Letter pubs.acs.org/JPCL
Conformational Entropy as Collective Variable for Proteins Ferruccio Palazzesi,†,‡ Omar Valsson,†,‡,§,∥ and Michele Parrinello*,†,‡,§ †
Department of Chemistry and Applied Biosciences, ETH Zurich c/o USI Campus, Via Giuseppe Buffi 13, CH-6900, Lugano, Switzerland ‡ Facoltà di Informatica, Instituto di Scienze Computationali, Università della Svizzera italiana, Via Giuseppe Buffi 13, CH-6900, Lugano, Switzerland § National Center for Computational Design and Discovery of Novel Materials MARVEL, Università della Svizzera italiana, Via Giuseppe Buffi 13, CH-6900, Lugano, Switzerland S Supporting Information *
ABSTRACT: Many enhanced sampling methods rely on the identification of appropriate collective variables. For proteins, even small ones, finding appropriate descriptors has proven challenging. Here we suggest that the NMR S2 order parameter can be used to this effect. We trace the validity of this statement to the suggested relation between S2 and conformational entropy. Using the S2 order parameter and a surrogate for the protein enthalpy in conjunction with metadynamics or variationally enhanced sampling, we are able to reversibly fold and unfold a small protein and draw its free energy at a fraction of the time that is needed in unbiased simulations. We also use S2 in combination with the free energy flooding method to compute the unfolding rate of this peptide. We repeat this calculation at different temperatures to obtain the unfolding activation energy.
E
CV and illustrate its efficiency in non trivial examples. Our guiding principle is that, in the behavior of proteins, and of many other systems,18 entropy plays an important role, and if we are able to identify a CV that measures conformational entropy even if in an approximate way, this could go some way toward being able to sample the complex landscape of proteins.19 In this search we shall be helped by the NMR literature in which several attempts have been made at converting the NMR observable into a measure of conformational entropy.20−26 Without going into the complex detail of the NMR technique, it suffices to say that in nuclear relaxation experiments, it is possible to measure the dynamical behavior of selected bonds, like N−H or C−H, and, within some approximations, extract the so-called order parameter S2. This parameter, that can vary between 0 and 1, can provide useful information on the degree of spatial motion of the system.27,28 From the knowledge of this parameter, several empirical relations between S2 and the conformational entropy have been proposed and their validity assessed in comparison with either experiments or molecular dynamics simulations.20,23−25 Despite being rather empirical,22 these relationships have been used to study several biomacromolecular processes, and to calculate protein heat capacity.21,26,29,30
nhanced sampling methods have received great attention since they offer the promise of overcoming the limited time scale that direct simulations can afford. Among the plethora of methods proposed in the literature,1−4 metadynamics (MetaD)5,6 and more recently variationally enhanced sampling (VES)7 are finding application in a vast array of problems.8−14 Like other similar approaches, they rely on the identification of appropriate collective variables (CVs) or order parameters. In MetaD or VES, the fluctuations of the CVs are enhanced such that transition between different metastable basins are favored.15,16 Brief descriptions of these two methods can be found in the Supporting Information (SI). A vast number of CVs have been suggested and are easily accessible in open source codes.17 However, the quest for new order parameters continues in the hope of finding CVs that are efficient and yet generic enough such that they can be applied to a larger class of problems, without prejudging the final results. Identifying appropriate CVs is not only a technical issue needed to accelerate sampling but offers a key to the understanding of the underlying physical processes. The need for such CVs is particularly pressing in the field of biomolecular systems, such as proteins, whose conformational changes are defined by a large number of degrees of freedom like the arrangement of backbone atoms, side-chains, and solvent molecules.14 At first this appears like a very demanding request. The purpose of this paper is to show that this is not necessarily so, at least for small proteins or selected regions of larger biosystems. To this effect we introduce a conceptually new © XXXX American Chemical Society
