Coarse-grained protein molecular dynamics ... - Semantic Scholar

THE JOURNAL OF CHEMICAL PHYSICS 126, 025101 !2007"

Coarse-grained protein molecular dynamics simulations Philippe Derreumauxa!

Laboratoire de Biochimie Théorique, UPR 9080 CNRS, Institut de Biologie Physico, Chimique et Université Paris 7, 13 Rue Pierre et Marie Curie, 75005 Paris, France

Normand Mousseaub!,c!

Département de Physique, Université de Montréal, C.P. 6128, Succursale Centre-ville, Montréal (Québec), Canada and Regroupement Québécois sur les Matériaux de Pointe, Université de Montréal, C.P. 6128, Succursale Centre-ville, Montréal (Québec), Canada

!Received 4 October 2006; accepted 15 November 2006; published online 10 January 2007" A limiting factor in biological science is the time-scale gap between experimental and computational trajectories. At this point, all-atom explicit solvent molecular dynamics !MD" are clearly too expensive to explore long-range protein motions and extract accurate thermodynamics of proteins in isolated or multimeric forms. To reach the appropriate time scale, we must then resort to coarse graining. Here we couple the coarse-grained OPEP model, which has already been used with activated methods, to MD simulations. Two test cases are studied: the stability of three proteins around their experimental structures and the aggregation mechanisms of the Alzheimer’s A!16–22 peptides. We find that coarse-grained isolated proteins are stable at room temperature within 50 ns time scale. Based on two 220 ns trajectories starting from disordered chains, we find that four A!16–22 peptides can form a three-stranded ! sheet. We also demonstrate that the reptation move of one chain over the others, first observed using the activation-relaxation technique, is a kinetically important mechanism during aggregation. These results show that MD-OPEP is a particularly appropriate tool to study qualitatively the dynamics of long biological processes and the thermodynamics of molecular assemblies. © 2007 American Institute of Physics. #DOI: 10.1063/1.2408414$ I. INTRODUCTION

All-atom explicit solvent molecular dynamics !MD" simulations are widely used for short-range protein motions. Because of their high computational costs, these simulations are generally limited to the study of phenomena occurring on time scales shorter than about 1 "s, leaving aside a wide range of fundamental biological processes. These processes include, among others, the long-range motion of specific protein regions1 and the folding of small proteins.2 The limits of this approach are even more visible when it is necessary to accumulate statistics, to construct free energy surfaces of molecular assembly, for example. The development of coarse-grained models as a fast alternative to united-atom representation, where all nonpolar hydrogens are eliminated, has a long history in the study of protein structure prediction,3–5 thermal fluctuations,6 and kinetics7 in the understanding of DNA supercoiling,8 and more recently in the study of RNA dynamics within the ribosome.9 When coarse graining is done at a very high level of simplification such as in discontinuous MD simulations,10 the accuracy of dynamics and thermodynamics description is less reliable, but such coarse-grained models allow us to explore the early steps of aggregation of amyloid-forming peptides.11 For more sophisticated representations, on the a"

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

0021-9606/2007/126"2!/025101/6/$23.00

other hand, it has become possible to study reliably the insertion and assembly of membrane proteins12 or the folding of small proteins for longer time scales with acceptable levels of detail.13–17 The OPEP force field can be placed in this second class of coarse-grained models. It was developed to describe the short-range and long-range interactions of proteins. It uses an off-lattice chain representation with six particles per amino acid. Its analytic energy form, which includes aqueous solution effects implicitly, was derived by maximizing the energy of the native structure and an ensemble of non-native states for six training peptides of 12–38 amino acids.18 It was reparameterized recently using a total of 23 proteins and %30 000 decoys.19 The OPEP force field, used with a Monte Carlo approach, predicts the native structure of a number of proteins within 3.0 Å C# root-mean-square deviations !RMSDs" from experiments.18,20 It was also used recently to study the aggregation mechanism of amyloid-forming peptides: using a combination of OPEP with the activationrelaxation technique !ART nouveau",21,22 a transition-state Monte Carlo !MC" approach, simulations on dimers and trimers of the Alzheimer’s fragment A!16–22,23,24 as well as heptamers of lys-phe-phe-glue !KFFE",25 revealed reptation moves of one strand with respect to the others, in agreement with recent isotope-edited IR spectroscopy study.26 Here, we present the first MD applications of the OPEP force field on proteins. This article is organized as follows. The OPEP coarse-grained energy model !version 3.0" is briefly reviewed along with the MD parameters. We then

