Quantitative Study of the Chiral Organization of the Phage Genome Induced by the Packaging Motor

Abstract

Molecular motors that translocate DNA are ubiquitous in nature. During morphogenesis of double-stranded DNA bacteriophages, a molecular motor drives the viral genome inside a protein capsid. Several models have been proposed for the three-dimensional geometry of the packaged genome, but very little is known of the signature of the molecular packaging motor. For instance, biophysical experiments show that in some systems, DNA rotates during the packaging reaction, but most current biophysical models fail to incorporate this property. Furthermore, studies including rotation mechanisms have reached contradictory conclusions. In this study, we compare the geometrical signatures imposed by different possible mechanisms for the packaging motors: rotation, revolution, and rotation with revolution. We used a previously proposed kinetic Monte Carlo model of the motor, combined with Brownian dynamics simulations of DNA to simulate deterministic and stochastic motor models. We find that rotation is necessary for the accumulation of DNA writhe and for the chiral organization of the genome. We observe that although in the initial steps of the packaging reaction, the torsional strain of the genome is released by rotation of the molecule, in the later stages, it is released by the accumulation of writhe. We suggest that the molecular motor plays a key role in determining the final structure of the encapsidated genome in bacteriophages.

Figure 1

Model for rotation and revolution of DNA within the packaging motor. The image illustrates (a) counterclockwise (CCW) rotation of the DNA around its axis and CCW revolving of the DNA with respect to the axis going through the center of the motor. The DNA (blue) is translocated in the upward direction, denoted by z, into the viral capsid. (b) The variables ϕ_rev, ϕ_rot, and r_rev measure revolving, rotation, and the distance between the axis of the DNA and the axis of the motor, respectively. CCW rotation and revolution are shown. To see this figure in color, go online.

Figure 2

The twist angle between normal vectors f_i and f_{i + 1} is equal to τ_i = α + β. To see this figure in color, go online.

Figure 3

Stochastic dynamics simulation results of average packaging velocities for various concentrations of ATP and ADP. Each data point is the average of the mean velocities of individual trajectories, with error bars indicating standard deviation of the samples. Data from Yu et al. (32) are included for comparison (solid lines). To see this figure in color, go online.

Figure 4

Histograms of bending energies showing equilibration. (a) A histogram of bending energies at the end of the packaging process for conformations generated with and without rotation of the DNA by the motor is given. The blue (dark) histogram shows the distribution when no rotation was included, and the orange (light) histogram shows it when CCW rotation was included. A t-test comparing the two means showed no statistical difference between the two (p-value = 0.55). (b) A histogram of bending energies for conformations generated with CCW rotation of the DNA by the motor is given. The orange (light) histogram shows the distribution immediately after the DNA packaging ends, and the green (dark) histogram shows it 5 s later. A t-test comparing the means showed no statistical difference between the two (p-value = 0.93). To see this figure in color, go online.

Figure 5

The results of packaging ∼3700 bp of DNA in an R_c = 15 nm capsid without rotation (left column) and with rotation (right column). Rows 1–2 show 3D renderings. Row 1 provides a side view (looking down the x axis), and row 2 provides a bottom view (looking up the z axis, directly at the motor). Color indicates the ordering of packaging: red is packaged first and violet last. Rows 3–4 show cross sections corresponding to the 3D renderings (cutting in the plane parallel to the page). The fifth row shows the “density” of the DNA, P(DNA), as a function of the radial distance from the center, showing concentric shelling with layers spaced 3–4 nm apart. The top left figure shows what might appear to be a self-intersection (the red end with the teal segment), but those vertices have been verified to have a distance greater than σ = 2.5 nm. To see this figure in color, go online.

Figure 6

Writhe (blue) and twist (orange) as a function of time under 12 different sets of parameters for simulations involving a capsid of radius R_c = 15 nm. In each panel, the x axis represents time and the y axis the value of writhe and twist. The top left corner of each graph indicates the mean writhe value with standard deviation (measured at 100 s) and the p-value for the one-sample t-test for the null hypothesis H₀: $\langle Wr\rangle$ = 0. The multiple points represent writhe and twist values measured every 2.5 s from different runs. The left column shows stochastic packaging; the right one shows deterministic (constant packaging speed) packaging. The first two rows show the effect of no rotation, the third and fourth the effect of CCW rotation, and the last two rows the effect of CW rotation. The small effect of revolving the DNA about the axis of the packaging direction as it passes through the motor is shown in alternate rows. For the deterministic motor with neither revolution nor rotation, the writhe value was 0.0 ± 1.6 (12 samples) and with revolution but no rotation was −0.9 ± 2.3 (six samples). The respective values for stochastic motors were −0.7 ± 1.6 (nine samples) and −0.2 ± 1.5 (10 samples). For deterministic motors implementing CCW rotation and no revolution, it was −3.0 ± 2.1 (11 samples), and for those with CCW rotation and revolution, it was −3.2 ± 1.3 (10 samples). For stochastic motors with a CCW rotation and no revolution, it was 3.1 ± 1.6 (14 samples), and for those with CCW rotation and CCW revolution, it was −4.1 ± 1.0 (eight samples). When there is no rotation, the p-values are all larger than 0.1, meaning that the null hypothesis is not rejected. Conversely, rotation during packaging yielded p-values smaller than 0.01, showing that the hypothesis is rejected and that in these cases, there likely exists a writhe bias. To see this figure in color, go online.