the appoah to utlze the oepondng eneget nfomaton a fully a poble whle keepng the omputatonal ode of the boundng expeon ontant. The eult a novel, hghly effetve boundng expeon that povde the ba fo the emanng B&T tehnque. The fom of the eultng expeon ha an addtonal advantage: t olate thoe pat of the expeon that ae dental fo otame on the ame level of a ubtee. Thu t poble to futhe neae the effeny of the eah by peomputng thee haed quantte a eah goup of node enounteed, athe than edundantly evaluatng the ente boundng expeon fo evey unque node. Th method debed n the Mateal and method eton. Temnaton The enhanement of the B&T algothm elatve to the B&B method ae baed on a poe alled temnaton. Beaue all the pawe nteaton ae peomputable, the oganzaton of the ombnatoal tee abtay (.e. thee no pef ode n whh dffeent amno ad poton mut be agned to dffeent level of the tee). The oganzaton of the tee an, howeve, have a gnfant nfluene on the peed of the alulaton. Fo example, a geate eduton n the ze of the eah deved fom punng a banh at the oot of the tee athe than punng a banh loe to the leave. Plang a banh at the leave that would be puned f plaed at the oot would be neffent beaue the ame punng tep would neealy be epeated fo evey leaf. In fat, t ommonly ou that all amno ad poton have ome otame that ould be puned f plaed at the oot of the tee. To umvent the potental lo of effeny, we mplement a pepoeng poedue befoe detemnng the tee oganzaton. Th poedue ont of tempoaly ondeng eah amno ad poton to be at the oot level and hekng f any of t otame an be mmedately puned. All otame puned fom oot poton may be ompletely daded fo the emande of the optmzaton, and ae dubbed temnated to eflet th fat. The eult an oveall eduton of the tee ze po to eahng, makng the optmzaton fate. The eleton of the wod temnate ntended to be ontated wth elmnate, whh ued to debe otame that ae analogouly daded by ung the D teon. Indeed, many of the ame otame ae

4 1092 Stutue 1999, Vol 7 No 9 daded. A wth D, temnaton may be pefomed teatvely untl no futhe otame ae temnated. Iteatve temnaton exeuted a the pepoeng tep befoe eah of the tee. Reuve temnaton Although temnaton eve a an effetve pepoeng tep, the hallmak of the B&T algothm that temnaton employed at evey level of euon. At any pont of the eah, the otame defned at level above the level of the uent amno ad poton may be ondeed a oot omped of a ngle, patally pefed upeotame. Temnaton, then, ont of tempoaly ondeng eah of the otame at all the emanng poton a anddate fo the next appendage of the upe-otame and applyng the boundng teon to eah one. All otame temnated th way may be daded fom the optmzaton of the ubtee wth th patally pefed upe-otame oot. In ontat, the euve tep n a B&B eah ont of the applaton of the boundng teon to the otame at only one amno ad poton. The beneft of the exta eduton n the ze of ubtee fa outwegh the ot of alulaton of exta bound fo temnaton. The eultng neae n effeny make the B&T eah gnfantly fate than a mlaly ontuted B&B eah. We have obeved that t not neeay to pefom teatve temnaton at evey level of euon, unlke temnaton pepoeng. A ngle teaton pe banh geneally yeld the bet pefomane. Seah ode When taveng the ombnatoal tee, t neeay to detemne the ode n whh to exploe otame at eah poton and the equene n whh to exploe the dffeent poton. Fo both ae, we utlze the boundng enege alulated fo eah otame dung temnaton. We have obeved an empal oelaton between low boundng enegy and membehp n the GMC; theefoe, the otame at eah poton ae eahed n ode of neang boundng enegy. Condutng the eah n th way neae the hane that oluton loe to