Efficient global biopolymer sampling with end-transfer configurational bias Monte Carlo

Transcription

1 THE JOURNAL OF CHEMICAL PHYSICS 126, Effcent global bopolymer samplng wth end-transfer confguratonal bas Monte Carlo Gaurav Arya and Tamar Schlck a Department of Chemstry and Courant Insttute of Mathematcal Scences, New York Unversty, 251 Mercer Street, New York, New York Receved 1 November 2006; accepted 5 December 2006; publshed onlne 29 January 2007 We develop an end-transfer confguratonal bas Monte Carlo method for effcent thermodynamc samplng of complex bopolymers and assess ts performance on a mesoscale model of chromatn olgonucleosome at dfferent salt condtons compared to other Monte Carlo moves. Our method extends tradtonal confguratonal bas by deletng a repeatng motf monomer from one end of the bopolymer and regrowng t at the opposte end usng the standard Rosenbluth scheme. The method s samplng effcency compared to local moves, pvot rotatons, and standard confguratonal bas s assessed by parameters relatng to translatonal, rotatonal, and nternal degrees of freedom of the olgonucleosome. Our results show that the end-transfer method s superor n samplng every degree of freedom of the olgonucleosomes over other methods at hgh salt concentratons weak electrostatcs but worse than the pvot rotatons n terms of samplng nternal and rotatonal samplng at low-to-moderate salt concentratons strong electrostatcs. Under all condtons nvestgated, however, the end-transfer method s several orders of magntude more effcent than the standard confguratonal bas approach. Ths s because the characterstc samplng tme of the nnermost olgonucleosome motf scales quadratcally wth the length of the olgonucleosomes for the end-transfer method whle t scales exponentally for the tradtonal confguratonal-bas method. Thus, the method we propose can sgnfcantly mprove performance for global bomolecular applcatons, especally n condensed systems wth weak nonbonded nteractons and may be combned wth local enhancements to mprove local samplng Amercan Insttute of Physcs. DOI: / I. INTRODUCTION Bopolymers such as RNA, DNA, protens, polysacchardes, and chromatn pose a consderable challenge to thermodynamc samplng. The sheer number of degrees of freedom, and hence, favorable confguraton regons nvolved has made molecular dynamcs, Brownan/Langevn dynamcs, and Monte Carlo smulatons all vable yet lmted approaches. Strong nonbonded ntramolecular nteractons k B T lke hydrogen bondng n protens and polysacchardes and strong electrostatcs n sngle-stranded RNA/DNA and chromatn confer a hghly corrugated free energy landscape that further exacerbates ther samplng. Hence, prohbtvely long smulatons are requred to sample the entre energy landscape of the bopolymer due to ther tendency to get trapped n deep local energy mnma separated by large energy barrers. In the case of chromatn, the nucleoproten complex n whch proten spools called nucleosomes are lnked to one another by wound DNA, structural complexty resultng from charged hstone tals protrudng from each nucleosome core whch lead to numerous possble foldng arrangements make confguratonal samplng of the thermally accessble confguratonal space challengng. Brownan dynamcs BD and Langevn dynamcs LD methods, whch replace the solvent molecules of molecular a Author to whom correspondence should be addressed. Electronc mal: schlck@nyu.edu dynamcs by a more computatonally affordable stochastc heat bath, offer a drect route to obtanng the confguratonal and dynamcal propertes of bopolymers. 1,2 The smulatons are, however, lmted by the magntude of the ntegraton tme step and generally lead to regonal samplng. Furthermore, the computaton of the hydrodynamc dffuson tensor n BD can be very demandng, lmtng the overall samplng. The BD/LD methods are, hence, more suted to smulatng bopolymers wth weak ntramolecular nonbonded nteractons, especally where hydrodynamc nteractons are mportant. Monte Carlo MC methods, on the other hand, offer more flexblty wth the type and sze of jumps attempted n the confguratonal phase space, though they are clearly recognzed as havng lmtatons as the number of varables ncreases. Several MC methods, many talored for polymer smulatons, have been developed to sample the complex energy landscape of bopolymers. The smplest nvolves local perturbatons to monomer postons or bond/torson angles wth a straghtforward Metropols acceptance crtera P acc = mn 1, exp U/k B T, where U s the energy dfference between the proposed and orgnal confguratons. Examples of such local moves nclude translatons, rotatons, crank-shaft rotatons, and pvot rotatons see Ref. 3 for a comprehensve revew. Such local moves are typcally combned wth larger perturbatons to /2007/126 4 /044107/12/$ , Amercan Insttute of Physcs

2 G. Arya and T. Schlck J. Chem. Phys. 126, sample space more globally, such as regrowng the entre or large portons of the bopolymer. The confguratonal-bas MC CBMC method 4 les at the heart of these global MC moves. It s well-known that the acceptance probablty of regrowng a self-avodng lattce chan va a MC procedure where each new segment s placed randomly at one of the neghborng stes of the prevously nserted segment whether occuped or not decreases exponentally wth the length of the polymer due to an ncreased tendency for segment/segment overlaps. 5 The orgnal CBMC scheme 6,7 was developed to overcome ths lmtaton by nsertng new segments at one of the unoccuped neghbor postons ether wth equal probabltes P j =1/n, where n s the number of tral stes and j s the selected ste or wth probabltes proportonal to the tral poston s Boltzmann weghts P j n =exp U j / k=1 exp U k, where U j s the selected poston s external energy. Ths bas toward successful chan nserton s then corrected n the acceptance crteron through the computaton of the Rosenbluth factor W, whch equals the product of the sum of the Boltzmann weghts of tral N n k=1 postons for each segmental nserton: W= =1 exp U k, where N s the number chan segments. The new acceptance crtera s then gven by P acc = mn 1, W new W old, where W new and W old are the Rosenbluth factors of generatng the new polymer confguraton and retracng the old confguraton usng the Rosenbluth scheme. The CBMC scheme was generalzed to contnuously deformable polymers 8,9 wth weak and strong ntramolecular nteractons bond stretchng, bendng, and torson by modfyng the tral segment generaton procedure to sample tral segments from a Boltzmann dstrbuton correspondng to the bonded nteracton energes. Other varants of the CBMC method nclude the end-brdgng algorthm, 10,11 whch regrows nteror portons of the polymer whle keepng end segments fxed; the recol growth algorthm, 12 whch prevents polymers from reachng dead-alleys by lookng several monomers ahead before attemptng a move; and the pruned-enrched Rosenbluth algorthm, whch selectvely enrches polymer conformatons wth hgh Rosenbluth weghts. 