Molecular Dynamics Simulations of Deformable Viral Capsomers

Abstract

Most coarse-grained models of individual capsomers associated with viruses employ rigid building blocks that do not exhibit shape adaptation during self-assembly. We develop a coarse-grained general model of viral capsomers that incorporates their stretching and bending energies while retaining many features of the rigid-body models, including an overall trapezoidal shape with attractive interaction sites embedded in the lateral walls to favor icosahedral capsid assembly. Molecular dynamics simulations of deformable capsomers reproduce the rich self-assembly behavior associated with a general T=1 icosahedral virus system in the absence of a genome. Transitions from non-assembled configurations to icosahedral capsids to kinetically-trapped malformed structures are observed as the steric attraction between capsomers is increased. An assembly diagram in the space of capsomer-capsomer steric attraction and capsomer deformability reveals that assembling capsomers of higher deformability into capsids requires increasingly large steric attraction between capsomers. Increasing capsomer deformability can reverse incorrect capsomer-capsomer binding, facilitating transitions from malformed structures to symmetric capsids; however, making capsomers too soft inhibits assembly and yields fluid-like structures.

Keywords: coarse-grained models; deformable nanostructures; elasticity; molecular dynamics simulations; self-assembly; soft capsomers; viral capsids.

Figure 1

Side view (a) and top view (b) of the deformable capsomer. (c,d) show the perspective and bottom views, respectively, highlighting the edges associated with the bending and stretching of the capsomer. Blue beads attract blue beads on other capsomers, and red beads repel red and blue beads on other capsomers. White and black edges represent bending and non-bending edges, respectively. All bending edges are identical and characterized with a bending modulus ${\kappa }_b$. Edges between nearest-neighbor beads can stretch and are characterized with a spring constant $k_s$.

Figure 2

Steady-state configurations associated with the assembly of rigid capsomers characterized with elastic moduli ${\kappa }_b=5000$, $k_s=5000$ (top row) and deformable (“soft”) capsomers characterized with ${\kappa }_b=10$, $k_s=1000$ (bottom row) at 298 K as the steric attraction between capsomers is increased. In both cases, rich variations in the self-assembly behavior are observed (from left to right): non-assembled fluid, partial capsid assembly, nearly complete capsids, and malformed structures.

Figure 3

(a) A diagram showing the assembly products for the capsomers characterized with capsomer–capsomer steric attraction ${ϵ}_{att}$ and capsomer deformability $({\kappa }_b,k_s)$. Legend at the top shows the assembly products: non-assembled fluid (triangles), nearly-complete capsid assembly (circles), and malformed structures (diamonds). The grayscale denotes the relative proportion of these products at a given statepoint (${ϵ}_{att},{\kappa }_b,k_s$). Assembling capsomers of higher deformability into capsids requires increasingly large steric attraction. (b) Maximum cluster size $N_{max}$ vs. ${ϵ}_{att}$ for capsomers of different elastic moduli $({\kappa }_b,k_s)$ noted in the legend.

Figure 4

(a) Representative simulation snapshots of fluid-like configuration, capsid assembly, and malformed structures observed for a system of deformable capsomers at the indicated steric attraction values. (b) Average number $N_{av}$ of capsomers aggregated in a cluster of size $N_c$ associated with each of the three assembly states shown in (a). (c) Steric energy per bead, $U_{LJ}$, vs. steric attraction between capsomers for different elastic moduli (${\kappa }_b$ in units of $k_{B}T$ and $k_s$ in units of $k_{B}T/{\mathrm{nm}}^2$).

Figure 5

(a) Representative snapshots of the assembly states formed by rigid, soft, and very soft capsomers characterized with elastic moduli $({\kappa }_b,k_s)=(5000,5000)$, $(10,1000)$, and $(5,300)$, respectively under the same steric attraction ${ϵ}_{att}=2$, where ${\kappa }_b$ is the bending modulus (in $k_{B}T$) and $k_s$ is the stretching modulus (in $k_{B}T/{\mathrm{nm}}^2$). (b) Average number $N_{av}$ of capsomers aggregated in a cluster of size $N_c$ associated with capsomers characterized with different $({\kappa }_b,k_s)$. (c) Average angle fluctuations (magenta clear bars) and edge length fluctuations (cyan filled bars) vs. capsomer elastic moduli. Fluctuations are normalized by their respective values for the rigid case characterized with $({\kappa }_b,k_s)=(5000,5000)$. Structural fluctuations increase with increasing capsomer deformability.

Figure 6

Assembly states of charged deformable capsomers characterized with $({\kappa }_b,k_s)=(10,300)$ for indicated values of steric attraction ${ϵ}_{att}$. Non-assembled fluid, nearly-complete capsids and malformed structures form at ${ϵ}_{att}=1.7,2.3,2.4$, respectively.