Mechanical Properties and DNA Organization of Viruses and Bacteria

(1)

Mechanical Properties and DNA

Organization of Viruses and Bacteria

Dissertation

-zur Erlangung des Doktorgrades der Naturwissenschaften

(Dr. rer. nat.)

dem Fachbereich Physik

der Philipps-Universit¨at Marburg

vorgelegt von

Mathias B¨unemann

aus Jena

(2)

Erstgutachter und Betreuer : Prof. Dr. P. Lenz

Zweitgutachter : Prof. Dr. F. Bremmer

(3)

Zusammenfassung

Viren sind nicht nur aufgrund ihrer medizinischen Relevanz seit jeher Gegenstand der biologischen Forschung. Insbesondere ihre erstaunliche Fähigkeit, sich ohne eigenen Stoff-wechsel zu reproduzieren und dabei aus wenigen Proteinen sehr komplexe Schalen (Kapside) aufzubauen, erregen starkes Interesse. Dieses Interesse wird nicht zuletzt durch zahlreiche Anwendungen genährt, die sich in jüngster Zeit auf dem Gebiet der Nanotechnologie ergeben haben.

Besonders die auffallende mechanische Robustheit viraler Kapside gegenüber externen wie internen Kräften hat in jüngerer Vergangenheit eine Reihe von bio-physikalischen SFM1 -Experimenten angeregt. In diesen Untersuchungen werden die Grenzen der mechanischen Belastbarkeit der Kapside quantitativ bestimmt. Sie geben daher Einblick in die Stärke der Proteinbindungen und tragen damit zum Verständnis des viralen Selbstzusammenbaus bei. Ein Anliegen der vorliegenden Arbeit ist es, durch eine theoretische und numerische In-terpretation der experimentellen Ergebnisse zur Aussagekraft dieser Untersuchungen beizu-tragen.

Nach einer Einführung in die Thematik und einer Diskussion der Grundlagen der Mecha-nik viraler Kapside in Kapitel 1 befassen wir uns in Kapitel 2 mit den analytischen Grundla-gen der Deformation elastischer Schalen. Die modellhafte Reduktion viraler Kapside auf ho-mogene Kugelschalen liefert uns einen ersten Einblick in die Physik ihrer Deformation. Für den Fall geringfügiger Verformungen gelingt uns mit einem Variationsansatz die Herleitung und die geschlossene analytische Lösung der Formgleichung. Insbesondere erhalten wir ex-akte Ausdrücke für die lineare Federkonstante und das Bruchverhalten solcher Schalen. Die Behandlung starker Eindellungen jedoch ist selbst in diesem stark vereinfachten Modell nur mittels Skalenabschätzungen möglich.

In den darauffolgenden Kapiteln 3 und 4 untersuchen wir die elastischen Eigenschaften und die mechanische Stabilität von viralen Kapsiden unter externen Kräften in Computersi-mulationen. Zu diesem Zweck entwickeln wir eine numerische Methode fort, die ursprünglich zur Untersuchung von axial-symmetrischen Vesikelformen eingesetzt wurde. Sie erlaubt die zuverlässige Simulation von Kapsiden selbst unter extremen Deformationen. Hierbei werden die Kapside durch eine triangulierte Oberfläche sphärischer Topologie repräsentiert. Derart diskretisierte Flächen weisen ein sehr ähnliches elastisches Verhalten auf wie zweidimension-ale, isotrope Medien. Für die Simulation viraler Kapside sind sie besonders geeignet, da sie im Gegensatz zu kontinuierlichen Modellen ähnliche strukturelle Eigenschaften wie vi-raler Schalen aufweisen. Insbesondere führt die Triangulierung sphärischer Oberfächen zum Auftreten topologischer Defekte.

Mit einer solchen diskretisierten Beschreibung gelingt es uns in Kapitel 3, experimentelle Phänomene wie das Auftreten einer bimodalen Verteilung der Federkonstanten auf die strukturellen Eigenheiten viraler Kapside zurückzuführen. Ihre numerische Repräsentation ermöglicht es außerdem, die Simulationsparameter an spezielle biologische Systeme anzu-passen. Insbesondere können spezifische Geometrien und Föppl-von-Kármán (FvK) Zahlen,

(4)

rials.

Aus dem Vergleich mit experimentellen Resultaten für φ29 mit numerischen Simulationen ermitteln wir eine kritische Verzerrung die zur Zerstörung der Schale führt. Anhand diese Kriteriums bestimmen wir die zur Zerstörung der Kapside von T 4 und CCMV nötigen Kräfte. Abweichungen von den experimentell beobachteten Werten lassen auf schwächere Kapsomerbindungen als in φ29 schließen. Allgemein beobachten wir eine erhöhte Stabilität für Kapside mit höherer FvK Zahl. Im Gesamtbild ergibt sich dabei ein Skalenverhalten der Festigkeit, das von der analytischen Voraussage abweicht und mithin Ausdruck der diskreten Struktur viraler Kapside ist.

Indem wir die Verteilung der Deformationsenergie in einem Ensemble von Punktde-formationen bestimmen, erstellen wir “Karten” der Bruchwahrscheinlichkeit auf viralen Schalen. In Kombination mit experimentellen Ensemblemessungen k¨onnen diese Karten einen Beitrag zur Bestimmung lokaler Proteinbindungsst¨arken liefern.

In Kapitel 4 studieren wir den Einfluss der DNA auf die Stabilität gefüllter Viren. Der mechanische Einfluss der DNA wird in erster Linie auf den enormen Druck zurückgeführt, den diese auf die Hülle ausübt. Im Allgemeinen beobachten wir eine erhöhte Festigkeit gegenüber externen Kräften. Experimentell kann die Robustheit gefüllter Viren auch über ihr Verhalten unter osmotischem Schock charakterisiert werden. Hierbei wird der Innen-druck über die Salzkonzentration der Pufferlösung kontrolliert. Während viele Kapside einen rapiden Druckanstieg unbeschadet überstehen, verlieren andere (z.B. T 4 Phagen) ihre strukturelle Integrität. In numerischen Simulationen finden wir, dass sich Schalen mit hohen FvK Zahl generell durch eine geringere Robustheit gegenüber osmotischem Schock auszeichnen.

Abschließend wenden wir uns in Kapitel 5 Bakterien und damit einem biologischen System zu, das im Vergleich zu viralen Kapsiden weitaus größere Längenskalen aufweist, als die bis dahin behandelten Viren. Fluoreszenzmikroskopische Untersuchungen beweisen jedoch auch hier einen hohen Organisationsgrad der DNA. Aufgrund der Längenskalendifferenz findet die Kompaktierung der DNA in Bakterien auf der Grundlage völlig anderer physi-kalischer Prinzipien statt als in Viren. Mit Monte Carlo Simulationen wenden wir uns der Frage zu, inwieweit aktive Organizationsprozesse notwendig sind, um die experimentellen Beobachtungen zu erklären.

(5)

CHAPTER 1 Introduction

V

iruses are in many ways astonishing biological systems. They are the smallest known self-replicating entities which use nucleic acids to store their biological information1_.

Their genome is highly optimized, being only 5 to 170 kilo base pairs long. It is protected against degradation by an outer shell, the “capsid”. Capsid diameters range from only 27nm to 300nm [1]. Apart from their high biochemical stability, capsids also exhibit remarkable mechanical properties. These properties arise from their fascinating, highly symmetric design which will be discussed in Section 1.1.

Despite their apparent simplicity viruses exhibit

Figure 1.1: Electron micrograph of T4

phages attached to E. coli. Their icosahe-dral capsids store the viral genome under high pressure.

a very complex life cycle. Though their genome en-codes all proteins needed to assemble a new capsid, viruses lack any metabolic components or ribosomes. Consequently, they can only replicate within a host cell. The viral genome is released into the cell which subsequently starts duplicating the viral nucleic acid and synthesizes the proteins necessary for viral repli-cation. After the assembly of sufficiently many (typ-ically hundreds of) copies, the new virus particles are released, often resulting in cell death. Outside their host cells the infectious particles remain passive until a new host is encountered.

