Monte Carlo simulations of polyelectrolytes inside viral capsids

Structural features of polyelectrolytes as single-stranded RNA or double-stranded DNA confined inside viral capsids and the thermodynamics of the encapsidation of the polyelectrolyte into the viral capsid have been examined for various polyelectrolyte lengths by using a coarse-grained model solved by Monte Carlo simulations. The capsid was modeled as a spherical shell with embedded charges and the genome as a linear jointed chain of oppositely charged beads, and their sizes corresponded to those of a scaled-down $T=3$ virus. Counterions were explicitly included, but no salt was added. The encapisdated chain was found to be predominantly located at the inner capsid surface, in a disordered manner for flexible chains and in a spool-like structure for stiff chains. The distribution of the small ions was strongly dependent on the polyelectrolyte-capsid charge ratio. The encapsidation enthalpy was negative and its magnitude decreased with increasing polyelectrolyte length, whereas the encapsidation entropy displayed a maximum when the capsid and polyelectrolyte had equal absolute charge. The encapsidation process remained thermodynamically favorable for genome charges ca. 3.5 times the capsid charge. The chain stiffness had only a relatively weak effect on the thermodynamics of the encapsidation.

Figure 1

Schematic illustration of the encapsidation process with the initial state composed of separated polyelectrolyte-free capsid (C) and polyelectrolyte (P) solutions and the final state of a polyelectrolyte-containing capsid solution (CP). The capsid and polyelectrolyte counterions are also displayed, and the shaded areas with embedded charges denote the capsid. Note, objects are not depicted to scale.

Figure 2

Capsid counterion (a) number density $\rho \left(r\right)$ and (b) running coordination number $n_{\mathit{rc}}\left(r\right)$ as a function of the radial distance $r$ for a polyelectrolyte-free capsid with cell radius $R_{\text{cell}}=100$ (solid curves) and $1000\mathrm{\AA }$ (dashed curves). The shaded areas denote the location of the capsid.

Figure 3

Difference of capsid counterion number densities $\Delta \rho \left(r\right)$ as a function of the radial distance $r$ for a polyelectrolyte-free capsid with a cell radius $R_{\text{cell}}=100\mathrm{\AA }$, where $\Delta \rho \left(r\right)=l_p^0\left(r\right)-l_p^0\left(r\right)$ (solid curve) and $\Delta \rho \left(r\right)=\rho ^{\mathrm{h}}{}_{\mathrm{PB}}\left(r\right)-\rho ^{\mathrm{h}}{}_{\mathrm{MC}}\left(r\right)$ (dotted curve) with superscripts “h” and “d” denoting a homogeneous and a discrete capsid charge distribution, respectively, and the subscript referring to the numerical solution method (see text for more details). The shaded area denotes the location of the capsid.

Figure 4

Snapshot showing an encapsidaded polyelectrolyte consisting of $N_b=250$ beads with a bare persistence length $l_p^0=8.5\mathrm{\AA }$ (connected gray spheres) and the capsid charges (dots) taken from the end of the production simulation. The small ions are omitted for sake of clarity.

Figure 5

Single-polyelectrolyte structure factor $S\left(q\right)$ as a function of the wave vector $q$ in a log-log representation for an encapsidaded polyelectrolyte with $N_b=250$ beads and a bare persistence length $l_p^0=8.5\mathrm{\AA }$ with a cell radius $R_{\text{cell}}=563.1\mathrm{\AA }$ (solid curve). The form factor $P\left(q\right)$ for a spherical layer with the inner radius $R_1=40\mathrm{\AA }$ and the outer one $R_2=50\mathrm{\AA }$, according to $P\left(q\right)=N_b{\{3∕\left[q^3(R_2{}^3-R_1{}^3)\right][{\left(q{R}_2\right)}^2{j}_1\left(q{R}_2\right)-{\left(q{R}_1\right)}^2{j}_1\left(q{R}_1\right)]\}}^2$ with $j_1$ denoting the spherical Bessel function of the first order, is also given (dotted curve).

