Biochemistry 2007, 46, 669-682
669
Polypeptide Motions Are Dominated by Peptide Group Oscillations Resulting from Dihedral Angle Correlations between Nearest Neighbors† James E. Fitzgerald,‡,§ Abhishek K. Jha,|,⊥ Tobin R. Sosnick,*,§,⊥ and Karl F. Freed*,| Department of Physics, Department of Mathematics, Department of Biochemistry and Molecular Biology, Department of Chemistry, Institute for Biophysical Dynamics, and The James Franck Institute, The UniVersity of Chicago, Chicago, Illinois 60637 ReceiVed August 3, 2006; ReVised Manuscript ReceiVed October 27, 2006
ABSTRACT: To identify basic local backbone motions in unfolded chains, simulations are performed for a variety of peptide systems using three popular force fields and for implicit and explicit solvent models. A dominant “crankshaft-like” motion is found that involves only a localized oscillation of the plane of the peptide group. This motion results in a strong anticorrelated motion of the Φ angle of the ith residue (Φi) and the Ψ angle of the residue i - 1 (Ψi-1) on the 0.1 ps time scale. Only a slight correlation is found between the motions of the two backbone dihedral angles of the same residue. Aside from the special cases of glycine and proline, no correlations are found between backbone dihedral angles that are separated by more than one torsion angle. These short time, correlated motions are found both in equilibrium fluctuations and during the transit process between Ramachandran basins, e.g., from the β to the R region. A residue’s complete transit from one Ramachandran basin to another, however, occurs in a manner independent of its neighbors’ conformational transitions. These properties appear to be intrinsic because they are robust across different force fields, solvent models, nonbonded interaction routines, and most amino acids.
All biological processes are perforce dynamical events. Local and global conformational changes are crucial to many of the biological functions performed by proteins. Since protein motions occur on a wide range of time scales (from subpicoseconds to seconds), they have been probed using many different experimental and theoretical techniques. However, relatively little attention has been devoted to elucidating the nature of the local changes (1-8), in spite of the fact such changes significantly contribute to most conformational transformations. The geometry of a protein backbone is largely described by a pair of backbone dihedral angles, Φ and Ψ. A third torsion angle ω exists between the amide nitrogen and carbonyl carbon, but this angle is essentially fixed at 180° due to the partial double bond character of the bond between the nitrogen and oxygen. If the protein backbone were rigid, a change in either of the Φ or Ψ angles would pivot a large portion of the protein, moving each atom by an amount proportional to its distance from the axis of rotation. In a † This work is supported by grants from the National Institutes of Health (GM55694 to T.R.S.) and the National Science Foundation (CHE-0416017 to K.F.F.) and Grant 0353989. J.E.F. is a University of Chicago Beckman Scholar and acknowledges the Arnold and Mabel Beckman Foundation. A.K.J. acknowledges the support of Burroughs Wellcome Fund Interfaces 1001774. * To whom correspondence should be addressed. T.R.S.: phone, (773) 834-0657; fax, (773) 702-0439; e-mail,
[email protected]. K.F.F.: phone,(773)702-7202;fax,(773)702-5863;e-mail,
[email protected]. ‡ Department of Physics and Department of Mathematics. § Department of Biochemistry and Molecular Biology. | Department of Chemistry and The James Franck Institute. ⊥ Institute for Biophysical Dynamics.
