Optimization of replica exchange molecular dynamics by fast mimicking

Transcription

1 THE JOURNAL OF CHEMICAL PHYSICS 127, Optimization of replia exhange moleular dynamis by fast mimiking Jozef Hritz and Chris Oostenbrink a Leiden Amsterdam Center for Drug Researh (LACDR), Division of Moleular Toxiology, Vrie Universiteit, Amsterdam, The Netherlands Reeived 8 August 2007; aepted 5 September 2007; published online 27 November 2007 We present an approah to mimi replia exhange moleular dynamis simulations REMD on a miroseond time sale within a few minutes rather than the years, whih would be required for real REMD. The speed of mimiked REMD makes it a useful tool for testing the effiieny of different settings for REMD and then to selet those settings, that give the highest effiieny. We present an optimization approah with the example of Hamiltonian REMD using soft-ore interations on two model systems, GTP and 8-Br-GTP. The optimization proess using REMD mimiking is very fast. Optimization of Hamiltonian-REMD settings of GTP in expliit water took us less than one week. In our study we fous not only on finding the optimal distanes between neighboring replias, but also on finding the proper plaement of the highest level of softness. In addition we suggest different REMD simulation settings at this softness level. We allow several replias to be simulated at the same Hamiltonian simultaneously and redue the frequeny of swithing attempts between them. This approah allows for more effiient onversions from one stable onformation to the other Amerian Institute of Physis. DOI: / I. INTRODUCTION Sine the introdution of the replia exhange method REM using the Monte Carlo algorithm, replia exhange moleular dynamis REMD or parallel tempering simulations in the late 1990s, 1 5 there has been a steep inrease in their popularity. In these simulations shemes the onformational sampling of a moleular system is greatly enhaned by onneting multiple simulations that are performed at different temperatures temperature REMD T-REMD or using different funtional forms of the potential interation energy Hamiltonian REMD H-REMD. 6 By maintaining a detailed balane requirement when individual simulations are swithed to a different temperature or Hamiltonian, the orret thermodynami ensemble will be obtained for eah of the simulation parameters. An intelligent hoie of REMD settings allows for the swift generation of anonial ensembles of systems in whih the potential energy barriers between stable onformations are too large to be rossed repeatedly in a normal moleular dynamis MD simulation. In this paper we will introdue an approah to effiiently optimize the settings for REMD simulation for systems with multiple stable onformations. Settings to optimize involve the number of simultaneous simulations replias and the optimal simulation settings for eah of these simulations temperature, Hamiltonian. Only researhers with aess to extraordinary omputational resoures an afford a trial and error approah when searhing for effiient REMD settings. Several studies desribe approahes leading to effiient REMD simulations, mostly for T-REMDs. Beause one MD step requires the same amount of CPU time for any temperature or Hamiltonian, the alloation of replias to CPUs an be trivial one replia per CPU. However, in replia exhange Monte Carlo simulations, the average wall lok time to omplete one Monte Carlo move varies with the temperature or the Hamiltonian, and the CPU alloation beomes an important issue. 7 Many authors laim that the highest effiieny is obtained when the swithing probability between neighboring replias is onstant at a value of approximately 20% This is still the most often used riterion in REMD appliations despite the fat that in 2004 Trebst et al. presented the feedbakoptimized parallel tempering approah. 14 It was shown that for the optimal temperature sets the swithing probabilities between neighboring replias are not onstant but rather depend on the temperature A signifiant pratial drawbak of this approah, however, is the simulation time required to obtain the optimized settings. Espeially for biomoleular simulations, this hampers the pratial appliability of the method. The appliation of feedbak-optimized parallel tempering for the 36-residue villin headpiee subdomain HP-36 required REMD simulations that overed swithing trials, whih takes many years of CPU time. 16 It is probably for this reason that less appliations of the method have been desribed. For this reason we have developed a set of effiient tools for optimizing the settings of REMD both T- and H-REMD simulations for systems, of whih multiple stable onformations are known. By generating the appropriate onformational ensemble of the system REMD gives insight into the relative populations of stable onformations. We present the pratial appliation of the proposed optimization sheme for two biologially relevant systems: GTP and 8-Br-GTP Fig. 1. GTP is an important omponent in, e.g., ellular signaling, while 8-Br-GTP is onsidered as promising antia Eletroni mail:.oostenbrink@few.vu.nl /2007/ /204104/13/$ , Amerian Institute of Physis Downloaded 30 Mar 2011 to Redistribution subet to AIP liense or opyright; see

2 J. Hritz and C. Oostenbrink J. Chem. Phys. 127, FIG. 1. Struture of GTP and 8-Br-GTP in syn onformation. Conformational transitions between the syn and anti states our by rotation around the glyosidi bond indiated. baterial agent. It inhibits FtsZ polymerization but does not affet tubulin polymerization. 17 GTP and 8-Br-GTP both have two stable orientations of the base toward the sugar: anti and syn Fig. 1. Nulear magneti resonane NMR studies show that while GTP prefers to be in the anti onformation with an estimated population of 70% from NMR experiments, 8-Br-GTP shows preferene for the syn onformation population 90%. Our MD study shows that there is a high energy barrier between the anti and syn onformations for both moleules. In two MD simulations of GTP starting from anti and syn onformation eah of 20 ns only one single transition from syn to anti was observed after 1 ns while no anti syn transition was seen at all. Analogous simulations of 8-Br-GTP did not reveal any transition indiating an even higher potential energy barrier between the anti and syn onformations. 