13 We propose a smple modfcaton of the tradtonal CBMC scheme, whch we call the end-transfer CBMC method, that leads to dramatc mprovement n ts samplng effcency. Rather than regrowng deleted porton of a polymer from the same cut end, as n the tradtonal CBMC, we regrow the deleted portons of the polymer at the opposte end n the same sprt as the reptaton algorthm. 3 After descrbng ths CBMC modfcaton n detal and showng that t conserves mcroscopc reversblty, we apply t to samplng olgonucleosomes modeled usng our recent coarse-graned flexble-tal model 14 and compare ts performance to other moves local, pvot rotatons, and tradtonal CBMC. We characterze samplng effcency usng parameters relevant to the dfferent degrees of freedom of the bopolymer. We fnd that, under all condtons tested here 2 medum to hgh salt, our method mproves samplng by several orders of magntude compared to the tradtonal CBMC scheme. Our approach also leads to better samplng than other moves under hgh salt condtons, but falls short of the pvot rotaton method at low to medum salt condtons. The dffculty at low salt can be explaned by reduced electrostatc salt-screenng effects whch lead to large electrostatc energes/barrers that dramatcally reduce the method s acceptance ratos. Thus, though the samplng of complex but moderately charged/neutral bopolymers may be substantally mproved through the end-transfer CBMC scheme, adequate samplng of hghly charged systems such as chromatn remans a challenge, especally when salt-screenng effects are small. II. METHOD DEVELOPMENT A. Concept The man concept of the end-transfer CBMC scheme deletng portons of the bopolymer from one end and regrowng them at the opposte end s sketched n Fg. 1. Wthout loss of generalty, t suffces to focus on the smulaton of sngle bopolymers where only ntramolecular nteractons are consdered; extensons to bopolymers nteractng wth the surroundngs s straghtforward. We denote the polymer s two ends by head and tal, arbtrarly. Each repeatng unt of the bopolymer depcted as crcles n Fg. 1 could represent a nucleosome plus the assocated lnker DNA n the case of chromatn, four actn monomerc subunts n the case of an F-actn flament, etc. In general, the repeatng unt may be asymmetrc e.g., n chromatn so the drecton of regrowth depends upon whch of the two end motfs s selected. The sze of the cut porton must be an nteger multple of the sze of the repeatng motf to preserve the polymer s ntegrty n the case of asymmetrc bopolymers. To preserve mcroscopc reversblty, both head-total H T and tal-to-head T H must be employed and appled wth equal probablty of 1/2. If the above cut and regrow scheme s performed randomly wthout any bases n the transton matrx and n a stepwse manner begnnng at the polymer termn, the acceptance probablty s gven by the standard Metropols crteron P acc = mn 1, exp U grow U cut k B T. 3 Here, U grow and U cut, respectvely, represent the nteracton energes of the regrown motf at ts new and orgnal locatons/confguratons wth the rest of the polymer. For example, n the frst end-transfer move n Fg. 1, U cut represents the energy of nteracton between motf 7 at the tal end wth motfs 1 6 plus ts nternal energy, and U grow represents the nteracton energy of the regrown motf 7 at the head end wth motfs 1 6 plus ts nternal energy. Unfortunately, such a random regrowth of polymer ends usually results n very low acceptance rates. As n CBMC, the acceptance probablty of the endtransfer moves may be consderably mproved by employng the Rosenbluth scheme for regrowng bopolymer motfs.

3 Global bopolymer samplng J. Chem. Phys. 126, FIG. 1. Comparson of end-transfer CBMC left box and regular CBMC rght box schemes for a bopolymer comprsng of seven repeatng motfs depcted as crcles. Empty and shaded crcles represent unsampled and sampled regrown motfs, respectvely. Bopolymers A1 and A2 represent the startng conformatons, and E1 and E2 represent fnal well-sampled conformatons. The followng sequence of moves are llustrated for the end-transfer CBMC scheme: transfer/regrowth of segments 1 from head to tal B1, transfer/regrowth of segment 1 from tal to head C1, and transfer/regrowth of segment 7 from tal to head D1. The followng sequence of moves are llustrated for the standard CBMC scheme: regrowth of segment 7 B2, regrowth of segments 1 and 2 C2, and regrowth of segments 1 3 D2. Consder a bopolymer as N repeats of each buldng block, tself of n R segments. The total n R N segments nteract wthn the bopolymer through a bonded force feld, U bond, comprsng of stretchng, bendng and torson terms, and a nonbonded force feld, U nonb, comprsng long-ranged nteractons such as van derwaals and Coulomb. Consder frst the mplementaton of the T H endtransfer move. A pseudocode for ths move s provded n the Appendx. Accordng to the Rosenbluth scheme, the n R segments of the motf are nserted one after the other at the head end n the followng order: =n R,...,1. Frst, k tral postons of segment n R are generated by samplng from the Boltzmann dstrbuton correspondng to the bonded nteracton energy between segments n R and n R +1,.e., the probablty of choosng a partcular poston j for segment s gven by j P bond = A norm,j exp -U bond /k B T, 4 where A norm ncludes the Jacoban and the dstrbuton s normalzaton constant. 4 The nonbonded external Boltzmann factor of each tral poston j s computed and summed to yeld the Rosenbluth weght for the nserton of segment, 15 as gven by: The remanng segments n R 1,..., 1 are also nserted usng the above procedure and ther Rosenbluth weghts are recorded to yeld the cumulatve Rosenbluth weght n R W H = w H. =1 The cumulatve Rosenbluth weght of the orgnal motf confguraton at the tal end, W T, s computed by retracng the old confguraton from segments n R N n R +1 onwards. For each segment, k 1 tral postons of segment are generated; the orgnal segment poston becomes the kth tral poston. The Rosenbluth weght for each segment s then computed as w T = exp U,o nonb k 1 /k B T + j=1 7,j exp U nonb /k B T, 8,o where U nonb represents the external energy of segment n the old conformaton o. The cumulatve Rosenbluth for the motf n the old conformaton/poston s gven by w H k = j=1,j exp U nonb /k B T. 5 Next, one of the tral poston s s selected wth a probablty of W T = n R N =n R N n R +1 w T. 9 Fnally, the regrown H motf s chosen wth a probablty s P nonb,s = exp U nonb k / j=1,j exp U nonb. 6 P T H acc = mn 1, W H W T. 10

