Compact Representation of Continuous Energy Surfaces for More Efficient Protein Design

Transcription

1 pub.ac.og/jctc Compact Repeentaton of Contnuou Enegy Suface fo Moe Effcent Poten Degn Mak A. Hallen,, Pablo Ganza, and Buce R. Donald*,,, Depatment of Compute Scence and Depatment of Chemty, Duke Unvety, Duham, Noth Caolna 27708, Unted State Depatment of Bochemty, Duke Unvety Medcal Cente, Duham, Noth Caolna 27710, Unted State ABSTRACT: In macomolecula degn, confomatonal enege ae entve to mall change n atom coodnate; thu, modelng the mall, contnuou moton of atom aound low-enegy well confe a ubtantal advantage n tuctual accuacy. Howeve, modelng thee moton come at the cot of a vey lage numbe of enegy functon call, whch fom the bottleneck n the degn calculaton. In th wok, we emove th bottleneck by conoldatng all confomatonal enegy evaluaton nto the pecomputaton of a local polynomal expanon of the enegy about the deal confomaton fo each low-enegy, otamec tate of each edue pa. Th expanon called enegy a polynomal n ntenal coodnate (EPIC), whee the ntenal coodnate can be de-chan dhedal, backub angle, and/o any othe contnuou degee of feedom of a macomolecule, and any enegy functon can be ued wthout addng any aymptotc complexty to the degn. We demontate that EPIC effcently epeent the enegy uface fo both moleculamechanc and quantum-mechancal enegy functon, and apply t pecfcally to poten degn fo modelng both de chan and backbone degee of feedom. 1. INTRODUCTION Computatonal degn algothm ae an effectve appoach to engneeng poten and dcoveng new dug fo many bomedcally elevant challenge, uch a dug etance pedcton, 1 peptde-nhbto degn, 2 and enzyme degn. 3 Poten degn algothm each though lage equence and confomatonal pace fo equence that wll fold to a deed tuctue and pefom a pecfc functon. One of the key challenge n poten degn modelng and eachng the many contnuou confomatonal degee of feedom nheent n poten and othe molecule. Poten degn algothm mut etmate optmal value fo all thee degee of feedom to optmze the equence of the poten and to optmze the chemcal tuctue of the lgand f ued fo dug degn. Molecula dynamc mulaton can be ued fo th pupoe f the poten equence and lgand ae known, becaue they can move all of the molecule degee of feedom, 4 but thee mulaton ae computatonally expenve and mut be un epaately fo each equence o lgand chemcal tuctue. Hence, th dect mulaton appoach unutable fo eachng lage combnatoal degn pace. Fo example, many poten degn poblem eque eachng ove tllon of equence fa too many fo ndvdual molecula dynamc un. To adde the combnatoally lage equence pace nheent to poten degn, dedcated poten degn algothm effcently chooe an amno-acd type and confomaton fo each edue n a poten that togethe mnmze ome enegy functon. 5 Becaue the de chan confomaton of each amnoacd type ae geneally found n clute, known a otame, 6 the poten degn poblem ha often been teated a a dcete optmzaton poblem. In th cae, the output a et of otame agnment (a otame, ncludng amno-acd type, agned to each edue). The objectve functon an enegy functon, whch map confomaton to the enege. Howeve, becaue poten ae contnuouly flexble and have backbone a well a de chan flexblty, ome of the poten ntenal coodnate wll lkely have functonally gnfcant vaaton fom the otame deal value (at the cente of the clute). Clahng deal otame can often be conveted to favoable confomaton by elatvely mall adjutment n the de chan confomaton. 7,8 Small adjutment n the backbone confomaton away fom the wld-type backbone can alo be functonally gnfcant A a eult, modelng of contnuou flexblty ha been hown to damatcally mpove the accuacy of tuctual modelng n degn, 8,11 even ung a lmted et of degee of feedom, and ha led to degn that pefom well expementally. 1 3,5,7,12 Futhemoe, attempt to mmc th effect by dcete amplng at fne eoluton have been hown to ethe pooly appoxmate the contnuou oluton o be computatonally pohbtve. 8 Modelng addtonal contnuou degee of feedom, wth the goal of modelng all confomatonal vaaton that gnfcantly mpact poten functon, expected to futhe nceae the accuacy of the degn. Modelng of contnuou flexblty n poten degn can tll explot ou knowledge of otame, becaue otame povde an excellent po etmate of whee enegy well ae lkely to be n the confomatonal pace of the poten. Redue de Receved: Novembe 19, 2014 Publhed: Mach 20, Amecan Chemcal Socety 2292

2 Fgue 1. (A) The enegy of each edue epeented by EPIC a a polynomal n the ntenal coodnate, uch a de chan dhedal χ. Lowdegee, nexpenve polynomal (blue) ae ted ft, and the degee nceaed a needed to acheve a good ft (black) to the actual enegy functon (ed). Thee polynomal ae then ued fo the degn n place of the full enegy functon. (B) Inteacton between pa of edue ae epeented n tem of both edue ntenal coodnate. chan wll uually be found n the egon of confomatonal pace faly cloe (e.g., wthn fo de chan dhedal) to an deal otame, even wth a elatvely mall otame lbay. 13 A a eult, f by ung a mnmzaton-awae each poce one can fnd the neaet deal otamec confomaton to the tue global mnmum-enegy confomaton (GMEC) of a poten, the GMEC telf can geneally be found by local mnmzaton ntalzed to that deal otamec confomaton. Thu, poten degn can fully account fo contnuou de chan flexblty whle tll functonng a a mnmzaton-awae each 7 ove a dcete otame pace. Th ame paadgm can be extended to contnuou backbone flexblty f deal confomaton that nclude backbone moton, known a edue confomaton (RC), 11 ae ncluded n the each. Such mnmzaton-awae eache can take multple fom. Fo example, the MnDEE algothm 8 poduce a gap-fee, povably accuate lt of otame agnment n ode of lowe bound on mnmzed enegy. MnDEE pefom enegy mnmzaton on each of thee otame agnment n that gap-fee ode untl the lowe bound exceed the bet mnmzed enegy E b enumeated to that pont. At th pont, any ubequently enumeated agnment would be guaanteed to have hghe mnmzed enege than E b ;thu,e b povably the global mnmum enegy. MnDEE enumeate otame agnment effcently ung the A* each algothm. 14,15 A Monte Calo each ove otame pace can alo ncopoate mnmzaton 16,17 but wthout any povable guaantee. A Monte Calo each wll be mnmzaton-awae f contnuou mnmzaton pefomed fo equence and confomaton dung the each, and the mnmzed enegy ued n the calculaton of acceptance pobablte fo new otame agnment, a n ef 16 and the fnal phae of ef 17. Nevethele, a method wthout povable guaantee wll lkely eque moe confomaton and equence to be mnmzed to obtan the ame gan n accuacy a MnDEE, becaue unlke n MnDEE, the confomaton beng mnmzed ae not guaanteed to be the mot pomng one. Futhemoe, thee no fnte numbe N fo a gven poten degn poblem uch 2293 that enumeatng N confomaton by Monte Calo guaanteed to yeld the GMEC. Any mnmzaton-awae method, howeve, wll eque a lage numbe of uboutne call to local mnmzaton. Contnuou enegy mnmzaton computatonally expenve, even wth molecula mechanc-type enegy functon that potze peed ove accuacy. Th caue the mnmzaton of the enegy functon to be the bottleneck n poten degn wth contnuou flexblty. Th bottleneck become moe evee when moe ophtcated enegy functon ae ntoduced. Computatonal poten degn typcally pefomed wth enegy functon that potze peed ove accuacy. Fo example, they typcally ue mplfed mplct olvaton model, uch a EEF1. 18 Vzcaa et al. have nvetgated the ue of the Poon-Boltzmann model, a much moe accuate mplct olvaton model, n poten degn. 19 They found t to be amenable to epeentaton a a um of edue-pa nteacton, whch the fom equed fo mot poten degn algothm, but ode of magntude moe expenve than EEF1. Othe method of mpovng enegy functon accuacy ae lkely to face the ame poblem. Fo example, quanttatvely accuate decpton of mot molecula nteacton eque computaton of the electonc tuctue ung quantum chemty, but the method that can do th ae vey computatonally ntenve. 20 Method to educe the numbe of call to an enegy functon needed n poten degn could allow moe accuate enegy functon to be ued and thu yeld moe accuate eult. In poten degn wth only dcete flexblty, pecomputaton method ae typcally ued to educe the numbe of enegy functon call needed, that, the numbe of confomaton fo whch the enegy mut be evaluated. Befoe the degn tated, the nteacton enegy of each pa of deal, gd otame at dffeent edue poton pecomputed and toed n an enegy matx. Then, an deal otame choen fo each edue baed on the enege n th matx, and no futhe call to the enegy functon ae needed dung the actual degn calculaton. The numbe of enegy functon call equed

