Theory of conformational transitions of viral shells

We propose a continuum theory for the conformational transitions of viral shells. Conformational transitions of viral shells, as encountered during viral maturation, are associated with a soft mode instability of the capsid proteins [F. Tama and C. L. Brooks, J. Mol. Biol. 345(2), 299 (2005)]. The continuum theory presented here is an adaptation of the Ginzburg-Landau theory of soft-mode structural phase transitions of solids to viral shells. The theory predicts that the conformational transitions are characterized by a pronounced softening of the shell elasticity in the critical region. We demonstrate that the thermodynamics of the conformational transition can be probed quantitatively by a micromechanical atomic force microscope study. The external force can drive a capsid into a state of phase coexistence characterized by a highly nonlinear force deformation curve.

Figure 1

Schematic Gibbs free energy $g_0\left(\eta \right)$ per unit area for the maturation reaction as a function of the reaction coordinate $\eta$. The first minimum ${\eta }_{-}$, near $\eta =0$, corresponds to the initial state and the second minimum, ${\eta }_{+}$, to the final state. The free energy barrier per unit area separating the two minima equals $\mathrm{\Delta }E∕a^2$.

Figure 2

Graphical construction for obtaining equilibrium values of the reaction coordinate. The solid line shows $d{g}_0∕d\eta$. Intercepts of $d{g}_0∕d\eta$ with a straight line (dashed) represent possible values for the reaction coordinates. Only intercepts with $d^2{g}_0∕d{\eta }^2$ positive can represent thermodynamically stable states. Intercepts with the straight line through the origin correspond to the case of zero osmotic pressure. If the pressure increases (lines labeled 1 and 2) then the solution corresponding to the initial state disappears above a threshold “spinodal” pressure.

Figure 3

Maxwell equal-area construction determining the radial displacement $f_b$ for which initial and final states can coexist. The horizontal axis is the reaction coordinate $\eta$. The vertical axis is the radial displacement $f$, with $\gamma$ the coupling constant between reaction coordinate and radial displacement. Radial displacements and reaction coordinates corresponding to the initial and final states are indicated. The dashed line shows the relation between reaction coordinate and displacement.

Figure 4

Indentation profiles $f\left(r\right)$ and $\eta \left(r\right)$ of a capsid subject to an applied point force $F$. The capsid is resting on a surface. The dimple has a force-independent width $\delta \sim (R{\mathcal{l}}_B{)}^{1∕2}$ with $R$ the radius and $l_B$ the buckling length.

Figure 5

Dimple profile in the linear response regime for large capsids $(R\gg {\xi }^2∕l_B)$. The horizontal axis is the distance $r$ from the probe in units of the characteristic in-plane width $\delta =\sqrt[4]{R^2\kappa ∕Y_{eff}}$ of the dimple. The vertical axis is the out-of-plane displacement $f$ in units of $d=\left(\frac{FR}{2\pi \sqrt{Y_{eff}\kappa }}\right)$, the characteristic size of the depth of the dimple. The dimensionless pressure osmotic pressure $\Pi ∕{\Pi }_g$, with ${\Pi }_g=4\sqrt{\kappa Y_{eff}}∕R^2$, ranges from 0 to 1.8.

Figure 6

Numerically computed linear response profiles in the crossover regime where the capsid dimple size $\delta$ is comparable to the characteristic length scale $\xi$ for the spatial variation of the reaction coordinate. The ratio of the effective and bare Young’s moduli was fixed at 1/4.

Figure 7

Phase coexistence induced by a force probe. The capsid is initially in a radially swollen state where $f=f_{+}$ and $\eta ={\eta }_{+}$. The point force produces a circular region of radius $\rho$ inside of which the capsid is in the contracted state with $f=f_{-}$ and $\eta ={\eta }_{+}$.

Figure 8

(a) Numerically computed energy $H\left(\rho \right)$ of a capsid that is in a state of phase coexistence as a function of the radius $\rho$ of the domain wall. The parameters were $R=14\mathrm{nm}$; $\kappa =800\mathrm{pN}\bullet \mathrm{nm}$; $\mathrm{\Delta }\mu =-2\mathrm{pN}∕\mathrm{nm}$; $\Lambda =20\mathrm{pN}$; $Y_{+}=200\mathrm{pN}∕\mathrm{nm}$, and $Y_{-}=40\mathrm{pN}∕\mathrm{nm}$. The radii of the swollen and contracted state differed by 10%. The horizontal axis is in nm and the vertical axis is in units of $k_{\mathrm{B}}T$, with the energy of the initial swollen state subtracted off. Curves 1–4 correspond to force levels $F$ in the range of 0 to $1.2\mathrm{nN}$. A second minimum, indicating phase coexistence, appears for force levels above, approximately, one nanoNewton. (b) Same as (a) except that $Y_{+}=200\mathrm{pN}∕\mathrm{nm}$ and $Y_{-}=400\mathrm{pN}∕\mathrm{nm}$. Phase coexistence is not encountered at any force level.

Figure 9

Phase diagram for the swollen $(+)$, contracted $(-)$, and coexistence $(+∕-)$ states of a capsid with $\mathrm{\Delta }\mu$ the free energy difference per unit area between the two states and $F$ the external point force. Swelling corresponds to a 10% increase of the radius. For lower values of $\mathrm{\Delta }\mu$, there is a direct transition between the swollen and contracted states (horizontal line marked 1), while for larger values of $\mathrm{\Delta }\mu$ the application of force can produce phase coexistence (horizontal line marked 2) for forces above a threshold $F_c$. The dotted line is the linear-response prediction for this transition line. The other parameters were $\Lambda =2.5\mathrm{kT}∕{\mathrm{nm}}^2$, $\kappa =100\mathrm{kT}$, $Y_{+}=220\mathrm{pN}{\mathrm{nm}}^{-1}$, $Y_{-}=44\mathrm{pN}{\mathrm{nm}}^{-1}$, $R=14\mathrm{nm}$, and ${\Pi }_0=0$.

Figure 10

Threshold force $F_{\pm }$, at given $\mathrm{\Delta }\mu$, for the onset of force-induced phase coexistence as a function of the ratio $Y_{+}∕Y_{-}$ of Young’s moduli of the swollen, respectively, contracted state for different values of the Föppl–Von Karman number ${\gamma }_{\mathrm{FVK}}=Y{R}^2∕\kappa$ with $Y$ the bare Young’s modulus. The threshold force is expressed in dimensionless units with $\kappa$ the bending modulus and $R$ the capsid radius. The critical force diverges at a threshold that depends on the value of ${\gamma }_{\mathrm{FVK}}$. The remaining parameters are $\Lambda =0.7kT∕\mathrm{nm}$, $\mathrm{\Delta }\mu =0.5kT∕{\mathrm{nm}}^2$, ${\Pi }_0=0$, $\mathrm{\Delta }R∕R=0.1$, $\kappa =100kT$ and $R=14\mathrm{nm}$.

Figure 11

Force-deformation curves corresponding to the two horizontal lines marked “1” and “2” in Fig. 9. Along curve 1, the capsid is in the contracted state $(-)$ at higher force levels. The vertical dotted line marks a large, first-order transition to the expanded state $(+)$. At $F_{c1}$, the contracted and expanded states have the same free energy. Both force-deformation curves are linear. Along curve 2, the capsid is in a state of phase coexistence (marked $+∕-$) at higher force levels. The force-deformation curve is nonlinear. At $F_{c2}$, the free energies of the expanded state and the coexistence state are equal.