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P. Derreumaux and N. Mousseau

examine two test cases: the stability of three proteins around their experimental structures within 50 ns and the aggregation mechanisms of four Alzheimer’s A!16–22 peptides starting from randomly chosen conformations. Our results show that MD-OPEP is a very appropriate tool to study qualitatively the dynamics of long biological processes and the thermodynamics of molecular assemblies.

oppositely charged residues; otherwise H!x" = 0 and the repulsive −6 potential is used. Following previous studies,29 the hydrogen-bond function is expressed by EHB =

OPEP

In this work, we use the new parameter set !version 3.0" refined on homemade and publicly available decoys of 23 proteins.19 This parameter set has already been used to predict the equilibrium conformations of the amyloid-! !21–30" fragment, consistent with NMR spectroscopy.27 OPEP uses an off-lattice chain representation with five particles for the backbone including H and O, and generally one particle for the side chain. In its most recent version, each side chain is defined by a position with respect to the amide N, C#, and C atoms. The positions coincide with the centers of mass of nonhydrogen atoms in the all-atom side chains of 2248 protein structures with sequence identity less than 30%.19 OPEP is expressed as a sum of local, nonbonded, and hydrogen-bond !H-bond" terms. The short-range potential is expressed by Kb!r − req"2 + & K#!# − #eq"2 & bonds angles

&

+

improper angles

k$!$ − $eq"2 + & k%!% − %o"2 %

+ & k &! & − & o" 2 ,

!1"

&

where req and #eq are taken from AMBER.28 %0 = % within the interval #%lower , %upper$ and %0 = min!% − %lower , % − %upper", otherwise, with %lower = −160° and %upper = −60°, respectively. Similarly, we use &lower = −60° and &upper = −160°. The nonbonded interactions are expressed by Enb = & ELJ + 1,4

+

&

C#,C#

&

ELJ +

M !,M !

ELJ +

&

ELJ

C#,M !

& ELJ + Sc,Sc & ELJ ,

!2"

M,Sc

where 1, 4 stands for 1–4 interactions along the torsions %, &, and ', M ! consists of the main chain !M" atoms excluding C# !i.e., N, C, O, H", Sc is the side chain, and ELJ is defined as ELJ = (ij

'' ( ' ( ( r0ij rij

12

r0 − 2 ij rij

&

ij,j-i+4

+hb1-4"!rij",!#ij"

+ & +hb-! exp!− !rij − ."2/2"exp!− !rkl − ."2/2"

force field

Elocal =

+hb1−4"!rij",!#ij" +

+ & +hb-# exp!− !rij − ."2/2"exp!− !rkl − ."2/2"/!ijkl"

II. MATERIALS AND METHODS A.

&

ij,j=i+4

6

H!(ij" − (ij

'( r0ij rij

6

H!− (ij". !3"

Here the heavyside function H!x" = 1 if x ) 0 and 0 if x * 0, rij is the distance between particles i and j, r0ij = !r0i + r0j " / 2 with r0i the van der Waals radius of particle i. H!x" is set to 1 and a 12-6 potential is used for all interactions, except between two side chains. For the side chains, H!x" is set to 1 if the interaction is hydrophobic in character or results from

0/!!ijkl", where

"!rij" = 5

,!#ij" =

!4"

'( '(

)

. rij

12

. rij

−6

10

cos2 #ij , #ij - 90 ° 0,

!5"

,

otherwise.