3 thu quadatc n the numbe of edue n the ytem (that, t cale a the numbe of pa of edue). A pecomputed enegy matx of lmted ue, howeve, f we want to model contnuou flexblty. No beneft n degn ganed by pefomng pot hoc mnmzaton on the bet confomaton found ung deal otame. 8 Th tue even f a hgh degee of flexblty ued fo mnmzaton, e.g., f molecula dynamc technque ae ued, becaue the degned equence aleady detemned befoe mnmzaton pefomed. In contat, a mnmzaton-awae each pefom local contnuou mnmzaton fo all otame agnment that mght be optmal to fnd the tue GMEC. 7 Thu, an analogou enegy matx pecomputaton method fo contnuou flexblty would be vey ueful. It would enue a polynomal numbe of enegy functon call fo mnmzatonawae poten degn, n contat to the exponental numbe of call that may ae n mnmzaton of all pobly optmal otame agnment (becaue the numbe of uch agnment may be exponental wth epect to the numbe of edue modeled). The each fo otame agnment telf unlkely to admt a polynomal-tme algothm, becaue t NP-had even to appoxmate. 21,22 But a method to pecompute pawe enege fo contnuouly flexble degn would change the oveall tme cot fom (a lage otame each cot) (the enegy functon cot) to (a lage otame each cot) + (the enegy functon cot). The otame each cot wll necealy be exponental n the wot cae f one want to obtan the GMEC o an appoxmaton to the GMEC wthn a fxed eo thehold. But the enegy functon cot wll meely be quadatc n the numbe of edue, ndcatng that the pawe enegy pecomputaton hft the bottleneck away fom the enegy functon call. Th bng the ame mpovement to mnmzaton-awae degn that enegy matx pecomputaton bought to non-contnuouly flexble degn. We now peent a pawe enegy pecomputaton method that admt contnuou flexblty: enegy a polynomal n ntenal coodnate (EPIC). EPIC compute a epeentaton of the pawe enegy fo each otame pa, not jut at the otame deal value of the ntenal coodnate but alo fo value wthn pecfed ange aound the deal value (Fgue 1). Th computaton pefomed befoe the otame each computaton begun. Th allow the otame each to ubttute the new, quckly evaluable epeentaton fo the ognal enegy functon. EPIC mplemented n the OSPREY 7,23,24 openouce poten degn package, whch ha yelded many degn that pefomed well expementally n vto, 1 3,25 28 n vvo, 1,2,25,26 and n nonhuman pmate. 25 EPIC povde a gnfcant peedup when ued wth OSPREY default AMBER 29,30 - and EEF1 18 -baed enegy functon, but t alo hown to be utable fo epeentng quantum-mechancal enege. Th pape make the followng contbuton: 1. A compact, cloed-fom epeentaton of enegy a a functon of contnuou ntenal coodnate of a poten ytem. 2. A modfed leat-quae method to compute th epeentaton. 3. A modfed mplementaton of the MnDEE 8 and DEEPe 11 poten degn algothm ntegated nto the OSPREY 1,3,7,23,24 open-ouce poten degn package that make ue of th epeentaton to acheve ubtantal peedup. Ou mplementaton avalable fo fee download at donaldlab/opey.php Computatonal expement howng that compact and accuate EPIC epeentaton ae poble both fo the tandad enegy functon n OSPREY and fo enege obtaned by quantum chemty at the SCF and MP2 level of theoy. 5. Computatonal expement howng that EPIC geatly acceleate mnmzaton-awae poten degn calculaton, thu allowng degn to nclude not only moe flexble edue but alo moe confomatonal flexblty at thoe edue. 2. METHODS 2.1. Pelmnae. EPIC, lke mot pevou poten degn algothm, degned fo pawe enegy functon. Pawe enegy functon ae um of nta+hell and pawe tem. Inta+hell tem ae functon of the amno-acd type and confomaton of one edue, and pawe tem ae functon of the amno-acd type and confomaton of two edue. Each pawe tem epeent the nteacton between a pa of flexble edue, and each nta+hell tem epeent the ntenal enegy of a edue plu t nteacton wth nonflexble hell edue (thoe that ae fozen n a ngle, fxed confomaton thoughout the ente calculaton). EPIC could be ealy modfed to nclude ome hghe-ode tem fo defned combnaton of moe than two edue: thee tem can alo be epeented a polynomal n the edue degee of feedom. To fnd the GMEC, we mut fnd an amno-acd type and confomaton fo each flexble edue uch that the um of the nta+hell tem fo all flexble edue, plu the um of pawe tem fo all pa of flexble edue, mnmzed. Th poblem efeed to a confomatonal each. Confomatonal each can alo compe a equence each by eachng fo the bet confomaton aco many equence confomatonal pace. Many algothm ae avalable fo th poblem, ncludng MnDEE 8 and DEEPe, 11 whch olve the poblem wth povable accuacy. EPIC can be ued n any confomatonal each algothm that model contnuou flexblty becaue t povde polynomal that can be dectly ubttuted fo nta+hell and pawe tem of the enegy functon. The eence of EPIC to explot the fact that fo each pawe o nta+hell enegy, the enegy n the vcnty of the mnmum can be decbed well by a elatvely low-degee polynomal (uually quadatc total degee, ometme hghe; ee Fgue 5B). Th decpton computed ung a modfed leat-quae method. We wll efe to the tate to whch a edue may be agned a edue confomaton (RC, cf. DEEPe 11 ). In the abence of backbone flexblty, each RC wll coepond to a de chan otame. Wthn a edue confomaton, the edue contnuou enegy vaaton can be decbed by a et of ntenal coodnate, whch ae ubject to box contant (.e., bound on each ntenal coodnate). Heen, the wod polynomal wll be ued n two vey dffeent ene n the decpton of EPIC below. Ft, EPIC a polynomal epeentaton of the enegy, namely, a polynomal functon wth epect to the ntenal coodnate that explctly contucted by the EPIC algothm. Second, a meaue m of the computatonal complexty of an algothm can be decbed a polynomal 5,31 f, fo nput ze n, m gow no fate than n d fo a fxed exponent d. In th cae, one can contuct a polynomal wth epect to the ze of the nput that wll be an uppe bound on the computatonal cot, no matte how lage the nput (n). Fo th pupoe, we conde ethe the tme o the numbe of enegy functon call a m thee

4 Fgue 2. The numbe of enegy well n a poten ytem cale exponentally wth the numbe of flexble edue, leadng to an exponental numbe of enegy functon call. EPIC can eplace mot of thee call wth quck evaluaton of low-degee polynomal. (Top) A poten may have an enegy well fo evey combnaton of otame (anbow) at dffeent edue. The global mnmum-enegy confomaton (GMEC) of a poten may be n any of thee well. We model the enegy a a um of pawe enegy tem. Each pawe tem wll have well fo pa of otame, but thee ae fa fewe well of th knd a numbe quadatc n the numbe of edue. We can ealy affod the enegy functon call needed to chaacteze each pawe well. (Bottom) By pecomputng a polynomal epeentaton (blue) of the enegy wthn each well of each pawe tem (ed), we enable the computaton of any pawe tem n any pawe well and thu of the full poten enegy n any enegy well of the poten olely by a quck evaluaton of polynomal. two meaue of computatonal complexty ae elated to each othe by a contant facto n cuent poten degn algothm wth contnuou flexblty becaue enegy functon call geneally domnate the cot of thee algothm. Fo example, pecomputaton of an EPIC epeentaton mut be pefomed only once fo evey pa of RC at dffeent edue, and thu the total numbe of polynomal ft (and thu the total tme fo pecomputaton of EPIC ft) doe not gow fate than the quae of the numbe of edue beng modeled. On the othe hand, poten degn telf ha been hown to be NP-had, 21,22 whch mean no polynomal-tme algothm lkely to ext fo t. In othe wod, fo evey polynomal p(n ) n the numbe n of edue, thee ae poten degn poblem of n edue that ae not expected to be olvable n tme p(n ). A poblem only olvable by exponental-tme algothm (.e., thoe that take tme calng a b n,wheeb a contant and n the ze of the nput) would typcally be condeed NP-had. Enegy functon call ae typcally the bottleneck n poten degn algothm that model contnuou flexblty. EPIC, howeve, enue that the numbe of enegy functon call n a poten degn calculaton lnea n the numbe of RC pa and thu polynomal n the ze of the nput (Fgue 2) Bac Leat-Squae Method. Conde two nteactng edue and j. Let u tat wth the well-behaved cae whee thee ext a low-degee polynomal epeentaton of a pawe enegy thoughout the allowed ange of both edue ntenal coodnate (of the fom n eq 1). We employ the notaton ntoduced n the DEEPe algothm. 11 Suppoe we have RC and j wth pawe enegy E(,j,x), whee x the vecto of ntenal coodnate (fo example, dhedal) affectng edue and j when they 2295 have amno-acd type coepondng to and j. Let E (, j ) = E(, j, x (, j )) = mn E(, j, x) 0 x, whee the mnmum taken wth epect to the ntenal coodnate ove the allowed ange fo the cuent RC, and x 0 (,j ) the et of ntenal coodnate that mnmze the pawe enegy. Th defnton of E content wth MnDEE 8 and DEEPe. 11 We eek a multvaate polynomal p,j (x) uch that E (, j ) + p ( x x (, j )), j 0 (1) a good appoxmaton to E(,j,x). Th multvaate polynomal appoxmately a fnte, low-degee Taylo expanon about the mnmum. Howeve, we ue leat-quae ft becaue we have found that they pefom much bette than Taylo expanon that ae baed on numecal devatve. The ft ae pefomed ung a tanng et wth ten tme a many ample a thee ae paamete (polynomal coeffcent) n the ft; the amplng pocedue decbed n ecton 2.7. The ft ae co-valdated wth an ndependent et of ample (ecton 2.6). The contant p(0) = 0 appled, o the eal enegy and the polynomal wll agee exactly at the mnmum-enegy pont. Th contant ealy mplemented by not ncludng a contant tem n the polynomal and eflect the need fo the hghet accuacy to be attaned fo the lowet-enegy and thu the mot bophycally eaonable confomaton. A a eult of th contant, all value of the polynomal on t doman wll be non-negatve. Fttng begn wth a multvaate quadatc ft and then move to hghe degee a needed (ee ecton 2.6). Becaue polynomal ae lnea wth epect to the coeffcent, the fttng a lnea leat-quae poblem.