the GMC ae found qukly, theeby povdng tngent efeene enege ealy n the alulaton. Wth epet to the odeng of the dffeent poton, we ontut a heut baed on both the temnaton boundng enege and the ze of the otame lt. In a onventonal tee eah, the poton hould be oganzed n ode of neang numbe of otame pe poton n ode to mnmze the total numbe of node n the tee. Howeve, n a B&T eah, thee ae othe oganzaton heme that favo hgh-level punng by temnaton that edue the tee ze moe gnfantly. We ue the boundng enegy of the top-anked (lowet boundng enegy) otame at eah poton to ndate whh poton ae lkely to ett the et of the ytem, and onequently favo hgh-level temnaton f plaed at the upe-otame oot. Beaue the mnmum opeato at a node ae appled ove a et nludng the ubet oepondng to the ubtee node, boundng enege of ubtee node mut be hghe than o equal to the paent node. Theefoe, plang poton wth hgh lowet-enege at the top of the tee pomote hgh boundng enege fo the deendent. Beaue the otame lt of a ubtee an be gnfantly dffeent fom thoe of t paent, edue odeng pefomed at evey level of euon depth. We have obeved that an optmal odeng an be obtaned by ombnng eneget and lt-ze otng tea ung the followng heut. Poton ae oted n deendng ode aodng to a ank ndex, a omputed by the expeon 1 Rank ndex (1 f ) f 1 lnn top,max top,mn top,mn whee N the numbe of otame at the poton, top the boundng enegy of the top-anked otame of that poton, and top,mn and top,max ae the mnmum and maxmum top-anked boundng enege of all poton, epetvely. The expeon 1/(1 lnn) ontuted to podue an attenuated weghtng nveely popotonal to the numbe of otame that evaluate to unty when N 1. The quantty f eleted to ontol the elatve weghtng of the two tea. A value of zeo fo f oepond to ot baed entely on the numbe of edue pe poton, and a value of one podue a ankng baed entely on boundng enege. Appoxmate algothm A oluton that vey loe to the GMC equene an be found vey apdly by ung an appoxmate vaaton of the B&T method. Appoxmate alulaton ae patulaly ueful fo povdng a fat way to obtan low efeene enege fo exat B&T optmzaton. Moeove, the appoxmate alulaton often uffent to podue the GMC enegy. The appoxmaton baed on the obevaton that the GMC otame ae often among thoe wth the lowet temnaton boundng enege aodng to the boundng expeon (quaton 21 n Mateal and method eton). Th ndate that the boundng expeon ha pedtve popete. To apdly fnd an appoxmate oluton, the anked otame lt ae abtaly tunated afte the pepoeng temnaton tep and top (5)

5 Reeah Atle Banh-and-Temnate Godon and Mayo 1093 the B&T eah onduted on the abbevated et of otame. A moe elable oluton an be found by epeatng the appoxmate optmzaton wth moe lenent tunaton, ung the oluton fom the peedng un fo the ntal efeene enegy. D pepoeng Pehap the mot patal ue of the B&T algothm to omplement D when dealng wth optmzaton poblem that ae too dffult to olve ung ethe algothm alone. In uh ae, the algothm ae ued n ueon. D ued to elmnate otame and to pefom unfaton untl the optmzaton eahe teaton that ae neffent. Ineffeny typally ou afte eveal unfaton when the total numbe of otame and unfed upe-otame beome vey lage (>5000) and vey few elmnaton eult even fom lengthy Goldten double alulaton. At th tage, the D optmzaton aboted, and the tate nfomaton tanfeed to a B&T mplementaton. Rotame lt and enegy table ae tanfeed detly, nludng efeene to unfed upeotame, whh ae tanpaently epeented a odnay otame n the B&T algothm. Fgue 1 (a) ƒ (b) 7 6 Tme (mn) An addtonal pefomane mpovement obtaned by alo pang the lt of dead-endng pa (DP). DP ae pa of otame (o upe-otame) the membe of whh annot