Multiple scaling replica exchange for the

THE JOURNAL OF CHEMICAL PHYSICS 127, 234102 共2007兲

Multiple scaling replica exchange for the conformational sampling of biomolecules in explicit water Hiqmet Kamberaj and Arjan van der Vaarta兲 Center for Biological Physics and Department of Chemistry and Biochemistry, Arizona State University, Tempe, Arizona 85287, USA

共Received 10 September 2007; accepted 16 October 2007; published online 18 December 2007兲 A multiple scaling replica exchange method for the efficient conformational sampling of biomolecular systems in explicit solvent is presented. The method is a combination of the replica exchange with solute tempering 共REST兲 technique and a Tsallis biasing potential. The Tsallis biasing increases the sampling efficiency, while the REST minimizes the number of replicas needed. Unbiased statistics can be obtained by reweighting of the data using a weighted histogram analysis technique. The method is illustrated by its application to a ten residue peptide in explicit water. © 2007 American Institute of Physics. 关DOI: 10.1063/1.2806930兴 I. INTRODUCTION

Molecular dynamics 共MD兲 is a powerful method to sample the conformational space of a molecular system. By integrating Newton’s equations of motion, the technique also provides the time-dependent behavior and evolution of the system. Molecular dynamics methods are commonly used for biomolecular systems and have provided a significant contribution to the understanding of the link between protein structure, dynamics, and function.1–3 Unfortunately, many interesting dynamical properties of large biological molecules cannot be studied using standard MD due to time scale limitations of the method 共up to tens of nanoseconds for large systems兲.4 Proteins are characterized by complex energy landscape topologies with large numbers of energy minima; in addition, their dynamics involve motions at different time scales.5–7 Collective motions or correlated fluctuations involving large portions of the structure are essentially important for the biological function of proteins.7–10 These correlated motions can act on microsecond to millisecond time scales and must be appropriately handled in molecular simulations. Thus, the development of numerical methods that are capable to efficiently determine the structure and population of all relevant conformations of the system of interest is a major goal of computational biochemistry. Significant efforts have been made to develop new atomistic MD methods for exploring slow conformation transitions.11,12 These methods can broadly be divided into two classes: one that employs multiple time step integrators13–16 and one that enhances the rate of barrier crossing events.12,17–28 Using multiple time step methods, disparate mass systems can be subdivided into slow and fast degrees of freedom, the fast oscillating degrees of freedom can be dissolved in a soft bath with slow varying degrees of freedom, or the forces can be split into slow and fast varying components.13 Methods that enhance barrier crossing use poa兲

Author to whom correspondence should be addressed. Electronic mail: [email protected]

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tential energy function transformations or perform random walks in energy space by generalized ensemble algorithms, which allow an efficient sampling of the configurational space and acceleration of barrier transitions.12,17–28 Examples of barrier crossing rate enhancing methods include hyperdynamics,20,21 conformational flooding,19 22 26 metadynamics, accelerated MD, umbrella sampling,17 self-guided MD,23 targeted MD,27,28 transition path sampling,29,30 and the replica exchange method 共REM兲.24,31–33 The REM is particularly promising for overcoming problems of quasi-ergodicity in the sampling of biomolecular systems. In the method, multiple copies of the system 共replicas兲 are run independently at different temperatures. At regular time intervals, the coordinates of the replicas are swapped, based on an energy criterion that preserves detailed balance. The high temperature replicas are used to enhance the barrier crossing and the low temperature replicas to sample the potential energy basins. The number of replicas needed scales as the square root of the number of degrees of freedom in the system.34 A large number of replicas require an increase of the simulation time to maintain a satisfactory rate of round trips between the two extremal points along the replicated coordinate 共e.g., temperature兲.35 These increases in the number of replicas and the simulation time for each replica restrict REM to small systems. Efforts have been made to avoid the increase in the number of replicas, for example, by eliminating the solvent degrees of freedom through the use of an implicit solvent model.36–39 Other approaches include the use of hybrid explicit/implicit solvent models, where each replica is simulated with explicit solvent models that are partially replaced by a continuum for the exchange criterion,40 and the use of separate heat baths at different temperatures for the solute and solvent, such that only the solute temperature varies along the replicas.41 These reductions in system size reduce the number of replicas, but it is not clear whether the methods accurately describe the structure and dynamics of the system.37–39,42 Fukunishi et al.34 introduced the so-called

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H. Kamberaj and A. van der Vaart