5 Th method can be genealzed wthout modfcaton to nta +hell enege a well a to any contnuou degee of feedom, uch a newly modeled backbone petubaton 11 o gd-body moton of lgand. In evey cae, the numbe of vaable fo the polynomal wll be the numbe of contnuou degee of feedom that defne the confomaton of the edue o edue pa of nteet. Fo example, n a pawe enegy computaton fo a otame of lyne and a otame of valne wth only de chan flexblty, the polynomal wll have fve vaable (the fou dhedal of lyne and one dhedal of valne). The polynomal coeffcent ae eal numbe. Let be an RC agnment, epeented a a tuple of RC wth one RC fo each edue. Let be the otame n at edue. To appoxmate the mnmzed enegy of an enumeated confomaton, ntead of mnmzng the full enegy E ( x) = E(, x) + E(, j, x) j< wth epect to the ytem contnuou degee of feedom x, we mply mnmze the polynomal appoxmaton q ( x) = E ( ) + p ( x x ( )) + E (, j ) 0 j< (2) + p ( x x (, j )), j 0 (3) wth epect to x. Thee leat-quae appoxmaton acheve hgh accuacy fo the low-enegy well of otame and local backbone moton (.e., the poton of confomatonal pace whee both the contnuou degee of feedom and the enegy ae elatvely cloe to the local mnmum of the pawe enegy). Hghe enege may alo be found cloe n confomatonal pace to the local mnmum, but thee enege ndcate taned confomaton unlkely to be een n natue. Thu, fo a wellbehaved enegy tem whoe enegy untaned thoughout the bound on contnuou degee of feedom that defne ou cuent RC, EPIC mply pefom a leat-quae ft of the enegy to epeent t a a multvaate polynomal wth epect to the contnuou degee of feedom. Many RC do howeve contan both egon wth feable enege and egon wth hghe enege that epeent bophycally nacceble confomaton, uch a tec clahe. Thee RC peent dffculte fo the bac leat-quae ft, but the followng algothmc modfcaton avod th poblem Modfed Leat-Squae Method. To handle RC wth hgh-enegy egon, we note that we do not necealy need the polynomal to be a good appoxmaton fo the enegy thoughout the ente egon allowed by the box contant. We meely eque that eq 2 be a good appoxmaton fo eq 3 when ued wth bophycally feable, mnmzed value of x. In patcula, we can expect that the optmal, mnmzed tuctue ha no clahe o othe patculaly lage local tan. We have the advantage that even though nteacton enege n poten can e teeply towad nfnty n the cae of tec clahe, thee ae no phycal phenomena that wll caue nteacton enege to deceae teeply towad negatve nfnty. Thu, enege ae elatvely well-behaved n low-enegy egon. We can thu effectvely patton the confomatonal pace nto elatvely mooth, low-enegy egon that we appoxmate accuately and hgh-enegy egon that can be uled out. Let u denote the enegy elatve to the mnmum a E (, j, x x (, j )) = E(, j, x) E (, j ) 0 n the pawe cae o E (, x x0( )) = E(, x) E ( ) n the nta hell cae. Ou equement fo a good appoxmaton of the enegy can be defned goouly n tem of two uppe bound b 1 and b 2 that we place on E. Fo each nta+hell o pawe enegy tem, we etmate an uppe bound b 1 on E that we expect to hold fo all mnmzed confomaton that we want to output (the GMEC, o the lowet-enegy c confomaton f we ae computng a c-confomaton enemble). The algothm wll be able to detemne f b 1 vald o not, o we can ty agan wth a hghe b 1 f neceay. Addtonally, we need a econd, pobly looe, uppe bound b 2 on E that we ae confdent wll be vald fo all mnmzed confomaton that we compute dung ou each, whethe they tun out to be the GMEC o not. The value of b 2 mut be the ame fo all nta+hell and pawe tem (b 1 can be tem-pecfc, though n pactce a ngle value fo b 1 convenent). If EPIC beng ued wth the MnDEE algothm fo the confomatonal each, 8 we can povably obtan the GMEC wthout condeng any confomaton whoe enege E exceed the punng nteval, 8 an uppe bound computed by MnDEE fo the dffeence between the lowet confomatonal enegy lowe bound (baed on pawe mnmum enege) and the GMEC. Thu, when unnng MnDEE, we can et b 2 equal to the MnDEE punng nteval. When b 2 et equal to the punng nteval, we know t a vald uppe bound on E fo all mnmzed confomaton computed dung the each, and thu ou GMEC calculaton povable. We can alo do th when unnng DEEPe, 11 whch eentally a backboneflexble veon of MnDEE. Fo othe algothm, we may want to et b 2 baed on knowledge of the ytem beng degned. Settng b 2 =2b 1 lkely to be an acceptable heutc. Ou polynomal only need to be a good ft toe fo value of the ntenal coodnate whee E b 1. Fo E > b 1, we wll eque that the polynomal le above b 1. Th wll enue that when we enumeate confomaton n ode of mnmzed enegy computed ung polynomal, a long a the thehold b 1 ae choen coectly, we wll obtan nonclahng confomaton befoe confomaton wth clahe, and thee nonclahng confomaton enege wll be accuately epeented by the polynomal ft. Futhemoe, we wll eque that fo b 1 < E < b 2 the polynomal hould be a lowe bound on E (Fgue 3). Th wll enue that egadle of what thehold b 1 wee ued, we neve oveetmate a confomatonal enegy that below the thehold b 2 and thu neve exclude t fom the enumeated lt of confomaton. The equement to be a lowe bound eay to atfy becaue clahng van de Waal nteacton ae vey teep and thu tend to e much moe quckly than the polynomal ft. Thu, when we pefom polynomal ft ung thehold, we know we wll be gettng a gap-fee lt of confomaton n ode of enegy. If the thehold b 1 wee choen to be too low, ome hghe-enegy confomaton wth undeetmated enegy mght be ncluded a well, but thee wll be lmted to mnmzed confomaton contanng enegy tem wth E > b 1. Th condton can ealy be checked. If deed, the un can be edone wth nceaed b 1 thehold to elmnate th eo. Thu, the choce of b 1 affect the ultmate peed of the algothm but not t coectne. Fo ou expement n th wok (ecton 3), we have et b 1 to 10 kcal/mol. Th thehold wa found to be uffcent fo all expement decbed n th wok a well a mot othe EPIC degn that we have ted. Phycally, any pa of edue whoe nteacton enegy 10 kcal/mol woe than the optmal nteacton fo t cuent RC pa lkely n a hghly taned confomaton, uch a a tec clah. Thu, a degn equng b 1

6 Fgue 3. (A) Fo each enegy value E, thee a ange of deal value fo the EPIC ft (geen). Fo enege below cutoff b 1, whch may be found n favoable confomaton, th ange jut the enegy (the ange ha zeo wdth). Fo hghe enege, the ange defned ung cutoff b 1 and b 2.Fofttng pupoe, EPIC ft value ae penalzed by the amount they le outde the deal ange (the puple pont epeent a ample confomaton fo a gven EPIC ft ncung the penalty ndcated n ed). (B) Example of cuve atfyng thee condton. The EPIC ft cloely matche the enegy up to cutoff b 1, afte whch t devate fom the enegy but tay n the taget egon hown n A by tayng below the enegy. Once the enegy ove b 2, the EPIC ft can be ethe above o below the tue enegy wthout leavng the taget egon. geate than 10 kcal/mol lkely to be bologcally nfeable. Fo example, the poten lkely to unfold o undego a lage and unexpected tuctual change athe than uffe th local tan. Let u ue z to denote a vecto n the doman of ou polynomal ft p, whch condeed a good epeentaton of the enegy f, fo ome mall ε > 0, the followng condton ae atfed: 1. Fo z uch that E (,j,z) b 1, p ( z) E (, j, z) < ε j, 2. Fo z uch that b < E (, j, z) < b, b ε < p ( z) 1 2 1, j < E (, j, z) + ε 3. Fo z uch that b2 E (, j, z), b ε < p ( z) 1, j Thee condton ae llutated n Fgue 3. They can be acheved ung a modfed leat-quae ft, ung pecal oneded penalte to enfoce the nequalte n condton 2 and 3 along wth uual (two-ded) leat-quae penalte to enfoce condton 1. The objectve functon the um of tem fom each ample n the tanng et. Fo a ample z uch that E (,j,z) 2 b 1, the objectve functon tem ( p ( z) E (, j, z)), j (a typcal fo leat-quae). A tem of th fom alo ued f the lowe-boundng condton volated (.e., f E < b 2 but p > E ). 