Role of Condensing Particles in Polymer Confinement: A Model for Virus-Packed "Minichromosomes"

Abstract

Confined mixtures of a polymer and nonspecifically binding particles (condensers) are studied as models for viruses containing double-stranded DNA (polymer) and condensing proteins (particles). We explore a model in which all interactions between the packed content (polymer and particles) and its confinement are purely repulsive, with only a short-range attraction between the condensers and polymer to simulate binding. In the range of physical parameters applicable to viruses, the model predicts reduction of pressure in the system effected by the condensers, despite the reduction in free volume. Condensers are found to be interspersed throughout the spherical confinement and only partially wrapped in the polymer, which acts as an effective medium for the condenser interactions. Crowding of the viral interior influences the DNA and protein organization, producing a picture inconsistent with a chromatin-like, beads-on-a-string structure. The model predicts an organization of the confined interior compatible with experimental data on unperturbed adenoviruses and polyomaviruses, at the same time providing insight into the role of condensing proteins in the viral infectious cycles of related viral families.

Figure 1

Pressure on the capsid from a confined mixture of DNA and condensing particles. (a) Pressure versus condenser-DNA interaction energy, $\mathit{ϵ}$, for a DNA volume fraction of ${\varphi }_{\text{p}}=0.3$ and condenser volume fraction of ${\varphi }_{\text{s}}=0.04$. Data are shown for three different condensing particle radii, $R_{\text{s}}$. The horizontal dashed line shows the pressure value without condensers $\left({\varphi }_{\text{s}}=0\right)$. (b) Pressure versus volume fraction, ${\varphi }_{\text{s}}$, of condensing particles for a DNA (polymer) volume fraction of ${\varphi }_{\text{p}}=0.3$ and attractive interaction energy between condensers and polymer of $\mathit{ϵ}=2$. Data are shown for three different condensing protein radii, $R_{\text{s}}$. (c) Pressure as a function of DNA (polymer) volume fraction, ${\varphi }_{\text{p}}$, for three different cases: strongly binding condensing particles $\left(\mathit{ϵ}=2\right)$ with sizes $R_{\text{s}}=1$ and $R_{\text{s}}=3$ and no condensing particles, ${\varphi }_{\text{s}}=0$. The shaded area corresponds to typical ranges of volume fractions found in viruses discussed in the text. To see this figure in color, go online.

Figure 2

Internal structure of confined mixtures in the weak and strong binding regime. (a) Radial probability density, $p_{\rho }\left(r\right)$, for finding a condenser in a spherical shell centered at r, for different condenser radii, $R_{\text{s}}$, with binding parameter $\mathit{ϵ}=1/2$ (weak binding). (b) RDF for condensers as a function of the interparticle distance, d, for different condenser radii, $R_{\text{s}}$, in the weak binding regime $\left(\mathit{ϵ}=1/2\right)$. (c) The same as in (a), but with binding parameter $\mathit{ϵ}=2$ (strong binding). The inset shows the probability density for DNA polymer beads. (d) The same as (b) but in the strong binding regime $\left(\mathit{ϵ}=2\right)$. Two schematic representations of direct contact between two condensing proteins and indirect (condenser-bead-condenser) contact are shown with arrows pointing from the corresponding correlation peak in the $R_{\text{s}}=1$ case. All plots (a–d) show data for ${\varphi }_{\text{p}}=0.3$, corresponding to $N_{\text{p}}=8221$ polymer beads, ${\varphi }_{\text{s}}=0.05$, and stiffness $K=25.$ To see this figure in color, go online.

Figure 3

Connectivity of DNA and condensing particles in confinement. (a) Probability distribution for the wrapping number, w, averaged over all representative configurations of the system in the weak binding regime. (b) Same as in (a), but for strong binding $\left(\mathit{ϵ}=2\right)$. The cartoon shows a schematic of the wrapping geometry for a polymer with $R_{\text{s}}=3$, corresponding to $w=4$ or about one-fourth of a full turn of DNA around the condenser. (c) Equilibrium state of the system $\left(R_{\text{s}}=3\right)$ in the strong binding regime. (d) Same as in (a), but after removal of the confinement. (e) Same as in (b), but after removal of the confinement. (f) Equilibrium state of the system $\left(R_{\text{s}}=3\right)$ in the strong binding regime and after removal of the confinement. Dashed lines in (d) $\left(R_{\text{s}}=3\right)$ and (e) $\left(R_{\text{s}}=\mathrm{1,3}\right)$ represent simulations done when starting from a random configuration of DNA and condensers in free space (i.e., configurations without a history of confinement). To see this figure in color, go online.