Because they interfere with the metabolic func-tions of complex organisms they are highly specified to their hosts. Accordingly, they can be classified by the types of hosts they infect. On the topmost level they are divided into bacteriophages, animal viruses, and plant viruses [1].

Bacteriophages, or shortly phages, infect prokaryotic host organisms. The phages attach to bacterium’s outer cell membrane which is subsequently penetrated by a spike-like exten-sion to inject the DNA into the cytoplasm (see Fig. 1.1). To provide the necessary force 1_{Prions, self-replicating proteins, are smaller. They encode biological information in their specific}

(8)

confor-for the injection, the DNA needs to be stored under high pressure. This is accomplished by an active packing process performed by a molecular motor after capsid assembly has completed.

In contrast, the host cells of animal and plant viruses are eukaryotic. Here, the viral genome has to be transferred to the nucleus and cannot be directly injected into the cy-toplasm. Therefore, many of these viruses possess certain surface proteins which facilitate binding to and fusion with the host cell membrane. Due to the ability of the capsid to fuse with the host cell, DNA does not need to be stored under pressure. Here, the DNA is directly included in the capsid during assembly.

The ability to build rather complex robust structures from simple constituents makes viruses unique among biological systems. It also accounts for the scientific interest that has grown in the last decades. The detailed knowledge of the mechanism of viral assembly becomes an increasingly relevant input to nano-technology [2]: The self-assembly of virus-like particles offers the possibility to enclose tailored cargo, which could not enter the host cell otherwise. Viruses have already been loaded experimentally with drugs, markers, and even nano-particles like quantum dots [3] or magnetic beads [4]. Even their application as nano-reactors for single enzyme experiments has been proven possible recently [5].

The biological importance of capsid mechanics as well as the wide range of applications in nano-technology have motivated a recent series of experiments exploring the mechanical limits of viral capsids. These studies revealed an astonishing robustness of viral shells against both large internal pressure and externally applied forces (see Section 1.3).

This work aims to contribute to the understanding of viral stability by analytical and numerical investigations. In the following introductory sections we will review earlier the-oretical and experimental results, that are important for the study of capsid mechanics.

1.1 Capsid Structure

Crick and Watson [6] argued, that due to their small genome size, viruses can encode just a small number of different proteins. Therefore, due to the limited number of possible protein bindings, the shell must be composed in a very regular fashion. From this principle of structural simplicity they concluded, that capsids have either a rod-like or a spherical shape. Indeed, many plant viruses (such as Tobacco mosaic virus) encapsulate their DNA in long tubular structures. However, most capsids are hollow spherical containers. In this work we will exclusively be concerned with the latter.

The design of spherical capsids was elucidated by Caspar and Klug’s “quasi-equivalence principle” [7]. In the simplest case, 60 identical proteins assemble into a perfect icosahedron. Here, every five proteins cluster into one larger “capsomer”. The twelve capsomers in turn form a regular icosahedron. Then, all proteins are located in chemically identical environments. Due to the limited size of a single capsomer, the diameter of an icosahedral shell is typically ∼ 25nm [8]. However, the genome of most viruses is too long to fit into such a small container. Hence, they must construct their shell from more than 60 proteins.

(9)

1.1 Capsid Structure

Caspar and Klug proposed that in larger shells the proteins arrange in such a way that the protein interactions are quasi equivalent to the ones found in the perfectly icosahedral shell. In their theory, the icosahedral symmetry of capsids is generalized to an icosadeltahedral arrangement of capsomers. In this particular geometry the local chemical environment of an individual protein is equivalent to that in a perfect icosahedron. The quasi-equivalence principle requires that the protein assemble into different types of capsomers: A capsomer may consist of either five or six proteins forming a “pentamer” or “hexamer”, respectively.

These capsomers are located on the vertex positions of an equilateral triangulation of a sphere. While a planar, regularly triangulated sheet consists exclusively of vertices with six neighbors (i.e. hexamers), the spherical topology of viral shells introduces local defects in the triangulation. Due to Euler’s formula exactly twelve vertices with five neighbors (i.e. pentamers) are required to close the triangulated sheet to a spherical shape.2 These pentamers lie on the vertex positions of a perfect icosahedron. The number of hexamers is in principle unlimited, provided it is compatible with Euler’s formula. They can arrange in multiple ways, extending the original icosahedron to an icosadeltahedron.

Figure 1.2: The pentamers (green) are

connected by h steps on the lattice fol-lowed by a 60◦ _{turn and k more steps.}

Image from VIPERdb [8, 9]. Quantitatively, an icosadeltahedron can be

character-ized by the Caspar-Klug numbers h and k [7]. These numbers have a direct geometrical interpretation (see Fig. 1.2): In order to get from one pentamer to its nearest neighboring pentamer, one must follow a straight chain of h capsomers, then make a 60 degrees turn and pro-ceed another straight chain of k capsomers. In the follow-ing, we will call capsids with h, k 6= 0 skew. Note that, skew capsids with h 6= k are chiral. The number of cap-somers needed to build a (h, k)-capsid is 10T + 2, where T = h2+ hk + k2 is the so called triangulation number [7]. The triangulation number has an intuitive geomet-rical meaning: It is just the squared (straight) distance between neighboring pentamers measured in units of the capsomer-capsomer spacing.

The approximately icosahedral symmetry of capsids was already experimentally evident by the time Caspar

and Klug developed their theory [10, 11]. Since then myriads of Cryo-electron microscopy (CryoEM) and X-ray studies have confirmed the equivalence principle and the icosadelta-hedralsymmetry of capsids (for references see Ref. [8]).

A given triangulation might have several realizations (h, k), see Tab. 1.1. In particular, for T = 49, 169 both skew and non-skew structures can be realized. Examples for ambiguous T numbers are the Paramecium bursaria chlorella virus type 1 (PBCV1) with triangulation

2

For a surface with spherical topology Euler’s formula reads V −E +F = 2, where V, E, F are the numbers of vertices, edges, and facets, respectively. Since all facets are triangles, E = 3F/2. Any of the V5(6)

(10)

T (h, k) 49 (5,3); (7,0) 91 (6,5); (9,1) 133 (9,4); (11,1) 147 (7,7); (11,2) 169 (8,7); (13,0) 1183 (26,13);(29,9);(31,6)

Table 1.1: Selected triangulation numbers T with more than one (h,k) realization. T=49 is the smallest

triangulation number for which more than one realization exists. Examples for such T-numbers are indeed found in nature: the capsid of PBC virus type 1 has T=169, (8,7). The CI virus has T=147, (7,7). To the best of our knowledge no virus with T=49 is known. The giant mimi-virus presumably has T=1183. It is unclear which (h,k) pair is realized.

number T = 169 and (h, k) = (8, 7) [12] and the Chilo iridescent virus (CIV) [13] with T = 147 and (h, k) = (7, 7). The triangulation number of the giant mimi virus [14] is estimated to be around T ≈ 1179. The closest theoretically valid triangulation number is T = 1183 which has even three possible (h, k) pairs, see Tab. 1.1.

The local environment of a single protein is quasi-equivalent to its environment not only in a pure icosahedron but also in other icosadeltahedra. Indeed, with the aid of special scaffolding proteins identical coat proteins can assemble to different T structures. Examples include phage P22 which can assemble into T = 4 and T = 7 capsids [15] and phages P2 (T = 7) and P4 (T = 4) which consist of the same coat protein but differ in their T -numbers [16]. Disrupted polyoma shells reassemble to virus like particles (VLP) with T = 2 or T = 3 depending on the volume of the enclosed cargo [17].

Helper proteins may certainly play a role in the assembly of many viruses but are ab-sent in others. Indeed, numerical studies which model the arrangement of capsomers as a Monte-Carlo process [18, 19] suggest that assembly into highly symmetric structures might also occur as a fluctuation-driven reorganization towards the configuration of minimal in-teraction energy.