2 Othewe, the objectve functon tem fo z ( p ( z) b ) fo p ( z) < b and 0 fo p, j 1 ( z) b., j 1, j 1 If the modfed leat-quae method appled to a et of ample that motly have E > b 1, then ovefttng to the few pont wth E < b 1 may occu egadle of the numbe of ample. A an exteme cae, f all ample have E > b 2, then almot any polynomal wth vey lage value thoughout t doman wll gve a 0 value fo the objectve functon, but th may tll povde a vey poo decpton of the enegy landcape. Fo th tuaton to be avoded, when a tet et of n ample beng dawn and n/2 ample wth E > b 1 have aleady been dawn, then f moe ample come up wth E > b 1, they ae edawn to enue that a uffcent numbe of ample 2297 wth E b 1 ae avalable (ecton 2.7). Mnmzaton-awae dead-end elmnaton punng 8 (both ngle and pa punng) pefomed befoe computaton of the polynomal ft becaue the puned otame and pa wll not be needed dung enumeaton. Th punng uually elmnate the clahng otame and pa, leavng otame and pa that ae well uted fo mple polynomal epeentaton. Th objectve functon can be optmzed effcently becaue t convex wth epect to the polynomal coeffcent (ee ecton 2.4). Howeve, we found geneal-pupoe convex mnmze to be athe tme-conumng fo the hghe-ode ft. To adde th, we developed the algothm decbed n ecton 2.4. It explot pecfc popete of the objectve functon to obtan a moe effcent and elable ft than a genealpupoe convex mnmze would be lkely to obtan A Fat Algothm fo Modfed Leat-Squae Fttng. The followng algothm pefom a modfed leatquae ft, povdng a ueful polynomal fo enegy tem that nclude both low-enegy egon, whee an accuate polynomal epeentaton of the enegy uface equed, and hghenegy egon that mut be excluded fom ou each. Let u epeent ou polynomal ft p(z) a p yz ( ), whee p the polynomal vecto of coeffcent, y(z) the coepondng vecto of monomal bult fom the degee-of-feedom value z, and the tandad nne poduct. Fo example, f z cont of the two dhedal z 1 and z 2, and we ae pefomng a quadatc ft, then y(z) wll have the element 1, z 1, z 2, z 2 1, z 2 2, and z 1 z 2. Fo each ample n ou tanng et of ample (ee ecton 2.7), let z be the vecto of degee-of-feedom value and y = y(z ) be the coepondng vecto of monomal. Let E be the enegy fo the ample, whee the mnmum-enegy pont defned to have 0 enegy. Then, a modfed leatquae ft cont of mnmzng the objectve functon f to obtan the bet-ft polynomal coeffcent p b 2 p = ag mn ( E py ) b p E b1 + ( E py ) + ( b py ) py E, b1< E < b2 2 E > b1, py < b (4) whee { E b1 } denote the et of ample whoe enege ae le than o equal to b 1.Ifwedefne P 1 to be the et of ample pont uch that ethe E b1 (5) o p y E, b1 < E < b2 (6) and we defne P 2 to be the et of ample pont uch that E > b1, py < b1 (7) then ou objectve functon f become 2 2 ( E py ) + ( b py ) 1 P1 P2 (8) Thu, f we know P 1 and P 2, mnmzng the objectve functon a bac leat-quae poblem and can be olved analytcally. Lke bac leat-quae, th algothm opeate on a ngle tanng et of ample and povably mnmze the objectve functon (.e., the eo) fo that tanng et.

7 We can how the objectve functon convex wth epect to p by notng that the contbuton fom each ample a functon of the ngle lnea combnaton u = p y of the element of p. Th contbuton depend on E, but t alway convex (and pecewe quadatc). If E b 1, the contbuton jut the paabola (E u) 2.Ifb 1 < E < b 2, t the tuncated o flat-bottomed paabola gven by (b 1 u) 2 fo u b 1, 0 fo b 1 u E, and (E u) 2 fo u E. Othewe (f b 2 E ), the contbuton the one-ded paabola gven by (b 1 u) 2 fo u < b 1 and 0 othewe. Hence, the objectve functon a um of convex functon, makng t convex telf. Thu, mnmzng the objectve functon to fnd p tactable, wth any local mnmum beng the global mnmum. A a eult, we know that f, fo any et of ample P 1 and P 2,wehavecoeffcent p that mnmze eq 8 and atfy the condton of eq 5 7, then the coeffcent p ae globally optmal. The algothm fnd P 1 and P 2 teatvely. A an ntal gue, P 1 can be ntalzed to uch that E b 1, and P 2 to be empty. Th coepond to aumng that the one-ded etant can all be atfed pefectly. Th followed by pefomng the bac leat-quae computaton of mnmzng eq 8, whch etun coeffcent p, and ecalculatng P 1 and P 2 fom p ung the condton fom eq 5 7. Th pocedue then epeated ung the new P 1 and P 2 untl a elf-content oluton found. Geneally, only a mall mnoty of the ample wll be moved n and out of the leat-quae poblem at each teaton; thu, the leat-quae matx can be updated quckly at each tep, whch ueful becaue fomng th matx the bottleneck. Typcally, only a few teaton ae needed. Th algothm actually a pecal cae of Newton method becaue t etmate fo the objectve-functon mnmum at each teaton the mnmum of the local quadatc Taylo expanon of the objectve functon. Th mnmum can be found analytcally becaue the local expanon convex. In ou mplementaton of th algothm, by fa the bulk of t tme cot pent n fomng the matx fo the ft bac leatquae ft (wth the ntal P 1 and P 2 ). The ubequent ft ae much fate becaue they ae only pae update. Thu, the modfed leat-quae fttng neglgbly moe expenve than the ft bac leat-quae fttng Spae Atom-Pa Enege (SAPE). SAPE a method to educe the degee of polynomal needed by EPIC by ncludng ome nonpolynomal tem n the epeentaton of the enegy. The need fo hghe-ode polynomal ft dven by lage value of hghe devatve. Thee value ae contbuted pmaly by a mall numbe of van de Waal (vdw) tem between pa of atom that ae vey nea each othe. It poble to obtan ubtantal tme and memoy avng by evaluatng thee tem explctly and fttng the et of the enegy functon to a polynomal. To elect atom pa whoe vdw tem ae to be evaluated explctly, we choe a cutoff dtance of 3 o 4 Å (ee ecton 2.6). Then, an atom pa vdw tem ae evaluated explctly f and only f the atom can be found wthn that dtance of each othe wthn the bound on ntenal coodnate fo the gven edue confomaton. Thee tem ae not polynomal n the degee of feedom becaue they ae nvee powe of dtance between atom, and the atom coodnate themelve ae n geneal not polynomal functon of the degee of feedom. Fo example, the expeon fo atom coodnate n a de chan n tem of the de chan dhedal angle wll nclude ne and cone of thoe angle. Once we decde to evaluate vdw tem fo a gven pa of atom, t cot neglgble exta tme and memoy to alo 2298 calculate the electotatc nteacton between thee atom (becaue we aleady have the dtance between the atom) Attanng the Requed Accuacy. We wll now decbe the method ued to chooe polynomal degee fo EPIC ft and enue that ft of uffcent accuacy ae obtaned. Ft accuacy checked and contolled ung co-valdaton. Fo co-valdaton pupoe, a mean-quae eo computed wth abolute eo ued below E = 1 kcal/mol and elatve eo above. Th can be een a a weghtng of the eo tem: the weght 1 fo E 1 and 1/mn ( E, b 1 ) fo E 1 (th level off at b 1 to avod exceve undeweghtng of the oneded contant). Thee weght, whch ae contnuou wth epect to E, ae alo ued dung the leat-quae fttng. Co-valdaton ued to elect the degee of the polynomal that ft. Low-degee polynomal ave tme and memoy both dung the A*/enumeaton tep and dung the pecomputaton tep but may not povde a uffcently good