*

!6"

The first two terms describe the two-body interactions. They are summed over all residues i and j separated by j = i + 4 and j - i + 4. rij is the O¯H distance between the carbonyl oxygen and amide hydrogen, #ij the NHO angle, and ., set to 1.8 Å, is the equilibrium value of the O¯H distance. The third and fourth terms define four-body effects. They represent the cooperative energies between hydrogen bonds ij and kl. Here, the parameter /!ijkl" is set to 1 if residues !k , l" = !i + 1 , j + 1", otherwise /!ijkl" = 0. The parameter /!!ijkl" is set to 1 if k and l satisfy either conditions: !k , l" = !i + 2 , j − 2" or !i + 2 , j + 2"; otherwise /!!ijkl" = 0. Thus, these conditions help stabilize # helices and ! sheets, independently of the !% , &" dihedral angles, but also any segment satisfying the conditions on ijkl. Full optimization of the OPEP 3.0 parameters led to two sets that could not be distinguished on the basis of the energy gap between native or nativelike structures and non-native structures: one with the parameters +hb−# and +hb−! dependent on the propensities of the residues to be in # or ! conformations !version 3.1", the other with the parameters +hb−#, and +hb−! constant !version 3.2, all propensities set to zero".19 In what follows, we use version 3.2, but rather similar results are obtained with version 3.1.

B. Molecular dynamics

Molecular dynamics simulations are performed using the velocity-Verlet algorithm30 for integrating Newton’s equations of motion with a time step sufficient to maintain the total energy in the microcanonical ensemble. Each main chain atom has its standard mass while the side-chain pseudoatoms have a mass equal to the total mass of their atomic constituents. Production runs are simulated at constant temperature, using the thermostat by Berendsen et al., with a coupling constant of 0.1 ps.31 They are performed either in an open space for single chains or in a large sphere with reflecting


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J. Chem. Phys. 126, 025101 "2007!

boundary conditions for multiple chains. In all cases, the total momentum and the total angular momentum are set to zero at t = 0 and at every 500 times steps. In the runs presented here, each initial conformation is first brought to a local energy minimum using a damped MD approach. The structure is then heated by steps and equilibrated at the desired temperature for 100 ps. The time step for the dynamics is set to 1 fs and all nonbonded interactions are updated at every step. C. Effects of a coarse-grained model with implicit solvent on protein dynamics

Moving from explicit to implicit solvent increases significantly the sampling of phase space. Rao et al., for example, have shown that the effective boost achieved by eliminating the friction associated with the solvent is about two orders of magnitude.32 A phenomenon taking 1 "s with explicit water would occur in 10 ns with an implicit description. We face a similar situation here where in addition to removing the solvent, we reduce the representation of the protein itself, replacing all side chains with a single bead. It is clear that this will further speed up the effective clock of our simulations in addition to introducing an additional dynamics perturbation over the standard molecular mechanics potentials combined with implicit solvents. As a result, the time scales presented below must be taken as indicative only. In what follows, we use the RMSD from the native structure and the rms fluctuations around the structure to determine whether the simulations accurately propagate the coarse-grained proteins. Although used by many, the fluctuations of the minimum structure are connected to the accuracy of the OPEP potential in describing the curvature of the potential of mean force near the equilibrium point. Thus, this parameter is not sufficient to determine whether the calculated OPEP trajectories are true trajectories. In addition, we know that, often, the gain is not uniform for all the modes in the system so that the details of the dynamics can be affected by moving from an explicit to an implicit solvent model.33,34 In spite of these biases, the experience of many reported simulations suggests that reduced potentials still lead to reasonable pathways, that is, although the clock for a simplified model cannot be transposed directly to a real system, the relation between the various events can still hold reasonably well.32,35,36 This is how we interpret the data presented below so that the OPEP trajectories must be taken as qualitative only.

