Proc. Natl. Acad. Sci. USAVol. 88, pp. 2773-2777, April 1991Biophysics

Rigid protein motion as a model for crystallographictemperature factors

(protein dynamics/x-ray refinement/molecular dynamics/protein crystallography)

JOHN KURIYAN*t AND WILLIAM I. WEIS**Howard Hughes Medical Institute, Laboratory of Molecular Biophysics, The Rockefeller University, 1230 York Avenue, New York NY 10021; and tHowardHughes Medical Institute, Department of Biochemistry and Molecular Biophysics, Columbia University, 630 West 168th Street, New York, NY 10032

Communicated by Joseph Kraut, December 20, 1990 (receivedfor review August 28, 1990)

ABSTRACT The extent to which the librations of rigid mol-ecules can model the crystallographic temperature factor profiles ofproteins has been examined. For all proteins considered, idinginfluenza virus tinn ahione ductase, myohem-erythrin, myoglobin, and streptavidin, a simple 10-paranetermodel [V. Schomaker and K. N. Trueblood (1968) Acda Crystal-logr. Sect. B 24, 63-761 is found to reproduce qualitatively thepatterns of maxima and minima in the isotropic backbone mean-square displacements. Large deviations between the rigid moleculeand individual atomic temperature factors are found to be corre-rated with a region in hemalutinin for which the refined struc-tural model is unatisfy and with errors in the structure in apartially incorrect model of myohemerythrin. For the high-resolution glutathione reductase suure, better results are ob-tained on treating each of the compact domains in the stcture asindependent rigid bodies. The method allows for the refinement ofreliable temperature factors with the introduction of minimalparameters and may prove useful for the evaluation of models inthe early stages of x-ray structure refinement. While these resultsby themselves do not the nature of the undeling dis-placements, the success of he rigid protein model in reproducingqualitative features of temperature factor profiles suests thatrigid body refinement results should be considered in any inter-pretation of crystallographic thermal parameters.

In their pioneering analysis of temperature factors in lyso-zyme, Sternberg et al. (1) pointed out that a model thatconsiders the protein molecule to be internally rigid doesquite well in explaining the broad features of the variation ofatomic temperature factors. In this extremely simple model,the atomic displacements are determined by a total of 10parameters that describe the rigid body translations andlibrations of the protein molecule. This model has not beenseriously pursued since their work, perhaps because there isample evidence from both experiments and molecular dy-namics simulations that proteins are by no means internallyrigid (2, 3). However, recent analyses of x-ray diffuse scat-tering data for insulin and lysozyme indicate that, in additionto internal fluctuations, individual molecules in the crystallattice undergo rigid body-like displacements, with magni-tudes comparable to those obtained from analysis of atomictemperature factors (4, 5). This is perhaps not surprising,considering the high solvent content of protein crystals, andit led us to investigate the extent to which the simple rigidbody model could account for the patterns of temperaturefactor variations in a wide range of proteins.

Following the general ideas described by Sternberg et al.(1), who based their work on the rigid body model derived bySchomaker and Trueblood (6), we have refined temperaturefactors for a number of proteins including influenza virushemagglutinin (7), glutathione reductase (8), myohemeryth-

rin (9), myoglobin (10), and streptavidin (11) by treating themas internally rigid bodies. Our results show that in all casesconsidered, the rigid body model is able to account for thebroad features of temperature factor variation in these pro-teins. It appears that this approach, which is very frugal in itsuse of free parameters, might be a reliable means of estimat-ing the pattern of temperature factor variation during x-raystructure refinement at low or medium resolution. As weshow below, large deviations between atomic temperaturefactors and those obtained by the rigid body treatment mayindicate regions of the protein that are disordered or forwhich the structural model is seriously in error.

