A temperature accelerated method for sampling free energy and determining reaction pathways in rare events simulations

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1 Chmcal Physcs Lttrs 426 (26) A tmpratur acclratd mthod for samplng fr nrgy and dtrmnng racton pathways n rar vnts smulatons Luca Maraglano *, Erc Vandn-Endn Courant Insttut of Mathmatcal Scncs, Nw York Unvrsty, 251 Mrcr Strt, Nw York, NY 112, Untd Stats Rcvd 11 March 26 Avalabl onln 25 May 26 Abstract A mthod for samplng ffcntly th fr nrgy landscap of a complx systm wth rspct to som gvn collctv varabls s proposd. Insprd by mtadynamcs [A. Lao, M. Parrnllo, Proc. Nat. Acad. Sc. USA 99 (22) 12562], w ntroduc an xtndd systm whr th collctv varabls ar tratd as dynamcal ons and show that ths allows to sampl th fr nrgy landscap of ths varabls drctly. Th samplng s acclratd by usng an artfcally hgh tmpratur for th collctv varabls. Th valdty of th mthod s stablshd usng gnral rsults for systms wth multpl tm-scals, and ts numrcal ffcncy s also dscussd va rror analyss. W also show how th mthod can b modfd n ordr to sampl th ractv pathways n fr nrgy spac, and thrby analyz th mchansm of a racton. Fnally, w dscuss how th mthod can b gnralzd and usd as an altrnatv to th Krkwood gnralzd thrmodynamc ntgraton approach for th calculaton of fr nrgy dffrncs. Ó 26 Elsvr B.V. All rghts rsrvd. 1. Introducton and man rsults A man ssu n molcular dynamcs smulatons of complx systms s mappng th fr nrgy landscap assocatd wth a fw collctv varabls whch ar of ntrst for a partcular racton. Mor spcfcally, consdr a systm whos confguratonal stat s spcfd by x 2 R N, and suppos w ar ntrstd n th statstcs of a st of collctv varabls h(x) = (h 1 (x),...,h m (x)) (whch, wthout losng n gnralty, w shall assum dmnsonlss). If z = (z 1,...,z m ) s a partcular ralzaton of ths varabls, th fr nrgy F(z) of th systm (f no molcular constrants ar prsnt) s dfnd as F ðzþ ¼ b 1 ln 1 dðh ðxþ z Þdx; ð1þ R N bv ðxþ Ym whr ¼ R R N bv ðxþ dx, V(x) s th potntal nrgy of th systm and b = 1/k B T, whr k B s th Boltzmann constant * Corrspondng author. Fax: E-mal addrsss: maragla@cms.nyu.du (L. Maraglano), v2@ cms.nyu.du (E. Vandn-Endn). and T th tmpratur. From (1), th probablty dnsty functon of h 1 (x),...,h m (x) s q(z) = bf(z). In ordr to rconstruct F(z), a drct approach s smply to smulat th voluton of x(t) usng.g. ovrdampd dynamcs c_x ¼ qffffffffffffffffffffff þ 2b 1 cg x ðtþ; whr c s th frcton coffcnt, and g x ðtþ s a wht-nos procss,.. a Gaussan procss wth man zro and covaranc hg x ðtþgx ðsþ ¼ d dðt sþ. From (2) w can montor th voluton of h 1 (x(t)),...,h m (x(t)) to comput drctly thr probablty dnsty functon, from whch th fr nrgy (1) s obtand. Ths mthod s clarly not usful as soon as th systm shows som mtastablty,.. th landscap F(z) has dp mnma n whch th systm stays trappd, whch s always th cas whn vnts such as ractons or conformatonal transformatons ar undr study. Spcfcally, suppos that th samplng of F(z) s hndrd by fr nrgy barrrs btwn mtastabl basns around local mnma of F(z). If th hght of th hghst rlvant barrr s DF and w hav bdf 1, thn gong ovr ths barrr s a rar vnt whos tm-scal s of th ordr of (usng th fact that th paramtr c n (2) fxs th unt of tm) ð2þ /$ - s front mattr Ó 26 Elsvr B.V. All rghts rsrvd. do:1.116/.cpltt

2 L. Maraglano, E. Vandn-Endn / Chmcal Physcs Lttrs 426 (26) t ð1þ ¼ Oðc bdf Þ: Ths s th standard Arrhnus formula of chmcal racton thory (t assums that th collctv varabls ar adquat to dscrb th racton: f th collctv varabls ar badly chosn, th actual tm-scal of gong btwn th mnma of F(z) sparatd by DF wll b vn longr). Th tm-scal ovr whch a smulaton of (2) must b run to obtan an accurat samplng of F(z) must b much longr than (3), and n most cass ths tm-scal wll b prohbtvly long. Ths concluson dos not dpnd on th spcfc dynamcs chosn for x(t) n (2) and t rmans tru f (2) s rplacd by th quatons of Langvn, or Nosé-Hoovr [1], tc. To gt around ths dffculty, n ths Lttr w propos to rplac (2) by th followng xtndd systm of quatons c_x ¼ Xm ðh ðxþ z Þ oh