4 G. Arya and T. Schlck J. Chem. Phys. 126, Note that the acceptance probablty expresson of Eq. 8 s typcal of all confguratonal bas methods. The acceptance probablty for the reverse move, namely the H T move, s gven by P H T acc = mn 1, W T W H. 11 The transfer and regrowth of polymer segments from one end to the other s what dfferentates the end-transfer CBMC scheme from tradtonal CBMC schemes. Standard CBMC schemes nvolve cuttng the polymer at a random poston along ts entre length and then regrowng the shorter half of the polymer va a Rosenbluth scheme rghthand sde of Fg. 1. The sze of the regrown porton therefore vares unformly between 1 and N/2 segments, where N s the total number of segments n the polymer. Assumng that the acceptance probablty of regrowng a sngle motf s gven by a, where a 1, the acceptance probablty of regrowng portons of polymer contanng n motfs decreases exponentally power law wth n as gven by a n. Ths means that polymer samplng by the tradtonal CBMC method s rate lmted by the samplng of the nnermost regons of the polymer whch requre the largest cut szes wth the smallest acceptance probabltes. The centermost segment of the polymer s therefore sampled approxmately wth an acceptance probablty of a N/2. If we defne samplng tme 1 as the average number of MC steps taken before the regrowth of a sngle motf s accepted, then 1 a 1, and the average MC steps requred to sample the entre length of the polymer, N, s gven by N 1 N/2. 12 Consequently, the samplng tme of polymers ncreases n a power-law fashon wth the length of the polymer, makng the standard GCMC method requre prohbtvely long smulatons for samplng long polymers chans. In the end-transfer move, nstead, regrowng such large polymer segments s not requred to sample the nnermost segments; the random transfer of polymer motfs from one end to the other routnely exposes nner regons of the bopolymer for samplng. Thus, the sze of the regrown portons remans short enough equal to one motf here to allow regrowth wth hgh acceptance probabltes. The polymer chan s exhaustvely sampled when each motf has been transferred from one end to the other at least once. It can be shown that for a polymer chan composed of N repeatng motfs, the average MC steps requred to sample the entre polymer s gven by N N + 1 N 1, 13 2 where 1 s defned above see dervaton of Eq. 13 n the Appendx. Note that the average tme needed to sample a polymer chan now scales quadratcally wth the chan length, as opposed to a power-law n the tradtonal CBMC. The end-transfer method resembles the reptaton algorthm commonly employed n polymer smulatons. In a reptaton move, a move smply nvolves deletng a segment from one end and randomly placng t at one of the neghborng lattce postons of the termnal segment at the other end; a confguratonal bas method s not requred, and the acceptance crtera s smply gven by Eq. 3. In complex bopolymers, the repeatng unt generally conssts of many segments n R ; the confguraton bas MC approach s preferable to mplementng Eq. 3 because of ts effcency n generatng an energetcally favorable confguraton. The end-transfer scheme s not lmted to transfer/ regrowth of sngle repeatng motfs. Larger portons comprsed of more than one motf may be regrown, though the acceptance probablty of regrowng larger porton grows exponentally wth sze. Thus, even though the entre polymer wll be sampled wth a fewer number of accepted moves, the number of MC steps taken to accept a sngle end-transfer move wll ncrease drastcally wth the sze of the regrown porton. Specfcally, f x motfs are transferred/regrown at each end-transfer step, the samplng tme N wll roughly vary as x N/x N/x + 1 N If each motf s dffcult to sample,.e., 1 s large, the end-transfer moves actually become less effcent as the sze of the transferred portons s ncreased. B. Proof of mcroscopc reversblty Mcroscopc reversblty for the acceptance crtera n Eqs. 10 and 11 can be proven by consderng the detaled balance between two states T and H. The two states are connected to one another through an end-transfer move H T and ts reverse T H as T T H acc T H = H H T acc H T, 15 where T and H are the thermodynamc probabltes of occurrence of states T and H, respectvely; T H and H T represent the probabltes of generatng the tral confguraton H when already n state T, and vce versa, respectvely; and acc T H and acc H T represent the probablty of acceptng the generated tral moves. We now assume that T and H are proportonal to ther Boltzmann factors,.e., n R N T = A norm exp U T,,o bond /k B T exp U T,,o nonb /k B T, =1 n R N H = A norm exp U H,,s bond /k B T exp U H,,s nonb /k B T. = In the above expressons, the overall Boltzmann weghts have been separated nto the Boltzmann weghts of ndvdual segments, and by bonded and nonbonded terms. Recall that the prefactor A norm n Eqs. 16 and 17 contans terms related to the normalzaton and Jacoban of the

