Mechanical limits of viral capsidsMathias Buenemann* and Peter Lenz*†‡

*Fachbereich Physik, Philipps-Universitat Marburg, D-35032 Marburg, Germany; and †Center for Theoretical Biological Physics, University of California at SanDiego, La Jolla, CA 92093-0374

Edited by David R. Nelson, Harvard University, Cambridge, MA, and approved April 23, 2007 (received for review December 22, 2006)

We studied the elastic properties and mechanical stability of viralcapsids under external force-loading with computer simulations.Our approach allows the implementation of specific geometriescorresponding to specific phages, such as �29 and cowpea chlo-rotic mottle virus. We demonstrate how, in a combined numericaland experimental approach, the elastic parameters can be deter-mined with high precision. The experimentally observed bimodal-ity of elastic spring constants is shown to be of geometrical origin,namely the presence of pentavalent units in the viral shell. Wedefine a criterion for capsid breakage that explains well theexperimentally observed rupture. From our numerics we find acrossover from �2/3 to �1/2 for the dependence of the rupture forceon the Foppl-von Karman number, �. For filled capsids, highinternal pressures lead to a stronger destabilization for viruseswith buckled ground states versus viruses with unbuckled groundstates. Finally, we show how our numerically calculated energymaps can be used to extract information about the strength ofprotein–protein interactions from rupture experiments.

biomaterials � membranes � thin shells

Bacteriophage capsids have astonishing elastic properties.They withstand extreme internal pressures exerted by their

densely packed DNA. DNA packaging experiments on phage�29 (1) and theoretical arguments (2–6) imply that phagecapsids store their DNA under �50 atm (1 atm � 101.3 kPa).This pressure is necessary to inject the DNA into the prokaryotichost cell (7). It has been shown experimentally that the ejectionpressure from �-phages is �20 atm (8). This remarkable robust-ness under high pressures motivated recent nanoindentationstudies (9) that showed that capsids also resist external pointforces up to �1 nN.

In contrast, viruses that infect eukaryotic cells, e.g., cowpeachlorotic mottle virus (CCMV) and polyoma, penetrate theirhost cell. Their DNA is released by subsequent disassembly ofthe shell. Correspondingly, these viruses do not store their DNAunder high pressure. Nevertheless, their resistance to externalforces is remarkably strong (10).

Viral capsids are composed of a small number of differentproteins that cluster to morphological units (‘‘capsomers’’).These units are put together in a highly regular fashion describedby the ‘‘quasiequivalence’’ principle of Caspar and Klug (11).Because of the presence of capsomers, the surface of capsids hasa discrete structure. Mathematically, these capsomers corre-spond to the vertices of a regular triangulation of a sphere. Forsuch a triangulation, the number of vertices, V, connecting edges,E, and associated faces, F, have to fulfill Euler’s theorem F � E� V � 2. This theorem implies that capsids of icosahedralsymmetry have 12 pentavalent morphological units (pentamers)embedded in an environment of hexavalent units (hexamers).

Viral capsids can undergo elastic and bending deformations.Generally, the elastic properties of such thin shells (of typicallength scale R) depend only on a single dimensionless parameter,the Foppl-von Karman (FvK) number � ' R2 �e/�b set by theratio between elastic modulus �e and bending rigidity �b. It wasshown in ref. 12 that at a critical �b � 154, spherical shellsundergo a buckling transition in which the shell acquires a morefaceted shape. This transition is caused by large strains associ-

ated with the pentamers that are reduced upon buckling into aconical shape.

Typical � values of viral shells lie in the range from below thebuckling threshold [e.g., alfalfa mosaic virus with � � 60 (13)] upto several thousand [e.g., phage T4 with � � 3,300 (14)]. Anextreme example is the giant mimivirus (diameter of � 600 nm)with a Fvk number of � � 20,000 (15). Even higher � values arepossible for artificial capsules, such as vesicles with crystallizedlipid membranes (16) (� � 105) or polyelectrolyte capsules (17)(� � 106).

The elastic properties of capsids can be probed in scanningforce microscopy (SFM) experiments (9, 10). Depending on thestrength of loading, two regimes are explored. Small forces leadto a completely reversible deformation of capsids. Here, thelinear and nonlinear regime of thin spherical shell elasticity canbe studied. In this regime, the deformations explore the globalelastic properties. Larger forces (�1 nN) cause irreversiblechanges in the shell structure commonly attributed to bondrupture. Rupturing studies therefore give information about themolecular interactions between capsomers, thus elucidatinglocal features of shell mechanics.