FIG. 1. Single-chain protein dynamic properties. C# rms deviations !in Å" with respect to the experimental structure of BETANOVA !a", protein A !b", and protein G !c". rms fluctuations !in Å" as a function of the residue number for BETANOVA !d", protein A !e", and protein G !f". BETANOVA was simulated at 283 K and the proteins A and G at 300 K.

an # / ! topology with the helix laying above the plane formed by two ! sheets.39 The initial structure is taken from the protein data bank !PDB" and each molecule is then minimized and thermalized, as discussed in the previous section. In Fig. 1 we show the time fluctuations of the C# RMSD for the three proteins as measured from their minimized-energy structures. Note that BETANOVA was simulated at 283 K and proteins A and G at 300 K, as done experimentally. The BETANOVA protein #Fig. 1!a"$ appears marginally stable and hops from a structure with 1.5 Å RMSD between 5 and 20 ns to an ensemble of structures with %3.0 Å RMSD, which are sampled during the second half of the simulation. The rms deviations for protein A #Fig. 1!b"$ oscillate between two states, centered at around 0.8 and 1.8 Å. We see a similar behavior for protein G #Fig. 1!c"$, which oscillates between two positions at 1.2 and 2.3 Å RMSDs, respectively. The rms fluctuations !RMSFs" with respect to the average MD structures are also shown in Figs. 1!c"–1!f". The high flexibility of BETANOVA in Fig. 1!d" is consistent with the NMR-derived estimated !-sheet population of %10% in water at 283 K.37 Figure 2 shows representative snapshots of BETANOVA at various times. We see that the first ! hairpin is rather stable while the second ! hairpin is very mobile, and conformations fully devoid of any ! structure are populated #Fig. 2!c"$. This ensemble of conformations in equilibrium is consistent with all-atom 100 ns MD simulations in explicit solvent.40 The RMSF profile for protein G #Fig. 1!f"$ exhibits four

III. RESULTS A. Protein stability

We first apply MD-OPEP to test the stability of proteins around their experimental structures within 50 ns time scale. Three proteins with 20, 50, and 56 residues and various secondary structures are subject to this analysis. The models include BETANOVA which adopts a three-stranded ! sheet in solution,37 the B domain of staphylococcal protein A, hereafter referred to protein A, which forms a three-helix bundle in solution,38 and the B1 domain of protein G, hereafter referred to protein G, which is characterized by NMR to adopt

FIG. 2. BETANOVA MD simulations at 283 K. Representative snapshots are taken at 14.6 ns !a", 22.3 ns !b", 28.1 ns !c", 28.4 ns !d", and 40.0 ns !e". The structures are drawn using the MOLMOL program.50


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peaks at positions 8–15 !loop between strands !1 and !2", position 20 !beginning of # helix", position 40 !loop between helix and strand !3", and position 47 !loop connecting strands !3 and !4". The positions of the major peaks agree well with the experimental data and OPLS all-atom MD simulations.41 The generated RMSF profile of protein A #Fig. 1!e"$ is also very similar to that obtained by CHARMM allatom MD simulations.42 To investigate whether these properties vary with different initial velocity distributions, simulations of proteins A and G were repeated for 10 ns using the same starting structure. We find that the RMSD and RMSF plots superpose well from one run to another !data not shown". Taken together, these results demonstrate that these proteins with #, !, and # / ! character are stable with native secondary and tertiary structures using the OPEP coarse-grained model, and the generated submicrosecond dynamics are qualitatively consistent with experimental data and all-atom explicit solvent studies. B. Aggregation of a tetramer of A!16–22