Hamiltonian REM in which the replicas are characterized by different strengths of hydrophobic interactions and in which a phantom chain is used that allows for various degrees of atomic overlap. A REM with specifically designed backbone dihedral biasing potentials also uses a limited number of replicas;43 however, this method requires prior knowledge of the backbone dihedral angle distribution, which might be computationally expensive to calculate for large peptides or proteins. In resolution REM,44,45 a chosen subset of configurational coordinates of a coarse grained model is exchanged in addition to the temperature; unfortunately, this method is restricted to implicit solvent models.44 Liu et al.42 proposed a promising replica exchange with solute tempering 共REST兲 method to overcome the poor scaling with system size. In their method, each replica m is subject to a scaled potential Em共Xm兲 = E p共Xm兲 + 共␤0 / ␤m兲Eww共Xm兲 + 共共␤0 + ␤m兲 / 2␤m兲E pw共Xm兲, with E p, Eww, and E pw the solute, solvent-solvent, and solute-solvent interaction potential energies, respectively, ␤m = 1 / kBTm with kB the Boltzmann constant and Tm the temperature of replica m, T0 the desired temperature, and Xm the configurational phase space vector. The number of replicas is significantly reduced due to the cancellation of the water self-interaction term in the replica exchange acceptance probability.42 A physical explanation for the increase of the acceptance probability for the exchange can be found from an analysis of the REST energy. Since ␤0 ⬎ ␤m, the scaling factors for E pw and Eww are greater than 1; therefore, the barriers in the Eww and E pw terms increase. This barrier increase suppresses large conformational changes in the solvent. Since the solvent has the largest number of degrees of freedom, the energy difference between replicas is decreased. Here, we combine REST with the Tsallis biasing potential46–52 yielding the multiple scaling REST 共MREST兲 method. Although Tsallis statistics lead to nonextensive entropy and other problems,53–57 the method can be used in biasing MD to increase the sampled phase space.50,51 Here, we merely use the Tsallis potential to bias the dynamics within the REST framework; all statistics from these biased simulations will subsequently be reweighted, such that proper unbiased properties corresponding to the Boltzmann distribution are obtained. The method is similar but not identical to the REST method of Liu et al.;42 we sample a nonBoltzmann configurational distribution, which increases the rate of high energy barrier crossings. To test the method, we present simulations of an amyloid peptide in explicit water and compare the results with those of the REST. Our comparison shows that the MREST provides a more efficient sampling of the conformational space than the REST method. Recent work shows that the REST may not work for every system: problems may occur when the energy gap between different replicas is too large 共not unlike the cause of problems in normal REM simulations兲.58 To solve for these problems, the interaction energy between the protein and the first solvent shell could be used for E pw.42,58 This would require monitoring the number of water molecules which leave and enter the first hydration shell. Since we had no problems with the REST in our application, such monitoring was not

pursued in the current method. Moreover, we expect that the chance of such problems is reduced in MREST because the energy gap is decreased in MREST.

II. COMPUTATIONAL METHODS A. Multiple scaling replica exchange

In combining REST with the Tsallis biasing potential, the configurational state of the mth replica is characterized by 共Xm , ˜Em共Xm ; qm , ␤m兲 , ␤m兲 with ˜E共X ;q ,T 兲 m m m =

冋

冉

冉冊

␤0 qm ln 1 + ␤m共qm − 1兲 E p共Xm兲 + ␤m共qm − 1兲 ␤m ⫻Eww共Xm兲 +

冉

冊

␤0 + ␤m E pw共Xm兲 − ␧ 2␤m

冊册

.

共1兲

Xm are the coordinates, ␤m is the inverse temperature of replica m, and qm the Tsallis weight of replica m; E p is the protein internal energy, Eww the solvent-solvent interaction energy, E pw the protein-solvent interaction energy, ␧ is equal to the global minimum potential energy, and ␤0 is the inverse target temperature. The acceptance probability for the exchange between replicas m and n follows from exp共−⌬Enm兲, where42 ˜ 共X ;q ,T 兲 − ˜E 共X ;q ,T 兲兲 ⌬Enm = ␤m共E m n m m m m m m ˜ 共X ;q ,T 兲 − ˜E 共X ;q ,T 兲兲. + ␤n共E n m n n n n n n

