Semiflexible polymer condensates in poor solvents: Toroid versus spherical geometries

Semiflexible polymers, such as DNA, in the presence of a condensing agent often form toroids. This is due to a balance between bending and surface area free energy penalties. Here we show why in experiments all the toroids have been found to have similar physical size. We also introduce a novel morphology, that of the hollow sphere, which is favored for long polymer chains. This offers the possibility of encapsulating material inside a “vesicle” made of semiflexible polymers. We also consider the case of many such polymer chains placed in a poor solvent. We show a transition between two morphologies occur on increasing concentration of polymer chains, from a thickened toroid to a spherical globule.

Figure 1

A diagram of the parameters used in the toroidal calculations. $R$ is the major radius, $r$ is the minor radius, and $\rho$, $\theta$ are the variables for integration over the cross section.

Figure 2

The free energy vs $\alpha$ for the toroid and hollow sphere. The solid line is the free energy of the toroid and the dashed that of the hollow sphere. The critical value of $\alpha \approx 0.8423$ occurs where the two free energies intersect.

Figure 3

The radii verses length for the toroid and hollow sphere. The solid line and the dotted lines are the major and minor radii of the toroid. The dashed and dot-dashed lines are the outer and inner radii of the hollow sphere. $P∕a=30$ and $g=3$ for this graph but these values only affect the scaling of the graph and not the actual shape. The minimum length value corresponds to the change from a straight rod to a prototorus. Note the major radius of the toroid and the outer radius of the hollow sphere appear to be nearly equal at the phase boundary. This is a universal result, but appears to be a coincidence.

Figure 4

(a) Schematic of a typical disc of semi-flexible chains. (Note the figure should be viewed as if all chains and disk are in the same plane. Not all chains in disk are drawn.) The chains pack the disc up to radius $R_0$ and then begin to wrap around each other in the same disk. (b) A schematic of the pseudospherical globule. The ends are chopped off so that the final disk (top and bottom) has $a_0$ chains and corresponding radius $R_m$. The radius of the sphere is $R$.

Figure 5

Dimensionless free energy $\mathcal{F}$ of a spherical globule (dashed line—$a_0=10$, dot-dashed line—$a_0=30$) and thick toroid (solid line) versus number of chains, $\mathcal{N}$. (a) Up to ${10}^6$ chains (the $x$ axis is in units of ${10}^5$ chains). (b) For fewer chains showing the crossover from toroids to spheres.