A Continuum Theory for the Natural Vibrations of Spherical Viral Capsids - Matematicamente

Viral Capsids

A Continuum Theory for the Natural Vibrations of

Spherical Viral Capsids

Francesco Bonaldi

Master’s Thesis in Mathematical Engineering

Advisors

Prof. Paolo Podio-Guidugli Prof. Chandrajit Bajaj Universit`a di Roma Tor Vergata The University of Texas at Austin

Viral Capsids

Outline

1 Viral Capsids

Functions, structure, geometry Density, material moduli

2 Shell Theory

Geometry Kinematics Field Equations

3 Natural Vibrations

Radial Vibrations without Thickness Changes Uniform Radial Vibrations with Thickness Changes Parallel-Wise Twist Vibrations

Parallel-Wise Shear Vibrations

F. Bonaldi

Viral Capsids

Functions, structure, geometry

Viral Capsids

Viral capsids: nanometre-sized protein shells that enclose and protect the genetic materials (RNA or DNA) of viruses in a host cell, transport and release those materials inside another host cell.

In most cases, their shape is either helical (nearly cylindrical) or icosahedral (nearly spherical).

Viral Capsids

Functions, structure, geometry

Viral Capsids

They consists of several structural subunits, the capsomers, made up by one or more individual proteins. In spherical capsids, the capsomers are classified as pentamers and hexamers.

STMV capsid: 60 copies of a single protein, clustered into 12 pentamers.

CCMV capsid: 180 copies of a single protein, clustered into 12 pentamers and 20 hexamers.

F. Bonaldi

Viral Capsids

Functions, structure, geometry

Triangulation Number

There are 12 pentamers in any spherical capsid. The number of hexamers depends on the T-number of the capsid (Caspar and Klug, 1962).

L (0,0) (1,0) (2,0) (3,0) (4,0) (5,0) h k (0,1) (0,2) (0,3) (1,1) (1,2) (1,3) (2,1) (3,1) T = √ 3 4 L 2 √ 3 4 = L2= h2+ hk + k2

Viral Capsids

Functions, structure, geometry

Triangulation Number

T = 1 =_{⇒ only pentamers (STMV capsid)} T > 1 =_{⇒ pentamers + hexamers} Example: T = 3 (CCMV capsid) h k (1,1) (0,0) F. Bonaldi

Viral Capsids

Functions, structure, geometry

Triangulation Number

Viral Capsids

Functions, structure, geometry

Geometry

The thickness of a spherical capsid is actually non-uniform. Ideal values of the inner and outer radii of the STMV capsid are R1= 55.4 ˚A and

R2= 86 ˚A (Yang et al., 2009). The thickness of the spherical shell is

then t_S = 30.6 ˚A and its middle surface has radiusρo = 70.7 ˚A.

F. Bonaldi

Viral Capsids

Density, material moduli

Density, material moduli

Mass density of the STMV capsid: δo = 823.82 kg/m3.

From the measured value of the longitudinal sound speed cl in

STMV crystals, Yang et al. determined the value of the Young’s modulus of the STMV capsid. The Poisson ratio is thought to be close to that of soft condensed matter, i.e.,ν = 0.3. In a generic three-dimensional isotropic elastic continuous body,

cl= s E (1_{− ν)} δo(1 +ν)(1− 2ν) . Remark

This formula does not account for the shell-like geometry of the STMV capsid: it involves only its density, but none of its geometrical features, such as the thickness and the radius of the middle surface.

Shell Theory

Part I

Linearly Elastic Spherical Shells

F. Bonaldi

Shell Theory

Features

The shell is capable of both transverse shear deformation and thickness distension.

The shell is transversely isotropic with respect to the radial direction, with fiber-wise constant elastic moduli, in order to account for the rotational symmetries of the capsomers. The thickness may vary over the middle surface.

Shell Theory Geometry

Middle Surface

Curvilinear coordinates z1_{=ϑ, z}2_=ψ. Let S := (0, π)_{× (0, 2π).} c1 c2 c3 o S x ψ ϑ e1 e2 n c(ψ) ρo 1 S_{3 (ϑ, ψ) ↔ x = x(ϑ, ψ) = o + x(ϑ, ψ) ∈ S,} x(ϑ, ψ) = ρo(sinϑ c(ψ) + cos ϑ c3), c(ψ) = cos ψ c1+ sinψ c2. F. Bonaldi