共2兲

Note that if qm = qn = 1, the standard REST method is obtained. For q ⬎ 1, the magnitude of the force close to the barrier regions is decreased.50,51 Thus, close to the barriers, the particles experience less resistance, which increases the probability of barrier crossings. The largest value of q is limited by the stability of the algorithm; therefore, a preliminary investigation is needed for each particular system. We found that the upper value of q is q ⬇ 1 + 1 / f 共where f is the number of degrees of freedom兲, as also suggested in Ref. 49. Although contrary to REST, Eww does not cancel in MREST, its magnitude is suppressed due to the ␤0 / ␤m factor 关Eq. 共1兲兴. The Tsallis biasing potential can also be used on a selected number of degrees of freedom, for example, certain dihedral angles. In that case, the protein internal energy equals ˜E = E⬘ + ˜E p p,dihe-cmap p = E p,bond + E p,angle + E p,vdW + E p,elec + ˜E p,dihe-cmap ,

共3兲

where E p,bond, E p,angle, E p,vdW, and E p,elec are the bond stretching, bend angle, van der Waals, and electrostatic potential energy terms, respectively. ˜E p,dihe-cmap is the Tsallis biasing potential of the proper and backbone dihedral angle terms, ˜E p,dihe−cmap =

q ln关1 + ␤共q − 1兲共E p,dihe + E p,cmap − ␧兲兴, ␤共q − 1兲共4兲

with ␧ the minimum energy of the sum E p,dihe + E p,cmap, and


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E p,cmap the CHARMM CMAP energy term for the protein backbone.59 The exchange between replicas m and n is then based on the effective energy difference, ⌬Enm = 共␤n − ␤m兲关共E⬘p共Xm兲 + 21 E pw共Xm兲兲

M 关␦2⍀ml兴−1⍀ml 兺m=1 M 兺m=1 关␦2⍀ml兴−1

␦ 2⍀ l =

− 共E⬘p共Xn兲 + 21 E pw共Xn兲兲兴

共10兲

,

1

共11兲

. M 兺m=1 关␦2⍀ml兴−1

Then, for MREST,

˜ + ␤m共E p,dihe−cmap共Xn ;qm,Tm兲

␦2⍀ml = ␦2Hml

− ˜E p,dihe−cmap共Xm ;qm,Tm兲兲 ˜ + ␤n共E p,dihe−cmap共Xm ;qn,Tn兲 − ˜E p,dihe−cmap共Xn ;qn,Tn兲兲.

⍀l =

共5兲

Similar to REST, this difference does not include Eww.

冋

q Zm exp共␤m˜Eml兲 ⌬ENm

册

2

,

共12兲

with ␦2Hml ⬇ gml具Hml典共Ref. 62兲 and gml the statistical inefficiency. The statistical inefficiency is obtained from gml = 1 + 2␶ml,62 where ␶ml is the correlation time of the simulation in bin l from replica m. The count expectation 具Hml典 is given by −q Nm⍀l exp共− ␤m˜Eml兲⌬E, 具Hml典 = Nm p共El ; ␤m,q兲⌬E = Zm

共13兲

B. Weighted histogram analysis method for MREST

With the weighted histogram analysis method 共WHAM兲, thermodynamic properties of the unbiased system at a desired temperature can be calculated from multiple biased simulations60 at one or more temperatures.61 In MREST, the configurational probability density for each replica m is given by p共E; ␤m,q兲 =

1 q Zm

⍀共E兲关1 + ␤m共q − 1兲共E − ␧兲兴q/共1−q兲 ,

共6兲

where E = E p + Eww + E pw is the unbiased energy, ⍀共E兲 is the potential energy density of states, and the configurational partition function 共Zm兲 is given by Zm =

冕

dE⍀共E兲关1 + ␤m共q − 1兲共E − ␧兲兴1/共1−q兲 .

共7兲

and thus, ⍀l =

−1 M 兺m=1 gml Hml −1 M q 兺m=1 gml Nm⌬EZm exp共− ␤m˜Eml兲

共14兲

.

Equations 共7兲 and 共14兲 are solved iteratively starting from an q q , e.g., Zm = 1. The uncertainty on the arbitrary choice of Zm density of state ⍀l is obtained from

␦ 2⍀ l =

⍀l −1 M q 兺m=1 gml Nm⌬EZm

exp共− ␤m˜El兲

.