METHODTemperature factors (B factors) are used to model the effectsof dynamics and disorder on x-ray scattering from crystals(12). The isotropic temperature factor, B, of an atom isone-third the trace of the anisotropic temperature factortensor, B, and is related to the mean-square displacement(Ar2) of the atom by

B = 8ir2(A&r2)/3. [1]

Note that (Ar2) = (u2) + (u2) + (u2), where ux, uy, and uz arethe displacements along the x, y, and z axes, respectively.Treatments of motional effects in proteins are usually

restricted to isotropic B factors (1 parameter per atom),although in exceptional cases anisotropic B factors may beused (6 parameters per atom). The rigid body model intro-duced by Schomaker and Trueblood (6) reduces the numberof displacement parameters from 6N or N (for a moleculewith N atoms), to 20 or 10, depending on whether anisotropicor isotropic temperature factors are used. In the more generalanisotropic case, commonly used for small molecules, therigid body parameters take the form of three 3 x 3 tensors.The symmetric tensors T and L describe the anisotropictranslational and rotational displacements of the rigid body,respectively, and a nonsymmetric tensor, S, describes thescrew motions or the coupling between the translations andthe rotations. One of the elements of S is redundant, leadingto a total of 20 parameters that are sufficient to describe themotions in the quadratic approximation (6). In the isotropiccase only the trace of the mean-square displacement tensor,(Ar2), is considered for each atom and for the rigid moleculemodel this is given by

(Ar2) = Tiso + L11(y2 + z2) + L22(x2 + z2) + L33(x2 + y2)

-2L12XY - 2L13xz - 2L23yz + 2S1x + 2S2y

+ 2S3z, [2]

tTo whom reprint requests should be addressed at: Box 3, TheRockefeller University, 1230 York Avenue, New York, NY 10021.

The publication costs of this article were defrayed in part by page chargepayment. This article must therefore be hereby marked "advertisement"in accordance with 18 U.S.C. §1734 solely to indicate this fact.

2773

Dow

nloa

ded

by g

uest

on

Nov

embe

r 28

, 202

0

2774 Biophysics: Kuriyan and Weis

where x, y, and z are the cartesian coordinates of the atom inquestion, relative to the molecular centroid of the rigid body.The parameters to be refined are the trace of the mean-squaredisplacement tensor, Tso, the components of the rotationaltensor, Lij, and three combinations of screw tensor elements,Si (1, 6). These 10 parameters specify the B factors for theentire protein molecule, with the B factors varying with theposition of atoms in the structure (Eq. 2). The parameters inEq. 2 may be determined by optimizing them against theindividual atomic temperature factors, as was done by Stern-berg et al. (1), or by refinement against the x-ray structurefactor data. We find that both approaches yield very similarresults, the advantage of the first being that it is an extremelyfast linear least-squares optimization, whereas the secondapproach may be advantageous in cases in which a reliableset of individual atomic temperature factors have not yetbeen obtained.

Sternberg et al. (1) showed that it is necessary, in the caseof lysozyme, to include' the three screw'parameters (Se; Eq.2) to best fit their refined atomic temperature factors. We useall three sets of parameters in the refinements described hereand refer to the resulting rigid body 'temperature factors asTLS B factors. We refer to B factors obtained by- refinementof a single parameter for every atom in the protein as atomicB factors. A limitation of the TLS parameters in the isotropiclimit is that even though L is fully determined, and themagnitudes of the angular displacements and the orientationsof the three rotation axes can thus be specified, the full detailof the S and T matrices is lost, preventing the determinationof the relative shifts between the three axes of rotation (6).We focus mainly on the rotation axes in the analysis thatfollows. We also follow Sternberg et al. (1) in using a squaredresidual based on residue-averaged B factors for the back-bone in order to judge the agreement between two sets ofBfactors. This residual, R2, is given by

residues

Ya [()calc - (Bidobs]'R2 [3residues

[Bmean - (Bi)obs]2

where (B),)ac is the average B factor for the ith residuecalculated from the TLS model, and (Bi)Obs is the average forthe individual atomic B factors. Bmean is the average over therigid body ofthe individual atomic B factors for the backboneatoms. R2 indicates the extent to which the rigid body modelaccounts for the deviation of the B factors from the mean,with an R2 value of zero corresponding to perfect agreement(1). The standard linear correlation coefficient between theTLS and the atomic B factors is also used, with a correlationcoefficient of 1.0 corresponding to perfect agreement. Aswith R2, the correlation coefficient is calculated for residue-averaged values. For simplicity, we focus on the backbonetemperature factors for most of the discussion that follows.Similar results are obtained on considering the' side chainatoms in'the analysis (results not shown).

