Effects of self-avoidance on the packing of stiff rods on ellipsoids

Using a statistical-mechanics approach, we study the effects of geometry and self-avoidance on the ordering of slender filaments inside nonisotropic containers, considering cortical microtubules in plant cells, and packing of genetic material inside viral capsids as concrete examples. Within a mean-field approximation, we show analytically how the shape of the container, together with self-avoidance, affects the ordering of the stiff rods. We find that the strength of the self-avoiding interaction plays a significant role in the preferred packing orientation, leading to a first-order transition for oblate cells, where the preferred orientation changes from azimuthal, along the equator, to a polar one, when self-avoidance is strong enough. While for prolate spheroids the ground state is always a polar-like order, strong self-avoidance results with a deep metastable state along the equator. We compute the critical surface describing the transition between azimuthal and polar ordering in the three-dimensional parameter space (persistence length, eccentricity, and self-avoidance) and show that the critical behavior of this system is in fact related to the butterfly catastrophe model. We calculate the pressure and shear stress applied by the filament on the surface, and the injection force needed to be applied on the filament in order to insert it into the volume. We compare these results to the pure mechanical study where self-avoidance is ignored, and discuss similarities and differences.

Figure 1

$\sigma$ vs $\delta$, for different values of the persistence length $\epsilon$ (rows) and the self-avoidance strength $u$ (columns). The vertical gray dashed line marks the case of a sphere ($\delta =0$). Longer persistence lengths drive the system into more ordered states along the large dimension of the ellipsoid. Stronger self-avoidance, however, both adds a metastable state (not shown, but discussed in [49]) and drives the lowest energy state to order along the poles. This is a purely entropic effect, caused by the existence of more ordered states along this direction. This happens even for oblate spheroids ($\delta <0$). Note that the $\delta =0$ case, corresponding to a sphere, should be viewed as the limit $\delta \to 0$, since the treatment here assumes from the beginning a broken symmetry (namely, an axial symmetry and not a spherical one).

Figure 2

$\sigma$ vs $u$, for different values of the flattening $\delta$ (rows) and persistence lengths $\epsilon$ (columns). In strong self-avoidance, the system is driven into a polar-oriented configuration even for oblate spheroids. In this case the transition is an abrupt one (namely, a first-order transition) due to entropy.

Figure 3

$\sigma$ vs $\epsilon$, for different values of the flattening $\delta$ (rows) and self-avoidance $u$ (columns). The longer the persistence length, the energy becomes more important (rather than entropy), and the filaments align with the principal axis of the ellipsoid. More specifically, for $\delta <0$ the transition (from entropy dominated to energy-dominated phases) is of first order.

Figure 4

The critical surface $u^{*}(\delta ,\epsilon )$. The volume above this surface describes the region in parameter space, where there are three physical solutions, namely, one stable, one metastable, and one unstable, while underneath it there is only one (stable) solution. Note that this surface is infinite, in the sense that for every $\delta$ and $\epsilon$ there exists a $u^{*}$, above which there are three solutions.

Figure 5

The global order parameter ${\sigma }_{\mathrm{rand}}$ of a randomly oriented director field on the surface of an ellipsoid, as a function of its flattening $\delta$. ${\sigma }_{\mathrm{rand}}(\delta \to -1)=\frac{1}{2}$ as expected from a random director field on a disk. ${\sigma }_{\mathrm{rand}}(\delta \to \infty )=\frac{1}{4}$ as expected from a random director field on a cylinder.

Figure 6

Pressure of the stable solution, exerted on the ellipsoid by the filament, as a function of the ellipsoid scale $\ell =\sqrt{\frac{A}{2\pi }}$, provided by Eq. (42). Left: $\delta =0.1$, right: $\delta =-0.1$. In both graphs $L=10, \tilde{u}=1$ and $\tilde{\epsilon }=1$. The small jump in the pressure in the right figure corresponds to the jump in the stable solutions from polar-like ($\sigma \sim 0$) to equatorial-like ($\sigma \sim 0.5$) configurations as $\ell$ gets closer to 1. This is similar to the transitions that can be seen in Figs. 2 and 3, as $u$ and $\epsilon$ decrease.

Figure 7

Left: Free energy $W$ of the ground state as function of $\delta$ given by Eq. (41); right: corresponding elongation shear stress $T$ exerted on the ellipsoid for deformations with the same area given by Eq. (43). In both graphs $\tilde{u}=\tilde{\epsilon }=1, L=10, \ell =\frac{1}{2}$, which correspond to $u=6.36$ and $\epsilon =2$. Note a cusp in the energy (and jump in the shear stress) at the critical ${\delta }_c$ where the solution jumps from an equatorial-like ($\sigma \sim 0.5$) to polar-like ($\sigma \sim 0$) configurations. Additionally, it is clear that in both cases, $\delta <0$ and $\delta >0$, there are some stable solutions, in which the energy is at a local minimum if the ellipsoid shape is allowed to change.

Figure 8

Injection force $F$ needed to push further the filament into the ellipsoid, given by Eq. (44), as a function of the total length of the filament, $L$, already inside the container. Left: $\delta =0.1$ (prolate), right: $\delta =-0.1$ (oblate). In both graphs $\ell =\frac{1}{2}$ (and hence $A=\frac{\pi }{4}$), $\tilde{u}=1$, and $\tilde{\epsilon }=1$ (thus, $\epsilon =2$). The jump in the force in the right figure corresponds to the jump in stable solutions from equatorial-like ($\sigma \sim 0.5)$ to polar-like ($\sigma \sim 0$) as $L$ increases. This is similar to the transitions that can be seen in Fig. 2 as $u$ increases.

Figure 9

$\sigma$ as function of $\delta$ for $u=0$ and $\epsilon =10,000$, approaching the limit of pure cold, non-self-avoiding filaments. That $\sigma$ has a local minimum at small $\delta$'s suggests that a transition similar to the one described in Ref. [23] occurs.