Proc. Nati. Acad. Sci. USAVol. 83, pp. 8147-8151, November 1986Biophysics

Ultrasonic absorption evidence for enhanced volume fluctuations inthe tobacco mosaic virus protein helical aggregate

(relaxation spectrometry/conformatlonal fluctuations/segnental mobility/protein assemblies/tobacco mosaic virus loop)

R. CERF AND Y. DORMOYLaboratoire de Spectrometrie et d'Imagerie Ultrasonores, Unite Associde au Centre National de la Recherche Scientifique, Universitt Louis Pasteur, 4, rueBlaise Pascal, 67000 Strasbourg, France

Communicated by Manfred Eigen, April 30, 1986

ABSTRACT The increased ultrasonic absorption broughtabout by self-assembly of biomolecules is analyzed for theassembly process from the 20S aggregate to the helical rod oftobacco mosaic virus protein in solution, designated here as the20S -* P-helix transition. The analysis is based on theoreticaldevelopments in ultrasonic relaxation spectrometry presentedpreviously and illustrates the possibility that this technique canbe used for characterizing fluctuations. The analysis makes useofNMR data for the system in solution and of x-ray diffractiondata for the closely related transition from the two-ring disk tothe vrion. These x-ray data comprise the high-resolutionstructures and the Debye-Waller temperature factors of themain chain atoms of both the two-ring disk in crystals and thevirion in oriented gel form. First, reduced ultrasonic spectraare obtained for the 4S, 20S, and helical rod aggregates. Thefluctuation-enhancement factor for the helical rod is deter-mined independently of any deconvolution into normal modesof relaxation and is shown not to depend on the particularprocedure of reduction employed. The increase of ultrasonicabsorption in the 20S -÷ P-helix transition primarily revealsenhancement of the relaxing system's normal-mode volumefluctuations. The observed relaxations probably involve oneconformational process per subunit. The normal-mode volumefluctuations are then estimated from a bimodal least-squaresbest fit to the data, and a lower bound for the reaction volumeassociated with the fast steps is obtained. Two mechanisms areconsidered as follows: (i) a destabilization process in which thefree-energy difference between two states is reduced and (ii) anincrease in reaction volumes of local conformation changes inthe helical aggregate, resulting from the formation of a"carboxyl cage-like" structure and from the change in envi-roument produced inside the cage. Increased reaction volumeswould not be detected with x-ray diffraction. The possibleoccurrence of fluctuations at the RNA binding site raises thequestion of whether a quaternary structure that exhibitssignificant conformational fluctuations must be present for thebinding of the nucleic acid.

Several physical techniques are being used to study fluctu-ations in biomolecules, among them fluorescence quenching(1) and relaxation (2), NMR (3, 4), and flash photolysis (5). Inx-ray crystallography, atomic motions have long been knownto contribute to a decrease in the average reflection intensitywith increasing scattering angle, as expressed by theDebye-Waller temperature factors, from which atomic meansquare displacements may be obtained.The measurement of a dissipative property-for example,

the attenuation of an ultrasonic longitudinal wave-can alsobe used to characterize the fluctuations ofa system, by virtueof the general relationship between transport properties andfluctuations, known as the fluctuation-dissipation theorem.

Frequencies currently accessible to ultrasonic methods-0.3-200 MHz-are similar to the frequencies of conforma-tional transitions between rotameric states in small moleculesor in polymeric chains (6, 7).The unexpected observation leading to the present work

was that in many biomolecular assemblies [spherical viruses(8), tobacco mosaic virus (TMV) coat protein (9, 10), frogvirus 3 (11), and microtubules and hemocyanins (12)] theultrasonic absorption is larger in an assembled system than inits dissociated parts. This result surprised us because thecommonly accepted sources of ultrasonic absorption areprotein-solvent interactions, which should be largest in thedissociated system.An interesting example of ultrasonic absorption enhanced

