1. INTRODUCTION. Received: October 21, Article. pubs.acs.org/jctc

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1 * UNKNOWN * MPSJCA JCA /W Ucode c d (R3.6.4 HF0: alpha 39) 203/0/2 02:46:00 PROD-JCAVA rq_ /9/204 05:48:33 4 JCA-DEFAUL pubs.acs.org/jcc Varaoal Approach o Molecular Kecs 2 Felks Nu ske, Bea G. Keller,* Gullermo Peŕez-Herańdez, Aoa S. J. S. Mey, ad Frak Noe * 3 Deparme for Mahemacs ad Compuer Scece, Fere Uversa of Berl, 495 Berl, Germay 4 ABSRAC: he egevalues ad egevecors of he molecular dyamcs 5 propagaor (or rasfer operaor) coa he esseal formao abou he 6 molecular hermodyamcs ad kecs. hs cludes he saoary dsrbuo, 7 he measable saes, ad sae-o-sae raso raes. Here, we prese a 8 varaoal approach for compug hese doma egevalues ad egevecors. 9 hs approach s aalogous he varaoal approach used for compug 0 saoary saes quaum mechacs. A correspodg mehod of lear varao s formulaed. I s show ha he marces eeded for he lear 2 varao mehod are correlao marces ha ca be esmaed from smple MD 3 smulaos for a gve bass se. he mehod proposed here s hus o frs defe 4 a bass se able o capure he releva coformaoal rasos, he compue he 5 respecve correlao marces, ad he o compue her doma egevalues ad 6 egevecors, hus obag he key gredes of he slow kecs.. INRODUCION 7 Bomolecules, parcular proes, ofe ac as small bu 8 hghly complex maches. Examples rage from alloserc 9 chages,2 o moor proes, such as kes, whch lerally 20 walks alog mcroubules,,3 ad he rbosome, a eormous 2 complex of RNA molecules ad proes resposble for he 22 syhess of proes he cell.,4 o udersad how hese 23 bomolecular maches work, does o suffce o kow her 24 srucure, ha s, her hree-dmesoal shape. Oe eeds o 25 udersad how he srucure gves rse o he parcular 26 coformaoal dyamcs by whch he fuco of he molecule 27 s acheved. Proe foldg s he secod feld of research 28 whch coformaoal dyamcs plays a major role. Proes are 29 log polymers of amo acds ha fold o parcular hree- 30 dmesoal srucure. he asoshgly effce search for hs 3 ave coformao he vas coformaoal space of he 32 proe ca be udersood erms of s coformaoal 33 dyamcs. Besdes me-resolved expermes, molecular dyamcs 34 smulaos are he ma echque o vesgae coformaoal 35 dyamcs. o dae, hese smulaos yeld formao o he 36 srucure ad dyamcs of bomolecules a a spaal ad emporal 37 resoluo, whch cao be paralleled by ay expermeal 38 echque. However, he exraco of kec models from 39 smulao daa s far from rval, sce kec formao cao 40 be ferred from srucural smlary. 5,6 Smlar srucures mgh be 4 separaed by large kec barrers, ad srucures ha are far apar 42 some dsace measure mgh be kecally close. 43 Aauralapproachowardmodelghekecsofmolecules 44 volves he parog of coformao space o dscree 45 saes. 77 Subsequely, raso raes or probables bewee 46 saes ca be calculaed, eher based o rae heores, 7,8,9 or 47 based o rasos observed MD rajecores. 6,3,5,6,20 22 he 48 resulg models are ofe called raso eworks, Maser 49 equao models, or Markov (sae) models (MSM), where 50 Markovay meas ha he kecs are modeled by a memoryless jump process bewee saes. I Markov sae models, 5 s assumed ha he molecular dyamcs smulaos used 52 represe a ergodc, reversble, ad measable Markov process Ergodcy meas ha every possble sae would be vsed a 54 fely log rajecory ad every al probably dsrbuo of 55 he sysem coverges o a Bolzma dsrbuo. Reversbly 56 reflecs he assumpo ha he sysem s hermal equlbrum. 57 Measably meas ha here are pars of he sae space whch 58 he sysem remas over me scales much loger ha he fases 59 flucuaos of he molecule. I order o cosruc a MSM, 60 he coformaoal space of he molecule s dscrezed o 6 ooverlappg mcrosaes, ad he observed rasos bewee 62 pars of mcrosaes are coued. Oe obas a square marx wh 63 raso probables, he so-called raso marx, from whch 64 awderageofkecadhermodyamcproperescabe 65 calculaed. he equlbrum probably dsrbuo ( he chose 66 sae space) s obaed as he frs egevecor of he raso 67 marx. Drecly from he marx elemes, oe ca fer kec 68 eworks ad raso pahs. 26,27 he doma egevecors of 69 he raso marx are used o defy measable saes Each doma egevecor ca be erpreed as a kec process, 7 ad he assocaed egevalue s relaed o he me scale o whch 72 hs process occurs. 25 All hs formao ca be combed o 73 recosruc he herarchcal srucure of he eergy ladscape. 3,33 74 Fally, raso marces represe a very useful framework o 75 coec daa from me-resolved expermes wh smulao 76 daa. 34,35 Over he pas decade, exesve kowledge o whch 77 facors deerme he qualy of a MSM has bee accumulaed. 78 For example, MSMs ha are cosruced usg he eral 79 degrees of freedom of he molecule ed o yeld beer resuls 80 ha hose ha were cosruced usg global descrpors of 8 he srucure (H-bod paers, umber of ave coacs) Receved: Ocober 2, 203 Amerca Chemcal Socey A

2 83 Also, degrees of freedom ha are o cluded he model should 84 decorrelae o shor me scales from hose ha are cluded Naurally, he samplg of he rasos lms he accuracy of a 86 MSM, ad ools o accou for hs error have bee 87 developed O he whole, he research feld has maured o 88 apoawhchwell-esedproocolsforhecosrucoof 89 MSMs from MD daa have bee esablshed, 25,40,4 ad sofware 90 o cosruc ad valdae Markov sae models from MD daa s 9 freely avalable. 42,43 MSMs have bee appled o aalyze he 92 coformaoal dyamcs of pepdes 5,3,44 ad of small proe 93 domas, such as Vll head pece, 45 p WW, 46 FP35 WW, Recely, has become possble o aalyze he foldg 95 equlbra of full fas-foldg proes MSMs have also 96 bee used o vesgae coformaoal chages, such as he 97 self-assocao sep he maurao of HIV-proease, lgad bdg, 5 or he olgomerzao of pepde fragmes 99 o amylod srucures A mpora aspec ha has lmed he roue use of 0 MSMs s he dffculy o oba a sae space dscrezao ha 02 wll gve rse o a MSM ha precsely capures he slow 03 kecs. he hgh-dmesoal molecular space s usually frs 04 dscrezed usg cluserg mehods some merc space. he 05 form ad locao of hese clusers, somemes called MSM 06 mcrosaes, are crucal for deermg he qualy of he 07 esmaed raso raes Varous mercs ad cluserg 08 mehods have bee aemped for dffere molecular sysems. 