In general, quasi-equivalence is a powerful theory to explain and predict capsid structure. However, there are many exceptional cases: A prominent example is the shell of Polyoma virus which consists of 360 identical proteins assembled to 72 pentamers [20]. Neverthe-less, these capsomers arrange on a regular icosadeltahedral (2, 1) lattice. The violation of quasi-equivalence principle implies, that the detailed structure of the capsomer-capsomer-interaction has a spatial dependence.

There is also a wide range of aspherical capsid shapes. For example, the human immu-nodeficiency virus (HIV) has a conical shape [21]. Some bacteriophages like φ29 [22] and T 4-mutants [23] have sphero-cylindrical shapes. Recent numerical studies [24, 25] suggest, that such complex shapes result from a spontaneous curvature. It may arise from the action of scaffolding proteins that force capsomers not to bind under their preferred binding angles. The previous discussion makes clear that the internal structures and shapes of capsids and even more their stability depends strongly on the protein interactions. The energetic

(11)

1.2 Defects and Buckling

landscape of individual protein-protein associations is highly complex. It not only reflects the distance dependence of the covalent binding but also depends on the conformational changes in protein structure under stress. The exact shape of these potentials of course differs from protein to protein. Only for some proteins the mechanical properties have been subject to experimental and theoretical studies [26, 27].

For the purpose of theoretical and numerical modeling it is convenient and (for small strains) sufficient to replace these complex interactions by simpler ones. The distance dependent part of the interaction can be captured by a harmonic potential,

Ee=

k

2(a − a0)

2 _, _(1.1)

where k is a generalized spring constant, a the length of the covalent bond and a0 its

rest length. For the modeling of the angle-dependent part different approaches have been proposed. In the simplest one, the normal vectors ni, nj of two adjacent triangles i and j

of the icosadeltahedron are compared via Eb=

˜ κ

2|ni− nj|

2 _. _(1.2)

Here, κ is a generalized bending rigidity of the covalent bonds. Other approaches are discussed in Appendix C.1.3. It is noteworthy that for large number of capsomers Eqs. (1.1) and (1.2) turn into the elastic free energy of a two dimensional continuous shell. This aspect is discussed in more detail in Appendices B.1 and C.1.3.

1.2 Defects and Buckling

The inevitable existence of disclinations in the triangular mesh of capsomer-capsomer bonds induces additional strain in the capsid surface. Pentamers can therefore be expected to have a strong influence on capsid elasticity. The physics of disclinations can be understood, when a single disclination in a planar triangulated sheet is considered [28].

A fivefold disclination, as shown in Fig. 1.3, induces strain: If the sheet is to remain flat, circumferential bonds are stretched while radial bonds are compressed, leading to a deformation energy according Eq. (1.1). It was shown in Ref. [28], that in such a flat sheet with radius R these strains give rise to a deformation energy which diverges with R2. If the sheet is allowed to buckle out [i.e. by introducing a bending energy as in Eq. (1.2)], the total deformation energy shows merely a logarithmic R dependence. The crossover between the flat and the buckled regime is determined by the competition between bending and stretching energy, which can be characterized by the F¨oppl-von-K´arman (FvK) number γ ≡ R2κe/κb [see also Eq. (B.18)]. It was shown, that above the buckling threshold

γb≃ 154 (1.3)

it becomes favorable for the sheet to buckle out of plane, thus forming a cone like surface (see Fig. 1.3).

(12)

Figure 1.3: A fivefold disclination in a hexagonal mesh causes additional strain. If the triangulated sheet is to remain flat, circumferential bonds are stretched and radial bonds are compressed. If the sheet is allowed to bend the disclination will buckle into a conical shape above a critical Föppl-von-Kármán number. Figure taken from Ref. [28].

In the theory of elasticity of thin shells (see Appendix B.1) the Föppl-von-Kármán number is related to the ratio between shell thickness and radius [see Eq. (B.19)]. For many viruses this value is known from CryoEM measurements or X-ray diffraction patterns. Typical γ values of viral shells lie in the range from below the buckling threshold (e.g. alfalfa mosaic virus with γ ≃ 60 [29]) up to several thousand (e.g. phage T 4 with γ ≃ 3300 [30]). An extreme example is the giant mimivirus (diameter ≃ 600nm) with a FvK number γ ≃ 20000 [14]. Even higher γ-values are possible for artificial capsules such as vesicles with crystallized lipid membranes (γ ∼ 105) [31] or polyelectrolyte capsules (γ ∼ 106) [32].

The importance of the buckling threshold for the shape of viral capsids has been pointed out in Ref. [33]. Using the phenomenological model of the capsid elasticity given in Eqs. (1.1) and (1.2), they found that the buckling threshold [Eq. (1.3)] marks the crossover from a spherical to a strongly faceted, icosahedral shape.

Analytically, disclinations in flat sheets can be incorporated into the von-K´arm´an plate equations [Ref. [28] and also Eq. (B.28)]. The extended plate equations Eq. (B.28) imply that disclinations may screen the Gaussian curvature of the buckled surface. Therefore, disclinations give rise to additional degrees of freedom, i.e. position, number, and charge by which the total deformation energy can be minimized [34]. The competition between cost of introducing a new disclination and gain from screening the Gaussian curvature is controlled by the ratio between radius R and capsomer spacing a [34]. Astonishingly, above a critical ratio R/a ≃ 0.2 it becomes favorable to introduce additional disclinations. In the icosadeltahedral geometry this critical ratio corresponds to a maximal number of capsomers V ≃ 305. For larger numbers one expects that the icosadeltahedral symmetry becomes unstable even in the zero-temperature limit. The newly introduced disclinations typically arrange in lines, so-called grain boundaries. Since V ≃ 305 corresponds to Caspar-Klug numbers above (5, 1), most viral capsids are to small to show grain boundaries.

Grain boundaries have experimentally been observed in artificial spherical crystals [35]. Large viruses like mimi virus, PBCV1, and CIV also exceed the threshold of V = 305. How-ever, no experimental evidence for grain boundaries in these viruses exists so far. Moreover, they are unlikely to occur, since additional disclinations would require the formation of

(13)

sev-1.3 Experimental Data

enfold capsomers in order to be compatible with the principle of quasi-equivalence.

1.3 Experimental Data

Bacteriophage capsids have extreme elastic properties. They withstand extreme internal pressures exerted by their densely packed DNA. This pressure is necessary to inject the DNA into the prokaryotic host cell [1]. Values as high as 6MPa have been reported for φ29 using the stall force of the packing machinery [36]. Experiments and analytical models on phage λ, in which the genome ejection is osmotically inhibited also report internal pressures of the order of 5MPa [37, 38]. The theoretical analysis of the energetics of packed DNA gives estimates of the same order of magnitude [39, 40, 41, 42].

The pressure arises mainly from two different

con-Figure 1.4: Experimental (left) and

numeri-cal (right) evidence for the spooling of DNA in T7 phages. The images show averaged DNA densities when viewed in axial direction. Figures from Refs. [43, 44].

tributions: The negative electrical charge of the DNA backbone gives rise to electrostatic repulsion between neighboring strands of DNA. Furthermore the per-sistence length of DNA (ξp≈ 50nm) [see [45, 46] and

references therein] exceeds the linear dimensions of most phages. Hence they are forced to bend DNA which yields another contribution to the pressure.

X-ray diffraction patterns [47] and CryoEM im-ages [44] of the packed DNA inside T 7 phim-ages show highly regular patterns (see Fig. 1.4, left). These experiments suggest, that DNA optimizes the

over-all packing energy by acquiring a spooled configuration. Direct numerical minimization of the conformational energy of the DNA [40, 43] supports this view (see Fig. 1.4, right).

The remarkable robustness under high internal pressure motivated recent scanning force microscopy (SFM) studies [48, 49], in which phage capsids are indented on the nanometer scale (see Fig. 1.5). In these experiments the shell is indented with a tip which is mounted to a flexible cantilever. The tip radius is typically < 20nm [48]. The measured data are force-distance curves as shown in Fig. 1.6.

For small forces one generally observes a reversible elastic behavior. Here, the linear and nonlinear regime of elasticity of spherical shells can be studied. The observed spring constants depend on the bulk elastic moduli, i.e. Young’s modulus and bending rigidity. In this sense, small forces explore the global elastic properties of the shell. Surprisingly, for φ29 a sharply peaked bimodal distribution of spring constants was reported [48], see Fig. 1.6 (right). As will be shown, this observation may be attributed to the internal inhomogeneities in the capsid structure.