multaneouly ext n the GMC. Thee pa may theefoe be afely omtted fom the mnmum opeato n quaton 21 (ee Mateal and method eton). Tme (mn) 5 4 Benhmak To ae the genealty of the B&T appoah, dffeent nanaton of the algothm wee appled to benhmak poblem epeentng dffeent tutual lae, a debed n the Mateal and method eton. Optmzaton tme wee heavly dependent on the otng heut, a hown n Fgue 1. The pefomane mpovement, a meaued by dvdng the total optmzaton tme, anged fom a fato of thee fo the ae of the β heet to a fato of ove 40 fo the mxed ae. Remakably, vey mla value of the otng fato f podued the fatet optmzaton tme fo all tutual lae. Intally, value at nteval of 0.1 wee teted, but a all benhmak ae exhbted mnma nea f 0.1, value at nteval of 0.01 wee ampled nea th value. At th level of efnement, the dffeent ae had dffeent optmal otng fato value, but a value of f 0.08 wa loe to optmal fo all of them. We alo obeve that optmzaton wth the fatet tme had the fewet node n the puned ombnatoal tee. The total alulaton tme fo the benhmak ung a otng fato of 0.08 ae ompettve ompaed to tme of ƒ Stutue Total optmzaton tme veu value of otng method fo (a) the mxed tutual type and (b) the β-heet ufae benhmak ae. Sotng detemned by the value of the fato f n quaton 5. The ae exhbt dffeent dependene on the value of the otng fato, but both have mnma n the vnty of f Th tend obeved fo all ae (not hown). a hghly optmzed D algothm, and ae gnfantly fate than the optmzed B&B eah (Table 1). Fo the β-heet ufae and the mall oe-bounday alulaton, the B&T method appoxmately 20 tme fate than D. Fo the mxed ae, t nealy eght tme fate. Fo the α-helal ae, howeve, the B&T method moe than two tme lowe. Th lkely to be a efleton of the lnea topologal aangement of the ytem, n whh t dffult to elet poton to plae at the tee oot

6 1094 Stutue 1999, Vol 7 No 9 Table 1 Benhmak tme. Benhmak ae Small oe bounday* α Sufae β Sufae Mxtue Coe bounday Total tme (mn) D B&B > 30,000 B&T # B&T omponent tme (mn) Pepoeng Seah Appoxmaton B&T total node ,634 *Refe to the benhmak omped of a mall et of oe and bounday poton. D wa pefomed ung the peed enhanement debed n [13] and [14]. The B&B algothm ue the novel boundng expeon and nlude temnaton pepoeng. Fo the dffult oe bounday ae, the nomplete B&B optmzaton wa aboted afte 30,000 mn. # Total B&T tme omputed a the um of the appoxmaton, pepoeng and eah tme. An appoxmate B&T algothm wa ued to obtan ntal bound fo the mxtue and dffult oe bounday ae. Thee alulaton ued only the top thty otame at eah poton aodng to the boundng enegy. that both ett lage pat of the ytem and ae themelve etted. The appoxmate fom of the algothm poved to be exeptonally effetve. Fo the fou ae above, B&T alulaton that ued only the 30 top-anked otame at eah poton all took le than 15 and podued the oet GMC oluton. Fo the moe dffult oebounday ae, the alulaton took 5 mn, and alo podued the oet GMC oluton. Fo th ae, a moe aggeve alulaton ung only the top 15 otame at eah poton took 25 and podued a oluton wth an enegy whh wa n eo by le than 1%. Th enegy wa ued a the ntal bound fo the emanng alulaton on the ytem. To llutate the potental fo ombnng D and B&T method by way of D pepoeng, we eleted a poblem omputable by ethe algothm to enable u to pefom quanttatve ompaon. In pate, howeve, the tehnque appled to poblem that ae not uently omputable n eaonable ompute tme by ethe algothm, fo whh the beneft obvouly muh geate. Fgue 2 llutate the total alulaton tme pattoned nto D and B&T tme fo optmzaton of the dffult benhmak ontng of oe and bounday edue. The alulaton dffe n the amount of tme allotted to D