Orienta(onal Phase Transi(ons and the Assembly of Viral Capsids i) Lorman-Rochal theory and the Landau theory of orienta5onal ordering in liquid crystals and glasses. ii) Extensions of Lorman-Rochal theory.
Orienta5onal ordering in liquid crystals and glasses • Nema5c liquid crystals. Order Parameter Nema5c Isotropic fluid 0 T c Temperature First-order phase transi5on Order parameter ( ) 0 cos θ P 2 Second Legendre polynomial P 2 0 (x) = (3x 2 -1)/2 ?
• Biaxial Nema5cs (Freiser, 1970) Two order parameters: S and T S = T = 0 isotropic ê S ≠ 0, T = 0 uniaxial nema5c ê S ≠ 0, T ≠ 0 biaxial nema5c Order parameters S = P 0 (cos θ ) 2 ( ) = P ( ) T = P 2 (cos θ )exp i ϕ − 2 (cos θ )exp − i ϕ 2 2 P 2 2 (x) = 3 (1-x 2 ) Associated Legendre polynomials ??
• General case: measure the density ρ(Ω) of molecules with orienta5on Ω. • ``Spherical Fourier decomposi5on”. ∞ + L ∑ ∑ ( ) = ( ) ρ Ω Q L , M Y L , M Ω L = 0 M = − L Spherical Harmonics • ( ) ∝ P ( ) exp( iM ϕ ) M cos θ Y L , M θ , ϕ L Expansion coefficients Q L, M Orienta:onal order parameter set • Infinite hierarchy of Orienta5onal Transi5ons • Biaxial nema5cs: L = 2, M = 0, ±2 • Cholesterics, blue phases: L = 2, M = 0, ±1, ±2 Hornreich & Shtrikman (1980, 1981) . • “Cuba5c” liquid crystal: L = 4 Nelson & Toner (1981)
Icosahedral Glasses/Quasicrystals: L=6 ( Steinhardt, Nelson, Ronchef 1983) • • ρ(Ω): surface density modula5on of cluster of atoms in liquids near mel5ng . (no posi5onal order) ( ) ρ Ω HCP cluster FCC cluster Icosahedral cluster • What is the best choice for the symmetry of the clusters? Lev Landau • Rules of Landau Theory
Rule # 1: Free energy = Sum of the scalar invariants of the order parameters Q L,m (under group SO(3) of rota5ons). Rule # 2: Use one ``irreducible representa5on”. Pick one single L. (but which L?) Rule # 3: Minimize free energy with respect to Q L,M . ``Wigner 3-j symbol” ⎛ ⎞ + L L L L ∑ ∑ 2 F = r + w + u { Q 4 } + .. ⎜ ⎟ Q L M 1 Q L M 2 Q L M 3 Q L M { } M 1 M 2 M 3 ⎝ ⎠ − L ≤ M 1,2,3 ≤ L M = − L M 1 + M 2 + M 3 = 0 quadra5c invariant cubic invariant quar5c invariant • r, w, u : Phenomenogical parameters. • Simpler: expand free energy in powers of orienta5onal density ρ (Ω) { } 2 + w ρ ! 3 + u ρ ! 4 + ... ( ) = ρ ! dS r ρ ! { } ( ) ( ) ( ) ( ) ∫ F r r r r Surface + L ∑ ( ) = ( ) ρ Ω Q L , M Y L , M Ω Insert M = − L
• Free energy minimum for L=6 has icosahedral symmetry ⎛ ⎞ 7 7 ( ) = Q 6 Y 0,0 + ρ Ω 11 Y 6, 5 − 11 Y 6, − 5 ⎜ ⎟ ⎝ ⎠ 2 + wQ 6 3 + uQ 6 6 = rQ 6 4 • L = 6 ``icosahedral spherical harmonic” F • Q 6 = Order parameter ``amplitude” Icosahedral Isotropic Q 6 0 Q 6 r 0 First-order transi5on (cubic term non-zero) • Icosahedral state is thermodynamically stable against fluctua5ons.