epeentaton. Hence, we poceed though a equence of nceangly expenve ft (Fgue 1A), and each tme a ft completed, t co-valdated wth an ndependently dawn et of ample. Lke the tanng et, th co-valdaton ample et ha ten tme a many ample a ft paamete. If the mean-quae eo below a pecfed thehold, the ft toed, and f t above, we poceed to the next method. The default thehold value et to Howeve, lmted nvetgaton ugget that lage thehold tll tend to keep the eo n the confomaton mnmzed enege mall compaed to themal enegy, and thu ae lkely acceptable a well. It alo ueful to avod dong ft wth a vey lage numbe of paamete, a thee have enomou tme and memoy cot n both the enumeaton and pecomputaton tep. Thu, OSPREY cuently et to efue to do ft wth ove 2000 paamete th way, computaton that would have pohbtve tme cot may tll be atfactoly completed wth a lghtly hghe eo thehold than uual. Some of the ft ue lowe-degee tem fo all degee of feedom and hghe-ode tem fo elected degee of feedom. Thee elected degee of feedom ae egenvecto v k of the Hean fom a modfed leat-quae quadatc ft (tep 1 n the lt of tep below). Lettng λ k be the egenvalue coepondng to v k,wedefne Dq = vk λk max λ q fo q > 0. Let u defne f n to be a polynomal ft of total degee d (e.g., f 2 a quadatc ft), f d (D q )tobeaftto a polynomal of total degee d n all degee of feedom plu tem of total degee d+1 and d+2 n the degee of feedom n D q, and (n,c) to be a polynomal ft of total degee d plu SAPE wth a cutoff of c Å. Ft wee ted n the followng ode: f 2, (2,3), f 2 (D 10 ), f 2 (D 100 ), f 4, (4,4), f 4 (D 10 ), f 4 (D 100 ), f 6, and (6,4). The Stone Weeta theoem 32 guaantee that a uffcently hgh-degee polynomal can appoxmate any contnuou functon on any cloed and bounded poton of Catean pace to any deed accuacy. In othe wod, t guaantee that any enegy functon can be epeented by EPIC to abtay accuacy f we allow uffcently hgh-degee polynomal. The ba of Benten polynomal can be ued to contuct uch appoxmaton wth guaanteed convegence to any functon. 33 Howeve, fo the pupoe of enegy epeentaton fo poten degn, modfed leat-quae lkely to povde good appoxmaton (9)

8 ung much lowe-degee polynomal than we would obtan ung the Benten ba, becaue we do not need cloe appoxmaton of the hgh enegy n clahng egon. In thee egon, we only need a eaonable lowe bound that much hghe than the otamec well. Th tategy keep the polynomal degee low enough to be pactcal Samplng To Tan and Valdate Leat-Squae Ft. Tanng and valdaton et fo EPIC ft cont of ample confomaton of the edue() nvolved, pecfed a vecto of ntenal coodnate, dawn fom thoughout the allowed egon of confomatonal pace. By default, ample fo both the tanng and valdaton et wee ampled unfomly (.e., each degee of feedom wa ampled unfomly and ndependently fom the nteval coepondng to the cuent otame o RC). Ten ample wee alway ued n each of thee tanng and valdaton et fo each paamete n a ft. Howeve, f mot of the ample coeponded to enege above the thehold b 1 (ee ecton 2.3), then ovefttng could eult becaue, fo uch ample, thee ae nfntely many polynomal value that yeld zeo eo. To avod th, we need uffcent ample fom the et B of confomaton wth enege below b 1 ; B the et of confomaton whee the polynomal need to be quanttatvely accuate. We enue uffcent ample fom B by ejectng ample outde of B wheneve we dee n ample n total and we aleady have n/2 ample outde B, and thu dawng the et of ou ample unfomly fom B by ejecton amplng. If ample ae ejected conecutvely, ndcatng that B too mall fo effcent ejecton amplng, then the Metopol algothm 34 ued to ample fom B. We have confdence n the paamete obtaned by fttng to the tanng ample fo thee eaon. Ft, a ueful meaue of the accuacy of a polynomal appoxmaton to the enegy uface that thee a low pobablty that any egon of the enegy uface devate gnfcantly fom the polynomal appoxmaton (except fo hgh-enegy egon appoxmated by mlaly hgh value of the polynomal). Becaue ou covaldaton of each polynomal ft ue a lage numbe of ndependent ample (ten tme the numbe of paamete), we ae left wth a vey low chance that ou co-valdaton ample wll m any uch egon. Thu, an nuffcently accuate polynomal uface wll be detected upon co-valdaton and emeded by an nceae n polynomal degee. Second, eo n the mnmzed enege obtaned ung polynomal appoxmaton ae contently low, a hown n ou computatonal expement (Table 1). Thd, we expect the enegy functon to be elatvely mooth n the vcnty of a mnmum becaue the gadent mut be zeo at the mnmum, and thu, we expect a polynomal of elatvely low ode (e.g., the Taylo ee of the enegy) to yeld a good appoxmaton n the vcnty of a mnmum Applcaton n Poten Degn Algothm. Once the polynomal ae computed, they can be ued n poten degn algothm wheeve an enegy functon would odnaly be called. The GMEC wll mply be the et of otame fo whch the mnmzed value of eq 3 wth epect to x ha the lowet poble value. The mplet method to povably fnd the GMEC ung EPIC to ue a poten degn algothm that enumeate confomaton n ode of a lowe bound, and then ntead of mnmzng the full enegy (eq 2), meely mnmzng the polynomal-baed enegy (eq 3) to compute the enegy fo each enumeated confomaton (Fgue 2). Fo example, MnDEE/A* 8 can be ued fo th enumeaton poce, and we ue th algothm n ou computatonal expement (ecton 2.10). EPIC can alo be appled n fee enegy calculaton ung the K* algothm, 7,12 whch appoxmate bndng contant a ato of patton functon computed fom low-enegy confomaton enumeated by A*. Dung thee calculaton, one can mply ue the polynomal ntead of the enegy functon to compute the patton functon, gven the enumeated RC agnment. Th method gve a contant-tme peedup detemned by the ato of tme to evaluate the enegy functon veu the EPIC enegy. An addtonal peedup poble fo banch-and-bound poten degn algothm (e.g., A* 14,15 ) that ue a tee tuctue fo the confomatonal each. Thee algothm buld node that each epeent a ubet of confomatonal pace and ae coed ung a lowe bound on the confomatonal enege n that pace. In each node confomatonal pace, ome edue ae etcted to a ngle RC; thee RC ae efeed to a agned to the epectve edue. At each level of the tee, an RC agned to one moe edue. One can ue the EPIC polynomal to mpove the lowe-bound enegy fo each of thee node. At each node, we need to compute a lowe bound L fo the confomatonal enegy q, whch defned n eq 3. That, we compute L uch that L q ( x) = E ( ) + p ( x x ( )) + E (, j ) 0 j< + p ( x x (, j )), j 0 (10) fo all RC agnment and all degee-of-feedom value x that ae pat of the node confomatonal pace. If known (.e., f RC ae fully agned at all edue poton), then a tght lowe bound can be computed tvally by local mnmzaton wth epect to x. Othewe, we let q ( x) = E ( ) + Ep(, x), whee E p cont only of EPIC polynomal E () = E ( ) + E (, j ) j< Ep(, x) = p ( x x0 ( )) + p ( x x (, j )), 0 j j< (11) (12) Now, f we compute lowe bound L and L p uch that L E ()and L p E p fo all, x n ou confomatonal pace, then L = L + L p wll atfy eq 10, gvng u a vald lowe bound. Computaton of L ha been decbed pevouly, becaue lowe bound of th fom ae computed n MnDEE 8 and DEEPe. 11 To compute L p, we ue the fact that EPIC polynomal ae alway non-negatve; thu, fo any and x and any ubet S of the edue we ae modelng, S p ( x x ( )) + p ( x x (, j )) E (, x) 0, j 0 j S, j< p (13) If we let S be the et of edue wth fully agned RC, then thee only one poble RC fo each edue S, and o we can fnd the mnmum of eq 13 mn p ( x x ( )) + p ( x x (, j )) 0 j x,, 0 S j S, j< (14) 2299