共15兲

In practice, the application of the MREST-WHAM for the REST method can be considered a special case of the same algorithm with q = 1. C. Implementation

The canonical distribution at the desired inverse temperature ␤0 can be constructed by reweighting Eq. 共6兲 as p共E; ␤0兲 = f m p共E; ␤m,q兲exp共␤m˜Em − ␤0E兲,

共8兲

q / Z 0. where ˜Em is the biased energy of replica m and f m = Zm Suppose that we performed M simulations at different temperatures Tm, m = 1 , . . . , M 共or equivalently ␤m兲 and different Tsallis scaling factors qm. We discretize E p, Eww, and E pw into Nb1, Nb2, and Nb3 bins, respectively, each with a spacing of ⌬E. To simplify the notation, we will use the index l = k + Nb1 · 共n + Nb2 · 共r − 1兲 − 1兲. The unbiased probability pm共El ; ␤0兲 of replica m is given by

pm共El ; ␤0兲 = f m

Hml exp共␤m˜Eml − ␤0El兲 ⌬ENm

⬅ Z−1 0 ⍀ml exp共− ␤0El兲,

共9兲

where Hml is the number of configurations from replica m with a potential energy that falls in bin l and Nm is the total number of configurations for replica m. Each independent canonical simulation m contributes to the estimate of the density of state ⍀ml. If we assume the Hml to be independent, then we can use the maximum likelihood formulas to estimate the energy density of states 共⍀l兲 and its uncertainty 共␦2⍀l兲,62

We implemented the REST, MREST, and MRESTWHAM methods in the CHARMM program.63 Decomposition of the potential energy into the protein, protein-water, and water-water interactions is done using the BLOCK module of the CHARMM program. BLOCK was modified to work with the particle-mesh Ewald method64 for the calculation of the electrostatic interactions. D. Simulation setup

Our test system consisted of the ten residue amyloid peptide Ser-Asn-Asn-Phe-Gly-Ala-Ile-Leu-Ser-Ser. The coordinates were obtained from the NMR structure in the Brookhaven Protein Data Bank 共PDB兲共PDB code of 1KUW兲;65 the peptide was solvated in a cubical box containing 3314 TIP3P 共Ref. 66兲 water molecules. Periodic boundary conditions were in effect, the particle-mesh Ewald64 method was used for the long-range electrostatic interactions, and bonds involving hydrogen atoms were constrained with the SHAKE algorithm.67 The temperature was controlled with two independent Nosé-Hoover chain thermostats of length of 3,68 one coupled to the solvent degrees of freedom and the other to the protein. During the equilibration an additional chain of thermostats was coupled to the barostat. A multiple time step integrator was employed to propagate


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FIG. 1. 共a兲 The probability distributions of potential energy 共in kcal/mol兲, E p + E pw + Eww for the standard MD simulations, 共b兲 of the potential energy term, E p + 共1 / 2兲E pw for REST, 共c兲 of the generalized effective potential ˜E = 关qm / ␤m共qm − 1兲兴ln关1 + ␤m共qm − 1兲共E p + 共␤0 / ␤m兲Eww + 关共␤0 + ␤m兲 / 2␤m兴E pw − ␧兲兴 for MREST, ˜ = E⬘ + 共1 / 2兲E + 关q / ␤ 共q − 1兲兴ln关1 + ␤ 共q − 1兲共E and 共d兲 of the potential energy term, E pw m m m m m p,dihe + E p,cmap − ␧兲兴 for MREST-DIHE at different values of the p temperature T and/or scaling factor q.

thermostat variables14 using three time steps. After minimizations, the system was slowly heated and equilibrated at constant N, P, and T using a temperature of 300 K and pressure of 1 atm. The equilibration involved the use of harmonic restraints on the protein that were slowly released; the unrestrained system was further equilibrated for 500 ps, and during the equilibration, the water box reached a density of 1 g cm−3. After equilibration, the simulations were continued in the NVT ensemble. We used the REST method for five replicas at 300, 350, 400, 450, and 500 K and the MREST method for five replicas at 共Tm , qm兲 = 共300, 1兲, 共350,1.000 04兲, 共400,1.000 045兲, 共450,1.000 05兲, and 共500,1.000 055兲. Two MREST simulations were performed: one in which all atoms were subjected to Tsallis biasing and one in which just the backbone ␸ and ␺ dihedral angles were subjected to Tsallis biasing. We will denote the former with “MREST” and latter with “MREST-DIHE.” To define the ground state energy, we first ran a short standard MD simulation, extracted the minimum energy from this run 共Emin兲,

and set ␧ = 1.05Emin. Each replica system was set up using the coordinates of five different preequilibrated systems. The configurations of neighboring replicas were exchanged every 1500 steps; if the configurations between two nearest replicas were exchanged, then the velocities were scaled to the 关i兴 ⬘ 冑 new temperatures according to Ref. 24: v关i兴 n = Tn / Tmvm and 关i兴关i兴 vm ⬘ = 冑Tm / Tnvn , where the index i goes over all particles. III. RESULTS AND DISCUSSION