+ 15 Capsid Canine Parvovirus ∑ ( ) ∝ ( ) ρ 15 Ω Q 15, M Y 15, M Ω M = − 15 L=15 Icosahedral spherical harmonic
For certain L there is a unique linear combina5on Y h (L) of the Y L,M that transforms as a scalar or as a pseudo-scalar under the icosahedral symmetry group. Even L L = 6, 10, 12, 16, .... • L=10 L=6 Y h (L) scalar density even under inversion r è - r Odd L L = 15, 21, 25, 27, ...... • pseudo-scalar density Y h (15) odd under inversion • Chiral pairs (isomers)
Rochal & Lorman: capsid density cannot be presented by the even L scalars amino-acids Capsid densi5es are not even under inversion: only odd L spherical harmonics . • Smallest viral capsids should correspond to Y h (L=15): T=1 viruses. • ( ) = Q 15 Y h 15 ( ) ρ Ω Parvovirus (T=1) :
2 + uQ 15 15 = r Landau free energy (L=15): 4 F 15 Q 15 F Cubic term zero for any odd L ! 15 r > 0 r Q 15 15 “Spontaneous chiral symmetry breaking transi5on”. r r < 0 r 15 0 15 − r Q 15 Q 0 = 15 − Q 0 0 Q 0 2 u
Ques5ons 1) Second-order transi5on? Collec:ve assembly pathway Cowpea Chloro5c Movle Virus (CCMV) R. Cadena-Nava, M. Comas-Garcia, R. Garmann, A. Rao C. M. Knobler, and W. M. Gelbart J. Virol. 86, 3318 (2012) . R. Garmann, M. Comas-Garcia, A. Gopal, C. M. Knobler, and W. M. Gelbart J. Mol. Biol. 426, 1050 (2013) • Fluorescence thermal shiw assay CCMV: first-order transi5on Guillaume Tresset, Jingzhi Chen, Maelenn Chevreuil, Naïma Nhiri, Eric Jacquet, and Yves Lansac Phys. Rev. Applied 7, 014005, 2017
2) Spontaneous chiral symmetry breaking transi5on at r 15 = 0. Uniform state is already chiral . 3) Thermodynamic stability against fluctua5ons: expand free energy around Y h (15) solu5on + 15 ∑ ( ) = const . + ρ 15 Y h (15) + ( ) ρ Ω δ Q 15, M Y 15, M Ω M = − 15 fluctua5on 15 15 15 = 1 ∑ ∑ δ F C M , M ' δ Q 15, M δ Q 15, M ' Fluctua5on free energy: 2 M = − 15 M ' = − 15 C M,M’ : 31 x 31 stability matrix should have posi5ve eigenvalues. • :
21 posi5ve eigenvalues 3 zero eigenvalues (rota5ons) 7 nega5ve eigenvalues ! saddle-point • L owest free energy state in the L = 15 sector: • Symmetry group D 5 : One five-fold symmetry axis. Five “odd” two-fold axes. • All odd icosahedral spherical harmonic states are unstable. Only L = 6, 10, 12, and 18 icosahedral states are stable. (L=16 is unstable, Mavhews). •
Should we perhaps take another look at the even L states ? Picornavirus: assembles from 12 iden5cal L=6 chiral pentagons composed of 15 proteins: capsomers • Interpret ρ (Ω) as the density of capsomer centers. Add chiral capsomers. CCMV: assembles from 32 chiral capsomers L=10 12 pentamers + 20 hexamers. • But .... no signature of chirality in the density of capsomer centers??
Extensions of Lorman-Rochal • Step 1 : Landau- Brazovskii theory • smec5c-nema5c transi5on • chiral liquid crystals (cholesterics) • weak solidifica5on (block co-polymers). { } 2 + r ρ ! 2 + w ρ ! 3 + u ρ ! ) ρ ! ( ∇ 2 + k 0 ( ) ( ) ( ) ( ) ∫ LB = 4 2 F d S r r r r • Wavenumber k 0 =2π/a. Dominant molecular length-scale: a = protein size. • First term minimized by density waves ρ ! n ⋅ ! ( ) ( ) ∝ exp i k 0 ˆ r r n = unit vector n ⋅ ! ∇ 2 + k 0 ( ) exp i k 0 ˆ ( ) = 0 2 r
• Look for combina:ons of density waves ( ) ρ ! 3 n j . ! ∑ ( ) = ρ 0 cos k 0 ˆ r r j = 1 3 ⎛ ! ⎞ ( ) k j . ! ∑ ρ 0 ≠ 0 cos r ⎜ ⎟ ⎝ ⎠ j ! k j = k 0 ˆ n j Pezzuf et al. • First-order transi5on: cubic term in Landau energy non-zero • Block co-polymers (L. Leibler): compe55on between hexagonal phases, lamellar phases and other symmetries.
