Pointlike Inclusion Interactions in Tubular Membranes

Membrane tubes and tubular networks are ubiquitous in living cells. Inclusions like proteins are vital for both the stability and the dynamics of such networks. These inclusions interact via the curvature deformations they impose on the membrane. We analytically study the resulting membrane mediated interactions in strongly curved tubular membranes. We model inclusions as constraints coupled to the curvature tensor of the membrane tube. First, as special test cases, we analyze the interaction between ring- and rod-shaped inclusions. Using Monte Carlo simulations, we further show how pointlike inclusions interact to form linear aggregates. To minimize the curvature energy of the membrane, inclusions self-assemble into either line- or ringlike patterns. Our results show that the global curvature of the membrane strongly affects the interactions between proteins embedded in it, and can lead to the spontaneous formation of biologically relevant structures.

Figure 1

The calculated energy cost of having two inclusions (as compared to none) for a membrane tube as a function of the distance between (a) two rings and (b) two rods. Inclusions impose either the same (dashed line) or opposite (solid line) curvatures.

Figure 2

The energy landscape for a membrane tube containing three rodlike inclusions I1, I2, and I3.

Figure 3

(a) The curvature energy ($\mathrm{\Delta }E/2\pi \kappa c^2$) of a membrane containing two inclusions, as a function of their angular ($\mathrm{\Theta }$) and longitudinal ($L$) separation, with $L$ in units of the tube radius $R$. (b) The line around the global minimum at which the energy equals the local minimum at large separations. For particles whose diameter exceeds the size of this region, the overall behavior is repulsive (settling in the local minimum at large separations). Smaller particles globally attract, but have a high energy barrier separating the attractive and repulsive regime. (c) Two identical inclusions placed at the same longitudinal coordinates ($L=0$) attract each other. (d) Pointlike inclusions behave like rings when they are situated on the same transversal coordinates ($\mathrm{\Theta }=0$); the inset shows the weak long-rage attraction.

Figure 4

Equilibrium configurations obtained by Monte Carlo simulation for a system containing (a) 10 inclusions with a hard-core radius of $a_0=0.2$, (b) 16 inclusions with a hard-core radius of $a_0=0.2$, (c) 16 inclusions with a hard-core radius of $a_0=1.1$, (d) 30 inclusions with a hard-core radius of $a_0=1.1$, and (e) 80 inclusions with a hard-core radius of $a_0=1.1$.