Viral shells have also been the subject of numerical investi-gations. The dependence of virus shape on the FvK number wasanalyzed in ref. 12. Only recently, the elasticity of capsids hascome into the focus of numerical studies (18) that are based ona discretization scheme introduced for crystalline membranes(19). In other approaches, the shells are directly built up ofproteins (20) or capsomers (21) with specified (spatially varying)interactions. On this basis, the stability of capsids against internalpressure has been studied (22).

Here, we generalize a numerical approach developed for theinvestigation of vesicles (23) to viral capsids. In our simulations,we can systematically vary the elastic moduli and geometry of thecapsids and probe their mechanical response to external distur-bances. We can even implement specific geometries, correspond-ing to specific phages and viruses, and determine (by a directcomparison with experimental indentation experiments) withhigh precision their elastic parameters, such as linear andnonlinear spring constants, and the FvK number. The bigadvantage of our method is that it produces highly stable andreliable results even under high local strain, which allows thesystematic study of rupture of mechanical shells in computersimulations that provide insights into the geometry-dependenceof viral stability. We also are able to access features numericallythat are not observable in experiments. For example, by mea-suring the local strain, we are able to determine numerically the

Author contributions: P.L. designed research; M.B. performed research; and M.B. and P.L.wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

Abbreviations: CCMV, cowpea chlorotic mottle virus; FvK, Foppl-von Karman; SFM, scan-ning force microscopy.

‡To whom correspondence should be addressed. E-mail: peter.lenz@physik.uni-marburg.de.

This article contains supporting information online at www.pnas.org/cgi/content/full/0611472104/DC1.

© 2007 by The National Academy of Sciences of the USA

www.pnas.org�cgi�doi�10.1073�pnas.0611472104 PNAS � June 12, 2007 � vol. 104 � no. 24 � 9925–9930

APP

LIED

PHYS

ICA

LSC

IEN

CES

Dow

nloa

ded

by g

uest

on

Mar

ch 1

5, 2

020

spatial distribution of rupture probabilities across the capsidsurface. We will show that experimental deviations from thisdistribution may be used to draw conclusions about the spatialvariation of protein binding strength.

We first present our study of the global response of capsids toexternally applied point forces. The numerical results for(empty) �29 and CCMV are compared with experimental data.Next, we analyze the local, irreversible response to indentationforces, and we discuss the dependence of local rupture proba-bility on the geometry of the capsid. Finally, we extend ouranalysis to filled capsids for which we have studied the elasticparameters and mechanical stability.

Results and DiscussionSpring Constants. As mentioned earlier, loading with small forcesprobes the global elastic behavior of viral shells. Here, weanalyze the influence of internal structure on the elastic responseof capsids by comparing the numerical simulations with analyt-ical results for small deformations. Furthermore, we extract thevalues of the elastic moduli of �29 and CCMV by a directcomparison of numerical and experimental force–distance re-lationships.

In a first step, we numerically calculated force–distance curves(see Methods for details). Viruses were indented radially by adistance X ' �R for 0 � � � 0.4 (varying with step-size �� �1/250). By minimizing the total energy, the (dimensionless)force, F, is obtained by F � ��E/(�b ��), where �E is the changein total energy caused by the increase �� in X. Both hexamersand pentamers were displaced. The indented hexamers discussedhere all lie in the center of the facet spanned by their threeneighboring pentamers.

As shown in Fig. 1, the elastic response of the capsid to thedeformation strongly depends on the ratio of bending and elasticenergy characterized by the FvK number �. For small �, capsidshave a nearly spherical shape (12) and behave like homogeneouscontinuous shells. For 10 � � � 500, the (dimensionless) linearspring constants Kl of both hexameric and pentameric regionsfollow a square root law (24). For sufficiently small � � �c(�) (seeFig. 2),

F � Kl� � 4���. [1]

This linear regime was already observed numerically (18). Thetransition to the nonlinear regime (at � � �c) takes place atsmaller �c the larger � (see Fig. 1 A and B). For small FvK number� � (R/h)2, corrections to thin shell theory of order h/R becomerelevant.

For � � 500, the capsid has a strongly faceted shape becausethe 12 disclinations buckle. Because of this structural inhomo-geneity, the elastic response of pentamers and hexamers isdifferent; i.e., pentamers generally become stiffer than hexamerswith increasing �. Furthermore, the abrupt shape transformationshown in Fig. 1C only occurs for loading on pentamers.