The second application of MD-OPEP is related to the aggregation mechanisms of A!16–22. This peptide was chosen because it comprises the central hydrophobic core that is thought to be important in full length Alzheimer’s !-amyloid assembly. It has also been subject of free energy calculations43 and aggregation simulations.23,24,44,45 Notably, the assembly of three A!16–22 chains has been investigated by ART-OPEP simulations free of any constraints23,24 and by MD at 300 K using an all-atom model of the peptides, an explicit solvent model and a bias to facilitate interactions between the chains.44 While the biased MD simulations emphasized the role of #-helical intermediates en route to a three-stranded ! sheet, ART-OPEP simulations pointed to the role of reptation moves. Since ART events bring a conformation from a fully relaxed state to a fully relaxed state, going through an activation barrier, it remains to be determined whether such moves are kinetically accessible at room temperature. We stress that this reptation mechanism was already found to be important during the folding of a 16-residue ! hairpin using ART-OPEP,29 and this was recently validated by MC simulations at room temperature.46 In contrast, recent extensive high-temperature MD simulations on the 12residue trpzip2 ! hairpin have failed to see the reptation move, raising the possibility that this move is not kinetically favored.47 Figure 3 presents the results of two 220 ns MD simulations at 330 K, a temperature often used to incubate the amyloid in experiments. The evolution of potential energy, radius of gyration, and !-strand content as a function of time is shown in Fig. 3. Representative snapshots for run 1 are shown in Fig. 4. Starting from a random organization of four peptides #Fig. 4!a"$, the chains are essentially disordered for 25 ns #Fig. 4!b"$ and form at 35 ns, an aggregate characterized by a two-stranded antiparallel ! sheet interacting with two chains #Fig. 4!c"$. Then, at 40 ns, a nonideal, out-ofregister three-stranded antiparallel ! sheet is created #Fig. 4!d"$. This is followed by a change in the pattern of registry of intermolecular H bonds by the reptation move of the

J. Chem. Phys. 126, 025101 "2007!

FIG. 3. Alzheimer’s A!16–22 peptide aggregation. The 220 ns MD simulations are carried out at 330 K, starting from randomly chosen conformations and orientations of the four chains. Evolution of potential energy Epot in kcal/mol #!a" and !d"$, radius of gyration Rg in Å #!b" and !e"$, and percentage of !-strand content as determined by the DSSP program #!c" and !f"$.51 Panels #!a"–!c"$ refer to run 1 and panels #!d"–!f"$ to run 2. For simplicity, the radius of gyration is calculated here using only the C# atoms.

strands with respect to each other. Snapshots of the details of the reptation move in run 1 is given in Fig. 5. Starting from a misaligned trimer !time 0 of the event", one of the outer strands partially detaches by breaking a number of its hydrogen bonds !at 50 and 100 ps", folding some of the free length. Subsequently, the strand continues to break its hydrogen bonds, although remaining in contact through its side chains with the rest of the system !at 150 ps", and brings back one of its extremities, pushing itself up by two residues !200 ps" before falling into place or in register !250 ps". The whole process lasts less than 300 ps. A few nanoseconds after the reptation move, at 50 ns, the chains adopt a three-stranded antiparallel ! sheet fully consistent with solid-state NMR analysis within the fibrils48 and the fourth chain forms a looplike conformation #Fig. 4!e"$. From there to 220 ns, the trimeric ! sheet remains formed, although the percentage of formed H bonds fluctuates, and the fourth chain is very dynamic, exploring various orientations and conformations above the trimer. This is the

FIG. 4. Representative steps in the aggregation of A!16–22. Dotted lines indicate the formation of H bonds. 0.0 ns !a", 25 ns !b", 35 ns !c", 40 ns !d", 50 ns !e", and 220 ns !f".


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FIG. 5. !Color" Details of the reptation move in run 1 of A!16–22. For simplicity, the time is reset to 0. See the text for a detailed description.

case in the final snaptshot #Fig. 4!f"$, where the fourth chain is fully extended and stretches across the trimer. Although the details of dynamics varies from one run to another !Fig. 3", the overall dynamics is very similar. For example, both runs show chain rearrangements occurring through a reptation move, although the time scales are different: the move takes less than 0.35 ns in run 1, while it spans 5 ns in run 2. These results show that reptation is a kinetically possible move at low temperatures, a result fully consistent with recent IR spectroscopy.26 Similarly, the final structure in both runs consists of a three-stranded ! sheet stabilized by the fourth chain stretched across it. This arrangement was observed previously in ART-OPEP simulations of four A!11–25 peptides.49 While it seems to be a purely entropic effect in the case of A!11–25, we find that the planar four-stranded ! sheet is also less favorable energetically for A!16–22. Although, there is no experimental evidence for such a structure during the early steps of amyloid formation because all structures are transient, we suggest that this arrangement could be an important intermediate. IV. CONCLUSIONS