5 Global bopolymer samplng J. Chem. Phys. 126, bonded nteracton of segment wthout double countng. Note that the confguraton/poston of motfs n the range 1 to n R N n R +1 n the ntal T confguraton remans unchanged durng the T H transton,.e., H,+n U R,s bond = U T,,o bonb,, when =, n R N n R + 1, H,+n U R,s nond = U T,,o nonb,, when =, n R N n R The probablty of attemptng a T H transton s then gven by T H = 1 2 =1 k j=1 n R A norm exp U nonb H,,s /k B T exp U H,,s bond /k B T exp U H,,j nonb /k B T, 19 where the prefactor 1/2 accounts for the probablty of choosng one of the two end motfs for transfer/regrowth; the second term accounts for samplng from the probablty dstrbuton correspondng to the bonded force feld, and the thrd term represents the Boltzmann-factor based probablty of pckng confguraton s from the k tral postons. Smlarly, the probablty of attemptng the reverse move T H s gven by H T = 1 2 k j=1 n R N =n R N n R +1 A norm exp U nonb T,,o /k B T exp U T,,j nonb /k B T. exp U T,,o bond /k B T 20 Substtutng Eqs. 16 and 20 nto Eq. 15, we obtan acc T H acc H T = =1 n R N A norm n R N =1 A norm n R N =n R N n R +1 n R =1 A norm exp U H,,s bond /k B T exp U H,,s nonb /k B exp U T,,o bond /k B T exp U T,,o nonb /k B T A norm exp U T,,o bond /k B T exp U T,,o k nonb /k B T / j=1 exp U H,,s bond /k B T exp U nonb exp U T,,j nonb /k B T H,,s k /k B T / j=1 exp U H,,j. 21 nonb /k B T Upon cancelng lke terms and substtutng Eqs. 5, 7 9, and 18, the above expresson smplfes to acc T H acc T H = =1 n R N n R k j=1 =nr N n R +1 j=1 exp U H,,j nonb /k B T k exp U T,,j nonb /k B T = W H. W T Fnally, usng the Metropols crteron, we obtan 22 acc T H = W H/W T, f W H W T 1, f W H W T, 23 whch s equvalent to our acceptance crteron n Eq. 10. The acceptance probablty of the reverse move acc H T presented n Eq. 11 can be proven smlarly not shown. III. APPLICATION TO CHROMATIN We test the effcency of the proposed end-transfer CBMC scheme n samplng short chromatn segments. Chromatn, the system that motvated our algorthm, s a sutable model system for two reasons. Frst, chromatn s composed of negatvely charged double-stranded DNA and mostly postvely charged hstone protens. Thus, ntramolecular nonbonded nteractons are strong, makng adequate samplng a challenge. Second, the complcated archtecture of chromatn, even n the coarse-graned formulaton that we use, 14,16 presents herarchcal nteractons that are dffcult to balance. We model short segments of chromatn olgonucleosomes usng the recently developed mesoscopc flexble-tal model, sketched n Fg. 2. The flexble tal model s brefly descrbed next; Ref. 14 contans full detals. A. Flexble tal model of an olgonucleosome An olgonucleosome conssts of a chan of repeatng unts, where each repeatng motf conssts of a nucleosome core, lnker DNA, and hstone tals. The nucleosome core s treated as a rgd body whose surface s unformly spanned by 300 dscrete charges; the charges are optmzed to reproduce the electrc feld around the atomstc nucleosome core. 17 Each charge s also assgned an effectve excluded volume usng a Lennard Jones potental to prevent overlap of nucleosome core wth the other chromatn components. Each lnker DNA s represented as a chan of n lb =6 charged beads for 60 base pars wth approprate excluded volume, stretchng, bendng and twstng terms n ts force feld The charges have been optmzed usng the procedure of Stgter 21 to reproduce the far-feld electrostatc potental of double-stranded DNA. The olgonucleosomal arrays are formed as the collecton of these nucleosomes, connected by lnker DNAs. The lnker DNAs enclose an angle of 90 about the center of the nucleosome core, and are separated by a dstance of 3.6 Å normal to the plane of the nucleosome core see Fg. 2, followng the crystal structure. Each nucleosome core also serves as the orgn for n h =10 hstone tals: two copes each of the N-termn of H2A, H2B, H3, and H4 hstones, and C-termnus of each H2A

6 G. Arya and T. Schlck J. Chem. Phys. 126, FIG. 2. Color Coarse-graned olgonucleosome. a Repeatng motf of an olgonucleosome consstng of a rgd nucleosome core, sx lnker DNA beads and ten hstone tals two hdden from vew, and b a typcal sx-unt olgonucleosome n a moderately unfolded state. The nucleosome tself s composed of 300 unformly dstrbuted sphercal charges. C-termnal doman. Each hstone tal s treated as a chan of coarse-graned beads wth a force feld comprsng of stretchng, bendng, charge/charge nteracton and excluded volume terms. 14 Brefly, the n hb =50 hstone tal beads per nucleosome core correspond to 4 for H2A N-termn n b1 =n b2 =4, 3 for H2A C-termn n b3 =n b4 =3, 5 for H2B n b5 =n b6 =5, 8 for H3 n b7 =n b8 =8, and 5 for H4 n b9 =n b10 =5. The stretchng and bendng potentals for nterbead lengths and bond angles defned by three consecutve beads are represented by harmonc potentals wth parameters that reproduce confguratonal propertes of the atomstc hstone tals. A Lennard Jones potental provdes excluded volume to each proten bead. Approprate charges are also assgned to each hstone tal bead to mmc ts electrostatcs. Each hstone chan s attached to the nucleosome core usng a stff sprng. The nucleosome core, lnker DNA, and hstone tal charges nteract electrostatcally wth each other through an effectve salt-dependent Debye Hückel potental. Usng N basc buldng blocks, the total number of major nucleosome core+lnker DNA components n the entre olgonucleosome s n lb +1 N, and the total number of hstone tal beads s n hb N. Hence, an N-unt olgonucleosome contans n lb +n hb +1 N nteractng partcles; a 12-unt olgonucleosome has 684 partcles. The array head corresponds to the frst unt and the array tal to the last N unt see Fg. 2. B. Implementaton of end-transfer CBMC To mplement the end-transfer scheme to olgonucleosomes, we consder the T H move. The frst step nvolves computng the Rosenbuth weght W T of the exstng tal motf n the olgonucleosome. Ths s done by retracng the poston and orentaton of the nucleosome core, the sx lnker DNA beads, and the hstone tals n that order, and computng the segmental Rosenbluth weghts at each step. FIG. 3. Color Implementaton of an T H end-transfer CBMC move n a trnucleosome llustratng the retrace and regrowth steps. Boxed regon encloses the orgnal and proposed conformaton of the trnucleosome. Fgure 3 llustrates ths retracng procedure for a trnucleosome. The overall Rosenbluth weght of the retraced end motf s gven by n lb W T = w C,T w L,T =1 n h =1 n b,j w T,T, j=1 24 where the frst term represents the Rosenbluth weght of nsertng the last nucleosome core at ts orgnal poston; the second term accounts for the Rosenbluth weght for the nserton of the last sx lnker beads; and the last term accounts for the nserton of n hb beads on the last nucleosome core. The hstone tal bead postons are retraced startng from the bead attached to the nucleosome core. The next step nvolves regrowth of a complete motf at the head end of the olgonucleosome. Agan, the sequence of nsertons s as follows. The sx lnker DNA beads are nserted frst, begnnng wth the lnker bead attached to the head nucleosome. Next, the nucleosome core s regrown from the last-nserted lnker DNA bead. Fnally, the hstone tals are regrown from the nserted nucleosome core, each tal regrowth begnnng wth the bead attached to the nucleosome core. The overall Rosenbluth weght of the regrown motf s then gven by

