Mechanics of bacteriophage maturationWouter H. Roosa, Ilya Gertsmanb,c,1, Eric R. Mayd,1, Charles L. Brooks IIId, John E. Johnsonb, and Gijs J. L. Wuitea,2

aNatuur- en Sterrenkunde and LaserLab, VU University, 1081 HV, Amsterdam, The Netherlands; bDepartment of Molecular Biology, The Scripps ResearchInstitute, La Jolla, CA 92037; cDepartment of Pediatrics, University of California, La Jolla, CA 92037; and dDepartment of Chemistry and BiophysicsProgram, University of Michigan, Ann Arbor, MI 48109

Edited by Charles M. Knobler, UCLA, Los Angeles, CA, and accepted by the Editorial Board December 20, 2011 (received for review June 27, 2011)

Capsid maturation with large-scale subunit reorganization occursin virtually all viruses that use a motor to package nucleic acid intopreformed particles. A variety of ensemble studies indicate that theparticles gain greater stability during this process, however, it is un-known which material properties of the fragile procapsids change.Using Atomic Force Microscopy-based nano-indentation, we studythe development of the mechanical properties during maturationof bacteriophage HK97, a λ-like phage of which the maturation-induced morphological changes are well described. We show thatmechanical stabilization and strengthening occurs in three indepen-dentways: (i) an increase of the Young’s modulus, (ii) a strong rise ofthe capsid’s ultimate strength, and (iii) a growth of the resistanceagainst material fatigue. The Young’s modulus of immature andmature capsids, as determined from thin shell theory, fit with thevalues calculated using a new multiscale simulation approach. Thismultiscale calculation shows that the increase in Young’s modulusisn’t dependent on the crosslinking between capsomers. In contrast,the ultimate strength of the capsids does increase even when a lim-ited number of cross-links are formedwhile full crosslinking appearsto protect the shell against material fatigue. Compared to phage λ,the covalent crosslinking at the icosahedral and quasi threefold axesof HK97 yields a mechanically more robust particle than the additionof the gpD protein during maturation of phage λ. These resultscorroborate the expected increase in capsid stability and strengthduring maturation, however in an unexpected intricate way, under-lining the complex structure of these self-assembling nanocontai-ners.

AFM ∣ elastic network model ∣ nanoindentation ∣ normal mode analysis ∣virus structural mechanics

HK97 is a double stranded DNA bacteriophage infectingEscherichia coli. The capsid protein of HK97 has a subunitfold that is shared by phages λ, P22, T4, and phi29, among others,as well as the eukaryotic virus, Herpes (1). In addition, the matu-ration pathways of these viruses show similar morphologicalchanges (2). HK97 provides a unique model system to study cap-sid maturation, as large quantities of isometric particles can beproduced in an E. coli expression system. These particles are ex-pressed without the portal and terminase proteins used for gen-ome packaging, but can be isolated as procapsids and matured invitro using chemical agents instead of DNA, and maturation canbe trapped in various expansion states (3, 4). In vitro expansioncan be effectively induced with a variety of chemical agents,including lowering the pH below 4. The particles expand fromthe Prohead II (P-II) form, which differs from Prohead I (P-I) bythe absence of the 104AA N-terminal polypeptide (termed theΔ-domain) from each subunit (gp5). The Δ-domain is presumedto function in scaffolding capsid assembly (5, 6). The Δ-domain iscleaved by an internally packaged protease (gp4) upon assembly,allowing the particle transition to the maturation competent P-IIparticle. During maturation the procapsid, with an average outerdiameter of approximately 50 nm, expands into a stable capsidof approximately 60 nm which possesses a twofold thinner shell(7, 8). In this process the skewed hexons of the P-II rearrange intothe symmetrical configuration that characterizes the expansionintermediate (EI) (8, 9). Upon EI formation, cross-links formcovalent isopeptide bonds between lysine 169 on one subunit