by an assembly process is provided by the polymerization ofthe 20S aggregate ofTMV protein into a long helical rod. Ithas been suggested that the 20S aggregate present in solutionsoflow ionic strength (13) is similar in structure to the two-ringdisk crystallized from solutions of high ionic strength. Thedisk is made of 34 subunits in two cylindrical closed rings,each containing 17 subunits. It has, however, been reported(14) that the 20S aggregate contains 39 + 2 subunits and may,in fact, be a helix rather than a closed disk.The structure of the crystallized two-ring disk has been

determined to 2.8-A resolution (15) using x-ray diffraction,while that of the virion in oriented gels has been determinedto 4-A (16) and to 3.6-A (17) resolution. The structures ofeachprotein subunit in the disk and in the virion differ in severalaspects, most remarkably in the region proximal to the RNAbinding site. In that region, 25 amino acid residues (residues89-113) form a disordered loop in the disk but are ordered inthe virion.We are concerned here with species that can be studied in

solution, and especially with changes in their dynamicsoccurring as a result of the transition from the 20S aggregateto the protein helix (whose commonly obtained A form has 161/3 subunits per turn). This transition will be referred tobelow as the 20S -* P-helix transition.X-ray diffraction studies on oriented gels have shown that

the protein helix is very similar in structure to the protein partof the virion (18). The disorder in the loop region of TMVprotein aggregates in crystals could in principle be structuralor dynamic in origin; in the case ofTMV protein in solution,1H NMR investigations at 360 MHz (19) suggested that theloop is highly mobile in the 20S aggregate. Significantly,further evidence from NMR investigations shows that thisobserved dynamic disorder persists to some extent in thehelical rod (19).We now consider the meaning of the experimentally

observed 4-fold increase of ultrasonic absorption producedby the 20S -) P-helix transition (9), in terms of the system'sfluctuations, and how they are changed in the transition. Thishas become a relevant question, since theoretically ultrasonic

Abbreviation: TMV, tobacco mosaic virus.

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8148 Biophysics: Cerf and Dormoy

spectroscopy can provide an absolute measurement of thesum offluctuations and of the corresponding relaxation modecontributions for a molecule, or an assembly (20, 21). Thistheory is summarized first.

Summary of Theory Used

The property of interest may be illustrated using the simplestmodel, i.e., a two-state model. In dilute aqueous solution thecontribution Aa to the ultrasonic absorption produced by crrelaxing systems per unit volume is given by (22, 23)

Aac p cc,(Av) 2 K T[co2 2kT (1 + K)2 1 +(o2,r2

where p is the density of the medium, c is the sound velocityin it, Av is the difference between the volumes of the twostates, K is the ratio between the unperturbed populations ofstates II and I, X is the relaxation time, w is the angularfrequency, k is Boltzmann's constant, and T is the absolutetemperature. For the mean-square volume fluctuation of therelaxing system we have

(8V2) = (AV)2K(1 + K)-2, [2]

and, from Eq. 1, Aa/w2 is proportional to (8v2). If a molecule,or an assembly, contains independent relaxing systems, thetotal fluctuation will appear in Eq. 1. Since one is mainlyinterested in the kinetic characteristics of the relaxing systemitself, in particular the fluctuation (8v2), the term "sum offluctuations" has been preferred in ref. 20 to total fluctuation.For ,A independent identical relaxing systems per molecule(or per assembly), this quantity is

S = pU(Sv2). [3]

Eq. 1 then reads

Aa PCCMS T [4]W2 2kT 1 +w02,r2

where CM = ,u-1c, is the known number of molecules (or ofassemblies) per unit volume.