09 Small pepdes ca be well descrbed by a drec dscrezao 0 of her backboe dhedrals. 3 I was suggesed ref 56 o use a dhedral prcpal compoe aalyss o reduce he dhedral 2 space o a low-dmesoal subspace ad subsequely cluser 3 hs space usg, for example, k-meas. A raher geeral merc 4 s he parwse mmal RMSD-merc cojuco wh some 5 cluserg mehod, such as k-ceers or k-medods. 25,30,4 6 Recely, he me-lagged depede compoe aalyss 7 (ICA) mehod was pu forward, a dmeso reduco 8 approach whch a slow low-dmesoal subspace s 9 defed, whch has bee show o provde mproved MSMs 20 over prevously employed mercs. 57,58 2 I rece years, has bee esablshed ha he precso of 22 a MSM depeds o how well he dscrezao approxmaes 23 he shape of he egefuco of he uderlyg dyamcal 24 operaor (propagaor or rasfer operaor) of he dyamcs Whe he dyamcs are measable, hese egefucos wll be 26 almos cosa o he measable saes, ad chage rapdly a 27 he raso saes. 59 hus, mehods ha have sough o cosruc 28 a maxmally measable dscrezao 30,60 have bee relavely 29 successful for measable dyamcs. However, he MSM ca be 30 mproved by usg a omeasable dscrezao, especally whe 3 fely dscrezes he raso saes, so as o race he varao 32 of he egefuco hese regos. 25,55 A alerave way of 33 achevg a good resoluo a he raso sae whou usg a 34 fe dscrezao s o use appropraely placed smooh bass 35 fucos, such as he smooh paro-of-uy bass fucos 36 suggesed refs he core-based dscrezao mehod 37 proposed ref effecvely employs a smooh paro-of-uy 38 bass defed by he commor fucos bewee ses All of he above mehods have commo ha hey aemp 40 o cosruc a approprae dscrezao based o he 4 smulao daa. hs has a wo-fold dsadvaage: () dffere 42 smulao rus wll produce dffere dscrezaos, makg 43 hem hard o compare; (2) daa-based clusers have o rsc 44 meag. Ierpreao erms of srucural rasos mus 45 be recovered by aalyzg he molecular cofguraos coaed specfc clusers. Wh all of he above mehods, choosg a approprae combao of he merc, he cluserg mehod, ad he umber ad he locao of clusers or cores s sll ofe a ral-ad-error approach. Followg he recely roduced varaoal prcple for measable sochasc processes, 65 we propose a varaoal approach o molecular kecs. Sarg from he fac ha he molecular dyamcs propagaor s a self-ado operaor, we ca formulae a varaoal prcple. Usg he mehod of lear varao we derve a Roohaa Hall-ype geeralzed egevalue problem ha yelds a opmal represeao of egevecors of he propagaor erms of a arbrary bass se. Boh ordary MSMs usg crspclusergadmsms wh a smooh dscrezao ca be udersood as specal cases of hs varaoal approach. I coras o prevous MSMs usg smooh dscrezao, our bass fucos do o eed o be a paro of uy, alhough hs choce has some mers. Besdes s heorecal aracveess, he varaoal approach has some advaages over MSMs. Frs, he daa-drve dscrezao s replaced by a user-seleco of a approprae bass se, ypcally of eral molecular coordaes. he chose bass se may reflec chemcal uo for example, bass fucos may be predefed o f kow raso saes of backboe dhedral agles or formao/dssocao of erary coacs bewee hydrophobcally or elecrosacally eracg groups. As a resul, oe may oba a precse model wh fewer bass fucos eeded ha dscree MSM saes. Moreover, each bass fuco s assocaed wh a chemcal meag, ad hus, he erpreao of he esmaed egefucos becomes much more sraghforward ha for MSMs. Whe usg he same bass se for dffere molecular sysems of he same class, oe obas models ha are drecly comparable coras o coveoal MSMs. he represeao of he propagaor egefucos ca sll be sysemacally mproved by addg more bass fucos or by varyg he bass se. Our mehod s aalogous o he mehod of lear varao used quaum chemsry. 66 he major dfferece s ha he propagaor s self-adjo wh respec o a o-eucldea scalar produc, whereas he Hamloa s self-adjo wh respec o he Eucldea scalar produc. he dervao of he mehod s dealed seco 2 ad Appedces A C. 2. HEORY 2.. Dyamcal Propagaor. Cosder he coformaoal space of a arbrary molecule cossg of N aoms, ha s, he 3N 6-dmesoal space spaed by he eral degrees of freedom of he molecule. he coformaoal dyamcs of he molecule hs space ca be represeed by a dyamcal process {x }, whch samples a a gve me a parcular po x. I hs coex, x s ofe called a rajecory. hs process s govered by he equaos of moo, ad ca be smulaed usg sadard molecular-dyamcs programs. We assume ha a mplemeao of hermosaed molecular dyamcs s employed, whch esures ha x s mehomogeeous, Markova, ergodc, ad reversble wh respec o a uque saoary desy (usually he Bolzma dsrbuo). We roduce a propagaor formulao of hese dyamcs, followg. 65 Readers famlar wh hs approach mgh wa o skp o seco 2.2. Nex, cosder a fe esemble of molecules of he same ype, dsrbued he coformaoal space accordg o some B

3 207 al probably desy ρ 0 (x). hs al probably desy 208 evolves me a defe maer ha s deermed by he 209 aforemeoed equaos of moo for he dvdual 20 molecules. We assume ha he me evoluo s Markova px (, y; τ )d y ( x yyx d x) () + τ ( x y d yx x) (2) τ 2 where τ s a fe me sep, ad p(x,y;τ) s he so-called 22 raso desy, whch s assumed o be depede of me 23 (me-homogeeous). Fgure shows a example of he Fgure. Illusrao of wo propagaors acg o a probably desy ρ (x). Gray surface: me evoluo of ρ (x). Black doed le: sap shos of ρ (x). Cya le: equlbrum desy (x) o whch ρ (x) eveually coverges. Red, blue: propagaors wh dffere lag mes τ, whch propagae a al desy by a me sep τ me. 24 me-evoluo of a probably desy a oe-dmesoal 25 wo-well poeal. Equao 2 mples ha he probably of 26 fdg a molecule coformao y dy a me + τ depeds 27 oly o he coformao x has occuped oe me sep 28 earler, ad o o he sequece of coformaos s has vsed 29 before. he ucodoal probably desy of fdg a 220 molecule coformao y a me + τ s obaed by 22 egrag over all sarg coformaos x ρ () τρ + τ y p ( x, y ; ) ()d x x 222 (3) 223 hs equao, fac, defes a operaor (τ) ha propagaes 224 he probably desy by a fe me sep τ ρ () τ ρ + τ x () () x 225 (4) ρ () x () τ ρ() x τ (5) (τ) s called a propagaor, ad he me sep τ s ofe called 227 he lag me of he propagaor. Oe says he propagaor s 228 paramerzed wh τ. Such as p(x,y;τ), he propagaor (τ) 229 eq 5 s me-homogeeous; ha s, does o deped o. he 230 way acs o a desy ρ(x,) s o a fuco of he me a 23 whch hs desy occurs bu oly a fuco of he me sep τ 232 by whch he desy s propagaed (Fgure ). 