qffffffffffffffffffffff ðxþ þ 2b 1 cg x ðtþ; p c_z ¼ ðh ðxþ z Þ þ ffffffffffffffffffffff 2b 1 c g z ðtþ; whr s a paramtr wth th dmnson of an nrgy (rcall that th collctv varabls h(x) ar dmnsonlss), c s an artfcal frcton coffcnt, b ¼ 1=k B T th nvrs of an artfcal tmpratur nd g z ðtþ a wht-nos ndpndnt of g x ðtþ. As n (2), th dynamcs of x(t) n (4) s actually not mportant and t could b rplacd by Langvn, Nosé-Hoovr, tc. s Scton 2 blow. Th systm (4) s an xtndd vrson of (2) whr V(x), th potntal nrgy of th orgnal systm, has bn rplacd by U ðx; zþ ¼ V ðxþ þ 1 2 Xm ðz h ðxþþ 2 ; and t s assumd that th nw varabls z(t) fl a dffrnt frcton coffcnt c and a dffrnt nvrs tmpratur b than th orgnal varabls x(t). Th obctv of ths Lttr s to xplan why (4) can b usd to sampl th fr nrgy F(z) and why t s mor ffcnt than (2) at dong so, but n a nutshll ths can b xpland as follows. Whn c s much largr than c so that th varabls z(t) volv much mor slowly than th x(t) and b s larg nough so that th componnts (h (x(t)) z (t))oh (x(t))/ of th forc nsur that z (t) h (x(t)) at all tms, t s shown blow (Scton 2) that th varabls z(t) soluton of (4) satsfy apprmatly th followng ffctv quaton qffffffffffffffffffffff c_z ¼ þ 2b oz 1 c g z ðtþ; ð6þ whr F(z) s prcsly th fr nrgy dfnd n (1) (that s, th fr nrgy at th physcal b, no mattr what th artfcal b s). Thrfor, th qulbrum probablty dnsty functon q ;c ðzþ of th varabl z(t) soluton of (4) s apprmatly th sam as th qulbrum probablty dnsty functon of (6),.. q ;c ðzþ 1 bf ðzþ whr ¼ bf ðzþ dz ð7þ R m ð3þ ð4þ ð5þ and ths apprmaton can b mad as prcs as ncssary by ncrasng c and, s (19) blow. As a rsult montorng th voluton of z(t) from (4) s an altrnatv way to obtan th fr nrgy F(z). But snc (7) holds at any tmpratur T, w can ncras ths artfcal tmpratur T untl bdf ¼ Oð1Þ, n whch cas th tm-scal of gong ovr DF bcoms smply (n (6) c fxs th unt of tm) t ð2þ ¼ OðcÞ; snc ths s not a rar vnt anymor. Ths mans that by adustng b n (4), tm-scals largr than t (2) wll rman accssbl n th smulatons vn whn thos largr than t (1) ar not, thrby ustfyng th usfulnss of (4) vrsus (2). How to choos th paramtrs c and n ths quaton s xpland n Scton 2 from th asymptotc analyss ladng to (6). Obsrv that what w propos hr shars das wth both mtadynamcs [2 4] and adabatc molcular dynamcs [5 7] mthods. Mtadynamcs s basd on th sam da of xtndng th systm by ncludng th st of collctv varabls to obtan a samplr n th collctv varabls spac. Howvr, whras t s smply postulatd n mtadynamcs that th tchnqu allows to sampl th fr nrgy, hr w drv th lmtng quaton (6) by asymptotc analyss. As a byproduct, ths allows us to dtrmn how to choos th paramtrs n th xtndd systm (4) to achv a prscrbd rror tolranc on F(z), somthng whch s byond th scop of mtadynamcs. In addton, n mtadynamcs, th samplng s mprovd by kpng th systm out of rgons that hav alrady bn vstd by addng a mmory-dpndnt trm n th quatons. Th acclraton that ths procdur offrs s dffcult to stmat, whras stmatng th acclraton achvd by artfcally ncrasng T n (4) s straghtfoward (smply compar (3) and ()). At ths stag, our mthod s closr to th adabatc molcular dynamcs (AMD) tchnqu, dscussd and appld n [5 7]. Th lattr s basd on th da of acclratng th samplng n th collctv varabls spac by makng us of an artfcal tmpratur n th quatons of moton. Howvr, AMD rqurs to rformulat th quatons of moton n a coordnat systm n whch th collctv varabls ar xplct coordnats. By nlargng th systm by ncludng th collctv varabls as dynamcal varabls, w avod ths rformulaton stp, whch s a consdrabl advantag whn on wants to us svral complcatd collctv varabls. An approach smlar to th on hr prsntd was dscussd n th hamltonan contxt n []. For an altrnatv tchnqu to nhanc th samplng of collctv varabls, whch xplots procton oprators, s [9]. Whn th numbr of collctv varabls s larg (manng largr than 3 or 4 n practc), samplng th fr nrgy by th mthod that w propos bcoms vry xpnsv bcaus th numbr of bns ndd to covr th spac of th collctv varabls bcoms normous. Ths dffculty s common to all th rcnt tchnqus that hav bn ntroducd to acclrat th samplng of th fr nrgy, such as for xampl mtadynamcs [2], th adaptv basng ðþ

