Encapsulation of a polymer by an icosahedral virus

Abstract

The coat proteins of many viruses spontaneously form icosahedral capsids around nucleic acids or other polymers. Elucidating the role of the packaged polymer in capsid formation could promote biomedical efforts to block viral replication and enable use of capsids in nanomaterials applications. To this end, we perform Brownian dynamics on a coarse-grained model that describes the dynamics of icosahedral capsid assembly around a flexible polymer. We identify several mechanisms by which the polymer plays an active role in its encapsulation, including cooperative polymer-protein motions. These mechanisms are related to experimentally controllable parameters such as polymer length, protein concentration and solution conditions. Furthermore, the simulations demonstrate that assembly mechanisms are correlated with encapsulation efficiency, and we present a phase diagram that predicts assembly outcomes as a function of experimental parameters. We anticipate that our simulation results will provide a framework for designing in vitro assembly experiments on single-stranded RNA virus capsids.

Figure C1

The fraction of trajectories that end in each outcome are shown in cumulative plots as a function of N_p for ε_cp = {3.5, 4.0, 4.5, 5.0}, for (a)–(d) respectively. The height of each color corresponds to the fraction of trajectories resulting in that outcome, color-coded according to the legend in figure 3b. The spike in at N_p ~ 300 in (c) corresponds to a large yield of size 30 defective capsids, examples of which are pictured in the bottom row of figure C3.

Figure C2

The driving force for subunits to adsorb on to the polymer is revealed by $c_1^{\text{eq}}$, the equilibrium one-dimensional concentration of subunits on the polymer in the absence of capsomer-capsomer attractions (ε_cc = 0). $c_1^{\text{eq}}$ is measured as the average number of adsorbed subunits divided by the polymer length, and shown as functions of ε_cp and log c₀.

Figure C3

Examples of common malformed but closed capsids. The top row shows the single dominant morphology for sizes 22, 24 and 26. For sizes 24 and 26, the dislocations (2 in the former case, 3 in the latter) relieve strain by arranging themselves at opposite poles of the 2 and 3 fold symmetry axes, respectively. In the bottom row are the 3 most prevalent morphologies for malformed capsids of size 30, for which more strain-relieving arrangements of hexamers are possible.

Figure C4

Typical disordered assembly products for high capsomer-polymer affinities ε_cp ≥ 5.0.

Figure C5

Visualization of the polymer density. The polymer density is averaged over a large number of successful assembly trajectories after completion, for a polymer with length N_p = 150. Densities are averaged over the threefold symmetry of the capsomer, but not over the 20-fold symmetry group of the completed capsid.

Figure 1

The model capsid geometry. (a) Two dimensional projection of one layer of a model subunit illustrating the geometry of the capsomer-capsomer pair potential, equation (3), with a particular excluder and attractor highlighted from each subunit. The potential is the sum over all excluder-excluder and complementary attractor-attractor pairs. (b) An example of a well-formed model capsid. (c) Cutaway of a well-formed capsid.

Figure 2

(a) Image of a trimer of dimers of the MS2 coat protein [25], which was generated from the crystal structure PDBID:1ZDH[25] using VMD [80]. The three proteins of the crystal structure asymmetric unit are shown along with the three symmetry-related subunits that complete the dimer subunits. The protein atoms are shown in van der Waals representation, RNA-stem loops are drawn in cartoon format and colored green, and positive charges on the proteins are colored blue. (b) The arrangement of polymer attractors on the model capsid subunit, as viewed from inside the capsid. The capsomer-polymer attractors are colored blue and the capsomer-capsomer attractors are colored green. (c) A cutaway view of a snapshot of a polymer with N_p = 200 segments encapsulated in a well-formed model capsid. Polymer subunits and capsomer-attractors are colored according to their interaction energy: red for non-interacting, green for optimal interaction and a gradient for intermediate states.

Figure 3

Kinetic phase diagram showing the dominant assembly product as a function of N_p and ε _cp for ε _cc = 4.0 and log c₀ = −7.38 at observation time t_obs = 2 × 10⁴t₀. The legend on the right shows snapshots from simulations that typify each dominant configuration. Data points indicate the majority outcome, except for the ‘malformed’ and ‘mixture’ points. For malformed points there was a plurality of malformed capsids and a majority of malformed plus well-formed capsids. For points labeled ‘mixed phase’ there was no clear plurality. The exact proportions of the outcomes are available in figure C1, Appendix C. Data points correspond to 20 independent assembly trajectories.

Figure 4

a) Residual chemical potential difference between a polymer grown inside a well-formed capsid and a free chain, ${\mu }_{\text{chain}}^{\text{cap}}-{\mu }_{\text{chain}}$, at indicated capsomer-polymer affinities ε _cp. b) The fraction of Brownian dynamics trajectories that end with a polymer completely encapsulated in a well-formed capsid is shown for the same capsomer-polymer affinities.

Figure 5

Two mechanisms for assembly around the polymer. (a,b) The number of capsomer subunits adsorbed onto the polymer (solid) and the size of the largest partial capsid (dashed) are shown as a function of time for (a) a trajectory with low (the sequential assembly mechanism) and (b) a trajectory exhibiting high (the en masse mechanism). Parameters are (a) N_p = 200, ε_cp = 3.0, log c₀ = −6.5, ε_cc = 4.5 and (b) N_p = 150, ε_cp = 4.5, log c₀ = −5, ε_cc = 3.25. (c) Snapshots from the simulation trajectory shown in (a) (points marked with arrows). (d) Snapshots corresponding to points marked with arrows in (b) showing the the mass adsorption of subunits onto the polymer followed by annealing of multiple intermediates and finally completion. Once the polymer is completely contained within the partial capsid (second to last frame), addition of the last subunit is relatively slow as discussed in the text.

Figure 6

Contour plots of (top panels) the yield, or fraction of trajectories that end with well formed capsids and (bottom panels) the assembly mechanism order parameter defined in the text. Plots are shown as functions of ε_cp and log c₀ for parameter values {ε_cc = 3.25, N_p = 150} (left), {ε_cc = 4.0, N_p = 150} (center), {ε_cc = 3.25, N_p = 200} (right).

Figure 7

a) The median growth times (time between nucleation and completion) as a function of N_p for indicated values of ε_cp, with ε_cc = 4.0 and log c₀ = −7.38. b) Snapshots from an assembly trajectory demonstrating both sliding, or one-dimensional diffusion of subunits along the polymer, and the ‘fly-casting’ mechanism described in the text. A free subunit binds the polymer (first frame) and slides towards the growing edge (second and third frame). It then binds to the growing edge of the capsid (fourth frame) while still attached to the polymer, forming a small loop. Note that fly-casting is not limited to such short loops.