and asparagine 356 of a different subunit at all threefold axes ofthe capsid. Each subunit takes part in such a reaction, ultimatelyresulting in an intertwined cross-link network in the mature HeadII (H-II) particle that resembles chainmail. It was shown thatupon approximately 60% cross-link formation, the EI particleexpands further to the penultimate balloon state, and then finallyto H-II. The maturation steps involve massive conformationalchanges between subunits (4, 8) (Fig. 1). A recent, crystallo-graphic model of P-II revealed, with near atomic resolution, thattwisting and bending of helix and β-sheet domains in the capsidprotein occur during maturation (10). Whereas these resultsshow that the structural basis of capsid maturation is increasinglywell understood, the mechanics of this process remain elusive. Itis, however, a mechanical model that has been put forward as thehypothesis to explain the need for capsid maturation: i.e., astabilization of the fragile procapsid (2, 11). This idea has a longhistory but it has never been tested in a direct experiment. Weexamine, here, the mechanical consequences of maturation byAtomic Force Microscopy (AFM) nanoindentation (12–15). Thistechnique has developed into a versatile tool to study particle me-chanical properties (16). Such studies have successfully includedFinite Element Methods (FEM) and/or full capsid MolecularDynamics (MD) simulations to analyze the material propertiesof protein nanocontainers (13, 15, 17, 18). However, MD simula-tions of entire capsids are computationally expensive and FEMrequires a priori knowledge of the elastic properties. In this workwe employed a different method; a multiscale approach combin-ing MD of only the asymmetric unit and normal mode analysis(NMA) of elastic network models (ENM) of the entire capsid(19, 20). This method allows for ab initio predictions of the me-chanical properties, independent from the experimental data andin a computationally efficient way. Using this multiscale methodwe calculate the Young’s modulus of the HK97 particles andcompare it to estimates obtained via thin shell continuum elastictheory. Combining these computational approaches with our ex-perimental AFM data on stiffness, material fatigue, and ultimatestrength we dissect the complex mechanics of viral capsid matura-tion and compare the properties of mature HK97 particles withthe known mechanical properties of mature phage λ (21, 22).

ResultsTo assess the change in material properties of HK97 duringmaturation we looked at both the prohead states, an intermediatestate, and the final matured particle. Before starting experimentswe image the viral particles by AFM (23, 24). The morphologiesof the particles in these images, as shown in Fig. 2, reveal sub-

Author contributions: W.H.R., C.L.B., J.E.J., and G.J.L.W. designed research; W.H.R., I.G.,and E.R.M. performed research; I.G., E.R.M., C.L.B., and J.E.J. contributed new reagents/analytic tools; W.H.R., I.G., E.R.M., C.L.B., J.E.J., and G.J.L.W. analyzed data; and W.H.R.,I.G., E.R.M., C.L.B., J.E.J., and G.J.L.W. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission. C.M.K. is a guest editor invited by the EditorialBoard.1I.G. and E.R.M. contributed equally to this work.2To whom correspondence should be addressed. E-mail: gwuite@nat.vu.nl.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1109590109/-/DCSupplemental.

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structures and the deposition orientation, and therefore the corre-sponding symmetry axes upon which indentation experiments werecollected. However, the orientation could not be determined forevery image, in particular not for the P-I and EI particles. Afterimaging, the particles are indented by pushing with the cantilevertip in the middle of the particle (16). Various orientations weresampled and the ensemble of corresponding force-indentationcurves (Fig. 3) yield the average particle spring constant and thecritical force for breakage during a single, large indentation; i.e.,the ultimate strength. Breakage (failure) of the particles is likelythe result of fracture or crack formation in the shell, however thereare generally two ways that such failure can occur: either by apply-ing a single large force, which yields the ultimate strength of theparticle, or by repeated loading at a lower force, which can result inmaterial fatigue (12, 16).

We studied P-I particles with an inactivated protease (6) as P-Iwith active protease is a transient, short-lived particle. ComparingP-I and P-II (Fig. 3, Fig. S1, and Fig. S2) shows that the springconstant of P-I (0.018� 0.001 N∕m) is much lower than thatof P-II (0.12 � 0.01 N∕m). Moreover, the deformability of P-I ismuch higher than that of P-II, the indentation at the critical forceis 16� 2 nm vs. 5.5� 0.6 nm, respectively. Maturation of P-IIto H-II also goes hand in hand with mechanical changes as is im-mediately clear by examining the indentation curves shown inFig. 3. The linear part of the H-II extends much further than theP-II state. This difference indicates that the H-II state is moreresilient to breakage than the P-II state. In Table 1 we summarizethe results and demonstrate that despite the increase in diameterof the particle during maturation, the spring constant k does notsignificantly change going from P-II via EI to H-II. These obser-vations indicate that the Young’s Modulus increases during thesuccessive maturation steps. Furthermore, the critical force Fcritrises considerably after the onset of maturation.