eduton befoe ompleton wth the B&T algothm. At the bet tmng, the ombned algothm omplete the optmzaton eght tme fate than D alone. Moeove, we have obeved that, n pate, the B&T method geneally effetve at ompletng lage poblem that D an edue to a hgh a emanng equene. Duon We have debed a detemnt eah method fo otame optmzaton and have demontated that fo ome ae t a fat a the uent tandad algothm fo poten degn, and fo othe ae t muh fate. The ue of the B&T method et on the ontuton of a novel pawe boundng expeon, whh ued both to pefom temnaton and to upply eneget nfomaton wth whh to detemne the eah ode. Although the algothm taloed to poten ytem, t an be genealzed to any poblem that an be mlaly debed. Although the B&T algothm qute effetve when ued alone, t pehap moe mpotant that t neae the poblem ze lmt of D alulaton by povdng an effent way to omplete optmzaton fo whh elmnaton tea have beome le effetve at emovng otame. Th make t poble to pefom optmzaton on lage poten and on ytem wth lage numbe of nteatng edue. The ze lmt may be aed even hghe one the lmtaton of the appoxmate fom of the algothm beome bette undetood. Fo the benhmak ae, the appoxmate algothm found the GMC oluton up to a thouand tme fate than ethe of the exat method. ven the D mplementaton to whh the B&T method ompaed nopoate ome onevatve appoxmaton n the fom of hgh-enegy thehold ejeton (HTR) tea [3]. Analogou tehnque may povde a way to ontut a fate, appoxmate B&T algothm wth lealy defned auay. Along the

7 Reeah Atle Banh-and-Temnate Godon and Mayo 1095 Fgue make t poble to etmate how muh tme the optmzaton wll eque. Th aomplhed ung a wellknown tee etmaton tehnque [25] n whh tatt ae ompled fo andom ample tajetoe though the tee. Th ha helped u to pedt when t bet to tanfe D poblem to B&T fo ompleton. Cumulatve tme (mn) In pate, we beleve that the bet way to ue the B&T method to ft attempt to optmze a poblem ung D. Upon obevng that D begn to podue vey few elmnaton o dead-endng pa, the tate nfomaton hould be tanfeed to an appoxmate fom of the B&T algothm. Ung the enegy fom th alulaton a the ntal uppe bound, the appoxmate algothm may be epeated agan wth uevely moe onevatve tunaton. The fnal enegy hould then be ued a the ntal bound fo the exat B&T alulaton Remanng equene on tanfe of optmzaton fom D to B&T (log ale) Stutue Optmzaton tme eultng fom the ombnaton of the B&T (whte ba) and D (old ba) algothm. The ba on the exteme left and ght of the fgue ae the tme fo lone B&T and D optmzaton, epetvely. The emanng ba ae the umulatve B&T and D optmzaton tme when the two algothm ae ued n ueon. The udden jump n D tme ae fom lengthy Goldten double alulaton. ame lne of eaonng, tunaton baed on boundng enege mght be an effetve eplaement fo HTR utoff n D. Thee alo oom fo mpovement n the heut fo detemnng eah ode. Heut that ae even moe effetve may ext that make ue of tutual nfomaton n addton to eneget and ze ondeaton. In addton, we ae uently explong featue of the B&T algothm that ae ommon to all baktak eahe. Ft, t poble to exhautvely ample the amno ad and otame equene pae nea the GMC. Th aomplhed by modfyng the algothm o that t efan fom loweng the ntal mnmum enegy upon fndng low-enegy ombnaton [24]. The eult a full enumeaton of all equene wth enege below the pefed ntal mnmum enegy, povded that th enegy loe enough to the GMC enegy that the alulaton eman tatable. Alo, t taghtfowad to adapt baktak algothm fo paallel omputaton by dpathng banhe to dffeent omputatonal node. We obeve a alng effeny between 60 and 80%, dependng on the type of poblem. Anothe advantage of the tee epeentaton that t