7 Global bopolymer samplng J. Chem. Phys. 126, n lb W H = w C,H w L,H =1 n h =1 n b,j w H,T, j=1 25 where the frst term agan s the Rosenbluth weght of nsertng the frst nucleosome core, the second term s the Rosenbluth weght for nsertng the sx lnker beads, and the last term s the Rosenbluth weght for the nserton of n b beads of n ht hstone tals at nucleosome core 1. The overall T H move s then accepted wth the probablty gven by Eq. 10. The reverse move H T s mplemented smlarly wth the sequence of retracng and regrowth reversed. C. Smulaton detals We use the end-transfer method n combnaton wth three local Monte Carlo moves to enhance effcency: translaton and rotaton of nucleosome cores and lnker DNA beads and hstone tal regrowths. Translaton s performed by choosng a randomly orented axs passng through a randomly selected lnker DNA bead/nucleosome core, and shftng that component and ts assocated hstone tals f the chosen component s a nucleosome core along the axs by a dstance sampled from a unform dstrbuton n the range 0, 0.6 nm. Rotaton nvolves rotatng the selected nucleosome core/lnker DNA bead about one of ts axs by an angle unformly sampled from the range 0, 36. Note that whle the nucleosome core may be rotated about ether of ts three axes, the lnker DNA bead may only be rotated about ts nterbead axs see Ref. 14. The translaton and rotaton moves are accepted/ rejected based on the standard Metropols crteron. Tal regrowth nvolves selectng a hstone tal randomly and regrowng t n a dfferent conformaton usng the standard CBMC algorthm by generatng N t =4 tral postons. The end-transfer moves are performed by transferrng portons of one repeatng motf usng N t =10 trals for lnker DNA and nucleosome core regrowth and N t =4 trals for tal regrowths. The four moves translaton, rotaton, tal regrowth, and end-transfer are performed wth frequences 0.1:0.1:0.6:0.2; the tals are sampled more frequently to account for ther larger numbers recall that there are 50 tals beads per nucleosomes. We call smulatons wth the above set of moves L+E for local+end transfer. For comparson, we perform three addtonal smulatons wth dfferent sets of Monte Carlo moves. The frst L nvolves only local moves translatonal, rotatonal, and tal regrowth n frequences 0.2:0.2:0.6, respectvely. The second L+ P employs pvot rotatons n addton to the three local moves; pvotng nvolves randomly choosng one lnker DNA bead or nucleosome core, selectng a random axs passng through the chosen component, and then rotatng the shorter part of the olgonucleosome about ths axs by an angle chosen from a unform dstrbuton wthn 0, 20. The relatve frequences of attemptng the translaton/rotaton/tal-regrowth/pvot moves s fxed at 0.1: 0.1: 0.6: 0.2, respectvely. The thrd type of smulatons L+C employs the standard CBMC algorthm to regrow large portons of the olgonucleosome n addton to the three local moves. The sze of the regrowth porton s chosen unformly between 1 and N/2 f N even or N/2+1 f N odd motfs, and the olgonucleosome end to be regrown s selected wth a probablty of 1/2. The mechancs of the regrowth procedure n the CBMC moves are smlar to that of the end-transfer moves, wth the excepton that the sze of the regrown portons n the CBMC method s not restrcted to sngle motfs. The relatve attempt frequences of the translaton/rotaton/tal regrowth/pvot/olgonucleosome regrowths are gven by 0.1: 0.1: 0.6: 0.2, respectvely. All smulatons are performed n the canoncal ensemble at T=293 K and at dfferent monovalent salt concentratons to assess the mpact of the strength of electrostatc nteractons on samplng effcency. We choose two salt concentratons: 0.2 M medum salt; close to physologcal salt concentraton, where olgonucleosomes are moderately folded; and 0.5 M hgh salt, where electrostatc nteractons are mostly screened and olgonucleosomes exhbt slghtly more extended confguratons as compared to those at 0.2 M. Olgonucleosome szes range from N=3 to N=12 nucleosomes. The length of the smulatons vares between 10 and 50 mllon MC moves. The ntal confguraton of the olgonucleosome corresponds to the square solenod nucleosomal pattern of Ref. 14. We found that the ntal state does not nfluence the results. All results reflect averages from a set of four runs started from a dfferent random number generator seed. D. Samplng effcency measures We quantfy the degree of samplng usng four parameters that characterze four separate degrees of freedom of the olgonucleosomes that should be properly sampled for thermodynamc convergence. To examne how well an olgonucleosome samples the surroundng volume,.e., ts translatonal degree of freedom, we compute the mean square devaton of the center of mass poston of the olgonucleosomes versus the number of MC steps t separatng the two olgonucleosomes, as gven by t = r CM t + t 0 r CM t 0 2, 26 where r CM t 0 and r CM t+t 0 are the center of mass coordnates of the olgonucleosome computed from ts nucleosome core postons at tmes t 0 and t+t 0, respectvely, and s an ensemble average over dfferent orgns t 0 and smulaton runs wth dfferent random number generator seeds. The rate of translatonal samplng, R, may be quantfed by takng the slope of t versus MC steps t as t, as s normally done for computng dffuson coeffcents usng the Ensten relaton t R lm. 27 t t To quantfy rotatonal samplng of olgonucleosomes, we compute an autocorrelaton of the end-to-end unt vector of the olgonucleosome, as gven by

