Fig 1.
(A) Each shell subunit contains ‘Attractors’ (green circles) on the perimeter, a ‘Top’ (tan circle, ‘T’) in the center above the plane, and a ‘Bottom’ (purple circle, ‘BH’ and ‘BP’ below the planes of the hexamer and the pentamer respectively). (B) Interactions between Attractors drive subunit binding, while Top-Top and Bottom-Bottom repulsions control the subunit-subunit angle and the shell bending modulus κs. Attractions are indicated by green arrows in (A) for the pentamer-hexamer interface and in (B) for the hexamer-hexamer interface. (C) Only hexamer Bottom psuedoatoms ‘BH’ bind cargo molecules (terra cotta circles, ‘C’). Excluder atoms (blue and brown pseudoatoms in (D)) placed in the plane of the ‘Top’ experience excluded volume interactions with the cargo. (D) The positions of excluder atoms in the preferred shell geometry for subunits with spontaneous curvature, a truncated icosahedron with 12 pentamers (blue) and 20 hexamers (brown). (E) Example of a shell that is larger than the preferred subunit geometry. (F) Subunits without spontaneous cuvature. (G) Example of hexamers without spontaneous curvature assembled around cargo (red).
Fig 2.
Snapshots from assembly trajectories of subunits with T = 3 preferred curvature.
(A) Small T = 3 shells (20 hexamers, 12 petamers) assembled without cargo at εHH = 2.6 and pentamer/hexamer stoichiometric ratio ρp/ρh = 0.5. Notice that the intermediate in the third frame contains a hexamer where a pentamer is required for icosahedral symmetry. This hexamer eventually dissociates. (B) One-step assembly with moderate cargo-cargo interaction strength, εCC = 1.5. A small nucleus of cargo and hexamer subunits forms, followed by simultaneous cargo coalescence, shell growth, and finally filling in of defects by pentamers subunits. The final structure has 68 hexamers, 12 pentamers, and 408 encapsulated cargo particles. Other parameters are hexamer-hexamer affinity εHH = 1.8, ratio of pentamer/hexamer affinity εPH/εHH = 1.3, and shell-cargo affinity εSC = 8.75, and ρp/ρh = 0.5. (C) Two-step assembly pathway for strong cargo-cargo affinity εCC = 1.65. Rapid cargo coalescence is followed by adsorption and assembly of shell subunits. The final structure has 167 hexamers, 12 pentamers, and 1520 encapsulated cargo particles. Other parameters are εHH = 1.8, εSC = 8.5, and ρp/ρh = 0.5. (D) Assembly and budding of shells from a cargo globule, for high pentamer/hexamer affinity ratio εPH/εHH = 2.0. Other parameters are εCC = 1.65, εHH = 1.8, εSC = 8.5 and ρp/ρh = 0.8. (We report energies in units of kBT throughout this article.) The shell bending modulus for all panels is κs = 10kBT. Animations corresponding to these trajectories are provided in S1, S2 and S3 Videos.
Fig 3.
Dependence of the mean shell size and most probable morphology on the cargo-cargo and subunit-cargo affinities (εCC & εSC).
(A) The mean shell size (number of hexamers + 12 pentamers) is indicated by the color bar, and the predominant morphology is indicated by symbols, with a snapshot corresponding to each morphology shown on the right. (B) The mean shell size maximized over εSC is shown as a function of εCC. Other parameters in (A) and (B) are εHH = 1.8, ρp/ρh = 0.5, εPH/εHH = 1.3, and κs = 10kBT.
Fig 4.
Dependence of shell size on the driving force for pentamer addition.
The mean shell size (number of hexamers + 12 pentamers) is shown as a function of the pentamer/hexamer stoichiometry ratio ρp/ρh for indicated values of the pentamer/hexamer affinity ratio εPH/εHH for simulations with cargo. Results from empty shell simulations are also shown for εPH/εHH = 1.3. Snapshots of typical assembly morphologies for indicated parameter values are shown around the plot. In these simulations the hexamer concentration, hexamer-hexamer affinity, and hexamer-shell affinity, and bending modulus were fixed at , εHH = 1.8, εCC = 1.65, εSC = 8.5, and κs = 10kBT.
Table 1.
Effect of parameters on shell size.
Fig 5.
Snapshots of assembly trajectories for hexamer subunits with zero spontaneous curvature.
(A) Assembly with no cargo, for εHH = 2.5, and shell bending modulus parameter εangle = 0.1 (shell bending modulus κs ≈ 20kBT). (B) Assembly with cargo, for εHH = 1.8, εSC = 7.0, and εangle = 0.08 (κs ≈ 18kBT). The final shell has 231 and 2261 hexamers and cargo particles respectively, as well as 12 pentameric vacancies. (C) Assembly with cargo in a small system with low shell bending modulus, for εHH = 1.8, εSC = 7.0, and εangle = 0.015 (κs ≈ 3kBT). The final shell has 71 and 361 hexamers and cargo particles respectively, 8 pentameric vacancies, and 2 double vacancies. An example of a double vacancy is visible in the front of the final frame. Animations of the trajectories in (B) and (C) are shown in S4 and S5 videos respectively.
Fig 6.
Size and morphology of shells assembled from subunits with no spontaneous curvature, for varying system sizes and shell bending modulus κs.
The y-axis gives the number of subunits in the largest cluster at the final simulation frame. Black diamonds correspond to Brownian dynamics simulation results for the smallest system size in which a complete shell formed, and the dashed line shows the best fit of Eq (1) to this data. The snapshots show examples of the final morphology at indicated parameter values. Two snapshots are shown of shells just below the threshold size for completion, with corresponding parameters indicated by circles. Other parameters are εCC = 1.7, εHH = 1.8, and εSC = 7.0.