Motions of an a-Helical Polypeptide: Comparison of Molecular and Harmonic Dynamics D. PERAHIA,1 R. M. LEVY2 and M. KARPLUS3 ~Laboratoire d’Enzymologie Physico-Chimique et Mol~culaire, Bat 430, Universit~ de Paris-Sud, 91405 Orsay, France; 2Department of Chemistry, Rutgers, The State University of New Jersey, New Brunswick. New Jersey 08903. USA 3Department of Chemistry, Harvard University, Cambridge, Massachusetts 02138, USA, and Laboratoire de Chimie des Interactions Mol~culaires, Coll~ge de France, 75005 Paris, France SYNOPSIS Molecular and harmonic dynamics simulations have been performed for a decaglycine a-helix. The extent of anharmonicity for various observables is studied by a direct comparison of the two types of simulations, at temperatures ranging from 5 to 300 K. The fluctuations of the cartesian, internal and normal mode coordinates, and their time dependence, are analyzed. The heat capacity of the a-helix is evaluated both from the temperature response of the system to an energy perturbation and from the fluctuations in the temperature of the system. It is shown that the anharmonicity depends on the kind of observable. The root mean square atomic fluctuations have significant anharmonic components at temperatures above 100 K. In contrast, the dihedral angle fluctuations are much closer to being harmonic at all the temperatures considered. The analysis of potentials of mean force experienced by individual atoms shows that atomic displacements have approximately Gaussian distributions from 50 to 300 K, with different force constants at each temperature (quasi-harmonic model). At 300 K, the force constants obtained by molecular dynamics are significantly lower than in the harmonic ease. The time dependence of the projection of the molecular dynamics displacements on the normal mode coordinates shows that mode mixing is important above 100 K. The motions of the helix associated with the low-frequency normal modes are described and illustrated. INTRODUCTION diffraction and Mossbauer studies of protein crys- tals.5-v Further, molecular dynamics analysis of A wide range of experimental and theoretical stud- the time scales of the atomic fluctuations have ies have shown the existence of internal motions in indicated that they have a local subpicosecond proteins)’2 Of particular importance have been component superposed on a larger scale more col- molecular dynamics simulations, which have lective motion that develops over 1-10 or more s,9 demonstrated that the root mean square (rms) of picoseconds, Two extreme models for such con- ° atom displacements in proteins are on the order of densed-phase dynamic behavior can be formulated) 0.5 ~,, with a range of 0.1-2/~,~.3.4 a magnitude in In the first, analogous to a liquid, the atomic overall agreement with estimates from x-ray fluctuations can be described as localized diffusion in which the collisions between atoms play the essential role; in the second, analbgous to a solid, the atomic fluctuations involve the coupling of a 1990 John Wiley & Sons, Inc. CCC 0(D6-3525/90/040645-33 $04.00 large number of normal modes. It is likely that the Biopolymers, Vol. 29, 645-677 (1990) interiors of proteins contain both fluid-like and 646 PE~AHIA, LEVY, AND KARPLUS solid-like regionsS’~°’11; e.g., it has been suggested the classical equations of motions for the atoms of that hydrophobic clusters are more fluid-like, which the helix is composed. Up to and including whereas parts with considerable secondary struc- 100 K (5, 50, and 100 K), a kinetic energy twice the ture (a-helices, fl-sheets) are more solid-like.1° Re- final desired temperature was imparted to the sys- sults on the collective character of the motions tem in a single step; in all cases the temperature indicate that the harmonic model may be useful for dropped to its mean value within 50-100 steps. At analysis, even if anharmonic contributions are pre- temperatures above 100 K, the system was heated sent. Further, it has been suggested that the har- 25-50 K every 200 steps until the desired tempera- monic model may be improved by the quasi- ture was reached. The total kinetic energy im- harmonic approximation, in which the force con- parted to the system for the 20 trajectories with a stants are obtained not by differentiating the static given temperature was the same. The values of the potential energy function, but by averaging over average temperature of individual trajectories var- the potential of mean force at a given ied by up to 20%. In all cases, the initial coordi- temperature.12-’4 nates for the system corresponded to the minimum In this paper we present the detailed results of a energy geometry of the decaglycine helix and the study of the full molecular dynamics of an isolated individual trajectories of a set differed only in the a-helix performed at a series of temperatures (5-300 initial velocity components chosen by using a set of K). The results are compared with those for the random numbers corresponding to the appropriate same system in the harmonic approximation.