Model for Curvature-Driven Pearling Instability in Membranes

A phase-field model for dealing with dynamic instabilities in membranes is presented. We use it to study curvature-driven pearling instability in vesicles induced by the anchorage of amphiphilic polymers on the membrane. Within this model, we obtain the morphological changes reported in recent experiments. The formation of a homogeneous pearled structure is achieved by consequent pearling of an initial cylindrical tube from the tip. For high enough concentration of anchors, we show theoretically that the homogeneous pearled shape is energetically less favorable than an inhomogeneous one, with a large sphere connected to an array of smaller spheres.

Figure 1

Polymer wedge effect inducing a spontaneous curvature in a bilayer. A bilayer formed by one kind of lipids with zero spontaneous curvature tends to be flat (a). When a certain amount of anchor groups of an amphiphilic polymer gets stuck in the outer leaflet of the bilayer, a spontaneous curvature is induced (b).

Figure 2

Onset of the pearling instability. Comparison of the experimental result from Ref. 7 (a) and the phase-field simulation (b). It is important to note that in the simulation there is no fitting parameter, but we just let the system evolve from an initial tubular shape, under a relatively low homogeneous induced spontaneous curvature, $C_0=0.48$, below the pearling instability limit.

Figure 3

Dynamic evolution (time goes by from upper images to down) of a tube with a polymer concentration on the membrane such that $C_0=0.68$. We show (a) the experimental results from 7 and (b) the phase-field simulation for comparison. Pearls start at points where the tube looses its perfect cylindrical geometry, namely, the cap. The simulation has been performed on a $200\times 40$ axisymmetric lattice. No-flux boundary conditions at the lateral walls have been implemented.

Figure 4

Inhomogeneous pearling. (a) Experimental result from Ref. 7 and (b) phase-field simulation using $C_0=1.1$. (c) Plot of the energy difference $\Delta E$ between the bending energy corresponding to a set of spheres and the one associated with a set of small spheres added to a bigger one, with respect to the increment of spontaneous curvature from the value $C_0=2/3$, when equally sized spheres have zero bending energy. For spontaneous curvatures bigger than a critical value, the homogeneous configuration is energetically less favorable than the inhomogeneous one. The area of the vesicle is finite, and chosen in such a way that the homogeneous pearled chain consists of 8 spheres. Magnitudes are measured in normalized units, given by $\kappa =1$ and $\lambda =1$.