A Continuum Theory for the Natural Vibrations of Spherical Viral Capsids

Transcription

1 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 Università di Roma Tor Vergata Prof. Chandrajit Bajaj The University of Texas at Austin Rome, July 24th, 2012

2 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

3 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). Typical examples of spherical capsids: STMV and CCMV capsids.

4 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.

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

6 Viral Capsids Functions, structure, geometry Triangulation Number T = 1 = only pentamers (STMV capsid) T > 1 = pentamers + hexamers Example: T = 3 (CCMV capsid) k (1,1) h (0,0)

7 Viral Capsids Functions, structure, geometry Triangulation Number T = 1 capsid (left) and T = 3 capsid (right) surfaces

8 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 R 1 = 55.4 Å and R 2 = 86 Å (Yang et al., 2009). The thickness of the spherical shell is then t S = 30.6 Å and its middle surface has radius ρ o = 70.7 Å.

9 Viral Capsids Density, material moduli Density, material moduli Remark Mass density of the STMV capsid: δ o = kg/m 3. From the measured value of the longitudinal sound speed c l 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, c l = E(1 ν) δ o (1 + ν)(1 2ν). 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.

10 Shell Theory Part I Linearly Elastic Spherical Shells

11 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.

12 Shell Theory Geometry Middle Surface Curvilinear coordinates z 1 = ϑ, z 2 = ψ. Let S := (0, π) (0, 2π). S c3 n x e2 e1 o ψ ϑ ρo c(ψ) c2 c1 S (ϑ, ψ) x = x(ϑ, ψ) = o + x(ϑ, ψ) S, x(ϑ, ψ) = ρ o (sin ϑ c(ψ) + cos ϑ c 3 ), c(ψ) = cos ψ c 1 + sin ψ c 2.

13 Shell Theory Geometry Local Bases Covariant basis Contravariant basis Physical basis e 1 = x ϑ = ρo(cos ϑ c sin ϑ c 3) e 2 = x = ρo sin ϑ c ψ e 3 = n = ρ 1 o x = sin ϑ c + cos ϑ c 3 e 1 = s ϑ = ρ 2 o e 1 e 2 = s ψ = (ρ o sin ϑ) 2 e 2 e 3 = n e<1> = e 1 e 1 e<2> = e 2 e 2 e<3> = n

14 Shell Theory Geometry Shell-like Region Curvilinear coordinates z 1 = ϑ, z 2 = ψ, z 3 = ζ. Let I := ( ε, +ε). n(x) S h S 2ε S h x + hn(x) x x hn(x) G(S,ε) S I (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).

15 Shell Theory Kinematics Kinematics Displacement field (0) u (x, t) = a(x, t) + w(x, t)n(x), u(x, ζ; t) = (0) u (x, t) + ζ (1) u (x, t), a(x, t) n(x) = 0, ϕ(x, t) n(x) = 0, (1) u (x, t) = ϕ(x, t) + γ(x, t)n(x), 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 )

16 Shell Theory Kinematics Kinematics hγn wn p u hϕ a a + hϕ T ph (S h) S h wn x u a T x(s) S h

17 Shell Theory Field Equations Weak Formulation Let S be the Piola stress tensor, d o the distance force per unit volume and c o the contact force per unit area. Define the internal virtual work W int (G) [δu] := S δu and the external virtual work W ext (G) [δu] := Principle of Virtual Work: G G d o δu + c o δu. G δu, W int (G) [δu] = W ext (G) [δu].

18 Shell Theory Field Equations Weak Formulation By integration over the thickness, ( ) W int (G) [δu] = s F s δ (0) u + s M s δ (1) u + f (3) δ (1) u, S where ( s F := I ) αsg β dζ e β, ( s M := f (3) := αsn dζ. I I ) αζsg β dζ e β,

19 Shell Theory Field Equations Weak Formulation By integration over the thickness, W ext (G) [δu] = S ( ) q o δ (0) u + r o δ (1) u, where q o := α d o dζ + α + c + o + α c o, I I r o := αζd o dζ + ε ( α + c + o α c ) o.