As Fig. 2 shows, the difference between the spring constantsfor pentamers and hexamers increases with increasing �. At � �2,000, the spring constants are already well separated, explainingthe experimentally measured bimodality of spring constants of�29 (9). From our numerical data, we find Kl � �1/3 for hexamersand Kl � �1 for pentamers. A similar stiffening of pentamers wasobserved for probing with extended objects (18). When loadingon pentamers is parallel to ridges, Kl � �5/6, in agreement withthe findings of refs. 25 and 26 (data not shown).

The elastic behavior of spherical capsids under axial pointforce can be understood by simple scaling arguments (24) of anelastic sphere with energy E � Ee � Eb. For small indentations,X, the top of the sphere is f lattened in a circular region ofdiameter d (Fig. 3A). In the flattened region, the radius ofcurvature is �d2/X and meridians are compressed by an amount

��. In equilibrium, one then finds d � R/�1/4, yielding a totalenergy E/�b � �1/2�2 of the deformed sphere. Thus, the force–distance relationship is linear with Kl � �1/2.

In fact, a more rigorous analysis of the linear regime yields Eq.1 for � � 10 [see supporting information (SI) Fig. 9]. For � �10, the scaling picture breaks down because the dominantbending energy is no longer concentrated in a small regionaround the pole (see SI Fig. 10).

At a critical indentation �c � h/R � ��1/2 the shell undergoesa larger shape transformation in which a circular region in theupper part inverts its shape (Fig. 3B) (24). Here, the deformationenergy is concentrated in a ring of radius r and width d. Theradius of curvature of the ring is �R d/r, and meridians areshortened by a factor �r d/R2. By again using d to equilibratebending and elastic energy (i.e., d � R/�1/4), one finds E/�b ��3/2�1/4, leading to a nonlinear force–distance relationship F ��1/2 (24). This scaling behavior is valid only for point forces.Loading with extended objects leads to a different � dependenceof �c (18).

A

B

C

Fig. 1. Force–distance relationships of locally deformed viral shells. (A) Forsmall � the force–distance relationship is linear F � � for the full range ofapplied forces. (B and C) Above the buckling threshold, the structural inho-mogeneity of the capsid material is reflected by the increasing differences inthe elastic response of pentamers and hexamers. Above � � 1,000, theinversion transition of pentamers (see Fig. 3B) causes softening, which leads toa decrease in F(�). The influence on hexamers is much weaker. At higher �,pentamers may even snap into a new stable inverted configuration, whichleads to discontinuities of F(�). The dotted blue line in C represents metastableconformations that have a higher energy than stronger deformed ones.Hexamers do not show a discontinuous inversion transition.

9926 � www.pnas.org�cgi�doi�10.1073�pnas.0611472104 Buenemann and Lenz

Dow

nloa

ded

by g

uest

on

Mar

ch 1

5, 2

020

For sufficiently large �, the inversion transition of pentamers(shown in Fig. 3B) is a first-order transition (in �), as can be seenfrom the F(�) curve in Fig. 1C. There, only the green linescorrespond to conformations of minimal energy. The blue linesrepresent intermediate (metastable or unstable) conformationsconnecting the weakly and strongly deformed capsid shapes. Forsmaller �, the transition is continuous (see Fig. 1B). Inversion-induced softening of capsids has been observed experimentallyand numerically (18, 27).

With our numerical simulations it also is possible to extractprecise material parameters of experimentally investigated vi-ruses. To do so, we have simulated the conditions correspondingto the SFM experiments on �29 (9). By using the FvK number� as fit parameter for the measured bimodality ratio Kl

pent/Klhex

� 1.88, one finds � � 1,778, in good agreement with the valueestimated above from the geometrical parameters of ref. 28.

Then, �b/R2 � 1.14 mN/m can be extracted from the springconstant Kl' kl R2/�b shown in Fig. 2 by using, e.g., the softestexperimentally measured value of kl � 0.16 N/m (with corre-sponding kl

pent � 0.296 N/m) (9). The force scale is �b/R � 24 pN.For empty CCMV, the bimodality cannot be resolved experi-mentally [kl � 0.15 0.01 N/m (10)], whereas numerically wefind kl

hex � 0.17 N/m and klpent � 0.12 N/m. Here, we extract �b/R2

� 2.4 0.2 mN/m and �b/R � 29 2 pN from the numericaldata.