protein consisting of hundreds of amino acids, this rotation would involve a huge displacement of the protein and would be strongly prohibited due to inertial and viscous drag. Hence, such large-scale motions do not occur. To alter the dihedral angles while avoiding a huge unfavorable displacement, several folding models have utilized crankshaft motions, cooperative motions involving three neighboring torsion angles in which all other torsion angles remain relatively unperturbed (1, 7, 9-12). Through this crankshaft motion, neighboring residues cooperate to eliminate large-scale motions, long-range steric clashes, and excessive viscous or inertial drag caused by moving large portions of the protein through the solvent. However, whether cooperative crankshaft motions that eliminate huge displacements exist remains unclear. Simulations of synthetic polymers demonstrate the lack of a link between dynamical chain cooperativity and crankshaft motions (3, 13, 14). Instead, the simulations find that some macromolecules achieve the chain cooperativity necessary to avoid excessive displacements without crankshaft motions (3, 13, 14). The conditions under which crankshaft motions might persist remain unanswered for proteins. Local moves are essential to any motion where long-range displacements must be avoided. Theoretical investigations have considered the types of movements required to leave long-range structure unperturbed (15, 16). These local movements have been used to model the dynamics of protein backbones in an effort to make Monte Carlo conformational searches more efficient (16, 17). In addition to strictly local crankshaft moves, these studies also utilize longer-range
10.1021/bi061575x CCC: $37.00 © 2007 American Chemical Society Published on Web 12/22/2006
670 Biochemistry, Vol. 46, No. 3, 2007 motions involving several residues to avoid large, unphysical displacements of the chain by rotating multiple interior residues’ dihedral angles in such a manner that the ends of the loop are constrained to their initial positions. Evidence of the presence of anticorrelated motions of the backbone torsion angles (∆Φi and ∆Ψi-1) has appeared in normal mode analyses for alanine and glycine in helical structures (1). More recently, these correlations have been observed in several simulations of proteins, which find that this anticorrelation is part of a crankshaft motion in which more distant torsion angles are uncorrelated (2, 4-7, 1820). However, these computational studies are performed using molecular dynamics simulations that are too short to ensure sufficient equilibration of the trajectories (presumably due to computational limitations). In addition, recent simulations demonstrate that different force fields yield strikingly different results for the backbone dynamics of small peptides (21-25). It is, therefore, important to reexamine these types of motions in a more general and systematic context. Very recently, structure refinement using heteronuclear NMR relaxation measurements has detected evidence of local crankshaft motions that involve anticorrelations between the motions of torsion angles Φi and Ψi-1 (26). Simulations using experimental constraints find that to a large degree, crankshaft motions exist across the entire protein backbone, with only a few regions exhibiting a lack of this correlated mode of motion. Here we delineate the conditions under which crankshaft motions are present by performing simulations with modern force fields and various solvent models to test the degree to which these motions are ubiquitous in protein backbone dynamics. Our simulations consider three popular force fields, a variety of nonbonded interaction routines, and both implicit and explicit solvent models. Simulations are performed using a variety of amino acid sequences, thereby enabling a comparison of how the different force fields and solvent models describe the correlated motions and other dynamic and equilibrium properties. Finally, we discuss the implications of our findings. METHODS
Fitzgerald et al. by Shen and Freed (28, 29) which has been tested by comparison with explicit solvent simulations. The all-atom LD simulations proceed by solving the Langevin equation
mi
∂u bi ) -∇ B ı U - σ ib ui + B Ai(t) ∂t
where b ui is the velocity of atom i, ζi is its friction coefficient, mi is its mass, B Ai(t) is the random force acting on atom i, and U is the energy function. These simulations are performed with several different energy functions as described below. In general, the energy function is a combination of an all-atom force field that represents the solutesolute interactions and implicit solvation terms which attempt to reproduce the influence of the solvent. These energy functions are given by
Usolute ) Ubond + Ubend + Utorsion + Uimproper + UVDW Usolvent ) Ucharge() + Usolvation(σ) such that the total potential is
U ) Usolute + Usolvent The energy function, Usolute, is henceforth termed the force field. We use several commonly used force fields, namely, CHARMM27 (30), all-atom OPLS (31), and Garcia and Sanbonmatsu’s GS-AMBER94 (24). Differences between these force fields result in different equilibrium Φ and Ψ distributions, as well as in the prediction of different dynamical properties (21). All three force fields utilize individual energy terms of the form
∑
Ubond )
/2Kb(b - bo)2
1
bonds
Ubend )
∑
/2Kθ(θ - θo)2
1
bond angles
∑
Utorsion )
Kφ[1 + cos(nφ - δ)]
torsion angles
Simulation Protocol. Computer simulations are run for a variety of sequences, force fields, solvation models, and nonbonded interaction routines. The sequences are chosen to test whether there is a dependence on the number and type of residues. Simulations of an alanine7 (Ala7) and an alanine20 (Ala20) chain are used to test the influence of peptide length. A heteropolymer (EDEVARLKKLLW) is investigated because it both has sequence heterogeneity and is short in length while possessing significant helical structure according to circular dichroism measurements (27). This helical structure is important for observing numerous transitions involving the R-basin, which are found to be much rarer than β T Polyproline II (PPII) transitions in the Ala7 and Ala20 oligomers. Because proline and glycine behave rather differently compared to other residues, simulations have also been run for a proline7 (Pro7) and a glycine7 (Gly7) chain. All peptides are amino-acetylated and carboxyamidated to eliminate charges at the termini. Langevin dynamics (LD) simulations of these sequences have been probed using the implicit solvent model developed
∑
Uimproper )
/2Kχ(χ - χo)2
1
improper torsion angles
UVDW )
∑
nonbonded pairs
[( ) ( ) ] σij
4ij
r
12
-
σij
6
r
where b is the bond length, θ is the bond angle, χ is the improper torsion angle, φ is the torsion angle, the K values are the respective force constants, and the subscript o indicates an equilibrium value. The parameter n in the torsion potential is the multiplicity, and δ is the phase. The van der Waals potential contains ij (the well depth), σij (the LennardJones diameter), and r (the nonbonded distance). The difference between force fields enters through the values used for the parameters in these energy terms. In particular, the coefficients used in the torsion potential differ strongly in these force fields. Previous work catalogues these differences and discusses them in detail (21).