18 In these ases REMD an be used to enhane the onformational sampling and obtain the orret onformational ensemble with the proper populations of syn and anti onformations. From this ensemble, the free energy differene between the two states as well as a variety of moleular properties an be alulated. II. METHODS A. MD settings All MD simulations were onduted using the GRO- MOS05 MD simulation pakage running on a linux luster. 19 All bonds were onstrained, using the SHAKE algorithm 20 with a relative geometri auray of 10 4, allowing for a time step of 2 fs used in the leapfrog integration sheme. 21 Periodi boundary onditions, with a trunated otahedral box, were applied. After the steepest desent minimization to remove bad ontats between moleules, the initial veloities were randomly assigned from a Maxwell Boltzmann distribution at 298 K, aording to the atomi masses. The temperature was ontrolled using a weak oupling to a bath of 298 K with a time onstant of 0.1 ps. 22 The solute moleules GTP or 8-Br-GTP and solvent i.e., expliit water moleules and 3 Na + ounterions were independently oupled to the heat bath. The pressure was ontrolled using isotropi weak oupling to atmospheri pressure with a time onstant 0.5 ps. 22 Van der Waals and eletrostati interations were alulated using a triple range utoff sheme. Interations within a short-range utoff of 0.8 nm were alulated every time step from a pair list that was generated every five steps. FIG. 2. Shemati figure of the energy landsape profile as funtion of the softness of nonbonded interations. In the presented H-REMD using softore interations, the replias at higher levels of softness have a higher probability for the onformational transition between two stable onformational states. Still, this transition requires some time. At these time points, interations between 0.8 and 1.4 nm were also alulated and kept onstant between updates. A reation-field ontribution was added to the eletrostati interations and fores to aount for a homogeneous medium outside the long-range utoff, using the relative permittivity of SCP water All interation energies were alulated aording to the GROMOS fore field, parameter set 53A6. 24 Fore field parameters used for GTP and 8-Br-GTP are listed in the supplementary material of Ref. 18. In this work we will fous on the optimization of a H-REMD approah in whih the modifiation of the Hamiltonian onsists of a softening of seleted interations. For this reason we have used the following form for Van der Waals and eletrostati soft-ore interations as a funtion of the interatomi distane r i : 25 E vdw r i = C 12 6 A + r C 1 i 6 6 A + r, 1 i E el r i = q iq 1 4 B + r, 2 2 i where A = vdw C 12 /C 6 2 and B = el 2. C 12 and C 6 are the Lennard Jones parameters, q i and q are the partial harges of partiles i and, and vdw and el are the softness parameters that an be set for every pair of atoms. In the urrent study we used in all simulations vdw = el =1 and the softness of the interations was ontrolled through the parameter. It an be seen that at longer distanes the soft-ore interation approximates the interation for normal atoms and that they differ mostly at short distanes between atoms. Potential energy barriers are mostly the result of short-ranged repulsion between atoms, whih an strongly be redued at higher levels of softness Fig. 2. In this study, only interations between the nuleotide base and sugar are treated using the soft-ore interation. The systems were first equilibrated for 100 ps MD at onstant pressure, where position restraints were applied on Downloaded 30 Mar 2011 to Redistribution subet to AIP liense or opyright; see

3 Optimization of replia-exhange moleular dynamis J. Chem. Phys. 127, heavy atoms of GTP/8-Br-GTP, after whih another 100 ps of equilibration at onstant pressure without any restraints were arried out. Finally, the systems were simulated for 1 ns at different values, where oordinates of the whole system were reorded every 1 ps. Ten independent 1 ns MD simulations at different levels of softness were performed for GTP =0.0,0.05,...,0.4,0.45 and 15 for 8-Br-GTP =0.0,0.05,...,0.65,0.7. B. REMD In a REMD simulation, a number of noninterating MD runs alled replias are simulated at different onditions. Let us label replias as 0,1,...,n. After a given time elementary period, t elem, an exhange between two neighboring replias is attempted, followed by another set of independent MD simulations. In the GROMOS05 implementation swithes are first attempted between pairs 0 1,2 3,... type I of replia exhange trials and after the next t elem of MD simulations swithes are subsequently attempted between pairs 1 2,3 4,... type II of replia exhange trials. This means that the effetive time between swithing attempts of the same type is twie the elementary period. In our study we used t elem =2.5 ps leading to an effetive period of 5 ps between idential swithing attempts. In ontrast to T-REMD, where the temperature is inreased to failitate the rossing of high energy barriers, we are modifying the Hamiltonian within H-REMD by making use of the soft-ore interations, Eqs. 1 and 2. This will lead to a derease of the potential energy barrier between onformations and thus also failitate the transition from one potential energy minimum into the other see Fig. 2. The proper ensemble of onformations at every value of an be obtained by applying the Metropolis riterion for the exhange probability w i, between neighboring replias running at Hamiltonians orresponding to the parameters i and, w i, = min 1,exp i,, 3 where i, = E q i E i q i + E i q E q. The oordinates q i represent a onformation that was obtained from a simulation at the Hamiltonian orresponding to parameter i and E is the potential energy alulated aording to the Hamiltonian orresponding to the parameter Eqs. 1 and 2. =1/k B T with k B as the Boltzmann onstant and T as the absolute temperature. C. Conept of double/multiple replias at the highest lambda/temperature The elementary period, t elem, in REMD should be long enough to relax the energy before the next swithing attempt. Otherwise one observes many reswithes at the next swithing attempt of the same type after a replia swith with low probability. Of ourse one an inrease t elem, but then the overall REMD effiieny dereases with dereasing number of replia swithes. Therefore, a reasonable balane will need to be struk. 