RESULTSThe first system we consider is the influenza virus hemag-glutinin glycoprotein, one of the largest proteins for which arefined' x-ray structure exists (13). The protein is a trimer ofclosely packed monomers, with each monomer consisting oftwo polypeptide chains that result from cleavage of a pre-cursor polypeptide. Hemagglutinin crystallizes with 'a trimerin the crystallographic asymmetric unit, with an unusuallylarge solvent content of '78% by weight. A model for theGly-146 -- Asp variant of the X:31 influenza virus hemag-

glutinin has been refined to' an R factor of 22.2% at 3 A

resolution, the diffraction limit ofthe crystals (13). Individualatomic B factors were refined in this work, as they candistinguish between regions of high and low mobility, even atthis moderate resolution limit (13). 'The Gly-146 Aspstructure is used throughout the following analysis.We first considered the entire hemagglutinin trimer, con-

sisting of 11,871 nonhydrogen protein atoms, to be a singlerigid body. The 10 parameters of the TLS model wereoptimized against the individual atomic B factors obtainedfrom the x-ray structure refinement (13). The results for thebackbone atom's are shown in Fig. 1, and it can be seen thatthe broad features of the temperature factor profile are wellreproduced by the rigid protein model, except for one regionaround residue 58 in the second polypeptide of each mono-mer (see below). For the backbone atoms, the R2 valuebetween the target B values and those derived from the TLSmodel is 0.52, with a correlation coefficient of 0.70. Thedisplacements about the three rotation axes vary greatly,ranging from rms values of 1.540 to 0.740. The axes of rotationare very closely aligned with the principal axes of inertia (15)of'the long cylindrical molecule (Table 1), with the largestrotational displacement being about the 'longest axis of themolecule-i.e., the one about which the radius of gyration isminimal.A natural extension of this approach is to consider the

individual monomersaas independent rigid bodies (Table 1).Each rigid body now consists of 3957 atoms,'and on refine-ment of TLS parameters against the atomic B factors theagreement is somewhat improved (R2 = 0.43; correlationcoefficient, 0.73). This value of R2 is about the same as thatobtained by Sternberg et al. (1) for the backbone atoms oflysozyme (R2 = 0.45), and it is interesting to note that by thiscriterion the rigid body model works equally as well forlysozyme (129 residues) as it does for hemagglutinin (503residues in the monomer). Once again, the largest rotationaldisplacements are about the longest axes of the monomers.Note that in this case the axis about which the displacementis smallest has a nonphysical negative value for the mean-square rotation (-0.886 degrees2). This represents a break-down of the rigid body model for the monomer and isprobably due to its inability to account for the damping ofthemotions at the surface of the molecule by intermonomercontacts. In some of the cases that we consider below, thesmallest mean-square rotational displacement is similarlynegative and nonphysical (Table 1), but in all cases theabsolute magnitude of the displacement about this axis issmaller by about a factor of 10 than the largest rotationaldisplacement, showing that this nonphysical effect is a rela-tively small contribution to the total B factor. It is possible toincorporate a constraint into the refinement that L be positivedefinite, but we have not examined the consequences ofdoing so.We have also carried out refinements for hemagglutinin

wherein the TLS parameters were optimized' against thex-ray structure factor data. For example, in one such refine-ment the monomers were treated'as independent rigid bodies,resulting in 30 TLS parameters that enter into the x-raystructure factor calculation in place ofthe 11,871 temperaturefactors in the conventional treatment. These 30 parameterswere optimized by a conjugate-gradient least-squares struc-ture factor refinement procedure implemented by us in theprogram X-PLOR (16). The 33'water and 285 sugar atoms in themodel were excluded'from the rigid body calculations', andtheir temperature factors were kept fixed at the originalvalues. The TLS parameters were refined in consecutivecycles that varied the T, L, and S parameters separately; thiswas necessitated by the strong correlation between theparameters, which led to unreliable results if they weresimultaneously varied. The implementation of structure fac-tor first derivatives by fast Fourier transformation in X-PLOR