Received: July 10, 2017 Accepted: September 14, 2017 Published: September 14, 2017 4752
DOI: 10.1021/acs.jpclett.7b01770 J. Phys. Chem. Lett. 2017, 8, 4752−4756
Letter
The Journal of Physical Chemistry Letters It is the existence of these relations and the notion that entropy plays an important role in proteins that have inspired us to use S2 as a CV. However, to order to proceed, one needs to express S2 as a function of the atomic coordinates. Luckily such relations are available for both the NH and CH3 groups.31−34 Since here we shall only bias the N−H bondsrelated order parameter, we solely report the relevant expression proposed by Zhang and Brüschweiler:35 S2 =
∑ Sn2
(1)
n
S2n
where is the order parameter for the nth amino acid residue defined as Sn2 = tanh(0.8 ∑ [exp( −rnO− 1, k) + exp( −rnH, kN)]) − 0.1 k
(2)
where k runs over all the heavy atoms with the exception of the residues n and n − 1, and rHn,kN and rOn−1,k are the distances from the heavy atom k from the amide hydrogen in residue n and the carboxyl oxygen in residue n − 1. In eq 2 the distances are to be expressed in ångstroms. Note that, according to a correction36 to ref 35, the distances rOn−1,k and rHn,kN used in eq 2 should be shortened by 1.2 Å. Here we have used eq 2 without shorting the distances. As can be seen in Figure S3 in the SI, the usage of shortened distances results in a rather similar free energy surface. Structurally speaking, eq 2 is a relation between protein databank structures and S2, and there is no guarantee that it will remain valid for any arbitrary conformation. With this in mind and well aware of the lack of rigor, we shall continue to assume that eq 2 can be used to estimate the conformation entropy of the system, within some approximation. Whether this is a valid assumption or not will have to be judged by the results. We started up by checking that S2 is able to distinguish between the folded and unfolded protein conformations by calculating the free energy surface (FES) along this CV utilizing the long unbiased trajectory for chignolin (CLN025)37 provided to us by the D.E. Shaw Research Group.38 As can be seen in Figure 1, the FES exhibits two well-defined basins corresponding to the folded and unfolded state. This in itself is an interesting result that suggests the attempt at using S2 as a collective variable is not totally devoid of merit. We therefore proceeded with biased simulations where the fluctuations of S2 are amplified by using an enhanced sampling technique, such as MetaD or VES. We tried to drive the folding transition using only S2 (hereafter denoted as sS) as CV but failed. Thus, guided by the fact that folding can be described as a trade off between entropy and enthalpy, we introduced a second CV that is meant to be a surrogate for enthalpy. This could have been accomplished by separating from the total energy of the systems those components that describe the protein−protein interactions and use these as CV. However, this is somewhat expensive and we preferred to use as surrogate for the enthalpy the native H-bonds contact map (hereafter sH) as detailed in the SI. At the end we shall reweight the FES39 and obtain its projection onto the protein enthalpy defined as the sum of the protein−protein contributions, Epp, to the total energy. We performed the simulation at T = 340 K and use the Charmm22*40 protein force field and the TIP3P41 water model in order to be able to compare directly with the long unbiased simulation (∼100.0 μs) from ref 38. We use the GROMACS
Figure 1. FES along the sS CV calculated from long unbiased molecular dynamics simulation of ref 38, with the ribbon representation of chignolin in two representative snapshots of the folded and unfolded conformations.