Fig. 4 demonstrates that at the EI state a relatively small num-ber of cross-links have been formed (9, 25). Nevertheless, thecritical force is already at the level of the H-II particle as can beseen in Fig. S3. However, there is a clear difference in pronenessto fatigue, when EI particles are compared to H-II ones. Duringimaging, a small force is exerted to the particles with the AFMcantilever tip, which for some particles induces material fatigue.We observed, that during high resolution imaging, considerablymore EI particles break (>80%) than H-II particles (

that accounts for the structural heterogeneity of the particle.NMA and ENM have been successful in describing equilibriumfluctuations and large scale conformation changes of proteins(29, 30) and virus capsids (19, 20, 31–33). Previous studies haveexplored a potential pathway for the maturation process of HK97by employing a coarse-grained description of the capsid fromwhich displacements along the lowest energy icosahedral normalmodes were projected onto the displacement vectors describingthe difference between the P-II and H-II structures (31, 33). Theemployment of ENM and NMA in the current study is fundamen-tally different in that we are not concerned with describing thematuration structural transition of HK97, but rather are inter-ested with how the maturation process influences the elasticproperties of the capsid. To this end, we examine only equilibriumproperties of a given structure, namely the energetic parametersrelevant to the energy cost of the AFM induced deformations.ENMs are attractive models for studying macromolecular sys-tems because they account for local interactions, but do so witha simplified interaction potential. Nearby atoms in the networkare connected via harmonic springs in the ENM. The springs are

the only potential terms in the model, making it computationalinexpensive. The normal modes of the ENM were determined bydiagonalizing a Hessian matrix of the potential energy. Thelowest frequency normal modes are low energy modes whichcharacterize the collective and often functional motions of bio-molecules (29, 30). In order to connect a molecular descriptionto the continuum scale it is sensible to focus on the long-wave-length (collective) motions of the system, which are governed bythe low frequency normal modes and hence are the modes mostrelevant to the continuum mechanical properties of the system.However, the magnitude of thermal fluctuations for the ENM-based modes are unknown, and here is where we utilize the multi-scale methodology to provide thermal amplitudes based on fullyatomic simulations of the capsid. It was recently shown that theAFM nanoindenation deformation is largely characterized bythe l ¼ 1 spherical harmonic mode (19). By examining the equi-librium fluctuations of the capsid surface, the 2D stiffness ofthe l ¼ 1 deformation mode (Y l¼1) can be calculated. The 3DYoung’s Modulus E is related to Y l¼1 by E ¼ Y l¼1∕h, which allowsan estimate of E independent from the experimental measure-ments. We performed the multiscale calculation on P-II, H-II,and Head I (H-I) to investigate the influence of crosslinking.There are no complete high resolution structures for the P-I orEI particle and therefore we cannot perform the multiscalecalculation for these systems. H-I is a particle which contains amutation to the crosslinking lysine. The lysine is mutated to tyr-osine (K169Y) to prevent the formation of cross-links. Howeverthe mutant particle can still undergo a structural transition to astate morphologically and structurally similar to wild-type H-II[1.4 Å rmsd of hexamers, 6 Å rmsd of pentamers (34, 35)]. Thecalculated Y l¼1 values are 1.3 GPa*nm and 1.6 GPa*nm for P-IIand H-II, respectively. Dividing the Y l¼1 values by the shell thick-ness, we calculate E values of 0.4 GPa and 0.9 GPa for P-II andH-II respectively. These values are in excellent agreement withthose obtained from the experimental data. In the case of thecross-link deficient H-I structure we obtained an E-value of1.1 GPa, which is slightly larger than that computed for H-IIbut given the simplicity of the ENM these values are very similar,and certainly the effect of stabilizing and strengthening the capsid(relative to P-II) is exhibited by both H-I and H-II. In Fig. 5, thespherical harmonic spectra are shown for all three structures. TheP-II structure has a larger magnitude of the l ¼ 1 mode thaneither H-II or H-I, indicating that P-II has larger fluctuations inthat mode and is therefore softer with respect to an l ¼ 1 (AFM)deformation. In this analysis the mode amplitudes are summedover the m-component of the spherical harmonics, and thereforethe l ¼ 1 deformation actually captures two modes (m ¼ 0, 1),the shapes of which are also shown in Fig. 5.

Fig. 4. SDS gel of Prohead II and Expansion Intermediate. The Prohead IIshows the absence of cross-links, while the EI band shows the presence ofhigher oligomers.