With these material parameters, a direct comparison withSFM experiments can be achieved. Fig. 4 shows as an examplethe force–distance curves for hexamers and pentamers of �29.

Within our numerical analysis, it is even possible to take intoaccount the elastic spring constant of the cantilever Kc (mea-sured in units of �b/R2). Then, a measured force F correspondsto a displacement X/R � F/Kc � � of the shell. Thus, as Fig. 4Inset shows, for sufficiently stiff cantilevers the shape inversionof the viral shell leads to a discontinuous force–displacementcurve. The numerical results (shown in Fig. 4) are in excellentagreement with the experimentally measured force–distancecurves (9). In particular, for both numerical and experimentaldeformations, the shape instability occurs at �c � 0.13

Rupture. We have also analyzed the local response of elastic shellsto loading by investigating the conditions under which ruptureoccurs. This process takes place in regions of high in-plane stress,where bonds break due to overstretching or compression. Thus,this analysis provides information about the interaction betweencapsomers. As experimentally observed, phage �29 breaks at acritical indentation Xr � 7.7 nm � 0.35R (9). To determine theassociated change in bond length, we numerically calculated theshape of a shell with � � 1778 at this indentation, yielding amaximal compression of �4.5%. The corresponding numericalrupture force is fr � 43�b/R � 1.0 nN in agreement withexperimental findings (9). By using this criterion, the numericsfor empty CCMV predicts a rupture force fr � 22 �b/R �0.57 nN, whereas the experimentally determined value is fr �0.63 nN (10).

Now we will analyze rupture of viral shells as a function of �.Fig. 5 shows the � dependence of fr. For � � 100, rupture occursin the linear regime (Fig. 3A). Here, rupture is caused bycompression of meridians (Fig. 5 Left Inset). In the linear regime,we observe that the rupture force scales approximately ��2/3.However, most viruses have higher FvK numbers and willtherefore rupture in the nonlinear regime after shape inversion(Fig. 3B). Here, rupture is caused by circumferential compres-sion in the highly bent ring (see SI Fig. 11). In this regime, therupture force scales (approximately) ��1/2.

We also have investigated the influence of the capsid geom-etry on rupture by comparing an icosahedral and a spherocy-lindrical capsid. This investigation is motivated by the fact thateven-numbered T-phages and �29 have additional capsomers inthe equatorial region elongating the icosahedral shape. Forexample, the head of T4 is an icosahedron with triangulationnumber 13 and one additional ring of hexamers (29). Thegeometry of the spherocylinder used in our simulations is that of

Fig. 2. Linear spring constants as a function of � for empty (solid lines) andfilled (dashed lines) icosahedral capsids. The black dashed line represents theanalytical solution (see Eq. 11 in SI Text). For � � �b, the spring constants of theempty capsid follow closely the theoretical prediction. Above �b, the springconstant of hexamers scales as �1/3. Pentamers become stiffer and (approxi-mately) ��1. Generally, filled capsids have higher spring constants. Data forthe filled capsids are for p � 0.06�e/R (the internal pressure of phage �29).

X

Rd

X

dR

rA B

Fig. 3. Global response of elastic spherical capsids to local deformations. (A)For small indentations, only the top of the shell is flattened, giving rise to alinear force–distance relationship. (B) At a critical indentation, �c � h/R, thecapsid partially inverts its shape.

Fig. 4. Numerical indentation experiment on �29 (with � � 1,778). At anindentation of 0.13R, the mechanical response of the pentamers suddenlydecreases due to shape inversion (see Fig. 3B). Linear spring constants aredetermined by fitting the region of small �, implying F � 258� for pentamersand F � 139� for hexamers. (Inset) The same curve as it would be measured inan experiment with an elastic cantilever with spring constant Kc � 100.

Buenemann and Lenz PNAS � June 12, 2007 � vol. 104 � no. 24 � 9927

APP

LIED

PHYS

ICA

LSC

IEN

CES

Dow

nloa

ded

by g

uest

on

Mar

ch 1

5, 2

020

�29, i.e., an icosadeltahedron with triangulation number 3elongated by two additional equatorial rings of radius R (28).