8 G. Arya and T. Schlck J. Chem. Phys. 126, t = e t 0 + t e t 0, 28 IV. RESULTS AND DISCUSSION where e t 0 s the end-to-end unt vector of the olgonucleosome after t 0 steps pontng from nucleosome 1 to nucleosome N, as gven by e t 0 = r N t 0 r 1 t 0 r N t 0 r 1 t 0, 29 where r N t 0 and r 1 t 0 are the postonal coordnates of the two end nucleosomes. The autocorrelaton can be ftted to an exponental curve t =exp t/ to obtan the rotatonal relaxaton tme n terms of number of MC steps. To characterze nternal samplng of olgonucleosomes, we compute translaton-rotaton corrected mean square devaton n the postons of nucleosome cores of olgonucleosomes separated by t MC steps durng the course of the smulaton, as gven by N t = mn r t + t 0 r t 0 2, =1 30 where mn mples that the two olgonucleosome have been supermposed onto each other to mnmze ther relatve devaton from each other before computng the mean square devaton; r t 0 s the coordnate vector of nucleosome of the olgonucleosome after t MC steps and r t+t 0 s the coordnate vector of nucleosome after t+t 0 steps after the superposton. The superposton nvolves determnng the rotaton matrx and translaton vector that gves the best ft supermposton of the two sets of molecules based on the program PDBSUP created by Rupp and Parkn. 22 Such a superposton therefore captures only the nternal structural arrangements wthn the olgonucleosome wthout corrupton from purely translatonal and rotatonal modes of the olgonucleosome. The characterstc tmescale assocated wth the samplng of the nternal structure of the olgonucleosome relaxaton tme can then be obtaned by fttng the t versus t plot to a stretched exponental functon of the form t = 1 exp t/, 31 where and are best ft parameters. Note that t does not dfferentate between the samplng of the nner portons of the bopolymer from the rest. Because the nnermost portons of the bopolymer are sampled nfrequently n tradtonal CBMC smulatons, we defne t to characterze the rate of samplng of the nnermost motf.e., the mth nucleosome core, where m =nt c n /2 +1, and the ensung n lb lnker DNA beads by computng devatons n ts nternal structure wth MC steps usng the same superposton formalsm as above n t = mn lb +1 r t + t 0 r t 0, =1 32 where r t 0 s the poston of the th nucleosome core/lnker DNA bead at tme t 0, and r t+t 0 s ts poston at tme t +t 0 after superposton. As above, we derve the characterstc samplng tme by fttng the t to Eq. 31. We compare samplng effcency n terms of the measures above for smulatons employng the end-transfer method L+E versus standard CBMC method L+C, pvot rotatons L+ P, and local moves only L at the two salt condtons 0.2 and 0.5 M Fg. 4. Fgure 4 a analyzes translatonal samplng for the dfferent sets of smulatons. Mean square devatons n the olgonucleosome center of masses t ncrease lnearly wth t as s typcal for a random walk a representatve -t plot s shown. The rate of samplng s computed from the slopes of t versus t and plotted on a logarthmc scale. As expected, samplng becomes more dffcult as the length of the polymers ncreases and/or the salt concentraton s lowered. Sgnfcantly, the effcency of the end-transfer method L+E compared to the other methods n samplng translatonal degrees of freedom of the olgonucleosomes s excellent: the L+E smulatons enhance samplng by a factor of 10 medum salt and 100 hgh salt over other methods. Ths advantage stems from the reptatonlke nature of the move where the bopolymer advances/recedes an entre repeatng motf at a tme. Indeed, reptaton moves commonly employed n lattce polymer smulatons are well known for provdng effcent samplng. 3 The L+ P approach provdes the second best samplng effcency, followed by L + C, and L smulatons. Surprsngly, the conventonal CBMC method L+C provdes poor translatonal samplng, especally for olgonucleosomes at low salt, due to ts extremely low acceptance rates. The conservatve local moves n L smulatons provde the slowest translatonal samplng as expected. The rotatonal samplng effcency n Fg. 4 b shows that the autocorrelaton of the end-to-end separaton vector t decays to zero wth a characterstc rotatonal relaxaton tme, see representatve plot. A small s characterstc of fast rotatonal samplng of the olgonucleosomes and vce versa. Such autocorrelatons are typcally ftted to an exponental, but we found that stretched exponentals wth exponents n the range ft the data better, possbly due to couplng of multple rotatonal modes n the autocorrelaton. The end-transfer moves provde the best rotatonal samplng at hgh salt and second to pvot moves at medum salt. As for translatonal samplng, local moves and standard CBMC regrowth of olgonucleosomes are worst. In fact, the rotatonal relaxaton tme correspondng to local moves for 12- unt olgonucleosomes exceeds the smulaton length, whch explans the mssng data. Fgure 4 c analyzes nternal degree of freedom samplng va mean square devatons t for a trnucleosome at hgh salt separated by a samplng tme t best ft superposton. We note that t ncreases monotoncally untl t plateaus at a characterstc nternal relaxaton tme,. The computed values ncrease dramatcally wth the length of the olgonucleosomes. At hgh salt, the end-transfer moves provdes the best nternal confguratonal samplng; at medum salt, pvotng mproves samplng by an order of magntude than end-transfer. The CBMC move provdes the next best

9 Global bopolymer samplng J. Chem. Phys. 126, FIG. 4. Samplng effcency of varous MC method n terms of four samplng measures: a translatonal, b rotatonal, c nternal entre olgonucleosome, and d nternal mddle motf. Each box contans a representatve correlaton functon for a trnucleosome at hgh salt used for computng the relevant samplng measure top, and the relevant measures for dfferent-szed olgonucleosomes at medum left and low salt rght ; the dashed lnes are gudes to the eye. Results from smulatons employng local l, pvot L+ P, standard CBMC L+C, and end-transfer CBMC L+E moves are represented by black, red, blue, and green symbols/lnes, respectvely. nternal samplng and the local moves provde the worst samplng for both neutral and charged systems. Fgure 4 d compares results n terms of samplng only the nnermost motf of the olgonucleosome. The evoluton of the mean square devaton of the center unt also follows a stretched exponental behavor. The correspondng relaxaton tmes show that the end-transfer moves agan provde the best nternal samplng at hgh salt, whle pvotng appears the best for olgonucleosomes at medum salt. Sgnfcantly, the standard CBMC method performs very poorly almost lke local