Eq. 4 is for our problem the explicit expression of thefluctuation-dissipation theorem, and its form is of generalvalidity (21). For a multistate system one has

Aa PCCMZ STj,c)2 2kT i 1 +w2rJ

where Tj is the jth relaxation time, and Sj is the normal-modecontribution to the sum of fluctuations S. In turn

S = Ay2, [6]

where y2 is the jth normal-mode contribution to the fluc-tuation (8v2) = Yj y2, and

S =XS. [7]

The most obvious consequence of Eq. 5 is that it makespossible an absolute measurement of the sum of fluctuationsS and of its normal-mode contributions Sp if the separaterelaxation times Tj and the corresponding relaxation ampli-tudes, i.e., the factors multiplying the frequency-dependentterm (1 + w 2j)-' in the right-hand side of Eq. 5, can beextracted from a relaxation spectrum. This is not alwayspossible for biological systems because the low frequencypart of their ultrasonic spectra may not be measurable with

existing techniques. In this case, deconvolution into evenonly two relaxation modes cannot be unequivocal, as itrequires the following five adjustable parameters: the highfrequency limit a. of the solution's absorption coefficient a,which must be measured to derive the experimental Aa = a- a,~ plus two relaxation times and two amplitudes.However, simple situations may present themselves, for

example when, in a process such as the assembly of proteinsubunits, all relaxation times rj are unchanged, whereas allnormal-mode contributions Sj are multiplied by the samefactorf. According to Eqs. 6 and 7, the fluctuations (5v2) andS then are multiplied by f. Whenf> 1, the fluctuations areenhanced. Furthermore, according to Eq. 5 the observableAa is uniformly multiplied by f over the whole frequencyrange, so that a composite curve, the "reduced" ultrasonicspectrum, may be constructed from the measurements car-ried out before and after the process considered has oc-curred. In this case, which is illustrated in Fig. 1, the valueofthe single parameters, the fluctuation-enhancement factor,may be determined safely.The parameters affecting the volume fluctuations are

manifest in Eq. 2 for the two-state system. For this model, thevolume fluctuations are enhanced when either Av is beingincreased, or K is made closer to unity, as the ground stateI is destabilized with respect to state II. In a destabilizationprocess the free-energy difference between states I and II isreduced, and the equilibrium population of state I is de-creased relative to that of state II.

In the analysis of the experimental data, reduced spectracontaining two relaxation modes must be considered. Rathergeneral conditions were given in ref. 20 under which athree-state system (having two normal modes of relaxation)may lead to a reduced spectrum through destabilization ofone of the states. This is the case when, for example, state Iis much stabler than state II and the transition rate I -- II isbeing increased (multiplied by y> 1) on destabilizing state I.Ifall other transition rates are kept fixed, the relaxation timesare unchanged and the normal-mode contributions Sj (I = 1,2)are each multiplied by y; the fluctuation-enhancement factoris then f = y.

If, in addition to an increase in the rate of I -- II, all othertransition rates are each multiplied by 8, the relaxation timesare multiplied by 3-1. A reduced spectrum may still beobtained in a semi-logarithmic plot of Ac against w when thefrequency scale is renormalized on shifting each spectrumalong the w axis by log106 and Aa is multiplied by 'y-2. Thefluctuation-enhancement factor in this case is f = y8'1.On the other hand, if only the reaction volume Av of either

the transition I -- II or the transition II -* III is important,both values of Sj (j = 1,2) are proportional to (Av)2. Thisstraightforwardly follows from equation 44 of ref. 21 and fromthe form of the normalized volumes therein. Multiplication ofJAvi by V'y then has the same effect as the destabilization ofstate I considered above.

Enhanced Volume Fluctuations in the Helical Aggregate ofTMV Protein

Reduced Ultrasonic Spectra and Fluctuation-EnhancementFactor. Ultrasonic absorption measurements have been car-ried out in collaboration with Michels, Schulz, and Witz (9,10) on the Strasbourg (vulgare) and Dahlemense strains ofTMV virions, as well as on the helical aggregate and smalleraggregates of the Strasbourg strain protein. We analyze herethe Strasbourg strain protein aggregates; details of the ma-terials and methods used may be found in refs. 9 and 10.For all particles and subparticles the values of Aa were

linear functions of the protein concentration, showing thatthe effects being characterized were only inside (and notbetween) particles and subparticles. The most striking fea-

Proc. Natl. Acad. Sci. USA 83 (1986)

Proc. Natl. Acad. Sci. USA 83 (1986) 8149

ture concerns the differing behaviors of the 20S and helicalrod aggregates.