9 Table 1. GMEC Calculaton Tet Cae a poten name PDB code mutable edue count mn. peedup wth SAPE mn. peedup wthout SAPE A* peedup wth SAPE A* peedup wthout SAPE ft tme (mn) tot. tme (mn) GMEC enegy eo (kcal/mol) copon toxn 1aho copon toxn 1aho copon toxn 1aho ft DNF >142.9 b n/a copon toxn 1aho ft DNF >4.30 b n/a cytochome c553 1c Atx1 metallochapeone 1cc bad ft bad ft bucandn 1f nonpecfc lpd-tanfe 1fk poten tancpton facto IIF ft DNF ft DNF feedoxn 1qz Tp epeo 1jhg bad ft bad ft fuctoe-6-phophate 1l6w aldolae cephalopon C 1l7a ft DNF >3170 b n/a deacetylae PA-I lectn 1l7l phophoene 1l7m phophatae α-d-glucuondae 1l8n ganulyn 1l9l γ-glutamyl hydolae 1l9x fetn 1lb cytochome c 1m1q hypothetcal poten YcI 1mwq ponn 2o bad ft >45.81 b n/a cytovn 2qk dhydofolate eductae 2h putatve monooxygenae 2l dpy-30-lke poten 3g HIV gp120 3u7y bad ft >14.33 b n/a a EPIC wth SAPE olved each of thee cae. In mot cae, A* wthout EPIC wa lowe o dd not fnh wthn the tme lmt b. Mnmzaton (mn.) peedup (ato of ngle-confomaton mnmzaton tme wthout EPIC elatve to tme wth EPIC) epoted fo EPIC wth and wthout SAPE. Smlaly, A* peedup denote the ato of total A* tme wthout EPIC elatve to tme wth EPIC. Ft tme (tme to calculate the EPIC polynomal) and total calculaton tme (tot. tme) ae epoted fo EPIC calculaton wth SAPE, calculated wthout paallelzaton of fttng. GMEC enegy eo the abolute dffeence between GMEC enege calculated wth and wthout EPIC. Ft DNF mean the tme lmt wa exceeded dung the polynomal ft pecomputaton, and bad ft mean that ft fo EPIC wthout SAPE, even at the maxmum allowed polynomal degee, dd not accuately epeent the enegy; th ue ecued by SAPE n thee cae. When nethe A* wthout EPIC no EPIC wthout SAPE wee ucceful, A* peedup wthout SAPE cannot be calculated and lted a n/a. EPIC wa pefomed wth mnmzaton of patal confomaton n all cae. A* wthout EPIC dd not fnh wthn the tme lmt, o a lowe bound on the peedup epoted: the ato of the tme lmt (17 day) to the A* tme wth EPIC and SAPE. exactly by local mnmzaton wth epect to x. Equaton 14 a lowe bound on E p (,x), and thu we et L p equal to t, gvng u a coe fo ou node. We note that L p tctly non-negatve becaue eq 13 and thu eq 14 ae alway non-negatve. Becaue th contnuou mnmzaton wth epect to x moe expenve and ha to be pefomed epaately at each node, t evaluated n a lazy 31 fahon n ou A* mplementaton. Node ae agned the tadtonal, dcete lowe bound L when they ae geneated; th bound fat to compute. The A* poty queue contan node both wth and wthout the polynomal contbuton L p ncluded. When a new node popped fom the queue, we check whethe o not L p peent. If t, we expand the node; f t not, we compute L p and net the node back nto the poty queue. Th enue that node come off the poty queue n ode of the complete lowe bound L + L p, a neceay fo A* to functon coectly. Howeve, t alo enue that we do not wate tme computng L p fo node whoe L hgh enough to peclude expanon. Th method gve a combnatoal peedup, becaue a hgh polynomal contbuton fo a patal confomaton can effectvely pune an ente banch of the A* tee. In pactce, though, the lage contant peedup fom EPIC mnmzaton of fully agned confomaton tend to be moe gnfcant (ecton 3.1, Table 2) Complexty of Enegy Evaluaton. The peedup due to EPIC can be explaned n tem of the aymptotc cot of EPIC polynomal evaluaton compaed to dect enegy functon call. The cot of evaluatng an EPIC polynomal cale a the numbe of tem n the polynomal. Th cot telf a polynomal (uually quadatc) n the numbe of ntenal coodnate of the edue pa (o ngle edue, n the nta +hell cae) of nteet. In contat, the cot of evaluatng a molecula mechanc-baed enegy functon geneally quadatc n the numbe of atom nvolved becaue dtance between all pa of atom need to be condeed. EPIC acheve a maked peedup becaue mot edue have fa moe atom than gnfcantly flexble ntenal coodnate. Fo example, mot poten edue have two o fewe de chan 2300

10 Table 2. Patton Functon Calculaton Reult a poten name PDB ID mutable edue count A* peedup due to EPIC b A* peedup due to patal c htdne tad poten 2c7 14 >48.48 d 1.52 ponn 2o9 14 >1131 d 0.96 tancptonal egulato 2p5k 11 >256.0 d 1.34 AhC cytovn 2qk hemolyn 22z 12 >217.7 d 1.24 dhydofolate eductae 2h2 14 >50.45 d 2.16 putatve monooxygenae 2l α-cytalln 2wj5 15 >29.06 d 0.80 cytochome c555 2zxy 14 >8.87 d 3.32 hgh-potental on 3a38 13 >1916 d 1.40 ulfu poten ClpS poteae adapto 3dnj 12 >50.25 d 0.93 putatve monooxygenae 3fgv 10 >9639 d 1.57 poten G 3fl 14 >1475 d 1.31 val capd 3g21 15 >7.84 d 1.21 dpy-30-lke poten 3g Hfq poten 3hfo 10 >194.8 d 1.45 cold hock poten 32z 14 >872.8 d 1.24 typn 3pwc 10 >624.2 d 1.04 typn 3pwc 11 >131.8 d 2.07 a EPIC wa pefomed wth SAPE n all cae. A n Table 1, EPIC olved all cae, and cae fo whch A* wthout EPIC dd not fnh ae denoted. d b Rato of total A* tme wthout EPIC elatve to total A* tme wth EPIC and mnmzaton of patal confomaton. Rato of total A* tme wth EPIC but no mnmzaton of patal confomaton elatve to total A* tme wth EPIC and mnmzaton of patal confomaton. A* wthout EPIC dd not fnh wthn the tme lmt, o a lowe bound on the peedup epoted: the ato of the tme lmt (17 day) to the A* tme wth EPIC. dhedal but ove ten atom. The emanng ntenal coodnate (e.g., bond length, angle, etc.) ae elatvely nflexble. Polynomal evaluaton ae alo pefomed entely by addton and multplcaton, whch ae much fate than the moe complcated elementay opeaton (e.g., tgonometc functon, quae oot, etc.) needed to evaluate molecula mechanc enegy tem. When quantum-mechancal enegy functon ae ntoduced, all of the electon mut be accounted fo explctly, and even faly appoxmate quantum-chemcal method have tme cot that ae hghe-ode polynomal wth epect to the numbe of electon. Fo example, any method that account fo epulon between all atomc obtal (e.g., Hatee-Fock and all pot-hatee-fock method) mut calculate epulon ntegal fo all quaduple of atomc obtal, and thee ae at leat a many atomc obtal a electon. Futhemoe, thee ae fa moe electon than thee ae atom and fa moe atom than ntenal coodnate, gvng EPIC an exteme pefomance advantage. Yet EPIC can epeent the ame enegy uface to a hgh degee of accuacy once the EPIC polynomal have been pecomputed. Whethe o not EPIC ued, thee type of pawe enegy evaluaton mut be pefomed fo evey pa of edue n a degn ytem. In geneal, th mean the numbe of pawe enegy evaluaton needed quadatc wth epect to the numbe of edue n the ytem. Th numbe can be educed f a cutoff appled to emove nteacton between dtant edue. Howeve, th peedup apple equally fo EPIC and non-epic calculaton Computatonal Expement. Poten degn calculaton wee pefomed n OSPREY 1,3,7,23,24 wth and wthout EPIC to nvetgate (a) what pevouly ntactable ytem become newly tactable wth EPIC, (b) what peedup EPIC bng to confomatonal enumeaton fo pevouly tactable ytem, and (c) what type of polynomal epeentaton ae needed fo thee pupoe. EPIC un wee pefomed wth SAPE and wth confomatonal mnmzaton fo patally agned confomaton dung A* each. Fo compaon, un wth each of thee featue omtted wee alo pefomed. Tme wee compaed fo the A* each, ncludng confomaton enumeaton and mnmzaton, becaue th the poton of the degn that not guaanteed to be completed n polynomal tme and thu the bottleneck. A pat of the EPIC un, GMEC enege wee alo computed ung the egula enegy functon fo compaon to the EPIC eult, and the ato of mnmzaton tme wth and wthout EPIC wa