Fig. 6 shows the rupture force mapped across the surface for� � 100 and � � 1,000. In the corresponding numericalsimulations, every vertex was indented in steps of R/250 followedby the calculation of the new equilibrium conformation (seeMethods for details). When stretching of some bond exceeded4.5%, the applied force was determined. For � � �b below thebuckling threshold, the shells have a uniform elastic behavioracross the surface. Because the force exerted on the caps has atangential component, part of the displacement directly leads tobond stretching. Therefore, the caps are the regions of highestinstability. Above the buckling threshold, the pentamers are rigidcones, but the space between them becomes more flexible. Whena pentameric region is pushed, the displacement is mainlytransformed into compression of the ridges. Pentamers are thusthe most unstable regions. On average, a larger (scale-free) forceis required to break the capsid for larger FvK number �.

The spatial distribution of deformation energy can serve as ameasure for the tendency to rupture. To mimic an ensemble ofindentation experiments, we carried out simulations in which

each vertex of the shell was indented by a constant � (� � 0.1).For all vertices, the deformation energy was recorded for allpoints of loading. To find the spatial distribution of deformationenergy in the ensemble, the elastic energy per vertex wasaveraged over the ensemble measurement. Fig. 7 shows a map ofthe average spatial distribution of elastic energy, normalized bythe total elastic energy put into the system during the measure-ment on the ensemble. The red regions correspond to theprobable rupture lines. One should note that in such an ensemblethe point of loading and the region of rupture do not necessarilycoincide.

In our simulations we assume a uniform distribution of bindingenergies between capsomers. In reality, binding energies willshow a spatial variation and an ensemble of indentation exper-iments on capsids will yield different rupture probabilities thanthe numerical data presented in Fig. 7. However, with a directcomparison of the two approaches, conclusions about the dis-tribution of binding potentials can be drawn.

Filled Capsids. Phages infect their host cells by penetrating the cellmembrane and rapidly injecting the DNA into the cell plasma.Therefore, their DNA is stored under high pressure. The pres-sure p of DNA inside �29 is of the order of p � 6 MPa (1, 5),or, in units of elastic parameters, p � 0.06�e/R. Other viruses,like CCMV, self-assemble inside the host cell and enclose theDNA. Correspondingly, their internal pressure is much lowerthan that of phages. Typically, p � 1 atm � 0.01�e/R is a goodestimate for the internal pressure of a filled CCMV capsid.

To investigate the influence of internal pressure on the elasticproperties, we have numerically simulated filled capsids with themethods described above. However, here the additional con-straint has to be taken into account that, because of the presenceof DNA inside the capsids, the enclosed volume is fixed. Tomimic a packed �29 phage, first the equilibrium conformationunder pressure was determined by minimizing the total energywith an additional pressure contribution Ep � pV with p �0.06�e/R. For simplicity we assume pressure to be isotropic. Asshown in SI Text (by using the approach developed in ref. 5), theratio between minimum and maximum pressure inside �29 is(under typical in vivo conditions) between 0.3 and 1. Thesevariations in pressure are too small to alter significantly theresults presented here (data not shown).

Experiments on full CCMV were performed in ref. 10. Adirect comparison of our numerical simulations with theseexperiments yields (for p � 0.01�e/R and constant volume V) the

Fig. 5. (Dimensionless) rupture force of an icosahedral capsid with (greencurve) and without (red curve) volume constraint. Rupture is assumed to occurat a critical bond length change of 4.5%. The rupture force increases with �

because it becomes harder to stretch/compress the material. The rupture forceshows a �� dependence. We find � � 0.65 0.03 � 2/3 for low � and � � 0.53 0.04 � 1/2 for large �. The crossover takes places at the inversion transitionshown in Fig. 3. (Insets) Ruptured structures for � � 10 (Left) and � � 3,162(Right). The bonds where rupture occurs first are shown as red diamonds.

γ =1000γ =100

1.368

7.276

13.185

10.493

20.595

30.696

Fig. 6. Rupture force in units of �b/R plotted across the surface of sphericaland spherocylindrical capsids with � � 100 and � � 1,000. Rupture is assumedto occur at a critical bond length change of 4.5%.

γ =1000γ =100

−.097

.040

.178

−.060

.016

.093

Fig. 7. Averaged distribution of elastic deformation energy for an ensemblemeasurement at which every vertex was indented by an amount � � 0.1. In thesimulations, loading was perpendicular to the axis of symmetry. The ensembleaverage shows that the elastic energy is concentrated in a small region justbelow the caps. The capsid will most likely break in this region.

9928 � www.pnas.org�cgi�doi�10.1073�pnas.0611472104 Buenemann and Lenz

Dow

nloa

ded

by g

uest

on

Mar

ch 1

5, 2

020

spring constants klhex � 0.19 N/m, kl

pent � 0.12 N/m, and a ruptureforce f r

full � 0.83nN. These values are again in good agreementwith the experimental findings (kl � 0.20 0.02 N/m and f r

full �0.81 0.04 nN). There are no SFM experiments on filled �29phages to which we could compare our numerical simulations.

