Useful scars: Physics of the capsids of archaeal viruses

We propose a physical model for the capsids of tailed archaeal viruses as viscoelastic membranes under tension. The fluidity is generated by thermal motion of scarlike structures that are an intrinsic feature of the ground state of large particle arrays covering surfaces with nonzero Gauss curvature. The tension is generated by a combination of the osmotic pressure of the enclosed genome and an extension force generated by filamentous structure formation that drives the formation of the tails. In continuum theory, the capsid has the shape of a surface of constant mean curvature: an unduloid. Particle arrays covering unduloids are shown to exhibit pronounced subdiffusive and diffusive single-particle transport at temperatures that are well below the melting temperature of defect-free particle arrays on a surface with zero Gauss curvature.

Figure 1

Electron microscopy images of archaeal viruses. (a) Sulfolobus spindle-shaped virus. (b) Acidianus two-tailed virus (ATV). (c) Acidianus bottle-shaped virus. (d) Sulfolobus neozealandicus droplet-shaped virus. Scale bars equal to $100\text{nm}$. Reprinted by permission from Macmillan Publishers [8]. Originally adapted from Refs. [9] and [10].

Figure 2

Cryoelectron micrographs of the conformational change of ATV and its reconstruction. Scale bars represent $50\text{nm}$. Reprinted by permission from Macmillan Publishers [8]. Originally adapted from Ref. [11].

Figure 3

Capsid with zero contact angle in the shape of an unduloid. The smallest diameter is $2a$, the largest diameter is $2c$.

Figure 4

Graphical construction for the capsid volume. Vertical axis: osmotic pressure $\mathrm{\Pi }$. Horizontal axis: dimensionless density $V_m/V$ with $V_m$ the minimum volume corresponding to densely packed double-stranded (ds) DNA. Solid black line: Schematic equation of state of ds DNA under physiological conditions in the absence of condensing agents. The osmotic pressure diverges at $V_m/V=1$. Dashed curves: $\tau /\left[\pi a{(3V/4\pi )}^{1/3}\right]$ for different values of the extension force $\tau$. At the minimum value ${\tau }_{\text{min}}$, the length $L$ of the central filament equals the diameter $2c$ of the unduloid

Figure 5

Force-extension curve of the central filament as a plot of filament length $L$ versus extension force $\tau$. For $\tau$ less than ${\tau }_{\text{min}}$, when the length $X$ of the tails is zero, the filament only indents the capsid and the force-extension curve is approximately linear. For larger values of the extension force, the capsid develops tails. The area of the capsid limits the maximum value of the filament length.

Figure 6

Spiral construction: (a) Initial unduloid. (b) Eight equidistant spirals covering the unduloid surface and constructed according to the method of Appendix pp2 (one spiral is colored in red to highlight spacing).

Figure 7

Sequential placement of particles on the eight helices covering the unduloid of Fig. 6. The arrays is constructed first on two half unduloids, which are then merged. Any holes left after merging are filled with particles

Figure 8

Voronoi construction corresponding to the initial particle configuration covering the unduloid. Cells with five edges are colored blue (pentamers), cells with six edges gray (hexamers), and cells with seven edges red (heptamers). (a) Side view showing isolated dislocations. (b) Front view showing eight spiral dislocations strings. (c) Front view overlaid with construction spirals.

Figure 9

Annealing of a cylinder. (a) Initial state; (b) intermediate state after the heating pulse; and (c) final state.

Figure 10

Monte Carlo annealing. (a) Initial state with spiral order. (b) Intermediate state after heating pulse (images corresponds to 1000 Monte Carlo steps, see also Fig. 11). (c) Final state.

Figure 11

Evolution of the energy (left vertical axis) and capsomer numbers (right vertical axis) versus Monte Carlo steps during the annealing of an unduloid. The vertical dashed line corresponds to the intermediate state shown in Fig. 10.

Figure 12

Annealing of a particle array covering a sphere. (a) Initial, (b) intermediate (image corresponds to about $1/8$ of the annealing simulation time), and (c) final configurations.

Figure 13

Mean-square displacement (MSD) $\langle u^2\rangle /r_m^2$ on a cylinder versus time on a log-log scale. Time is expressed in units of the inverse step frequency ${\nu }_0$ and distance in units of $r_m$, the equilibrium spacing of the LJ pair potential. The dimensionless inverse potential depth $k_{B}T/\epsilon$ is increased in multiples of 0.75 as indicated for some of the curves. The horizontal line is the MSD of a hexagonal lattice at the melting point. The black solid line is the MSD of an isolated particle.

Figure 14

Mean-square displacement $\langle u^2\rangle /r_m^2$ (MSD) of a particle array on an unduloid, with $r_m$ the equilibrium spacing of the ordered lattice. The dimensionless inverse potential depth $k_{B}T/\epsilon$ is increased in multiples of 0.75 as indicated for some curves. The horizontal red solid line is the late-time MSD of the particles on a flat Lennard-Jones hexagonal lattice at the melting point. The upper black solid line has a slope for an MSD proportional to $t$ (diffusion) and the lower black solid line has a slope for an MSD proportional to $t^{\alpha }$ with $\alpha \simeq 0.5$ (subdiffusion). For $k_{B}T/\epsilon =7$ subdiffusion crosses over to diffusion at a scale $t^{*}$ determined by the intersection of these black lines.

Figure 15

Average late-time slope of the MSD vs time as a function of $k_{B}T/\epsilon$ with $\epsilon$ the LJ binding energy. The average slope together with its standard deviation (error bars) were obtained from six different KMC analyses per $k_{B}T/\epsilon$ and per geometry. Black line: unduloid. Red line: sphere. Blue line: cylinder. Vertical dashed line: estimated melting point for the cylinder.

Figure 16

Late-time snapshot of a simulation with periodic boundary conditions at $k_{B}T/\epsilon =2.5$

Figure 17

Average MSD slope vs time as a function of $k_{B}T/\epsilon$ for an unduloid with periodic boundary conditions. Black line: MSD of particles in the center (central third). Red line: MSD of particles in the tails (initial and final thirds). Blue line: all particles.

Figure 18

(a) Averaged normal forces per unit area computed with a particle array under tension on an unduloid. The system was under an applied tension of approximately 5% while $k_{B}T/\epsilon =0.05$. Note that there are significant pressure variations. (b) Corresponding defect array. Note that there is some correlation between low-pressure regions and heptamers. (c) Capsid rupture under a larger applied tension of $\approx 6$%.

Figure 19

The two possibilities for the parameters in the quadrature equations leading to a minimum surface of revolution with fixed volume. The case illustrated on the left relates to the work reported here.

Figure 20

A minimizing surface corresponding to the conditions illustrated on the left-hand side of Fig. 19.

Figure 21

Schematic of helices on a surface of revolution.

Figure 22

Helix uniformly covering the sphere.

Figure 23

Five helices covering a sphere. The $T=7$ construction is recovered when they are decorated with particles.

Figure 24

A zoomed-in view of the underlying lattice on which the particles move.

Figure 25

Concentration vs time for diffusion problem with no particle interactions and absorbing boundary conditions.

Figure 26

Concentration vs time for diffusion problem with annihilation reaction A+A $\to 0$

Figure 27

Initial, intermediate (at 1100 Monte Carlo steps) and final capsomers configuration during Monte Carlo annealing of a $T=7$ virus structure. Due to the particles' motion during the annealing procedure, the initial and final capsomers configurations are rotated with respect to each other.