5.1.4 MD package42 patched with the PLUMED 2 plug-in17 in which we have implemented the sS CV. In the MetaD calculations we integrate the equation of motion using a time step of 2.0 fs, and the temperature is controlled by the stochastic velocity rescaling thermostat.43 Gaussians of initial height 2.82 kJ/mol and width 0.05 for sS and for sH were deposited every picosecond. The value of the bias factor γ was set equal to 8.6 To speed up the calculation and make use of parallelism we used 4 multiple walkers.44 We evaluate the convergence using the error metric previously used in refs 6, 45, 46 and using the unbiased data of ref 38 as reference. It can be seen from Figure 2 that the convergence in the FES is reached in about 1.0 μs, which is a much shorter time relative to ∼100.0 μs reported in ref 38. This reflects the fact that in the MetaD run the rate of transition between folded and unfolded states is accelerated about 100 times. In Figure 2 we also express the FES as a function of entropy and enthalpy. The entropy is estimated from sS using the relation given by Wand and co-workers,23 while as a measure of the protein enthalpy we use Epp, which is defined as the protein−protein contribution to the total energy. As can be expected, entropy and enthalpy are highly correlated; however, to obtain a correct description of the FES and to ensure reversible transitions between the two states, we need both of these CVs. This calculation clearly proves the usefulness of using entropy and enthalpy to drive reversible transitions between folded and unfolded states. We have repeated the calculations using VES, obtaining statistically indistinguishable results. The VES simulations are performed using the VES code module for PLUMED 2. The results for this calculation are reported in the SI. We would like to underline the great conceptual simplification that has been achieved. We have been able to reversibly fold and unfold chignolin without the use of a special purpose machine, replica exchange techniques, or a large number of CVs.14,38,47−50 Our results also point to the value of going off the beaten track in distinguishing new CVs. An analysis of the structures lying at the bottom of the folded basin in Figure 1 shows that they deviate from the experimental 4753
DOI: 10.1021/acs.jpclett.7b01770 J. Phys. Chem. Lett. 2017, 8, 4752−4756
Letter
The Journal of Physical Chemistry Letters
Figure 2. (a) Reweighted FES calculated from the MetaD simulation using entropy and enthalpy (Epp) as CVs. The entropy is estimated from sS using the relation given in ref 23, while the protein enthalpy Epp is the protein−protein contribution to the total energy. (b,c) Monodimensional FESs for entropy and enthalpy, respectively. The data from the unbiased simulation of ref 38 is shown in red, while the one from MetaD simulation is in green. (d) Free energy error of the 2D FES along the simulation time.
structure37 on average by around 0.8 Å. This suggests that if we are only interested in studying the unfolding process, sS alone could be used to promote unfolding transition and calculate unfolding rates using the approach introduced in ref 51. Such calculation has been already performed on a mutated form of chignolin using the root-mean square deviation with respect to the folded structure as biased CV and the infrequent MetaD approach.52 The success of this calculation indicates that by using a single variable it is possible to obtain the unfolding rates. This suggests that driving the unfolding process is much easier than the reverse folding process that requires the addition of an extra CV as seen above. If this is the case, it is of great interest to investigate whether sS has the ability to induce protein unfolding. In this respect we would put in evidence that the use of sS is more subtle and elegant than using the rootmean square deviation. To this aim we make use of the VES analogous of the infrequent MetaD method of Tiwary and Parrinello,53 that in turn is based on the ideas of Grubmuller3 and Voter.1 In all these approaches one relates the rates of a rare event, as calculated in a biased system, to the physical unbiased rates by a simple relation. The requirement that the bias does not act in the transition region is crucial for this relation to hold. The methods mentioned earlier differ in the way this is achieved. In ref 51 the variational approach is used to truncate the bias up to a preassigned free energy level. If this cutoff value is smaller than the free energy barrier this latter region remains free of bias and the condition put forward by Grubmuller3 and Voter1 applies. Computational details can be found in the SI. The results of our calculation are shown in Figure 3. The value obtained at T = 340 K is in good agreement with both the unbiased estimation of Lindorff-Larsen et al.38 and the aforementioned infrequent MetaD study by Tung and Pfaendtner from ref 52 on a mutated form of our system.
Figure 3. Arrenhius plot of the unfolding process of the chignolin. The green square is the unfolding time calculated from the unbiased data of ref 38. The black line is the linear regression used to calculated the activation energy (R2 is equal to 0.97). The acceleration factors are 10, 50, and 100 for temperature equal to 340, 320 and 300, respectively.