Generally, the volume constraint leads to increased linearspring constants (Fig. 2). This can be understood in the scalingpicture: To preserve constant cross-sectional area, the localcompression at the point of loading must be compensated by anexpansion of the equatorial area. Therefore, the influence of thevolume constraint becomes more apparent for larger �.

At high pressures, the circumferential stress at the fivefolddisclinations is balanced by the volume contribution pV. Thus,pentamers do not form rigid buckles and hexamers remain flat.For example, the DNA pressure of �29 phages reduces itsaspherity a from a �1.4 � 10�3 for an empty capsid to a � 0.6 �10�3. This similarity between hexamers and pentamers is re-f lected by the fact that the elastic response of hexamers andpentamers becomes similar for high internal pressure (Fig. 2).

The rupturing behavior of filled capsids is shown in Fig. 5. Therupture force is larger for filled capsids than for empty ones.Because of the internal pressure, the nonindented shell is alreadystretched, which compensates the force-induced compression inthe meridional direction at the poles, thus reducing their ten-dency to rupture. Rupture of filled capsids is caused by circum-ferential expansion at the equator.

Osmotic Shock. One way to extract DNA from viral capsids is toput them under osmotic shock. Under these conditions someviral capsids (e.g., T-even phages) rupture, whereas others (e.g.,T-odd phages) stay intact.§ Thus, these experiments also eluci-date details about capsomer–capsomer interactions, and we haveanalyzed whether rupture induced by internal pressure and byexternal force are related. To do so, we have numericallydetermined the shape of the capsid under internal pressure p andmeasured the bond length between neighboring vertices. It wasassumed that, at rupture pressure, the critical bond stretching is4.5%, as it is for rupturing due to bond compression. Fig. 8 showsthe corresponding rupture pressure pr (in units of �e/R) asfunction of � for an icosahedral shell. Again, we assume thatpressure is isotropic. This is a strong assumption because underosmotic shock conditions salt concentrations are very high,which in principle can give rise to a strongly anisotropic pressureinside the capsids (see SI Fig. 12). However, the results presentedbelow are valid even in this case, provided that pr denotes themaximum pressure (see below).

As can be seen from Fig. 8, capsids with buckled ground states(� � �b) rupture at lower pr than capsids with unbuckled groundstates. One should note that, for rupture induced by an externalforce, rupture forces are generally higher for � � �b than for � ��b (Fig. 5). However, as can be seen from Fig. 5 Inset, theforce-induced deformation propagates over a larger area in thebuckled configuration than it does in the unbuckled state.Therefore, it is possible that the external rupture pressure forforce-induced rupture also is lower for buckled capsids, but thisquestion needs to be addressed in a more detailed analysis.

For small �, the rupture pressure reaches a constant value(Fig. 8). Within this limit, deviations from spherical shape aresmall and the internal pressure simply leads to a form-invariantup-scaling of the viral shell. Thus, changes in bending energy canbe neglected. By equilibrating the expanding pressure force andthe restoring force arising from bond stretching for a triangu-lated sphere, the relative change of bond length is found to bepR/(3�e). The numerically found value is somewhat lower be-cause of the small deviations from spherical shape. The samecalculation shows that a triangulated sphere with a harmonicstretching energy is only stable for pressures p � 0.75�e/R,independent of � (see SI Text). Anisotropic pressures inside thecapsid typically lead to deviations in pr on the order of 10–20%(data not shown).

Summary and Conclusions. Recent SFM measurements have re-vealed the elastic properties and mechanical limits of viralcapsids. Here, we have shown that numerical simulations are apowerful tool in complementing these experimental efforts. Wehave examined the reversible and irreversible mechanical be-havior on local and global scales. In particular, we have shownthat the elastic parameters characterizing the mechanical prop-erties of phages and viruses can be determined very precisely bya direct comparison of numerical and experimental data. Withthe numerical methods presented here, one is able to resolve themechanical response of viral shells to an external force in highdetail. In particular, because we are able to distinguish theresponse of hexamers and pentamers, we can identify the originof the experimentally observed bimodality of the elastic springconstants. We also make predictions for the elastic response andrupturing behavior of both empty and filled capsids as functionof the FvK number �. Furthermore, in our simulations themechanical limits of viral shells can be probed in an ensemblewhere every capsomer is indented. The comparison of thecorresponding rupture map of the shells with experimental dataon SFM-induced rupturing offers methods that can be used toexperimentally probe local protein–protein interactions. We canalso use the rupture criterion to predict the maximal sustainableinternal pressure of capsids.