computed. Fo un wth multple confomaton vey cloe n enegy to each othe (wthn the eo ange of EPIC, typcally <0.1 kcal/mol), the tme ato wee aveaged. All mnmzaton wee pefomed ung a cyclc coodnate decent mnmze, whch now ncluded n OSPREY. Default OSPREY enegy functon ettng wee ued whee applcable: AMBER wth EEF1 olvaton and a dtance-dependent delectc contant of 6. Rotame wee detemned ung the Penultmate otame lbay. 13 Tet ytem wee choen to evaluate both patton functon and GMEC calculaton and to nclude all thee type of contnuou degee of feedom ued n OSPREY: de chan dhedal, backbone petubaton (hea and backub) paamete, 11 and gd-body otaton and tanlaton of tand. Some of the tet ae ntended to be wthn the cope of pevou method, allowng a quanttatve compaon of unnng tme, wheea othe ae ntended to how that EPIC can compute pevouly ntactable GMEC and patton functon wth povable accuacy. Fo GMEC calculaton (Table 1), the ft et of ytem ued wa taken fom Ganza, Robet, and Donald, 8 and featued only de chan dhedal flexblty. The tuctue fo thee coepond to PDB ID 2o9, 2qk, 2h2, 2l, and 3g36. The econd et of ytem wa taken fom Hallen, Keedy, and Donald, 11 and ncluded both de chan and backbone flexblty. The tuctue PDB code wee 1aho, 1c75, 1cc8, 1f94, 1fk5, 127, 1qz, 1jhg, 1l6w, 1l7a, 1l7l, 1l7m, 1l8n, 1l9l, 1l9x, 1lb3, 1m1q, and 1mwq. Thee vaant of the 1aho ytem wth moe edue wee ted a well. Fnally, a GMEC calculaton wa pefomed fo the complex of the HIV uface poten gp120 wth the boadly neutalzng antbody NIH45-46 (PDB code 3u7y 35 ). To nvetgate the applcaton of EPIC to patton functon calculaton (Table 2), we ft choe ytem wth only de chan dhedal flexblty fom Ganza, Robet, and Donald 8 and calculated a patton functon fo the unlganded poten, wth wld-type amno acd at all edue poton, to wthn 97% guaanteed accuacy. Patton functon calculaton uch a thee ae the key opeaton n K* 7,12 calculaton. The tuctue fo thee coepond to PDB ID 2c7, 2o9, 2p5k, 2qk, 22z, 2h2, 2l, 2wj5, 2zxy, 3a38, 3dnj, 3fgv, 3fl, 3g21, 3g36, 3hfo, and 32z. Futhemoe, a K* un peented fo typn wth a mall-molecule nhbto (PDB ID: 3pwc); the un tactable wth EPIC but fal to fnh wthout t. Unlke the calculaton fo the othe, monomec tuctue, the K* 2301

11 un fo typn nvolve the calculaton of thee patton functon: one each fo the poten, lgand, and complex. Each degn wa allowed 17 day of total untme, afte whch thoe that had not fnhed wee deemed to have exceeded the tme lmt and wee temnated. A* tme wth EPIC anged fom 0.7 to 4 day, and the peedup due to EPIC ae hown n Table 1 and 2. In thee expement, fttng wa pefomed wthout paallelzaton. Howeve, the computaton of the EPIC polynomal fo each pa of RC an ndependent opeaton, o each can be done n paallel, meanng that paallelzaton to p poceo wll gve a p-fold peedup a long a p doe not appoach the numbe of RC pa. OSPREY cuently uppot computaton of each edue pa n paallel, o the peedup hold a long a p doe not appoach the numbe of edue pa. In pactce, howeve, punng and A* take longe than the polynomal fttng fo lage ytem, whch mean that th paallelzaton may not be neceay. Addtonally, once the EPIC ft have been computed fo a ytem, thee may be a lage numbe of computaton that can be pefomed ung t, ncludng calculaton of patton functon fo many equence, computaton of GMEC fo vaou ubet of the equence pace, etc. Thee extenve eue of the ft may be epecally deable when degnng a lbay of equence fo expemental tetng f one pefom vaou optmzaton wth dffeent aumpton and tet top equence fom each optmzaton, the eult wll be moe obut to eo n the aumpton. To nvetgate the ablty of EPIC to epeent quantummechancal enegy functon, we pefomed EPIC calculaton on the apatame dpeptde (extacted fom PDB ID: 1a8j 36 ) wth the enege fo EPIC ample evaluated ung NWChem 37 ntead of OSPREY uual enegy functon. Calculaton wee pefomed at the SCF level of theoy wth STO-3G and wth 6-31G** ba et and alo at the MP2 level of theoy wth a STO-3G ba et. 20 Fo each otame of each edue, dhedal wee ampled wthn the allowed ange fo the otame, and the total enegy of the dpeptde wa ft to a polynomal Applcaton of EPIC to Othe Algothm. Fo th tudy, EPIC wa mplemented n the context of the poten degn package OSPREY 1,3,7,23,24 to un along wth the algothm (MnDEE, DEEPe, and K*) and pawe enegy functon aleady mplemented n OSPREY. Howeve, EPIC would enable ome othe capablte n dffeent mplementaton. Ft, EPIC can be appled n the context of othe poten degn algothm. Fo example, t can be appled to an teatve algothm, uch a FASTER 38 o Monte Calo, 34 that te to fnd a utably low-enegy confomaton by acceptng o ejectng otame change baed on the enege of confomaton wth thee change. Wheneve the enegy needed fo a otame agnment, the EPIC enegy fo the poten can be locally mnmzed tatng at the deal ntenal coodnate value fo that otame agnment. In th cae, the matx of EPIC polynomal ubttute dectly fo the matx of pawe otame enege commonly ued to calculate confomatonal enege fo thee algothm n the abence of contnuou flexblty. EPIC could even be ued fo molecula dynamc becaue mot edue pa n a molecula dynamc tajectoy 4 wll pend mot of the tme n faly elaxed confomaton, whch the egon of confomatonal pace modeled by EPIC. In all thee cae, EPIC enegy evaluaton would be makedly fate than egula enegy functon call, patculaly fo expenve enegy functon. Thu, EPIC would povde Fgue 4. Mutable edue n the edegn of the uface of the HIV uface poten gp120 n complex wth the boadly neutalzng antbody NIH45-46 (PDB ID: 3u7y 35 ). Th degn fnhed only when EPIC wa ued. Mutable edue, blue backbone and pnk de chan; gp120, black backbone; NIH45-46 heavy chan, geen backbone; NIH45-46 lght chan, bown backbone. a ubtantal peedup fo any algothm whoe bottleneck enegy functon call. Fo dockng algothm 39 that eque de chan optmzaton o local backbone optmzaton, EPIC can be ued fo both confomaton cong and optmzaton ung any of the above algothm EPIC Can Accommodate Hghe-than-Pawe Enege. EPIC wa mplemented n th wok to handle pawe enegy functon (n the ene of a um of 1-body and 2-body enege wth no tem dependent on thee o moe edue degee of feedom) becaue thee ae cuently typcal fo poten degn and nclude the AMBER, CHARMM, and EEF1 enegy functon we ue n OSPREY. Howeve, the tue enegy of poten not exactly pawe decompoable, and EPIC could ealy accommodate hghe-ode tem. EPIC mply eque that each enegy tem coepond to a et D of degee of feedom, contaned to a egon n whch they ae elatvely well-behaved (e.g., dhedal at each edue contaned to a ngle otame); we can ample D ubject to the contant and then ft the enege a a polynomal functon wth doman D. Thu, fo any et R of moe than two nteactng edue, fo each RC agnment to thoe edue, we can ft an EPIC polynomal and thu decbe the enegy tem fo R. In pactce, the numbe of et of edue that can nteact gnfcantly qute lmted, becaue edue typcally mut be phycally nea each othe to have gnfcant hghe-thanpawe nteacton. Fo example, f we have a Ramachandanbaed potental, t tem each depend on the ψ and ϕ backbone dhedal of a cetan edue and thu depend on the confomaton of the thee edue 1,, and + 1. Lkewe, the confomaton of a edue can nduce polazaton effect n a neaby edue j that wll affect the nteacton of j wth anothe edue k, and th effect can be quantfed ung quantum chemty, but and j have to be phycally vey cloe to each othe ( 1 nm) fo th effect to be gnfcant (and j and k have to be faly cloe too, pobably ubnanomete a well, becaue 2302

