Nonspecific interactions in spontaneous assembly of empty versus functional single-stranded RNA viruses

We investigate the influence of salt concentration, charge on viral proteins and the length of single-stranded RNA (ssRNA) molecule on the spontaneous assembly of viruses. Only the nonspecific interactions are assumed to guide the assembly, i.e., we exclude any chemical specificity that may lock the viral proteins and ssRNA in preferred configurations. We demonstrate that the electrostatic interactions screened by the salt in the solution impose strong limits on viral composition that can be achieved by spontaneous assembly. In particular, we show that viruses whose ssRNA carries more than twice the amount of charge that is located on the viral proteins, cannot be assembled spontaneously. We find that the spatial distribution of protein charge is important for the energetics of the assembly. We also show that the pressures that act on the viruses as a result of attractive protein-ssRNA electrostatic interactions are at least an order of magnitude smaller than is the case with bacteriophage viruses that contain double-stranded DNA molecule.

Figure 1

(Color online) Panel (a) represents the reference state, i.e., the empty capsid and the dissolved ssRNA, while panel (b) represents the filled capsid that contains the ssRNA. The free energies of the reference and the filled states are $F\left(0\right)$ and $F\left(N\right)$, respectively [see Eq. (5) and the discussion following it].

Figure 2

(Color online) Total [$F\left(N\right)$, squares], entropic [$F_{\mathrm{ent}}\left(N\right)$, circles], and polyelectrolyte self-interaction [$F_{\mathrm{int}}\left(N\right)$, triangles] free energies of the polyelectrolyte-capsid-salt system as functions of the number of monomers. The parameters used are $p=1$, $a=0.5\mathrm{nm}$, $v=0.05{\mathrm{nm}}^3$, $T=300\mathrm{K}$, $\sigma =0.4e∕{\mathrm{nm}}^2$, and $R=12\mathrm{nm}$. The results are shown for three different bulk concentrations of salt ions, $c_0=10$ [panel (a)], 100 [panel (b)], and $700\mathrm{mM}$ [panel (c)]. The vertical dashed line denotes the number of monomers for which the polyelectrolyte charge equals in magnitude the capsid charge $(N\approx 724)$.

Figure 3

(Color online) The complexation free energies of the polyelectrolyte-capsid-salt system as functions of the number of monomers. The parameters used are $p=1$, $a=0.5\mathrm{nm}$, $v=0.05{\mathrm{nm}}^3$, $T=300\mathrm{K}$, $\sigma =0.4e∕{\mathrm{nm}}^2$, and $R=12\mathrm{nm}$. The results are shown for three different bulk concentrations of salt ions, $c_0=10$, 100, and $700\mathrm{mM}$. The vertical dashed line denotes the number of monomers for which the polyelectrolyte charge equals in magnitude the capsid charge $(N\approx 724)$.

Figure 4

(Color online) The polyelectrolyte concentration profile for $c_0=100\mathrm{mM}$. The curves displayed correspond to $N=100$, 300, 500, 700, 900, 1100, 1300, and 1500. The lines are styled so that the length of their dashes is proportional to $N$. Other parameters of the calculation are the same as in Fig. 3.

Figure 5

(Color online) (a) Density of negative monovalent counterions. (b) Density of positive monovalent counterions. The bulk salt concentration is $c_0=100\mathrm{mM}$. The curves displayed correspond to different polyelectrolyte lengths ($N=100$, 300, 500, 700, 900, 1100, 1300, and 1500). The lines are styled so that the length of their dashes is proportional to $N$. Other parameters of the calculation are the same as in Fig. 3. Thick vertical line denotes the position of the capsid (capsid radius $r=R$).

Figure 6

(Color) One quarter of the cucumber mosaic virus capsid (strain FNY). The image was constructed by applying the group of icosahedral transformations to the RCSB Protein Databank [38] entry 1F15 and all atoms in the resulting structure were represented as spheres of radius $3.4\mathrm{\AA }$ (which is the experimental resolution [37]). They were colored in accordance with their distance from the geometrical center of the capsid, so that the atoms that are farthest away from the center are orange, while those that are closest to the center (and belonging to the capsid protein tails) are light blue.

Figure 7

(Color online) The complexation free energies for the capsid charge modeled as in Eq. (7) with $Q_c=724$ $e$ and $\xi =2.5\mathrm{nm}$. The other parameters of the calculation are the same as in Fig. 3.

Figure 8

(Color online) The polyelectrolyte concentration profile for $c_0=100\mathrm{mM}$ and for the capsid density model represented in Eq. (7) and used in the calculation in Fig. 7. The curves displayed correspond to $N=100$, 300, 500, 700, 900, 1100, 1300, and 1500. The lines are styled so that the length of their dashes is proportional to $N$. A vertical line at $r=9.5\mathrm{nm}$ indicates the position of the $N$-termini of the capsid proteins. The other parameters of the calculation ($p,a,v,T$, and $R$) are the same as in Fig. 3.

Figure 9

(Color online) (a) Osmotic pressure acting on the viral capsid as a function of the number of monomers. (b) Osmotic pressure for fixed number of monomers $N=420$, $c_0=100\mathrm{mM}$ [the point denoted by a circle in panel (a)] as a function of the capsid charge density $\sigma$. The other relevant parameters of the calculation are the same as in Fig. 3.