So far, we have focused on shells with the shape of anicosadeltahedron. However, many viruses, such as phage HK97(30) or the animal virus polyoma (31), are chiral. Therefore, theinfluence of chirality on the elastic properties and mechanicalstability should be studied.

MethodsWe performed numerical minimization simulations of triangu-lated surfaces representing the surface of capsids with elastic(stretching) and bending energy. In our (small mesh size)triangulation, every capsomer is represented by several vertices.Therefore, these units also have some flexibility and elasticity.Such a discretization is suitable to determine the shape of viralshells (12) and yields results that do not depend on the numberof vertices [in contrast to coarse triangulations in which eachcapsomer is represented by a single vertex (18)]. The triangula-tion represents the underlying icosahedral (quasi)symmetry ofthe capsid. In particular, the pentavalent vertices correspond tothe centers of the pentamers.§Anderson, T. F. (1950) J. Appl. Phys. 21:70 (abstr.).

Fig. 8. Rupture pressure (i.e., pressure at which the relative bond stretchingreaches 4.5%) for an icosahedral shell as function of �.

Buenemann and Lenz PNAS � June 12, 2007 � vol. 104 � no. 24 � 9929

APP

LIED

PHYS

ICA

LSC

IEN

CES

Dow

nloa

ded

by g

uest

on

Mar

ch 1

5, 2

020

In our model, the vertices are connected by harmonic springs.Any deviation from the preferred intervertex distance l gives riseto an elastic energy (19),

Ee ��� e

2 ��i, j

� l �ri rj� 2, [2]

where vertex i is at position ri and the sum extends over nearestneighbor pairs only. �e is related to the 2D Young modulus �evia �e � �3�e/2. The bending energy is given by

Eb � �b �i

Hi2

Ai2 , [3]

where the sum extends over all vertices i. Here, Ai is the areaassigned to the ith vertex given by one-third of the area of alladjacent triangles. Hi is the mean curvature of vertex i (withcoordination number zi) given by (23)

Hi �12 �

j�1

zi eij � �nj,l nj,r

�nj,r� �nj,l� � nj,r � nj,l, [4]

with eij' ri � rj, and nj,l and nj,r are the nonnormalized surfacevectors of the facets left and right of the edge connecting vertexi and j. This discretization of the bending energy is chosen herebecause it is independent of the underlying geometry. It allowsus to numerically determine the elastic moduli of differentcapsids.¶

Starting from a triangulated sphere, the shape of minimalenergy is found by minimizing the total energy E � Ee � Eb by

using a conjugate gradient algorithm (32). To simulate the SFMexperiments mentioned in the Introduction, vertices i at thepoint of loading were moved away from their equilibriumposition ri. We therefore work in an ensemble of prescribedindentation X rather than in an ensemble of applied force.Generally, all vertices (except the indented vertex i) are free tomove; they are only constrained to lie above a (virtual) plane atz � 0. The shape of the triangulated surface under theseconstraints is again found by minimizing the total energy. In theensemble measurements, the force was applied perpendicularlyto the axis of symmetry. To avoid tilting of the structure, thelowest 10% of vertices of the shell were kept fixed in thesimulations.

The shapes of bacteriophage �29 and the CCMV plant virushave been quantitatively characterized by cryoelectron micros-copy and x-ray studies (28, 33). Phage �29 has an averageequatorial radius R � 21 nm, a shell thickness of h � 1.6 nm, anda FvK number of � � 11(R/h)2 � 2,000, implying that the shapeof �29 is noticeably buckled. For CCMV, the correspondingvalues are R � 12 nm and h � 2.5 nm. Thus, � � 300, and CCMVis slightly buckled.

¶The simpler discretization scheme used in refs. 12, 18, and 19 also has the drawback thatit does not yield the correct small � values of Kl (data not shown).

This work was supported by the Fonds der Chemischen Industrie and theCenter for Theoretical Biological Physics through National ScienceFoundation Grants PHY0216576 and PHY0225630 (to P.L.).