3 17 L. Maraglano, E. Vandn-Endn / Chmcal Physcs Lttrs 426 (26) forc mthod [1] or adabatc molcular dynamcs [5]. But vn whn a larg numbr of collctv varabls s usd, (4) rmans a usful tool to xplor th fr nrgy landscap, that s, mov wthn ths landscap by a dynamcs consstnt wth th qulbrum probablty dnsty functon (7). Ths may b an usful way to dtrmn th transton pathways of som racton dscrbd by th collctv varabls h(x), but t rqurs to modfy slghtly th xtndd systm (4) so as to proprly account for dynamcal ffcts. As shown n Scton 4, nstad of (4), t amounts to usng c_x ¼ þ Xm ðz h ðxþþ oh ðxþ þ qffffffffffffffffffffff c_z ¼ Pm ~M k ðxþðz k h k ðxþþ þ 2b 1 c qffffffffffffffffffffff 2b 1 cg x ðtþ; P N ¼1 p 1 ffffffff l oh ðxþ g x ðtþ; ð9þ whr g x ðtþ s a wht-nos n RN ndpndnt of g x ðtþ, l s th mass of x and w hav ntroducd th quantty ~M k ðxþ ¼ XN ¼1 1 oh ðxþ oh k ðxþ : ð1þ l How (9) coms about s xpland n Scton 4, whr w rlat ths quaton to th formula of th rcntly dvlopd transton path thory [11,12]. Fnally, n Scton 5, w show how to gnralz (4) to comput th fr nrgy n som xtrnal control paramtrs th systm s subct to. Ths problm frquntly arss n th contxt of fr nrgy dffrncs calculatons btwn two dffrnt stats of a systm, as for xampl a molcul n th gas phas and n soluton. Many yars ago, Krkwood [13] proposd th da of prformng ths calculatons va a thrmodynamc ntgraton along a path dscrbd by an xtrnal control paramtr k. Snc thn, th tchnqu has bn xtnsvly usd n many dffrnt applcatons [14 16]. Mor rcntly [17,1], t has bn suggstd that th ffcncy of ths calculatons can b mprovd by tratng k as a dynamcal varabl, n addton to thos of th orgnal systm. In Scton 5, w show how th rsultng xtndd systm can b smulatd wth a schm analogous to (4). 2. Proprts of th xtndd systm (4) In ths scton, w ustfy why th xtndd systm (4) lads to th lmtng quaton (6) for z(t) whn c c and b 1, and hnc allows to sampl (7). W wll procd through two stps: frst, w show that whn c c th st of quatons (4) rducs to a lmtng quaton for th voluton of th z varabls alon. Thn, w show that whn b 1 ths lmtng quaton rducs to (6) Th c c lmt bhavor of quatons (4) Th analyss of (4) whn c c can b don usng a standard asymptotc analyss tchnqu for systms wth multpl tm-scals [19 21]. Ths tchnqu s brfly rvwd n th Appndx, but n ssnc t gos as follows. Whn c c, on th O(1) tm-scal n c 1, th varabls x(t) volv at fxd z(t) and thrmalz on th condtonal probablty dnsty functon for th quaton for x(t) n (4) at z(t) = z fxd: q ðxzþ ¼ 1 ðzþ bu ðx;zþ ; ð11þ whr U (x,z) s th xtndd potntal (5) and ðzþ ¼ R R N bu ðx;zþ dx. Th voluton of th varabls z(t) thn occurs on th much longr OðcÞ tm-scal and on ths tm-scal thy only fl th avrag valu of th varabls x(t). In othr words, th varabls z(t) volv accordng to an ffctv quaton obtand by avragng th quatons for z(t) n (4) wth rspct to th probablty dnsty q k (xz) n (11). Th only trm n th quaton whch s affctd by ths avrag s th trm (z h (x)), for whch w hav 1 ðzþ R N ðz h ðxþþ bu ðx;zþ dx ¼ b 1 o ln ðzþ ; ð12þ oz oz whr w hav ntroducd th ffctv potntal F ðzþ ¼ b 1 ln R 1 bu ðx;zþ dx ; ð13þ N wth ¼ R R N bv ðxþ dx (ths factor s addd for dmnsonal consstncy but s othrws rrlvant). Aftr rplacng (z h (x)) by ts avrag (12) n th quaton for z(t) n (4), w obtan th followng ffctv quaton for z(t) vald whn c c: c_z ¼ of qffffffffffffffffffffff ðzþ þ 2b oz 1 c g z ðtþ: ð14þ s Obsrv that th qulbrum dnsty functon for (14) 1 bf ðzþ whr ¼ bf ðzþ dz: R m ð15þ Ths alrady ndcats that by smulatng (4) wth c c, on can xplor th landscap of th ffctv potntal F (z) and t smply rmans to show that F (z) F(z) whn b 1. Ths s don n th nxt scton Th b 1 lmt: calculaton of th man forc Gvn th xprsson for th ffctv potntal n (13), w want to show that of ðzþ lm ¼ ; ð16þ b!1 oz oz whr F(z) s th fr nrgy dfnd n (1). To prov ths, obsrv P that (usng th Fourr rprsntaton of 1 2 b m ðz hðxþþ2 vwd as a functon of z and prmutng th ordr of som ntgrals)