4 FIG. 3. Shemati example of real REMD with three replias at max 2,0 = 2,1 = 2,2 = max and one middle 1 illustrating global onformational transitions, indiated by a diamond symbol. Crosses indiate the presene of syn onformation. Swithes between replias at 2,0 and 2,1 are attempted only at times, t, whih are a multiple of 5 t elem and between 2,1 and 2,2 if t+t elem is a multiple of 5 t elem. It ensures that a replia that swithes from 2,0 to 2,1 will spend the required time, t max =10t elem at 2,1 and 2,2 altogether. However, there are more relaxation proesses that play a role. Even at values of where the barrier has been removed or at temperatures where it is readily rossed, the system still needs time to move from one onformation to the other. Note: Even for a very small onformational barrier the average transition time is signifiantly longer than times generally used in REMD between swithing trials. This is mainly true for transitions to the less favorable onformational states. The ourrene of syn states for a set of simulations starting at anti under the onditions of the highest value or temperature and the ourrene of anti states for simulations starting from a syn onformation as a funtion of time, will typially be represented by a sigmoidal urve. transit We define the transition time t max as the time at whih this sigmoidal urve reahes saturation. The height of the urve allows us to estimate the relative populations of syn and anti at this value and from that the transition probabilities at times larger than t transit max, P anti syn max, and P syn anti max. The onformational transition time, t transit max, is for the maority of systems muh longer than t elem.ift elem would be extended to t transit max, then the effiieny gain of REMD beomes very low. In order to ombine frequent swithing trials with long enough relaxation times to allow for anti syn onformational transitions to our at max we introdue a new sheme, alled degenerated max sheme, involving multiple n replias at max or at the highest temperature for T-REMD. This an be done by degenerating max into max,0, max,1,..., max,n 1. In the swithing sheme all values are ordered as 0, 1,..., max 1, max,0, max,1,..., max,n 1, and only swithes are attempted between neighboring values. The swithing frequeny between replias at max is redued by only allowing swithing attempts after a given multiple of the elementary period Fig. 3. We define the time t max as the simulation time be- Downloaded 30 Mar 2011 to Redistribution subet to AIP liense or opyright; see

4 J. Hritz and C. Oostenbrink J. Chem. Phys. 127, FIG. 4. Shemati figure showing the appliation of degenerate s. By limiting the number of swithing attempts between replias at the same value, the systems are allowed more time to relax toward different onformations at this value of. tween the swith max,0 max,1 until the return swith max,1 max,0. Note that the swithing probability between replias at max equals 1 beause they orrespond to the same Hamiltonian and in Eq. 4 beomes 0 by definition. The proedure to find the optimal t max is desribed in Se. II D 2. The swithing trial frequeny between all other pairs of replias is muh higher with an elementary period of 2.5 or 5 ps between two swithing trials of the same type, whih maintains an effiient diffusion of replias between the lowest and highest value. An inreasing number of replias at max inreases the onvergene of onformational populations at max, whih subsequently leads to a faster onvergene of onformational populations at max 1, max 2,... and finally to the faster onvergene of populations over the whole REMD system. The REMD effiieny gain when using more replias at max omes from the fat that multiple onformational transition attempts are performed in parallel. With n replias at max, a replia that has had suffiient time to show onformational transition beomes available at max,0 with a period of 1/n 1 t max. This allows for values of n up to n max =t max /t elem +1, although the effiieny is limited in this ase beause replias tend to stay for several periods of t max at max rather than get an opportunity to swith down to max 1. For this reason, the maximal overall effiieny is obtained at n w max 1, max,0 n max. Espeially for omplex systems the required time t max at max an be high, so a higher number of replias at max an signifiantly inrease the overall REMD effiieny, almost by a fator n 1. For the examples desribed in this work, a onedimensional setup with multiple replias at max seems suffiient, but the onept an easily be extended to having a degenerate set of replias at a ertain value or temperature by going to a seond or third dimension of or T. See Fig. 4. D. Searhing for the optimal REMD settings The effiieny of REMD to sample onformational spae depends mainly on the effiieny of onformational transitions at max and an effiient diffusion of replias between the lowest and highest values. In this setion, we desribe the tools that are used to optimize the hoie of values and the number of replias at max. Setion II D 1 desribes the algorithm we propose for the fast mimiking of REMD without performing atual MD. This algorithm requires knowledge of the optimal value of max, the optimal onformational onversion time at max, t max, as well as estimates of the saturated onformational transition probabilities, P anti syn max, P syn anti max. In addition, the algorithm makes use of swithing probabilities between replias at different values. Setion II D 2 desribes the seletion of max, the alulation of t max, and onversion probabilities. Setion II D 3 desribes how swithing probabilities are sampled from prealulated probability distributions and Se. II D 4 finally desribes the approah we take to use the mimiking algorithm to obtain the most effiient settings for REMD simulations. 