Contrary to these previous estimations, we push our calculation to lower temperatures (320 and 300 K), finding for the unfolding of this simple peptide an Arrhenius behavior, with an activation energy of around 50 kJ/mol. This value is in the right ballpark when compared to the estimation based on a similar β-peptide.54 In conclusion, we have shown that the idea of using entropy and enthalpy as collective variables, or rather surrogate expressions for them, is a useful one. One of the secret of the success in using these CVs is not only that it is founded on physical ideas and concepts but also on the fact that is nonlocal and thus sensitive to the whole structure of the protein. We 4754
DOI: 10.1021/acs.jpclett.7b01770 J. Phys. Chem. Lett. 2017, 8, 4752−4756
Letter
The Journal of Physical Chemistry Letters would like to add that the S2 defined in eq 2 reflects the backbone structure. If one were interested in the role of the side-chains the use of S2 order parameter based on the methyl groups might prove useful.55 We believe that here we have introduced a powerful new concept in the simulation of protein. How far it can be pushed when going to larger systems remains to be seen. Already at this stage, it can be safely said that, without any modification, the folding of small proteins, the study of intrinsically disordered proteins, and the conformational flexibility of proteins segments can be handled as described above.56 We are confident that, with an appropriate adaptation, much larger proteins can be similarly handled.
■
(7) Valsson, O.; Parrinello, M. Variational Approach to Enhanced Sampling and Free Energy Calculations. Phys. Rev. Lett. 2014, 113, 090601. (8) Palazzesi, F.; Barducci, A.; Tollinger, M.; Parrinello, M. The Allosteric Communication Pathways in KIX Domain of CBP. Proc. Natl. Acad. Sci. U. S. A. 2013, 110, 14237−14242. (9) Martoňaḱ , R.; Donadio, D.; Oganov, A. R.; Parrinello, M. Crystal Structure Transformations in SiO2 from Classical and Ab Initio Metadynamics. Nat. Mater. 2006, 5, 623−626. (10) Pavan, G. M.; Barducci, A.; Albertazzi, L.; Parrinello, M. Combining Metadynamics Simulation and Experiments to Characterize Dendrimers in Solution. Soft Matter 2013, 9, 2593−2597. (11) Branduardi, D.; Gervasio, F. L.; Parrinello, M. From A to B in Free Energy Space. J. Chem. Phys. 2007, 126, 054103. (12) Tiwary, P.; Limongelli, V.; Salvalaglio, M.; Parrinello, M. Kinetics of Protein-Ligand Unbinding: Predicting Pathways, Rates, and Rate-Limiting Steps. Proc. Natl. Acad. Sci. U. S. A. 2015, 112, E386− E391. (13) Bottaro, S.; Banás,̌ P.; Sponer, J.; Bussi, G. Free Energy Landscape of GAGA and UUCG RNA Tetraloops. J. Phys. Chem. Lett. 2016, 7, 4032−4038. (14) Shaffer, P.; Valsson, O.; Parrinello, M. Enhanced, Targeted Sampling of High-Dimensional Free-Energy Landscapes Using Variationally Enhanced Sampling, with an Application to Chignolin. Proc. Natl. Acad. Sci. U. S. A. 2016, 113, 1150−1155. (15) Barducci, A.; Bonomi, M.; Parrinello, M. Metadynamics. WIREs Comput. Mol. Sci. 2011, 1, 826−843. (16) Valsson, O.; Tiwary, P.; Parrinello, M. Enhancing Important Fluctuations: Rare Events and Metadynamics from a Conceptual Viewpoint. Annu. Rev. Phys. Chem. 2016, 67, 159−184. (17) Tribello, G. A.; Bonomi, M.; Branduardi, D.; Camilloni, C.; Bussi, G. PLUMED 2: New Feathers for an Old Bird. Comput. Phys. Commun. 