Thermodynamic behaviour of the capsid and lattice gas models, with dashed lines indicating the state points considered in this paper. (a) Capsid model: The solid line is an approximation to the “critical capsomer concentration” ρcc, where the equilibrium state of the model has half of the particles in complete capsds. (b) Lattice gas model: The solid line is the binodal, which separates regimes of one-phase and two-phase behaviour.
Assembly from a disordered state. (a) Dynamical capsid assembly yields in the NVT ensemble. The fraction of subunits in well-formed capsids, n capsid(t) is shown for t = 210 000t 0 as a function of the binding interaction parameter εb. Snapshots exemplify typical clusters at the circled points. Green attractor pseudoatoms are experiencing favorable interactions, while gray attractors are not. The size of the attractors indicates the length scale of their interaction. The system contains N = 500 trimer subunits in a box of sidelength L = 74σb. (b) Phase change in the lattice gas at density ρ = 0.1. The binodal is located at εb/T = 1.86. The assembly yield is the fraction of particles that have four bonds n 4(t). The snapshots show representative configurations at time t = 105. The dashed line shows the yield that would be obtained by equilibrating a very large system: this “thermodynamic yield” is monotonic in the bond strength εb while the yield at fixed time is non-monotonic. The lattice size is L = 128.
Steady state ensemble results for the capsid model. We show the yield Y ss, the product “quality” and the production rate R ss. Four snapshots corresponding to product clusters from the circled perimeter set, εb = 6.0 are shown on top of the plots. The steady state simulations had N = 1000 trimer subunits in a box with sidelength L = 93.23σb. The red arrows indicate the location of hexameric defects discussed in Sec. V C.
Steady state ensemble in the lattice gas with n max = 100. At two different densities, we show the yield Y ss, the product “quality” and the “production rate” R ss. At ρ = 0.1, we compare with the yield ⟨n 4(t)⟩ at a time t = 105 after a “quench” from a disordered state (data from Fig. 2, rescaled for comparison). In the central panels, we show example “product” clusters, to indicate their morphologies.
(a) Measurement of cluster equilibration in the lattice gas model at ρ = 0.1, by comparison of steady state and umbrella-sampled ensembles. The data points show ⟨Δn 4(n)⟩ss, while the solid lines show ⟨Δn 4(n)⟩umb. For εb/T = 2.5, the system is far from global equilibrium, but deviations from the cluster equilibration condition are small (compare the black symbols with the solid black lines). As εb/T increases, deviations from cluster equilibration increase. (b) Similar data for the viral capsid model. The deviation in the number of bonds between clusters and their ground states is shown as a function of the cluster size n for indicated values of εb. All data points correspond to results from the steady state ensemble except for the curve with black ▲ symbols, which were obtained from umbrella sampling.
Clusters of sizes (top left to bottom-right) 5, 8, 10, and 12 in which every capsomer has at least two bonds. For numbers of subunits in between the sizes shown, there are no structures in which every capsomer has at least two unstrained bonds. This pattern gives rise to the sawtooth form for ⟨ΔB(n)⟩ in Fig. 5.
(a) Rate R ss vs for kinetic rate equations, showing non-monotonic behaviour due to “kinetic trapping” in states with many intermediates and few monomers. We take M = 50, m* = 10 and the rate is normalised by its value as . (b) The fraction of particles in the assembling steady state that are free monomers, further emphasising that the small rate for large arises from the states with a small number of monomers, and hence a small rate of bond formation.
The model capsid geometry. (a) Two dimensional projection of one layer of a model subunit illustrating the geometry of the capsomer-capsomer pair potential, Eq. (A1), 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 from a simulation trajectory.
(a) The binding free energy g b to add an additional subunit is shown as a function of intermediate size for εb = 4.5. The ▲ symbols denote values computed from umbrella sampling simulations, while the □ symbols were calculated based on the cluster configurational entropy, as described in the text. (b) The change in the configurational entropy, Δs c, computed from the ground state cluster geometries is shown as a function of intermediate size.
Article metrics loading...
Full text loading...