We have tested the coarse-grained OPEP model using constant temperature molecular dynamics simulations with a Berendsen thermostat. Application of OPEP version 3.0 to study protein stability and aggregation demonstrates that this coarse-grained potential is robust and efficient: !1" MD simulations on three 20-56 residue proteins with #, !, or # / ! character show that these proteins, comprising various secondary structures, are stable over 50 ns; !2" MD simulations of four amyloid-forming A!16–22 peptides starting from random conformations locate a three-stranded ! sheet stabilized by a fourth chain stretched across the sheet, a structure already observed in other systems.49 Moreover, we find that reptation of one strand over the others, observed previously in Monte Carlo simulations of single-chain peptides,29,46 is a kinetically relevant rearrangement mechanism, in agreement with ART simulations23,24,49 and experiments.26

This result suggests that the absence of reptation mechanism in high-temperature MD simulations of tripzip2 !Ref. 47" is likely due to the tightness of the loop in this short 12-residue peptide. Based on our current MD and ART results, reptation, which allows the structure to reorganize without having to fully dissociate, should therefore be an important mechanism during both single-chain protein folding and amyloid fibril formation. Compared with all-atom simulations, MD-OPEP is fast. Using a 1 fs time step, one 1 ns MD trajectory on protein A !consisting of 50 residues" takes approximately 1 h on a 2.7 GHz Intel microprocessor. Although CPU improvement can be achieved by eliminating the high-frequency motions of bonds and using cutoff distances for the nonbonded interactions, the current computational speed will allow to cover longer time scales and run replica-exchange MD-OPEP simulations to construct free energy surfaces of single proteins and molecular assemblies. ACKNOWLEDGMENTS

The authors are supported in part by the Alzheimer Society of Canada. One of the authors !N.M." also acknowledges partial support from Natural Sciences and Engineering Research Council of Canada, the Canada Research Chair Fund, the Fonds québécois de recherche sur la nature et les technologies, and a poste rouge from CNRS. The other author !P.D." acknowledges fundings from CNRS, Université of Paris 7 and 6ème Européen PCRD. 1

P. Derreumaux and T. Schlick, Biophys. J. 74, 72 !1998". Y. Duan and P. Kollman, Science 282, 740 !1998". 3 M. Levitt and A. Warshel, Nature !London" 253, 694 !1975". 4 A. Kolinski, A. Godzik, and J. Skolnick, J. Chem. Phys. 98, 7420 !1993". 5 C. Micheletti, F. Seno, J. R. Banavar, and A. Maritan, Proteins 42, 422 !2001". 6 S. Nicolay and Y. H. Sanejouand, Phys. Rev. Lett. 96, 078104 !2006". 7 Y. Zhou and M. Karplus, Nature !London" 401, 400 !1999". 8 T. Schlick and W. K. Olson, Science 257, 1110 !1992". 9 J. Trylska, V. Tozzini, and J. A. McCammon, Biophys. J. 89, 1455 !2005". 10 F. Ding, S. V. Buldyrev, and N. V. Dokholyan, Biophys. J. 88, 147 !2005". 11 H. D. Nguyen and C. K. Hall, Proc. Natl. Acad. Sci. U.S.A. 101, 16180 !2004". 12 P. J. Bond and M. S. Sansom, J. Am. Chem. Soc. 128, 2697 !2006". 13 M. Khalili, A. Liwo, F. Rakowski, P. Grochowski, and H. A. Scheraga, J. Phys. Chem. B 109, 13785 !2005". 14 A. Liwo, M. Khalili, and H. A. Scheraga, Proc. Natl. Acad. Sci. U.S.A. 102, 2362 !2005". 15 I. A. Hubner, E. J. Deeds, and E. I. Shakhnovich, Proc. Natl. Acad. Sci. U.S.A. 102, 18914 !2005". 16 S. Takada, Proteins 42, 85 !2001". 17 Y. Fujitsuka, S. Takada, Z. A. Luthey-Schulten, and P. G. Wolynes, Proteins 54, 88 !2004". 18 P. Derreumaux, J. Chem. Phys. 111, 2301 !1999". 19 J. Maupetit, P. Tuffery, and P. Derreumaux !unpublished". 20 P. Derreumaux, Phys. Rev. Lett. 85, 206 !2000". 21 R. Malek and N. Mousseau, Phys. Rev. E 62, 7723 !2000". 22 G.-H. Wei, N. Mousseau, and P. Derreumaux, J. Chem. Phys. 117, 11379 !2002". 23 S. Santini, G. Wei, N. Mousseau, and P. Derreumaux, Structure !London" 12, 1245 !2004". 24 S. Santini, N. Mousseau, and P. Derreumaux, J. Am. Chem. Soc. 126, 11509 !2004". 25 A. Melquiond, N. Mousseau, and P. Derreumaux, Proteins 65, 180 !2006". 2


