Numerical simulation of endocytosis: Viscous flow driven by membranes with non-uniformly distributed curvature-inducing molecules

Abstract

The formation of membrane vesicles from a larger membrane that occurs during endocytosis and other cell processes are typically orchestrated by curvature-inducing molecules attached to the membrane. Recent reports demonstrate that vesicles can form de novo in a few milliseconds. Membrane dynamics at these scales are strongly influenced by hydrodynamic interactions. To study this problem, we develop new diffuse interface models for the dynamics of inextensible vesicles in a viscous fluid with stiff, curvature-inducing molecules. The model couples the Navier-Stokes equations with membrane-induced bending forces that incorporate concentration-dependent bending stiffness coefficients and spontaneous curvatures, with equations for molecule transport and for a Lagrange multiplier to enforce local inextensibility. Two forms of surface transport equations are considered: Fickian surface diffusion and Cahn-Hilliard surface dynamics, with the former being more appropriate for small molecules and the latter being better for large molecules. The system is solved using adaptive finite element methods in 3D axisymmetric geometries. The results demonstrate that hydrodynamics can indeed enable the rapid formation of a small vesicle attached to the membrane by a narrow neck. When the Fickian model is used, this is a transient state with the steady state being a flat membrane with a uniformly distributed molecule concentration due to diffusion. When the Cahn-Hilliard model is used, molecule concentration gradients are sustained, the neck stabilizes and the system evolves to a steady-state with a small, compact vesicle attached to the membrane. By varying the membrane coverage of molecules in the Cahn-Hilliard model, we find that there is a critical (smallest) neck radius and a critical (fastest) budding time. These critical points are associated with changes in the vesicle morphology from spherical to mushroom-like as the molecule coverage on the membrane is increased.

Keywords: Clathrin; Endocytosis; Helfrich energy; Navier-Stokes flow; Numerical Simulation; Phase-field model.

Figure 1

Schematic of the processes involved in endocytosis. Curvature inducing molecules attach to the membrane and induce out-of-plane deformations. Once enough molecules cover the membrane a vesicle is formed that provides the vehicle to transport extracellular cargo into the cell.

Figure 2

The cylindrical configuration used in the numerical simulations. The membrane Γ is defined as the zero-level set of the phase field ϕ and separates the intracellular from the extracellular fluid region where ϕ = 1 or ϕ = −1, respectively. A part of the membrane, Γ_c, is covered with CIM.

Figure 3

Comparison of membrane evolution and CIM concentrations c when the curvature-inducing molecules are transported by the surface Cahn-Hilliard model (top) or Fickian surface diffusion (bottom). The Cahn-Hilliard model maintains sharp gradients of CIMs between the distinct values of 1 (covered) and 0 (uncovered) and produces a steady-state configuration consisting of a compact vesicle connected to the membrane by a small neck. Fickian diffusion decreases the gradients of the CIMs that transiently produces longer necks and more deformed vesicles.

Figure 4

Comparison of the membrane evolution and CIM concentrations with (bottom) and without flow (top) at times t = 2.90µs, 7.46µs, 14.02µs, 16.98µs from left to right. After forming a neck the membrane pinches off without flow. This is not seen when flow is involved and the neck radius stops decreasing at a certain point. The arrows indicate the velocity direction and magnitude on a logarithmic scale.

Figure 5

Stationary shapes of the membrane and CIM concentrations for different sizes of the initial CIM region using the surface Cahn-Hilliard model.

Figure 6

Diameter of the vesicle and neck radius of the stationary membrane for different sizes of the initial CIM region.

Figure 7

Time required for a bud to form for different initial radii of the CIM region.