Fig. 1 shows a reduced spectrum for the A protein (amixture of species typically containing 3-10 subunits), the20S aggregate, and the helical rod that was obtained onmultiplying the values ofAa for the A protein and for the 20Saggregate by thef values in Table 1. The protein concentra-tion was 5 mg/ml, and the temperature was 31TC. No changesof relaxation times are involved in this reduction, andfis theenhancement factor both of the fluctuations and of theobservable Aa.A second reduced spectrum was obtained by allowing the

relaxation times to differ from one aggregate to the other.This is shown in Fig. 2 in which the spectra for the A proteinand for the 20S aggregate were shifted using the 8 factors inTable 1. However, the values ofthe fluctuation-enhancementfactor f obtained from both the reduced spectra are essen-tially the same, as can be seen by comparing the figures inTable 1. This shows that meaningful values of the parameterf can be measured with ultrasonic techniques.From results obtained at 20TC only an imperfectly reduced

spectrum was obtained. Fluctuations are enhanced in thehelix at 20'C also, though the process of enhancement doesnot fully fulfill the simple conditions considered above. Thoseresults further show that obtaining a reduced spectrum, as inFigs. 1 and 2, is not merely a manifestation of insensitivity ofthe analysis.The 20S -+ P-Helix Transition Enhances the Relaxng

System's Fluctuations. A mere increase of the number ,u ofrelaxing systems brought about by the polymerization to thehelix might in principle explain enhancement of the values ofthe Sp, in accordance with Eq. 6. Thus, each enhancementfactor of the ultrasonic absorption Aa in Fig. 1 could beexplained in terms of an increase of /i by a constant factor,equal to 4.6 in the polymerization of the A protein to the helixand equal to 4 in the polymerization of the 20S aggregate.Therefore, one must wonder, at first, whether in an ultrasonicexperiment the value of 1L is not merely expressing thenumber of intersubunit contacts.

Considering only a linear trimer component of the Aprotein with two nearest-neighbor contacts or "bonds"among the three protein subunits (assumptions that tend tounderestimate the number of contacts in the A protein), therewould be two-thirds of a bond per protein subunit. Comparedwith the three bonds per subunit in the helix, this wouldindeed lead to an increase in the number ofbonds per subunitby a factor of 4.5 in the polymerization to the helix, which

300

0 ,

1 10 102Frequency, MHz

FIG. 1. Ultrasonic absorption of the TMV protein aggregates at31C and 5 mg/ml against the frequency N = e/2)r. 0, Values for theA protein multiplied by 4.6; A, values for the 20S aggregate multipliedby 4; o, values for the helical aggregate. Adjustment is visual;reduced spectrum is hand drawn.

Table 1. TMV protein helix versus smaller protein aggregatesAggregate f= 5at y,_2t f= 76-1t

A protein 4.6 0.67 6 420S aggregate 4 0.89 4.5 4

The TMV protein helix was compared to smaller protein aggre-gates. Parameters were obtained from the reduced spectra of Figs. 1and 2, respectively. Protein concentration, 5 mg/ml; temperature,31'C. Note that the fluctuation-enhancement factor f is largelyindependent ofthe reduction procedure employed; yand _y-2 are theenhancement factors of Aa/N2; log108 is the shift factor in thefrequency scale.*From reduced spectrum of Fig. 1.tFrom reduced spectrum of Fig. 2.

compares well with the increase of Aa. However, if thestructure of the 20S aggregate resembled that of the two-ringdisk, which has two bonds per subunit, the increase innumber of bonds produced by the 20S -+ P-helix transitioncould not exceed 50%, instead ofthe 300% increase ofAa thatwas measured.An increase in ,u through creation of new relaxing systems

in the transition, at locations other than the contacts betweensubunits, is unlikely also. This follows from the highlyconserved characteristics of the ultrasonic spectra for allthree protein aggregates, as manifested in the fact thatreduced ultrasonic spectra were obtained.