Proc. Natl. Acad Sci. USA 88 (1991)

Dow

nloa

ded

by g

uest

on

Nov

embe

r 28

, 202

0

Proc. Natl. Acad. Sci. USA 88 (1991) 2775

0

U0b

Residue Number

JB0 l IlI__ ____I__I.I I

20

10 ;W

0 100 200 300 400 50(Residue Number

30

FIG. 1. Backbone temperature factors. Backbone temperature fac- 60

tors (in A2), averaged over residues, are plotted as a function of residue Cnumber for various proteins. In all cases, the solid lines are the B factors 50obtained by refinement of individual atomic temperature factors against 0. lx-ray data. The broken lines are B factors obtained from the TLS model. 40

(A) Hemagglutinin: B factors are shown only for monomer 1 (see Table _____l1). The profiles for the other monomers are very similar. Each monomer

consists of two polypeptide chains. The residue numbers for the second m 20 .i A1 I

polypeptide chain have been incremented by 350. ,Atomic B factors U'f(13); -.-.-, TLS B factors with the trimer treated as a rigid body;., TLS 10

B factors with the monomers treated as independent rigid bodies. TheTLS parameters were obtained by optimization against the x-ray data at 03 A resolution (13). Arrow marks the region of largest discrepancy 0 20 40 60 80 100 120between atomic and TLS B factors. (B) Glutathione reductase: B factors Residue Numberfor one monomer. Atomic B factors taken from Protein Data Bank,Entry 3GRS (8); ----, TLS B factors with the monomer treated as a single rigid body and refinement against the B factor residual. (C)Myohemerythrin model with errors: The structure used is the Al structure described by Kuriyan et al. (14). Solid circles on top mark thoseresidues that have backbone positions that deviate from the correct structure by 0.4 A or more. The rms deviation for all backbone positionsis 0.61 A. -, Atomic temperature factors obtained by unrestrained refinement. The final R factor was 29.9o. ----, TLS B factors using theincorrect structure and refinement against the same x-ray data. The R factor was 34.0%o. X-ray data were obtained from the Protein Data Bank,Brookhaven National Laboratory, entry R2MHRSF.

does not lend itself to convenient use of more sophisticatedtreatments of the correlation. In each refinement cycle,parameters for all three monomers were varied simultane-ously and no positional coordinates were allowed to vary. Afinal crystallographic R factor of 0.249 was obtained by thisprocedure, compared to the value of 0.222 obtained by theuse of independently varying atomic temperature factors and0.258 with a single overall B factor (13). The R2 values aresomewhat higher (0.48-0.56) than that obtained by optimi-zation against the B factors, but the overall results are verysimilar, with a rms rotation of 2.70, about the longest axis formonomer 1 in both cases (Table 1).The patterns of temperature factor variation obtained by

using the rigid body model for the trimer are very similar tothose obtained by treating each monomer separately. Thissuggests that the dominant contribution to the temperaturefactor variation may arise from displacements of trimerichemagglutinin units in the crystal lattice. In both cases, theindividual atomic B factors in the region around residue 58 inthe second polypeptide of each monomer are significantlyhigher than for the rigid body model. This is a region ofqualitative discrepancy between the two models, where theuse ofindividualB factors results in values that are the largestfor the second subunit (except for the disordered C terminus),whereas the TLS B factors are lowest in this region. Inter-estingly, there is no electron density for residue 58 (see figure12 of ref. 13), even though the polypeptide chain is known tobe chemically intact (13). While it is possible that the proteinadopts more than one conformation in this region, the limitedresolution of the data precluded the construction of a satis-factory model (13). It is suggestive that the region of moststriking qualitative disagreement between the TLS B factorsand atomic B factors corresponds to one where the structuralmodel is unsatisfactory.