Bologal mplaton Poten degn and poten homology alulaton typally ue ombnatoal optmzaton algothm to ompute the optmal plaement of amno ad dehan otame on poten bakbone. The apablte of exhautve eah algothm ae uently lmted by poten ze and enegy landape. The Banh-and-Temnate vaaton of the Banh-and-Bound eah algothm debed hee povde a way to optmze thee poblem, both alone and ued n onjunton wth well-etablhed algothm baed on the dead end elmnaton theoem. Mateal and method Benhmak ae We teted the genealty of the algothm by applyng t to a ute of optmzaton poblem epeentatve of dffeent poten tutual lae. Rotame wee eleted fom a bakbone-dependent lbay [26]. To tet α-helal ufae poton, the 12 edue oupyng the (b), (), and (f) loaton n the heptad epeat of one helx of the oledold GCN4-p1 dme [27] wee optmzed fom the et of otame oepondng to hydophl amno ad (A, D,, H, K, N, Q, R, S and T). Thee wee otame ombnaton. The β1 doman of teptooal poten G [28] wa ued fo the emanng ae. A a epeentatve of oe and bounday optmzaton poblem, a ubet of poton detemned to be n the oe and bounday aodng to ou edue lafaton heme (poton 3, 5, 7, 12, 23, 25, 26, 30, 34, 43, 45, 52 and 54) wee optmzed fom the ombnaton of hydophob otame (amno ad A, F, I, L, M, V, W and Y). Fo β-heet ufae, a ubet of the β-heet ufae edue (poton 4, 6, 15, 17, 42, 44, 53 and 55) wee optmzed fom the ombnaton of hydophl otame. To epeent poblem ontng of a mxtue of dffeent tutual type, nludng tun, we alo nluded the optmzaton of the edue ontanng any atom wthn 10 Å of the dehan atom of Val21. Of thee 14, the oe edue (poton 3, 20 and 36) wee allowed to have any of the hydophob dentte, the ufae edue (poton 2, 19, 21, 22 and 24) had hydophl dentte, and the emanng bounday edue (poton 1, 18, 23, 25, 27 and 29) wee eleted fom a goup of hydophl and hydophob edue, exludng methonne (amno ad A, D,, F, H, I, K, L, N, Q, R, S, T, V, W and Y). Thee wee poble otame ombnaton.

8 1096 Stutue 1999, Vol 7 No 9 The mot dffult benhmak onted of all 18 nonglyne oe and bounday edue [2]. The oe edue (poton 3, 5, 7, 20, 26, 30, 34, 39, 52 and 54) wee eleted fom the et of hydophob amno ad and the bounday edue (poton 1, 12, 23, 33, 37, 45, 50 and 56) wee eleted fom the ompote lt of hydophl and hydophob edue. Thee wee poble otame ombnaton. negy expeon We employ an enegy expeon that ont of van de Waal, eletotat and olvaton tem. Fo van de Waal, a Lennad Jone 6 12 potental wa ued wth ad aled by a fato of 0.9 [29]. letotat wee omputed ung a dtane-dependent delet and a hybdzaton-dependent hydogen bondng tem [30]. Solvaton effet wee appoxmated fom hydophob ufae aea bual [31]. Atom ad and hydogen bond well depth wee baed on the DRIDING foe feld [32]. Calulaton Fo efeene, alulaton tme wee eoded ung a fully optmzed D algothm nopoatng HTR [3], and mag bullet and othe double optmzaton [14]. Calulaton wee alo pefomed ung an enhaned B&B mplementaton that employed the effent boundng tea and temnaton pepoeng. Fo the ft thee benhmak ae, all alulaton wee pefomed ung an ntal uppe bound of 0.0 kal/mol, a ou enegy expeon typally eult n optmal equene wth negatve enege. Fo the emanng two ae, ntal bound wee obtaned by ft unnng the appoxmate veon of the algothm, n whh the otame lt wee tunated to the 15 otame wth the lowet boundng enege at eah edue poton. Thee povded ntal bound of and kal/mol, epetvely. The genealty of the otng tea wa demontated by pefomng optmzaton wth value of f n quaton 5 angng fom 0 to 1. To llutate the elablty of the appoxmate fom of the algothm, optmzaton wee alo pefomed ung only the top 30 otame at eah poton a anked afte a