Therefore, the enhanced ultrasonic absorption that hasbeen observed in the 20S -- P-helix transition of the TMVprotein primarily reveals enhancement of the relaxing sys-tem's fluctuations yj in Eq. 6.The absence of any pH dependence in the ultrasonic

spectra, reported earlier (9), is probably not compatible withlocal fluctuations associated with proton-transfer reactions.The two salient features, which we stressed, i.e., theintraparticle nature of the effect and the success of thereduction procedures resulting in the spectra shown in Figs.1 and 2, then lead to the conclusion that the observedrelaxations probably involve one conformational process persubunit. Several independent processes could, indeed, hard-ly be all modified in a way that would produce a reducedspectrum. As the spectra of Figs. 1 and 2 contain at least tworelaxation modes, the process must involve at least three

400

E 300

p200

100

01 10 102Frequency, MHz

FIG. 2. Same data as in Fig. 1. The values of Aa/N2 for the Aprotein and the 20S aggregate are multiplied by a constant factor aftereach spectrum has been shifted along the frequency axis (see Table1). o, Values for the A protein multiplied by 6; A, values for the 20Saggregate multiplied by 4.5; o, values for the helical aggregate.Adjustment is visual; reduced spectrum is hand drawn.

Biophysics: Cerf and Dormoy

8150 Biophysics: Cerf and Dormoy

conformational states, i.e., two steps, according to thefollowing scheme: I = II III.

Estimates of the Normal-Mode Volume Fluctuations. Thedata for the three protein aggregates were used to compute byleast-squares best fitting a reduced bimodal spectrum. TheAa values for the A protein and for the 20S aggregate weremultiplied by thef factors (4.76 and 4.005 for A protein and20S aggregate, respectively), which turned out to be nearlyidentical to those (given in Table 1) derived by visualadjustment (see Figs. 1 and 2). The relaxation times T1 and T2thus determined were 0.14 and 0.01 Ius, respectively, and thevalues of (S,)1/2 and (S2)112 for the helical aggregate were 25and 43 cm3/mol of subunit, respectively.

Neglecting contributions from relaxation modes that couldoccur at frequencies below those investigated, we find for thesum of fluctuations in the protein helix S112 - 49 cm3/mol ofresidue. Assuming one conformational process per subunit (/I= 1 in Eq. 3), in accordance with the model proposed above,(8v2)1"2 has the same value: 49 cm3/mol of subunit. Therefore,(8V2)1"2/Vs, where V, is the volume of the subunit, is less than4 x 10-3. The preceding value of S rests on the choice , =1, but is otherwise independent of the assignment of therelaxations.As the relaxation times ri and T2 differ by at least a factor

of 10, the conditions described by Eigen and de Maeyer (22)are met, under which the fast time and correspondingamplitude in a three-state system depend exclusively on thekinetic and thermodynamic parameters ofthe faster reaction,e.g., I II. The fast mode may thus be described as anindependent single-step reaction (22). This makes it possibleto derive from the value of S2 a lower bound of the reactionvolume Av associated with I II. Using Eq. 2, in which thefactor K(1 + K)h2 is given its highest value 1/4 by setting K= 1, we find for the reaction volume in the protein helix thatJAvJ > 86 cm3/mol of subunit.

Discussion

Destabilization Versus Volume Change. Destabilization of aconformational state results in enhanced conformationalfluctuations. For example, in a two-state system the mean-square fluctuation in occupancy is

(8p2) = K(1 + K2)-1, [8]

and, if the x coordinates of an atom differ by Ax in the twostates, the mean-square displacement of the atom is

(8x2) = (AX)2K(1 + K2)-. [9]

The conformational fluctuations are enhanced when K -* 1through destabilization of the ground state.The Debye-Waller temperature factors of the main-chain

atoms of the two-ring disk have been determined byMondragon and Bloomer (as cited in ref. 24) and those of thevirion by Namba and Stubbs (25). The mean-square displace-ment of an atom (in any direction) in the virion is higher thanin the disk in two regions of the subunit: at high radius in mostof the 32 residues starting at the N terminus, and whereresidues 56-64 link two of the main a-helices, the RS and RRhelices.

