Effects of Rim Fluctuations in Classical Nucleation Theory of Virus Capsids

Abstract

Most spherical viruses exhibit icosahedral symmetry, yet the growth of viral shells remains poorly understood due to the short lifetimes and broad size distribution of assembly intermediates. Classical nucleation theory has been widely applied to describe this process, but it treats the boundary of a growing shell as rigid and structureless. Here, we extend classical nucleation theory by incorporating thermal fluctuations of the capsid rim using both discrete and continuum descriptions. Allowing the rim of a partially formed capsid to undergo small geometric undulations, we show that these fluctuations generate an entropic contribution that renormalizes the effective line tension. As a result, rim fluctuations can either promote or hinder capsid closure, depending on the subunit-subunit binding free energy, temperature, and fluctuation amplitude. We find that fluctuations generally lower the nucleation barrier when the binding free energy is below a threshold value, while for sufficiently strong binding, they can instead raise the barrier by stabilizing incomplete capsids through a finite-size entropy penalty associated with rim closure. By moving beyond the idealized capillarity approximation, our results provide a controlled extension of classical nucleation theory that clarifies how boundary fluctuations influence capsid nucleation and growth.

Figures (7)

• Figure 1: A comparison of an actual rim fluctuation to the discrete model. A randomly generated rim height profile $h(s/L)$, with maximum deviation $|b|=0.25$, along a rim segment of arc length $s$ (orange dashed line) is compared to the best-fit three-state discrete step model (solid green line). The blue dotted line indicates a reference rim with no fluctuations.
• Figure 2: Contributions to the dimensionless fluctuation free energy $\beta \Delta F$ from Eq. \ref{['eq:deltaF_def']} are shown as a function of the number of rim subunits $N$ for $\beta \varepsilon^{\mathrm{D}} = 3 > \beta \varepsilon^{\mathrm{D}}_\ast$. The blue curve represents the extensive contribution $\beta \Delta F_{\mathrm{ext}}$, while the red curve represents the non-extensive contribution $\beta \Delta F_{\mathrm{non}}$ arising from the closure constraint. The purple curve shows the total fluctuation correction $\beta \Delta F$. The thin vertical dashed gray lines mark $N_0$ (principal Lambert branch, left) and $N_{-1}$ (lower Lambert branch, right), the two values of $N$ for which $\Delta F = 0$ according to Eq. \ref{['eq:discrete_lambert_solution']}. As a result, $\beta \Delta F > 0$ only within the finite interval $N_0 < N < N_{-1}$.
• Figure 3: Dimensionless fluctuation free-energy contributions in the continuum rim model from Eq. \ref{['eq:cont_deltaF']} as a function of the rim size $N$ at the threshold parameter value $\beta \varepsilon^{\text{C}}/2\pi=1/e$. The solid blue curve shows the extensive contribution $\beta \Delta F_{\mathrm{ext}}$, the solid red curve shows the non-extensive contribution $\beta \Delta F_{\mathrm{non}}$ arising from the closure constraint, and the solid purple curve shows the total fluctuation correction $\beta \Delta F$. The dashed vertical gray line at $N=1$ marks the unique solution of the zero-condition equation, where the two Lambert branches coalesce and $\beta \Delta F=0$. For all other values of $N$, the total fluctuation correction is negative, indicating that rim fluctuations lower the free energy. Note that only the part $N\geq1$ is physical (or relevant).
• Figure 4: Continuum rim fluctuation contributions to the dimensionless free energy $\beta \Delta F$ from Eq. \ref{['eq:cont_deltaF']} as a function of the rim size $N$ for the parameter value $\beta \varepsilon^{\text{C}}/2\pi=1.1>1/e$. The solid blue curve shows the extensive contribution $\beta \Delta F_{\mathrm{ext}}$, the solid red curve shows the non-extensive contribution $\beta \Delta F_{\mathrm{non}}$, and the solid purple curve shows the total fluctuation correction $\beta \Delta F$. The dashed vertical gray line at $N=1$ corresponds to the single real solution of the zero-condition equation. In this regime, the total fluctuation correction remains positive for all $N>1$, indicating that rim fluctuations uniformly raise the free-energy barrier. The regime $N<1$ is unphysical, as the rim must consist of at least one bond.
• Figure 5: Dimensionless fluctuation free-energy contributions in the continuum rim model from Eq. \ref{['eq:cont_deltaF']} for the parameter value $\beta \varepsilon^{\text{C}}/2\pi=0.25<1/e$. The solid blue curve shows the extensive contribution $\beta \Delta F_{\mathrm{ext}}$, the solid red curve shows the non-extensive contribution $\beta \Delta F_{\mathrm{non}}$, and the solid purple curve shows the total fluctuation correction $\beta \Delta F$. The two dashed vertical gray lines mark the two real solutions of the zero-condition equation, corresponding to the two Lambert branches. In this regime, the fluctuation correction is positive only over a narrow interval of small rim sizes with $N_0<N<N_{-1}\leq 1$; since $N<1$ is non-physical, rim fluctuations lower the free energy for all physical rim sizes $N\geq1$.