Influenza virus hemagglutinin is unusual for its high solventcontent, and it is not unreasonable to imagine that the limitednumber of crystal packing interactions results in rigid body-like motions dominating the observed atomic displacements.As a contrast, we consider the dimeric flavoprotein glu-tathione reductase, which is almost as big (478 residues in themonomer, which comprises the asymmetric unit of the crys-tal), but which forms crystals that are more tightly packedand that diffract to high resolution (8). A highly refinedstructural model including individual atomic B factors hasbeen refined against x-ray data to 1.5 A resolution (8). Thedimeric enzyme has 6998 nonhydrogen protein atoms in themodel and we first considered this to be a single rigid bodyand optimized TLS parameters against the isotropic atomic Bfactors obtained from the x-ray structure. While this treat-ment does result in a B factor variation that qualitativelyreproduces the broad features of the atomic B factor profile(correlation coefficient, 0.65), the quantitative agreement forthe backbone atoms is quite poor (R2 = 1.30; Table 1). Someimprovement is obtained on considering the monomer ratherthan the dimer as the rigid body (R2 = 1.08; correlationcoefficient, 0.74), and the resulting B factor profile is shownin Fig. 1B. Despite the high value of R2, the variation of Bfactors in the TLS model is in qualitative agreement with thatof the atomic B factors (Fig. 1B), with no regions of strikingdissimilarity such as that found in hemagglutinin. It is inter-esting that the angles between the rotation axes of the dimerand its principal axes are very small (-3°, 3°, 00), whereasthese angles are much larger for the monomer rigid body(=10°, 33°, 32°).

Glutathione reductase has a well defined internal domainstructure (8), and much better agreement is obtained on

treating each of the domains as independent rigid bodies.These are the FAD binding domain (residues 18-157 and

so

0U.0b

Biophysics: Kuriyan and Weis

III

OI A

Dow

nloa

ded

by g

uest

on

Nov

embe

r 28

, 202

0

2776 Biophysics: Kuriyan and Weis

Table 1. Rotational displacements and agreement factors for TLS parametersRotational

displacements Orientation of axes Radii of gyration

Method of Atoms in (w2)x, (CO2)Y, (W2)z, Ox, OYI OZ R, R Rzrefinement rigid body R2 deg2 deg2 deg2 deg deg deg A X A

HemagglutininTrimer B factor 11,871 0.52 2.40 0.72 0.56 10.4 12.8 12.5 31.2 38.3 43.0Monomer 1 B factor 3,957 0.43 7.22 2.23 -0.89 10.0 11.3 5.4 11.3 37.8 37.9Monomer 1 X-ray 3,957 0.48 7.22 1.77 -0.59 11.0 20.5 20.0 11.3 37.8 37.9Monomer 2 X-ray 3,957 0.56 8.57 1.90 -0.52 3.5 28.0 28.2 11.3 37.8 37.9Monomer 3 X-ray 3,957 0.54 8.86 1.31 -0.33 4.6 23.4 23.3 11.3 37.8 37.9

MyohemerythrinCorrect model X-ray 979 0.69 13.03 2.79 -0.59 18.9 39.3 35.6 9.1 11.8 12.4Incorrect model X-ray 979 0.% 7.48 3.02 2.00 3.0 28.3 28.4 9.2 11.9 12.5

Glutathione reductaseDimer B factor 6,998 1.30 3.35 1.51 -0.23 2.9 2.9 0.0 19.6 27.3 27.5Monomer B factor 3,499 1.08 3.54 2.82 0.16 10.0 33.6 32.5 17.2 19.9 21.8FAD domain B factor 1,609 0.97 9.55 1.74 1.31 5.9 43.6 44.0 11.3 19.2 19.5FAD domain,

without residues59-109 B factor 1,190 0.62 5.61 3.91 1.18 34.4 25.8 23.9 11.7 11.4 12.4