1. Mimiked REMD Mimiked REMD assumes a number of disreet stable onformational states exists, labeled, =0,1,...,N 1 in our ase anti and syn. In the present ase these onformations are desribed as anti and syn. Instead of performing MD we estimate the probability of a onformational transition as well as the REMD swithing probabilities and subsequently propagate the onformational states through time. This requires knowledge of the probability distributions of swithing probabilities w k, l i, for all neighboring pairs of values: i, as well as approximate probabilities of the onformational hanges at eah value from one stable onformational region to the other ones. We would like to note that if preise onformational transition probabilities would be known then no REMD mimiking is needed, as one ould easily derive the individual onformational populations as well as the free energy differene between the stable onformations. The merit of the urrent approah omes from the fat that the transition probabilities an be estimated for max fairly easily, while they are 0 for all other values. As explained in Se. II C the transition between onformational states is a dynami proess in whih the onformational transitions depend sigmoidally on time. From these urves at max, one an estimate the probability of the transition i, after the prolonged residene time, P i max. Beause max will be seleted suh that the orresponding t max is shortest see Se. II D 2 and beause for all other replias the time between swithing attempts 2t elem is muh shorter than the orresponding t,itis reasonable to assume that the probabilities of onformational transitions at any value other than max approah zero P i max 2t elem =0. In our REMD mimiking algorithm we begin by assigning starting onformational states to all replias. This an be done either randomly, sampled from the orret ensemble, or biased toward one of the states. Subsequently four steps make up the main yle of our algorithm: Downloaded 30 Mar 2011 to Redistribution subet to AIP liense or opyright; see

5 Optimization of replia-exhange moleular dynamis J. Chem. Phys. 127, Intralambda onformational transitions. We predit if the onformational state hanges during the elementary period in our ase t elem =2.5 ps at eah value by the given probabilities. As desribed above all onformational transition probabilities are set to zero for all values, exept for max, where P i max t max = P i max is nonzero one per number of yles orresponding to the t max spent at max,1,..., max,n 1. Replia exhanges of type I between 0 1,2 3,.... Aording to the atual onformational state at eah value we take the orresponding probability distributions of swithing probabilities w k, l i, for all neighboring pairs of type I. A swithing probability is ran- domly hosen from the distribution of swithing probabilities w k, l i, and a seond random number de- termines if the swith ours. Intralambda onformational transitions. The same as the first step. 4 Replia exhanges of type II between 1 2, 3 4,.... The same as the seond step but now for the pairs of type II. The length of the whole yle orresponds to the double of elementary swithing attempt period in our ase ps=5 ps. Sampling of a half million yles orresponding to 10 6 t elem of real REMD takes typially 15 min depending on the exat number of replias using an unoptimized python sript. 2. Seletion of max The average onformational transition time at the unperturbed Hamiltonian =0.0, t 0.0 is by far too long to transit sample suffiient transitions reversibly. In H-REMD, the Hamiltonian is parameterized suh that the potential energy barrier assoiated to the onformational transition is redued for higher values of, thereby also reduing t transit i. To obtain the optimal value of max, we perform ten short MD simulations 200 ps starting from anti and 10 short MD runs starting from syn onformation at different values for GTP =0.4;0.45;0.5 and for 8-Br-GTP =0.6;0.65;0.7;0.8;0.9. Note: max values lower than 0.4 respetively. 0.6 were not onsidered based on the results of 1 ns runs at different values used for the alulation of swithing probability distributions as desribed in the next paragraph. Our aim is to find the value of t max for whih we would get the highest number of onformational transitions at max during a given length of REMD simulation see also Se. II C. While the time dependeny of the number of simulations where the onformation has hanged with respet to the initial onformation has an S-urve profile, it is lear that the most effiient t max will be in the interval between the midpoint of the S urve and the saturated region. Quantitatively t max is obtained as the time orresponding to the maximum of the number of onformational hanges divided by time. Beause these maxima our at different times for the anti syn transition and the syn anti transitions, and beause we need to have a large enough number of both types of transitions, we selet the longer time of both transitions at one max value. We selet the value of max for whih t max is shortest and where enough transitions our in both diretions. The set of ten MD runs starting from different onformations is subsequently prolonged for the seleted max, in order to refine the onverged values of the onformational onversion probabilities P anti syn max and P syn anti max. Beause the exat shape of the S urve is not properly onverged from only ten simulations, it is not wise to diretly read P transit max t max from these urves. Rather, we set P transit max t max = P transit max, a value whih onverges also for a limited number of simulations. Using P transit max also ensures the generating the proper onformational populations at max within the mimiked REMD. 3. Generation of probability distributions of swithing probabilities Let us onsider the system, whih has several different stable onformational sates syn, anti, for whih standard MD simulations do not produe enough transitions due to the onformational barriers between these states. We assume that the onformational spae within eah of these stable regions is sampled reasonably well by a relatively short MD simulation 1 ns. The desription below follows the shemati representation in Fig. 5. To obtain the probability distributions of the swithing probabilities needed for the REMD mimiking we performed a set of 1 ns MD simulations starting from different stable onformers at different values of alternatively, one