Landau-Brazovskii on a spherical surface. ∞ + L ∑ ∑ ( ) = ( ) • Insert ρ Ω Q L , M Y L , M Ω in Landau-Brazovskii free energy. L = 1 M = − L ⎛ ⎞ + L L 1 L 2 L 3 ∑ ∑ ∑ 2 LB = + w + u { Q 4 } ⎜ ⎟ Q L 1 , M 1 Q L 2 , M 2 Q L 3 , M 3 F r L Q L , M { } M 1 M 2 M 3 ⎝ ⎠ M = − L L 1 L 2 L 3 M 1 M 2 M 3 L M 1 + M 2 + M 3 = 0 • Sum over all L. r L L=16 L=15 2 ⎡ ⎤ r L = r + L ( L + 1) − ( k 0 R ) 2 • r 16 r 15 ⎣ ⎦ r k 0 R 15.5 16.5 Predict the L value of a capsid ! 16 r L has minimum when k 0 R ≈ L
Step 2 : Include chirality directly in free energy Chiral liquid crystals: add lowest-order chiral pseudoscalar to the Landau free energy of the achiral liquid crystal. P-G De Gennes scalars lowest non-zero pseudo-scalar • variant of ``Helfrich-Prost’’ • Sanjay Dharmavaram • • Now let’s try again
Icosahedral Isotropic • Stable minima for L=6, 10, 12, and 16. Q L L=10 L=6 r L 0 • Thermal fluctua:ons around the isotropic and Y h (L) states are chiral. • No spontanous chiral symmetry breaking at r L = 0 . è Conclusion: L = 6, 10, 12, and 16 icosahedral spherical harmonics are possible representa5ons for collec5ve capsid assembly. ç • But .... odd L icosahedral spherical harmonics states remain unstable!
L=15 icosahedral state: L=16 icosahedral state: unstable unstable L = 16 L = 15 k 0 R 15.5 16.5 16 • r 15 = r 16 when k 0 R = 16 • Could there be a stable icosahedral order parameter composed of two one-dimensional ``irreducible representa5ons” of SO(3) near k 0 R = 16 ? • L = 15 + 16 space has 31+33 = 64 dimensions
r I sotropic State k 0 R 15.5 16 ≈ 16.5 0 Non-Icosahedral States Non-Icosahedral States Second-Order Transi/ons Stable L = 15+16 Icosahedral States Lev Landau You violated my #2 rule!! k 0 R = 15.5 k 0 R = 16.5 15 + 16 stable mixture, mostly L = 15, 15 + 16 stable mixture, mostly L = 16 chiral, 60 maxima chiral, 72 maxima Dharmavaram, S., Xie, F., Klug, W., Rudnick, J., & Bruinsma, R. (2017). Orienta:onal phase • No spontaneous chiral symmetry breaking! transi:ons and the assembly of viral capsids . Physical Review E, 95(6), 062402.
Examples k 0 R = 16.5 k 0 R = 15.5 Parvovirus Polyomavirus Maxima: single capsid protein s 72 maxima: capsid protein pentamers
Numerical simula5ons: 72 par5cles on a spherical surface D 5 D 3 Icosahedral Tetrahedral D 2 • Icosahedral state: very fragile . Competes with other states that have D 3 , D 5 , T symmetry. S. Paquay, H. Kusumaatmaja, D. Wales, R. Zandi, and P. van der Schoot, Energe5cally favoured defects in dense packings of par5cles on spherical surfaces hvp://arxiv.org/abs/1602.07945 WHY WAS ICOSAHEDRAL SYMMETRY SELECTED FOR VIRUSES?
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