4 L. Maraglano, E. Vandn-Endn / Chmcal Physcs Lttrs 426 (26) R N bu ðx;zþ dx ¼ C ¼ C ¼ C R N bv ðxþ R N bv Ym bv ðxþ R N P m 1 2 ðbþ 1Pm g 2 þ g ðz h ðxþþ R ð1þoððbþ 1 ÞÞ R m whr C = (2pb) m/2. As a rsult of ðzþ ¼ b 1 o ln 1 bu ðx;zþ dx oz oz R N ¼ b 1 o oz ln 1 b 1 o lnc oz Ym bv ðxþ R N ¼ þ OððbÞ 1 Þ; oz P m 1 dgadx g ðz h ðxþþ dðh ðxþ z ÞdxþOððbÞ 1 Þ! dðh ðxþ z Þdx þ OððbÞ 1 Þ 1 dgadx ; ð17þ!! ð1þ whr w usd oc /oz =. Ths mpls (16). Notc that th O((b) 1 ) trm can b computd xplctly by gong to nxt ordr n th xpanson n powrs of (b) 1 n (17). Th rsult n (16), togthr wth (14), mpls that (6) s th lmtng quaton govrnng th voluton of th varabls z(t) soluton of (4) whn c c and b 1. As a rsult, th qulbrum probablty dnsty of th varabls z(t) soluton of (4) s apprmatly th sam as th on of (6),.. (7) holds. Th rror n ths apprmaton can b stmatd as q ;c ðzþ ¼ 1 bf ðzþ ð1 þ Oðc=cÞ þ OððbÞ 1 ÞÞ; ð19þ and th sam stmat thrfor holds for xpctatons wth rspct to ths dnsts. Th prfactor n th trm of ordr c=c can n prncpl b stmatd by gong to nxt ordr n th xpanson n powrs of c=c prformd n th Appndx; th prfactor n th trm of ordr (b) 1 can b stmatd by gong to nxt ordr n th xpanson n powrs of (b) 1 don n (17). Explct xprssons for ths prfactors wr drvd sparatly and can b found n [22] and [23] for th trm of ordr c=c and n [24] for th trm of ordr (b) 1. It should b notd that, n practc, thr wll also b som statstcal rrors du to fnt-tm avragng on th tractory for z(t) gnratd from (4) and on only nds to adust c and n such a way that th rrors n (19) b of th sam ordr as ths statstcal rrors. pffffffffffffffffff Snc th statstcal rrors dcay rathr slowly as Oð c=t sþ, whr Ts s th lngth of th tractory, ths mans that, gvn a ralstc T s, c and b wll not hav to b takn vry larg. How to choos T s and adust c and accordngly s nvstgatd n mor dtals n Scton 3. W conclud ths scton wth a fw mportant rmarks. Frst notc that th argumnt abov sms to rqur that 1 b c=c (snc to drv (6) w took frst th lmt as c=c! 1 thn th on as b! 1) but a smlar argumnt (not ncludd hr) shows that ths rstrcton s n fact unncssary (.. th two lmts commut). Scond, th argumnt abov shows that w can asly lft th assumpton that th dynamcs for th x(t) varabls n (4) b ovrdampd. Indd, th only rqurmnt for th x(t) dynamcs s that, at fxd z, t gnrats th canoncal probablty dstrbuton functon (11). Hnc dffrnt typs of dynamcs, such as Langvn or Nosé-Hoovr can b usd as wll, provdd only that thy lad to (11) whn th xtndd potntal U (x,z) s usd nstad of V(x). Fnally, notc that th analyss abov ndcats that (4) can b rplacd by any xtndd systm whos lmtng quaton s (6), and usng othr systms than (4) may lad to vn gratr ffcncy gan. On such othr xtndd systm s c_x r ¼ ov ðxr Þ Xm ðh ðx r Þ z Þ oh qffffffffffffffffffffff ðxþ þ 2b 1 cg x;r ðtþ; P c_z ¼ R p ðh R ðx r Þ z Þ þ ffffffffffffffffffffff 2b 1 c g z ðtþ; r¼1 ð2þ whr x 1 (t),...,x R (t) ar R ndpndnt rplca of th orgnal systm. Usng (2) nstad of (4) s convnnt on a multprocssor computr snc (2) s mbarrassngly paralllzabl, and t may allow to tak c smallr by a factor up to 1/R from what t s n (4) (bcaus th slf-avragng along th tractory n (4) whch occurs bcaus c c s rplacd by an nstantanous nsmbl-avrag ovr R n (2)). Anothr xtndd systm ladng to (6) s c_x ¼ Xm ðh ðxþ z Þ oh qffffffffffffffffffffff ðxþ þ 2b 1 cg x ðtþ; R c_z ¼ ðh p ðxðt t ÞÞ z ðtþþdt þ ffffffffffffffffffffff 2b 1 c g z ðtþ; ð21þ whr must b takn such that s c s c, wth s c = O(c) and s c ¼ OðcÞ. (21) s wll-sutd for a numrcal schm n whch on would frst volv x(t) at z(t) fxd on a tm-ntrval of lngth and comput th tm-avrag at th rght-hand sd of th quaton for z(t) n (21), thn updat z(t) from t to t + usng ths tm-avrag. Ths schm works bcaus z(t) s ndd apprmatly constant on a tm-ntrval of lngth snc s c ¼ OðcÞ. In addton, whn s c s c, w hav ðh ðxðt t ÞÞ z ðtþþdt of ðzðtþþ : ð22þ oz Thus (21) allows to comput drctly th man forc along th tractory z(t) (n contrast, n (4), th nstantanous valu of (h (x(t)) z (t)) s far from that of of (z(t))/oz and th lmtng quaton only mrgs bcaus of th slf-avragng of (h (x (t) z (t)) along th tractory).