ould use different temperatures between 0.0 and max usually at inrements of 0.05 in priniple one an use also REMD at these values for this purpose. Every 1 ps we store one onformational onfiguration frame leading to 1000 different onfigurations. It is possible that during these MD simulations onformational transitions our. For this reason, we ollet onformations belonging to the same stable onformational region and the same value of at whih the simulation was performed into one set of onformations q, = 0,0.05,0.1,..., max ; =0,1,...,N 1. For every ombination of and we obtain about 1000 different strutures. The rth strutural frame from this traetory is expressed as q t r. For eah of these strutures we alulate its energy not only using the Hamiltonian at whih it was simulated represented by i, but also the energy aording to the Hamiltonian orresponding to all other values, and expressed as: E q i t r. For eah pair of onfigurations q k i t r ; q l t s we alulate swithing energy differene k, l i, t r,t s = E q k i t r E i q k i t r + E i q l t s E q l t s and the orresponding swithing probability 5 Downloaded 30 Mar 2011 to Redistribution subet to AIP liense or opyright; see

6 J. Hritz and C. Oostenbrink J. Chem. Phys. 127, FIG. 5. Sheme desribing the steps for obtaining the probability distribution of the swithing probabilities. w k, l i, t r,t s = min 1,exp k, l i, t r,t s, 6 where =1/k B T, k B is the Boltzmann onstant, and T is the absolute temperature. The distributions of k, l i, and w k, l i, values are thus alulated from altogether 10 6 values, whih are marked as k, l syn,anti i, and w 0.0,0.3 after normalization. This gives us the probability distribution for a randomly hosen pair of onfigurations, q k i, q l with the swithing probability between them given by w k, l i,. For the REMD mimiking we found it onvenient to produe Cum_ w k, l i,, whih ontains the umulants of the probability distribution of swithing prob- abilities. 4. Searhing algorithm for the parameters of the most effiient REMD yles of REMD mimiking orresponding to 10 6 t elem of real REMD, in our ase 2.5 s takes 15 min whih allows us to perform REMD for hundreds of different ombinations of values and numbers of replias at max produing time sequenes of the ourrene of onformational states at different w syn,anti 0.0,0.3 =min 1,exp syn,anti 0.0,0.3 values. From these one an deide for different riteria from whih to prefer one set of parameters over the other. For ases where different onformational states an be defined e.g., the anti and syn states, we propose to measure effiieny of REMD, through the number of global onformational transitions, N gt. We ount the number of onformational transitions for eah partiular replia at the lowest value, 0. The shemati REMD example shown in Fig. 3, ontains three replias at max 2,0 = 2,1 = 2,2 = max and one middle value of lambda, 1. The replia represented by a solid urve, is in the anti onformation at 0 after 103 t elem and then exhanges to higher values. At 2,1 it makes the transition to the syn onformation after 119 t elem and subsequently exhanges with replias at lower values until it finally returns to 0, but now in the different onformational state syn as before. The ombination of replia exhanges and a onformational transition is ounted as one global anti syn transition. From this moment onwards, we will monitor for the next syn anti global onformational transition for the same replia, et. From this desription one an see that for a global onformational transition to our it is not neessary for the replia to reah max, beause the onformational transition an our also at lower values see, e.g., in Fig. 3 the replia represented by a dotted line. At 131 t elem an anti syn transition ours at 1 leading to the global onformational transition at 133 t elem or even at 0 within real REMD. This is not the ase for the mimiked REMD, where onformational transition an our only at max. On the other hand, the replia represented by the dotted urve is in a syn onformation at 0 at 102 t elem, then went up to max where it hanges its onformation to anti and later bak to syn again. When it exhanges bak to 0 at 119 t elem, this event is not ounted as a global onformational transition, beause the onformations at 102 t elem and 119 t elem are the same. In the ase of MD as a speial ase of REMD with only one replia at 0 N gt is equal to the number of onformational transitions within one MD run. This allows for a diret omparison of the effiieny between very different REMDs or MDs performed under different onditions. Our aim is to find the REMD settings whih give us the maximum value of N gt per CPU. In the following, we assume that in real REMD simulations every replia is assigned one CPU. Having deided on the optimal value of max and t max we vary: 1 the number and plaement of middle s between 0.0 and max and 2 the number of replias at max. Whether it is effiient to invest into more replias at max or not depends on the ratio of t max relative to the average time needed for global onformational transitions to our. If more time is needed for a onformational hange at max, it beomes more likely that an inreased number of replias at max improves the overall effiieny per CPU. Downloaded 30 Mar 2011 to Redistribution subet to AIP liense or opyright; see