233 he way he propagaor acs o he desy ca be 234 udersood erms of s egefucos { l (x)} ad 235 assocaed egevalues {λ }, whch are defed by he followg 236 egevalue equao ( τ) l () x λ l () x 237 (6) 238 For he class of processes whch are dscussed hs 239 publcao, he egefucos form a complee se of N 3 3N. 240 Hece, ay probably desy ( fac ay fuco) hs 0 space ca be expressed as lear combao of {l (x)}. Equao 5 ca be rewre as ρ () x c λ l () x (7) + τ c e l ( x) (8) τ/ where s he umber of dscree me seps τ. he 243 egefucos ca be erpreed as kec processes ha 244 raspor probably desy from oe par of he coforma- 245 oal space o aoher ad hus modulae he shape of he 246 overall probably desy. See ref 25 for a dealed explaao 247 of he erpreao of egefucos. he egevalues are 248 lked o he me scales o whch he assocaed kec 249 processes ake place by 250 τ l( λ) (9) 25 hese me scales are of parcular eres because hey may 252 be accessble usg varous kec expermes. 35, Gve he aforemeoed properes of he molecular 254 dyamcs mplemeao, (τ) s a operaor wh he 255 followg properes. A more dealed explaao ca be 256 foud Appedx A. 257 (τ) has a uque saoary desy; ha s, here s a 258 uque soluo (x) o he egevalue problem 259 (τ) (x) (x). Is egevalue specrum s bouded from above by λ. 260 Also, λ s he oly egevalue of absolue value equal 26 o oe. 262 (τ) s self-adjo wh respec o he weghed scalar 263 produc f g Ω f(x)g(x) (x)dx. Cosequely, 264 s egefucos l (x) form a orhoormal bass of he 265 Hlber space of square-egrable fucos wh respec 266 o hs scalar produc. Is egevalues are real ad ca be 267 umbered descedg order: 268 λ > λ λ... (0) Varaoal Prcple ad he Mehod of Lear 270 Varao. A varaoal prcple ca be derved for ay 27 operaor whose egevalue specrum s boud (eher from 272 above or from below) ad whose egevecors form a complee 273 bass se ad are orhoormal wh respec o a gve scalar 274 produc. he varaoal prcple for propagaors was derved he dervao s aalogous o he dervao of he varaoal 276 prcple of he quaum-mechacal Hamlo operaor. 66 For 277 coveece, we gve a compac dervao Appedx B. 278 he varaoal prcple ca be summarzed hree seps. Frs, 279 for he exac egefuco l (x), hefollowgequalyholds: 280 τ/ () 28 l () τ l λ () τ e he expresso f (τ) f s he aalogue of he quaum- 282 mechacal expecao value ad has he erpreao of a me- 283 lagged auocorrelao (c.f. seco 2.3). he auocorrelao of he 284 -h egefuco s decal o he -h egevalue. 285 Secod, for ay ral fuco f ha s ormalzed accordg 286 o eq 64, he followg equaly holds: 287 f () τ f f() x () x ()()d τ f x x (2) λ (3) C

4 288 where equaly f (τ) f x λ s acheved f ad oly f f l. 289 hs s a he hear of he varaoal prcple. 290 hrd, hs equaly s applcable o oher egefucos: 29 Whe f s orhogoal o he frs egefucos, he 292 varaoal prcple wll apply o he -h egefuco/ 293 egevalue par: f ( τ) f 294 λ (4) fl β 0 β,..., 295 (5) 296 hs varaoal prcple allows o formulae he mehod of 297 lear varao for he propagaor. Aga, he dervao 298 dealed ref 65 s aalogous o he dervao of he mehod 299 of lear varao quaum chemsry. 66 he ral fuco 300 f s learly expaded usg a bass of bass fucos 30 { φ } f a φ 302 (6) 303 where a are he expaso coeffces. We oly choose bass 304 ses cossg of real-valued fucos because all egevecors 305 of (τ) are real-valued fucos. Cosequely, he expaso 306 coeffces a are real umbers. However, he bass se does o 307 ecessarly have o be orhoormal. I he mehod of lear 308 varao, he expaso coeffces a are vared such ha he 309 rgh-had sde of eq 3 becomes maxmal, whle he bass 30 fucos are kep cosa. he varao s carred ou uder 3 he cosra ha f remas ormalzed wh respec o 32 eq 64 usg he mehod of Lagrage mulplers. For deals, 33 see Appedx C. he dervao leads o a marx formulao 34 of eq 6: 35 Ca λsa (7) 36 a s he vecor of expaso coeffces a, C s he (me- 37 lagged) correlao marx wh elemes C φ () τ φ j 38 j (8) 39 ad S s he overlap marx of he bass se, where he overlap 320 s calculaed wh respec o he weghed scalar produc S j φφ 32 j (9) 322 Solvg he geeralzed egevalue problem eq 7, oe 323 obas he frs egevecors of (τ) expressed he bass 324 { φ } ad he assocaed egevalues λ Esmag he Marx Elemes. o solve he 326 geeralzed egevalue equao (eq 7), we eed o calculae 327 he marx elemes C j. I he quaum chemcal verso of 328 he lear varao approach, he marx elemes H j for he 329 Hamloa (see Appedx A) are calculaed drecly wh 330 respec o he chose bass, eher aalycally or by solvg 33 he egral H j φ φ j umercally. Such a drec 332 reame s o possble for he marx elemes of he 333 propagaor. However, we ca use a rajecory x of a sgle 334 molecule, as s geeraed for examplebymdsmulaos,o 335 sample he marx elemes ad hus oba a esmae for 336 C j.forhs,weroduceabassse{χ }cossgofhe 337 cofucos of he orgal bass se {ϕ }byweghghe 338 orgal fucos wh χ() x () x φ() x φ () x χ() x 339 (20) Iserg eq 20 o he defo of he marx elemes C j 340 (eq 8), we oba 34 C φ () τ φ j j χ () τ χ j χ()(, zpy z, τ)() yχ()d y ydz (2) he las le of eq 2 has he erpreao of a me-lagged cross-correlao bewee he fucos χ ad χ j cor( χ, χ, τ): χ( z) ( x zx y) (22) j +τ χ () y ( x y)dy d z (23) j whch ca be esmaed from a me-couous me seres x of legh as or from a me-dscrezed me seres x as where N /Δ, τ τ/δ, ad Δ s he me sep of he me-dscrezed me seres. I he lm of fe samplg ad for a ergodc process, he esmae approaches he rue value Noe ha he secod le eq 2 ca also be read as he marx represeao of a operaor whch acs o he space spaed by {χ }, he cofucos of {φ } (eq 20). hs s he socalled rasfer operaor (τ). wh C () τ χ () τ χ (27) j j χ () τ χ (28) χ () τ χ () τ f() z () z j j py (, z, τ )()()d yf y y (28) (29) I parcular, (τ) has he same egevalues as he propagaor, ad s egefucos are he cofucos of he propagaor egefucos: r () x () x l () x (30) We wll somemes refer o he fucos r as rgh egefucos. For more deals o he rasfer operaor he reader s referred o ref Crsp Bass Ses Coveoal MSMs. Markov sae models (MSMs), as hey have bee dscussed up o ow he leraure, 23 25,28,30,3,40 43,55,70 arse as a specal case of he proposed mehod. Namely, he choce of bass ses coveoal MSMs s resrced o dcaor fucos, ha s, fucos ha have he value o a parcular se S of he coformaoal space ad he value 0 oherwse MSM f x S χ () x 0 oherwse (3) D