Larger forces may cause the shell to weaken irreversibly in a small region (see Fig. 1.6, left). This softening is commonly attributed to the failure of individual capsomer-capso-mer bonds and local disintegration of the shell. Hence, such rupture experiments elucidate the local elastic properties, i.e. the strength of individual protein interactions. Rupture experiments carried out on phage φ29 showed that these capsids resist external point forces

(14)

Figure 1.6: (left) Force distance curves for SFM rupture experiments on phage φ29 reported in Ref. [48]. During probing along the cyan line “2” the force experiences a sudden jump. A successive measurement on the same locus of indentation (“3”) shows, that this jump caused an irreversible change of the local elastic response. The red lines illustrate the effect of a repeated force application. Here, in a first measurement the elasticity of the intact shell is probed. Line “2” shows the elastic behavior of the shell which apparently disintegrated after a large number of SFM contacts. (right) Phage φ29 exhibits a bimodal distribution of spring constants. The different colors correspond to different SFM modes of operation. Figures taken from Ref. [48].

up to ∼1nN.

In contrast to bacteriophages, viruses that infect

eu-~60nm

Figure 1.5: SFM experiment in which

a phage φ29 capsid is successively in-dented until the shell eventually breaks. Image by Wuite Lab.

karyotic cells, e.g. CCMV and polyoma, penetrate their host cell. Their DNA is released by subsequent disas-sembly of the shell. Correspondingly, these viruses do not store their DNA under high pressure. Nevertheless, their resistance to external forces is remarkably strong [50].

More recently, the effect of maturation3and DNA pack-ing on the elastic properties have attracted much exper-imental interest [51, 52, 53, 54]. Filled capsids generally exhibit increased linear spring constants. It is unclear, if this effect can be attributed to the internal pressure alone or if additional biochemical interactions are needed to explain the data. Similarly, during maturation capsids develop an increased stiffness. This effect might be due to the alteration of capsid geometry (e.g. decreasing wall

thickness) or the bio-chemically induced change of bulk elastic moduli.

Most SFM experiments on viral capsids carried out so far are ensemble measurements in the sense that the exact locus of indentation is poorly resolved and no information about the type of symmetry of the indented sites is available. Moreover, the size of the cantilever is comparable to the capsid size. Hence, the average elastic response of a larger region is measured. Recent experiments [52, 55] have largely improved this situation by resolving the points of 2-, 3-, and 5-fold icosahedral symmetry. These high resolution experiments offer 3_{Maturation is the last stage of capsid assembly and involves conformational changes of the capsomers}

(15)

1.4 Questions and Methods

the possibility to test directly the influence of quasi-symmetric design on capsid stability.

1.4 Questions and Methods

Nano-indentation experiments help to understand the nature of protein interactions inside capsids and thereby shed light on the mechanisms of viral self assembly. Appropriate models of the underlying physics help to give stronger explanatory power to these experiments.

So far, the stability of viral shells has been the subject of several numerical investigations. The dependence of virus shape on the FvK number was analyzed in Ref. [33]. The effect of DNA packing and maturation on viral shape has been studied in Ref. [56]. Only recently, the elasticity of capsids has come into the focus of numerical studies [57] which are based on a discretization scheme introduced for crystalline membranes in Ref. [28]. In other approaches, the shells are directly built up of proteins [58] or capsomers [18] with specific, spatially varying interactions. On this basis the stability of capsids against internal pressure has been studied [59].

Finite element methods have widely been used to estimate the bulk elasticity of the specific capsids [48, 50, 60]. In these methods a homogeneous capsid material is assumed. In particular, elastic effects emerging from the discrete composition of the viral shell are not captured by these approaches.

More recent works which analyze the properties of

dis-Figure 1.7: The effect of stress on

icosahedral shells in an early experiment by Caspar [61].

cretized shell models take defects into account. These studies include the normal mode analysis (NMA) of shape transformations during capsid maturation [62]. NMA has also been used to address the significance of the buckling threshold to the elastic response of capsids [63]. The shape transformation of icosahedral shells under an ex-ternal load was studied in Ref. [57].

In this work, we aim to understand capsid stability from the quasi-equivalent design principle. In particular, we will stress the importance of topological defects on the elastic response. Inspired by the nano-indentation exper-iments discussed in the previous section, we aim for an understanding of the behavior of icosadeltahedral shells under point loading. We will generalize the

experimen-tal results to make predictions about the reversible and irreversible (rupture) behavior for different FvK numbers and capsid geometries.

In a second part we consider bacteria as biological systems which also show a high de-gree of DNA organization. Here, length scales are 10 to 100 times larger than for viruses. Therefore, genome compaction relies on completely different physical mechanisms. The distribution of DNA has many biological consequences, e.g. for intracellular transport and regulation as well as for the mechanics of cell division. Thus the theoretical understanding of the packing of the bacterial genome will be a valuable contribution to ongoing biological

(16)

research.

This work is organized as follows: In Chapter 2 we derive exact shape equations for arbi-trarily shaped, continuous shells subjected to small deformations. In particular they allow the analytical treatment of icosahedral shells. We restrict their evaluation to the analyti-cally tractable case of spherical shells. We obtain results for the reversible elastic regime as well as for the rupture behavior. The response of spherical shells to large deformations is qualitatively investigated in a scaling analysis.

Chapters 3 and 4 are concerned with the numerical analysis of capsid stability. There, we extend an approach developed for the investigation of vesicles [64]. Compared to the standard method [28, 33] it has the big advantage that it produces highly stable and reliable results even under high local strain. These discretized models are very well suited for the simulation of viral capsids since they possess an internal structure similar to that of viral capsids.

In Chapter 3 we focus on empty capsids. In our simulations, we systematically vary the elastic moduli and geometry of the capsids and probe their mechanical response to external deformations. This allows the systematic study of rupture of mechanical shells. The simulations provide insights into the geometry-dependence of viral stability and allow us to relate the observed macroscopic elasticity to the strength of capsomer bindings and the underlying capsid design. By comparing our numerical results with experimental data we are able to give reasonable estimates for the bulk elastic moduli.

In our simulations we can even access features which are not observable in experiments, such as e.g., the local strain. We determine the distribution of strain and rupture forces across capsid surfaces. By comparing with ensemble rupture measurements these rupture maps offer the possibility to draw conclusions about the spatial variation of protein binding strength.

The stability of filled capsids is discussed in Chapter 4. The DNA is taken into account by an appropriate pressure contribution to the total energy of the capsid. This allows us to simulate nano-indentation experiments on loaded phage heads with high internal pressure. In this context, we will also discuss the influence of anisotropic packing. We extend our numerical scheme to study the stability under extreme internal pressure. These results can be related to osmotic shock experiments in which capsids are osmotically swollen.

In Chapter 5 we finally study random configurations of the bacterial genome in Monte Carlo simulations. We extend random walk methods known from polymer physics to cope with the geometrical constraints imposed by the bacterial cell. These simulations indicate to which extend active mechanisms are involved in DNA compaction.

(17)

CHAPTER 2 Elastic Shells

T

he SFM experiments mentioned in the introduction are far too complex to be un-derstood analytically in every detail. Apart from the potential of individual protein interactions, which is simply not known, the discreteness of the viral capsid poses a chal-lenging analytical problem. Moreover, in experiments the geometry of loading cannot be controlled with high precission. This leads us to consider a homogeneous, spherical shell under point load as an idealized, analytically tractable model. We will see, that some of the experimentally observed features are explained by this simplified system. However, exper-imental observations which reflect the discrete structure of viral capsids will be addressed in the following chapters.

In the first section of the chapter we will derive the equations which govern the shape of a continuous surface (of arbitrary geometry) under external load. In Sections 2.2 and 2.3 we consider the simplest case, a spherical shell under point load. Still, even the elasticity of this reduced model is only partly understood analytically. In Section 2.2 we employ scaling arguments to gain qualitative insights into the response of a spherical shell to the applied point forces. The scaling results for small indentations are put on a solid analytical basis in Section 2.3.