7 Optimization of replia-exhange moleular dynamis J. Chem. Phys. 127, In ases where only a few replias are needed one an test various settings in a systemati manner and selet the optimal ombination. In spite of the effiieny of mimiked REMD, a larger number of required replias will make the optimization problem more omplex. We propose three useful approahes: The number and plaement of values between 0.0 and max middle s and the number of replias at max an be optimized independently. Middle s are essential for the diffusion of replias between 0.0 and max, whih will be the same for any number of replias at max. The optimal setting of middle s will therefore be independent of the number of replias at max. However, the gain in effiieny due to optimization of the middle s is more pronouned if the onformational hange at max is not the bottlenek of the simulation. Therefore, we propose to searh for the optimal settings with a high number of replias at max, initially 5 in the ase of 8-Br-GTP. After this initial optimization, the number of optimal replias at max should be obtained after whih one should hek if having one more middle does not improve results even further. One the optimal number of replias at max has been obtained for an optimal set of m middle values, this n opt m an be used to redue the searhing possibilities for different numbers of middle values by taking into aount the following inequalities: n opt k m n opt m and n opt k m n opt m. Searhing for the optimal settings of middle s. When expanding or reduing the number of middle s in the optimization proess, we do not need to onsider the omplete range 0.0, max for the new values. If we denote the optimal values in a sheme with n middle s as n opt i n opt 0 =0.0; n opt n+1 = max, then it is most likely that in a sheme with n+1 middle s, the optimal values are in the following range: n+1 opt i n opt i 1, n opt i. Similarly, one an generally write that for a redution of the number of middle s, the optimal values are restrited to n opt i n+1 opt i, n+1 opt i+1. In many ases, already a short simulation indiates if a ertain set of settings is promising or not. In an approah we all ontinuous filtering we disregard possible settings on the fly, thereby only spending omputer time on the most promising settings. We perform a mimiked REMD simulation for all potentially relevant REMD settings for yles. We subsequently disregard all settings that show less than 10% of the at that point maximum value of N gt. After yles this threshold is inreased to 20% and by another 10% every yles until after yles only those settings are kept that lead to 80% of the maximum value of N gt.we subsequently refine the searh by inreasing the threshold by 2.5% every yles and selet the optimal setting after yles. FIG. 6. a Number of syn onformations observed in a set of ten MD simulations of GTP as funtion of time at different values starting from ten different anti and syn onformations. Runs for =0.45 were prolonged up to 400 ps in order to reah onvergene. b Population of onverted onformations per time syn anti or anti syn. For larity, running averages over 20 time points have been taken. III. RESULTS Both GTP and 8-Br-GTP have two stable onformations: anti and syn, depending on the dihedral angle around the glyosidi bond Fig. 1. Based on the distributions of this dihedral angle we define for GTP to be in the syn onformation, when its dihedral angle around the glyosidi bond is within the interval 25,150, and otherwise to be in the anti onformation. For 8-Br-GTP the syn onformational interval is 35, A. Seletion of max MD simulations of GTP in expliit water were performed for 200 ps at different values of 0.4; 0.45; 0.5. Ten simulations started from anti and ten simulations started from syn onformations. During the MD runs onformational transitions our. Figure 6 a show the number of simulations in whih GTP is in a syn onformation as funtion of time, when starting from anti and when starting from syn onformation. We obtained t max as the larger of two times orresponding to the maximas of the time dependene of the number of hanged onformations divided by time. From Fig. 6 b follows: t 0.4 =155 ps, t 0.45 =100 ps, and t 0.5 =135 ps. As the optimal value of max we hoose 0.45 beause it gives the shortest t max 100 ps together with a suffiient number of onformational transitions after this time. Another advantage of max =0.45 over 0.5 is the more effiient diffusion of replias between 0 and max. For max =0.45, the set of ten MD simulations starting from syn and anti was prolonged leading Downloaded 30 Mar 2011 to Redistribution subet to AIP liense or opyright; see

8 J. Hritz and C. Oostenbrink J. Chem. Phys. 127, FIG. 8. Probability distributions of swithing energy differenes for GTP: i, 0.0,0.3= E0.3 q i 0.0 r E 0.0 q i 0.0 r +E 0.0 q 0.3 s E 0.3 q 0.3 s, where i, are different onformational states: syn, anti and r, s are different onfigurations from MD traetories. FIG. 7. Color a Number of syn onformations observed in a set of ten MD simulations of 8-Br-GTP as funtion of time at different values starting from ten different anti and syn onformations. Runs for =0.7 and =0.8 were prolonged up to 400 ps in order to reah onvergene. b Population of onverted onformations per time syn anti or anti syn. For larity, running averages over 20 time points have been taken. to estimate of the onverged onformational transition probabilities P anti syn =0.45 =0.3 and P syn anti =0.45 =0.7 Fig. 6 a. The same kind of analysis was performed for 8-Br-GTP using =0.6;0.65;0.7;0.8;0.9 and leading to the population time dependenes shown in Figs. 7 a and 7 b. We onsidered as max andidates max =0.7 and max =0.8 for whih the prolonged MD simulations were performed. Based on these we deided to take max =0.7 for 8-Br-GTP with the orresponding t max =100 ps and saturated transition probabilities P anti syn =0.7 =0.2 and P syn anti =0.7 =0.8. For =0.9 the transition syn anti is quite effiient. However, the onversion probability from anti to syn has beome very low whih means that these transitions would our only very rarely at =0.9. examples of the swithing energy differenes and swithing probability distributions are given in Figs. 8 and 9. C. REMD mimiking REMD mimiking as desribed in the Se. II D 1 produes as output the time evolutions of onformational states anti and syn at all values that are inluded. Beause of its speed we an run mimiked REMD for a very long time and thus obtain onverged populations of anti and syn onformations for GTP and 8-Br-GTP. Typially, we simulate yles of mimiked REMD orresponding to 10 6 t elem in real REMD 2.5 s. Syn populations at individual values of mimiked REMD for GTP yles with two replias at max =0.45 ombined with no or a single middle value are shown in Table I. Theoretially, syn populations at =0.0 should be the same for all mimiked REMDs. We do not suffer here from insuffiient lengths of REMD simulations, but inauraies rather stem from the probability distributions of swithing probabilities. Disrepanies in the syn B. Generation of probability distribution of swithing probabilities To obtain the probability