It is likely that in some segments of the polypeptide chainpositional fluctuations are larger in the helical rod than in the20S aggregate. It is likely also that certain conformations aredestabilized by the 20S -, P-helix transition.

If the outermost segment of the subunit were involved inthe ultrasonic relaxations we observe, one would expect theeffect to differ from TMV Strasbourg to TMV Dahlemense.In the latter strain, the outer end has been shown to havespatially periodic perturbations, and subunits seem to seek

contact with a subunit in the next turn (26). However,although the ultrasonic spectra of the two strains showdifferences, the relaxation amplitudes, which measure the Si,are the same (9).On the other hand, the ultrasonic data impose restrictive

conditions to a destabilization process that may bring aboutthe enhanced ultrasonic absorption observed in the 20SP-helix transition. With the simple model of one singleprocess per subunit including one slow step and one fast step,considered above, the reaction volume Av of the fast stepindeed reaches an irrealistic value. Some of the largestvolume changes, associated with the transfer ofapolar groupsinto water, are of the order of 20 cm3/mol (27). It isconceivable that in the transfer of a buried amino acid sidechain into an aqueous environment the volume change couldreach 2-3 times the preceding value, but hardly the value lAvi> 86 cm3/mol of subunit found above.The following model would, nevertheless, agree with our

suggestion that the data require a single (multistep) confor-mational process per subunit. Consider a part of the poly-peptide chain along which a few pairs of rapidly exchangingconformational states exist. Assume that this part ofthe chainmay undergo a slower (larger scale) conformation changewhen, and only when, one well-defined conformational stateamong the two possible ones is occupied in each of the fastpreequilibria.For two identical fast preequilibria, the system comprises

five states. Two relaxation times are equal to the relaxationtime of the fast preequilibria, as expected. In addition, onetime equal to twice the preceding time and one slow time arefound. With this model, the value required for the volumechange lAvi attached to each preequilibrium is divided by thesquare root of the number of preequilibria.The high-frequency component of the bimodal spectrum

considered above is defined less accurately than its low-frequency component, as shown by the fact that the relativeerror on r2 is about 5.7 times larger than on -1. This mayindicate that several processes are present at high frequency,in accordance with the model above. However, if enhancedultrasonic absorption occurred through a destabilizationprocess, the population of the state of lower energy shoulddecrease, upon destabilization, by roughly the same factor ineach pair of rapidly exchanging states. As this factor is anexponential function of the free energy, all high-frequencysystems should be destabilized by nearly the same amount ofextra free energy.

It is more likely, then, that fast-step reaction volumes areinvolved. These volumes need to increase in the 20S -*P-helix transition by only a factor of 2, and by only roughlythe same factor each. This follows because the relaxationterms in Eq. 5 are directly proportional to the squared normalvolumes y, and because each fast step simply adds its owncontribution to the ultrasonic absorption.A Possible Mechanism. Perhaps the enhanced volume

fluctuations detected with ultrasonics occur in the vicinity ofthe RNA binding site (20). This interpretation of the ultra-sonic data was suggested by the evidence, based on NMRmeasurements (19), that dynamic disorder exists in the loopregion in all TMV protein aggregates in solution, includingthe helix. The determination of the Debye-Waller factors byNamba and Stubbs (25) most impressively shows that thelargest fluctuations that persist in the virion are located in theloop region. It is likely that even larger fluctuations exist inthis region of the protein-helix in solution. The possibleoccurrence of fluctuations at the RNA binding site in TMVraises the question of whether a quaternary structure thatexhibits significant conformational fluctuations must be pres-ent for the binding of the nucleic acid (20, 21).The disk -- virion transition is known to bring in closer

proximity several carboxyl and two guanidyl groups, as an

Proc. Natl. Acad. Sci. USA 83 (1986)