2.1 Shape Equations

In the following derivation of the shape equations disclinations are not explicitly taken into account. As discussed in Section 1.2, disclinations cause additional strain when introduced in a previously disclination-free surface. If the bending rigidity is zero, the surface will acquire a shape, in which any strain vanishes. If the geometry of this strain-less surface is known, the formalism presented in this section can be used to determine the shape of surfaces with disclinations. In the case of an icosahedral distribution of disclinations the strain-less surface is an icosahedron. Thus, the parameterization of the icosahedron might serve as a starting point for the analysis of viral capsids under load. Such a parameteriza-tion will be given in Appendix B.3.

(18)

As discussed in detail in Section B.1, for thin shells the full 3D elastic energy decomposes into two separate contributions.

The deformation energy is the sum of a contribution Ee from the in-plane deformation

and another term Eb arising from the bending of the material,

E = Ee+ Eb . (2.1)

The in-plane term is given by Ee= Z dS λ 2(u i i)2+ µuiju j i . (2.2)

Here, ui_j is the two dimensional strain tensor, which measures the deviation from the reference metric of the undeformed shell,

2uij = g′ij− gij , (2.3)

where g_ij′ is the metric of the deformed and gij that of the undeformed capsid, see Appendix

A. The bending contribution has the form Eb = Z dS κ 2(b i i)2+ κGdet b . (2.4) Here, bi

j is curvature tensor, see Appendix A. Its trace is related to the mean curvature

H = −bii/2, while det b is the Gaussian curvature K. Due to the Theorem of Gauss-Bonnet

the surface integral over the Gaussian curvature is a topological constant for closed surfaces [65]. Hence, the second term in Eq. (2.4) does not contribute to the elastic response of a shell, since it does not change under deformations which do not introduce any holes in the surface. Note, that the thin shell expression Eq. (2.4) is a special case of the Helfrich free energy for vanishing spontaneous curvature [66]. The general bending energy for material with a preferred mean curvature H0 is

Eb =

Z

dS 2κ(H − H0)2 . (2.5)

In viral capsids a preferred curvature may arise from preferred bond angles between cap-somers.

In order to derive a shape equation for a shell under external forces we investigate the change of the total deformation energy under deformations of the form [67]

R′ = R + ηiRi+ ψN . (2.6)

Conveniently, the tangential deformation field ηi is decomposed into an irrotational and a solenoidal part,

ηi= gji∂jχ + ǫij∂jξ , (2.7)

(19)

2.2 Scaling Results for a Spherical Shell

We want to calculate the total energy under a prescribed (local) deformation which imposes geometrical constraints on the fields ψ and ηi. For example, in the case of a shell under axial point load we have a fixed normal displacement at the poles

ψ(θ, ϕ)|θ=0= ψ(θ, ϕ)|θ=π = −

∆

2 . (2.8)

To illustrate the problem we are facing, let’s omit the ηi _{dependence for a moment and}

assume that the deformation ψ is known. Then, the variation of the deformation energy with respect to some additional displacement δψ must vanish. Hence

δE(ψ) = E1(ψ)δψ + O(δψ2)= 0 .! (2.9)

Here, E1(ψ) is a linear operator which contains information about the geometry of the

deformed manifold such as b′_ij and g_ij′ . The primed quantities b′_ij and g′_ij depend on ψ and on the undeformed geometry. Thus, Eq. (2.9) is a differential equation for ψ, which describes the deformation under the constraint Eq. (2.8). However, these boundary conditions are not easy to meet. The variation δψ must retain the constraint Eq. (2.8) unaltered, i.e.

δψ(θ, ϕ)|θ=0= δψ(θ, ϕ)|θ=π = 0 . (2.10)

Note, that since the functionals Eqs. (2.2) and (2.4) are surface integrals, the Euler-Lagrange-equations do not take into account any boundary terms. Therefore it is not straightforward to include Eq. (2.8) into the variational analysis.

2.2 Scaling Results for a Spherical Shell

The mechanical behavior of a spherical shell under point load can already be understood qualitatively from scaling arguments [68]. The scaling analysis is based on two assumptions: (i) A small force leads to a flattening of the polar region. (ii) Above a critical force the shell inverts its shape locally at the point of loading. We will see that this inversion transition causes a crossover from a linear to a non-linear elastic response.

2.2.1 Spring Constants

The linear regime Under a small load the shell will be flattened in an area of order d2 (see Fig. 2.1). The deformation is up-down symmetric, since the spherical shell has also a point contact to the substrate at the bottom. Thus the total indentation ∆ is shared equally between the two hemispheres. The indentation of one hemisphere is

∆

2 = R(1 − cos θ) − ρ(1 − cos θ

′

) ≈ 1₂(Rθ2_{− ρθ}′2) , (2.11)

where we assume that the diameter d of the deformed area is small, i.e. θ ≪ 1. The radius of the flattened region is

d ≈ ρθ′ ≈ Rθ . (2.12)

Thus the mean curvature in the deformed region is H ≈ 1_ρ ≈ _R1 + ∆

(20)

R

θ

d

ρ

∆/2

Figure 2.1: The shell is flat-tened by a small point load. Therefore, in presence of a spontaneous curvature 1/R the

contri-bution to the Helfrich bending energy is of the order Eb ≈ 4πκbd2κbd2 1 ρ − 1 R 2 = 8πκb ∆2 d2 . (2.14)

When the material was initially flat, i.e. H0 = 0,

Eb ≈ 4πκbd2 1 ρ2 − 1 R2 = 8πκb 1 2 ∆2 d2 + ∆ R . (2.15)

In Eqs. (2.14) and (2.15) it has been assumed that the mean curvature is constant on the deformed area (with surface area πd2_).

Another factor 2 accounts for the up-down symmetry.

The in-plane stretching energy arises from the compression in meridional direction inside the flattened area. The mean in-plane

displacement is u ∼ 1 − d/l0, where d ∼ l0cos θ. Thus the strain tensor is of the order

u ≈ 1 − cos θ ≈ θ

2

2 ≈

∆

R . (2.16)

For the in-plane elastic energy it follows that Ee≈ 2πd2κeu2= π 2κe ∆ R 2 d2 . (2.17)

The size d of the flattened area follows from the condition that the total deformation energy must be in equilibrium with respect to variations in d. It follows that

d ≈ 23/4Rγ−1/4 . (2.18)

The total energy without spontaneous curvature is then E ≈ 8πκb ∆ R + 2 √ 2πκb√γ ∆ R 2 . (2.19)

The force distance law follows from

F = dE d∆ ≈ 8π κb R + 4 √ 2π√γκb R ∆ R . (2.20)

The ∆-independent term appearing in Eq. (2.20) vanishes when a spontaneous curvature of 1/R is assumed. It therefore represents the force needed to bend the material without spontaneous curvature into the spherical shape during assembly. Hence, the force needed to deform the shell is only the ∆-dependent part of Eq. (2.20). Measuring F in units of κb/R and making use of the Föppl-von-Kármán number γ = R2κe/κb the above relation

reads ˜ F ≡ _κR bF ∼ √_γ ∆ R . (2.21)

In the above reasoning we assumed d ≪ R. Because of the relation Eq. (2.18) this condition translates to γ ≫ 1, i.e. Eq. (2.21) holds only for shells with an in-plane stiffness at least of the order of its bending stiffness divided by R2_{. In particular, Eq. (2.21) is not valid for}

(21)

∆

θ ρ

d

R

θ

R

r

d

Figure 2.2: (left) Under an increasing force the shell eventually inverts its curvature at the top. The deformation

energy is mainly located in the circular rim around the inversion. In absence of a spontaneous curvature the inverted region does not contribute to the deformation energy. (right) A detailed view of the highly bent rim. Locally, the material has a radius of curvature ρ which is much smaller than the radius of the shell.