5 373 I effec, hs s a dscrezao of he coformaoal space, 374 for whch he esmao of he marx C (eq 25) reduces o 375 coug he observed rasos z j bewee ses S ad S j C j N N z j τ τ N τ χ MSM MSM j + τ ( x) χ ( x ) (32) (33) 376 I s easy o verfy, 65 ha he overlap marx S s a dagoal 377 marx, wh eres equal o he saoary probables of he 378 ses: S ()d x x : 379 S (34) 380 hus, he egevalue problem eq 7 becomes 38 Ca λπa (35) 382 a λa (36) 383 where C s he correlao marx, S dag{,..., } s he 384 dagoal marx of saoary probables, ad C s he 385 MSM raso marx. hus, a s a rgh egevecor of 386 he MSM raso marx. As he equaos above provde he 387 lear varao opmum, usg MSMs ad her egevecors 388 correspods o fdg a opmal sep-fuco approxmao 389 of he egefucos. Moreover, we ca use he weghed 390 fucos b Πa 39 (37) 392 ad see ha hey are lef egefucos of : 393 Π b λπ b (38) 394 b Π C λb (39) 395 b λb (40) 396 Noe ha he crsp bass fucos form a paro of uy, 397 meag ha her sum s he cosa fuco wh value oe, 398 whch s he frs exac egefuco of he rasfer operaor 399 (τ). For hs reaso, ay sae space paro ha s a paro 400 of uy solves he approxmao problem of he frs 40 egevalue/egevecor par exacly: he frs egevalue s 402 exacly λ, he expaso coeffces a l of he frs 403 egevecor r are all equal o oe. he correspodg frs lef 404 egevecor b a fulflls he saoary codo: 405 b b (4) 406 ad s, herefore, whe ormalzed o a eleme sum of, he 407 saoary dsrbuo of Saoary Probably Dsrbuo he Vara- 409 oal Approach. All prevous MSM approaches cludg 40 he mos commo crsp cluser MSMs bu also he smooh 4 bass fuco approaches used refs 24, 6, ad 64 have 42 drecly or drecly used bass fucos ha are a paro of 43 uy. he reaso for hs s ha usg such a paro of uy, 44 oe ca recover he exac frs egevecor ad, hus, a 45 meagful saoary dsrbuo. 46 I he prese corbuo, we gve up he paro of uy 47 codo, order o be able o fully explo he varaoal 48 prcple of he propagaor wh a arbrary choce of bass ses. 49 herefore, we mus vesgae wheher hs approach s sll meagful ad ca gve us somehg lke he saoary dsrbuo. Revsg he MSM case, he saoary probably umbers ca be erpreed as saoary probables of he ses S, or, oher words, hey measure he corbuo of hese ses o he full paro fuco Z: Z Z vx () MSM () Z e dx χ ( x)e dx (42) vx S (43) 427 Z Z (44) where v(x) s a reduced poeal. If we move o o a geeral bass, we ca maa a smlar erpreao of he vecor b Sa,aslogashefrs esmaed egevalue λ remas equal o oe. If we use he geeral defo of Z as he local desy of hebassfucoχ : vx () Z χ ()e x dx he, we sll have b Z C for all, where vx () C χ ()e x dx (45) (46) (47) Ieresgly, hs relao also becomes approxmaely rue f he esmaed egevalue λ approaches oe, as proved Appedx D. As a resul, he cocep of he saoary dsrbuo s sll meagful for bass ses ha do o form a paro of uy. Moreover, s compleely cosse wh he varaoal prcple, because he vecor b becomes a probably dsrbuo he opmum λ Esmao Mehod. We summarze by formulag a compuaoal mehod o esmae he egevecors ad egevalues of he assocaed propagaor from a me seres (rajecory) x usg a arbrary bass se.. Choose a bass se {χ }. 2. Esmae he marx elemes of he correlao marx C ad of he overlap marx S usg eq 25 wh lag mes τ ad 0, respecvely. 3. Solve he geeralzed egevalue problem eq 7. hs yelds he -h egevalue λ of he propagaor (ad he rasfer operaor) ad he expaso coeffces a of he assocaed egevecor. 4. he egevecors of he rasfer operaor are obaed drecly from he expaso coeffces a va r (48) 460 a χ 5. If a esmae of he saoary desy s avalable, he egevecors of he propagaor (τ) are obaed from l a φ a χ (49) E

6 3. MEHODS Oe-Dmesoal Dffuso Models Smu- 465 laos. We frs cosder wo examples of oe-dmesoal 466 dffuso processes x govered by Browa dyamcs. he 467 process s he descrbed by he sochasc dffereal equao 468 d x v( x)d + 2D db (50) 469 where v s he reduced poeal eergy (measured us of 470 k B, where k B s he Bolzma cosa ad s he 47 emperaure), D s he dffuso cosa, ad db deoes he 472 dffereal of Browa moo. For smplcy, we se all of he 473 above cosas equal o oe. he poeal fuco s gve by 474 he harmoc poeal v() x 0.5 x, x (5) 476 he frs case, ad by he perodc double-well poeal 477 v() x + cos(2), x x [, ) (52) 478 he secod case. I order o apply our mehod, we frs 479 produced fe smulao rajecores for boh poeals. o 480 hs ed, we pcked a (also arfcal) me-sep Δ 0 3, ad 48 he used he Euler Maruyama mehod, where poso x k+ s 482 compued from poso x k as xk+ xk Δ v( xk) + 2DΔy 483 (53) y (0, ) 484 (54) 485 I hs way, we produced smulaos of me-seps 486 for he harmoc poeal ad 0 7 me-seps for he double- 487 well poeal Gaussa Model. We apply our mehod wh Gaussa 489 bass fucos o boh problems. o hs ed, 2,3,...,0 490 ceers are chose a uform dsace bewee x 4 ad x 49 4 for he harmoc poeal ad bewee x ad x for 492 he double-well poeal. I he laer case, he bass fucos 493 are modfed o be perodc o [,). Subsequely, a 494 opmal wdh of he Gaussas s pcked by smply ryg ou 495 several choces for he sadard devaos bewee 0.4 ad ad usg he oe whch yelds he hghes secod egevalue. 