Proc. Natl. Acad. Sci. USA 83 (1986) 8151

electrostatic subunit interface, the carboxyl cage (16) isformed. Although the electrical charges are partly neutralizedin the 20S -- P-helix transition, a similar structure may beformed in the helical aggregate. If, then, the mobility of someof the amino acid side chains in solution allows them to movein and out of a cage in the protein helix, the conditions are metfor the process just described to occur. The volume changeassociated with each fast process would, indeed, be larger inthe helix, when the cage has been formed, and a side chainmoves back and forth between a mainly hydrophobic envi-ronment and an aqueous environment. The slower process,on the other hand, would consist in a larger scale conforma-tional rearrangement. Possible electrostatic effects, e.g.,contributions to the volume change Av, are a subject forfurther investigation.The preceding mechanism straightforwardly explains why

ultrasonic absorption is smaller in the virion, when the RNAis encapsulated, and restrains motions in the loop region. Oursuggested mechanism need not be ruled out if the mean-square displacement of the main-chain atoms in the loopregion is larger in the 20S aggregate than in the protein helix,as seems likely from the data for the Debye-Waller factors forthe two-ring disk and for the virion. The greater mobility inthe loop indeed implies that long range deformation modes ofthis segment, which should be forbidden in the protein helix,contribute to the motion ofeach atom. This makes it possiblefor overall mean-square displacements to be larger in the loopregion in the 20S aggregate than in the helix, even if localvolume changes (or possibly local conformational modes) areenhanced in the helix. Furthermore, ifa large volume changeis associated with the local conformation changes, theselatter processes will be detected preferentially in ultrasonicrelaxation spectrometry, whereas an increase in a reactionvolume would not be in x-ray diffraction.

Condcluding Remarks. The value of dynamic measurementsin solution is accentuated by the x-ray diffraction data for theprotein helix that are not as detailed as for the virion and thatare much less detailed than for the two-ring disk, while nodata at all are available for the 20S aggregate.Comparison of the dynamic properties in solution of the

20S aggregate and of the protein helix is of further interest asthe former aggregate is the nucleating unit in the assembly ofTMV. Using the previously presented theory (21), we canconclude that the normal-mode volume fluctuations Sj areenhanced by the 20S -* P-helix transition. However, thesuggestion that the fluctuations detected with ultrasonicsoccur in the loop region rests in particular on the assumptionsthat (i) one (multistep) conformational process per subunit isresponsible for the effect, and (ii) fast steps involving nearlyequally populated states (K = 1) exist in solution.With our suggested assignment, the function of the en-

hanced fluctuations that we detected in the helical rodaggregate could be to speed up the exacting adjustment oftheRNA string into the protein structure. Segmental mobilitythat has been reported to make it easier for an antigenicdeterminant of a protein to adjust to a preexisting antibodysite (ref. 24 and refs. cited therein) may, therefore, also beimportant for the binding of a nucleic acid to a virion in theassembly process. It is noteworthy that fluctuations remaingreater in the virion than in the 20S aggregate, as shown bythe comparison of their ultrasonic spectra, and may, there-fore, fulfill some function in the assembled virion, possibly indecapsidation.Although the example of TMV protein aggregates was

taken here, the processes considered-i.e., a destabilizationof one or several conformational states and an increase inreaction volumes upon assembly-are sufficiently general

that they may possibly exist in other systems in whichenhanced fluctuations have been observed in assemblies (8,11, 12).The example of TMV protein aggregates shows that ultra-

sonics may yield information not available by other tech-niques. The present study provides the first example of acalculation of normal-mode volume fluctuations using ultra-sonic relaxation spectrometry. Even when deconvolutioninto normal relaxation modes is not unequivocally possible,under favorable conditions a fluctuation-enhancement factormay still be measured when some parameter, here the stateof aggregation of a protein, has been varied.