The nonlinear regime When subjected to a larger point load in radial direction the shell is deformed as sketched in Fig. 2.2. The curvature of the shell is inverted inside a region of radius r ∼ Rα. In absence of a spontaneous curvature the Helfrich bending energy depends only on the squared mean curvature. In this case the inverted region does not contribute to the deformation energy. Rather the main deformation takes place inside a small rim of width d, where the shell is highly bent. The indentation depth is of the order

∆ ∼ R(1 − cos θ) ∼ Rθ2 ∼ r

2

R . (2.22)

Thus the radius r of the inversion is fixed by R and ∆. The only remaining parameter is the width d of the rim. The radius of curvature in the deflection zone ρ follows from the relation d ∼ ρθ, see Fig 2.2 (right). Thus the bending energy is of the order

Eb ∼ κbrd 1 ρ2 = κb 1 d ∆3/2 R1/2 , (2.23)

where the factor rd is attributed to the area of the bent rim. The contribution to the in-plane elastic energy comes from the strain of the circles of latitude. The radius of a circle changes in the order dθ2. The relative compression is thus

u ≈ dθ2/r ∼ rd/R2 . (2.24)

Therefore the elastic energy due to the in-plane deformation results in Ee∼ κerd rd R2 2 = κe d 3 R5/2∆ 3/2 _. _(2.25)

From the equilibrium condition one obtains the width of the deflected rim d ∼ R21 κb

κe

1₄

. (2.26)

The total deformation energy is then

E ∼ ∆32κ

1/4 e κ3/4_b

(22)

Thus, the force-distance-law is no longer that of an ideal spring but rather has square root dependence on ∆, ˜ F ∼ γ1/4 ∆_R 1/2 . (2.28)

Obviously, this approximation is only valid if the radius of the highly bent rim is larger than its width, r > d. Because of Eqs. (2.22) and (2.26) this means that the inversion threshold is given by

∆inv

R ∼ γ

−1/2 _. _(2.29)

Together with the linear force distance law Eq. (2.21) the above inequality states, that the shell is inverted at a critical force

˜

Finv ∼ 1 . (2.30)

From the thin shell relations Eqs. (B.14) and (B.17) it follows that (κb/κe)1/2 has the

meaning of the thickness h of the sheet. For this reason Eq. (2.29) states, that the transition to the nonlinear behavior sets in when the displacement ∆ becomes comparable to the shell thickness.

Fig. 2.3 summarizes the results of this section. The plot shows qualitatively the force distance behavior expected for bacteriophage φ29. Radius and thickness of φ29 are taken to be R=35nm and h=3.5nm, in agreement with experimental values [22]. The experimental spring constant kl ∼ 0.3N/m was used to estimate the bending stiffness via Eq. (2.20).

The estimated crossover from a linear to a non-linear elastic response at ∆ ≈ 10nm agrees quite well with the experimentally measured [48].

0 0.5 1 1.5 2 2.5 3 3.5 0 2 4 6 8 10 12 14 16

Force ( nN )

Distance ( nm )

Figure 2.3: _{At a displacement ∆ ∼}pκb/κe the linear force-distance-law turns over to a square root shaped

curve. The numbers used to generate this plot are the experimentally determined values for φ29 (linear spring constant kl=0.3N/m, R=35nm, h=3.5nm). The onset of nonlinearity at an indentation of 10nm which compares

(23)

2.2.2 Rupturing

Using the above scaling arguments we can make qualitative estimates about the rupturing of the shell due to stretching or compression of the material. We assume that the critical strain needed to rupture the material is some fixed ur. Using Eq. (2.16) to estimate the

strain in the linear regime we find that the shell will eventually break at an indentation depth given by

∆r

R ∼ 1 . (2.31)

Note, that the dimensionless rupture indentation is independent of γ. From the scaling relation Eq. (2.21) the rupture force is found to be

˜

Fr∼ γ1/2 . (2.32)

After shape inversion the main deformation is the compression of the circles of latitude. Here, the strain is given by Eq. (2.24). The critical indentation is thus given by

∆r

R ∼ γ

1/2 _. _(2.33)

From the nonlinear force distance law Eq. (2.28) one finds the same scaling of the rupture force as in the linear regime, Eq. (2.32).

2.2.3 Relation to Experimental Results

In the following section, we will test whether the above scaling analysis is sufficient to capture the physics of capsid deformation. Ivanovska et al. [48] probed the elasticity of viral capsids by SFM. The experimental setup is schematically depicted in Fig. 2.4. When comparing our results to the experiments we have to take into account that the indentation of the shell is not directly measured. Rather, the sample holder is elevated by a distance dv until a given set force is reached. The force is determined by measuring the deflection

d

_g

r−

∆

d

_v

r

0000000 0000000 1111111 1111111 0000000 0000000 1111111 1111111

Figure 2.4: The indentation ∆ does not directly correspond to the elevation of the sample holder because the

(24)

of the cantilever, which has a spring constant kc. Thus, the experimental data is rather

a deflection-elevation curve, which we will denote with F′(dv). The elevation dv of the

sample holder is used to indent the shell by ∆ and to inflect the cantilever by a distance dg. Knowing kc, the inflection and hence ∆ can be calculated via

∆ = dv− dg = dv−

F′(dv)

kc

, (2.34)

Then, the actual force-indentation relation F (∆) can be calculated from the experimentally measured data F′_(d v) via F (∆) = F′ ∆ + F ′_(d v) kc . (2.35)

Together with the slope k′_v of the experimental force-elevation curve F′(dv) the spring

constant kv of the viral capsid is determined by

kv = dF (∆) d∆ = k ′ vkc 1 kc− k′v . (2.36)

Reversely, taking into account the spring constant kc of the cantilever, the linear

force-indentation relation F (∆) = kv∆ can be transformed to a force-elevation relation via

F′(dv) = F (dv− dg) = kv dv− F′(dv) kc . (2.37) Hence, F′(dv) = kckv kc+ kv dv . (2.38)

Similarly, for the non-linear law F (∆) = Kv√∆ we get

F′(dv) = K_v2 2kc s 4k2 c K2 v dv+ 1 − 1 ! . (2.39)

The curves Eqs. (2.38) and (2.39) are shown schematically in Fig. 2.5 and are compared with the experimental results of Ref. [48].

2.3 Exact Solution for a Spherical Shell

As discussed in Section 2.1 exact results for small deformation can be obtained from a variational analysis of the deformation energy. This variational approach requires to expand the energy of a shell up to second order in the deformation fields. The general expansion, valid for arbitrary geometries, is given in Appendix B.2. As will be shown in this section, the energy expansion takes a relatively simple form for spherical shells. In this particular case the exact shape equations can be solved analytically.

(25)

2.3 Exact Solution for a Spherical Shell 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 20 40 60 80 100 120 Force ( nN ) Distance ( nm )

Figure 2.5: Qualitative comparison of the experimentally observed force-distance relation (left) and the results

from scaling results (right) when the finite stiffness of the SFM cantilever (dotted line) is taken into account. The values for R, h, and klused to calculate the right plot are those of φ29, given in the caption of Fig. 2.3.

2.3.1 Analytical Solution of the Linear Regime

Metric and curvature tensor are given by Eqs. (A.4) and (A.17), respectively. For a sphere, parameterized in spherical coordinates, they take the form

gij = R2 1 0 0 sin2θ ! and bij = − 1 Rgij . (2.40)

Mean curvature H and Gaussian curvature K are defined in Eqs. (A.19) and (A.20), respectively. For a sphere they are given by

H = 1

R and K =

1

R2 . (2.41)

The general expression for the deformation given by Eqs. (2.6) and (2.7) can be simpli-fied considerably. Since the problem of a radial point force is axially symmetric, any ϕ dependence of the deformation Eq. (2.6) vanishes. Hence, we can set

ηϕ _{= −∂}θξ = 0 . (2.42)

Furthermore, from Eq. (2.7) and the diagonal structure of gij we conclude that

ηθ = gθθ∂_θχ = ∂_θχ . (2.43)

Hence we can ignore all contributions of ξ to the total energy. The general expansion of the bending energy Eq. (B.30) thus becomes

Eb = 1 2 κb R2 Z dS 4 + 2ψψ_ii+ R2ψψij_ij . (2.44)

Here, dS denotes the area element of the undeformed sphere. Indices of the scalar quantities χ and ψ denote covariant derivatives, i.e. ψij ≡ Djψi (see also Appendix A). Similarly, the

(26)

general expression of in-plane stretching energy given by Eqs. (B.39) and (B.40) reduces to Ee= 1 R2 Z dSµ(χχi_i+ 2Rχψ_ii+ R2χχij_ij + 2ψ2) + λ 2(4Rχψ i i+ R2χχijij+ 4ψ2) . (2.45)

In the following we will set λ = µ, corresponding to a Poisson ratio σ = 1/3. This value is a reasonable approximation for most materials [69, 70]. Then, the above equation simplifies further to Ee= 3 16 κe R2 Z dS χχi_i+3 2R 2_χχij ij+ 4Rχψii+ 4ψ2 , (2.46)

where we have used the 2D Young’s modulus given by Eq. (B.11).

