Shape selection and mis-assembly in viral capsid formation by elastic frustration

  1. Carlos I Mendoza  Is a corresponding author
  2. David Reguera
  1. Instituto de Investigaciones en Materiales, Universidad Nacional Autónoma de México, Mexico
  2. Departament de Física de la Matèria Condensada, Universitat de Barcelona, Spain
  3. Universitat de Barcelona Institute of Complex Systems (UBICS), Universitat de Barcelona, Spain
7 figures, 1 table and 1 additional file


Sketch of the different structures considered in this study.

(a) A hexagonally-ordered spherical cap of radius R and geodesic radius ρ0 without defects; (b) a spherical cap with a single disclination at the center (as shown) or multiple disclinations; (c) a rectangular ribbon with length L and width W, that it is called belt when the length becomes L=2πR; and (d) a cylindrical patch with size L, that eventually becomes a cylinder of radius R. In the bending-dominated regime, R=R0.

Comparison of free energy landscapes for different structures.

Free energy of formation Δg versus the radius of the patch x in the bending-dominated regime for a defectless spherical shell (blue line), a spherical shell with defects (green), and a ribbon/belt (orange) for λ=0.0001 and Δμ~=0.001. The optimal structure is the one with the minimum energy, which is the belt in this case. The dashed line represents the unfrustrated decrease of energy expected by the classical nucleation picture for the defectless spherical cap in the absence of elastic stresses. The inset zooms the nucleation barrier located at small patch sizes.

Assembly phase diagrams.

Phase diagrams of the most stable structures in terms of the scaled chemical potential Δμ~ and the scaled line tension λ for different values of the FvK number: a) γ=0, corresponding to the bending-dominated regime, (b) γ=80, (c) γ=200, and d) γ=250. Three possible equilibrium regions are present: belts (i.e. closed ribbons, in orange), frustrated capsids with one defect (red), closed shells with defects (green), and cylinders (purple). Additionally, a region corresponding to a metastable spherical cap without defects (blue) is shown only in (a). In the white region, the equilibrium state corresponds to disaggregated individual capsomers.

Comparison of simulation results with the theoretical phase diagram.

(a) Phase diagram of the most stable structure in terms of the scaled chemical potential Δμ~ and the scaled line tension λ for γ=80. Snapshots of the final outcome of the simulation for different initial capsomer concentrations for: b) λ=0.00037, obtained with ν=1.45 and a m=36, n=18 potential. By increasing the concentration of capsomers one goes from a dissassembled state to the formation of a ribbon to a spherical shell with defects. (The ribbon and spherical shell are not closed in the snapshots due to the limited number of capsomers and finite simulation time). (c) λ=0.000525, obtained with ν=1.38 and a m=36, n=18 potential. As concentration increases, one goes from a shell with one defect to a complete shell with many defects. By increasing the FvK number to γ=1280 (by setting α=0.4), the simulations clearly forms cylindrical tubes. (d) λ=0.00084, obtained with ν=1.45 and a m=24, n=12 potential. In this case, the sequence is: disagreggated, metastable defectless shell, and closed spherical shell with many defects (a partial shell is shown). The simulation results agree qualitatively with the predictions from the theoretical phase diagram.

Appendix 1—figure 1
Free energy of formation Δgcap of a spherical cap without defects, Equation A4, versus the radius of the patch x for λ=0.0001 and different values of the scaled chemical potential Δμ~, illustrating the situations in which no assembly is possible (short dashed line, Δμ~=0.0002), a geometrically frustrated metastastable cap (dashed line, Δμ~=0.0005) or stable (solid line, Δμ~=0.001) finite shell.
Appendix 1—figure 2
Scaled optimal radius r of a spherical cap without defects as a function of the patch size x for a FvK number γ=80.

The solid line is the exact solution, Equation A29, the short dashed line is the approximation for large shells or FvK, Equation A31, and the dashed line is the approximation for small shells or FvK, Equation A30.

Appendix 1—figure 3
Other structures obtained in the simulations.

(a) Bullet-shaped shell and (b) conical shell obtained in two different repetitions for γ=224, λ=0.00069, with m=36, n=18, ν=1.345, α=0.4 and ρ=0.02; (c) T = 13 icosahedral shell obtained for γ=100, λ=0.0005246, with m=36, n=18, ν=1.40, α=0.1 and ρ=0.02; (d) coexistence between a cylinder and a partial spherocylindrical shell obtained for γ=900, λ=0.0005246, with m=36, n=18, ν=1.40, α=0.3 and ρ=0.02; (e) branched ribbon-like structure (Köhler et al., 2016) obtained at low values of the scaled line tension for γ=71, λ=0.00031, with m=36, n=18, ν=1.47, α=0.1 and ρ=0.0125.


Table 1
Estimates of the main geometric and elastic properties of different non-enveloped empty viral capsids.

The Young’s modulus E has been evaluated from AFM nanoindentation experiments (Mateu, 2012; Michel et al., 2006; Ivanovska et al., 2007; Sae-Ueng et al., 2014) and, for SV40, from the experimental spring constant (van Rosmalen et al., 2018) using the standard thin shell formula k=2.25Eh2/R (Ivanovska et al., 2004). The 2D Young’s Modulus was calculated as Y=Eh; the effective diameter of the capsomers as (Santolaria, 2011) σ=R/53π(T+13cot(π5)-1), where T is the triangulation number; the line tension as (Luque et al., 2012) λ=2ϵ03σ considering a typical binding energy ϵ010kBT; and the FvK number as γ=12(1νp2)(R/h)2, with νp=0.3 (Buenemann and Lenz, 2008).

VirusT-numberDiameter (nm)Thickness h (nm)E (Gpa)Y (N/m)σ (nm)Scaled line tension λFöppl-von Karman γ
λ Procapsid7504.00.160.646.80.000436427
λ Capsid7631.

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  1. Carlos I Mendoza
  2. David Reguera
Shape selection and mis-assembly in viral capsid formation by elastic frustration
eLife 9:e52525.