497 From hs choce, he marces C ad S are esmaed ad he 498 egevalues, fucos, ad mpled me scales are compued Markov Models. As a referece for our mehods, we 500 also compue Markov sae models for boh processes. o hs 50 ed, he smulao daa s clusered o 2,3,...,0 dsjo 502 clusers usg he k-meas algorhm. Subsequely, he EMMA 503 sofware package 43 s used o esmae he MSM raso 504 marces ad o compue egevalues ad me scales Alae Dpepde MD Smulaos. We 506 performed 20 smulaos of 200 s of all-aom explc solve 507 molecular dyamcs of alae dpepde usg he AMBER 508 ff-99sb-ildn force feld. 7 he dealed smulao seup s 509 foud Appedx E Gaussa Model. Smlar o he prevous example, we 5 use perodc Gaussa fucos ha oly deped o oe of he 52 wo sgfca dhedral agles of he sysem (see seco 4.2) 53 o apply our mehod. For boh dhedrals, we separaely perform 54 a preseleco of he Gaussa ral fucos. o hs ed, we 55 frs projec he daa oo he coordae, he we solve he 56 projeced opmzao problem for all possble choces of 57 ceers ad wdhs, ad he pck he oes yeldg he hghes 58 egevalues. I every sep of he opmzao, we selec hree 59 ou of four equdsrbued ceers bewee ad, ad oe of eleve sadard devaos bewee 0.04 ad 0.4. I hs way, we oba hree Gaussa ral fucos per coordae, resulg a full bass se of sx fucos. Havg deermed he parameers for boh agles, we use he resulg ral fucos o apply our mehod as before. A boosrappg procedure s used o esmae he sascal uceray of he mpled me scales. Noe ha he varaos of bass fucos descrbed here o fd a good bass se could be coduced oce for each amo acd (or shor sequeces of amo acds) for a gve force feld ad he be reused Markov Models. hs me, we cluser he daa o 5,6,0,5,20,30,50 clusers, aga usg he k-meas algorhm. From hese cluser-ceers, we buld Markov models ad esmae he egevalues ad egevecors usg he EMMA sofware Deca-alae MD Smulaos. We performed sx 500 s all-aom explc solve molecular-dyamcs smulaos of deca-alae usg he Amber03 force feld. See Appedx E for he dealed smulao seup Gaussa Model. As before, we use Gaussa bass fucos ha deped o he backboe dhedral agles of he pepde, whch meas ha we ow have a oal of 8 eral coordaes. A preseleco of he ral fucos s performed for every coordae depedely, smlar o he alae dpepde example. I order o keep he umber of bass fucos accepably small, we selec wo ral fucos per coordae. As before, her ceers are chose from four equdsrbued ceers alog he coordae, ad her sadard devaos are chose from eleve dffere values bewee 0.04 ad 0.4. We also buld a secod Gaussa model usg fve fucos per coordae, wh equdsrbued ceers ad sadard devaos opmzed from he same values as he frs model. Havg deermed he ral fucos, we esmae he marces C ad S ad compue he egevalues ad egevecors ad aga use boosrappg o esmae uceraes Markov Models. We cosruc wo dffere Markov models from he dhedral agle daa. he frs s bul usg kmeas cluserg wh 000 cluser ceers o he full daa se, whereas for he secod, we dvde he ϕ ψ plae of every dhedral par alog he cha o hree regos correspodg o he -helx, β-shee, ad lef-haded -helx coformao, see seco 4.2. hus, we have hree dscrezao boxes for all dhedral pars, whch yelds a oal of 8 3 dscree saes o whch he rajecory pos are assged. 4. RESULS We ow ur o he resuls obaed for he four sysems preseed he prevous seco. 4.. Oe-dmesoal Poeals. he wo oe-dmesoal sysems are oy examples where all mpora properes are eher aalycally kow or ca be compued arbrarly well from approxmaos. For he harmoc poeal, he saoary dsrbuo s jus a Gaussa fuco 2 x () x l() x exp 2 2 he exac egevalues λ (τ) are gve by (55) λ( τ) exp( ( )) τ (56) F

7 Fgure 2. Illusrao of he mehod wh wo oe-dmesoal poeals, he harmoc poeal he lef half ad a perodc double-well poeal he rgh half of he fgure. (A) Poeal v ogeher wh s vara dsrbuo (shaded) ex o wo possble choces of bass fucos: a hree-eleme crsp bass ad a se of hree Gaussa fucos. (B) Exac rgh ad lef secod egefucos, r 2 ad l 2. (C) Approxmao resuls for hese secod egefucos obaed from he bass ses show. 576 ad he assocaed rgh egefuco r s gve by he 577 ( )-h ormalzed Herme polyomal 2 2 x d x r() x H() x ( ) exp exp 2 dx (57) 579 he lef halves of paels A ad B Fgure 2 show he 580 harmoc poeal ad s saoary dsrbuo, as well as he 58 secod rgh ad lef egefuco. he sg chage of l dcaes he oscllao aroud he poeal mmum, whch 583 s he slowes equlbrao process. Noe, however, ha here s 584 o eergy barrer he sysem; ha s, hs process s o 585 measable. O he rgh-had sdes of pars A ad B Fgure 2, 586 we see he same for he perodc double-well poeal. he 587 vara desy s equal o he Bolzma dsrbuo, where he 588 ormalzao cosa was compued umercally. he secod 589 egefuco was compued by a very fe fe-eleme 590 approxmao of he correspodg Fokker Plack equao, 59 usg 000 lear elemes. he slowes raso he sysem s 592 he crossg of he barrer bewee he lef ad rgh mmum. 593 hs s refleced he characersc sg chage of he secod 594 egefuco. Pars A ad B of Fgure 2 also show wo choces of 595 bass ses ha ca be used o approxmae hese egefucos: A 596 hree eleme Gaussa bass se ad a hree sae crsp se. he 597 resulg esmaes of he rgh ad lef egefucos are 598 dsplayed Fgure 2C. Already wh hese small bass ses, a 599 good approxmao s acheved. 600 Le us aalyze he approxmao qualy of boh mehods 60 more deal. o hs ed, we frs compue he 602 L 2 -approxmao error bewee he esmaed secod 603 egefuco ad he exac soluo r 2, ha s, he 604 egral We expec hs error o decay f he bass ses grow. Ideed, hs s he case, as ca be see he upper graphcs of Fgure 3A ad B, bu he error produced by he Gaussa bass ses decays faser. Eve for he 0-sae MSM, we sll have a sgfca approxmao error. Aoher mpora dcaor s he mpled me scale (τ), assocaed o he egevalue λ (τ). I s he verse rae of expoeal decay of he egevalue, gve by (τ) τ/λ (τ) ad correspods o he equlbrao me of he assocaed slow raso. he exac value of s depede of he lag me τ. However,fweesmaehe me scale from he approxmae egevalues, he esmae wll be oo small due o he varaoal prcple. However, wh creasg lag me, he error s expeced o decay, as he approxmao error also decays wh he lag me. he faser hs decay occurs, he beer he approxmao wll be. I he lower graphcs of Fgure 3A ad B, we see he lag me depedece of he secod me scale 2 for growg crsp ad Gaussa bass ses. We observe ha akes oly four o fve Gaussa bass fucos o acheve much faser covergece compared eve o a 0-sae Markov model. For seve or more Gaussa bass fucos, we acheve precse esmaes eve for very shor lag mes, whch cao be acheved wh Markov models wh a reasoable umber of saes Alae Dpede. Alae dpepde (Ac-Ala-NHMe,.e. a alae lked a eher ed o a proeco group) s desged o mmc he dyamcs of he amo acd alae a pepde cha. Ulke he prevous examples, he egefucos ad egevalues of alae dpepde cao be calculaed drecly from s poeal eergy fuco bu have o be G

