Membrane Mediated Sorting

Inclusions in biological membranes may interact via deformations they induce on the shape of that very membrane. Such deformations are a purely physical effect, resulting in nonspecific forces between the inclusions. In this Letter we show that this type of interaction can organize membrane domains and hence may play an important biological role. Using a simple analytical model we predict that membrane inclusions sort according to the curvature they impose. We verify this prediction by both numerical simulations and experimental observations of membrane domains in phase separated vesicles.

Figure 1

Comparison of the potential energies of the completely mixed and completely demixed state of a vesicle with domains of two different sizes. The freely adjustable parameter $\varphi$ denotes the fraction of the vesicle’s surface area claimed by the big domains. The dashed blue (dark gray) line indicates the case in which the “big” and “small” domains are equal in size (and hence have equal contact angles). The solid red, yellow, and green (light gray) lines indicate contact angle ratios ${\alpha }_b/{\alpha }_s$ of 1.5, 2.0, and 2.5, respectively. Domain demixing occurs for any value of $\varphi$ for which the potential ratio is less than 1 (black horizontal line). For comparison the number fraction $\gamma$ of the big domains is indicated by the gray vertical line. Insets: typical distributions of domains for small (left) and big (right) values of $\varphi$. For small $\varphi$, the big domains are packed closely together and the small domains claim the largest area fraction, for large $\varphi$ the situation is reversed.

Figure 2

Monte Carlo relaxation of a random configuration of 70 small (red, light gray) and 30 big (blue, dark gray) domains on a spherical vesicle. (a)–(j) A folded-open view of the entire vesicle, with the azimuthal angle along the horizontal direction and the polar angle along the vertical direction. The pictures show configurations after (a) 1000  (b) 50 000  (c) 100 000  (d) 200 000  (e) 500 000  and (f) 1, (g) 2, (h) 3, (i) 4, and (j) $5\times {10}^6$ steps. (k) Evolution of the mean nearest-neighbor distance over time, split out to large (red, light gray) and small (blue, dark gray) domains. (l) Evolution of the mean nearest-neighbor size over time, split out to large (red, light gray) and small (blue, dark gray) domains. Saturation appears to be reached after approximately $4\times {10}^6$ steps. (m) Schematic drawing of a cross section of the vesicle, showing the contact angles ${\alpha }_s$ and ${\alpha }_b$ of the small and big domains, respectively. (n) The configuration on a sphere after $5\times {10}^6$ time steps. (o) Occupied area fraction of the big domains $\varphi$ as a function of their number fraction $\gamma$. The plot shows the effect of different number fractions and domain size ratios on the sorting process. As predicted by the analytical model, larger size ratios result in larger differences in occupied area fractions. Moreover, we find that the speed of demixing increases with decreasing number fraction.

Figure 3

Correlations between the size of a domain and that of its nearest neighbors. (a) Correlation plot of a simulation in which the domain sizes follow an exponential distribution. The simulation ran for $15\times {10}^6$ steps; error bars obtained using the last $5\times {10}^6$ steps. Inset: Example of the actual distribution of domains on the vesicle after $15\times {10}^6$ steps. (b) Correlation plot averaged over 21 experimental vesicles; the dashed line corresponds to the average $3.3\mu \mathrm{m}$. Domain sizes are grouped in equally sized bins. Inset: Two sides of the same vesicle showing very different domain sizes. Scale bar $20\mu \mathrm{m}$.