Fig. 5. Spherical harmonic analysis of capsid fluctuations. (A) The equilibrium fluctuations of Prohead II, Head II, and cross-link deficient Head I structuresprojected onto a spherical harmonic basis. The l ¼ 1 mode magnitudes are shown as solid data points, these are the values which are used in calculating theEnormal mode values presented in Table 1. The mode amplitudes are in units of nm

2 and defined as jâl j2 ¼ ∑maml am�

l , where aml is the magnitude of the spherical

harmonic Yml and * indicates the complex conjugate (20). (B and C) The shapes of the l ¼ 1 spherical harmonics.

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DiscussionTo describe the mechanical properties of protein shells we inves-tigated the Young’s modulus, the breaking force for a single, largedeformation and the occurrence of fatigue for multiple small de-formations (16). Both the Young’s modulus and the breaking forceincrease during maturation. The breaking force can be directlydeduced from the nanoindentation experiments, but the Young’smodulus requires a theoretical and/or modeling approach.We useda combined experimental/thin shell theory approach and a purelymodeling approach, based on MD and 2D elasticity theory, todetermine the particle’s Young’s modulus. While these approacheswere conducted independently, it is reassuring that they showstrong agreement. This correspondence supports the use of thinshell theory and MD/normal mode analysis to determine the 3DYoung’s modulus of protein shells. In our calculations, the effectof cross-links were evaluated by simulations of H-II and the cross-link deficient H-I structure. Interestingly, the resulting Young’smodulus for these two systems was very similar, and both systemswere in agreement with E as calculated from the experimental/thinshell theory approach for H-II. These results indicate that theincrease in Young’s modulus, from P-II to H-II, is mainly a resultof the maturation process in general [topological reorganization;e.g., twisting of capsid proteins, among others (10)], but not spe-cifically of the crosslinking. So the question is raised of the role ofthe cross-links. The Young’s modulus does not give the full pictureof the mechanical properties of the particle. In particular, E doesnot address the maximum deformation that the protein shells canwithstand. This ultimate strength, where failure occurs, is increasedthrough maturation and likely due to the effect of crosslinking. Toexamine these observations in detail we compare the mechanicsof HK97 with that of phage λ, where no crosslinking occurs.

The bacteriophages HK97 and λ are closely related in proteinfold and capsid architecture (36–38). Both phages possess icosa-hedral shells with a T ¼ 7l triangulation number and expandduring maturation to heads with approximately the same size. Itturns out that there are also mechanical similarities, comparingthe mature, empty heads of both particles. The spring constant ofHK97 is only slightly lower and the breaking force slightly higherthan that of phage λ, but both values are the same within the error(21). One obvious difference between the maturation process ofHK97 and λ is the formation, at all threefold axes, of cross-links inthe former and the addition of the gpD protein in the latter.These differences apparently play similar roles in stabilizing theirrespective protein shells (38). So each phage has followed adifferent route to provide capsid stabilization, and they result incomparable Young’s moduli. However, there is a significant dif-ference in the indentation at breakage: 8.5� 0.7 nm for HK97while for the empty λ particles this is 5.7� 0.6 nm (21). Eventhough we have not revealed the exact molecular mechanism offailure, it indicates that the cross-links in the HK97 capsid resultin a particle that can withstand higher deformations than the gpDdecorated phage λ capsids. This difference suggests that the cross-linking yields a strong but flexible particle, whereas the gpD-gpEconnection inside phage λ yields a comparable elasticity, but amore brittle particle. To summarize, our experiments show thatHK97 has found a mechanically more robust solution for capsidstabilization and strengthening than phage λ.

In order to examine whether the whole shell needs to be cross-linked to increase its ultimate strength, we looked at EI particlesand evaluated their cross-link state by SDS gel (Fig. 4). Theseparticles are cross-linked to approximately 50% or less based onvisual comparisons to carefully quantified gels of cross-linked par-ticle states done by Lee, et al. (9). The Young’s modulus of theseparticles is much lower than that of the fully cross-linked H-IIparticles, but their breaking force is already at the level of H-II(Table 1). Apparently crosslinking stabilizes the particle but com-plete crosslinking is to a large extent redundant to resist breakageupon large deformations, as is the case during AFM nanoindenta-

tion. However, calorimetric results show that EI denatures at sig-nificantly lower temperatures than H-II, indicating that full cross-linking does increase the stability of the particles (39). These resultsfit well with the observation that EI breaks after repetitive, smalldeformations, but that H-II can withstand these repetitive defor-mations. This observation shows that not the breaking force, butthe proneness to fatigue is an indicator for the stability of viralshells that is in accordance with calorimetric data (39). Fatigueon the molecular level occurring in viruses is likely associated withcrack formation, as is the case for macroscopic structures (16).Cracks in viral shells are expected when the protein-protein inter-action strength is reduced (40). Such a process is indeed muchmore likely for EI than for H-II as in the former approximately50% of the cross-links still have to be formed.