8 Fgure 3. Aalyss of he dscrezao error for boh D-poeals. I he upper fgure of boh paels, we show he L 2 -approxmao error of he secod egefuco from boh crsp bass fucos ad Gaussa bass fucos, depede o he sze of he bass se. he lower fgures show he covergece of he secod mpled me scales 2 (τ) depede o he lag me τ. Doed les represe he crps bass ses ad sold les he Gaussa bass ses. he colors dcae he sze of he bass. 635 esmaed from smulaos of s coformaoal dyamcs. 636 However, alae dpepde s a horoughly suded sysem; 637 may mpora properes are well-kow, hough her 638 esmaed values deped o he precse poeal eergy 639 fuco (force feld) used he smulaos. Mos 640 mporaly, s kow ha he dyamcal behavor ca be 64 esseally udersood erms of he wo backboe dhedral 642 agles ϕ ad ψ: Fgure 4A shows he free eergy ladscape 643 obaed from populao verso of he smulao, where 644 whe regos correspod o opopulaed saes. We fd he 645 hree characersc mma he upper lef, ceral lef, ad 646 ceral rgh par of he plae, whch correspod o he β-shee, 647 -helx, ad lef-haded -helx coformao of he amo 648 acd. he wo slowes rasos occur bewee he lef half 649 ad he lef haded -helx, ad from β-shee o -helx wh 650 he ma well o he lef, respecvely. 65 Fgure 4B shows he weghed secod ad hrd egefuc- 652 os. hey are obaed from applyg our mehod wh a oal 653 of sx bass fucos (hree for each dhedral), ad from a 654 MSM cosruced from 30 cluser-ceers. he resulg 655 esmaes of r 2 ad r 3 are he weghed wh he populao 656 esmaed from he rajecory order o emphasze he regos 657 of phase space whch are relaed o he srucural rasos. 658 Almos decal resuls are acheved, ad he sg paer of 659 boh approxmaos clearly dcaes he aforemeoed 660 processes. 66 Lasly, Fgure 4C, we aga vesgae he covergece of 662 he slowes mpled me scales. Dffere MSMs wh a growg Fgure 4. Illusrao of he mehod usg he 2D dhedral agle space (ϕ,ψ) of alae dpepde rajecory daa. (A) Free eergy ladscape obaed by drec populao verso of he rajecory daa. (B ad B2) Color-coded coour plos of he secod ad hrd egefucos of he propagaor ( l 2, l 3 ), obaed by approxmag he fucos r 2 ad r 3 by a Gaussa bass se wh sx fucos, cf eq 48, ad weghg he resuls wh he esmaed saoary dsrbuo from par A. (C ad C2) Color-coded coour plos of he secod ad hrd egefucos of he propagaor ( l 2, l 3 ), obaed by approxmag he fucos r 2 ad r 3 by a Markov sae model wh 30 cluser-ceers, c.f. eq 48, ad weghg he resuls wh he esmaed saoary dsrbuo from par A. (D ad D2) Covergece of mpled me scales (τ) ( pcosecods) correspodg o he secod ad hrd egefuco, as obaed from Markov models usg 5,6,0,5,20,30,50 cluser-ceers (h les), compared o he me scales obaed from he Gaussa model wh a oal of sx bass fucos (hck gree le). h vercal bars dcae he error esmaed by a boosrappg procedure. H

9 663 umber of crsp bass fucos (cluser-ceers) were used ad 664 compared o he sx bass fuco Gaussa model. he colors 665 dcae he umber of bass fucos used; he her les 666 correspod o he Markov models, whereas he hck sold le 667 s obaed from he Gaussa model. I agreeme wh he 668 prevous resuls, we fd ha 30 or more crsp bass fucos 669 are eeded o reproduce a approxmao qualy smlar o 670 ha of a sx-gaussa bass se Deca-alae. As a hrd ad las example, we sudy 672 deca-alae, a small pepde ha s abou fve mes he sze 673 of alae dpepde. A skech of he pepde s dsplayed 674 Fgure 5A. 675 he slow srucural processes of deca-alae are less obvous 676 compared o alae dpepde. he Amber03 force feld used 677 our smulao produces a relavely fas raso bewee Fgure 5. Illusrao of he mehod usg dhedral agle coordaes of he deca alae molecule. (A) Graphcal represeao of he sysem. (B) Covergece of he esmaed secod mpled me scale ( aosecods) depedg o he lag me. We show he resuls of boh Gaussa models ad of boh he kmeas based MSM ad he adaped MSM. h vercal bars dcae he error esmaed by a boosrappg procedure. (C) Assgme of represeave srucures for he secod slowes process: he hsogram shows how he values of he secod esmaed egefuco r 2 of he smaller model are dsrbued over all smulao rajecores. Udereah, we show a overlay of srucures ake a radom from he vcy of he peaks a 2.7,.6, 0.7, ad.3. (D) Overlays of srucures correspodg o he mos egave (lef) ad mos posve (rgh) values of he secod Markov model egevecor, ake from he k-meas MSM. he elogaed ad he helcal sae of he sysem, wh a assocaed me scale of 50 s. As we ca see Fgure 5B, we are able o recover hs slowes me scale wh our mehod, 2 coverges o roughly 6.5 s for boh models. Comparg hs o he wo Markov models cosruced from he same smulao daa, we see ha boh yeld slghly hgher me scales: he k-meas based MSM reurs a value of abou 8 s ad he fely dscrezed oe eds up wh 8.5 s. Noe ha he uderesmae of he prese Gaussa bass se s sysemac, lkely due o he fac ha all bass fucos were cosruced as a fuco of sgle dhedral agles oly, hereby eglecg he couplg bewee mulple dhedrals. Despe hs approxmao, we are able o deerme he correc srucural raso. I order o aalyze hs, we evaluae he secod egefuco r 2, obaed from he smaller model, for all rajecory pos, ad plo a hsogram of hese values as dsplayed Fgure 5C. We he selec all frames ha are wh close dsace of he peaks of ha hsogram ad produce overlays of hese frames as show udereah. Clearly, large egave values of he secod egefuco dcae ha he pepde s elogaed, whereas large posve values dcae ha he helcal coformao s aaed. hs s accord wh a smlar aalyss of he secod rgh Markov model egevecor: I Fgure 5D, we show overlays of srucures ake from saes wh he mos egave ad mos posve values of he secod egevecor, ad we fd ha he same raso s dcaed, alhough he mos egave values correspod o a slghly more be arrageme of he sysem. I summary, s possble o use a comparavely small bass of 36 Gaussa fucos o acheve resuls abou he slowes srucural raso whch are comparable o hose of MSMs cosruced from abou 000 ad 6500 dscree saes, respecvely. However, he dffereces he me scales po o a weakess of he mehod: he fac ha creasg he umber of bass fucos does o aler he compued me scale dcaes ha coordae correlao cao be appropraely capured usg sums of oe-coordae bass fucos. I order o use he mehod for larger sysems, we wll have o sudy ways o overcome hs problem. 