~5-18 Maxwellian distribution. Once the final tempera- The magnitudes of the atomic fluctuations and ture was reached in each case, the analysis of the their time dependence are analyzed. Also, the po- trajectory was begun. tentials of mean force seen by the individual atoms Twenty short trajectories with different initial at the various temperatures are calculated. The conditions were used in an attempt to cover a magnitudes and the time correlation functions of larger portion of phase space than would a single the fluctuations in cartesian and internal coordi- extended trajectory in this relatively harmonic sys- nates are compared with the corresponding har- tem. A set of 20 trajectories was chosen after a monic dynamics results. The contributions of de- series of test calculations performed at 5 K in phasing and population relaxation to the form of which the system was expected to be harmonic. It the correlation functions are examined. The heat was found that this number gave consistent results capacity of the a-helix is evaluated both from the when repeated with a different distribution of ini- temperature response of the system to an energy tial velocities. Sets of 5-10 trajectories showed perturbation and from the fluctuations in the tem- deviation between different samples that were too perature of the system. The motions of the helix large. We found that the differences between the associated with the low-frequency normal modes rms displacements of the C~ atoms of the helix are described. between sets of 5 trajectories at 5 K were about 0.005 /~, which represents almost 10% of the rms value at this temperature. Similar deviations were found with sets of 10 trajectories. The difference METHODOLOGY decreased to 0.001 ~, when two sets of 20 trajecto- ries were compared. In addition, a single longer The system considered is the model a-helix trajectory was calculated at 300 K; it was equili- decaglycine, with the amino and carboxyl ends brated for 9 ps and the analysis period lasted 10 ps. blocked by the CH3CO and NHCH3 groups, re- The numerical integration of the equations of mo- spectively; the decaglycine helix contains eight hy- tions was carried out with the Verlet algorithm~9 drogen bonds. All atoms and degrees of freedom of with a time step of 5.10’~6 s. The program the molecule are treated explicitly, except that the CHARMM2° was employed for all of the computa- CH2 and CH3 groups are represented as extended tions° atoms. The details of the molecular model and the The mean square amplitude of the fluctuations potential functions used in the present simulation of a given atom, (hr2)D, along the direction D, is have been presented previously.~5 A series of tem- evaluated by peratures between 5 and 300 K defined by the mean kinetic energy of the system were studied. ~_, ((r~D- (r)D)2) For each temperature, 20 trajectories, each 1 ps in (hr2)D= ~-1 (1) length, were calculated by solving simultaneously n MOTIONS OF AN a-HELICAL POI,YPEPTIDE 647 in which riD is the projection of the instantaneous was evaluated. The components of this function coordinate of the atom in the Ith trajectory onto along given directions were also considered. In the the D direction and n is the number of trajectories harmonic case, the time correlation function of in the set; the angle brackets represent time aver- positional fluctuations was evaluated by the equa- ages over individual trajectories and (r)D = tion (ET.~(r~D))/n is the mean value of the displace- ment in the D direction for the 20 trajectories. 3 N - 6 The harmonic approximation to the amplitudes k~T ~ aj~ cos (wit) of atomic vibrations are obtained from the eigen- j=~ values and eigenvectors of the decaglycine a-helix15; <~r(O). ~r(t))h~~ = that is (2) Spectral densities corresponding to the variation of a given va~able in the molecular dynamics simu- lation were obtained by taking the Fourier trans- in which k is the Boltzmann constant, T is the e form of its autocorrelation function. The fast absolute temperature, N is the number of atoms, Fourier transform technique was used for these wj is the frequency of the jth normal mode, and calculationsfl2 In the harmonic case, the spectral aid is the component of the jth eigenvector along densities were obtained by considering the contri- D direction for the atom of interest. To obtain the bution of normal modes to the mean square fluc- normal modes, the energy of the a-helix was mini- tuation of the given variable. mized by using the full potential, and the second- derivative matrix of the potential energy with re- spect to the mass-weighted cartesian coordinate displacements relative to the minimal-energy RESULTS AND DISCUSSION structure was determined.