The Young’s modulus of H-II is comparable to the Young’smodulus of mature phage λ capsids and the capsid of HerpesSimplex Virus type 1 (21, 41). However, it is at least three timeslarger than the moduli of viral shells that self-assemble aroundthe viral genome: for instance the capsids of Cowpea ChloroticMottle Virus (CCMV) and Hepatitis B Virus (HBV) (13, 16, 18).These observations support the hypothesis that capsids of virusesthat use a packaging motor for genome encapsidation are stron-ger than capsids of viruses that encapsidate their genome duringthe self-assembly process (42). Interestingly, the Young’s modu-lus of the HK97 P-II particles is comparable to the capsids ofCCMVand HBV, which do not undergo a maturation step invol-ving genome packaging. HK97 is of particular interest formaturation studies as its precursor particle can be purified in twodifferent states, P-I and P-II. The increased stiffness of P-II withrespect to P-I is likely linked to strong threefold interactionsbetween the hexamers of P-II, which are absent in P-I (6). TheΔ-domain in P-I is critical to maintain the weak reversible inter-actions allowing self-correction during assembly, explaining whyP-I can reversibly be associated/dissociated whereas P-II cannot.Our results show that the stability and strength of mature capsidsdepends on the particle quaternary structure. Strengtheningand stabilization requires a maturation process, as is clear by thegradual increase in Young’s modulus. Furthermore, the data re-veal that particles that have not (yet) undergone a maturationprocess have a significantly lower Young’s modulus. A logicalexplanation for these results is that the self-assembly processshould be reversible, allowing for the correction of any aberrantsubunit or capsomer (hexamers and pentamers) additions thatmay trap the particle in a nonfunctional state. However, for DNApackaging with a molecular motor, involving high forces (43), thecapsid needs to be strong to withstand the inner pressure of thegenome (2, 11) and thus relies on a staged assembly and matura-tion process that transforms the initially weakly connected pro-capsid particle to the more robust mature capsid.

To summarize, combining our data shows that the maturation-induced changes in the capsid morphology increases the stabilityand strength of the particles. The presence of the full set ofcross-links in H-II makes the particle considerably stronger thanmature phage λ and also stronger than the partially cross-linkedEI, as it is much less prone to fall apart as a result of fatigue.However, the EI particles are capable of as large deformationsas the H-II particles, showing that the resistance against majoranisotropic deformations is already existent when only approxi-mately 50% cross-links are available. These data provide directexperimental evidence at the single particle level of the previouslyhypothesized capsid stabilizing effects of maturation. Our resultsdemonstrate the profound effects of the change in quaternarystructure on the flexibility and strength of bacteriophage capsidsand likely of capsids of related eukaryotic viruses like herpes.

Materials and MethodsViral Particles. Expression of the gp5 capsid protein and the gp4 proteasewas performed in an E. coli T7 expression system and the purification of

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HK97 P-II was performed as previously described (4). P-I particles with inacti-vated protease were purified as described in ref. 6. All studied particlesare empty (i.e., no genome inside) and they are expressed without the portaland terminase proteins. The substitution of the portal by a regular vertexpentamer is unlikely to measurably affect the mechanical properties asresults on Herpes Simplex Virus type 1 and phage phi29 indicate (41, 44).P-II and H-II particles were stored and probed in Buffer A, containing:20 mM Tris, 40 mM NaCl, 0.02% azide, pH 7.5. P-I were stored and probedin P-I buffer (10 mM Tris-HCl, 50 mMmonosodium glutamate, pH 7.5). For thestudy of EI particles, P-II particles were expanded by diluting them 10-fold inlow pH Buffer (50 mM Sodium acetate, 200 mM KCl, pH 3.9, 0.02% azide).Particles were left in this solution overnight at room temperature beforeAFM experimentation. Experiments on EI were performed in low pH buffer.Changing the pH has previously been shown to affect the temperaturestability of HK97 particles (39) and the mechanical properties of CowpeaChlorotic Mottle Virus (45). Therefore the comparison between the results ofEI and of P-II and H-II should be considered with care.