5. CONCLUSIONS We have preseed a varaoal approach for compug he slow kecs of bomolecules. hs approach s aalogous o he varaoal approach used for compug saoary saes quaum mechacs, bu uses he molecular dyamcs propagaor (or rasfer operaor) raher ha he quaummechacal Hamloa. A correspodg mehod of lear varao s formulaed. Sce he MD propagaor s o aalycally racable for praccally releva cases, he marx elemes cao be drecly compued. Foruaely, hese marx elemes ca be show o be correlao fucos ha ca be esmaed from smple MD smulaos. he mehod proposed here s hus, o frs defe a bass se able o capure he releva coformaoal dyamcs, he compue he respecve correlao marces, ad he o compue her doma egevalues ad egevecors, hus obag he key gredes of he slow kecs. Markov sae models (MSMs) are foud o be a specal case of he varaoal prcple formulaed here, amely for he case ha dcaor fucos (also kow as crsp ses or sep fucos) o he MSM clusers are used as a bass se. We have appled he varaoal approach usg Gaussa bass fucos o a umber of model examples, cludg I

10 739 oe-dmesoal dffuso sysems ad smulaos of he alae 740 dpepde ad deca-alae explc solve. Here, we have used 74 oly oe-dmesoal bass ses ha were cosruced o sgle 742 coordaes (e.g., dhedral agles), bu s clear ha muldme- 743 soal bass fucos could be sraghforwardly used. Despe he 744 smplcy of our bases, we could recover, ad mos cases 745 mprove he resuls of -sae MSMs wh much less ha bass 746 fucos he applcaos show here. 747 Noe ha praccally all MSM approaches preseed hus far 748 use daa-drve approaches o fd he clusers o whch hese 749 dcaor fucos are defed. Such a daa-drve approach 750 mpars he comparably of Markov sae models of dffere 75 smulaos of he same sysem, ad eve more so of Markov 752 sae models of dffere sysems. (Esseally, every Markov 753 sae model ha has bee publshed so far has bee 754 paramerzed wh respec o s ow uque bass se). I 755 coras, he mehod proposed here allows o defe bass ses 756 ha are, prcple, rasferable bewee dffere molecular 757 sysems. hs mproves he comparably of models made for 758 dffere molecular sysems. he secod ad possbly 759 decsve advaage of he proposed mehod s ha he 760 bass ses ca be chose such ha hey reflec kowledge abou 76 he coformaoal dyamcs or abou he forcefeld wh whch 762 x has bee smulaed. I s hus cocevable ha opmal bass 763 ses are cosruced for cera classes of small molecules or 764 molecule fragmes (e.g., amo acds or shor amo acd 765 sequeces) ad he combed for compug he kecs of 766 complex molecular sysems. 767 As meoed earler, fuure work wll have o focus o a 768 sysemac bass se seleco ad o a effce use of 769 muldmesoal ral fucos. Relaed o hs s he queso 770 of model valdao ad error esmao. Due o he use of 77 fe smulao daa, use of a very fe bass se ca lead o a 772 growg sascal uceray of he esmaed egevalues ad 773 egefucos. I order o mprove he bass se whle 774 balacg he model error ad he sascal ose, a procedure 775 o esmae hs uceray s eeded. Whle he specal case of 776 a Markov model allows for a sold error-heory based o he 777 probablsc erpreao of he model, 72 hs s a ope opc 778 here ad wll have o be reaed he fuure. 779 APPENDI A 780 Propagaors of Reversble Processes 78 I he followg, we expla more deal he properes of he 782 dyamcal propagaor (τ), as roduced seco Saoary Desy. For ay me-homogeeous propagaor, 784 here exss a leas oe saoary desy (x), whch does 785 o chage uder he aco of he operaor: (τ) (x) 786 (x). Aoher way of lookg a hs equao s o say ha 787 (x) s a egefuco of (τ) wh egevalue λ. I s 788 guaraeed ha (x) 0 everywhere as he rasfer desy s 789 ormalzed. We addoally assume ha (x) > 0. I molecular 790 sysems, (x) s a Bolzma desy ad (x) > 0 s obaed 79 whe he emperaure s ozero ad he eergy s fe for all 792 molecular cofguraos. 793 Boud Egevalue Specrum. he egevalue λ always 794 exss for ay propagaor. I s also he egevalue wh he 795 larges absolue value λ ; ha s, he egevalue specrum 796 of (τ) s boud from above by he value. hs s due o he 797 fac ha he rasfer desy s ormalzed px (, y, τ )dy 798 (59) ha s, he probably of gog from sae x x o aywhere 799 he sae space (cludg x) durg me τ has o be. 73, Ergodcy. If he dyamcs of he molecule are ergodc, he 80 λ s odegeerae. As a cosequece, here s oly oe 802 uque saoary desy (x) assocaed o (τ). 803 Reversbly. If he dyamcs of he dvdual molecules 804 he esemble occur uder equlbrum codos, hey fulfll 805 reversbly (also somemes called dealed balace or mcro- 806 reversbly ) whrespecohesaoarydsrbuo 807 ()(, xpx y; τ) ()(, ypy x; τ) x, y (60) 808 Equao 60 mples ha f he esemble s equlbrum, ha s, s sysems are dsrbued over he sae space accordg o (x), he umber of sysems gog from sae x o sae y durg me τ s he same as he umber of sysems gog from y o x. Or,he desy flux from x o y s he same as he oppose dreco, ad hs s rue for all sae pars {x,y}. For reversble processes, he saoary desy becomes a equlbrum desy ad s equal o he Bolzma dsrbuo. I he followg, we wll oly cosder operaors of reversble processes. A cosequece of reversbly s ha λ s he oly egevalue wh absolue value. ogeher wh he prevous properes, he egevalues ca be sored by her absolue value λ > λ λ... (6) Self-adjo Operaor. Aoher cosequece of reversbly s self-adjoess of he propagaor, ha s, f () τ g g () τ f (62) 824 wh respec o he weghed scalar produc fg gx () ()()d xf x x ad he orm f ff (63) (64) where (x) /(x) s he recprocal fuco of (x) ad he bar deoes complex cojugao. hs s verfed drecly: () τ fg [ px (, y, τ)()d] f x x ()()d ygy y (65) τ y py (, x, ) () x f x x ()d ()()d () y g y y (66) py (, x, f )() x ()()dd xgy yx (67) f () τ g (69) f() x p ()[ x p(, y x, )()d]d g y y x (68) I he secod le, we have used reversbly (eq 60) o replace p(x,y,τ) by p(y,x,τ)(y)/(x). Noe ha we could om he complex cojugae eq 63 because f, (τ), ad g are realvalued fucos. Self-adjoess of (τ) mples ha s egevalues are real-valued, ad s egefucos form a complee bass of 3N whch s orhoormal wh respec o he weghed scalar produc l l δ β β (70) Comparso o he QM Hamlo Operaor. Wh hese properes of he propagaor, eq 6 ca be compared o he J