ngle ound of temnaton. The lage benhmak poblem ontng of oe and bounday edue wa ued to demontate how D and B&T method an wok n onet. The poblem wa optmzed ung a D algothm, and upon evey eduton of omplexty by at leat an ode of magntude, the tate of the dmnhed poblem wa eoded. A B&T algothm wa ued to omplete the alulaton fo eah edued tate. The alulaton wee pefomed ung the optmal otng fato a detemned fom the pevou benhmak. Fo all alulaton, the total CPU tme wa eoded, a well a the poton of that tme pent pefomng temnaton pepoeng and the atual euve eah. The total numbe of node ompng the fnal puned tee wa alo eoded by tallyng the numbe of node emanng afte temnaton at evey level of euon. Calulaton wee pefomed on a ngle R10000 CPU of a Slon Gaph Ogn Pawe boundng expeon Th eton debe the ontuton of a tngent expeon fo a lowe bound fo a ytem ompoed only of one- and two-body nteaton n tem of both a patally pefed equene and the et of otame avalable at t unpefed poton. Fo a ytem ontng only of two-body nteaton, the total potental enegy an be expeed a the um of enege between all pa: In a poten, and j efe to amno ad poton and (,j) the enegy of nteaton between amno ad at thoe poton. A poten ytem alo ont of ngle-body nteaton. Beaue eah body an amno ad dehan at a patula poton on the poten bakbone, thee an enegy ontbuton both fom dehan nteaton wth othe dehan a well a nteaton wth the poten template affoldng. Both enege of nteaton depend on the dehan poton, amno ad dentty, and onfguaton. Thu the total potental enegy an be expeed: whee a poton-pef ndex debng a dehan otame of a patula amno ad type and onfguaton. Fo the pupoe of devng an expeon fo a lowe bound, t deable to alte the nde to allow edundany. To enue that the boundng expeon atfe the ondton n quaton 3, we ue the followng nequalte: and (7) (8) (9) (10) n whh the nde and efe to all of the poble otame avalable at eah poton, and the mnmum opeato elet the ngle otame that mnmze the ubexpeon The ndex g denote the otame found at the pefed poton n the global mnmum ombnaton. A mple expeon fo the lowe bound theefoe obtaned by ummng mnmal nteaton enege between poton by doveng mnmal otame pa. (0) bound mn[ (,template)] (,template) mn[ (,j )] (,j ) mn[ (,template)] 1 2 g total (,template) (,j) j j> total (,template) 1 2 (, j ) j mn mn[ (, j)] j (11) The devaton above epeent a gene tategy fo podung a boundng expeon fom any total enegy expeon. Fo example, moe ettve bound an be obtaned fom enegy expeon that um ove thee- o fou-body nteaton. Howeve, the omputatonal ot to mplement uh bound on a poten ytem vey hgh. Fotunately, thee ae vaaton of quaton 7 that ae equvalent n tem of omputatonal ot yet yeld bette bound. An altenatve way to expe the total enegy of the ytem to dtbute the template enege nto the pa alulaton. Gven an enegy quantty fo a pa of otame (,template) ( j,template) (,j ) pa(,j) 2 p 2 2 (12) n whh p the numbe of amno ad poton, the total enegy an be expeed thu: g g g n 1 n total (,j) (,j) 1 j 1 j j> (6) total pa (,j ) j (13)

9 Reeah Atle Banh-and-Temnate Godon and Mayo 1097 whh, n tun, an be ued to podue the followng boundng expeon (14) Beaue the mnma mut be evaluated wth epet to ngle-body and pa enege multaneouly, th boundng expeon neealy geate than o equal to the expeon n quaton 11. Theefoe, the new bound moe ettve. The omputatonal equement fo both expeon, howeve, ae of the ame ode. ah eque n 2 p 2 alulaton, whee n the aveage numbe of avalable otame pe poton and p the numbe of poton. One an deve a lowe bound that even moe ettve by pefomng an expanon of quaton 13 befoe applyng the mnmzaton