Thermodynamics of nanospheres encapsulated in virus capsids

We investigate the thermodynamics of complexation of functionalized charged nanospheres with viral proteins. The physics of this problem is governed not only by electrostatic interaction between the proteins and the nanosphere cores (screened by salt ions), but also by configurational degrees of freedom of the charged protein $N$ tails. We approach the problem by constructing an appropriate complexation free-energy functional. On the basis of both numerical and analytical studies of this functional we construct the phase diagram for the assembly which contains the information on the assembled structures that appear in the thermodynamical equilibrium, depending on the size and surface charge density of the nanosphere cores. We show that both the nanosphere core charge and its radius determine the size of the capsid that forms around the core.

Figure 1

(Color online) Illustration of the portion of the spherical complex of proteins with flexible tails and a charged core. The parameters discussed in the text are indicated. The configuration of the tails corresponds to the case when the core is poorly charged. For the sake of a clear insight, the illustration shows a discretized model of the polyelectrolyte, although our theory is based on the continuous polyelectrolyte representation (see text).

Figure 2

(Color online) Comparison of the numerically exact results with the analytical DH expression (A9) for the free energy of the capsid with the charged core (complex free energy). The free energy per protein subunit is plotted as a function of the core surface charge density ${\sigma }_1$. The parameters of the calculation are denoted in the figure.

Figure 3

(Color online) Numerically exact results for the assembly free energy [Eq. (21)] assuming tail-less protein subunits $\left(c_0=100\text{mM}\right)$. The free energies were plotted for capsids with three different $\mathcal{T}$ numbers as functions of the core surface charge density ${\sigma }_1$ and radius $R_1$. The number of subunits and corresponding capsid radii are $N_p=30$ $\left(\mathcal{T}=1\right)$, 60 $\left(\mathcal{T}=“2”\right)$, and 90 $\left(\mathcal{T}=3\right)$, and $R_2=9.24$ $\left(\mathcal{T}=1\right)$, 13.06 $\left(\mathcal{T}=“2”\right)$, and 16 $\left(\mathcal{T}=3\right)$ nm.

Figure 4

(Color online) Assembly free energies calculated as functions of the core radius $R_1$, its charge ${\sigma }_1$, and for three different capsid radii $R_2$ (and $N_p$), as quantified by three different $\mathcal{T}$ numbers. The parameters of this calculation are $a=0.5\text{nm}$, $p=1$, $v=0.5{\text{nm}}^3$, $c_0=100\text{mM}$, and $N_t=18$.

Figure 5

(Color online) Assembly free energies as functions of the core radius $R_1$, its charge ${\sigma }_1$, and for three different capsid radii $R_2$ (and $N_p$), as quantified by three different $\mathcal{T}$ numbers. The maximal extensibility ansatz was used to obtain these results. The parameters of this calculation are $a=0.5\text{nm}$, $p=1$, $v=0.5{\text{nm}}^3$, $c_0=100\text{mM}$, and $N_t=18$.

Figure 6

(Color online) Comparison of the unconstrained monomer density (thick dashed line) and constrained monomer density (full line) approaches. The parameters of the calculation are ${\sigma }_1=0.5e/{\text{nm}}^2$, $c_0=100\text{mM}$, $R_1=9.875\text{nm}$, $R_2=16\text{nm}$, $N_p=90$, and $N_t=18$. The maximal extensibility limit [Eq. (10)] is denoted by a thin dashed line.

Figure 7

(Color online) Comparison of the unconstrained monomer density (thick dashed line) and constrained monomer density (full line) approaches. The parameters of the calculation are ${\sigma }_1=1.5e/{\text{nm}}^2$, $c_0=100\text{mM}$, $R_1=8\text{nm}$, $R_2=16\text{nm}$, $N_p=90$, and $N_t=18$. The maximal extensibility limit [Eq. (10)] is denoted by a thin dashed line.

Figure 8

(Color online) Coverage of the core defined by Eq. (24) as a function of ${\sigma }_1$ and $R_1$ for a complex with $\mathcal{T}=3$ capsids $\left(R_2=16\text{nm}\right)$. The calculation was performed with the maximal extensibility constraint on the polyelectrolyte amplitude.

Figure 9

(Color online) The difference of complex free energy of $\mathcal{T}=2$ and $\mathcal{T}=3$ structures for three different core radii in the transition region as a function of core surface charge density. The parameters of the polyelectrolyte are the same as before and the calculations were performed for the constrained polyelectrolyte model (maximal extensibility limit obeyed). Gray regions denote the surface charge densities ${\sigma }_1$ where the $\mathcal{T}=3$ structure has lower free energy.

Figure 10

Regions in the $R_1\text{−}{\sigma }_1$ space in which a particular $\mathcal{T}$-number structure has the lowest free energy, as denoted. (a) and (b) show the results for monovalent concentrations of $c_0=100\text{mM}$ and $c_0=10\text{mM}$, respectively. The two vertical dashed-dotted lines denote the radii of $\mathcal{T}=1$ and “2” structures $\left(R_2\right)$.