11 842 saoary Schro dger equao χ E χ. Boh equaos 843 are egevalue equaos of self-adjo operaors wh a boud 844 egevalue specrum. he equaos dffer some mahema- 845 cal aspecs: (τ) s a egral operaor, whereas s a 846 dffereal operaor; (τ) s self-adjo wh respec o a 847 weghed scalar produc, whereas s self-adjo wh respec 848 o he Eucldea scalar produc. Also, hey are o aalogous 849 her physcal erpreao. I coras o he quaum- 850 mechacal Hamlo operaor, whch acs o complex-valued 85 wave fucos, (τ) propagaes real-valued probably 852 deses. Moreover, he egefucos of he propagaor do 853 o represe quaum saes, such as he groud ad exced 854 saes, hey represe he saoary dsrbuo ad he 855 perurbaos o he saoary dsrbuo from kec 856 processes. Noeheless, he mahemacal srucures of eq ad he saoary Schro dger equao are smlar eough ha 858 some mehods whch are appled quaum chemsry ca be 859 reformulaed for he propagaor. 860 APPENDI B 86 Varaoal Prcple 862 he varaoal prcple for propagaors s derved ad 863 dscussed deal ref 65. We expad a ral fuco 864 erms of he egefucos of (τ) f c l 865 (7) 866 where he h expaso coeffces s gve as c l a f 867 (72) 868 he orm (eq 64) of he ral fuco f s he gve as ff cc βll β c β (73) 870 We herefore requre ha f s ormalzed ff 87 (74) 872 Wh hs, a upper boud for he followg expresso ca be 873 foud f () τ f c c l () τ l (75) 874 ad hece ccλ l l β β 2 c λ β β β β β c λ f f λ (78) 2 (76) (77) λ f ( τ) f 875 (79) 876 he above fucoal of ay ral fuco s smaller ha or 877 equal o oe, where he equaly oly holds f ad oly f 878 f l. 879 Furhermore, from he equaos above drecly follows 880 ha for a fuco f ha s orhogoal o egefucos 88 l,..., l : fl 0 j,..., 882 j (80) he varaoal prcple resuls 883 f () τ f λ (8) APPENDI Mehod of Lear Varao Gve he varaoal prcple for he rasfer operaor (eq 79), he fuco f ca be learly expaded usg a bass of bass fucos { φ } 884 C f a φ (82) 890 where a are he expaso coeffces. All bass fucos are real fucos, bu he bass se s o ecessarly orhoormal. Hece, he expaso coeffces are real umbers. I he mehod of lear varao, he expaso coeffces a are vared such ha he rgh-had sde of eq 79 becomes maxmal, whle he bass fucos are kep cosa. he dervao leads o marx formulao of eq 6. Solvg he correspodg marx dagoalzao problem, oe obas he frs egevecors of (τ) expressed he bass { φ } ad he assocaed egevalues. Iserg eq 6 o eq 79 obas aφ aφ (83) j aa φ φ j, j j aa φ φ j, j j j j (84) (85) where we have roduced he marx eleme of he correlao marx C C j j φ φ (86) he maxmum of he expresso of rgh-had sde eq 79 s foud by varyg he coeffces a, ha s, a f f a k k j aac j j (87) 0 k, 2,... (88) uder he cosra ha f s ormalzed ff aa φφ aas (89) j j j j j j (90) S j s he marx eleme of he overlap marx S defed as S j φφ φφ j j (9) o corporae he cosra he opmzao problem, we make use of he mehod of Lagrage mulplers K

12 aa φ φ j j j λ[ aa φ φ ] (93) j j j aac λ[ aas ] (94) j j j j j j 90 he varaoal problem he s 2 a k L ac j j + ac 2 2 j (92) λ as + as 2 [ j j j ] (96) j ac λ j as j (95) (97) 0 (98) k, 2,... (99) 9 where, he hrd le, we have used ha C j C j ad S j S j 92 (eqs 62 ad 9). Equao 95 ca be rewre as a marx 93 equao 94 Ca λsa (00) 95 whch s a geeralzed egevalue problem, ad decal o 96 S Ca λa (0) 97 where a s a vecor whch coas he coeffces a. he 98 soluos of eq 0 are orhoormal wh respec o a er 99 produc whch s weghed by he overlap marx S: f g a S a δ 920 fg (02) 92 where δ fg s he Kroecker dela. he, ay wo fucos f 922 a f φ ad g a f φ are orhoormal wh respec o he 923 -weghed er produc, as s expeced for he 924 egefucos of he rasfer operaor fg a φ aφ f j j g j a S a δ fg 925 APPENDI D f g 926 Lef Egevecors ad Saoary Properes j (03) (04) (05) 927 We wa o show ha he frs lef egevecor b Sa 928 approxmaes he saoary dsrbuo eve for bass ses 929 ha do o form a paro of uy. 930 Le us assume we have a sequece of bass ses {χ } j, such 93 ha he correspodg frs egevalue λ j coverges o. Le us 932 deoe he local deses of bass se j by Z j, he oal desy 933 from eq 47 by C j, ad he eres of he ormalzed frs lef 934 egevecor of bass se j by b j. We show j Z b j j C (06) as j, or oher words, 936 j j j (07) j j χ χ Z bc Z a s ( j j j ) k k k vx () vx () e e (08) j j lj j Z ( a s ) Z lk, k lk l χ χ j a k k j kj χ χ (09) j lj j a χ χ lk, k lj kj l 940 bc Z 0 o do so, we mulply by he verse paro fuco /Z ad rewre hs expresso as We ca use eq 48 o pull he summao over k o he secod argume of he brackes: χ χ χ χ Z bc Z r ( j j j j j ) lj j l lj l (0) From he covergece of he egevalue λ j oward, follows ha he approxmae frs egefuco r j coverges o he rue frs egefuco, he cosa fuco wh value oe, he scace L 2. hs ca be show usg a orhoormal bass expaso. Cosequely, we ca use he Cauchy Schwarz equaly o esmae he expresso χ r χ χ r j j j j j χ j r j () (2) As he secod erm eds o zero by he L 2 -covergece, he complee expresso lkewse decays o zero, provded ha he L 2 -orms of he bass fucos rema bouded, whch s reasoable o assume. By a smlar argume, we ca show ha he remag fraco χ l lj χ r l lj j (3) coverges o, provded ha he L 2 -orm of he sum of all bass fucos also remas bouded. Combg hese wo observaos, we ca coclude ha eq 0 eds o 0, whch was o be show. APPENDI Smulao Seups Alae dpepde. We performed all-aom molecular dyamcs smulaos of aceyl-alae-mehylamde (Ac-Ala- NHMe), referred o as alae dpepde he ex, explc waer usg he GROMACS smulao package, he AMBER ff-99sb-ildn force feld, 7 ad he IP3P waer model. 76 he smulaos were performed he caocal esemble a a emperaure of 300 K. he eergy-mmzed sarg srucure of Ac-Ala-NHMe was solvaed o a cubc box wh a mmum dsace bewee solve ad box wall of m,correspodgoaboxvolumeof2.72m 3 ad 65 waer molecules. Afer a al equlbrao of 00 ps, 20 produco rus of 200 s each were performed, yeldg a oal smulao me of 4 μs. Covale bods o hydroge aoms were cosraed usg he LINCS algorhm 77 (lcs_er, lcs_order 4), allowg for a egrao me sep of 2 fs. he leapfrog egraor was used. he emperaure was maaed by he velocy-rescale hermosa 78 wh a me cosa of 0.0 ps. Leard-Joes eracos were cu off a m. Elecrosac eracos were reaed by he Parcle Mesh Ewald (PME) E L