20 Shell Theory Field Equations Balance Equations The Principle of Virtual Work reads ( ) s F s δ (0) u + s M s δ (1) u + f (3) δ (1) u = S By localization, S ( ) q o δ (0) u + r o δ (1) u. s Div s F + q o = 0, s Div s 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.

21 Natural Vibrations Conclusions and Directions for Future Research Part II Analysis of Natural Vibrations

22 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 r o (d o d in o = δ o ü). No contact forces per unit area: c o 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 of ψ. Notation: ( ) = ( ) ϑ

23 Natural Vibrations Conclusions and Directions for Future Research Radial Vibrations without Thickness Changes Radial Vibrations without Thickness Changes Governing equation G(w + cot ϑ w ) u = wn w = 0 = ẅ + ω 2 0 w = 0, ω2 0 = ) 2E (1 + ν)(1 2ν) w = ρ2 oδ o (1 + ε2 3ρ 2 ẅ o ρ 2 oδ o ( 1 + ε2 3ρ 2 o 2E ) (1 + ν)(1 2ν) w(ϑ, t) = c cos ϑ cos(ω 1 t) ω1 2 = E(3 2ν) ( ) ρ 2 oδ o 1 + ε2 (1 + ν)(1 2ν) ω 2 1 > ω 2 0 3ρ 2 o

24 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(ω 1 t) S ϑ

25 Natural Vibrations Conclusions and Directions for Future Research Uniform Radial Vibrations with Thickness Changes Uniform Radial Vibrations with Thickness Changes Governing equations w = ŵ cos(ωt), u = (w + ζγ)n γ = γ cos(ωt) (( ) ) ω 2 ρ 2 o δo 1 + ε2 3ρ 2 ŵ + 2ε2 2E ( γ + (1 + ν)ŵ + (1 + ν 2 o 3ρ o (1 + ν) 2 )ρ ) o γ = 0, (1 2ν) (2) ( ( 2 1 ω 2 ε 2 δ o 3 ŵ + ρo 3 + ε2 5ρ 2 o + E 1 2ν ) ) γ + { 2ε 2 3ρ o γ ν [ ) ]} 2νŵ + ((1 ν)ρ o + (1 + ν) ε2 γ = 0 3ρ o (3)

26 Natural Vibrations Conclusions and Directions for Future Research Uniform Radial Vibrations with Thickness Changes Uniform Radial Vibrations with Thickness Changes (2) : ω 2 = aŵ + b γ cŵ + d γ = γ = K± ŵ, K ± = K ± (E, ν, δ o, ρ o, ε) (3) : ω 2 = eŵ + g γ hŵ + k γ ω± 2 2E ( ) (1 + ν) + (1 + ν 2 )ρ o K ± = (( ) ) ρ 2 oδ o (1 + ν) 2 (1 2ν) 1 + ε2 3ρ + 2ε2 2 o 3ρ o K ±

27 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> + cot ϑ a<2> cot 2 ϑ a<2> = ρ2 oδ o G a<2>(ϑ, t) = c sin ϑ cos ϑ cos(ωt) = ω 2 = (1 + ε2 3ρ 2 o ) ä<2> 5G ( ) ρ 2 oδ o 1 + ε2 3ρ 2 o

28 Natural Vibrations Conclusions and Directions for Future Research Parallel-Wise Twist Vibrations Parallel-Wise Twist Vibrations a<2>(0, t) = 0 S ϑ a<2> ( π 2, t) = 0 a<2>(π, t) = 0

29 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 ε 2 ρ 2 o ( ϕ<2> + cot ϑ ϕ<2> + (1 cot 2 ϑ) ϕ<2>) 3 ϕ<2> = ε 2 δ o G ( ε 2 ) ρ 2 ϕ<2> o

30 Natural Vibrations Conclusions and Directions for Future Research Parallel-Wise Shear Vibrations Parallel-Wise Shear Vibrations ε = 15.3 Å, ρ o = 70.7 Å (Yang et al., 2009) 3ε2 5ρ 2 o approximate frequency ω 2 = 3G ε 2 δ o Without approximation, ϕ<2>(ϑ, t) = c sin ϑ cos ωt = ω 2 G = 3ε 2 δ o ( ) ε 2 ρ 2 o Remark Both ω 2 and ω 2 diverge as ε 0.

31 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