Probing a self-assembled $\mathit{fd}$ virus membrane with a microtubule

The self-assembly of highly anisotropic colloidal particles leads to a rich variety of morphologies whose properties are just beginning to be understood. This article uses computer simulations to probe a particle-scale perturbation of a commonly studied colloidal assembly, a monolayer membrane composed of rodlike $\mathit{fd}$ viruses in the presence of a polymer depletant. Motivated by experiments currently in progress, we simulate the interaction between a microtubule and a monolayer membrane as the microtubule “pokes” and penetrates the membrane face-on. Both the viruses and the microtubule are modeled as hard spherocylinders of the same diameter, while the depletant is modeled using ghost spheres. We find that the force exerted on the microtubule by the membrane is zero either when the microtubule is completely outside the membrane or when it has fully penetrated the membrane. The microtubule is initially repelled by the membrane as it begins to penetrate but experiences an attractive force as it penetrates further. We assess the roles played by translational and rotational fluctuations of the viruses and the osmotic pressure of the polymer depletant. We find that rotational fluctuations play a more important role than the translational ones. The dependence on the osmotic pressure of the depletant of the width and height of the repulsive barrier and the depth of the attractive potential well is consistent with the assumed depletion-induced attractive interaction between the microtubule and viruses. We discuss the relevance of these studies to the experimental investigations.

Figure 1

Schematic illustration of the process modeled in this paper. A microtubule of length $L_{\text{t}}$ approaches (a) and penetrates (b) a two-dimensional membrane of rods of length $L$. The center of mass separation of the microtubule and membrane is denoted by $d$.

Figure 2

Simulation results for the interaction potential $\varphi$ (in units of $k_{\text{B}}T$) of a microtubule ($L_{\text{t}}=150$) and a membrane ($N=224,L=100$) as a function of their center of mass separation $d$ for three different pressures: $p=0.07,0.1,0.15$ (in units of $k_{\text{B}}T/{\sigma }^{-3}$). Lengths are measured in units of $\sigma$. The microtubule fully penetrates the membrane for $d\le 25$.

Figure 3

Force $F$ (in units of $k_{\text{B}}T/\sigma$) exerted by the membrane ($N=224,L=100$) on a microtubule ($L_{\text{t}}=150$) as a function of their center of mass separation $d$ (in units of $\sigma$) for $p=0.07,0.1,0.15$ ($p$ measured in units of $k_{\text{B}}T/{\sigma }^{-3}$) as calculated from the potential shown in Fig. 2. A positive force indicates the microtubule is being pushed out of the membrane.

Figure 4

Idealized membrane configuration used in the theoretical model, viewed along the $z$ axis, normal to the plane of the membrane. The red circles are the cross section of the virus rods and the blue solid circle at the center of the pattern is the microtubule's cross section. The blue dashed circle is the outer edge of the depletion zone of the microtubule and has radius $(1+\delta )/2=1.25$ for $\delta =1.5$, with lengths measured in units of $\sigma$. The overlap volume yielding the depletion potential between the microtubule and membrane is the part of the microtubule's depletion zone that overlaps with any of the rods' depletion zones.

Figure 5

Comparison between the simulation results (solid lines) for $\varphi$ (Fig. 2) and the theoretical excluded volume calculation (dashed lines) of ${\varphi }_{\text{ideal}}$ (based on the ideal configuration shown in Fig. 4) for a microtubule ($L_{\text{t}}=150$) piercing a membrane ($N=224,L=100$) as a function of the separation $d$ for pressures: (a) $p=0.15$; (b) $p=0.1$; (c) $p=0.07$ . The jump in ${\varphi }_{\text{ideal}}$ at $d=125$ is the free-energy cost of creating the vacancy for the microtubule to penetrate the membrane.

Figure 6

Comparison of the results for $\varphi$ as a function of the separation $d$ at pressures: (a) $p=0.07$, (b) $p=0.1$, (c) $p=0.15$, obtained from the full simulation (dotted-dashed black line; Fig. 2), the theoretical excluded volume calculation ${\varphi }_{\text{ideal}}$ (blue dashed line), and a simulation where rotations of the rods are excluded (solid red line).