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S. A. Petty and S. M. Decatur, J. Am. Chem. Soc. 127, 13488 !2005". W. Chen, N. Mousseau, and P. Derreumaux, J. Chem. Phys. 125, 084911 !2006". 28 D. A. Case, T. E. Cheatham 3rd, T. Darden, H. Gohlke, R. Luo, K. M. Merz, Jr., A. Onufriev, C. Simmerling, B. Wang, and R. J. Woods, J. Comput. Chem. 26, 1668 !2005". 29 G. Wei, N. Mousseau, and P. Derreumaux, Proteins 56, 464 !2004". 30 W. C. Swope, H. C. Andersen, P. H. Berens, and K. R. Wilson, J. Chem. Phys. 76, 637 !1982". 31 H. J. C. Berendsen, J. P. M. Postma, W. F. van Gunsteren, A. DiNola, and J. R. Haak, J. Chem. Phys. 81, 3684 !1984". 32 F. Rao, G. Settanni, E. Guarnera, and A. Caflisch, J. Chem. Phys. 122, 184901 !2005". 33 B. Zagrovic and V. Pande, J. Comput. Chem. 24, 1432 !2003". 34 A. Baumketner and J.-E. Shea, Phys. Rev. E 68, 051901 !2003". 35 P. Ferrara, J. Apostolakis, and A. Caflisch, Proteins 46, 24 !2002". 36 C. Chen and Y. Xiao, Phys. Biol. 3, 161 !2006". 37 T. Kortemme, M. Ramirez-Alvarado, and L. Serrano, Science 281, 253 !1998". 38 H. Gouda, H. Torigoe, A. Saito, M. Sato, Y. Arata, and I. Shimada, 27

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Biochemistry 31, 9665 !1992". T. Gallagher, P. Alexander, P. Bryan, and G. L. Gilliland, Biochemistry 33, 4721 !1994". 40 P. Soto and G. Colombo, Proteins 57, 734 !2004". 41 O. F. Lange, H. Grubmuller, and B. L. de Groot, Angew. Chem., Int. Ed. Engl. 44, 3394 !2005". 42 B. P. Pandey, C. Zhang, X. Yuan, J. Zi, and Y. Zhou, Protein Sci. 14, 1772 !2005". 43 S. Gnanakaran, R. Nussinov, and A. E. Garcia, J. Am. Chem. Soc. 128, 2158 !2006". 44 D. K. Klimov and D. Thirumalai, Structure !London" 11, 295 !2003". 45 G. Favrin, A. Irback, and S. Mohanty, Biophys. J. 87, 3657 !2004". 46 H. Imamura and J. Z. Chen, Proteins 63, 555 !2006". 47 J. W. Pitera, I. Haque, and W. C. Swope, J. Chem. Phys. 124, 141102 !2006". 48 A. T. Petkova, G. Buntkowsky, F. Dyda, R. D. Leapman, W. M. Yau, and R. Tycko, J. Mol. Biol. 335, 247 !2004". 49 G. Boucher, N. Mousseau, and P. Derreumaux, Proteins 65, 877 !2006". 50 R. Koradi, M. Billeter, and K. Wuthrich, J. Mol. Graphics 14, 29 !1996". 51 W. Kabsch and C. Sander, Biopolymers 22, 2577 !1983". 39