In order to meet the constraints Eq. (2.8) on ψ, it is useful to expand the deformation field ψ and χ into spherical harmonics Ylm [71]. As long as the shell has not yet undergone

the shape inversion transition, the problem is reflection symmetric with respect to the x-y-plane. Hence, we only need to take into account Ylm with even l. Noting also that any

ϕ-dependence vanishes we can restrict the expansion to Ylm with m = 0. Thus, for ψ and

χ we can write ψ = R ∞ X l=0 AlY2l,0 and χ = R2 ∞ X l=0 BlY2l,0 . (2.47)

Then, the indentation at the poles Eq. (2.8) fixes A0 as the sum over the remaining Al.

Since Y2l,0(0, 0) = Y2l,0(π, 0) =p(4l + 1)/4π we have A0 = −√π ∆ R − ∞ X l=1 √ 4l + 1Al . (2.48)

Now, any variation of the coefficients Al will obey the constraints on ψ, since A0 is adjusted

appropriately via Eq. (2.48). The total energy reads

E = κb ∞ X l=0 2l2(2l + 1)2_{− 2l(2l + 1) +} 3 2γ A2_l +3 4γ3l 2_{(2l + 1)}2_{− l(2l + 1)}_B2 l − 3γl(2l + 1)AlBl (2.49)

Introducing appropriate abbreviations we write

E ≡ κb

∞

X

l=0

αlA2l + βlBl2+ δlAlBl . (2.50)

The coefficient B0 does not contribute because β0 = γ0 = 0. This is clear from Eq. (2.43),

since the energy depends only in the derivative of χ. Taking into account relation Eq. (2.48) the total energy now reads

E = κbα0π ∆ R 2 + κbα0 ∞ X l=0 Al √ 4l + 1 !2 +κb ∞ X l=1 2∆ R √ π√4l + 1αlAl+ αlA2l + βlB2l + δlAlBl , (2.51)

(27)

2.3 Exact Solution for a Spherical Shell

The first variation is found by varying the expansion coefficients as Al → Al+ al, Bl →

Bl+ bl, δE = κb ∞ X l=1 al 2α0 √ 4l + 1 ∞ X m=1 Am√4m + 1 ! +κb ∞ X l=1 al 2Alαl+ 2α0 √ 4l + 1√π∆ R + δlBl + bl(δlAl+ 2βlBl) = κb ∞ X l=1 al 2Alαl− 2α0 √ 4l + 1A0+ δlBl + bl(δlAl+ 2βlBl) . (2.52)

The system of linear equations which follows from δE _{= 0, ∀a}! l, bl yields

Al= K0 Kl √ 4l + 1A0 and Bl= − δl 2βl Al , (2.53) where Kl≡ 2αl− δ_l2 2βl and K0 = 2α0. (2.54)

From Eq. (2.48) one has

A0= − √_π K0 ∆ R ∞ X l=0 4l + 1 Kl !−1 . (2.55)

Together with Eq. (2.53) the total energy Eq. (2.51) simplifies to

E = κb π 2 ∆ R 2 ∞ X l=0 4l + 1 Kl !−1 . (2.56)

The linear spring constant for an radial point force is found from

k = −_∆1 ∂E_∂∆ = −_Rκb₂π ∞ X l=0 4l + 1 Kl !−1 . (2.57)

The infinite sum appearing in Eq. (2.57) can be expressed with the help of Mathematica as ∞ X l=0 4l + 1 Kl = ln 16 − 4 + 2Γ 8 + 3γ + 1 2 X i −Ψ(−xi) + 3Ψ(−xi)xi+ 6Ψ(−xi)x2i −2 + 3γ + 10xi+ 20x2_i , (2.58)

where Γ ≃ 0.577216 is Euler’s constant and Ψ(x) ≡ d log Γ(x)/dx denotes the logarithmic derivative of the gamma function [71]. The xi are the zeros of the fourth order polynomial

P (x) = −3γ − 4x + 4x2+ 48x3+ 48x4 . (2.59)

With the explicit representation Eq. (2.58) the sum in Eq. (2.57) can be numerically evaluated without having to truncate it at some large (but finite) l. The only limiting

(28)

1e0 1e1 1e2 1e3 1e4 1e5

1e0 1e1 1e2 1e3 1e4 1e5 1e6 1e7 1e8

kl [ κb /R 2 ] γ

3.99γ

1/2

Figure 2.6: Numerical evaluation of the spring constant as function of γ for an axially indented spherical

shell. For Föppl-von-Kármán numbers below 10 the spring constant deviates noticeably from the γ1_/2

behavior. Asymptotically the curve fits kl=3.99√γ (fit was made for γ = 10

3

...108

).

factor is thus the numerical precision with which the zeros of the polynomial Eq. (2.59) can be determined numerically.

The spring constant, determined in this way, is plotted as function of the Föppl-von-Kármán number in Fig. 2.6. Asymptotically the curve agrees well with the square root behavior expected from the scaling analysis given above,

k = 3.99κb R2

√_{γ .} _(2.60)

This relation has been found by Reissner [72] in the thin shell limit of a 3D elastic theory. However, Eq. (B.19) implies that the thin shell approximation is equivalent to γ → ∞. The considerable deviations from the square-root law occurring for smaller γ are only captured by our more general expression Eq. (2.57).

2.3.2 Critical Review of the Scaling Arguments

The scaling arguments in Section 2.2.1 rely on the assumption that the deformation takes place in a concentrated region around the poles θ = 0, π. With the aid of our analytical results we can review the validity of this assumption. For this purpose we plot the density of deformation energy ee+ eb as function of θ and γ (see Fig. 2.7). This density is given

by the integrand of the total energy defined in Eqs. (2.1)-(2.4), i.e. E =

Z π

0

dθ (eb(θ) + ee(θ)) . (2.61)

The flattening of the polar region leads to a local decrease of bending energy. In case of dominating bending energy, i.e. γ < 1, the deformation energy concentrates around the equator, contradicting our assumptions. For higher γ the flattening leads to an increase in the density of the total deformation energy at the poles.

(29)

2.3 Exact Solution for a Spherical Shell

Π

4 Π

2 3 Π

4 - 2

0

2

4 - 0.2

0

0.2

0.4 Π

4 Π

2 3 Π

4

eb+ee E

θθ

log

₁₀

γ

Figure 2.7: Distribution of deformation energy along the θ coordinate as a function of the Föppl-von-Kármán

number γ. For the evaluation of the energy density ψ and η have been expanded into spherical harmonics up to Y40,0. For γ < 1 deformation energy concentrates along the equator.

2.3.3 Shape Inversion

The shell is expected to invert its curvature around the pole, when the indentation exceeds some critical ∆inv. There, the mean curvature of the pole changes sign. Therefore ∆inv is

implicitly given by the zero of the mean curvature at θ = 0,

H(θ, ∆inv(γ), γ)|θ=0= 0 . (2.62)

Making use of the analytical solutions for ψ and χ the solution ∆inv(γ) has been found by

expanding ψ and χ into spherical harmonics up to Y20000,0. Since H′(γ) = H + H1(γ)∆ +

O(∆2_{), the ∆}

inv which sets H′ to 0 is easily found. The function ∆inv(γ) is shown in Fig.