NADPH domain B factor 1,007 0.66 6.14 2.26 -0.66 17.3 13.2 11.4 11.9 10.9 12.8Interface domain B factor 1,513 0.47 4.46 3.51 0.52 39.3 52.8 41.6 9.2 12.8 13.3Interface domaindimer B factor 3,026 0.48 3.48 2.69 0.00 19.9 0.0 19.9 13.1 14.3 16.5

Method of refinement: B factor refers to optimization of TLS parameters against the individual atomic temperatures in the crystallographicmodel. X-ray refers to refinement ofTLS parameters by least-squares minimization of the standard crystallographic residual. For hemagglutinin,results are only shown for one monomer in the optimization against B factors. The results for the other two monomers are similar to those shownhere for the optimization against x-ray data. Rotational displacements: Diagonal elements ofthe rotational displacement matrix L, after the matrixis rotated into diagonal form. deg, Degrees. Orientation of axes: The angles between the axes of rotation and the principal axes of inertia ofthe molecule. Subscripts x, y, and z refer to the rotational axes, with x being the axis about which the rotation is greatest. Oi is the angle betweenthe ith rotational axis and the closest principal axis of inertia. In each case, the axes of rotation and the moments of inertia are calculated forthe rigid bodies being considered in the particular refinement. Unit masses were used in the calculation. Rx, Ry, Rz: Radii of gyration about theprincipal axes of inertia closest to the x, y, and z rotational axes. A small radius of gyration corresponds to a small molecular cross-sectionperpendicular to the axis in question.

294-364), the NADPH binding domain (residues 158-293),and the interface domain (residues 365-478), which is in-volved in extensive contacts that hold the dimeric enzymetogether. While the NADPH and interface domains are fairlycompact structures, the FAD domain is unusual in that itconsists of a compact structure from which two long helices(residues 59-109) form a large protrusion. When the entireFAD domain is considered to be a rigid body with 1609 atoms,the agreement with atomic B factors is not much better thanbefore (R2 = 0.97; correlation coefficient, 0.72). However,when the 419 atoms of the two protruding helices are re-moved from the rigid body, the agreement is considerablyimproved (R2 = 0.62; correlation coefficient, 0.83). The intactNADPH and interface domains both yield reasonable results(R2 of0.47 and 0.48, respectively, and correlation coefficientsof 0.84 and 0.92, respectively) when considered to be rigidbodies. It is perhaps not appropriate to consider the interfacedomain as a rigid body by itself, as it makes extensivecontacts with the equivalent domain of the other molecule inthe dimer. This can be seen clearly from results obtained byconsidering the two interface domains ofthe dimer as a singlerigid body. Even though this increases the number of atomsin the rigid body by a factor of 2 (from 1513 to 3026), theagreement between the TLS B factors and individual Bfactors isjust as good (R2 = 0.47 for a single interface domain;R2 = 0.48 for the dimeric interface domains; correlationcoefficient, 0.92 in both cases). Thus for glutathione reduc-tase, a model that considers each of the compact units in thestructure to be moving as independent rigid bodies providesa much better fit to the atomic B factor profile than modelsthat consider either the dimeric or the monomeric proteins tobe rigid bodies. Nevertheless, it is clear from Fig. 1B that therigid protein model is able to reproduce, in a qualitative

sense, the pattern of minima and maxima that are observedin the atomic B factor profile.We have also examined the effects of refining rigid body