distributions of swithing probabilities we performed MD simulations for 8-Br- GTP at s in the interval 0.0, max with a inrement of For eah two MD simulations of 1 ns were performed, one starting from an anti and the seond from a syn onformation. Configurations of traetories were stored every ps leading to 1000 frames for eah traetory. Configurations have been separated into sets aording their onformational state q anti, q syn. The distributions for the swithing energy differenes k, l i, as well as for the swithing probabilities w k, l i, and their umulative values were obtained as desribed in Se. II D 3 and shematially outlined in Fig. 5. Representative FIG. 9. Probability distribution of replia exhange swithing probabilities for GTP: w syn,anti 0.0,0.3 =min 1,exp syn,anti 0.0,0.3 for GTP. Notie that probability w syn,anti 0.0,0.3 is muh lower than for the opposite replia swith w anti,syn 0.0,0.3. Downloaded 30 Mar 2011 to Redistribution subet to AIP liense or opyright; see

9 Optimization of replia-exhange moleular dynamis J. Chem. Phys. 127, TABLE I. Syn populations at 0, 1 1, max,0, max,1 depending on the plaement of 1 1 from mimiked REMD of GTP with two replias at max =0.45 over yles. 1 1 None syn syn syn max, syn max, populations at =0.0 an be explained from an insuffiient sampling during the 1 ns MD simulation from whih the probabilities are obtained. For values that are evident outliers ompared to the other values, this simulation should probably be extended. Largest deviations are expeted for REMDs involving middle values, for whih we multiply very small and very high probabilities i.e., 1 1 =0.05 and 1 1 =0.4. The root-mean-square deviation of syn at 0 divided by its average value when exluding the values obtained with 1 1 =0.05 and 1 1 =0.4 is alulated to be indiating the relative inauray in the distributions of swithing probabilities. The average of syn populations at 0 for runs with 1 1 between 0.1 and 0.35 amounts to syn aver =0.048, whih orresponds to a free energy differene G syn-anti GTP =7.40 kj mol 1 at 298 K. As shown in Ref. 18 the obtained population of syn onformations from real H-REMD was GTP for GTP orresponding to a value of G syn-anti =7.63 kj mol yles of mimiking REMD using the same settings as the real REMD reported there =0.0;0.25,0.45,0.45,0.45,0.45 gives syn =0.043, orresponding to G syn-anti =7.69 kj mol 1. It shows that if most GTP of the transitions an indeed our only at max, then REMD mimiking an produe very reasonable values of populations of the individual onformational states. Therefore, REMD mimiking is not only useful as a tool for the optimization of REMD settings but an also as be used in itself for alulating the populations of onformations. We an see an analogy between our method and a free energy method that makes use of alternative pathways to alulate free energy differenes more effiiently along unphysial thermodynami yles. 26 We want to note, however, that mimiked REMD overs very many of suh pathways simultaneously. D. Finding the optimal settings for REMD In the previous paragraph we demonstrated how long mimiked REMD an approximate syn and anti populations for GTP and 8-Br-GTP. These simulations would orrespond to several s of real REMD. In real REMD, however, we an afford to simulate only limited time lengths tens ns, therefore it is ruial to find settings whih orresponds to the highest possible REMD effiieny. We have used mimiked REMD, beause it allows us to attempt REMD for hundreds of different ombinations of s. From the time sequenes of onformational states at different values we an hoose the ombination of s that produes the highest number of global onformational transitions, N gt, whih will ensure the fastest onvergene of populations at GTP Mimiked REMD yles without middle replia and two replias at max =0.45 gives 5221 global onformational transitions, whih is 5221/ 3 = 1740 global onformational transitions per CPU. The results for mimiked REMD with an inreasing number of replias at max =0.45 are summarized in the Table II. It shows that the highest effiieny per CPU 7506/4=1877 is obtained for three replias at max. In the next step we performed mimiked REMD with different numbers of replias at max =0.45 and one middle -value ,0.1,0.15,0.2,0.25,0.3,0.35,0.4 for yles Fig. 10 a. With two replias at t max =10t elem the highest values of N gt are obtained for 1 1 = , 1 1 = , and for 1 1 = , whih are higher than the values of N gt obtained from REMD without middle having the same two replias at max =0.45. It means that putting one middle 1 1 an inrease the effiieny of diffusion between 0 =0.0 and max =0.45. The effiieny per CPU for mimiked REMD ontaining one middle 1 1 =0.25, however, is lower 1522 ompared to Still the effiieny per CPU may be higher for REMD ontaining one middle lambda in ombination with a higher number of replias at max. Figure 10 presents the effiieny of REMD ontaining one middle lambda with an inreasing number of replias at t elem. It reveals that per CPU the most effiient set of s is 0.0, 0.25, 0.45, 0.45, 0.45, 0.45 with 12646/ 6 = 2108 global onformational transitions per CPU. It also shows that the differenes due to the plaement of 1 1 are more pronouned for settings with a higher number of replias at max. Beause the optimal REMD setting ontaining one 1 1 is more effiient per CPU than a set of s without any middle value, we ontinued to test REMD ontaining two middle values. As is explained in the methods Se. II D 4 the optimal setting of REMD with two middle s will still have t+t elem replias at max =0.45, beause this is the optimal TABLE II. Number of global onformational transitions, N gt, during mimiked REMD of GTP with n=2,3,4,5 replias at max =0.45 and no middle value after yles. n replias at max = N gt N gt /CPU Downloaded 30 Mar 2011 to Redistribution subet to AIP liense or opyright; see