moves. The rarty of acceptng regrowth of nnermost motfs consttutes the most severe drawback of the tradtonal CBMC method. Thus, the end-transfer CBMC approach provdes excellent samplng at hgh salt but ts performance degrades below that of pvot moves at medum salt. To understand the orgn of ths salt-dependent effcency, we compute n Fg. 5 the mean acceptance probabltes of the pvot, standard CBMC and end-transfer CBMC moves wth respect to the chan length at medum and hgh salt. At hgh salt, the acceptance probablty of the end-transfer move s roughly two orders of magntude smaller that that of pvot moves. As the salt concentraton s lowered, ths dsparty between the two acceptance probabltes wdens. In fact, at medum salt, ths gap s already larger than three orders of magntude. At even lower salt concentratons 0.01 M, the acceptance of an endtransfer move becomes smaller than one n a mllon. Ths drastc reducton n the acceptance probablty of the endtransfer move compared to pvot moves explans why the end-transfer method s less effcent than the pvot moves at medum salt. The same reasonng also explans why the standard CBMC wth even lower acceptance probabltes than the end-transfer CBMC for long chans yelds poor samplng. The low probablty of acceptance of the end-transfer moves s expected gven the number of hstone tals than

10 G. Arya and T. Schlck J. Chem. Phys. 126, FIG. 5. Acceptance probabltes of the pvot, standard CBMC, and endtransfer CBMC moves for dfferent olgonucleosome szes at a moderate and b hgh salt. Dashed lnes are gudes to the eye. need to be regrown at one step wthn the hghly corrugated electrostatc landscape surroundng each nucleosome core and lnker DNA. The fact that a sngle hstone tal regrowth s accepted on an average wth a probablty of about 0.25 clearly ponts to the dffculty n nsertng all ten tals at a tme. Surprsngly, despte the low acceptance probabltes of the end-transfer moves, the method stll yelds remarkably good samplng effcency. It s also nstructve to examne the scalng of the structural relaxaton tmes of the central motf obtaned from the end-transfer and regular CBMC methods wth chan length N and compare them wth analytcal predctons based on scalng arguments Eqs. 12 and 13. Fgure 6 a plots the standard CBMC relaxaton tmes for both the neutral and fully charged tal olgonucleosomes versus the chan length n a log-lnear plot; those correspondng to the end-transfer CBMC method are plotted na log-log plot n Fg. 6 b. The lnear relatonshp between log and N observed n standard CBMC confrms that the samplng tme of the centermost motf of the olgonucleosome scales n a power-law fashon wth the length of the olgonucleosome. The lnear relatonshp wth a slope of roughly 2 n the bottom fgure confrms the quadratc dependence of samplng tme wth chan length. Hence, for large N, the end-transfer CBMC approach quckly becomes superor to the regular CBMC n samplng the nteror portons of the bopolymer. The added computatonal cost n mplementng the endtransfer moves compared to pvot moves stems from generatng multple tral postons and energy computatons for each segment of the regrown and retraced motf. The CPU factor s about 3 4. We have not attempted to optmze the end-transfer moves, but t may be possble to reduce the CPU cost through careful optmzaton of parameters nvolved e.g., the number of trals N t and through reusage of computed Rosenbluth weghts untl an end-transfer move s rejected. Also, we have used the end-transfer CBMC method only alongsde local moves. Effcency can be further mproved, especally for bomolecules wth strong ntramolecular nonbonded nteractons, by combnng ths approach wth FIG. 6. Characterstc samplng tme of the olgonucleosome s nnermost motf wth a regular CBMC, and b end-transfer CBMC moves at dfferent olgonucleosome lengths ndcatng a power-law and quadratc dependence of samplng tme wth chan length. global pvot rotatons. Such a combnaton would provde superor translatonal samplng as well as good rotatonal and ntramolecular samplng. In sum, the developed end-transfer CBMC method provdes excellent samplng of structurally complex bopolymers when nonbonded nteractons of an electrostatc orgn are relatvely weak. The method, however, loses ts superorty over pvot moves when electrostatcs effects begn to domnate, as n low salt condtons, though t stll provdes good samplng of the translatonal degrees of freedom. Stll, the method consstently provdes orders of magntude better samplng than the tradtonal CBMC method. Ths exceptonal translatonal samplng and good rotatonal/nternal samplng makes t hghly sutable for samplng condensed polymers and bopolymers wth weak ntermolecular and nonbonded ntramolecular nteractons to study ther collectve propertes as opposed to sngle-molecule propertes lke phase behavor, thermodynamcs and structure. V. CONCLUSION Our extenson of the well-known confguratonal bas Monte Carlo methodology for better samplng of phase space n complex bopolymers, motvated by reptaton moves, nvolves randomly transferrng a repeatng motf n a complex bopolymer from one end to the other and regrowng t usng the effcent Rosenbluth scheme. We assessed effcency on mesoscale smulatons of olgonucleosomes compared to tradtonal CBMC and pvot rotatons for four degrees of freedom of the bopolymer translaton, rotaton, and ntramolecular samplng of the entre chan/mddle porton. Our method yelds very effcent samplng when charge-charge attractons wthn the olgonucleosomes are not too strong. When nonbonded charge-charge attractons become strong, however, samplng suffers due to ts low acceptance probabltes. Stll, the end-transfer method provdes several orders of magntude better samplng than the tradtonal CBMC method. Ths s partcularly due to the quadratc scalng between the smulaton tme requred to sample the n-