2014, 185, 604−613. (18) Piaggi, P. M.; Valsson, O.; Parrinello, M. Enhancing Entropy and Enthalpy Fluctuations to Drive Crystallization in Atomistic Simulations. Phys. Rev. Lett. 2017, 119, 015701. (19) Chavez, L. L.; Onuchic, J. N.; Clementi, C. Quantifying the Roughness on the Free energy Landscape: Entropic Bottlenecks and Protein Folding Rates. J. Am. Chem. Soc. 2004, 126, 8426−8432. (20) Akke, M.; Brueschweiler, R.; Palmer, A. G., III NMR Order Parameters and Free Energy: an Analytical Approach and its Application to Cooperative Calcium (2+) Binding by Calbindin D9k. J. Am. Chem. Soc. 1993, 115, 9832−9833. (21) Yang, D.; Mok, Y.-K.; Forman-Kay, J. D.; Farrow, N. A.; Kay, L. E. Contributions to Protein Entropy and Heat Capacity from Bond Vector Motions Measured by NMR Spin Relaxation. J. Mol. Biol. 1997, 272, 790−804. (22) Stone, M. J. NMR Relaxation Studies of the Role of Conformational Entropy in Protein Stability and Ligand Binding. Acc. Chem. Res. 2001, 34, 379−388. (23) Sharp, K. A.; O’Brien, E.; Kasinath, V.; Wand, A. J. On the Relationship Between NMR-Derived Amide Order Parameters and Protein Backbone Entropy Changes. Proteins: Struct., Funct., Genet. 2015, 83, 922−930. (24) Yang, D.; Kay, L. E. Contributions to Conformational Entropy Arising from Bond Vector Fluctuations Measured from NMR-Derived Order Parameters: Application to Protein Folding. J. Mol. Biol. 1996, 263, 369−382. (25) Li, Z.; Raychaudhuri, S.; Wand, A. J. Insights into the Local Residual Entropy of Proteins Provided by NMR Relaxation. Protein Sci. 1996, 5, 2647−2650. (26) Marlow, M. S.; Dogan, J.; Frederick, K. K.; Valentine, K. G.; Wand, A. J. The Role of Conformational Entropy in Molecular Recognition by Calmodulin. Nat. Chem. Biol. 2010, 6, 352−358. (27) Lipari, G.; Szabo, A. Model-Free Approach to the Interpretation of Nuclear Magnetic Resonance Relaxation in Macromolecules. 1. Theory and Range of Validity. J. Am. Chem. Soc. 1982, 104, 4546− 4559.
ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.7b01770. Further computational details and VES results (PDF)
■
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
Ferruccio Palazzesi: 0000-0003-1534-4781 Omar Valsson: 0000-0001-7971-4767 Present Address
∥ Max Planck Institute for Polymer Research, Ackermannweg 10, D-55128 Mainz, Germany
Notes
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS We acknowledge funding from the European Union Grant No. ERC-2014-AdG-670227/VARMET and by the NCCR MARVEL, funded by the Swiss National Science Foundation. This work was supported by a grant from the Swiss National Supercomputing Centre (CSCS) under project ID s721 and u1. The authors thank D. E. Shaw Research for sharing the simulation data for chignolin. Molecular graphics and analyses were performed with the UCSF Chimera package. Chimera is developed by the Resource for Biocomputing, Visualization, and Informatics at the University of California, San Francisco (supported by NIGMS P41-GM103311).57
■