The energy distributions 共E p + Eww + E pw兲 using normal MD simulations at 300, 350, and 400 K are shown in Fig. 1共a兲; the energy distributions of E p + 共1 / 2兲E pw from REST, 关qm / ␤m共qm − 1兲兴 ln 关1 + ␤m共qm − 1兲共E p + 共␤0 / ␤m兲Eww + 关共␤0 + ␤m兲 / 2␤m兴E pw − ␧兲兴 from MREST, and E⬘p + 共1 / 2兲E pw + 关qm / ␤m共qm − 1兲兴ln关1 + ␤m共qm − 1兲共E p,dihe + E p,cmap − ␧兲兴 for MREST-DIHE are shown in Figs. 1共b兲–1共d兲, respectively. In contrast to the distributions from MD, there is significant overlap of the REST and MREST distributions because the exchange is based on a much smaller energy difference. The


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FIG. 2. The probability distributions of the energy term Eww at different replica runs for the MREST method.

overlap in energy distributions for MREST 关Fig. 1共c兲兴 is higher compared to REST 关Fig. 1共b兲兴, due to the broader energy distribution of MREST. This broader energy distribution is caused by the Tsallis biasing potential: in MREST, the probability to escape an energy basin is increased due to both the use of high temperatures and the q factors. Since there is practically no overlap between the energy distributions of the MD simulations, no exchange between replicas at these temperatures would occur in the standard REM method. In MREST, Eww does not cancel for the acceptance criterion 共it does cancel for REST and MREST-DIHE兲; however, the distribution of this energy is very similar among the various replicas 共Fig. 2兲. This is due to the factor ␤0 / ␤m, which greatly reduces these energy terms. Therefore, in practice, the Eww contribution to ⌬Enm is small. The overlap in probability distributions for the neighboring replicas is quantified in Table I. The overlap probability increases with increasing temperature, from 11.4% to 48.4% for REST and from 32.2% to 50.1% for MREST, while MREST shows the higher overlap for all replicas. A higher overlap compared to REST is also observed with MREST-DIHE, varying from 23.9% to 44.4%. A measure of the ergodicity D共t兲 was calculated from the backbone root mean square deviations 共RMSDs兲 for two independent trajectories 关indicated by the superscripts 共1兲 and

FIG. 3. The RMSD variance between two independent trajectories 关see Eq. 共16兲兴 as a function of the simulation time. The data from all replicas are used to construct the histograms of the RMSD.

共2兲兴.52,69 The RMSDs were divided into bins, and for each of these bins, the population HRMSD was calculated as a function of simulation time t. From these populations, D共t兲 was obtained as D共t兲 =

1 Nbins

Nbins

共1兲共2兲共t兲 − HRMSD,i 共t兲兲2 . 兺i 共HRMSD,i

共16兲

For long simulations, D共t兲 decays to zero if the sampling is ergodic. In Fig. 3, D共t兲 is shown for the REST, MREST, and MREST-DIHE methods; D共t兲 was calculated from the contributions of all replicas. D共t兲 goes to zero about seven times faster in MREST and MREST-DIHE than in REST; therefore, the sampling in MREST and MREST-DIHE is more efficient than in REST. MREST converges slightly faster than MREST-DIHE, since in MREST, the bias is applied to the entire system rather than just on the protein backbone dihedral atoms. Although for MREST, the Eww term appears in the balance criterion, the magnitude of this term is sup-

TABLE I. Overlap probability of the energy distribution between two neighboring replicas calculated as the surface area of overlap Sm−n between the two energy distribution curves: E p + 共1 / 2兲E pw for REST method, ˜E = 共q / ␤共q − 1兲兲ln关1 + ␤共q − 1兲共E p + 共␤0 / ␤兲Eww + 共共␤0 + ␤兲 / 2␤兲E pw − ␧兲兴 for MREST, and ˜E = E⬘p + 共1 / 2兲E pw + 关qm / ␤m共qm − 1兲兴ln关1 + ␤m共qm − 1兲共E p,dihe + E p,cmap − ␧兲兴 for MREST-DIHE, of the replicas m and n. Overlap probability 共%兲 Replicas T1-T2 T2-T3 T3-T4 T4-T5

REST

MREST

MREST-DIHE

11.4 15.4 17.0 48.4

32.2 41.1 48.3 50.1

23.9 26.8 42.8 44.4

FIG. 4. Fractions of replicas visiting the lowest and the highest temperatures in the MREST-DIHE, MREST, and REST methods of simulations.