opeato. When tetng a patula node dung taveal of the ombnatoal tee, the poton oepondng to node above (and nludng) the uent node have unquely pefed otame, wheea the emanng, deepe node ae not yet unquely pefed. The et of all amno ad poton an theefoe be deompoed nto two ubet, fxed (F) and vaable (V). quaton 13 an be ewtten Next, we expand the ummaton: (15) (16) Applaton of the mnmum opeato to th expeon would yeld a boundng expeon equvalent to quaton 14. To neae the tngeny, we utlze the nequalty (17) The mddle two tem of quaton 16 dffe only n the nde, and they ae theefoe equvalent to one anothe. Howeve, thee a dffeene one the mnmum opeato ae appled, a the otame of the fxed ubet (F) wll ett the eleton of the mnmum enegy otame pa fo the mnmzed thd tem, but not fo the eond. Theefoe, we evee the ode of the ummaton fo the eond tem and ombne t wth the thd tem to make ue of quaton 17 uh that the mnmum wll be a lage a poble: total pa(,j ) 2pa(,j) pa(,j ) (18) Now we apply the mnmum opeato to all um ove poton that ae not unquely pefed: (2) bound total 2 mn mn pa( j) j pa (,j ) pa pa pa (,j ) (,j ) (,j ) (1) bound mn mn[ pa(, j)] j total pa(,j) { F, V } j { F, V } mn pa( j) j mn pa pa (,j ) (,j ) mn pa (,j ) (19) To aheve futhe tngeny, we eaange quaton 18 befoe applyng the mnmum opeato: total pa(,j) 2 pa(,j) pa(,j) (20) fom whh we obtan (fnal) bound (, j ) pa mn 2 pa (,j ) (,j)] (21) The expeon an be genealzed to any ytem ontng only of two-body nteaton uh that the total enegy of the ytem an be expeed a n quaton 13. ffent mplementaton of the boundng expeon The omputatonal ot of evaluatng quaton 21 popotonal to p 2 n 2, whee p the numbe of poton and n the aveage numbe of otame at eah poton. When pefomng temnaton, the bound evaluated fo all pn otame, o that the total alulaton ode fo a ound of temnaton p 3 n 3. Temnaton ont of evaluatng the boundng expeon fo otame at all the unpefed poton. Theefoe, a poton tempoaly ondeed a membe of et F whle t otame ae beng evaluated. A the expenve eond tem of the fnal ummaton dependent only on V, t poble value may be peomputed fo all otame one pe poton and plaed nto a table fo lookup dung the evaluaton of quaton 21. The ot of pefomng p 2 n 2 alulaton fo aemblng the table fo the temnaton of all p poton ale a p 3 n 2. The boundng expeon now only eque ode pn alulaton fo eah of the pn tme t pefomed, fo an oveall ode of p 2 n 2. The oveall alulaton tme theefoe ale appoxmately a p 3 n 2, whh nealy n tme fate than the det mplementaton. Beaue n often a lage a , the peed neae an be dat. Aknowledgement We wh to thank AG Steet fo nvaluable feedbak thoughout the poe of developng the algothm. We alo wh to thank M Rengold, R Manoha, and N Pee fo helpful duon. Th wok wa uppoted by the Howad Hughe Medal Inttute (SLM), tanng gant GM 07616C-19 fom the Natonal Inttute of Health (DBG), the Rta Allen Foundaton, and the Davd and Lule Pakad Foundaton. Refeene 1. Dahyat, B.I. & Mayo, S.L. (1997). De novo degn: fully automated equene eleton. Sene 278, Malakauka S. & Mayo, S.L. (1998). Degn, tutue and tablty of a hypethemophl poten vaant. Nat. Stut. Bol. 5, De Maeye, M., Demet, J. & Late, I. (1997). All n one: a hghly detaled otame lbay mpove both auay and peed n the modelng of de-han by dead-end elmnaton. Fold. De. 2, Jann, J., Wodak, S., Levtt, M. & Maget, D. (1978). Confomaton of amno ad de-han n poten. J. Mol. Bol. 125, Ponde, J. & Rhad, F. (1987). Tetay template fo poten ue of pakng tea n the enumeaton of allowed equene fo dffeent tutual lae. J. Mol. Bol. 193, mn[ pa

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