Shell Theory Geometry

Local Bases

Covariant basis e1= ∂x ∂ϑ= ρo(cos ϑ c − sin ϑ c3) e2= ∂x ∂ψ= ρosin ϑ c 0 e3= n = ρ−1o x = sin ϑ c + cos ϑ c3 Contravariant basis e1=s∇ϑ = ρ−2 o e1 e2=s∇ψ = (ρosin ϑ)−2e2 e3_{= n} Physical basis e<1>= e1 |e1| e<2>= e2 |e2| e<3>= n

Shell Theory Geometry

Shell-like Region

Curvilinear coordinates z1_{=ϑ, z}2_{=ψ, z}3_=ζ. Let I := (_{−ε, +ε).} G(S, ε) S 2ε n(x) x x + hn(x) x− hn(x) Sh S−h 1 S_{× I 3 (ϑ, ψ, ζ) ↔ p = p(ϑ, ψ, ζ) = o + p(ϑ, ψ, ζ) ∈ G(S, ε),} p(ϑ, ψ, ζ) = x(ϑ, ψ) + ζn(ϑ, ψ).

We can define analogous local bases for any p_{∈ G(S, ε).}

F. Bonaldi

Shell Theory Kinematics

Kinematics

Displacement field u(x, ζ; t) =(0)u (x, t) + ζ(1)u (x, t), (0) u (x, t) = a(x, t) + w (x, t)n(x), (1)u (x, t) = ϕ(x, t) + γ(x, t)n(x), a(x, t)· n(x) = 0, ϕ(x, t) · n(x) = 0, ∀x ∈ S, ∀t ∈ (0, +∞) a = a<1>e<1>+ a<2>e<2> ϕ = ϕ<1>e<1>+ϕ<2>e<2>

Six scalar parameters: a<1>, a<2>, ϕ<1>, ϕ<2>, w , γ

Strain tensor

E =sym∇u = 1

2 ∇u + ∇u T

Shell Theory Kinematics

Kinematics

x Tx(S) u a wn p a hϕ wn hγn _u Tph(Sh) a + hϕ S Sh h 1 F. Bonaldi

Shell Theory

Field Equations

Weak Formulation

Let S be the Piola stress tensor, do the distance force per unit volume

and co the contact force per unit area. Define the internal virtual work

Wint₍

G) [δu] := Z

G

S_{· ∇δu} and the external virtual work

Wext₍ G) [δu] := Z G do· δu + Z ∂G co· δu.

Principle of Virtual Work: ∀δu, Wint₍

G) [δu] = Wext₍

Shell Theory

Field Equations

Weak Formulation

By integration over the thickness, Wint₍ G) [δu] = Z S  s_F ·s∇δ(0)u +sM_·s_∇δ(1)u + f(3)_{· δ}(1)u  , where s_{F :=}Z I αSgβdζ  ⊗ eβ, sM := Z I αζSgβdζ  ⊗ eβ, f(3):= Z I αSn dζ . F. Bonaldi

Shell Theory

Field Equations

Weak Formulation

By integration over the thickness, Wext₍ G) [δu] = Z S  q_{o· δ}(0)u + ro· δ (1) u  , where qo:= Z I α dodζ + α+c+o +α−c−o, ro:= Z I αζdodζ + ε α+c+o − α−c−o
.

Shell Theory

Field Equations

Balance Equations

The Principle of Virtual Work reads Z S  s_F ·s ∇δ(0)u +sM_·s_∇δ(1)u + f(3)_{· δ}(1)u  = Z S  qo· δ (0) u + ro· δ (1) u  . By localization, s_Divs_{F + q} o= 0, s_Divs_M − f(3)_{+ r} o= 0. (1) ∂_{S = ∅ =⇒ no boundary conditions}

On inserting S = C[E], with E = sym∇u, into the previous definitions, (1) yields a system of six scalar equations in terms of the six kinematical parameters.

F. Bonaldi

Natural Vibrations Conclusions and Directions for Future Research

Part II

Natural Vibrations Conclusions and Directions for Future Research

Assumptions

1 The only distance actions per unit area involved are the inertial parts of q_o and ro (do≡ dino =−δo¨u). No contact forces per unit

area: co≡ 0.