5 172 L. Maraglano, E. Vandn-Endn / Chmcal Physcs Lttrs 426 (26) Numrcal consdratons Any standard numrcal schm can b usd to smulat (4). For smplcty of prsntaton, w wll dscuss forward Eulr, but a smlar analyss can b don f othr schms ar usd or f th quaton for th x(t) varabls n (4) s rplacd by Langvn, Nosé-Hoovr, tc. and on modfs th numrcal schm accordngly. Th forward Eulr dscrtzaton of (4) rads: cx ;nþ1 ¼ cx ;n ov ðx nþ Dt þ Xm ðz ;n h ðx n ÞÞ oh ðx n Þ Dt qffffffffffffffffffffffffffffffff þ 2b 1 cdtn x ;n ; p cz ;nþ1 ¼ cz ;n ðz ;n h ðx n ÞÞDt þ ffffffffffffffffffffffffffffffff 2b 1 cdt n z ;n ; ð23þ whr Dt s th tm-stp, x n x(ndt), z n z(ndt) (as bfor x(t) and z(t) dnot th solutons of (4)), and n x ;n and n z ;n ar ndpndnt Gaussan varabls wth man and varanc 1. Lt us stmat th numrcal gan n usng (23) rathr than a smlar schm for th orgnal systm (2). If Ds s th tm-stp usd n th algorthm for th orgnal systm, th numbr of tm-stps ndd for ffcnt samplng of F(z) wth (2) s of th ordr of N stps ¼ O cbdf ; ð24þ Ds snc ths s th numbr of stps t taks to smulat (2) on a tm-ntrval of lngth O(t (1) ), whr t (1) s th Arrhnus tm-scal n (3). In contrast, th numbr of stps n th xtndd systm wth b adustd at a valu such that bdf ¼ Oð1Þ s of th ordr of N stps ¼ O c : ð25þ Dt To compar N stps and N stps t rmans to stmat how Dt and Ds compar. Suppos that s such that th couplng trm n (23) dos not ntroduc n th systm a hghr frquncy than thos charactrstc of th potntal V(x). Thn, w can choos Dt = Ds. Whn ths s not th cas, w wll hav to tak Dt < Ds. In th worst cas scnaro, w wll hav to tak Dt = Ds/(b), so lt us consdr ths cas hr. It mans that N stps ¼ O bc bdf N stps c : ð26þ In othr words, w wll hav a gan as long as bc < c bdf : ð27þ Snc bdf ncrass vry fast wth bdf, ths condton s asly ralzabl vn whn b 1 and c c as rqurd n ordr that (15) holds. In [22] (s also [23,25]), mor sophstcatd numrcal tchnqus and algorthms for systms lk (4) wr ntroducd, and thr consstncy and accuracy wr analyzd n dtals. In ssnc, ths schms ar not basd on (4) tslf, but thy us th fact that ths systm lads to th sam lmtng quaton (6) for z(t) as systms lk (2) or (21). Th us of such schms may rsult n ffcncy gan supror to thos obtand usng (4). Howvr, th dscusson of ths tchnqus gos byond th am of ths lttr, and so w rfr th ntrst radr to ths Lttrs. Fnally, obsrv that, as alrady statd n Scton 2.1, dffrnt typs of dynamcs can b usd for th x(t) varabls n (23), provdd only that thy lad to th probablty dnsty functon (11). Ths rqur dffrnt ntgraton schms, but th analyss abov can b asly xtndd to ths cass and th stmat (26) wll rman vald aftr approprat rdfnton of c (t must b rplacd by som othr paramtr ntrnsc to th dynamcs for x(t)). As a rsult, (27) wll also hold. 4. Explorng pathways n fr nrgy spac In ths scton, w dscuss why (9) can b usd to xplor th pathways of a racton whos mchansm can b dscrbd by th collctv varabls h(x) (n a sns mad prcs blow). W do so by analyzng th lmtng quaton that (9) lads to n th lmt as c c and b 1 n contxt of th rcntly dvlopd transton path thory (TPT) [11,12]. But frst, lt us obtan what ths lmtng quaton s Lmtng quaton Whn c c n (9), w can procd as n Scton 2.1 and avrag th quaton for z(t) n (9) wth rspct to (11). Ths opraton s prformd n th Appndx and lads to c_z ðtþ ¼ Xm M k ðzþ oz k qffffffffffffffffffffff X m þ 2b 1 c Xm 1 om k ðzþ ðm Þ 1=2 k ðzþgz kðtþ; ð2þ whr w hav ntroducd th followng quantty M ðzþ ¼ 1 R ðzþ ~M ðxþ bu ðx;zþ dx: ð29þ N Ths s th nsmbl avrag of ~M ðxþ dfnd n (1) wth rspct to (11). (2) s vald at any. Consdrng now th lmt b 1 and procdng as n Scton 2.2, t can b sn that ths quaton rducs to c_z ðtþ ¼ Xm M k ðzþ qffffffffffffffffffffff þ 2b 1 c X m Xm 1 om k ðzþ M 1=2 k ðzþgz kðtþ; ð3þ whr F(z) s th fr nrgy dfnd n (1) and M k (z) s th lmt of M kðzþ as b! 1: M k ðzþ ¼ bf ðzþ XN Ym l¼1 R N ¼1 1 oh ðxþ l dðh l ðxþ z l Þdx: oh k ðxþ bv ðxþ ð31þ

