Forces and pressures in DNA packaging and release from viral capsids

Abstract

In a previous communication (Kindt et al., 2001) we reported preliminary results of Brownian dynamics simulation and analytical theory which address the packaging and ejection forces involving DNA in bacteriophage capsids. In the present work we provide a systematic formulation of the underlying theory, featuring the energetic and structural aspects of the strongly confined DNA. The free energy of the DNA chain is expressed as a sum of contributions from its encapsidated and released portions, each expressed as a sum of bending and interstrand energies but subjected to different boundary conditions. The equilibrium structure and energy of the capsid-confined and free chain portions are determined, for each ejected length, by variational minimization of the free energy with respect to their shape profiles and interaxial spacings. Numerical results are derived for a model system mimicking the lambda-phage. We find that the fully encapsidated genome is highly compressed and strongly bent, forming a spool-like condensate, storing enormous elastic energy. The elastic stress is rapidly released during the first stage of DNA injection, indicating the large force (tens of pico Newtons) needed to complete the (inverse) loading process. The second injection stage sets in when approximately 1/3 of the genome has been released, and the interaxial distance has nearly reached its equilibrium value (corresponding to that of a relaxed torus in solution); concomitantly the encapsidated genome begins a gradual morphological transformation from a spool to a torus. We also calculate the loading force, the average pressure on the capsid's walls, and the anisotropic pressure profile within the capsid. The results are interpreted in terms of the (competing) bending and interaction components of the packing energy, and are shown to be in good agreement with available experimental data.

FIGURE 1

The cohesive energy per unit length of DNA packed in a hexagonal array, as a function of the interstrand distance. The inset illustrates hexagonal packing of dsDNA rods.

FIGURE 2

The profile function of the DNA condensate.

FIGURE 3

DNA packing profiles within the viral capsid. The figure shows the contour lines corresponding to the top-right quarter of the condensate's cross section, for different values of the loading fraction, L_in/L, as labeled.

FIGURE 4

The total free energy of the DNA chain, F_tot = F_cap + F_sol, as a function of the ejected length, L_out (F_tot ≡ 0 at L_out = 0). The dashed and dashed-dotted curves describe F_tot for a DNA chain whose encapsidated part, F_cap, is treated as a perfect spool or a perfect torus (but adjustable d), respectively. The inset shows F_tot for the entire range of possible L_out values and the corresponding variation in the interaxial spacing, d.

FIGURE 5

The surface, bending, and DNA-DNA repulsion components of the loading force, as a function of the loaded genome length. The total force curve overlaps the repulsive component. The dashed curves describe the repulsion and bending forces corresponding to a model calculation in which d is not allowed to fall below 27 Å: the effect of this constraint on the surface term is negligible and therefore not shown.

FIGURE 6

(Solid line), The average (thermodynamic) pressure on the capsid wall, P = −∂F_cap/∂V_c, as a function of the length of DNA loaded into the capsid. (Dashed line), the average pressure calculated for a capsid wall represented by a harmonic restoring force with k = 10⁷k_BT/ξ⁴; for k ≥ 10⁹k_BT/ξ⁴ the calculated pressure is indistinguishable from the thermodynamic pressure (solid curve). (Dotted line), The osmotic pressure in a macroscopic phase of hexagonally packed DNA.

FIGURE 7

The pressure profile along one hemisphere of the viral capsid, for L_in = 290ξ.

FIGURE 8

The effect of osmotic pressure in solution on the total chain free energy, as a function of the ejected genome length. A minimum in F_tot appears for Π ≥ 0.5 atm. Assuming that DNA ejection stops (or at least delayed) at corresponding to the minimum in F_tot, the inset shows as a function of the external osmotic pressure.