Fine structure of viral dsDNA encapsidation

Unraveling the mechanisms of packing of DNA inside viral capsids is of fundamental importance to understanding the spread of viruses. It could also help develop new applications to targeted drug delivery devices for a large range of therapies. In this article, we present a robust, predictive mathematical model and its numerical implementation to aid the study and design of bacteriophage viruses for application purposes. Exploiting the analogies between the columnar hexagonal chromonic phases of encapsidated viral DNA and chromonic aggregates formed by plank-shaped molecular compounds, we develop a first-principles effective mechanical model of DNA packing in a viral capsid. The proposed expression of the packing energy, which combines relevant aspects of the liquid crystal theory, is developed from the model of hexagonal columnar phases, together with that describing configurations of polymeric liquid crystals. The method also outlines a parameter selection strategy that uses available data for a collection of viruses, aimed at applications to viral design. The outcome of the work is a mathematical model and its numerical algorithm, based on the method of finite elements, and computer simulations to identify and label the ordered and disordered regions of the capsid and calculate the inner pressure. It also presents the tools for the local reconstruction of the DNA “scaffolding” and the center curve of the filament within the capsid.

Figure 1

The graph represents the DNA density of bacteriophage T5. It shows the radial density distribution of DNA inside the viral capsid. The dashed line shows the density of an empty capsid. Figure reproduced from Ref. [16].

Figure 2

Plot of the isotropic penalty function $w$. The asterisks indicate points of zero slope.

Figure 3

The graph a schematic representation of the level surfaces $\omega =\mathrm{const}$, $\vartheta =\mathrm{const}$, designated as ${\mathcal{S}}_m$ and ${\mathcal{S}}_p$, respectively, whose intersections provide the model scaffolding that supports the DNA filament.

Figure 4

The graphs show the role of the parameters $\tilde{b}$ and ${\tilde{\gamma }}_0$ in the model. In both cases, the curves indicate the decrease in core radius with increasing parameter values, as a mechanism to lower the energy of the system. We recall that $\tilde{b}$ represents the surface energy of the transition layer between the spool-like and the disordered regions of the capsid, and ${\tilde{\gamma }}_0$ corresponds to the bulk energy of the disordered inner core.

Figure 5

The graphs show two different profiles of the order parameter in the capsid domain. Left plot is of $s$ evaluated along a line through the equator of the capsid. The center of the capsid is located at 0, and the capsid radius is normalized to 1. Since the maximum order corresponds to $s=0.8$, we take the disordered threshold at $s=0.4$, which gives the ratio $r_0/R_c\approx 0.496$. Right plot: The order profile of the capsid transitioning from ordered DNA (red) to disordered (blue) as a 2D slice through the capsid equator.

Figure 6

Three-dimensional visualization of the director field $\mathbf{n}$ showing a spooling configuration. Left is a side view; right is a top view. The coloring of the arrows corresponds to the degree of orientation $s$, i.e., red indicates well ordering (spooling) of the DNA strand and blue is disordered.

Figure 7

Plots of core radius and capsid pressure. Left plot show curves of constant normalized core radius ($\frac{r_0}{R_c}$ for different choices of $\tilde{b}$ and ${\tilde{\gamma }}_0$ overlaid on the pressure values). In the region where the core radius matches experiment, the pressure values are also comparable to experimental values (from a minimum of about 26 atms to a maximum of about 95 atms). The right plot is a 3D view of the pressure. Note: $\alpha$ is chosen as 75, which fully determines $k_3$, to give a decent fit for all four viruses in Table 1.