6 L. Maraglano, E. Vandn-Endn / Chmcal Physcs Lttrs 426 (26) Lnk wth transton path thory Lt us now xplan brfly how th lmtng quaton n (3) coms about n th contxt of transton path thory. Consdr a systm govrnd by th Langvn quaton l x ðtþ ¼ ov ðxðtþþ qffffffffffffffffffffff c_x ðtþ þ 2cb 1 g ðtþ ð32þ whr l s th mass of x. Unlk (2), whch was usd for samplng purposs only, (32) s supposd to b th dynamcal quaton govrnng th voluton of x(t) n th systm undr consdraton. Suppos that on s ntrstd n undrstandng th mchansm of racton btwn two sts A 2 R 2N and B 2 R 2N n phas-spac whch may, for nstanc, b th rgons assocatd wth th ractant and product stats of a racton. It has bn show n [11,12] (s also [26]) that th bst racton coordnat to dscrb th mchansm of racton btwn A and B, n fact th racton coordnat, s th commttor functon q(x, v). q(x, v) gvs th probablty that th tractory soluton of (32) ntatd at th pont (x,v) wll rach (.. gt commttd to) B frst rathr than A. As shown n [11,12], q(x,v) allows ons to comput many statstcal quantts of ntrst rgardng th ractv tractors by whch th racton from A to B occurs lk, n partcular, th probablty dnsty of ractv tractors. It s also wll-known that q(x, v) satsfs th followng partal dffrntal quaton, rfrrd to as th backward Kolmogorov quaton n th probablty ltratur [27], ¼ Lq PN oqðx; vþ v ¼1 þ c PN oqðx; vþ l 1 v ¼1 ov q ðx;vþ2a ¼ ; q ðx;vþ2b ¼ 1: l 1 1 l 2 oqðx; vþ ov o 2 qðx; vþ ; o 2 v ð33þ (33) s a vry complcatd partal dffrntal quaton n 2N dmnsons. To smplfy ths quaton consstnt wth th assumpton that th collctv varabls (h 1 (x),...,h n (x)) ar good varabls to captur th mchansm of th racton btwn A and B, lt us suppos that q(x,v) dos not dpnd on v and dpnds on x only va th collctv (h 1 (x),...,h n (x)),.. qðx; vþ Qðh 1 ðxþ;... ; h m ðxþþ; ð34þ for som Q to b dtrmnd. In ordr to ncorporat th apprmaton (34) n (33), w frst rformulat ths quaton as a varatonal prncpl. Spcfcally, lt I ¼ bh Lq 2 dxdv; ð35þ R N R N whr Hðx; vþ ¼ 1 2 P N ¼1 l v 2 þ V ðxþ s th Hamltonan. Snc I P and I = f q s th soluton of (33), ths soluton mnmzs I ovr all tst functons q satsfyng q (x,v) 2 A =, q (x,v) 2 B = 1. Lt us now us (34) n (35): ntgratng out th momnta, aftr som algbra w obtan that (35) rducs to I ¼ Rm Xm bf ; oq M k ðzþ oq dz; oz ð36þ whr M (z) s dfnd n Eq. (31). Hnc, w ar lft wth mnmzng (36) ovr all functons Q(z) subct to Q z2a =, Q z2b = 1, whr a and b ar th rprsntaton n z-spac of th sts A and B, rspctvly. Th Eulr Lagrang quaton assocatd wth th mnmzaton of (36) s ¼ Pm o bf M ðzþ oq ; oz oz ð37þ Q z2a ¼ ; Q z2b ¼ 1: Th ky obsrvaton now s that Eq. (37) s th backward Kolmogorov quaton assocatd wth th stochastc dffrntal quaton (3) [27]. Snc ths argumnt holds for any sts A and B, w ar thus ld to th followng concluson: f th collctv varabls (h 1 (x),...,h m (x)) ar good varabls to dscrb th mchansm of th racton, thn ths mchansm can also b undrstood by lookng at ractons n th systm govrnd by (3). Obsrv that th tm n (3) (or (9)) s artfcal and cannot b rlatd to th physcal tm n (32); ths s why w spok about usng (9) to undrstand th mchansm of th racton but not ts tmng Som numrcal aspcts In contrast wth (4), (9) nvolvs th physcal b n th quaton for z(t) and not th artfcal b. Ths s bcaus w nd th physcal tmpratur n (3) and (37) (th transton pathways can chang f b s altrd n ths quatons vn f F(z) and M k (z) rmans th quantts computd at th physcal tmpratur). But ths mans that a drct numrcal smulaton of (9) s dffcult or mpossbl bcaus t s mpdd by th sam tm-scal lmtaton as th on ncountrd wth (2). Thrfor, (9) must b usd n conuncton wth som othr tchnqu. On possblty s to us (9) wthn transton path samplng [2]. Indd, t would allow to ffctvly sampl th ractv tractors soluton of (3) wthout havng to xplctly know th coffcnts n ths quaton. Ths may b usful whn transton path samplng cannot b appld n th orgnal stat-spac bcaus th systm undr consdraton s too larg. Anothr possblty s to us only th quaton for x(t) n (4) or (9),.. us c_x ¼ þ Xm ðz h ðxþþ oh ðxþ qffffffffffffffffffffff þ 2b 1 cg x ðtþ ð3þ P wth z fxd to comput locally of(z)/oz, M k (z) and m om kðzþ= at z by rconstructng ths quantts from th followng stmators