parameters for a structural model that has serious errors in it.A partially incorrect model of myohemerythrin, obtained atan early stage ofthe structure determination ofthis protein byHendrickson and co-workers (9), was used for this test. Thismodel has previously been used in a test of x-ray refinementby simulated annealing and details regarding this structureare given in the report by Kuriyan et al. (14). While thisstructural model is correct in its chain topology and theplacement of the four helices, several local errors in chaintracing had prevented further refinement. These includederrors in the backbone positions of up to 3.5 A in severalloops. The partially incorrect model corresponding to struc-ture Al of Kuriyan et al. (14) was used for the refinement ofTLS parameters by using x-ray data to 2 A resolution (9), andthe entire protein was considered to be a single rigid body.For comparison, TLS parameters were also refined by usingthe same x-ray data but with the correct structural model (9).Unrestrained individual atomicB factors were also optimizedagainst the same x-ray data, using both structural models andstarting with a uniform B factor of 20 A2 assigned to eachatom and performing the same number of refinement steps inboth cases. No water molecules or disordered conformationswere present in any of the refinements, which were carriedout by the program X-PLOR (16).The unrestrained atomic B factors are well behaved when

the correct structural model is used, and these agree quali-tatively with the temperature factors obtained by using therigid body model (R2 = 0.69; correlation coefficient, 0.72;data not shown). The atomic B factors for the incorrectmodel, however, deviate strongly from the TLS results for

Proc. Natl. Acad. Sci. USA 88 (1991)

Dow

nloa

ded

by g

uest

on

Nov

embe

r 28

, 202

0

Proc. Natl. Acad. Sci. USA 88 (1991) 2777

the same structure in many places, and the quantitativeagreement is poorer (R2 = 0.97; correlation coefficient, 0.46).As shown in Fig. 1C, most of the residues for which thedeviations are large are those that have significant errors intheir backbone positions. The TLS B factors are similar forboth structural models since the TLS parameters are rela-tively insensitive to local features in the structure. Theseresults suggest that the rigid body model may be a usefulmeans ofestablishing the reasonableness ofa particular set oftemperature factors derived by using a structural model ofuncertain accuracy. It has long been recognized that high Bfactors are correlated with increased uncertainties in atomicpositions, and it may be useful to compare these with TLS Bfactors during the early stages of refinement.We have applied the method to a number of the proteins,

including myoglobin (all helical) (10) and streptavidin (all,8-sheet) (11), with results that are similar to those describedabove. For example, the values of R2 and the correlationcoefficient for the backbone atoms of myoglobin are 0.51 and0.71, respectively, in the case in which the rigid bodyparameters for 1265 atoms are determined by refinementagainst x-ray structure factor data. Likewise, for streptavi-din, the values of R2 and the correlation coefficient are 0.43and 0.81, with 17% atoms in the crystallographic asymmetricunit that was treated as the rigid body.

CONCLUSIONSFor most proteins, the limited resolution of the x-ray dataprecludes the refinement of independent anisotropic temper-ature factors for each atom as too many parameters arerequired. The rigid protein model introduces so few param-eters that it is not necessary to restrict refinements to theisotropic case, even at low resolution. Sheriff and Hendrick-son have shown, for example, that the refinement ofjust therigid body translational tensor (T) can significantly improvethe agreement with x-ray data in many cases (17). However,the introduction of the L and S tensors in the anisotropic casewould complicate drastically the fast Fourier transformationsused in the computation of structure factors and derivativesand we have not pursued this. While we have focused on theapplication of the TLS model to entire protein molecules ordomains, Howlin et al. (18) have refined anisotropic temper-ature factors by using the TLS model for selected side chaingroups in ribonuclease and have noted an improved agree-ment with the x-ray data. Similar results have been observedby Kim and co-workers for nucleic acid bases and phosphategroups (19).Diamond has recently used normal modes in the refinement

oftemperature factors for the small protein bovine pancreatictrypsin inhibitor (20). He finds that the 892 isotropic temper-ature factors for this molecule can be well reproduced by 19parameters. Ten of these are low-frequency normal modesthat account for internal fluctuations ofthe molecule, and theother 9 include rigid body translations and rotations about theprincipal axes of inertia of the molecule, similar to the T andL tensors. No account is taken of screw motions (the Stensor). The largest contribution to the temperature factorsarises from the rigid body motions rather than the internalmodes (20).