1. Smith DE, Tans SJ, Smith SB, Grimes S, Anderson DL, Bustamante C (2001)Nature 413:748–752.

2. Riemer SC, Bloomfield VA (1978) Biopolymers 17:785–794.3. Kindt J, Tzlil S, Ben-Shaul A, Gelbart WM (2001) Proc Natl Acad Sci USA

98:13671–13674.4. Tzlil S, Kindt JT, Gelbart WM, Ben-Shaul A (2003) Biophys J 84:1616–1627.5. Purohit PK, Kondev J, Phillips R (2003) Proc Natl Acad Sci USA 100:3173–

3178.6. Purohit PK, Inamdar MM, Grayson PD, Squires TM, Kondev J, Phillips R

(2005) Biophys J 88:851–866.7. Atlas RM (1998) Principles of Microbiology (Brown, Dubuque, IA).8. Evilevitch A, Lavelle L, Knobler CM, Raspaud E, Gelbart WM (2003) Proc

Natl Acad Sci USA 100:9292–9295.9. Ivanovska I, de Pablo P, Ibarra B, Sgalari G, MacKintosh F, Carrascosa J,

Schmidt C, Wuite GL (2004) Proc Natl Acad Sci USA 101:7600–7605.10. Michel J, Ivanovska I, Gibbons M, Klug W, Knobler C, Wuite GL, Schmidt C

(2006) Proc Natl Acad Sci USA 103:6184–6189.11. Caspar D, Klug A (1962) Cold Spring Harb Symp Quant Biol 27:1–24.12. Lidmar J, Mirny L, Nelson DR (2003) Phys Rev E 68:051910.13. Kumar A, Reddy VS, Yusibov V, Chipman PR, Hata Y, Fita I, Fukuyama K,

Rossmann MG, Loesch-Fries LS, Baker TS, Johnson JE (1997) J Virol71:7911–7916.

14. Olson NH, Gingery M, Eiserling FA, Bakera TS (2001) Virology 279:385–391.15. Xiao C, Chipman PR, Battisti AJ, Bowman VD, Renesto P, Raoult D,

Rossmann MG (2005) J Mol Biol 353:493–496.16. Dubois M, Deme B, Gulik-Krzywicki T, Dedieu, J-C, Vautrin C, Desert S,

Perez E, Zemb T (2001) Nature 411:672–675.

17. Elsner N, Dubreuil F, Weinkamer R, Wasicek M, Fischer F, Fery A (2006) ProgColloid Polym Sci 132:117–123.

18. Vliegenthart GA, Gompper G (2006) Biophys J 91:834–841.19. Seung H, Nelson DR (1988) Phys Rev A 38:1005–1018.20. Reddy V, Giesing H, Morton R, Kumar A, Post C, Brooks C III, Johnson J

(1998) Biophys J 74:546–558.21. Zandi R, Reguera D, Bruinsma RF, Gelbart WM, Rudnick J (2004) Proc Natl

Acad Sci USA 101:15556–15560.22. Zandi R, Reguera D (2005) Phys Rev E 72:021917.23. Wintz W (1997) Ph.D. thesis (Universitat Potsdam, Potsdam, Germany).24. Landau L, Lifshitz E (1975) Theory of Elasticity (Pergamon, New York).25. Lobkovsky AE, Witten T (1997) Phys Rev E 55:1577–1589.26. DiDonna B, Witten T (2001) Phys Rev Lett 87:206105.27. Klug WS, Bruinsma RF, Michel, J-P, Knobler CM, Ivanovska IL, Schmidt CF,

Wuite GJL (2006) Phys Rev Lett 97:228101.28. Tao Y, Olson NH, Xu W, Anderson DL, Rossmann MG, Baker TS (1998) Cell

95:431–437.29. Fokine A, Chipman PR, Leiman PG, Mesyanzhinov VV, Rao VB, Rossmann

MG (2004) Proc Natl Acad Sci USA 101:6003–6008.30. Helgstrand C, Wikoff W, Duda R, Hendrix R, Johnson J, Liljas L (2003) J Mol

Biol 334:885–899.31. Stehle T, Harrison S (1996) Structure (London) 4:183–194.32. Press W, Teukolsky SA, Vetterling WT, Flannery BP (1992) Numerical Recipes

in C (Cambridge Univ Press, New York).33. Speir JA, Munshi S, Wang G, Baker TS, Johnson JE (1995) Structure (London)

3:63–78.

9930 � www.pnas.org�cgi�doi�10.1073�pnas.0611472104 Buenemann and Lenz

Dow

nloa

ded

by g

uest

on

Mar

ch 1

5, 2

020