AFM. AFM measurements were performed with a Nanotec microscope, usingjumpingmode for imaging. Olympus OMCL-RC800PSA cantilevers with a nom-inal spring constant of 0.05 N∕m were used and calibrated by the method ofSader, et al. (46), yielding an averaged spring constant of 0.056� 0.005 N∕m(�SD). All measurements were performed in buffer solution with particlesadhered to hydrophobic glass cover slips (12). The particles were incubatedin droplets of 100 μL, at a concentration of approximately 0.1 mg∕mL onthe surface for 20min before the start of ameasurement. The height measure-ments of the P-II particles showed a distribution with peaks below and above57 nm, indicating that due to the contact with the surface or cantilever someof the particles had partially expanded to a size >57 nm. It was however notpossible to deduce to what degree the particles had expanded and this datawas discarded. See Fig. S1 for a distribution of the heights of P-II and H-II par-ticles. Molecular graphics images (Fig. 2 B and D) were made with Chimerasoftware (see Fig. S1).

Multiscale Modeling Approach. To generate the equilibrium capsid dynamicsrepresentative of a particular temperature, we employ a multiscale approach(20), in which we construct an ENM, propagate the network along the lowestfrequency normal modes, and project the renormalized equilibrium thermalfluctuations onto a spherical harmonic basis. To renormalize the fluctuationsin the ENM dynamics, we compute a MD trajectory of the capsid asymmetricunit under icosahedral boundary conditions using all atom force field basedmolecular models, and project these fluctuations onto the same sphericalharmonic basis. The spherical harmonic spectrum from the ENM is then re-

normalized based on the ratio of the sum of the spherical harmonic coeffi-cients between the MD and ENM simulations. This renormalization ensuresthat our ENM spectrum is scaled to the appropriate thermal scale.

The initial configurations of the asymmetric units for the MD simulationswere taken from the crystal structures in the VIPER database (47) (P-II:3e8k;H-II:1ohg, H-I:2fs3). These structures were energy minimized and thenequilibrated at 300 K, using the GBSW implicit solvent model (48) under ico-sahedral boundary conditions (49). The CHARMM simulation package andCHARMM19 force field were utilized to perform the MD simulations (50).To include the effects of crosslinking in H-II, adjacent protein units in thestructure were cross-linked by introducing bonds, angle, and torsional termsbetween the side chains of the appropriate residues (K169 and N356). Theseinteractions occur between the asymmetric unit which is being explicitly si-mulated and neighboring (image) proteins, and the force field terms weremodeled after the terms describing the peptide backbone. All systems wereequilibrated for 10 ns, production simulations were conducted for at least9 ns for all systems. The magnitude of the equilibrium fluctuations were cal-culated by performing block averaging over 4 ns blocks of data; the blockswere offset by 1 ns. There were seven blocks used for P-II and H-I and sixblocks for H-II, for which we had slightly less production simulation data.

The ENMmodels were constructed from the average structure of the pro-duction MD simulations. The icosahedral symmetry operators were imposedon the asymmetric unit to construct the full capsid structure. The ENMmodelsconsisted of only heavy (nonhydrogen) atoms and the rotation-translation ofblocks method (51) was used to determine the normal modes of the system.In this method, segments of the structure are treated as rigid units, we choseto make two consecutive residues a rigid unit in our analysis. The networkswere propagated along the 999 lowest frequency modes (excluding the firstsix modes which correspond to rigid body motions), to construct a trajectorylong enough to observe approximately 20 oscillations of the slowest modes.

ACKNOWLEDGMENTS. R. Hendrix and R. Duda (University of Pittsburgh) arethanked for discussions, I. Ivanovska (Vrije Universiteit) is thanked for tech-nical support and David Veesler (Scripps) is thanked for purifying the P-I par-ticles. G.J.L.W. acknowledges the support by the Nederlandse Organisatievoor Wetenschappelijk Onderzoek (NWO) for a CW-ECHO and Vici grant andthe Stichting voor Fundamenteel Onderzoek der Materie (FOM) for supportunder the “Physics of the Genome” research program.W.H.R. is supported byNWO through a Rubicon grant. I.G. and J.E.J. were supported by NationalInstitutes of Health (NIH) Grant R01 AI040101. E.R.M. is supported by aNational Science Foundation (NSF) postdoctoral fellowship. C.L.B. is sup-ported through the NIH (RR012255) and NSF (PHY0216576).

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