2.8.

From Eq. (2.29) one expects inversion when ∆inv/R ∼ γ−1/2. For Föppl-von-Kármán

numbers above 10 the analytical results roughly confirm the scaling arguments. For small γ we observe a considerable deviation from the γ−1/2-behavior. In this regime the equilibrium configuration is governed by the bending stiffness. Let’s consider the extreme case γ = 0. Since the bending energy is scale invariant, the shell responds to a prescribed ∆ with an appropriate shrinking of its diameter. Thus, the shell will always have a spherical shape and ∆inv is infinite. Moreover, for γ < 10 the deformation ∆inv is no longer a small

(30)

1e-2 1e-1 1e0

1e-1 1e0 1e1 1e2 1e3

∆r [R] γ meridional compression circumferential compression meridional expansion circumferential expansion

Figure 2.8: The black solid line gives the γ-dependence of the critical indentation ∆invat which the shell inverts

its shape. For γ > 10 the analytical curve roughly confirms the γ−1/2_{behavior expected from scaling arguments}

(black dashed line). The colored lines show the γ dependence of the rupture indentation ∆rat which the strain

exceeds a threshold of δi= ±4.5% (i = ϕ, θ). Different colors correspond to different rupture scenarios. Clearly,

the homogeneous shell ruptures first due to meridional compression at a constant depth. However, the analysis is only valid in the non-inverted regime, i.e. it is not valid above the black solid line. For the calculation of the inversion threshold spherical harmonics up to lmax=20000 have been taken into account. For the rupture

thresholds lmax=200.

2.3.4 Rupture

Using the analytical results for the deformation fields ψ and η, we can also determine the γ-dependence of the rupture indentation ∆r. For a spherical shell the strain tensor uij has

elements [see also Eq. (B.38)]

uθθ= ψ, (2.63)

and

uϕϕ= sin θ(ηθcos θ + ψ sin θ) . (2.64)

For axial-symmetric loading the off-diagonal terms vanish. The strain tensor is closely related to the local change of the first fundamental form [see Eq. (2.3)] and we can express the relative length change in direction of the i-th coordinate via

δi ≡√1 + 2uii− 1 , (2.65)

where i = θ, ϕ. Then, at the rupturing indentation ∆r the relative length change δi reaches

a critical strain ur

max

θ∈[0,π](±δi(θ, ∆r)) = ur . (2.66)

Here, (+) corresponds to critical expansion and (−) to critical compression. Furthermore, i = θ corresponds to meridional and i = ϕ to circumferential in-plane deformation. The four different scenarios are compared in Fig. 2.8. In accordance with the scaling analysis of Section 2.2 the shell ruptures most likely due to meridional compression at a constant critical indentation ∆r which is independent of γ. As mentioned, the analytical results are

(31)

2.3 Exact Solution for a Spherical Shell 1e0 1e1 1e2 1e3 1e4 1e5

1e0 1e1 1e2 1e3 1e4 1e5 1e6 1e7 1e8

kl [ κb /R 2 ] γ

3.99γ

1/2 1e-2 1e-1 1e0

1e-1 1e0 1e1 1e2 1e3

∆r [R] γ meridional compression circumferential compression meridional expansion circumferential expansion

Figure 2.9: (left) Spring constant as function of γ for a shell with spontaneous curvature H0= −1/R. Like the

curve for H0= 0 the spring constant deviates from the predicted γ1/2 scaling for small γ. Asymptotically both

curves are identical. (right) Different rupture scenarios and inversion threshold as function of γ for a shell with spontaneous curvature H0 = −1/R. Asymptotically, the inversion threshold follows roughly the γ−1/2 scaling

expected from the scaling analysis. For smaller γ the inversion threshold becomes constant at ∆inv ≈0.255R.

The rupture behavior shows qualitative agreement with the H0= 0 case.

we assume a critical strain of ur∼ 4.5%. This particular choice will be justified in Chapter

3, where we compare experimental data with our numerical simulations. For this choice of ur the rupture indentation ∆r exceeds the inversion threshold only above γ ≃ 20, see Fig.

2.8, where our analytical approach does not hold. 2.3.5 Sphere with Spontaneous Curvature

The general Helfrich energy Eq. (2.5) takes into account a spontaneous curvature H0. Using

the expansion method described above we find for the deformed sphere

Eb = 1 2 κb R2 Z dS 16√π(H0+ RH02)ψ + ψψi_i+ R2ψψij_ij _{− 4RH}0ψψii+ 4H02(ψ2− 1 2R 2_ψψi i) −_R4(H0+ H02)χχii+ 8(H0+ RH02)χψii . (2.67)

In general, the term linear in ψ does not vanish and the spherical surface is only stable for H0 = 0 and H0 = −1/R. In both cases the second order expansion of the curvature energy

depends on the normal displacement ψ only. For H0 = −1/R we have

Eb = 1 2 κb R2 Z dS 4 R2ψ 2_{+ R}2_ψψij ij + 4ψψii . (2.68)

The analysis of this case is completely analogous to the one given above for H0= 0, leading

only to different coefficients αl, βl and δl.

The linear spring constant as function of the Föppl-von-Kármán number is plotted in Fig. 2.9. The spring constant shows the same γ-dependence as for H0= 0 with deviations

(32)

The inversion behavior, shown in Fig. 2.9, also shows the asymptotic γ−1/2behavior but deviates for small γ from both the scaling result and the H0 = 0 case. For H0 = −1/R

and small γ (i.e. negligible stretching contribution) the inversion threshold approaches a constant ∆b ≈ 0.255R. In this regime, the shape is completely dominated by bending energy

consisting of a term linear in curvature and a scale invariant quadratic term of the bending energy. Since both contributions depend on κb alone, the equilibrium is independent of κb

and hence independent of γ.

The rupture behavior is qualitatively the same as for the H0 = 0 case. For

F¨oppl-von-K´arm´_{an numbers below γ ∼ 30 the shell breaks in the non-inverted regime. The} corresponding rupture indentation is constant at ∆r≈ 0.1R.

(33)

CHAPTER 3 Empty Capsids

T

he outstanding mechanical robustness of viral capsids and their increasing importance in nano-technology have inspired a recent series of nano-indentation experiments. From capsid breakage experiments conclusions about the strength of protein interactions can be drawn, giving these studies also biological relevance.

Up to now, the mechanical properties have been studied only for a handful of different viral capsids, including phage φ29 [48] and the plant virus cowpea chlorotic mottle virus (CCMV) [50]. Other studied viruses are minute virus of mice (MVM) [73, 52] and phage λ [74, 54]. The influence of particle maturation on capsid elasticity has been studied for murine leukemia virus (MLV) [51].

In this chapter we aim to relate the experimentally observed robustness of capsids to their specific geometry. To do so, we numerically investigate the mechanical behavior of elastic shells under external forcing. The numerical scheme is an extension of the discretization of fluid vesicles to elastic membranes introduced in Ref. [64]. The details of this discretization and its extension are discussed in Appendix C.

In this scheme, a capsid is represented by a triangulated surface with elastic (stretching) and bending energy. We employ a small mesh-size triangulation where every capsomer is represented by several vertices. Therefore, these units also have some flexibility and elas-ticity. Such a discretization is suitable to determine the shape of viral shells [33] and yields results which do not depend on the number of vertices, in contrast to coarse triangulations in which each capsomer is represented by a single vertex [57]. The triangulation represents the underlying icosadeltahedral symmetry of the capsid. In particular, the pentavalent vertices correspond to the centers of the pentamers.

In the first section of this chapter the importance of disclinations to viral shape is demon-strated. Particular emphasis is placed on the comparison of skew and non-skew capsid de-signs. In Section 3.2 we explore the reversible elastic response of capsids. In Section 3.3 we give, based on these results, estimates of the bulk elastic moduli of viral capsids, for which experimental data is available. Material failure under large forcing is studied in Section 3.4. Here, for some of the experimentally investigated viruses predictions about rupture

Mechanical Properties and DNA Organization of Viruses and Bacteria