REFERENCES
(1) Voter, A. F. Hyperdynamics: Accelerated Molecular Dynamics of Infrequent Events. Phys. Rev. Lett. 1997, 78, 3908. (2) Huber, T.; Torda, A. E.; van Gunsteren, W. F. Local Elevation: a Method for Improving the Searching Properties of Molecular Dynamics Simulation. J. Comput.-Aided Mol. Des. 1994, 8, 695−708. (3) Grubmüller, H. Predicting Slow Structural Transitions in Macromolecular Systems: Conformational Flooding. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 1995, 52, 2893. (4) Torrie, G. M.; Valleau, J. P. Nonphysical Sampling Distributions in Monte Carlo Free-Energy Estimation: Umbrella Sampling. J. Comput. Phys. 1977, 23, 187−199. (5) Laio, A.; Parrinello, M. Escaping Free-Energy Minima. Proc. Natl. Acad. Sci. U. S. A. 2002, 99, 12562−12566. (6) Barducci, A.; Bussi, G.; Parrinello, M. Well-Tempered Metadynamics: a Smoothly Converging and Tunable Free-Energy Method. Phys. Rev. Lett. 2008, 100, 020603. 4755
DOI: 10.1021/acs.jpclett.7b01770 J. Phys. Chem. Lett. 2017, 8, 4752−4756
Letter
The Journal of Physical Chemistry Letters (28) Lipari, G.; Szabo, A. Model-Free Approach to the Interpretation of Nuclear Magnetic Resonance Relaxation In Macromolecules. 2. Analysis of Experimental Results. J. Am. Chem. Soc. 1982, 104, 4559− 4570. (29) Spyracopoulos, L.; Sykes, B. D. Thermodynamic Insights into Proteins from NMR Spin Relaxation Studies. Curr. Opin. Struct. Biol. 2001, 11, 555−559. (30) Bracken, C. NMR Spin Relaxation Methods for Characterization of Disorder and Folding in Proteins. J. Mol. Graphics Modell. 2001, 19, 3−12. (31) Bowman, G. R. Accurately Modeling Nanosecond Protein Dynamics Requires at Least Microseconds of Simulation. J. Comput. Chem. 2016, 37, 558−566. (32) Chatfield, D. C.; Szabo, A.; Brooks, B. R. Molecular Dynamics of Staphylococcal Nuclease: Comparison of Simulation with 15N And 13C NMR Relaxation Data. J. Am. Chem. Soc. 1998, 120, 5301−5311. (33) Peter, C.; Daura, X.; Van Gunsteren, W. F. Calculation of NMRRelaxation Parameters for Flexible Molecules from Molecular Dynamics Simulations. J. Biomol. NMR 2001, 20, 297−310. (34) Maragakis, P.; Lindorff-Larsen, K.; Eastwood, M. P.; Dror, R. O.; Klepeis, J. L.; Arkin, I. T.; Jensen, M. Ø.; Xu, H.; Trbovic, N.; Friesner, R. A.; et al. Microsecond Molecular Dynamics Simulation Shows Effect of Slow Loop Dynamics on Backbone Amide Order Parameters of Proteins. J. Phys. Chem. B 2008, 112, 6155−6158. (35) Zhang, F.; Brüschweiler, R. Contact Model for the Prediction of NMR NH Order Parameters in Globular Proteins. J. Am. Chem. Soc. 2002, 124, 12654−12655. (36) Zhang, F.; Brüschweiler, R. Contact Model for the Prediction of NMR NH Order Parameters in Globular Proteins. J. Am. Chem. Soc. 2008, 130, 9178−9178. (37) Honda, S.; Akiba, T.; Kato, Y. S.; Sawada, Y.; Sekijima, M.; Ishimura, M.; Ooishi, A.; Watanabe, H.; Odahara, T.; Harata, K. Crystal Structure of a Ten-Amino Acid Protein. J. Am. Chem. Soc. 2008, 130, 15327−15331. (38) Lindorff-Larsen, K.; Piana, S.; Dror, R. O.; Shaw, D. E. How Fast-Folding Proteins Fold. Science 2011, 334, 517−520. (39) Tiwary, P.; Parrinello, M. A Time-Independent Free Energy Estimator for Metadynamics. J. Phys. Chem. B 