DNA packaging and ejection forces in bacteriophage

Footnotes

• ↵ § To whom reprint requests should be addressed. E-mail: gelbart{at}chem.ucla.edu.

• ↵ †† Beads are fed into the cavity at a rate of 3 × 10⁻⁵ σ/Δt, or at about 7500 μm/s. In actual packaging situations [see, for example, the in vitro measurements described in ref. 24] the loading rate decreases significantly with the force resisting loading. As explained later in this paper, however, we are careful to calculate our loading forces by stopping our loading at each of several different internal chain lengths and then averaging over long times the outward radial force on the last monomer.

• ↵ §§ Calculation details of our continuum theory results, reported in the present paper, are provided elsewhere [S.T., J.K., W.M.G., and A.-B.S., unpublished work], where we also treat more thoroughly the kinetics of loading/ejection and estimates of capsid pressures.

• ↵ ¶¶ The bending modulus κ is set to 50 k _B T nm, consistent with the experimentally determined persistence length for double-stranded DNA of ξ = 50 nm.

• ↵ ‖‖ The dependence of d on osmotic pressure in the presence of condensing agent, as measured in the experiments of Rau and Parsegian (22) was fit to the form P = F₀ {exp[−(d − d₀)/c]−1}, with F ₀ = 0.12 k _B T/nm³, c = 0.14 nm, and d ₀ = 2.8 nm. Integration of the pressure with respect to two-dimensional compression of the hexagonal lattice gives e _a(d) to within a constant. The cohesive energy per unit length at d ₀ was chosen as −0.74 k _B T/nm to give the known dimensions of toroidal condensates of DNA in solution (23).

• ↵ ¶ Gabashvili and Grosberg (15) have treated several scenarios for the progressive kinetics of ejection, allowing for different sources of friction being dominant in the capsid and tail. Although they also make estimates of the ejection forces, our primary concern in the present work is to provide a more realistic and systematic theory of the thermodynamics underlying these forces and of the structures of packaged DNA in the capsid.

• ↵ ‖ Later stages of the injection have been shown in many instances to be driven by bacterial-cell transcription of the leading portion of the translocated viral DNA; see, for example, ref. 16.

• ** The “beads” are Lennard–Jones particles, linked to their nearest neighbors along the chain by harmonic stretching potentials centered at an interbead distance of σ, the Lennard–Jones diameter (corresponding approximately to the diameter of hydrated DNA, or about 2.5 nm). A harmonic bending potential, applied to the angle between neighboring bonds, dictates the intrinsic persistence length of the chain. The step size of our Brownian dynamics simulation, which corresponds to DΔt/k _B T and where D is a diffusion constant, is 3 × 10⁻⁴ σ ²/k _B T. By the Stokes–Einstein relation for the diffusion constant of a sphere of diameter σ = 2.5 nm in a medium with the viscosity of water, this result translates to a time-step Δt of 10 ps. Hydrodynamic interactions between beads are neglected.

• ↵ ‡‡ The first crystallographic structure determination of the capsid of an icosahedral double-stranded DNA bacteriophage (HK97), including information about the distribution of charge on the interior wall, was reported only recently (see ref. 20).

• Copyright © 2001, The National Academy of Sciences