7 174 L. Maraglano, E. Vandn-Endn / Chmcal Physcs Lttrs 426 (26) and X m Xm ~M k ðxðtþþðh k ðxðtþþ z k Þdt M k ðzþ 1 om k ðzþ ðh ðxðtþþ z Þdt oz ð39þ ð4þ ~M k ðxðtþþdt M k ðzþ ð41þ whr s chosn such that s c, wth s c = O(c). Ths quantts can thn b usd.g. wthn th strng mthod as n [24] or to smulat (3) drctly by artfcally ncrasng th tmpratur n ths quaton,.. rplacng b by b < b. Ths wll acclrat th dynamcs n (3) and allows to dtrmn transton pathways obtand by movng at an artfcally hgh tmpratur n th landscap F(z) usng M k (z), both ths quantts bng computd at th physcal tmpratur. Obsrv howvr that, as xpland bfor, ths may chang th transton pathways n a nontrval way f ntropc ffcts ar stll mportant vn wthn th fr nrgy landscap. 5. Gnralzatons Th mthod that w propos can b asly gnralzd to handl th followng stuatons. Suppos that th potntal nrgy of th systm undr consdraton dpnds on som control paramtrs, z = (z 1,...,z m ), and w ar ntrstd n computng th fr nrgy n ths paramtrs,.. GðzÞ ¼ b 1 ln R 1 bv ðx;zþ dx ð42þ N whr ¼ R R N bv ðx;z¼þ dx (ths constant s addd for dmnsonal consstncy and t s othrws rrlvant). Ths problm arss whn, n ordr to calculat th fr nrgy dffrnc for a spcfc transformaton, th couplng paramtrs ar ntroducd to artfcally ralz a spcfc path along whch th transformaton occurs. Ths da gos back to Krkwood [13], and t has bn rcntly rformulatd by consdrng th z s as dynamcal varabls as wll [17,1]. In ths contxt, w show hr how a schm lk (4) can b usful, provdd that U (x,z) s rplacd by V(x,z). Gnralzng th das abov, lt us ntroduc th followng systm of quatons: qffffffffffffffffffffff ov ðx; zþ c_x ¼ þ 2b 1 cg x ðtþ; qffffffffffffffffffffff ð43þ ov ðx; zþ c_z ¼ þ 2b oz 1 c g z ðtþ: (If w ar ntrstd n G(z) for z n som rang, say, z 2 D, whr D s som doman n R m, thn th quaton for z(t) should b ntgratd on ths doman wth rflctng boundary condtons at th dg of th doman). Procdng as bfor usng th asymptotc tchnqu n th Appndx, t can b shown that n th lmt as c c, th voluton of th varabls z(t) soluton of (43) can b apprmatd by th lmtng quaton c_z ¼ ogðzþ oz qffffffffffffffffffffff þ 2b 1 c g z ðtþ: ð44þ As a rsult, whn c c th qulbrum probablty dnsty q c ðzþ of th varabls z(t) soluton of (43) s apprmatly th dnsty of (44),.. q c ðzþ ¼ 1 bgðzþ ð1 þ Oðc=cÞÞ ð45þ whr ¼ R R m bgðzþ dz. Ths s th quvalnt of (19), xcpt that thr s no quvalnt to n th prsnt problm. Thus, smulatng (43) and montorng th varabls z(t) to comput thr qulbrum probablty dnsty allows on to comput G(z). And, as bfor, ths opraton can b acclratd (wthout affctng G(z)) by takng b < b n (43). Acknowldgmnts W ar gratful to Govann Cccott and Tommy Mllr for many usful dscussons and suggstons. Th da to us (9) to samlssly apply TPS n collctv varabls s part of a ont proct wth Tommy Mllr. Ths work s partally supportd by NSF grants DMS and DMS , and by ONR grant N Appndx A. Asymptotc analyss of (4) and (9) In ths Appndx, w show how to drv th lmtng quatons n (14) and (2) from (4) and (9), rspctvly, n th lmt whn c c usng standard asymptotc tchnqus for sngularly prturbd Markov procsss. W wll drv xplctly quaton (2), whl as for (14) t smply amounts to rplacng ~M k ðxþ and b by d k and b, rspctvly, n th xprssons for th oprator L z blow. Gvn a tst functon /(z), valuat ths functon along th tractory z(t) gnratd from (9) wth th ntal condton z() = z and x() = x and consdr th obsrvabl E/ðzðtÞÞ ¼ uðx; z; tþ; ða:1þ whr E dnots xpctaton wth rspct to th whtnoss n (9) (u(x,z,t) s a functon of x and z snc z(t) dpnds on both ths ntal condtons). It s a standard rsult of stochastc procss thory [27] that th voluton of u(x,z,t) s govrnd by th backward Kolmogorov quaton assocatd to (9). Rscalng-tm as s ¼ tc=c (rcall that th lmtng dynamcs occurs on ths slow tm-scal snc th varabls z(t) ar quas-frozn on th orgnal tm-scal), ths quaton can b wrttn as c ou os ¼ L zu þ 1 L x u; uðx; z; s ¼ Þ ¼ /ðzþ ða:2þ

