Membrane lipid segregation in endocytosis

We explore the equilibrium mechanics of a binary lipid membrane that wraps around a spherical or cylindrical particle. One of the lipid membrane components induces a positive spontaneous curvature, while the other induces a negative local curvature. Using a Hamiltonian approach, we derive the equations governing the membrane shape and lipid concentrations near the wrapped object. Asymptotic expressions and numerical solutions for membrane shapes are presented. We determine the regimes of bending rigidity, surface tension, intrinsic lipid curvature, and effective receptor binding energies that lead to efficient wrapping and endocytosis. Our model is directly applicable to the study of invagination of clathrin-coated pits and receptor-induced wrapping of colloids such as spherical virus particles.

Figure 1

Endocytosis can involve intermediates of qualitatively different shapes. (a) A clathrin-coated pit with tight wrapping and spherical enclosure. (b) Endocytosis can also involve intermediates with long-tube-like processes. Both figures were taken from [56].

Figure 2

(Color) (a) Membrane wrapping of a cylindrical particle of radius $a$. Here, $s$ is the arclength from “south pole” of the particle to a point on the membrane, $q\left(s\right)$ is the angle that the membrane at position $s$ makes with the horizontal, and $s^{*}$ is the point where the membrane detaches from the particle. The horizontal distance from the midline is $r$, and the height of the membrane from the south pole is $z$. (b) Membrane wrapping of a spherical particle of radius $a$.

Figure 3

(Color online) Wrapping of cylindrical particles. (a) For certain values of $W$ and $\Sigma$ (here, $\Sigma =1$, $W=5.81$), the solution is unphysical because the membrane curve crosses itself. (b) The solution curve for a particle that is just fully wrapped is plotted. The parameters $\Sigma =1$ and $W=5.65$ [which were calculated using Eq. (A20)] were used. (c) There are three possible phases for a cylindrical particle in the weak coupling limit (delineated by solid curves) describing particles that are unwrapped (when $W<1$), partially wrapped, or fully wrapped. The boundary between the partially wrapped and fully wrapped phases was found numerically using Eq. (A20). In the strong coupling limit $(\alpha \to \infty )$, only two phases survive as shown by the dashed curve. For parameter values that lie above the dotted curve, we expect the particle to be partly wrapped; for parameters below the curve, the particle will be fully wrapped. Compared to the weak coupling limit, wrapping is enhanced by lipid segregation when the coupling $\alpha$ increases.

Figure 4

(Color online) Wrapping of spherical particles. (a)–(d) The total energy of a membrane bound to a spherical particle is plotted as a function of $Z^{*}=z^{*}∕a$. Surface tension was set to $\Sigma =1$, and the binding energy $W$ was varied. (a) In the first phase, the binding energy is very low, and energy increases monotonically with $Z^{*}$. (b) At higher binding energies, an unstable equilibrium emerges, and both the fully wrapped state and the unwrapped state are energetically stable. (c) As $W$ increases further, a stable equilibrium emerges as the unbound state becomes unstable. (d) When $W$ is sufficiently large, the system is driven towards a fully wrapped state. (e) Phase diagram for spherical particles with no curvature coupling (solid curves) were determined numerically. The dashed curve shows the boundary between the two phases—partial wrapping and full wrapping—that occur in the limit where coupling $\alpha$ between concentration and spontaneous curvature is very large. Above the dashed curve, the particles are partly wrapped; below, they are fully wrapped. Strong coupling enhances wrapping.

Figure 5

(Color online) Membrane shapes and concentration profiles for a partially wrapped sphere. (a) The profile of two free membranes are plotted. The detachment point $S^{*}=1.3$ was fixed. With increased coupling constant $\alpha$ (and decreased $C_2$ and $\Gamma$), the membrane bends more sharply at the detachment point and becomes flat more quickly. The binding energy necessary to wrap the specified amount was calculated and is significantly lower for larger $\alpha$ ($W=2.46$ vs 5.45). In (b), $\Phi \left(S\right)$ is plotted for the two profiles shown in (a). With increased $\alpha$ (and decreased $C_2$, $\Gamma$), $\Phi$ and ${\Phi }^{\prime }$ reach larger values. The dashed line marks the point at which the membrane detaches from the sphere.

Figure 6

Membrane shapes near full wrapping. The parameters $\Sigma =1$ and a detachment point at $q\left(S^{*}\right)=3$ (to approximate full wrapping) were used. Notice that when ${\kappa }_G$ is variable, the relaxation of the membrane to a flat surface is more gradual and the region with negative Gaussian curvature is extended.