Electrostatic self-energy of a partially formed spherical shell in salt solution: Application to stability of tethered and fluid shells as models for viruses and vesicles

We investigate the electrostatics of a partially formed, charged spherical shell in a salt solution. We solve the problem numerically at the Poisson-Boltzmann level and analytically in the Debye-Hückel regime. From the results on energetics of partially formed shells we examine the stability of tethered (crystalline) and fluid shells toward rupture. We delineate different regimes of stability, where, for fluid shells, we also include the effects of bending elasticity of the shells. Our analysis shows how charging of the shell induces its instability toward rupture but also provides insight regarding growth of charged shells.

Figure 1

(a) The geometry of a ruptured spherical shell. (b) The mode of rupture of a tethered (solid) shell. (c) The mode of rupture of a fluid shell. The amount of shell material in both modes of rupture is conserved in the process but is partitioned between the original shell and its progeny in the case of a tethered shell, or is contained completely in the progeny in the case of a fluid shell.

Figure 2

The results of the fit of Eq. (15) to the DH solution (pluses). The full line shows the $b_0/\rho$ behavior with $b_0\approx 0.161$. The empty symbols show the behavior of the fit to Eq. (15) of the full PB solution calculated for $R=10.04$ nm, for different surface charge densities: $\sigma =$ 0.1 $e_0/{\mathrm{nm}}^2$ (squares), 0.4 $e_0/{\mathrm{nm}}^2$ (circles), 0.8 $e_0/{\mathrm{nm}}^2$ (triangles), and 1.2 $e_0/{\mathrm{nm}}^2$ (diamonds). Dashed lines are guides to the eye. The numerical solution of the PB solution become less reliable with increasing $\rho$ as the errorbars indicate. A noticeable effect in PB case is the dependence of the line tension coefficient $b$ on the surface charge density.

Figure 3

The solution of the PB equation for the dimensionless electrostatic potential $\beta e_0\phi$. The shell radius is $R=10.04$ nm, and the opening angle ${\vartheta }_0=0.235\pi$. The shell surface charge density is $\sigma =0.4$ $e_0/{\mathrm{nm}}^2$, and the bulk concentration of salt ions is $c_0=10$ mM.

Figure 4

Electrostatic free energy of the system as a function of the opening angle, for a capsid of radius $R=10.04$ nm and several values of surface charge densities and salt concentrations, as denoted in the body of the figure. The dashed lines show the solution obtained in the DH approximation; the black dashed line aligns almost completely with the numerical PB solution. The star symbols show the PB results for a closed capsid (${\vartheta }_0=0$) taken from Ref. [20].

Figure 5

Fit of the angular dependence of the solution of the PB theory to the form $F_{\mathrm{fit}}({\vartheta }_0)=a[(1+\cos {\vartheta }_0)/2-b\sin {\vartheta }_0]$, for a shell of radius $R=10.04$ nm and surface charge density $\sigma =0.4$ $e_0/{\mathrm{nm}}^2$. Salt concentration is $10$ mM. The inset shows a separate fit of $-b\sin {\vartheta }_0$ to the numerical solution, from which the cosine dependence was removed, i.e., $F_{\mathrm{num}}({\vartheta }_0)/F_{\mathrm{num}}(0)-(1+\cos {\vartheta }_0)/2$. Note that in the latter case, a slight effect of numerical error can be discerned.

Figure 6

Symbols: Instability line of tethered shells toward rupture obtained by numerical PB calculation with $\gamma =1$ $k_{B}T$/nm and shell radius $R=$10.04 nm. Dashed line: Debye-Hückel approximation, $\sigma =e_0\sqrt{2\beta \gamma c_0/b_0}$. In PB case, the points $(\sigma ,c_0)$ of zero edge energy were obtained by interpolation of numerical results for several surface charge densities $\sigma$ and salt concentrations $c_0$, hence the error bars.

Figure 7

The “phase diagram” of the ruptured capsids that shows the sign of $\partial F_{\mathrm{total}}/\partial {\vartheta }_0$ (gray regions). In light-gray (dark-gray) regions, the fully opened, disk state of the membrane has lower (higher) free energy than the spherical state. In the white regions, the free energy always increases with ${\vartheta }_0$ and the vesicle is stable toward pore formation.

Figure 8

Different regions in an incomplete shells, selected on the criterion of their distance from the edge. An isolated point charge is shown in region 2 with the surrounding that contributes to its electrostatic energy.