10 J. Hritz and C. Oostenbrink J. Chem. Phys. 127, FIG. 10. Number of global onformational transitions, N gt observed in yles of mimiked REMD of GTP with one middle replia 1 1 at different positions and with different numbers of replias at max =0.45: a absolute value and b value per CPU. setting of REMD with one middle value. Table III shows N gt per CPU for n=4,5,6 and two additional values 2 1 and 2 2. It shows that the optimal setting for REMD in this ase is 0.0, 0.15, 0.3, 0.45, 0.45, 0.45, 0.45, However, its effiieny is slightly lower than for the optimal setting using only one middle ompared 2108 of global onformational transitions per CPU. In onlusion the most effetive setting predited by mimiked REMD is 0.0, 0.25, 0.45, 0.45, 0.45, 0.45 with 12646/ 6 = 2108 global onformational transitions per CPU. This number orresponds to REMD mimiking yles or 10 6 elementary swithing periods of real REMD. This means that one an expet 13 global onformational transitions per 1000 t elem of real REMD, whih amounts to 2.5 ns with t elem =2.5 ps. From an extrapolation of the linear inrease of N gt as a funtion of time, over ten individual 5000 yles mimiked REMD simulations we estimate the REMD equilibration period to be about 200 t elem 0.5 ns in real REMD. For omparison, in real REMD we reported that N gt inreases by 17 N gt per 1000 t elem and that a REMD equilibration of 249 t elem was required Br-GTP 8-Br-GTP has a higher onformational barrier between the anti and syn onformations than GTP, therefore we have found a higher value of max =0.7 and the optimal REMD will probably require more middle replias. For this reason we will not use a systemati searh for the optimal settings as for GTP, but rather use a more effiient approah as desribed in Se. II D 4. We initiate the optimization of the middle s using five replias at max =0.7 and obtain 1 opt 1 =0.5 as the optimal value if only one middle value of is taken into aount, with N gt =1079 per CPU Table IV. When inreasing the number of middle values to two, we restrited the searhing ranges to ,0.5 and ,0.65. Similarly the optimizations were extended to three and four middle values. The N gt per CPU for different numbers of middle values in ombination with five replias at max are summarized in Table IV, where arrows indiate the steps in the optimization. As an be seen from this table, the optimal setting ontaining four middle values in ombination with five replias at max is less effiient 1252 global onformational transitions per CPU than the optimal three middle -values setting Beause this may still be aused by too few onformational transitions at max, the same optimal settings of four middle values 0.2; 0.4; 0.55; 0.65 was tested in ombination with n=6,7,8 and indeed for n opt m=4 =7 we obtained a more effiient setting 1315 N gt per CPU. Taking into aount the inequalities: n opt m 4 n opt m=4 =7 and n opt m 4 n opt m=4 =7 that are desribed in Se. II D 4, the optimization proedure was ontinued by inluding five middle values in ombination with n=7 and subsequently with n=8. It appears that n opt m=5 =7, but that its effiieny is lower 1305 than for the optimal four middle -values setting 1315 N gt per CPU, meaning that the overall optimal sheme has 4 middle values with n 7. Beause the optimal three middle -values setting n opt m=3 =6 is also not giving a higher effiieny, the overall most effiient setting for 8-Br-GTP has thus been determined as 4 opt i 0.0;0.2;0.4;0.55;0.65;0.7;0.7;0.7;0.7;0.7;0.7;0.7. N gt per CPU 1315 is lower for 8-Br-GTP than for GTP 2108 indiating that REMD of 8-Br-GTP will be omputationally more demanding. Using the optimal set for 8-Br- GTP of 12 replias we estimate 16 global onformational transitions during 1000 t elem of real REMD 2.5 ns and the equilibration period to be 180 t elem based on a linear extrapolation of time dependene of N gt as funtion of time from ten individual runs of 5000 mimiked REMD yles. IV. DISCUSSION The present study on the GTP/8-Br-GTP two state model systems deepens our understanding about REMD effiieny. For simpliity, we assumed that onformational transitions our only at the highest lambda, max in the ase of H-REMD, or T max for T-REMD. To obtain a different onformational state for a given replia at 0, the replia has to diffuse from 0 through the middle values to max, where a onformational hange has to our. The system subsequently needs to diffuse from max bak to 0. Suh proess we named a global onformational transition. The advantage Downloaded 30 Mar 2011 to Redistribution subet to AIP liense or opyright; see

11 Optimization of replia-exhange moleular dynamis J. Chem. Phys. 127, TABLE III. Number of global onformational transitions, N gt per CPU for mimiked REMD of GTP using n=4,5,6 replias at max =0.45 and two additional -values 2 1, 2 2. The maximum is obtained for 2 1 =0.15 and 2 2 =0.3, at n=5. n= n= n= of monitoring the N gt ompared to the number of roundtrips between the lowest and highest temperature as proposed by Trebst et al omes from the fat that many roundtrips do not neessarily ensure any onformational hanges at 0. For example for GTP, we have observed simulations in whih there were many roundtrips thanks to relatively high values of w anti,anti i,, while w anti,syn i, were very low. In suh ases a syn onformation that ours at max will not be able to effiiently diffuse down to 0. Another advantage of ounting N gt is that if P max transit t elem 0 it does not require a replia to diffuse all the way up to max beause a onformational transition within real REMD an already our at any value. The disadvantage of monitoring N gt is that defined stable onformational states are required. We desribed the approah for finding the optimal max at whih onformational transitions our in suffiiently short time, whih is, however, typially still muh longer than the elementary swithing period, t elem, between swithing attempts. The ourrene of onformational transitions shows a sigmoidal time dependeny, from whih a relatively long t transit max, t max an be estimated. The presented approah using n multiple replias at max allows several replias to TABLE IV. Number of global transitions, N gt per CPU for the mimiked REMD of 8-Br-GTP, having optimal set of m middle lambdas m i opt ombined with n replias at max =0.7. The maximum is obtained for m=4 and n=7. The arrows indiate the sequene of steps during the optimization proedure. m Set of m opt i None , ,0.5, ,0.4,0.55, ,0.4,0.45,0.55,0.65 N gt /CPU, n= N gt /CPU, n= N gt /CPU, n= N gt /CPU, n= Downloaded 30 Mar 2011 to Redistribution subet to AIP liense or opyright; see