We would like to thank Keiichi Namba and Gerald Stubbs forkindly making available their preliminary data for the Debye-Wallertemperature factors of TMV in advance of publication. We aregrateful to Anthony Durham and Eric Westhof for critically readingthe manuscript.

1. Lakowicz, J. R. & Weber, G. (1973) Biochemistry 12,4171-4179.

2. Munro, I. M., Pecht, I. & Stryer, L. (1979) Proc. Natl. Acad.Sci. USA 76, 56-60.

3. Jones, W. C., Rothgeb, T. M. & Gurd, F. R. N. (1976) J. Biol.Chem. 251, 7452-7460.

4. Wuthrich, K. & Wagner, G. (1978) Nature (London) 275,247-248.

5. Beece, D., Eisenstein, L., Frauenfelder, H., Good, D.,Marden, M. C., Reinisch, L., Reynolds, A. H., Sorensen,L. B. & Yue, K. T. (1980) Biochemistry 19, 5147-5157.

6. Rogez, D. & Cerf, R. (1976) J. Phys. Lett. 37, 105-109.7. Rogez, D., Cerf, R. & Bader, M. (1980) C. R. Hebd. Seances

Acad. Sci. Ser. B 290, 377-379.8. Cerf, R., Michels, B., Schulz, J. A., Witz, J., Pfeiffer, P. &

Hirth, L. (1979) Proc. Natl. Acad. Sci. USA 76, 1780-1782.9. Michels, B., Dormoy, Y., Cerf, R., Schulz, J. A. & Witz, J.

(1985) J. Mol. Biol. 181, 103-110.10. Dormoy, Y. (1984) Dissertation (Universitd Louis Pasteur,

Strasbourg, France).11. Robach, Y., Michels, B., Cerf, R., Braunwald, J. & Tripier-

Darcy, F. (1983) Proc. Natl. Acad. Sci. USA 80, 3981-3985.12. Dormoy, Y. (1985) J. Chim. Phys. 82, 433-439.13. Durham, A. C. H., Finch, J. T. & Klug, A. (1971) Nature

(London) New Biol. 229, 37-42.14. Raghavendra, K., Adams, M. L. & Schuster, T. M. (1985)

Biochemistry 24, 3298-3304.15. Bloomer, A. C., Champness, J. N., Bricogne, G., Staden, R.

& Klug, A. (1978) Nature (London) 276, 362-368.16. Stubbs, G., Warren, S. & Holmes, K. (1977) Nature (London)

267, 216-221.17. Namba, K. & Stubbs, G. (1985) Acta Crystallogr. Sect. A 41,

252-262.18. Mandelkow, E., Stubbs, G. & Warren, S. (1981) J. Mol. Biol.

152, 375-386.19. Jardetsky, O., Akasaka, K., Vogel, D., Morris, S. & Holmes,

K. (1978) Nature (London) 273, 564-566.20. Cerf, R. (1984) C. R. Hebd. Seances Acad. Sci. Ser. B 299,

759-762.21. Cerf, R. (1985) Biophys. J. 47, 751-756.22. Eigen, M. & de Maeyer, L. (1963) in Technique of Organic

Chemistry, ed. Weissberger, A. (Wiley Interscience, NewYork), Vol. 8, Part 2, pp. 895-1054.

23. Lamb, J. (1963) in Scuola Internazionale di Fisica "EnricoFermi", ed. Sette, D. (Academic, New York), Vol. 27, pp.101-132.

24. Westhof, E., Altschuh, D., Moras, D., Bloomer, A. C.,Mondragon, A., Klug, A. & Van Regenmortel, M. H. V.(1984) Nature (London) 331, 123-126.

25. Namba, K. & Stubbs, G. (1986) Science 231, 1401-1406.26. Caspar, D. L. D. & Holmes, K. C. (1969) J. Mol. Biol. 46,

99-133.27. Davis, J. S. & Gutfreund, H. (1976) FEBS Lett. 72, 199-207.

Biophysics: Cerf and Dormoy