8 L. Maraglano, E. Vandn-Endn / Chmcal Physcs Lttrs 426 (26) whr ¼ c=c and w dfnd L z ¼ Pm ~M k ðxþðz k h k ðxþþ o Xm 1 o ~M 2 k ðxþ ; oz oz ;!! L x ¼ PN þ Xm ðz h ðxþþ oh ðxþ o 1 o 2 : ¼1 2 ða:3þ W ar ntrstd n fndng th voluton quaton for th lmt functon u ¼ lm! u ða:4þ snc ths quaton wll tll us how obsrvabls volv n th lmt as ¼ c=c!. In ordr to obtan an quaton for u, lt us mak th followng ansatz u ¼ u þ u 1 þ 2 u 2 þ : ða:5þ By nsrtng (A.5) n (A.2) and quatng trms of th sam ordr n on both sds w obtan th hrarchy of quatons L x u ¼ L x u 1 ¼ c ou L os zu ða:6þ L x u 2 ¼ Th frst quaton n (A.6) tlls that u blongs to th nullspac of th oprator L x. Obsrv that ths oprator, vwd as an oprator n x wth z ntrng as a paramtr s th gnrator of a Markov procss whch s rgodc wth rspct to th qulbrum probablty dnsty functon (11). As a rsult, t can b shown that th null-spac of L x s mad of functons that ar constant n x,.. w dduc that u (x,z,s) u (z,s). Snc th null-spac of L x s non-trval, ths oprator s nonnvrtbl n gnral, and th scond quaton n (A.6) rqurs a solvablty condton. To s what ths condton actually s, multply both sds of th scond quaton n (A.6) by th qulbrum probablty dnsty functon (11) and ntgrat ovr x: q ðxzþl x u 1 dx ¼ R N Snc ¼ L x q ðxzþ XN ¼1 o XN 1 ¼1 R N q ðxzþ c ou os L zu dx ða:7þ!! þ Xm ðz h ðxþþ oh ðxþ q ðx; zþ o 2 2 q ðx; zþ ða:þ ntgratng by parts, t follows that th ntgral at th lfthand sd of (A.7) s dntcally zro and ths quaton rducs to ¼ R N q ðxzþ c ou os L zu dx ¼ : ða:9þ Ths quaton s th solvablty condton for th scond quaton n (A.6) (ths condton s obvously ncssary from th argumnt abov; t can b shown that s also suffcnt to solv th scond quaton n (A.6) for u 1 ). On th othr hand, (A.9) s also an quaton for u whch can b computd xplctly snc u s ndpndnt of x, and hnc t can b pulld out of th ntgral. It can b chckd that ths quaton s xplctly c ou os ¼ Xm Xm Xm 1 ; M k ðzþ of Xm kðzþ 1 M k ðzþ o2 u oz : om k ðzþ! ou oz ða:1þ Gong back to th physcal tm t ¼ sc=c, ths quaton s prcsly th backward Kolmogorov quaton assocatd wth (2). Rfrncs [1] S. Nosé, Mol. Phys. 57 (196) 17. [2] A. Lao, M. Parrnllo, Proc. Nat. Acad. Sc. USA 99 (22) [3] M. Iannuzz, A. Lao, M. Parrnllo, Phys. Rv. Ltt. 9 (23) [4] C. Mchltt, A. Lao, M. Parrnllo, Phys. Rv. Ltt. 92 (24) [5] L. Rosso, P. Mnáry, S. Shou, M.E. Tuckrman, J. Chm. Phys. 116 (21) 439. [6] L. Rosso, M.E. Tuckrman, Mol. Sm. 2 (22) 91. [7] L. Rosso, J.B. Abrams, M.E. Tuckrman, J. Phys. Chm. B 19 (25) [] J. Vand Vondl, U. Rothlsbrgr, J. Phys. Chm. B 16 (22) 23. [9] S. Mlchonna, Phys. Rv. E 62 (2) 762. [1] E. Darv, A. Pohorll, J. Chm. Phys. 115 (21) [11] W. E, E. Vandn-Endn, J. Stat. Phys. n prss, do:1.17/s [12] E. Vandn-Endn, n: M. Frraro, G. Cccott, K. Bndr (Eds.), Computr Smulatons n Condnsd Mattr: From Matrals to Chmcal Bology, Sprngr, Brln, to appar. [13] J.G. Krkwood, J. Chm. Phys. 3 (1935) 3. [14] T.P. Straatsma, J.A. McCammon, Ann. Rv. Phys. Chm. 43 (1992) 47. [15] T. Smonson, G. Archonts, M. Karplus, Acc. Chm. Rs. 35 (22) 43. [16] M. Frraro, G. Cccott, E. Spohr, T. Cartallr, P. Turq, J. Chm. Phys. 1 (22) 117. [17] X. Kong, C.L. Brooks III, J. Chm. Phys. 15 (1996) [1] R. Bttt-Putzr, W. Yang, M. Karplus, Chm. Phys. Ltt. 377 (23) 633. [19] G.C. Papancolaou, Rocky Mt. J. Math. 6 (1976) 653. [2] G.C. Papancolaou, n: R.C. D Prma (Ed.), Lct. Appl. Math., 16, Amrcan Mathmatcal Socty, [21] V.M. Volosov, Usph Mat. Nauk. 17 (1962) 3. [22] E. Vandn-Endn, Comm. Math. Sc. 1 (23) 35. [23] W. E, D. Lu, E. Vandn-Endn, Commun. Pur Appl. Math. 5 (25) [24] L. Maraglano, A. Fschr, E. Vandn-Endn, G. Cccott, J. Chm. Phys. n prss. [25] I. Fatkulln, E. Vandn-Endn, J. Comp. Phys. 2 (24) 65. [26] W. E, W. Rn, E. Vandn-Endn, Chm. Phys. Ltt. 413 (25) 242. [27] R. Durrtt, Stochastc Calculus, CRC Prss, Washngton DC, [2] C. Dllago, P.G. Bolhus, P.L. Gsslr, Transton path samplng, Adv. Chm. Phys. 123 (22).