2015, 119, 736−742. (40) Piana, S.; Lindorff-Larsen, K.; Shaw, D. E. How Robust are Protein Folding Simulations with Respect to Force Field Parameterization? Biophys. J. 2011, 100, L47−L49. (41) Jorgensen, W. L.; Chandrasekhar, J.; Madura, J. D.; Impey, R. W.; Klein, M. L. Comparison of Simple Potential Functions for Simulating Liquid Water. J. Chem. Phys. 1983, 79, 926−935. (42) Abraham, M. J.; Murtola, T.; Schulz, R.; Páll, S.; Smith, J. C.; Hess, B.; Lindahl, E. GROMACS: High Performance Molecular Simulations Through Multi-Level Parallelism from Laptops to Supercomputers. SoftwareX 2015, 1−2, 19−25. (43) Bussi, G.; Donadio, D.; Parrinello, M. Canonical Sampling Through Velocity Rescaling. J. Chem. Phys. 2007, 126, 014101. (44) Raiteri, P.; Laio, A.; Gervasio, F. L.; Micheletti, C.; Parrinello, M. Efficient Reconstruction of Complex Free Energy Landscapes by Multiple Walkers Metadynamics. J. Phys. Chem. B 2006, 110, 3533− 3539. (45) Branduardi, D.; Bussi, G.; Parrinello, M. Metadynamics with Adaptive Gaussians. J. Chem. Theory Comput. 2012, 8, 2247−2254. (46) Valsson, O.; Parrinello, M. Well-Tempered Variational Approach to Enhanced Sampling. J. Chem. Theory Comput. 2015, 11, 1996−2002. (47) Zhang, T.; Nguyen, P. H.; Nasica-Labouze, J.; Mu, Y.; Derreumaux, P. Folding Atomistic Proteins in Explicit Solvent Using Simulated Tempering. J. Phys. Chem. B 2015, 119, 6941−6951. (48) Kührová, P.; De Simone, A.; Otyepka, M.; Best, R. B. ForceField Dependence of Chignolin Folding and Misfolding: Comparison with Experiment and Redesign. Biophys. J. 2012, 102, 1897−1906. (49) Seibert, M. M.; Patriksson, A.; Hess, B.; Van Der Spoel, D. Reproducible Polypeptide Folding and Structure Prediction Using Molecular Dynamics Simulations. J. Mol. Biol. 2005, 354, 173−183.
(50) Miao, Y.; Feixas, F.; Eun, C.; McCammon, J. A. Accelerated Molecular Dynamics Simulations of Protein Folding. J. Comput. Chem. 2015, 36, 1536−1549. (51) McCarty, J.; Valsson, O.; Tiwary, P.; Parrinello, M. Variationally Optimized Free-Energy Flooding for Rate Calculation. Phys. Rev. Lett. 2015, 115, 070601. (52) Tung, H.-J.; Pfaendtner, J. Kinetics And Mechanism of IonicLiquid Induced Protein Unfolding: Application to the Model Protein HP35. Mol. Syst. Des. Eng. 2016, 1, 382−390. (53) Tiwary, P.; Parrinello, M. From Metadynamics to Dynamics. Phys. Rev. Lett. 2013, 111, 230602. (54) Scian, M.; Shu, I.; Olsen, K. A.; Hassam, K.; Andersen, N. H. Mutational Effects on the Folding Dynamics of a Minimized Hairpin. Biochemistry 2013, 52, 2556−2564. (55) Ming, M.; Brüschweiler, R. Prediction of Methyl-Side Chain Dynamics in Proteins. J. Biomol. NMR 2004, 29, 363−368. (56) Kokh, D. B.; Czodrowski, P.; Rippmann, F.; Wade, R. C. Perturbation Approaches for Exploring Protein Binding Site Flexibility to Predict Transient Binding Pockets. J. Chem. Theory Comput. 2016, 12, 4100−4113. (57) Pettersen, E. F.; Goddard, T. D.; Huang, C. C.; Couch, G. S.; Greenblatt, D. M.; Meng, E. C.; Ferrin, T. E. UCSF Chimera - a Visualization System for Exploratory Research and Analysis. J. Comput. Chem. 2004, 25, 1605−1612.
4756
DOI: 10.1021/acs.jpclett.7b01770 J. Phys. Chem. Lett. 2017, 8, 4752−4756