2 As in the majority of the literature about capsids

We consider the subcase of homogeneous and isotropic response

We assume the thickness uniform over the middle surface

3 We restrict attention on axisymmetric vibrations: kinematical parameters independent ofψ. Notation: ∂

∂ϑ(·) = (·)0

F. Bonaldi

Natural Vibrations Conclusions and Directions for Future Research

Radial Vibrations without Thickness Changes

Radial Vibrations without Thickness Changes

u = w n Governing equation G (w00+ cotϑ w0)₋ 2E (1 +ν)(1− 2ν)w =ρ 2 oδo  1 + ε 2 3ρ2 o  ¨ w w0= 0 =_{⇒ ¨}w +ω2 0w = 0, ω20= 2E ρ2 oδo  1 +_3ρε22 o  (1 +ν)(1_{− 2ν)} w (ϑ, t) = c cos ϑ cos(ω1t) ⇒ ω21= E (3_{− 2ν)} ρ2 oδo  1 + _3ρε22 o  (1 +ν)(1− 2ν) ω2 1> ω 2 0

Natural Vibrations Conclusions and Directions for Future Research

Radial Vibrations without Thickness Changes

Radial Vibrations without Thickness Changes

w (ϑ, t) = c cos ϑ cos(ω1t)

S

ϑ

1

F. Bonaldi

Natural Vibrations Conclusions and Directions for Future Research

Uniform Radial Vibrations with Thickness Changes

Uniform Radial Vibrations with Thickness Changes

u = (w +ζγ)n w =w cos(ωt),_b γ =_{bγ cos(ωt)} Governing equations − ω2_ρ2 oδo  1 + ε 2 3ρ2 o  b w +2ε 2 3ρob γ  + 2E (1 + ν)2_{(1 − 2ν)} (1 + ν)bw + (1 + ν 2_)ρ obγ
= 0, (2) −ω2_ε2_δ o  2 3w + ρb o  1 3+ ε2 5ρ2 o  b γ  + + E 1 − 2ν  2ε2 3ρob γ + 1 1 + ν  2νw +b  (1 − ν)ρo+ (1 + ν) ε2 3ρo  b γ  = 0 (3)

Natural Vibrations Conclusions and Directions for Future Research

Uniform Radial Vibrations with Thickness Changes

Uniform Radial Vibrations with Thickness Changes

(2) :ω2₌ aw + bb bγ cw + d_{b bγ}

=

_⇒

_{bγ = K}±wb, K± = K±(E, ν, δo, ρo, ε) (3) :ω2₌ ew + gb bγ hw + k_{b bγ} ω_±2 = 2E (1 +ν) + (1 + ν 2_)ρ oK±
ρ2 oδo(1 +ν)2(1− 2ν)  1 + _3ρε22 o  +_3ρ2ε2 oK±  F. Bonaldi

Natural Vibrations Conclusions and Directions for Future Research

Parallel-Wise Twist Vibrations

Parallel-Wise Twist Vibrations

u = a<2>e<2> Governing equation a<2>00+ cotϑ a<2>0− cot2ϑ a<2>=ρ 2 oδo G  1 + ε 2 3ρ2 o  ¨ a<2>

a<2>(ϑ, t) = c sin ϑ cos ϑ cos(ωt) =⇒ ω2= 5G ρ2 oδo  1 +_3ρε22 o 

Natural Vibrations Conclusions and Directions for Future Research

Parallel-Wise Twist Vibrations

Parallel-Wise Twist Vibrations

S ϑ a<2> π₂, t
= 0 a<2>(0, t) = 0 a<2>(π, t) = 0 1 F. Bonaldi

Natural Vibrations Conclusions and Directions for Future Research

Parallel-Wise Shear Vibrations

Parallel-Wise Shear Vibrations

u =ζϕ<2>e<2> e<2> S Governing equation ε2 ρ2 o ϕ<2>00+ cot ϑ ϕ<2>0+ (1 − cot2ϑ) ϕ<2>
− 3 ϕ<2>=ε 2_δ o G  1 +3 5 ε2 ρ2 o  ¨ ϕ<2> F. Bonaldi

Natural Vibrations Conclusions and Directions for Future Research

Parallel-Wise Shear Vibrations

Parallel-Wise Shear Vibrations

ε = 15.3 ˚A,ρo= 70.7 ˚A (Yang et al., 2009)⇒ 3ε2 5ρ2 o ≈ 0.024 ⇒ approximate frequency eω2₌ 3G ε2_δ o Without approximation, ϕ<2>(ϑ, t) = c sin ϑ cos ωt =⇒ ω2= G 3ε2_δ o  1 + 3 5 ε2 ρ2 o  Remark Bothω2_and e ω2_{diverge asε} → 0. F. Bonaldi

Natural Vibrations Conclusions and Directions for Future Research

Conclusions and Directions for Future Research

Conclusions

We have set forth some simple cases of natural vibrations that might be considered as a reference to infer a correct evaluation of Young’s

modulus and Poisson’s ratio for a spherical capsid, when thought of as an isotropic body, by carrying out experiments that induce the relative vibrational modes.

Directions for Future Research

Multiscale modeling of spherical capsids Full capsids in a hydrostatic environment