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FIG. 5. 共Color online兲 Free energy contour map vs the backbone dihedral angles 共␾ and ␺兲 of the peptide from REST 共top兲, MREST 共middle兲, and MREST-DIHE 共bottom兲 methods, weighted at the temperature T = 300 K and collected for 0.3, 0.75, and 6.0 ns, as depicted in figure. The contours are spaced at intervals of 0.2kBT. The energy scale is in kcal/mol. The white regions denote areas not visited during the course of simulations.

pressed due to the scaling ␤0 / ␤m; therefore, the presence of this term does not affect the efficiency of the algorithm 共Fig. 3兲. A measure of the computational efficiency is the rate of round trips that each replica performs between the two extremal states.35,70 For REST, the extremal states are the replicas with the lowest and the highest temperatures; for MREST and MREST-DIHE, these are the states with the lowest and highest combinations of temperature and Tsallis weights. This rate can be obtained from the fraction f共Ti兲 =

nup共Ti兲 , nup共Ti兲 + ndown共Ti兲

共17兲

where ndown is the number of replicas that last visited the lowest extremal state and nup is the number of replicas that last visited the highest extremal state.35,70 In Fig. 4, f共T兲 is plotted for a 6.5 ns simulation for each method. f共T兲 fluctuates at around 0.5 for all methods, indicating that all replicas visit two extremal states, all simulations are well equilibrated, and a good computational efficiency is obtained for all methods. Although this was not needed in our simulations, the computational efficiency can be optimized by adapting the values of the temperature and Tsallis weights for the replicas during simulations.49 Figure 5 shows the free energy surfaces as a function of the 共␾ , ␺兲 backbone dihedral angles at 300 K, as obtained from MREST-WHAM analyses at various times of the simulations. There are five well-defined basins corresponding to the ␤, PII, ␣⬘, ␣R, and ␣L states 共as defined in Ref. 71兲, with multiple paths connecting the ␣⬘, ␣R, ␤, and PII, and a single path connecting PII and ␣L. The free energy surfaces indicate that the space is faster explored in the MREST and MRESTDIHE methods than in the REST method, since more areas are visited in the former. Moreover, longer simulations show

that the free energy surface is nearly converged at 3 ns for the MREST and MREST-DIHE methods 共data not shown兲. IV. CONCLUSIONS

We present an efficient replica exchange method for the sampling of conformational space. The method combines the REST method with a Tsallis biasing potential. The Tsallis biasing enhances the rates of barrier crossing, which increases the number of conformation transitions, while the REST limits the number of replicas needed. Normal, unbiased statistics can be obtained by reweighting of the simulated data using a weighted histogram analysis method. For a ten residue peptide in explicit water, the new method converges about seven times faster than the normal REST. We expect the method to be useful for the local sampling of protein or peptide conformations as well, since the Tsallis biasing potential can also be applied to a selected subset of degrees of freedom 共for example, the ␸ and ␺ dihedral angles as shown here兲. ACKNOWLEDGMENTS

This material is based on work supported by the National Science Foundation under the following NSF programs: Partnerships for Advanced Computational Infrastructure, Distributed Terascale Facility, and Terascale Extensions: Enhancements to the Extensible Terascale Facility. Computer time was also provided by the Fulton High Performance Computing Initiative at Arizona State University. M. Karplus and J. A. McCammon, Nat. Struct. Biol. 9, 646 共2002兲. T. E. Cheatham III and B. R. Brooks, Theor. Chem. Acc. 99, 279 共1998兲. 3 W. F. van Gunsteren, D. Bakowies, R. Baron, I. Chandrasekhar, M. Christen, X. Daura, P. Gee, D. P. Geerke, A. Glaättli, P. H. Hnenberger, M. A. Kastenholz, C. Oostenbrink, M. Schenk, D. Trzesniak, N. F. A. van 1 2


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