Our results are consistent with those of Diamond (20) andshow that the rigid body model is ofgeneral applicability. Thesuccess of the TLS model reflects the fact that atoms in theinterior ofa protein molecule generally have smaller displace-ments than those on the exterior. A rigid body model explic-itly accounts for the dependence of the magnitude of thedisplacements on distance from the center of mass of theprotein. However, analyses of molecular dynamics simula-tions where rigid body motions are removed also showincreases in mean-square displacement with increasing dis-tance from the centroid of the protein (3). X-ray diffractiondata provide no information on the nature of correlations inthe displacements, and our results should not be taken toimply that rigid body motions or displacements are in actualfact the major contributors to the temperature factor profiles.Further experiments using techniques such as inelastic neu-tron scattering or diffuse scattering are required to quantifythe actual contributions of true rigid body displacements (2).

We would like to thank Wayne A. Hendrickson, Martin Karplus,Joseph Kraut, and Don C. Wiley for helpful suggestions. This workwas supported in part by a grant from the National Institutes ofHealth to J.K. (GM 43094). J.K. is a Pew Scholar in the BiomedicalSciences. W.I.W. is a Howard Hughes Medical Institute Fellow ofthe Life Sciences Research Foundation.

1. Sternberg, M. J. E., Grace, D. E. P. & Phillips, D. C. (1979) J.Mol. Biol. 130, 231-253.

2. Neinhaus, G. U., Heinzl, J., Huenges, E. & Parak, F. (1989)Nature (London) 338, 665-666.

3. Brooks, C. L., Karplus, M. & Pettitt, B. M. (1988) Adv. Chem.Phys. 71, 1-249.

4. Doucet, J. & Benoit, J. P. (1987) Nature (London) 325, 643-646.

5. Caspar, D. L. D., Clarage, J., Salunke, D. M. & Clarage, M.(1988) Nature (London) 332, 659-662.

6. Schomaker, V. & Trueblood, K. N. (1968) Acta Crystallogr.Sect. B 24, 63-76.

7. Wilson, I. A., Skehel, J. J. & Wiley, D. C. (1981) Nature(London) 289, 366-373.

8. Karplus, P. A. & Schulz, G. E. (1987) J. Mol. Biol. 195,701-729.

9. Sheriff, S., Hendrickson, W. A. & Smith, J. L. (1987) J. Mol.Biol. 197, 273-2%.

10. Kuriyan, J., Wilz, S., Karplus, M. & Petsko, G. A. (1986) J.Mol. Biol. 192, 133-154.

11. Hendrickson, W. A., Pahler, A., Smith, J. L., Satow, Y.,Merritt, E. A. & Phizackerley, R. P. (1989) Proc. Natl. Acad.Sci. USA 86, 2190-2194.

12. Willis, B. T. M. & Pryor, A. W. (1975) Thermal Vibrations inCrystallography (Cambridge Univ. Press, Cambridge, U.K.).

13. Weis, W. I., Brunger, A. T., Skehel, J. J. & Wiley, D. C.(1990) J. Mol. Biol. 212, 737-761.

14. Kuriyan, J., Brunger, A. T., Karplus, M. & Hendrickson,W. A. (1989) Acta Crystallogr. Sect. A 45, 396-409.

15. Arfken, G. (1970) Mathematical Methods for Physicists (Aca-demic, New York).

16. Brunger, A. T. (1988) X-PLOR (Version 1.5) Manual (HowardHughes Med. Inst. and Dept. of Mol. Biophys. and Biochem.,Yale Univ., New Haven, CT).

17. Sheriff, S. & Hendrickson, W. A. (1987) Acta Crystallogr.Sect. A 43, 118-121.

18. Howlin, B., Moss, D. S. & Harris, G. W. (1989) Acta Crystal-logr. Sect. A 45, 851-861.

19. Holbrook, S. R., Wang, A. H.-J., Rich, A. & Kim, S.-H. (1988)J. Mol. Biol. 199, 349-357.

20. Diamond, R. (1990) Acta Crystallogr. Sect. A 46, 425-435.

Biophysics: Kuriyan and Weis

Dow

nloa

ded

by g

uest

on

Nov

embe

r 28

, 202

0