Dihedral Angle Correlations between Nearest Neighbors
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The solvation potential Usolvent defines the implicit solvent model. The first term of this energy function is a screened Coulomb potential of the form
qiqj Ucharge() ) ∑ i σ(∆Φi,∆Ψi), but the inequality is not quite as strong as for the implicit solvent and vacuum simulations. More specifically, the explicit solvent calculations yield a slightly stronger correlation between torsion angles on the same amino acid [explicit solvent gives σ(∆Φi,∆Ψi) in the range [-0.36,-0.17], while implicit solvent gives σ(∆Φi,∆Ψi) in the range [0.03, 0.09]] and a slightly weaker correlation between ∆Φi and ∆Ψi-1 [explicit solvent gives σ(∆Φi,∆Ψi) in the range [-0.67,-0.56], while implicit solvent gives σ(∆Φi,∆Ψi) in the range [-0.72,-0.71]]. In all cases, all correlations die out at longer distances, leaving the peptide group’s oscillatory motion as the dominant (anti)correlated motion. Because the dominance of the anticorrelation of (∆Φi,∆Ψi-1) also persists in the vacuum simulations and is independent of force field, the prevalence of this anticorrelated motion must arise from structural properties of the peptide, namely, from inertial and/or rigidity effects, rather than viscous drag. Simulations of the heteropolymer EDEVARLKKLLW have also been run using N2 and N∞ interactions. Figure 1 of the Supporting Information demonstrates that the pattern of correlations is almost identical for both of these trajectories. Further discussion of these simulations is contained in the text. SUPPORTING INFORMATION AVAILABLE Graphical representation of the correlation coefficients for movements of the torsion angles of the heteropolymer EDEVARLKKLLW using the implicit solvent model, the Garcia force field, and N2 interactions and N∞ interactions. This material is available free of charge via the Internet at http://pubs.acs.org. REFERENCES 1. Go, M., and Go, N. (1976) Fluctuations of an R-Helix, Biopolymers 15, 1119-1127. 2. Mccammon, J. A., Gelin, B. R., and Karplus, M. (1977) Dynamics of Folded Proteins, Nature 267, 585-590. 3. Helfand, E., Wasserman, Z. R., and Weber, T. A. (1979) Brownian Dynamics Study of Polymer Conformational Transitions, J. Chem. Phys. 70, 2016-2017. 4. Levitt, M. (1983) Molecular-Dynamics of Native Protein. 2. Analysis and Nature of Motion, J. Mol. Biol. 168, 621-657. 5. Dellwo, M. J., and Wand, A. J. (1989) Model-Independent and Model-Dependent Analysis of the Global and Internal Dynamics of Cyclosporine-A, J. Am. Chem. Soc. 111, 4571-4578. 6. Garcia, A. E. (1992) Large-Amplitude Nonlinear Motions in Proteins, Phys. ReV. Lett. 68, 2696-2699. 7. Fadel, A. R., Jin, D. Q., Montelione, G. T., and Levy, R. M. (1995) Crankshaft Motions of the Polypeptide Backbone in Molecular-
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