levitt jmb85 normal modes

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J Mo l . i o l ( 1 9 8 5 ) 181, 423447

Protein Normal-mode Dynamics: Trypsin Inhibitor,Crambin Ribonuclease and LysozymeMichael Levitt, Christian Sander? and Peter S. Stern

Chemi cal Ph ysics Depart mentWeizwa.nn In stit ut e of Sci ence76100 Rehovot, I srael

(Recei ved 2 N ovember 1983, and i n r evi sed f orm 25 September 1984)

We have developed a new method for modelling protein dvnamics using normal-mode

analysis in internal co-ordinates. This method. normal-mode dynamics. is particularly wellsuited for modelling collective motion, makes possible direct visualization of biologically

interesting modes, and is complementary to the more time-consuming simulation ofmolecular dynamics trajectories.

The essential assumption and limitation of normal-mode analvsis is that the molecular

potential energy varies quadratically. Our study starts with energy minimization of theX-rav co-ordinates with respect to the single-bond torsion angles. The main technical task

is the calculation of second derivative matrices of kinetic and potential energy with respectto the torsion angle co-ordinates. These enter into a generalized eigenvalue problem. and

the final eigenvalues and eigenvectors provide a complete description of the motion in thebasic 0.1 to 10 picosecond range. Thermodynamic averages of amplitudes, fluctuations andcorrelations can be calculated efficiently using analytical formulae.

The general method presented here is applied to four proteins, trypsin inhibitor, crambin,ribonuclease and lysozyme. When the resulting atomic motion is visualized by computergraphics, it is clear that the motion of each protein is collective with all atoms participatingin each mode. The slow modes, with frequencies of below 10 cm -(a period of 3 ps). are t/hemost nneresting in that the motion in these modes is segmental. The rootmea,n-squareatomic fluetItrations, which are dominated by a8 few slow modes. agree well wit111experiment,al temperature factors (B values). The normal-mode dynamics of these fourproteins have many features in common, although in the larger molecules, lysozyme andribonuclease, there is low frequency domain motion about the active site.

1. Introduction

Internal dvnamics, especiallv collective motion, isimportant for protein function (see review byHuber & Bennet t, 1983). Bv collective motion wemean a process in which a substantial part of theprotein moves as a unit relative to other parts. A

recent beautiful example provided by X-raycrystallography is citrate synthase (Remington etal, 1982; Chothia & Lesk, 1984). In each catalyticcycle, the la,rge and small domain of the enzymefirst close around the substrate for catalvticaction?and then open a cleft between them for productrelease and the next round of substrate binding. A

number of other proteins also have (a,t least) twodifferent1conformations in two functionallv distinctstates. The tra,nsition from one to the other must be

tPresent address: Department of Biophysics. MaxPlanck Institute of Medical Research, J ahnstrasse 29,6900 Heidelberg. West Germany.

a major mode of internal motion, i.e. there must be

a low-energy pathway of collective motionconnecting the two, made possible by the overa.three-dimensional structure. Neither the stages

along this pathway nor the time-scale of the

transition have yet been observed experimentally.

It is clear, however, that such processes, whichinvolve the correlated motion of large masses, mustbe slower than vibrations involving onlv localizedatomic motion.E xperi mental studv of protein dynamics bvspectroscopic observation of low-frequency vibra-

tions has recently become possible (see review bvPeticolas, 1979). For example, the neutron time-of-flight spectrum of lysozyme below 100 cm- shows a broad peak at 75 cm- and a shoulder at25 cm- (Bartunik et al., 1982); a new Ramantechnique (Genzel et al., 1976) also gives peaks atthese two frequencies. Similar Raman peaks areseen in cc-chymotrypsin at about 29 cm- (Brown etaZ.,1972) and in acid phosphatase at about 25, 50

0022-2836/85/030423-25 $03.00/o 4 2 3 01985Academic Press Inc. (London) Ltd.


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2 0 IO

0 2 0 3 0 4 0 50 60 0 I O 2 0 3 0 4 0 5 0

6 0

60 . , . , . , , , , . , . , . , . , ,

Residue number

Figure 5. Comparing the experimental (9 . . .) and calculated variations with residue number of the a-carbon11 values (t emperature factors). The ith calculated B value is derived from the r.m.s. fluctuation of the ithz-carbon a,torn when all modes are excited at 300 IL The 4 graphs show: (a) BPTI . (b) crambin, (c) ribonuclease and (d)lysozvme. Regions of a-helix and P-sheet secondarv structure (Kabsch & Sander. 1983) are shown as unbroken lines.

.

I I I I I I0 3 0 6 0 9 0 120 15 0

Frequency (cm-)

Figure 6. Showing for BPTI how the average r.m.s. fluctuations of the cc-carbon atoms. 0:. and single-bond torsionangles. af, depend on the frequency of the mode L-. The value of at drops rapidly as the frequency increases. with az/gi= 0*005. This means that the low frequencies make the major contribution to 0,. the -average r.m.s. C4 fluctuation dueto all modes. On tlheother hand, 4 drops rather slowly as the frequency increases. with ~f~/g, = O-08. This means thatthe average r.m.s. t orsion angle fluctuation due to all modes. 04. depends on vibrations over a wide range of frequencies.


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Protein. Normal-mode Dynamics 4 3 5

Fig. 8


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Fig. 8, cont.


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Fig. 8, cont.


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M .L evitt , C. Sander and P. S. Stern

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Figure 9. Showing the domain motion due to the lowest frequencv modes in ribonuclease and lvsozyme. The arrowson the a-carbon backbone are drawn as described in the leg&d to Fig. 8. (a) Ribonuclease , v 1 = 2-43 cm - ,period =13-Y ps . (b) Lysozyme, v 1 = 2.98 cm- , period = 1 l-2 SI

there are significant negative correlations betweenresidues as far apart as 20 a.(ii) Correht ion of torsion angl es

Figure 11 shows the correlation coefficients for all

208 torsion angles of BPTI. Because the angles areordered @q $iy$i+, $i+, . . ., xi, . . . . the upperblock gives the main-chain/main-chain correlations,the lower block gives the side-chain/side-chaincorrelations, and the off-diagonal block gives themain-chain/side-chain correlations. The pattern ofmain-chain correlations is very like the contact map

(Fig. 10, lower triangle), showing that the main-chain torsion angles are correlated only for residuesclose in space. This same pattern is seen for the

main-chain/side-chain correlations but not for theside-chain/side-chain correlations. This indicates

that the torsion angles of side-chains packed closelvtogether do not vibrate in unison.

The correlated vibration of torsion angles in theP-hairpin and a-helix are considered in more detail.For the P-hairpin (Table 4) there are two types of

strong correlations: (1) intrastrand, in which thecorrelated angles are from adjacent residues in the


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levitt jmb85 normal modes

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