Figure 6

Number density $\rho \left(r\right)$ of (a) beads and polyelectrolyte counterions and (b) polyelectrolyte counterions and capsid counterions as a function of the radial distance $r$ for an encapsidaded polyelectrolyte with $N_b=250$ beads and a bare persistence length $l_p^0=8.5\mathrm{\AA }$ with a cell radius $R_{\text{cell}}=563.1\mathrm{\AA }$. In (a), the density of the polyelectrolyte counterions is multiplied by 100.

Figure 7

Bead number density $\rho \left(r\right)$ as a function of the radial distance $r$ for an uncharged spherical capsid confining one polyelectrolyte with $N_b=250$ beads and a bare persistence length $l_p^0=8.5\mathrm{\AA }$, where the capsid is nonpermeable for the polyelectrolyte counterions (system I), the capsid is permeable for polyelectrolyte counterions and a cell radius $R_{\text{cell}}=563.1\mathrm{\AA }$ (system II), and the capsid is permeable for the polyelectrolyte counterions and infinite $R_{\text{cell}}$ (system III).

Figure 8

Snapshots showing encapsidated polyelectrolyte beads (connected gray spheres) and capsid charges (dots) at increasing number of beads $N_b$ (bottom to top) and increasing bare persistence length from $l_p^0$ (left to right) taken from the end of the production simulations. Same length scale is used among the panels. In (b.3) and (c.3), the right snapshots show final configuration obtained from simulations starting with a spool-like initial chain configuration. The small ions are omitted for sake of clarity.

Figure 9

Single-polyelectrolyte structure factor $S\left(q\right)$ as a function of the wave vector $q$ in a log-log representation at increasing number of beads $N_b$ (bottom to top) and increasing bare persistence length from $l_p^0$ (left to right) with random (solid curves) and spool-like (dashed curves) initial chain configuration.

Figure 10

Polyelectrolyte bead number density $\rho \left(r\right)$ (in units of ${10}^{-4}{\AA }^{-3}$) as a function of the radial distance $r$ (in units of Å) at increasing number of beads $N_b$ (bottom to top) and increasing bare persistence length from $l_p^0$ (left to right) with random (solid curves) and spool-like (dashed curves) initial chain configuration. The shaded areas denote the location of the capsid.

Figure 11

Normalized capsid counterion number density $\rho \left(r\right)∕{\rho }_{\mathrm{av}}$ as a function of the radial distance $r$ (in units of Å) at increasing number of beads $N_b$ (bottom to top) and increasing bare persistence length from $l_p^0$ (left to right) with random (solid curves) and spool-like (dashed curves) initial chain configuration. The numbers of counterions inside the capsid are also given.

Figure 12

Normalized polyelectrolyte counterion number density $\rho \left(r\right)∕{\rho }_{\mathrm{av}}$ as a function of the radial distance $r$ (in units of Å) at increasing number of beads $N_b$ (bottom to top) and increasing bare persistence length from $l_p^0$ (left to right) with random (solid curves) and spool-like (dashed curves) initial chain configuration. The numbers of counterions inside the capsid are also given.

Figure 13

Change in reduced free energy $\Delta A∕kT$, reduced enthalpy $\Delta U∕kT$, and reduced entropy $-\Delta S∕k$ for the polyelectrolyte encapsidation process shown in Fig. 1 and evaluated as described in Appendix as a function of the ratio of the polyelectrolyte and capsid charges $\beta$ at the bare persistence length $l_p^0=8.5\mathrm{\AA }$.

Figure 14

Change in reduced free energy $\Delta A∕kT$ for the polyelectrolyte encapsidation process shown in Fig. 1 and evaluated as described in Appendix as a function of the ratio of the polyelectrolyte and capsid charges $\beta$ at the bare persistence lengths $l_p^0=8.5$ (circles), 50 (squares), and $550\mathrm{\AA }$ (triangles). Results obtained using random (filled symbols and solid lines) and spool-like (open symbols and dashed lines) initial chain conformations in the CP systems are given.