11 Global bopolymer samplng J. Chem. Phys. 126, nermost regons of the bopolymer and the chan length, as compared to the drastc exponental scalng observed for the tradtonal CBMC method. Our extenson to the standard CBMC method may thus be further coupled to other technques lke parallel temperng or replca exchange where confguratons are swtched between ensembles at dfferent temperatures or stochastc tunnelng 26 where large energy barrers are smoothed out to allow barrer crossngs to better sample the vast rugged energy landscape of bopolymers. ACKNOWLEDGMENTS The authors gratefully acknowledge support by NSF Award No. MCB and NIH Grant No. R01 GM55164 to T. Schlck. Acknowledgment s also made to the donors of the Amercan Chemcal Socety Petroleum Research Fund for support or partal support of ths research Award No. PRF39225-AC4 to T. Schlck. They also thank Dr. Cha Chang for useful dscussons that led to the proof for the scalng of samplng tme wth chan length n the end-transfer scheme. APPENDIX: PSEUDOCODE FOR IMPLEMENTING T\H END-TRANSFER CBMC MOVE! N: total number of repeatng unts n the bopolymer! n_r: number of segments n each repeatng unt! n_t: number of tral postons generated for each segment! k_b: Boltzmann constant! T: temperature! R n_r*n : current coordnates of bopolymer segments! W_old: Rosenbluth weght of old confguraton! W_new: Rosenbluth weght of new confguraton! RETRACE OLD CONFIGURATION AT TAIL END W_old=1 Do =n_r* N-n_R+1, n_r* N w=0 Do t=1, n_t 1 call genpos r t! generate tral poston of segment! based on nternal constrants e.g., bond call energy r t,u t! calculate external energy of tral w=w+exp U t /k_b/t call energy r_old, U_old w=w+exp U_old/k_B/T W_old=W_old* w! REGROW NEW CONFIGURATION AT HEAD END W_new=1 Do =n_r 1, 1 w=0 Do t=1, n_t call genpos r t! generate tral poston of segment! based on nternal constrants e.g., bond call energy r t,u t! calculate external energy of tral w=w+exp U t /k_b/t W_new=W_new* w call rand q! generate a random number between 0 and 1 p 0 =0 p 1 =exp U 1 /k_b/t /w do t=2, n_t p t =p t 1 +exp U t /k_b/t /w do t=1, n_t f q p t 1 and q t then select =t! tral poston t has been selected Endf r_new =r t! store new poston n a temporary array! ACCEPT OR REJECT NEW CONFIGURATION BASED ON ROSENBLUTH CRITERIA call rand q f q W_new/W_old then! accept new confguraton do =n_r*n n_r+1,1! shft ndces of polymer R +n_r =R do =1, n_r 1 R =r_new Endf Scalng of the samplng tme as a functon of polymer length The end-transfer scheme appled to a bopolymer can be smplfed to the followng mathematcal model. The bopolymer wth N repeatng motf represents a sngle fle of balls, where each ball represents a sngle repeatng motf Fg. 7. Intally, all balls are whte unsampled ; at each step, analogous to an accepted end-transfer move, a ball s selected randomly from one end and placed at the opposte end; n ths process the ball s color s changed from whte to black sampled. Note that f a ball s already black, t remans black rrespectve of ts transfer. The process s repeated untl all the balls become black. The problem of determnng the average tme to sample the entre polymer then s the equvalent of determnng the average number of steps requred to turn all the balls black. FIG. 7. Breakdown of the end-transfer procedure nto ntermedate steps. The black balls represent bopolymer motfs that have been transferred at least one from one end to another and the whte balls represent motfs that are yet to be transferred.

12 G. Arya and T. Schlck J. Chem. Phys. 126, The followng key nsght helps solve the problem more readly: for a transton from n to n+1 black balls to occur, the black and whte balls must be separated out on opposte ends of the fle see Fg. 7. Only such a stuaton allows the possblty of the transfer wth probablty 1/2 of a whte ball from one end to the other, and consequently, the color change from whte to black. If ths s not the case, both ends of the fle are capped by a black ball and the n to n+1 transton cannot occur. Ths nsght helps us understand why the process of changng zero black balls to N black balls can be broken down nto N 1 ntermedate stages contanng 1,..., N 1 black balls, n sequence, arranged together at one of the ends of the fle, as shown n Fg. 7. The total number of steps taken to reach the fnal confguraton wth N black balls n s gven by the cumulatve sum of average number of steps requred to go from one ntermedate stage to the next, gven by n 1, where =1,..., N 1. Clearly, n 0 1 =1, as t takes a sngle end-transfer move to turn the frst ball black. It can be shown that n 1 2 s gven by the nfnte seres n 1 2 = , A1 where each term represents the steps taken to transfer a whte ball from one end to other and ts assocated bnomal probablty of occurrence. The above sum represents a combned arthmetc/geometrc progresson whose sum s gven by n 1 2 =2. Smlarly, n 2 3 =3 and, n general, n 1 =. Ths mples that the total number of steps needed to change all whte balls to black s gven by N n = n 1 = N = =1 N N + 1. A2 2 Hence, all motfs of a bopolymer get sampled, on average, n N = 1 N N+1 /2 MC steps, f each transfer move takes 1 MC steps for acceptance. 1 D. L. Ermak and J. A. McCammon, J. Chem. Phys. 69, E. Dcknson, Chem. Soc. Rev. 14, K. Bnder, n Computer Smulaton of Polymers, edted by E. A. Colbourn Longmans Scentfc and Techncal, London, 1994, p D. Frenkel and B. Smt, Understandng Molecular Smulaton: From Algorthms to Applcatons Academc, San Dego, J. Batouls and K. Kremer, J. Phys. A 21, J. I. Sepmann, Mol. Phys. 70, J. I. Sepmann and D. Frenkel, Mol. Phys. 75, J. J. de Pablo, M. Laso, and U. W. Suter, J. Chem. Phys. 96, D. Frenkel, G. C. A. M. Mooj, and B. Smt, J. Phys.: Condens. Matter 4, M. Vendruscolo, J. Chem. Phys. 106, Z. Chen and F. A. Escobedo, J. Chem. Phys. 113, S. Consta, N. B. Wldng, D. Frenkel, and Z. Alexandrowcz, J. Chem. Phys. 110, P. Grassberger, Phys. Rev. E 56, G. Arya, Q. Zhang, and T. Schlck, Bophys. J. 91, M. N. Rosenbluth and A. W. Rosenbluth, J. Chem. Phys. 23, G. Arya and T. Schlck, Proc. Natl. Acad. Sc. U.S.A. 103, Q. Zhang, D. A. Beard, and T. Schlck, J. Comput. Chem. 24, J. Sun, Q. Zhang, and T. Schlck, Proc. Natl. Acad. Sc. U.S.A. 102, D. A. Beard and T. Schlck, Structure London 9, S. A. Allson, R. Austn, and M. Hogan, J. Chem. Phys. 90, D. Stgter, Bopolymers 16, B. Rupp and S. Parkn, PDBSUP A FORTRANP Program That Determnes the Rotaton Matrx and Translaton Vector for Best Ft Superposton of Two PDB Fles by Solvng the Quaternon Egenvalue Problem Lawrence Lvermore Natonal Laboratory, Lvermore, CA, U. H. E. Hansmann, Chem. Phys. Lett. 281, Q. Yan and J. J. de Pablo, J. Chem. Phys. 111, Y. Sugta and Y. Okamoto, Chem. Phys. Lett. 314, A. Schug, T. Herges, and W. Wenzel, Phys. Rev. Lett. 91,