12 Fgue 5. (A) A* tme wth and wthout EPIC. Fve degn that dd not fnh wthout EPIC ae hown on the ght n ed. (B) Popoton of each type of ft (ee ecton 2.6) equed n EPIC calculaton. The quatc* categoy nclude both full quatc ft and quadatc ft wth quatc tem added fo D 10 o fo D 100. Ft wee all made a hgh-degee a needed to obtan a edual below , a decbed n ecton 2.6. Some ft have ubtantally lowe edual, epecally quadatc ft wthout SAPE, becaue no lowe ft degee wee allowed. (C) Speedup due to dffeent EPIC method compaed to A* baed on pawe lowe-bound enege; tandad EPIC nclude both SAPE and mnmzaton of patal confomaton. PF denote patton functon calculaton; the othe ae GMEC calculaton. the potental of an nduced dpole fall off fate than 1/d 2 wth dtance d). Hence, EPIC can be ued to model any ealtc enegy functon by accountng fo all et of edue wth gnfcant enegetc nteacton. 3. RESULTS Computatonal expement wee pefomed to meaue what knd of polynomal ft ae neceay to accuately model dffeent poten wth dffeent degee of feedom and enegy functon, and what peedup EPIC bng to DEE/A* and K* calculaton. The eult demontate that EPIC povde a ubtantal peedup to degn calculaton when poten ae modeled n the ame manne a pevou OSPREY degn. 1,3,8,11,23 They alo how that EPIC effcently epeent enege calculated by quantum chemty and a potentally decve tool fo ung both ealtc, contnuou flexblty and quantum-mechancal enegy functon n poten degn Applcaton to Poten Degn. Ft, computatonal expement wee pefomed to compae GMEC each wth and wthout EPIC, a decbed n ecton Key poton of the degn calculaton wee tmed wth and wthout EPIC to detemne the peedup fo thee poton n patcula (Table 1). On aveage, mnmzaton of fully enumeated confomaton wa 79-fold fate ung EPIC than wth tadtonal enegy functon call. Oveall, A* peedup due to EPIC aveaged 167-fold (Fgue 5A). The oveall A* peedup lkely geate than the mnmzaton peedup becaue of the way OSPREY tandad enegy functon mplemented. Each tme the enegy functon un on a new equence, etup tme (e.g., ntalzaton of the enegy functon) equed to dentfy electotatc, van de Waal, and olvaton tem that wll be neceay fo that equence. Th etup tme elmnated by EPIC and not counted a pat of the mnmzaton tme hee, but t may be pefomed an exponental numbe of tme wthout EPIC becaue mnmzaton may be equed fo an 2303

13 exponental numbe of equence. Run that dd not fnh wthout EPIC ae not ncluded n thee aveage. Eghty-fve pecent of the ft n thee expement wee quadatc wth no SAPE needed (Fgue 5B). GMEC fom EPIC un howed good ageement between enege fom mnmzaton of EPIC enege and enege fom the actual enegy functon. The aveage enegy dffeence wa 0.04 kcal/mol, whch le than one-tenth of themal enegy at oom tempeatue (0.592 kcal/mol, calculated a the unveal ga contant tme a oom tempeatue of 298 K) and thu functonally ngnfcant. Fve of the 27 ytem fnhed only wth EPIC, demontatng that EPIC allow fo the degn of lage and moe dvee ytem than wee pevouly poble. Fo example, a edegn of the complex of HIV uface poten gp120 wth the antbody NIH45-46 dd not fnh when un wthout EPIC but fnhed wth EPIC ung about one day of A* tme (Fgue 4). Th edegn allowed the mutaton of 16 edue all ove the gp120 uface n the nteface fve n the D-loop of gp120, 40 whch cental to the nteacton wth NIH45-46, and the othe 11 catteed though othe pat of the nteface n vaou type of econday tuctue. Redegn of the gp120 uface to acheve pecfc bndng to patcula antbode ha been ntumental n the development of pobe to olate thee antbode fom ea. 28 Redegnng the antbody uface of a gp120 antbody complex ha alo been effectve n optmzng antbody affnty, 25 whch ueful fo pave mmunzaton and mmunogen degn. Inteetngly, the edegn of the NIH45-46 complex yelded 12 top confomaton wthn 0.06 kcal/mol of each othe two fom the top equence and ten fom a double mutant. Th hgh denty of favoable confomaton ugget that the complex entopcally favoed, a eult content wth the obeved hgh affnty of NIH45-46 fo gp120, attaned though extenve affnty matuaton of the antbody. Two vaaton of EPIC wee alo ted fo thee ytem. It wa found that mnmzaton of patal confomaton dung A* (ecton 2.8) povde a peedup of 2.3-fold on aveage, though t not nealy a geat a the peedup fom fate mnmzaton of fully agned confomaton (Fgue 5). Futhemoe, EPIC wthout SAPE wa often effectve; howeve, unde ome ccumtance, t wa unable to povde accuate ft (a uual, tyng only the polynomal degee decbed n ecton 2.6). In ytem whee EPIC wthout SAPE wa effectve, t aveaged ngnfcantly (1.1-fold) lowe than EPIC wth SAPE fo A*. Howeve, thee wee fou ytem that equed SAPE to gve accuate eult (out of 27; Table 1), ncludng the HIV gp120 complex wth antbody NIH45-46 (Fgue 4). Thee wee alo fou ytem that exceeded the tme lmt dung fttng. Thee lmtaton on EPIC wthout SAPE do not ndcate a fundamental theoetcal bae, becaue n pncple the Stone-Weeta theoem guaantee an accuate ft f the polynomal degee nceaed uffcently (ee ecton 2.6). Howeve, they do ndcate that EPIC wthout SAPE may ometme eque polynomal degee that ae pohbtvely tme-conumng fo typcal poten degn and/o a hghe numecal pecon than the double pecon effcently uppoted n Java and thu ued n OSPREY. Expement wee alo pefomed to compae patton functon calculaton wth and wthout EPIC (Table 2). To obtan a povably good appoxmaton to the patton functon, 7,12 many moe confomaton mut be enumeated and 2304 mnmzed than fo GMEC calculaton. A a eult, maked peedup wee acheved by EPIC (Fgue 5C). Out of the 19 ytem fo whch EPIC fnhed, only thee fnhed wthout EPIC (2000-fold aveage peedup). The peedup n EPIC degn fom mnmzaton of patal confomaton wa modet at 1.4-fold. Wth an A* peedup of 2 3 ode of magntude, degn that would pevouly take yea can be pefomed wth EPIC n day. Th wll allow many degn that would othewe be condeed ntactable to be completed ung EPIC Quantum-Mechancal Enege. In addton to clacal mechanc-baed enegy functon, we alo ued EPIC to ft confomatonal enege of the apatame dpeptde calculated ung quantum-mechancal model of electonc tuctue. EPIC ft fo apatame howed that quantummechancal enege and AMBER and EEF1 enege can be epeented by polynomal of vey mla degee (.e., enegy uface fom quantum chemty ae jut a polynomal-lke a enegy uface fom molecula mechanc). SAPE wa not found to gnfcantly nceae the accuacy of the ft, and thu wee not ncluded, though t lkely that epaametezed van de Waal and/o electotatc tem (o othe pecally ft functon of the atom-pa dtance) would be able to mpove the ft qualty. Th dcepancy ndcate that the atom-pa enege ued n SAPE ae a poo appoxmaton to the nteacton between the ame atom pa pedcted by quantum mechanc and, thu, that enege etuned by quantummechancal and molecula-mechanc method ae ubtantvely dffeent. Fo Phe 2 of apatame, the ame type of polynomal ft wee needed fo Hatee-Fock wth a STO-3G ba et, Hatee- Fock wth a 6-31G** ba et, and the uual AMBER/EEF1 enegy functon (Fgue 6). Thee wee quadatc ft fo thee otame and a quadatc ft plu quatc ft on D 10 fo the fouth. MP2 wth an STO-3G ba et alo equed quadatc ft fo the ft thee otame and equed a quadatc ft plu quatc ft on D 100 fo the fouth. Fgue 6. Inta-edue enegy calculated fo Phe 2 of apatame ung Hatee-Fock theoy wth an STO-3G ba et, and quadatc EPIC ft, a a functon of the two de chan dhedal. The ft vey cloe to the enegy uface, though a lght dcepancy vble n the uppe ghthand cone (χ 1 70, χ ).